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transmission-justification-warrant	## 1. Introduction Arthur has the measles and stays in a closed environment with his sister Mafalda. If Mafalda ends up contracting the measles herself because of her staying in close contact with Arthur, there must be something--the measles virus--which is transmitted from Arthur to Mafalda in virtue of a relation--the relation of staying in close contact with one another--instantiated by the two siblings. Epistemologists have been devoting their attention to the fact that epistemic properties--like being justified or known--are often transmitted in a similar way. An example (not discussed in this entry) is the transmission of knowledge via testimony. A different but equally important phenomenon (which occupies center stage in this entry) is the transmission of epistemic justification across an inference or argument from one proposition to another. Consider proposition $q$ constituting the conclusion of an argument having proposition $p$ as premise (where $p$ can be a set or conjunction of propositions). If $p$ is justified for a subject $s$ by evidence $e$, this justification may transmit to $q$ if $s$ is aware of the entailment between $p$ and $q$. When this happens, $q$ is justified for $s$ in virtue of her justification for $p$ based on $e$ and her knowledge of the inferential relation from $p$ to $q$. Consider this example from Wright (2002): Toadstool $\rE\_1$. Three hours ago, Jones inadvertently consumed a large risotto of Boletus Satana. $\rP\_1$. Jones has absorbed a lethal quantity of the toxins that toadstool contains. Therefore: $\rQ\_1$. Jones will shortly die. Here $s$ can deductively infer $\rQ\_1$ from $\rP\_1$ given background information.[1] Suppose $s$ acquires justification for $\rP\_1$ by learning $\rE\_1$. $ s$ will also acquire justification for $\rQ\_1$ in virtue of her knowledge of the inferential relation from $\rP\_1$ to $\rQ\_1$ and her justification for $\rP\_1$.[2] Thus, $s$'s justification for $\rP\_1$ will transmit to $\rQ\_1$. It is widely recognized that epistemic justification sometimes fails to transmit across an inference or argument. In interesting cases of transmission failure, $s$ has justification for believing $p$ and knows the inferential link between $p$ and $q$, but $s$ has no justification for believing $q$ in virtue of her justification for $p$ and her knowledge of the inferential link. Here is an example from Wright (2003). Suppose $s$'s background information entails that Jessica and Jocelyn are indistinguishable twins. Consider this possible reasoning: Twins $\rE\_2$. This girl looks just like Jessica. $\rP\_2$. This girl is actually Jessica. Therefore: $\rQ\_2$. This girl is not Jocelyn. In Twins $\rE\_2$ can give $s$ justification for believing $\rP\_2$ only if $s$ has independent justification for believing $\rQ\_2$ in the first instance. Suppose that $s$ does have independent justification for believing $\rQ\_2$, and imagine that $s$ learns $\rE\_2$. In this case $s$ will acquire justification for believing $\rP\_2$ from $\rE\_2$. But it is intuitive that $s$ will acquire no justification for $\rQ\_2$ in virtue of her justification for believing $\rP\_2$ based on $\rE\_2$ and her knowledge of the inferential link between $\rP\_2$ and $\rQ\_2$. So, Twins instantiates transmission failure when $\rQ\_2$ is independently justified. An argument incapable of transmitting to its conclusion a specific justification for its premise(s)--e.g., a justification based on evidence $e$--may be able to transmit to the same conclusion a different justification for its premise(s)--e.g., one based on different evidence $e^\$. Replace for instance $\rE\_2$ with $\rE\_2^\.$ This girl's passport certifies she is Jessica. in Twins, $\rE\_2^\$ can provide $s$ with justification for believing $\rP\_2$ even if $s$ has no independent justification for $\rQ\_2$. Suppose then that $s$ has no independent justification for $\rQ\_2$, and that she acquires $\rE\_2^\$. It is intuitive that $s$ will acquire justification from $\rE\_2^\$ for $\rP\_2$ that does transmit to $\rQ\_2$. Now the inference from $\rP\_2$ to $\rQ\_2$ instantiates epistemic transmission. Although many of the epistemologists participating in the debate on epistemic transmission and transmission failure speak of transmission of warrant, rather than justification, almost all of them use the term 'warrant' to refer to some kind of epistemic justification (in Sect. 3.3 we will however consider an account of warrant transmission failure in which 'warrant' is interpreted differently.) Most epistemologists investigating epistemic transmission and transmission failure--e.g., Wright (2011, 2007, 2004, 2003, 2002 and 1985), Davies (2003, 2000 and 1998), Dretske (2005), Pryor (2004), Moretti (2012) and Moretti & Piazza (2013)--broadly identify the epistemic property capable of being transmitted with propositional* justification.[3] Only a few authors explicitly focus on transmission of doxastic justification--e.g., Silins (2005), Davies (2009) and Tucker (2010a and 2010b [OIR]). In this entry we follow the majority in discussing transmission and transmission failure of justification as phenomena primarily pertaining to propositional justification. (See however the supplement on Transmission of Propositional Justification versus Transmission of Doxastic Justification.) Epistemologists typically concentrate on transmission of (propositional or doxastic) justification across deductively valid (given background information) arguments. The fact that justification can transmit across deduction is crucial for our cognitive processes because it makes the advancement of knowledge--or justified belief--through deductive reasoning possible. Suppose evidence $e$ gives you justification for believing hypothesis $p$, and you know that $p$ entails another proposition $q$ that you haven't directly checked. If the justification you have for believing $p$ transmits to its unchecked prediction $q$ through the entailment, you acquire justification for believing $q$ too. Epistemologists may analyze epistemic transmission across inductive (or ampliative) inferences too. Yet this topic has received much less attention in the literature on epistemic transmission. (See however interesting remarks in Tucker 2010a.) In the remaining part of this entry we will focus on transmission and transmission failure of propositional justification across deductive inference. Unless differently specified, by 'epistemic justification' or 'justification' we always mean 'propositional justification'. ## 2. Epistemic Transmission As said, $s$'s justification for $p$ based on evidence $e$ transmits across entailment from $p$ to $p$'s consequence $q$ whenever $q$ is justified for $s$ in virtue of $s$'s justification for $p$ based on $e$ and her knowledge of $q$'s deducibility from $p$. This initial characterization can be distilled into three conditions individually necessary and jointly sufficient for epistemic transmission: > > > $s$'s justification for $p$ based on $e$ transmits to > $p$'s logical consequence $q$ if and only if: > > > > (C1) > $s$ has justification for believing $p$ based on $e$, > (C2) > $s$ knows that $q$ is deducible from $p$, > (C3) > $s$ has justification for believing $q$ in virtue of > the satisfaction of > (C1) > and > (C2). > > Condition (C3) is crucial for distinguishing transmission of justification across (known) entailment from closure of justification across (known) entailment. Saying that $s$'s justification for believing $p$ is closed under $p$'s (known) entailment to $q$ is saying that: > > > If > > > > (C1) > $s$ has justification for believing $p$ and > (C2) > $s$ knows that $p$ entails $q$, then > (C3$^{\textrm{c}}$) > $s$ has justification for believing $q$. > > One can coherently accept the above principle--known as the principle of epistemic closure--but deny a corresponding principle of epistemic transmission, cashed out in terms of the conditions outlined above: > > > If > > > > (C1) > $s$ has justification for believing $p$ and > (C2) > $s$ knows that $p$ entails $q$, then > (C3) > $s$ has justification for believing $q$ in virtue of > the satisfaction of (C1) and (C2). > > The principle of epistemic closure just requires that when $s$ has justification for believing $p$ and knows that $q$ is deducible from $p, s$ have justification for believing $q$. In addition, the principle of epistemic transmission requires this justification to be had in virtue of her having justification for $p$ and knowing that $p$ entails $q$ (for further discussion see Tucker 2010b [OIR]). Another important distinction is the one between the principle of epistemic transmission and a different principle of transmission discussed by Pritchard (2012a) under the label of evidential transmission principle. According to it, > > > If $s$ perceptually knows that $p$ in virtue of evidence set > $e$, and $s$ competently deduces $q$ from $p$ (thereby coming > to believe that $q$ while retaining her knowledge that $p)$, then > $s$ knows that $q$, where that knowledge is sufficiently supported > by $e$. (Pritchard 2012a: 75) > > > Pritchard's principle, to begin with, concerns (perceptual) knowledge and not propositional justification. Moreover, it deserves emphasis that there are inferences that apparently satisfy Pritchard's principle but fail to satisfy the principle of epistemic transmission. Consider for instance the following triad: Zebra $\rE\_3$. The animals in the pen look like zebras. $\rP\_3$. The animals in the pen are zebras. Therefore: $\rQ\_3$. The animals in the pen are not mules cleverly disguised to look like zebras. According to Pritchard, the evidence set of a normal zoo-goer standing before the zebra enclosure typically includes, above and beyond $\rE\_3$, the background proposition that $(\rE\_B)$ to disguise mules like zebras would be very costly and time-consuming, would bring no comparable benefit and would be relatively easy to unmask. Suppose $s$ knows $\rP\_3$ on the basis of her evidence set, and competently deduces $\rQ\_3$ from $\rP\_3$ thereby coming to believe $\rQ\_3$. Pritchard's evidential transmission principle apparently accommodates the fact that $s$ thereby comes to know $\rQ\_3$. For her evidence set--because of its inclusion of $\rE\_B$--sufficiently supports $\rQ\_3$. But the principle of epistemic transmission is not satisfied. Although (C1$\_{\textit{Zebra}}$) $s$ has justification for $\rP\_3$ and (C2$\_{\textit{Zebra}}$) she knows that $\rP\_3$ entails $\rQ\_3$, she has justification for believing $\rQ\_3$ in virtue of, not (C1$\_{\textit{Zebra}}$) and (C2$\_{\textit{Zebra}}$), but the independent support lent to it by $\rE\_B$. Condition (C3) suffices to distinguish the notion of epistemic transmission from different notions in the neighborhood. However, as it stands, it is still unsuitable for the purpose to completely characterize epistemic transmission. The problem is that there are cases in which it is intuitive that the justification for $p$ based on $e$ transmits to $q$ even if (C3), strictly speaking, is not satisfied. These cases can be described as situations in which only part of the justification that $s$ has for $q$ is based on her justification for $p$ and her knowledge of the entailment from $p$ to $q$. Consider this example. Suppose you are traveling on a train heading to Edinburgh. At 16:00, as you enter Newcastle upon Tyne, you spot the train station sign. Then, at 16:05, the ticket controller tells you that you are not yet in Scotland. Now consider the following reasoning: Journey $\rE\_4$. At 16:05 the ticket controller tells you that you are not yet in Scotland. $\rP\_4$. You are not yet in Scotland. Therefore: $\rQ\_4$. You are not yet in Edinburgh. As you learn $\rE\_4$, given suitable background information, you get justification for $\rP\_4$; moreover, to the extent to which you know that not being in Scotland is sufficient for not being in Edinburgh, you also acquire via transmission justification for $\rQ\_4$. This additional justification is transmitted irrespective of the fact that you already have justification for $\rQ\_4$, acquired by spotting the train station sign 'Newcastle upon Tyne'. If (C3) were read as requiring that the whole of the justification available for a proposition $q$ were had in virtue of the satisfaction of (C1) and (C2), cases like these would become invisible. A way to deal with this complication is to amend the tripartite analysis of epistemic transmission by turning (C3) into (C3$^{+}$), saying that at least part of the justification that $s$ has for $q$ has been achieved by her in virtue of the satisfaction of (C1) and (C2). Let us say that a justification for $q$ is an additional justification for $q$ whenever it is not a first-time justification for it. Condition (C3$^{+}$) can be reformulated as this disjunction: (C3$^{+}$) $s$ has first-time justification for $q$ in virtue of the satisfaction of (C1) and (C2), or $s$ has an additional justification for $q$ in virtue of the satisfaction of (C1) and (C2). Much of the extant literature on epistemic transmission concentrates on examples of transmission of first-time justification. These examples include Toadstool. We have seen, however, that what intuitively transmits in other cases is simply additional justification. Epistemologists have identified at least two--possibly overlapping--kinds of additional justification (cf. Moretti & Piazza 2013). One is what can be called independent justification because it appears--intuitively--independent of the original justification for $q$. This notion of justification can probably be sharpened by appealing to counterfactual analysis. Suppose $s$'s justification for $p$ based on $e$ transmits to $p$'s logical consequence $q$. This justification transmitted to $q$ is an additional independent justification just in case these three conditions are met: (IN1) $s$ was already justified in believing $q$ before acquiring $e$, (IN2) as $s$ acquires $e, s$ is still justified in believing $q$, and (IN3) if $s$ had not been antecedently justified in believing $q$, upon learning $e, s$ would have acquired via transmission a first-time justification for believing $q$. Journey instantiates transmission of justification by meeting (IN1), (IN2), and (IN3). Thus, Journey exemplifies a case of transmission of additional independent justification. Consider now that justification for a proposition or belief normally comes in degrees of strength. The second kind of additional justification can be characterized as quantitatively strengthening justification. Suppose again that $s$'s justification for $p$ based on $e$ transmits to $p$'s consequence $q$. This justification transmitted to $q$ is an additional quantitatively strengthening justification just in case these two conditions are satisfied: (STR1) $s$ was already justified in believing $q$ before acquiring $e$, and (STR2) as $s$ acquires $e$, the strength of $s$'s justification for believing $q$ increases. Here is an example from Moretti (2012). Your background information says that only one ticket out of 5,000 of a fair lottery has been bought by a person born in 1970, and that all other tickets have been bought by older or younger people. Consider now this reasoning: Lottery $\rE\_5$. The lottery winner's passport certifies she was born in 1980. $\rP\_5$. The lottery's winner was born in 1980. Therefore: $\rQ\_5$. The lottery's winner was not born in 1970. Given its high chance, $\rQ\_5$ is already justified on your background information only. It is intuitive that as you learn $\rE\_5$, you acquire an additional quantitatively strengthening justification for $\rQ\_5$ via transmission. For your justification for $\rP\_5$ transmitted to $\rQ\_5$ is intuitively quantitatively stronger than your initial justification for $\rQ\_5$. In many cases when $q$ receives via transmission from $p$ an additional independent justification, $q$ also receives a quantitatively strengthening justification. This is not always true though. For there seem to be cases in which an additional independent justification transmitted from $p$ to $q$ intuitively lessens an antecedent justification for $q$ (cf. Wright 2011). An interesting question is whether it is true that as $q$ gets via transmission from $p$ an additional quantitatively strengthening justification, $q$ also gets an independent justification. This seems true in some cases--for instance in Lottery. It is controversial, however, if it is the case that whenever $q$ gets via transmission a quantitatively strengthening justification, $q$ also gets an independent justification. Wright (2011) and Moretti & Piazza (2013) describe two examples in which a subject allegedly receives via transmission a quantitatively strengthening but not independent additional justification. To summarize, additional justification comes in two species at least: independent justification and quantitatively strengthening justification. This enables us to lay down three specifications of the general condition (C3$^{+}$) necessary for justification transmission, each of which represents a condition necessary for the transmission of one particular type of justification. Let's call these specifications (C3$^{\textrm{ft}}$), (C3$^{\textrm{ai}}$), and (C3$^{\textrm{aqs}}$). (C3$^{\textrm{ft}}$) $s$ has first time justification for $q$ in virtue of the satisfaction of (C1) and (C2). (C3$^{\textrm{ai}}$) $s$ has additional independent justification for $q$ in virtue of the satisfaction of (C1) and (C2). (C3$^{\textrm{aqs}}$) $s$ has additional quantitatively strengthening justification for $q$ in virtue of the satisfaction of (C1) and (C2). Transmission of first-time justification makes the advancement of justified belief through deductive reasoning possible. Yet the acquisition of first-time justification for $q$ isn't the sole possible improvement of one's epistemic position relative to $q$ that one could expect from transmission of justification. Moretti & Piazza (2013), for instance, describe a variety of different ways in which s's epistemic standing toward a proposition can be improved upon acquiring an additional justification for it via transmission. We conclude this section with a note about transmissivity as resolving doubts. Let us say that $s$ doubts $q$ just in case $s$ either disbelieves or withholds belief about $q$, namely, refrains from both believing and disbelieving $q$ after deciding about $q$. $s$'s doubting $q$ should be distinguished from $s$'s being open minded about $q$ and from $s$'s having no doxastic attitude whatsoever towards $q$ (cf. Tucker 2010a). Let us say that a deductively valid argument from $p$ to $q$ is able to resolve doubt about its conclusion just in case it is possible for $s$ to be rationally moved from doubting $q$ to believing $q$ solely in virtue of grasping the argument from $p$ to $q$ and the evidence offered for $p$. Some authors (e.g., Davies 1998, 2003, 2004, 2009; McLaughlin 2000; Wright 2002, 2003, 2007) have proposed that we should conceive of an argument's epistemic transmissivity in a way that is very closely related or even identical to the argument's ability to resolve doubt about its conclusion. Whereas some of these authors have eventually conceded that epistemic transmissivity cannot be defined as ability to resolve doubt (e.g., Wright 2011), others have attempted to articulate this view in full (see mainly Davies 2003, 2004, 2009). However, there is nowadays wide agreement in the literature that the property of being a transmissive argument doesn't coincide with the one of being an argument able to resolve doubt about its conclusion (see for example, Beebee 2001; Coliva 2010; Markie 2005; Pryor 2004; Bergmann 2004, 2006; White 2006; Silins 2005; Tucker 2010a). A good reason to think so is that whereas the property of being transmissive appears to be a genuinely epistemic property of an argument, the one of resolving doubt seems to be only a dialectical feature of it, which varies with the audience whose doubt the argument is used to address. ## 3. Failure of Epistemic Transmission ### 3.1 Trivial Transmission Failure versus Non-Transmissivity It is acknowledged that justification sometimes fails to transmit across known entailment (the acknowledgment dates back at least to Wright 1985). Moreover, it is no overstatement to say that the literature has investigated transmission failure more extensively than transmission of justification. As we have seen, justification based on $e$ transmits from $p$ to $q$ across the entailment if and only if (C1) $s$ has justification for $p$ based on $e$, (C2) $s$ knows that $q$ is deducible from $p$, and (C3$^{+}$) at least part of $s$'s justification for $q$ is based on the satisfaction of (C1) and (C2). It follows from this characterization that no justification based on e transmits from $p$ to $q$ across the entailment if (C1), (C2), or (C3$^{+}$) are not satisfied. These are cases of transmission failure. Some philosophers have argued that knowledge and justification are not always closed under competent (single-premise or multiple-premise) deduction. In the recent literature, the explanation of closure failure has often been essayed in terms of agglomeration of epistemic risk. This type of explanation is less controversial when applied to multi-premise deduction. An example of it concerning justification is the deduction from a high number $n$ of premises, each of which specifies that a different ticket in a fair lottery won't win, which are individually justified even though each premise is somewhat risky, to the conclusion that none of these $n$ tickets will win, which is too risky to be justified. (For more controversial cases in which knowledge or justification closure would fail across single-premise deduction because of risk accumulation, see for example Lasonen-Aarnio 2008 and Schechter 2013; for responses, see for instance Smith 2013 and 2016.) These cases of failure of epistemic closure can be taken to also involve failure of justification transmission. For instance, it can be argued that even if (C1$^{\}$) $s$ has justification for each of the $n$ premises stating that a ticket won't win, and (C2$^{\}$) $s$ knows that it follows from these premises that none of the $n$ tickets will win, $s$ has no justification--and, therefore, no justification in virtue of the satisfaction of (C1$^{\}$) and (C2$^{\}$)--for believing that none of the $n$ tickets will win. In these cases, transmission failure appears to be a consequence of closure failure, and it is therefore natural to treat the former simply as a symptomatic manifestation of the latter. For this reason, we will follow the current literature on transmission failure in not discussing these cases. (For an analysis of closure failure, see the entry on epistemic closure, especially section 6.) The current literature treats transmission failure as a self-standing phenomenon, in the sense that it focuses on cases in which transmission failure is not imputed--and thus not considered reducible--to an underlying failure of closure. For the rest of this entry, we shall follow suit and investigate transmission failure in this pure or genuine form. The most trivial cases of transmission failure are such that (C3$^{+}$) is unsatisfied because (C1) or (C2) are unsatisfied, but it is also true that (C3$^{+}$) would have been satisfied if both (C1) and (C2) have been satisfied (cf. Tucker 2010b [OIR]). It deserves emphasis that these cases involve arguments that aren't unsuitable for the purpose to transmit justification depending on evidence $e$: had the epistemic circumstances been congenial to the satisfaction of (C1) and (C2), these arguments would have transmitted the justification based on $e$ from $p$ to $q$. As we could put the point, these arguments are transmissive of the justification depending on $e$.[4] Cases of transmission failure of a more interesting variety are those in which, regardless of the validity of closure of justification, (C3$^{+}$) isn't satisfied because it could not have been satisfied, no matter whether or not (C1) and (C2) have been satisfied. These cases concern arguments non-transmissive of justification depending on a given evidence, i.e., arguments incapable of transmitting justification depending on that evidence under any epistemic circumstance. An example in point is Twins. Suppose first that $s$ has independent justification for $\rQ\_2$. In those circumstances (C1$\_{\textit{Twins}}$) $s$ has justification from $\rE\_2$ for $\rP\_2$. Furthermore (C2$\_{\textit{Twins}}$) $s$ does know that $\rP\_2$ entails $\rQ\_2$. However, $s$ hasn't justification for $\rQ\_2$ in virtue of the satisfaction of (C1$\_{\textit{Twins}}$) and (C2$\_{\textit{Twins}}$). So (C3$^{+}$) is not met. Suppose now that $s$ has no independent justification for $\rQ\_2$. Then, it isn't the case that (C1$\_{\textit{Twins}}$) $s$ has justification from $\rE\_2$ for $\rP\_2$. So, (C3$^{+}$) is not met either. Since there is no other possibility--either $s$ has independent justification for $\rQ\_2$ or she doesn't--it cannot be the case that $s$'s belief that $\rQ\_2$ is justified in virtue of her justification for $\rP\_2$ from $\rE\_2$ and her knowledge that $\rP\_2$ entails $\rQ\_2$. Note that none of the cases of transmission failure of justification considered above entails failure of epistemic closure. For in none of these cases $s$ has justification for believing $p, s$ knows that $p$ entails $q$, and $s$ fails to have justification for believing $q$. Unsurprisingly, the epistemologists contributing to the literature on transmission failure have principally devoted their attention to cases involving non-transmissive arguments. Epistemologists have endeavored to identify conditions whose satisfaction suffices to make an argument non-transmissive of justification based on a given evidence. The next section reviews the most influential of these proposals. ### 3.2 Varieties of Non-Transmissive Arguments Some non-transmissive arguments explicitly feature their conclusion among their premises. Suppose $p$ is justified for $s$ by $e$, and consider the premise-circular argument that deduces $p$ from itself. This argument cannot satisfy (C3$^{+}$) even if (C1) and (C2) are satisfied. The reason is that no part of $s$'s justification for $p$ can be acquired in virtue of, among other things, the satisfaction of (C2). For if (C1) is satisfied, $p$ is justified for $s$ by $e$ independently of $s$'s knowledge of $p$'s self-entailment, thus not in virtue of (C2). Non-transmissive arguments are not necessarily premise-circular. A different source of non-transmissivity instantiating a subtler form of circularity is the dependence of evidential relations on background or collateral information. This type of dependence is a rather familiar phenomenon: the boiling of a kettle gives one justification for believing that the temperature of the liquid inside is approximately 100degC only if one knows that the liquid is water and that atmospheric pressure is the one of the sea-level. It doesn't if one knows that the kettle is on the top of a high mountain, or the liquid is, say, sulfuric acid. Wright argues, for instance, that the following epistemic set-up, which he calls the information-dependence template, suffices for an argument's inability to transmit justification. > > > A body of evidence, $e$, is an information-dependent justification > for a particular proposition $p$ if whether $e$ justifies $p$ > depends on what one has by way of collateral information, $i$. > [...] Such a relationship is always liable to generate examples > of transmission failure: it will do so just when the particular $e, > p$, and $i$ have the feature that needed elements of the relevant > $i$ are themselves entailed by $p$ (together perhaps with other > warranted premises). In that case, any warrant supplied by $e$ for > $p$ will not be transmissible to those elements of $i$. (Wright > 2003: 59, edited.) > > > The claim that $s$'s justification from $e$ for $p$ requires $s$ to have background information $i$ is customarily understood as equivalent (in this context) to the claim that $s$'s justification from $e$ for $p$ depends on some type of independent justification for believing or accepting $i$.[5] Instantiating the information-dependence template appears sufficient for an argument's inability to transmit first-time justification. Consider again this triad: Twins $\rE\_2$. This girl looks just like Jessica. $\rP\_2$. This girl is actually Jessica. Therefore: $\rQ\_2$. This girl is not Jocelyn. Suppose $s$'s background information entails that Jessica and Jocelyn are indistinguishable twins. Imagine that $s$ acquires $\rE\_2$. It is intuitive that $\rE\_2$ could justify $\rP\_2$ for $s$ only if $s$ had independent justification for believing $\rQ\_2$ in the first instance. Thus, Twins instantiates the information-dependence template. Note that $s$ acquires first-time justification for $\rQ\_2$ in Twins only if (C1$\_{\textit{Twins}}$) $\rE\_2$ gives her justification for $\rP\_2$, (C2$\_{\textit{Twins}}$) $s$ knows that $\rP\_2$ entails $\rQ\_2$ and (C3$^{\textrm{ft}}\_{\textit{Twins}}$) $s$ acquires first-time justification for believing $\rQ\_2$ in virtue of (C1$\_{\textit{Twins}}$) and (C2$\_{\textit{Twins}}$). The satisfaction of (C3$^{\textrm{ft}}\_{\textit{Twins}}$) requires $s$'s justification for believing $\rQ\_2$ not to be independent of $s$'s justification from $\rE\_2$ for $\rP\_2$. However, if (C1$\_{\textit{Twins}}$) is true, the informational-dependence template requires $s$ to have justification for believing $\rQ\_2$ independently of $s$'s justification from $\rE\_2$ for $\rP\_2$. Thus, when the information-dependence template is instantiated, (C1$\_{\textit{Twins}}$) and (C3$^{\textrm{ft}}\_{\textit{Twins}}$) cannot be satisfied at once. In general, no argument satisfying this template together with a given evidence will be transmissive of first-time justification based on that evidence. One may wonder whether a deductive argument from $p$ to $q$ instantiating the information-dependence template is unable to transmit additional justification for $q$. The answer seems affirmative when the additional justification is independent justification. Suppose the information-dependence template is instantiated such that $s$'s justification for $p$ from $e$ depends on $s$'s independent justification for $q$. Note that $s$ acquires additional independent justification for $q$ only if (C1) $e$ gives her justification for $p$, (C2) $s$ knows that $p$ entails $q$, and (C3$^{\textrm{ai}}$) $s$ acquires an additional independent justification in virtue of (C1) and (C2). This additional justification is independent of $s$'s antecedent justification for $q$ only if, in particular, condition (IN3) of the characterization of additional independent justification is satisfied. (IN3) says that if $s$ had not been antecedently justified in believing $q$, upon learning $e, s$ would have acquired via transmission a first-time justification for believing $q$. (IN3) entails that if $q$ were not antecedently justified for $s, e$ would still justify $p$ for $s$. Hence, the satisfaction of (IN3) is incompatible with the instantiation of the information-dependence template, which entails that if $s$ had not antecedent justification for $q, e$ would not justify $p$ for $s$. The instantiation of the information-dependence template then precludes transmission of independent justification. As suggested by Wright (2007), the instantiation of the information-dependence template might also appear sufficient for an argument's inability to transmit additional quantitatively strengthening justification. This claim might appear intuitively plausible: perhaps it is reasonable that if the justification from $e$ for $p$ depends on independent justification for another proposition $q$, the strength of the justification available for $q$ sets an upper bound to the strength of the justification possibly supplied by $e$ for $p$. However, the examples by Wright (2011) and Moretti & Piazza (2013) mentioned in Sect. 2 appear to undermine this intuition. For they involve arguments that instantiate the information-dependent template yet seem to transmit quantitatively strengthening justification to their conclusions. (Alspector-Kelly 2015 suggests that these arguments are a symptom that Wright's explanation of non-transmissivity is overall inadequate.) Some authors have attempted Bayesian formalizations of the information dependence template (see the supplement on Bayesian Formalizations of the Information-Dependence Template.) Furthermore, Coliva (2012) has proposed a variant of the same template. In accordance with the information-dependence template, $s$'s justification from $e$ for $p$ fails to transmit to $p$'s consequence $q$ whenever $s$'s possessing that justification for $p$ requires $s$'s independent justification for $q$. According to Coliva's variant, $s$'s justification from $e$ for $p$ fails to transmit to $q$ whenever $s$'s possessing the latter justification for $p$ requires $s$'s independent assumption of $q$, whether this assumption is justified or not. Pryor (2012: SS VII) can be read as pressing objections against Coliva's template, which are addressed in Coliva (2012). We have seen that non-transmissivity may depend on premise-circularity or reliance on collateral information. There is at least a third possibility: an argument can be non-transmissive of the justification for its premise(s) based on given evidence because the evidence justifies directly the conclusion--i.e., independently of the argument itself (cf. Davies 2009). In this case the argument instantiates indirectness, for $s$'s going through the argument would result in nothing but an indirect (and unneeded) detour for justifying its conclusion. If $e$ directly justifies $q$, no part of the justification for $q$ is based on, among other things, $s$'s knowledge of the inferential relation between $p$ and $q$. So (C3$^{+}$) is unfulfilled whether or not (C1) and (C2) are fulfilled. Here is an example from Wright (2002): Soccer $\rE\_6$. Jones has just kicked the ball between the white posts. $\rP\_6$. Jones has just scored a goal. Therefore: $\rQ\_6$. A game of soccer is taking place. Suppose $s$ learns evidence $\rE\_6$. On ordinary background information, (C1$\_{\textit{Socc}}$) $\rE\_6$ justifies $\rP\_6$ for $s$, (C2$\_{\textit{Socc}}$) $s$ knows that $\rP\_6$ entails $\rQ\_6$, and $\rE\_6$ also justifies $\rQ\_6$ for $s$. It seems false, however, that $\rQ\_6$ is justified for $s$ in virtue of the satisfaction of (C1$\_{\textit{Socc}}$) and (C2$\_{\textit{Socc}}$). Quite the reverse, $\rQ\_6$ seems justified for $s$ by $\rE\_6$ independently the satisfaction of (C1$\_{\textit{Socc}}$) and (C2$\_{\textit{Socc}}$). This is so because in the (imagined) actual scenario it seems true that $s$ would still possess a justification for $\rQ\_6$ based on $\rE\_6$ even if $\rE\_6$ did not justify $\rP\_6$ for $s$. In fact suppose $s$ had noticed that the referee's assistant raised her flag to signal Jones's off-side. Against this altered background information, $\rE\_6$ would no longer justify $\rP\_6$ for $s$ but it would still justify $\rQ\_6$ for $s$. Thus, Soccer is non-transmissive of the justification depending on $\rE\_6$. In general, no argument instantiating indirectness in relation to some evidence is transmissive of justification based on that evidence. The information-dependence template and indirectness are diagnostics for a deductive argument's inability to transmit the justification for its premise(s) $p$ based on evidence $e$ to its conclusion $q$, where $e$ is conceived of as a believed proposition capable of supplying inferential and (typically) fallible justification for $p$. (Note that even though $e$ is conceived of as a belief, the collateral information $i$, which is part of the template, doesn't need to be believed on some views.) The justification for a proposition $p$ might come in other forms. For instance, it has been proposed that a proposition $p$ about the perceivable environment around the subject $s$ can be justified by the fact that $s$ sees that $p$ (where 'sees' is taken to be factive). In this case, $s$ is claimed to attain a kind of non-inferential and infallible justification for believing $p$. This view has been explored by epistemological disjunctivists (see for instance McDowell 1982, 1994 and 2008, and Pritchard 2007, 2008, 2009, 2011 and 2012a). One might find it intuitively plausible that when $s$ sees that $p, s$ attains non-inferential and infallible justification for believing $p$ that doesn't rely on $s$'s background information. Since this justification for believing $p$ would be a fortiori unconstrained by $s$'s independent justification for believing any consequence $q$ of $p$, in these cases the information-dependence template could not possibly be instantiated. Therefore, one might be tempted to conclude that when $s$ sees that $p, s$ acquires a justification that typically transmits to the propositions that $s$ knows to be deducible from $p$ (cf. Wright 2002). Pritchard (2009, 2012a) comes very close to endorsing this view explicitly. The latter contention has not stayed unchallenged. Suppose one accepts a notion of epistemic justification with internalist resonances saying that a factor $J$ is relevant to $s$'s justification only if $s$ is able to determine, by reflection alone, whether $J$ is or is not realized. On this notion, $s$'s seeing that $p$ cannot provide $s$ with justification for believing $p$ unless $s$ can rationally claim that she is seeing that $p$ upon reflection alone. Seeing that $p$, however, is subjectively indistinguishable from hallucinating that $p$, or from being in some other delusional state in which it merely seems to $s$ that she is seeing that $p$. Hence, one may find it compelling that $s$ can claim by reflection alone that she's seeing that $p$ only if $s$ has an independent reason for ruling out that it merely seems to her that she does (cf. Wright 2011). If this is true, for many deductive arguments from $p$ to $q, s$ won't be able to acquire non-inferential and infallible justification for believing $p$ of the type described by the disjunctivist and transmit it to $q$. This will happen whenever $q$ is the logical negation of a proposition ascribing to $s$ some delusionary mental state in which it merely seems to her that she directly perceives that $p$. Wright's disjunctive template is meant to be a diagnostic of transmission failure of non-inferential justification when epistemic justification is conceived of in the internalist fashion suggested above.[6] According to Wright (2000), for any propositions $p, q$ and $r$ and subject $s$, the disjunctive template is instantiated whenever: (D1) $p$ entails $q$; (D2) $s$'s justification for $p$ consists in $s$'s being in a state subjectively indistinguishable from a state in which $r$ would be true; (D3) $r$ is incompatible with $p$; (D4) $r$ would be true if $q$ were false. To see how this template works, consider again the following triad: Zebra $\rE\_3$. The animals in the pen look like zebras. $\rP\_3$. The animals in the pen are zebras. Therefore: $\rQ\_3$. The animals in the pen are not mules cleverly disguised to look like zebras. The justification from $\rE\_3$ for $\rP\_3$ arguably fails to transmit across the inference from $\rP\_3$ to $\rQ\_3$ because of the satisfaction of the information-dependence template. For it seems true that $s$ could acquire a justification for believing $\rP\_3$ on the basis of $\rE\_3$ only if $s$ had an independent justification for believing $\rQ\_3$. Now suppose that $s$'s justification for $\rP\_3$ is based on, not $\rE\_3$ but $s$'s seeing that $\rP\_3$. Let's call Zebra\* the corresponding variant of Zebra. Given the non-inferential nature of the justification considered for $\rP\_3$, Zebra\* could not instantiate the information-dependence template. However, it is easy to check that Zebra\* instantiates the disjunctive template. To begin with, (D1$\_{\textit{Zebra}}$) $\rP\_3$ entails $\rQ\_3$; (D2$\_{\textit{Zebra}}$) $s$'s justification for believing $\rP\_3$ is constituted by $s$'s seeing that $\rP\_3$, which is subjectively indistinguishable from the state that $s$ would be in if it were true that $\rR\_{\textit{Zebra}}$. the animals in the pen are mules cleverly disguised to look like zebras; (D3$\_{\textit{Zebra}}$) $\rR\_{\textit{Zebra}}$ is incompatible with $\rP\_3$; and, trivially, (D4$\_{\textit{Zebra}}$) if $\rQ\_3$ were false $\rR\_{\textit{Zebra}}$ would be true. Since Zebra\* instantiates the disjunctive template, it is non-transmissive of at least first-time justification. In fact note that $s$ acquires first-time justification for $\rQ\_3$ in this case if and only if (C1$\_{\textit{Zebra}}$) $s$ has justification for $\rP\_3$ based on seeing that $\rP\_3$, (C2$\_{\textit{Zebra}}$) $s$ knows that $\rP\_3$ entails $\rQ\_3$, and (C3$^{\textrm{ft}}\_{\textit{Zebra}}$) $s$ acquires first-time justification for believing $\rQ\_3$ in virtue of (C1$\_{\textit{Zebra}}$) and (C2$\_{\textit{Zebra}}$). Also note that (C3$^{\textrm{ft}}\_{\textit{Zebra}}$) requires $s$'s justification for believing $\rQ\_3$ not to be independent of $s$'s justification for $\rP\_3$ based on seeing that $\rP\_3$. However, if (C1$\_{\textit{Zebra}}$) is true, the disjunctive template requires $s$ to have justification for believing $\rQ\_3$ independent of $s$'s justification for $\rP\_3$ based on her seeing that $\rP\_3$. (For if $s$ could not independently exclude that $\rQ\_3$ is false, given (D4), (D3) and (D2), $s$ could not exclude that the incompatible alternative $\rR\_{\textit{Zebra}}$ to $\rP\_3$, which $s$ cannot subjectively distinguish from $\rP\_3$ on the ground of her seeing that $\rP\_3$, is true.) Thus, when the disjunctive template is instantiated, (C1$\_{\textit{Zebra}}$) and (C3$^{\textrm{ft}}\_{\textit{Zebra}}$) cannot be satisfied at once. The disjunctive template has been criticized by McLaughlin (2003) on the ground that the template is instantiated whenever the justification for $p$ is fallible, i.e. compatible with $p$'s falsity. Here is an example from Brown (2004). Take this deductive argument: Fox $\rP\_7$. The animal in the garbage is a fox. Therefore: $\rQ\_7$. The animal in the garbage is not a cat. Suppose $s$ has a fallible justification for believing $\rP\_7$ based on $s$'s experience as if the animal in the garbage is a fox. Take now $\rR\_{\textit{fox}}$ to be $\rP\_7$'s logical negation. Since the justification that $s$ has for $\rP\_7$ is fallible, condition (D2) above is met by default. As one can easily check, conditions (D1), (D3), and (D4) are also met. So, Fox instantiates the disjunctive template. Yet it is intuitive that Fox does transmit justification to its conclusion. One could respond to McLaughlin that his objection is misplaced because the disjunctive template is meant to apply to infallible, and not fallible, justification. A more interesting response to McLaughlin is to refine some condition listed in the disjunctive template to block McLaughlin's argument while letting this template account for transmission failure of both fallible and infallible justification. Wright (2011) for instance suggests replacing (D3) with the following condition:[7] (D3$^{\}$) $r$ is incompatible with some presupposition of the cognitive project of obtaining justification for $p$ in the relevant fashion. According to Wright's (2011) characterization, a presupposition of a cognitive project is any condition such that doubting it before carrying out the project would rationally commit one to doubting the significance or competence of the project irrespective of its outcome.[8] For a wide class of cognitive projects, examples of these presuppositions include: the normal and proper functioning of the relevant cognitive faculties, the reliability of utilized instruments, the obtaining of the circumstances congenial to the proposed method of investigation, the soundness of relevant principles of inference utilized in developing and collating one's results, and so on. With (D3$^{\}$) in the place of (D3), Fox no longer instantiates the disjunctive template. For the truth of $\rR\_{\textit{fox}}$, stating that the animal in the garbage is not a fox, appears to jeopardize no presupposition of the cognitive project of obtaining perceptual justification for $\rP\_7$. Thus (D3$^{\}$) is not fulfilled. On the other hand, arguments that intuitively don't transmit do satisfy (D3$^{\}$). Take Zebra\. In this case $\rR\_{\textit{fox}}$ states that the animals in the pen are mules cleverly disguised to look like zebras. Since $\rR\_{\textit{fox}}$ entails that conditions are unsuitable for attaining perceptual justification for believing $\rP\_3$, $\rR\_{\textit{fox}}$ looks incompatible with a presupposition of the cognitive project of obtaining perceptual justification for $\rP\_3$. Thus, $\rR\_{\textit{fox}}$ does satisfy (D3$^{\}$) in this case. ### 3.3 Non-Standard Accounts In this section we outline two interesting branches in the literature on transmission failure of justification and warrant across valid inference. We start with Smith's (2009) non-standard account of non-transmissivity of justification. Then, we present Alspector-Kelly's (2015) account of non-transmissivity of Plantinga's warrant. According to Smith (2009), epistemic justification requires reliability. $s$'s belief that $p$, held on the basis of $s$'s belief that $e$, is reliable in Smith's sense just in case in all possible worlds including $e$ as true and that are as normal (from the perspective of the actual world) as the truth of $e$ permits, $p$ is also true. Consider Zebra again. Zebra $\rE\_3$. The animals in the pen look like zebras. $\rP\_3$. The animals in the pen are zebras. Therefore: $\rQ\_3$. The animals in the pen are not mules cleverly disguised to look like zebras. $s$'s belief that $\rP\_3$ is reliable in Smith's sense when it is based on $s$'s belief that $\rE\_3$. (Disguising mules to make them look like zebras is certainly an abnormal practice.) Thus, all possible $\rE\_3$-worlds that are as normal as the truth of $\rE\_3$ permits aren't worlds in which the animals in the pen are cleverly disguised mules. Rather, they are $\rP\_3$-worlds--i.e., worlds in which the animals in the pen are zebras. Smith describes two ways in which a belief can possess this property of being reliable. One is that a belief that $p$ has it in virtue of the modal relationship with its basis $e$. In this case, $e$ is a contributing reliable basis. Another possibility is when it is the content of the belief that $p$, rather than the belief's modal relationship with its basis $e$, that guarantees by itself the belief's reliability. In this case, $e$ is a non-contributing reliable basis. An example of the first kind is $s$'s belief that $\rP\_3$, which is reliable because of its modal relationship with $\rE\_3$. There are obviously many sufficiently normal worlds in which $\rP\_3$ is false, but no sufficiently normal world in which $\rP\_3$ is false and $\rE\_3$ true. An example of the second kind is $s$'s belief that $\rQ\_3$ as based on $\rE\_3$. It is this belief's content, and not its modal relationship with $\rE\_3$, that guarantees its reliability. As Smith puts it, there are no sufficiently normal worlds in which $\rE\_3$ is true and $\rQ\_3$ is false, but this is simply because there are no sufficiently normal worlds in which $\rQ\_3$ is false. According to Smith, a deductive inference from $p$ to $q$ fails to transmit to $q$ justification relative to $p$'s basis $e$, if $e$ is a contributing reliable basis for believing $p$ but is a non-contributing reliable basis for believing $q$. In this case the inference fails to explain $q$'s reliability: if $s$ deduced one proposition from another, she would reliably believe $q$, but not--not even in part--in virtue of having inferred $q$ from $p$ (as held on the basis of $e)$. Zebra fails to transmit to $\rQ\_3$ justification relative to $\rP\_3$'s basis $\rE\_3$ in this sense.[9] Let's turn to Alspector-Kelly (2015). As we have seen, Wright's analysis of failure of warrant transmission interprets the epistemic good whose transmission is in question as, roughly, the same as epistemic justification.[10] By so doing, Wright departs from Plantinga's (1993a, 1993b) influential understanding of warrant as the epistemic quality that (whatever it is) suffices to turn true belief into knowledge. One might wonder, however, if there are deductive arguments incapable of transmitting warrant in Plantinga's sense. (Hereafter we use 'warrant' only to refer to Plantinga's warrant.) Alspector-Kelly answers this question affirmatively contending that certain arguments cannot transmit warrant because the cognitive project of establishing their conclusion via inferring it from their premise is procedurally self-defeating. Alspector-Kelly follows Wright (2012) in characterizing a cognitive project as a pair of a question and a procedure that one executes to answer the question. An enabling condition of a cognitive project is, for Alspector-Kelly, any proposition such that, unless it is true, one cannot learn the answer to its defining question by executing the procedure associated with it. That a given object $o$ is illuminated, for example, is an enabling condition of the cognitive project of determining its color by looking at it. For one cannot learn by sight that $o$ is of a given color unless $o$ is illuminated. Enabling conditions can be opaque. An enabling condition of a cognitive project is opaque, relative to some actual or possible result, if it is the case that whenever the execution of the associated procedure yields this result, it would have produced the same result had the condition been unfulfilled. The enabling condition that $o$ be illuminated of the cognitive project just considered is not opaque, since looking at $o$ never produces the same response about $o$'s color when $o$ is illuminated and when it isn't. (In the second case, it produces no response.) Now consider the cognitive project of establishing by sight whether ($\rP\_3)$ the animals enclosed in the pen are zebras. An enabling condition of this project states that ($\rQ\_3)$ the animals in the pen are not mules disguised to look like zebras. For one couldn't learn whether $\rP\_3$ is true by looking, if $\rQ\_3$were true. $\rQ\_3$ is opaque with respect to the possible outcome that $\rP\_3$. In fact, suppose $\rQ\_3$ is satisfied because the pen contains (undisguised) zebras. In this case, looking at them will attest they are zebras, and the execution of the procedure associated with this project will yield the outcome that $\rP\_3$. But this is exactly the outcome that would be generated by the execution of the same procedure if $\rQ\_3$ were not satisfied. In this case too, looking at the animals would produce the response that $\rP\_3$. We can now elucidate the notion of a procedurally self-defeating cognitive project. The distinguishing feature of any project of this type is that it seeks to answer the question whether an opaque enabling condition--call it '$q$'--of the project itself is fulfilled. A project of this type has the defect that it necessarily produces the response that $q$ is fulfilled when $q$ is unfulfilled. Given this, it is intuitive that it cannot yield warrant to believe $q$. As an example of a procedurally self-defeating project, consider trying to answer the question whether your informant is sincere by asking her this very question. Your informant's being sincere is an enabling condition of the very project you are carrying out. If your informant were not sincere, you couldn't learn anything from her. This condition is also opaque with respect to the possible response that she is sincere. For your informant would respond that she is sincere both in case she is so and in case she isn't. Since the execution of this project is guaranteed to yield the result that your informant is sincere when she isn't, it is intuitive that it cannot yield warrant to believe that the informant is sincere. Alspector-Kelly contends that arguments like Zebra don't transmit warrant because the cognitive project of determining inferentially whether their conclusion is true is procedurally self-defeating in the sense advertised. Let's apply the explanation to Zebra. Imagine you carry out Zebra. This initially commits you to executing the project of establishing whether $\rP\_3$ by looking. Suppose you get $\rE\_3$ and hence $\rP\_3$ as an outcome. As we have seen, $\rQ\_3$ is an opaque enabling condition relative to the outcome $\rP\_3$ of the cognitive project you have executed. Imagine that now, having received $\rP\_3$ as a response to your first question, you embark in the second project which carrying out Zebra commits you to: establishing whether $\rQ\_3$ is true by inference from $\rP\_3$. This project appears doomed. $\rQ\_3$ is an enabling condition of the initial project of determining whether $\rP\_3$, which is opaque with respect to the very premise $\rP\_3$. It follows from this that $\rQ\_3$ is also an enabling condition of the second project. For if $\rQ\_3$ were unfulfilled, you couldn't learn $\rP\_3$ and, a fortiori, you couldn't learn anything else--$\rQ\_3$ included--by inference from $\rP\_3$. Another consequence is that $\rQ\_3$ is an enabling condition of the second project that is opaque relative to the very outcome that $\rQ\_3$. Suppose first that $\rQ\_3$ is true and that, having visually inspected the pen, you have verified $\rP\_3$. You then execute the inferential procedure associated with the second project and produce the outcome that $\rQ\_3$. Now consider the case in which $\rQ\_3$ is false. Looking into the pen still generates the outcome that $\rP\_3$ is true. Thus, when you execute the procedure associated with the second project, you still infer that $\rQ\_3$ is true. Since you would get the result that $\rQ\_3$ is true even if $\rQ\_3$ were false, the second project is procedurally self-defeating. In conclusion, carrying out Zebra commits you to executing a project that cannot generate warrant for believing its conclusion $\rQ\_3$. This explanation of non-transmissivity generalizes to all structurally similar arguments. ## 4. Applications The notions of transmissive and non-transmissive argument, above and beyond being investigated for their own sake, have been put to work in relation to specific philosophical problems and issues. An important problem is whether Moore's infamous proof of an external world is successful, and whether so are structurally similar Moorean arguments directed against the perceptual skeptic and--more recently--the non-believer. Another issue pertains to the solution of McKinsey paradox. A third issue concerns Boghossian's (2001, 2003) explanation of our logical knowledge via implicit definitions, criticized as resting on a non-transmissive argument schema (see for instance Ebert 2005 and Jenkins 2008). As the debate focusing on the last topic is only at an early stage of its development, it is preferable to concentrate on the first two, which will be reviewed in respectively Sect. 4.1 and Sect. 4.2 below. ### 4.1 Moorean Proofs, Perceptual Justification and Religious Justification Much of the contemporary debate on Moore's proof of an external world (see Moore 1939) is interwoven with the topic of epistemic transmission and its failure. Moore's proof can be reconstructed as follows: Moore $\rE\_8$. My experience is in all respects as of a hand held up in front of my face. $\rP\_8$. Here is a hand. Therefore: $\rQ\_8$. There is a material world (since any hand is a material object existing in space). Evidence $\rE\_8$ in Moore is constituted by a proposition believed by $s$. One might suggest, however, that this is a misinterpretation of Moore's proof (and variants of it that we shall consider shortly). One might argue that what is meant to give $s$ justification for believing $\rP\_8$ is $s$'s experience of a hand. Nevertheless, most of the epistemologists participating in this debate implicitly or explicitly assume that one's experience as if $p$ and one's belief that one has an experience as if $p$ have the same justifying power (cf. White 2006 and Silins 2007). Many philosophers find Moore's proof unsuccessful. Philosophers have proposed explanations of this impression according to which Moore is non-transmissive in some of the senses described in Sect. 3 (see mainly Wright 1985, 2002, 2007 and 2011).[11] A different explanation is that Moore's proof does transmit justification but is dialectically ineffective (see mainly Pryor 2004). According to Wright, there exist cornerstone propositions (or simply cornerstones), where $c$ is a cornerstone for an area of discourse $d$ just in case for any proposition $p$ belonging to $d, p$ could not be justified for any subject $s$ if $s$ had no independent justification for accepting $c$ (see mainly Wright 2004).[12] Cornerstones for the area of discourse about perceivable things are for instance the logical negations of skeptical conjectures, such as the proposition that one's experiences are nothing but one's hallucinations caused by a Cartesian demon or the Matrix. Wright contends that the conclusion $\rQ\_8$ of Moore is also a cornerstone for the area of discourse about perceivable things. Adapting terminology introduced by Pryor (2004), Wright's conception of the architecture of perceptual justification thus treats $\rQ\_8$ conservatively with respect to any perceptual hypothesis $p$: if $s$ had no independent justification for $\rQ\_8$, no proposition $e$ describing an apparent perception could supply $s$ with prima facie justification for any perceptual hypothesis $p$. It follows from this that Moore instantiates the information-dependence template considered in Sect. 3.2. For $\rQ\_8$ is part of the collateral information which $s$ needs independent justification for if $s$ is to receive some justification for $\rP\_8$ from $\rE\_8$. Hence Moore is non-transmissive (see mainly Wright 2002). Note that the thesis that Moore's proof is epistemologically useless because non-transmissive in this sense is compatible with the claim that by learning $\rE\_8, s$ does acquire a justification for believing $\rP\_8$. For instance, Wright (2004) contends that we all have a special kind of non-evidential justification, which he calls entitlement, for accepting $\rQ\_8$ as well as other cornerstones in general.[13] So, by learning $\rE\_8$ we do acquire justification for $\rP\_8$. Wright's analysis of Moore's proof and Wright's conservatism have mostly been criticized in conjunction with his theory of entitlement. A presentation of these objections is beyond the scope of this entry. (See however Davies 2004; Pritchard 2005; Jenkins 2007. For a defense of Wright's views see for instance Neta 2007 and Wright 2014.) As anticipated, other philosophers contend that Moore's proof does transmit justification and that its ineffectiveness has a different explanation. An important conception of the architecture of perceptual justification, called dogmatism in Pryor (2000, 2004), embraces a generalized form of liberalism about perceptual justification. This form of liberalism is opposed to Wright's conservatism, and claims that to have prima facie perceptual justification for believing $p$ from an apparent perception that $p, s$ doesn't need independent justification for believing the negation of skeptical conjectures or non-perceiving hypotheses like $\notQ\_8$. This is so, for the dogmatist, because our experiences give us immediate and prima facie justification for believing their contents. Saying that perceptual justification is immediate is saying that it doesn't presuppose--not even in part--justification for anything else. Saying that justification is prima facie is saying that it can be defeated by additional evidence. Our perceptual justification would be defeated, for example, by evidence that a relevant non-perceiving hypothesis is true or just as probable as its negation. For instance, $s$'s perceptual justification for $\rP\_8$ would be defeated by evidence that $\notQ\_8$ is true, or that $\rQ\_8$ and $\notQ\_8$ are equally probable. On this point the dogmatist and Wright do agree. They disagree on whether $s$'s perceptual justification for $\rP\_8$ requires independent justification for believing or accepting $\rQ\_8$. The dogmatist denies that $s$ needs that independent justification. Thus, according to the dogmatist, Moore's proof transmits the perceptual justification available for its premise to the conclusion.[14] The dogmatist argues (or may argue), however, that Moore's proof is dialectically flawed (cf. Pryor 2004). The contention is that Moore's is unsuccessful because it is useless for the purpose to convince the idealist or the external world skeptic that there is an external (material) world. In short, neither the idealist nor the global skeptic believes that there is an external world. Since they don't believe $\rQ\_8$, they are rationally required to distrust any perceptual evidence offered in favor of $\rP\_8$ in the first instance. For this reason they both will reject Moore's proof as one based on an unjustified premise.[15] Moretti (2014) suggests that the dogmatist could alternatively contend that Moore's proof is non-transmissive because $s$'s experience of a hand gives $s$ immediate justification for believing both the premise $\rP\_8$ and the conclusion $\rQ\_8$ of the proof at once. According to this diagnosis, Moore's proof is epistemically flawed because it instantiates a variant of indirectness (in which the evidence is an experience of a hand). Pryor's analysis of Moore's proof has principally been criticized in conjunction with his liberalism in epistemology of perception. (See Cohen 2002, 2005; Schiffer 2004; Wright 2007; White 2006; Siegel & Silins 2015.) Some authors (e.g., Wright 2003; Pryor 2004; White 2006; Silins 2007; Neta 2010) have investigated whether certain variants of Moore are transmissive of justification. These arguments start from a premise like $\rP\_8$, describing a (supposed) perceivable state of affairs of the external world, and deduce from it the logical negation of a relevant skeptical conjecture. Consider for example the variant of Moore that starts from $\rP\_8$ and replaces $\rQ\_8$ with: $\rQ\_8^\$. It is not the case that I am a handless brain in a vat fed with the hallucination of a hand held up in front of my face. Let's call Moore\** this variant of Moore. While dogmatists a la Pryor argue that Moore\* is transmissive but dialectically flawed (cf. Pryor 2004), conservatives a la Wright contend that it is non-transmissive (cf. Wright 2007). Although it remains very controversial whether or not Moore is transmissive, epistemologists have found some prima facie reason to think that arguments like Moore\* are non-transmissive. An important difference between Moore and Moore\* is this: whereas the logical negation of $\rQ\_8$ does not explain the evidential statement $\rE\_8$ ("My experience is in all respects as of a hand held up in front of my face") adduced in support of $\rP\_8$, the logical negation of $\rQ\_8^\$--$\notQ\_8^\$--somewhat explains $\rE\_8$. Since $\notQ\_8^\$ provides a potential explanation of $\rE\_8$, is it intuitive that $\rE\_8$ is evidence* (perhaps very week) for believing $\notQ\_8^\$. It is easy to conclude from this that $s$ cannot acquire justification for believing $\rQ\_8^\$ via transmission through Moore\* upon learning $\rE\_8$. For it is intuitive that if this were the case, $\rE\_8$ should count as evidence for $\rQ\_8^\$. But this is impossible: one and the same proposition cannot simultaneously be evidence for a hypothesis and its logical negation. By formalizing intuitions of this type, White (2006) has put forward a simple Bayesian argument to the effect that Moore\** and similar variants of Moore are not transmissive of justification.[16] (See Silins 2007 for discussion. For responses to White, see Weatherson 2007; Kung 2010; Moretti 2015.) From the above analysis it is easy to conclude that the information-dependence template is satisfied by Moore\* and akin proofs. In fact note that if $\rE\_8$ is evidence for both $\rP\_8$ and $\notQ\_8^\$, it seems correct to say that $s$ can acquire a justification for believing $\rP\_8$ only if $s$ has independent justification for disbelieving* $\notQ\_8^\$ and thus believing* $\rQ\_8^\$. Since $\notQ\_8^\$ counts as a non-perceiving hypothesis for Pryor, this gives us a reason to doubt dogmatism (cf. White 2006).[17] Coliva (2012, 2015) defends a view--baptized by her moderatism--that aims to be a middle way between Wright's conservatism and Pryor's dogmatism. The moderate contends--against the conservative--that cornerstones cannot be justified and that to possess perceptual justification $s$ needs no justification for accepting any cornerstones. On the other hand, the moderate claims--against the dogmatist--that to possess perceptual justification $s$ needs to assume (without justification) relevant cornerstones.[18] By relying on her variant of the information-dependent template described in Sect. 3.2, Coliva concludes that neither Moore nor any proof like Moore\* is transmissive. (For a critical discussion of moderatism see Avnur 2017; Baghramian 2017; Millar 2017; Volpe 2017; Coliva 2017.) Epistemological disjunctivists like McDowell and Pritchard have argued that in paradigmatic cases of perceptual knowledge, what is meant to give $s$ justification for believing $\rP\_8$ is, not $\rE\_8$, but $s$'s factive state of seeing that $\rP\_8$. This seems to have consequences for the question whether Moore\* transmits propositional justification. Pritchard explicitly defends the claim that when $s$ sees that $\rP\_8$, thereby learning that $\rP\_8, s$ can come to know by inference from $\rP\_8$ the negation of any skeptical hypothesis inconsistent with $\rP\_8$, like $\rQ\_8^\$ (cf. Pritchard 2012a: 129-30). This may encourage the belief that, for Pritchard, Moore\** can transmit the justification for $\rP\_8$, based on s's seeing that $\rP\_8$, to $\rQ\_8^\$ (see Lockhart 2018 for an explicit argument to this effect). This claim must however be handled with some care. As we have seen in Sect. 2, Pritchard contends that when one knows $p$ on the basis of evidence $e$, one can know $p$'s consequence $q$ by inference from $p$ only if $e$ sufficiently supports $q$. For Pritchard this condition is met when $s$'s support for believing $\rP\_8$ is constituted by $s$'s seeing that $\rP\_8, s$'s epistemic situation is objectively good (i.e., $s$'s cognitive faculties are working properly in a cooperative environment) and the skeptical hypothesis ruled out by $\rQ\_8^\$ has not been epistemically motivated. For, in this case, $s$ has a reflectively accessible factive support for believing $\rP\_8$ that entails--and so sufficiently supports--$\rQ\_8^\$. Thus, in this case, nothing stands in the way of $s$ competently deducing $\rQ\_8^\$ from $\rP\_8$, thereby coming to know $\rQ\_8^\$ on the basis of $\rP\_8$. If upon deducing* one proposition from another, $s$ comes to justifiably believe $\rQ\_8^\$ for the first-time, the inference from $\rP\_8$ to $\rQ\_8^\$ presumably transmits doxastic justification. (See the supplement on Transmission of Propositional Justification versus Transmission of Doxastic Justification.) It is more dubious, however, that when $s$'s support for $\rP\_8$ is constituted by $s$'s seeing that $\rP\_8$, Moore\* is also transmissive of propositional justification. For instance, one might contend that Moore\* is non-transmissive because it instantiates the disjunctive template described in Sect. 3.2 (cf. Wright 2002). To start with, $\rP\_8$ entails $\rQ\_8^\$, so (D1) is satisfied. Let $\rR^{\}\_{\textit{Moore}}$ be the proposition that this is no hand but $s$ is victim of a hallucination of a hand held up before her face. Since $\rR^{\}\_{\textit{Moore}}$ would be true if $\rQ\_8^\$ were false, (D4) is also satisfied. Furthermore, take the grounds of $s$'s justification for $\rP\_8$ to be $s$'s seeing that $\rP\_8$. Since this experience is a state for $s$ indistinguishable from one in which $\rR^{\}\_{\textit{Moore}}$ is true, (D2) is also satisfied. Finally, it might be argued that the proposition that one is not hallucinating is a presupposition of the cognitive project of learning about one's environment through perception. It follows that $\rR^{\}\_{\textit{Moore}}$ is incompatible with a presupposition of $s$'s cognitive project of learning about one's environment through perception. Thus (D3$^{\}$) appears fulfilled too. So, Moore\** won't transmit the justification that $s$ has for $\rP\_8$ to $\rQ\_8^\$. To resist this conclusion, a disjunctivist might insist that Moore\*, relative to $s$'s support for $\rP\_8$ supplied by $s$'s seeing that $\rP\_8$, doesn't always instantiate the disjunctive template* because (D2) isn't necessarily fulfilled (cf. Lockhart 2018.) By invoking a distinction drawn by Pritchard (2012a), one might contend that (D2) isn't necessarily fulfilled because $s$, though unable to introspectively discriminate seeing that $\rP\_8$ from hallucinating it, may have evidence that favors the hypothesis that she is in the first state, which makes the state reflectively accessible. For Pritchard, this happens--as we have seen--when $s$ sees that $\rP\_8$ in good epistemic conditions and the skeptical conjecture that $s$ is having a hallucination of a hand hasn't been epistemically motivated. In this case, $s$ can come to know $\rQ\_8^\$ by inference from $\rP\_8$ even if she is unable to introspectively discriminate one situation from the other. Pritchard's thesis that, in good epistemic conditions, $s$'s factive support for believing $\rP\_8$ coinciding with $s$'s seeing that $\rP\_8$ is reflectively accessible is controversial (cf. Piazza 2016 and Lockhart 2018). Since this thesis is essential to the contention that Moore\** may not instantiate the disjunctive template and may thus be transmissive of propositional justification, the latter contention is also controversial. So far we have focused on perceptual justification. Now let's switch to religious justification. Shaw (2019) aims to motivate a view in religious epistemology--he contends that the 'theist in the street', though unaware of the traditional arguments for theism, can find independent rational support for believing that God exists through proofs for His existence. These proofs are based on religious experiences and parallel in structure Moore's proof of an external world interpreted as having the premise supported by an experience. A Moorean proof for the existence of God is one that deduces a belief that God exists from what Alston (1991) calls a 'manifestation belief' (M-belief, for short). An M-belief is a belief based non-inferentially on a corresponding mystical experience. Typical M-beliefs are about God's acts toward a given subject at a given time: for example, beliefs about God's bringing comfort to me, reproving me for some wrongdoing, or demonstrating His love for me, and so on. Here is Moorean proof for the existence of God: God $\rP\_9$. God is comforting me just now. Therefore: $\rQ\_9$. God exists. Note that the 'good' case in which my experience that $\rP\_9$ is veridical has a corresponding, subjectively indistinguishable, 'bad' case, in which it only seems to me that $\rP\_9$ because I'm suffering from a delusional religious experience. A concern is, therefore, that God may instantiate the disjunctive template. In this case, my experience as if $\rP\_9$ could actually justify $\rP\_9$ only if I had independent justification for $\rQ\_9$ (cf. Pritchard 2012b). If so, God isn't transmissive. Shaw (2019) concedes that God is dialectically ineffective. To defend its transmissivity, he appeals to religious epistemological disjunctivism, which says that an M-belief that $p$ can enjoy infallible rational support in virtue of one's pneuming that $p$, where this mental state is both factive and accessible on reflection. Shaw intends 'pneuming that $p$' to stand as a kind of religious-perceptual analogue to 'seeing that $p$' (cf. Shaw 2016, 2019). When it comes to God, my belief that $\rP\_9$ is supposed to be justified by my pneuming that God is comforting me just now. As we have seen in the case of Moore\, a worry is that conceiving of the justification for $\rP\_9$ along epistemological disjunctivist lines may not suffice to exempt God* from the charge of instantiating the disjunctive template and being thus non-transmissive. ### 4.2 McKinsey's Paradox McKinsey (1991, 2002, 2003, 2007) has offered a reductio argument for the incompatibility of first-person privileged access to mental content and externalism about mental content. The privileged access thesis roughly says that it is necessarily true that if $s$ is thinking that $x$, then $s$ can in principle know a priori (or in a non-empirical way) that she is thinking that $x$. Externalism about mental content roughly says that predicates of the form 'is thinking that $x$'--e.g., 'is thinking that water is wet'--express properties that are wide, in the sense that possession of these properties by $s$ logically or conceptually implies the existence of relevant contingent objects external to $s$'s mind--e.g., water. McKinsey argues that $s$ may reason along these lines: Water $\rP\_{10}$. I am thinking that water is wet. $\rP\_{11}$. If I am thinking that water is wet, then I have (or my linguistic community has) been embedded in an environment that contains water. Therefore: $\rQ\_{10}$. I have (or my linguistic community has) been embedded in an environment that contains water. Water produces an absurdity. If the privileged access thesis is true, $s$ knows $\rP\_{10}$ non-empirically. If semantic externalism is true, $s$ knows $\rP\_{11}$ a priori by conceptual analysis. Since $\rP\_{10}$ and $\rP\_{11}$ do entail $\rQ\_{10}$ and knowledge is presumably closed under known entailment, $s$ knows $\rQ\_{10}$--which is an empirical proposition--by simply competently deducing it from $\rP\_{10}$ and $\rP\_{11}$ and without conducting any empirical investigation. Since this is absurd, McKinsey concludes that the privileged access thesis or semantic externalism must be false. One way to resist McKinsey's incompatibilist conclusion that the privileged access thesis and externalism about mental content cannot be true together is to argue that Water is non-transmissive. Since knowledge is presumably closed under known entailment, it remains true that $s$ cannot know $\rP\_{10}$ and $\rP\_{11}$ while failing to know $\rQ\_{10}$. However, McKinsey's paradox originates from the stronger conclusion--motivated by the claim that Water is a deductively valid argument featuring premises knowable non-empirically--that $s$, by running Water, could know non-empirically the empirical proposition $\rQ\_{10}$ that she or members of her community have had contact with water. This is precisely what could not happen if Water is non-transmissive: in this case $s$ couldn't learn $\rQ\_{10}$ on the basis of her non-empirical justification for $\rP\_{10}$ and $\rP\_{11}$, and her knowledge of the entailment between $\rP\_{10}$, $\rP\_{11}$, and $\rQ\_{10}$ (see mainly Wright 2000, 2003, 2011).[19] A first possibility to defend the thesis that Water is non-transmissive is to argue that it instantiates the disjunctive template (considered in Sect. 3.2) (cf. Wright 2000, 2003, 2011). If Water is non-transmissive, $s$ could acquire a justification for, or knowledge of, $\rP\_{10}$ and $\rP\_{11}$ only if $s$ had an independent justification for, or knowledge of, $\rQ\_{10}$. (And to avoid the absurd result that McKinsey recoils from, this independent justification should be empirical.) If this diagnosis is correct, one need not deny $\rP\_{10}$ or $\rP\_{11}$ to reject the intuitively false claim that $s$ could know the empirical proposition $\rQ\_{10}$ in virtue of only non-empirical knowledge. To substantiate the thesis that Water instantiates the disjunctive template one should first emphasize that the kind of externalism about mental content underlying $\rP\_{10}$ is compatible with the possibility that $s$ suffers from illusion of content. Were this to happen with $\rP\_{10}, s$ would seem to introspect that she believes that water is wet whereas there is nothing like that content to be believed by $s$ in the first instance. Consider: $\rR\_{\textit{water}}$. 'water' refers to no natural kind so that there is no content expressed by the sentence 'water is wet'. $s$'s state of having introspective justification for believing $\rP\_{10}$ is arguably subjectively indistinguishable from a situation in which $\rR\_{\textit{water}}$ is true. Thus condition (D2) of the disjunctive template is met. Moreover, $\rR\_{\textit{water}}$ appears incompatible with an obvious presupposition of $s$'s cognitive project of attaining introspective justification for believing $\rP\_{10}$, at least if the content that water is wet embedded in $\rP\_{10}$ is constrained by $s$ or her linguistic community having being in contact with water. Thus condition (D3$^{\}$) is also met. Furthermore, $\rP\_{10}$ entails $\rQ\_{10}$ (when $\rP\_{11}$ is in background information). Hence, condition (D1) is fulfilled. If one could also show that (D4) is satisfied in Water, in the sense that if $\rQ\_{10}$ were false $\rR\_{\textit{water}}$ would be true, one would have shown that the disjunctive template* is satisfied by Water. Wright (2000) takes (D4) to be fulfilled and concludes that the disjunctive template is satisfied by Water. Unfortunately, the claim that (D4) is satisfied in Water cannot easily be vindicated (cf. Wright 2003). Condition (D4) is satisfied in Water only if it is true that if $s$ (or $s$'s linguistic community) had not been embedded in an environment that contains water, the term 'water' would have referred to no natural kind. This is true only if the closest possible world $w$ in which this counterfactual's antecedent is true is like Boghossian's (1997) Dry Earth--namely, a world where no one has ever had any contact with any kind of watery stuff, and all apparent contacts with it are always due to multi-sensory hallucination. If $w$ is not Dry Earth, but Putnam's Twin Earth, however, the counterfactual turns out false, as in this possible world people usually have contact with some other watery stuff that they call 'water'. So, in this world 'water' refers to a natural kind, though not to H2O. Since determining which of Dry Earth or Twin Earth is modally closer to the actual world (supposing $s$ is in the actual world)--and so determining whether (D4) is satisfied in Water--is a potentially elusive task, the claim that Water instantiates the disjunctive template appears less than fully warranted.[20] An alternative dissolution of McKinsey paradox--also based on a diagnosis of non-transmissivity--seems to be available if one considers the proposition that $\rQ\_{11}$. $s$ (or $s$'s linguistic community) has been embedded in an environment containing some watery substance (cf. Wright 2003 and 2011). This alternative strategy assumes that $\rQ\_{11}$, rather than $\rQ\_{10}$, is a presupposition of $s$'s cognitive project of attaining introspective justification for $\rP\_{10}$. Even if Water doesn't instantiate the disjunctive template, a new diagnosis of what's wrong with McKinsey's paradox could rest on the claim that the different argument yielded by expanding Water with $\rQ\_{11}$ as conclusion--call it Water\--instantiates the disjunctive template. If $\rR\_{\textit{water}}$ is the same proposition as above, it is easy to see that Water\** satisfies conditions (D4), (D2), and (D3$^{\}$) of the disjunctive template. Furthermore (D1) is satisfied at least in the sense that it seems a priori* that $\rP\_{10}$ via $\rQ\_{10}$ entails $\rQ\_{11}$ (if $\rP\_{11}$ is in background information) (cf. Wright 2011). On this novel diagnosis of non-transmissivity, what would be paradoxical is that $s$ could earn justification for $\rQ\_{11}$ in virtue of her non-empirical justification for $\rP\_{10}$ and $\rP\_{11}$ and her knowledge of the a priori link from $\rP\_{10}$, $\rP\_{11}$ via $\rQ\_{10}$ to $\rQ\_{11}$. If one follows Wright's (2003) suggestion[21] that $s$ is entitled to accept $\rQ\_{11}$--namely, the presupposition that there is a watery substance that provides 'water' with its extension--Water becomes innocuously transmissive, and the apparent paradox surrounding Water vanishes. This is so at least if one grants that it is a priori that water is the watery stuff of our actual acquaintance, once it is presupposed that there is any watery stuff of our actual acquaintance. For useful criticism of responses of this type to McKinsey's paradox see Sainsbury (2000), Pritchard (2002), Brown (2003, 2004), McLaughlin (2003), McKinsey (2003), and Kallestrup (2011).
identity-transworld	## 1. What is transworld identity? ### 1.1 Why transworld identity? Suppose that, in accordance with the possible-worlds framework for characterizing modal statements (statements about what is possible or necessary, about what might or could have been the case, what could not have been otherwise, and so on), we treat the general statement that there might have been purple cows as equivalent to the statement that there is some possible world in which there are purple cows, and the general statement that there could not have been round squares (i.e., that it is necessary that there are none) as equivalent to the statement that there is no possible world in which there are round squares. How are we to extend this framework to statements about what is possible and necessary for particular individuals--what are known as de re modal statements ('de re' meaning 'about a thing')--for example, that Clover, a particular (actually existing) four-legged cow, could not have been a giraffe, or that she could have had just three legs? A natural extension of the framework is to treat the first statement as equivalent to the claim that there is no possible world in which Clover is a giraffe, and the second as equivalent to the claim that there is some possible world in which Clover is three-legged. But this last claim appears to imply that there is some possible world in which Clover exists and has three legs--from which it seems inescapably to follow that one and the same individual--Clover--exists in some merely possible world as well as in the actual world: that there is an identity between Clover and some individual in another possible world. Similarly, it appears that the de re modal statements 'George Eliot could have been a scientist rather than a novelist' and 'Bertrand Russell might have been a playwright instead of a philosopher' will come out as 'There is some possible world in which George Eliot (exists and) is a scientist rather than a novelist' and 'There is some possible world in which Bertrand Russell (exists and) is a playwright and not a philosopher'. Again, each of these appears to involve a commitment to an identity between an individual who exists in the actual world (Eliot, Russell) and an individual who exists in a non-actual possible world. To recapitulate: the natural extension of the possible-worlds interpretation to de re modal statements involves a commitment to the view that some individuals exist in more than one possible world, and thus to what is known as 'identity across possible worlds', or (for short) 'transworld identity'. (It is questionable whether the shorthand is really apt. One would expect a 'transworld' identity to mean an identity that holds across (and hence within) one world, not an identity that holds between objects in distinct worlds. (As David Lewis (1986) has pointed out, our own Trans World Airlines is an intercontinental, not an interplanetary, carrier.) Nevertheless, the term 'transworld identity' is far too well established for it to be sensible to try to introduce an alternative, although 'interworld identity' or even 'transmodal identity' would in some ways be more appropriate.) But is this commitment acceptable? ### 1.2 Transworld identity and conceptions of possible worlds To say that there is a transworld identity between A and B is to say that there is some possible world w1, and some distinct possible world w2, such that A exists in w1, and B exists in w2, and A is identical with B. (Remember that we are treating the actual world as one of the possible worlds.) In other words, to say that there is a transworld identity is to say that the same object exists in distinct possible worlds, or (more simply) that some object exists in more than one possible world. But what does it mean to say that an individual exists in a merely possible world? And--even if we accept that paraphrases of modal statements in terms of possible worlds are in general acceptable--does it even make sense to say that actual individuals (like you and your neighbour's cat and the Eiffel Tower) exist in possible worlds other than the actual world? To know what a claim of transworld identity amounts to, and whether such claims are acceptable, we need to know what a possible world is, and what it is for an individual to exist in one. Among those who take possible worlds seriously (that is, those who think that there are possible worlds, on some appropriate interpretation of the notion), there is a variety of conceptions of their nature. On one account, that of David Lewis, a non-actual possible world is something like another universe, isolated in space and time from our own, but containing objects that are just as real as the entities of our world; including its own real concrete objects such as people, tables, cows, trees, and rivers (but also, perhaps, real concrete unicorns, hobbits, and centaurs). According to Lewis, there is no objective difference in status between what we call 'the actual world' and what we call 'merely possible worlds'. We call our world 'actual' simply because we are in it; the inhabitants of another world may, with equal right, call their world 'actual'. In other words, according to Lewis, 'actual' in 'the actual world' is an indexical term (like 'here' or 'now'), not an indicator of a special ontological status (Lewis 1973, 84-91; Lewis 1986, Ch. 1). On Lewis's 'extreme realist' account of possible worlds, it looks as if, for Clover to exist in another possible world as well as the actual world would be for her to be a part of such a world: Clover would somehow have to exist as a (concrete) part of two worlds, 'in the same way that a shared hand might be a common part of two Siamese twins' (Lewis 1986, 198). But this is problematic. Clover actually has four legs, but could have had three legs. Should we infer that Clover is a part of some world at which she has only three legs? If so, then how many legs does Clover have: four (since she actually has four legs), or seven (since she has four in our world and three in the alternative world)? Worse still, we appear to be ascribing contradictory properties to Clover: she has four legs, and yet has no more than three. Those who believe in the 'extreme realist' notion of possible worlds may respond by thinking of Clover as having a four-legged part in our world, and a three-legged part at some other world. This is Yagisawa's (2010) view (cf. Lewis 1986, 210-220). Yagisawa thinks of concrete entities--everyday things such as cats, trees and macbooks--as extended across possible worlds (as well as across times and places), in virtue of having stages (or parts) which exist at those worlds (and times and places). Ordinary entities thus comprise spatial, temporal and modal stages, all of which are equally real. Metaphysically, modal stages (and the worlds at which they exist) are on a par with temporal and spatial stages (and the times and places at which they exist). (This view is the modal analogue of the 'perdurance' account of identity over time, according to which an object persists through time by having 'temporal parts' that are located at different times.) Thus, when we say that Clover has four legs in our world but only three in some other world, we are saying that she has a four-legged modal stage and a distinct three-legged modal stage. Clover herself is neither four-legged nor three-legged. (However, there is a sense in which Clover herself--the entity comprising many modal stages--has awfully many legs, even though she actually has only four.) Another option for the 'extreme realist' about possible worlds is to hold that Clover is four-legged relative to our world, but three-legged relative to some other world. In general, qualities we would normally think of as monadic properties are in fact relations to worlds. McDaniel (2004) defends a view along these lines. A feature of this account is that one and the same entity may exist according to many worlds, for that entity may bear the exists-at relation to more than one world. Accordingly, the view is sometimes called genuine modal realism with overlap (McDaniel 2004). This view, transposed to the temporal case, is precisely what the endurantist says: objects do not have temporal parts; each object is wholly present at each time. (See the separate entry on Temporal Parts.) Lewis rejects both of these options. He rejects the overlap view because of what he calls 'the problem of accidental intrinsics'. On the overlap view, having four legs is a relation to a world, and hence not one of Clover's intrinsic properties. In fact, any aspect of a particular that changes across worlds turns out to be non-intrinsic to that particular. As a consequence, every particular has all its intrinsic properties essentially, which Lewis thinks is unacceptable (1986, 199-209). Lewis himself combines his brand of realism about possible worlds with a denial of transworld identities. According to Lewis, instead of saying that George Eliot (in whole or in part) inhabits more than one world, we should say that she inhabits one world only (ours), but has counterparts in other worlds. And it is the existence of counterparts of George Eliot who go in for a career in science rather than novel writing that makes it true that she could have been a scientist rather than a novelist (Lewis 1973, 39-43; 1968; 1986, Ch. 4). However, Lewis's version of realism is by no means the only conception of possible worlds. According to an influential set of rival accounts, possible worlds, although real entities, are not concrete 'other universes', as in Lewis's theory, but abstract objects such as (maximal) possible states of affairs or 'ways the world might have been'. (See Plantinga 1974; Stalnaker 1976; van Inwagen 1985; Divers 2002; Melia 2003; Stalnaker 1995; also the separate entry on Possible Worlds. A state of affairs S is 'maximal' just in case, for any state of affairs S\, either it is impossible for both S* and S\* to obtain, or it is impossible for S to obtain without S\: the point of the restriction to the maximal is just that a possible world should be a possible state of affairs that is, in a relevant sense, complete.) On the face of it, to treat possible worlds as abstract entities may seem only to make the problem of transworld identity worse. If it is hard to believe that you (or a table or a cat) could be part of another Lewisian possible world, it seems yet harder to believe that a concrete entity like you (or the table or cat) could be part of an abstract* entity. However, those who think that possible worlds are abstract entities typically do not take the existence in a merely possible world of a concrete actual individual to involve that entity's being literally a part of such an abstract thing. Rather, such a theorist will propose a different interpretation of 'existence in' such a world. For example, according to Plantinga's (1973, 1974) version of this account, to say that George Eliot exists in some possible world in which she is a scientist is just to say that there is a (maximal) possible state of affairs such that, had it obtained (i.e., had it been actual), George Eliot would (still) have existed, but would have been a scientist. On this (deflationary) account of existence in a possible world, it appears that the difficulties that accompany the idea that George Eliot leads a double life as an element of another concrete universe as well as our own (or the idea that she is partially present in many such universes) are entirely avoided. On Plantinga's account, to claim that an actual object exists in another possible world with somewhat different properties amounts to nothing more risque than the claim that the object could have had somewhat different properties: something that few will deny. (Note that according to this account, if the actual world is to be one of the possible worlds, then the actual world must be an abstract entity. So, for example, if a merely possible world is 'a way the world might have been', the actual world will be 'the way the world is'; if a merely possible world is a maximal possible state of affairs that is not instantiated, then the actual world will be a maximal possible state of affairs that is instantiated. It follows that we must distinguish the actual world qua abstract entity from 'the actual world' in the sense of the collection of spatiotemporally linked entities including you and your surroundings that constitutes 'the universe' or 'the cosmos'. The sense in which you exist in this concrete universe (by being part of it) must be different from the sense in which you exist in the abstract state of affairs that is in fact instantiated (cf. Stalnaker 1976; van Inwagen 1985, note 3; Kripke 1980, 19-20).) The discussion so far may suggest that whether the notion of transworld identity (that an object exists in more than one world) is problematic depends solely on whether one adopts an account of possible worlds as concrete entities such as Lewis's (in which case it is) or an account of possible worlds as abstract entities such as Plantinga's (in which case it is not). However, matters are not so simple, for a variety of reasons (to be discussed in Sections 3-5 below). ## 2. Transworld identity and Leibniz's Law There may seem to be an obvious objection to the employment of transworld identity to interpret or paraphrase statements such as 'Bertrand Russell could have been a playwright' or 'George Eliot might have been a scientist'. A fundamental principle about (numerical) identity is Leibniz's Law: the principle that if A is identical with B, then any property of A is a property of B, and vice versa. In other words, according to Leibniz's Law, identity requires the sharing of all properties; thus any difference between the properties of A and B is sufficient to show that A and B are numerically distinct. (The principle here referred to as 'Leibniz's Law' is also known as the Indiscernibility of Identicals. It must be distinguished from another (more controversial) Leibnizian principle, the Identity of Indiscernibles, which says that if A and B share all their properties then A is identical with B.) However, the whole point of asserting a transworld identity is to represent the fact that an individual could have had somewhat different properties from its actual properties. Yet does not (for example) the claim that a philosopher in the actual world is identical with a non-philosopher in some other possible world conflict with Leibniz's Law? It is generally agreed that this objection can be answered, and the appearance of conflict with Leibniz's Law eliminated. We can note that the objection, if sound, would apparently prove too much, since a parallel objection would imply that there can be no such thing as genuine (numerical) identity through change of properties over time. But it is generally accepted that no correct interpretation of Leibniz's Law should rule this out. For example, Bertrand Russell was thrice married when he received the Nobel Prize for Literature; the one-year-old Bertrand Russell was, of course, unmarried; does Leibniz's Law force us to deny the identity of the prize-winning adult with the infant, on the grounds that they differ in their properties? No, for it seems that the appearance of conflict with Leibniz's Law can be dispelled, most obviously by saying that the infant and the adult share the properties of being married in 1950 and being unmarried in 1873, but alternatively by the proposal that the correct interpretation of Leibniz's Law is that the identity of A and B requires that there be no time such that A and B have different properties at that time (cf. Loux 1979, 42-43; also Chisholm 1967). However, it seems that exactly similar moves are available in the modal case to accommodate 'change' of properties across possible worlds. Either we may claim that the actual Bertrand Russell and the playwright in some possible world (say, w2) are alike in possessing the properties of being a philosopher in the actual world and being a non-philosopher in w2, or we may argue that Leibniz's Law, properly interpreted, asserts that the identity of A and B requires that there be no time, and no possible world, such that A and B have different properties at that time and world. The moral appears to be that transworld identity claims (combined with the view that some of an individual's properties could have been different) need no more be threatened by Leibniz's Law than is the view that there can be identity over time combined with change of properties (Loux 1979, 42-43). It should be mentioned, however, that David Lewis has argued that the reconciliation of identity through change over time with Leibniz's Law suggested above is oversimplified, and gives rise to a 'problem of temporary intrinsics' that can be solved only by treating a persisting thing that changes over time as composed of temporal parts that do not change their intrinsic properties. (See Lewis 1986, 202-204, and for discussion and further references, Hawley 2001; Sider 2001; Lowe 2002; Haslanger 2003.) In addition, it is partly because Lewis regards the analogous account of transworld identity in terms of modal parts as an unacceptable solution to an analogous 'problem of accidental intrinsics' that Lewis rejects transworld identity in favour of counterpart theory (Lewis 1986, 199-220; cf. Section 1.2 above). ## 3. Is 'the problem of transworld identity' a pseudo-problem? In the discussion of transworld identity in the 1960s and 1970s (when the issue came to prominence as a result of developments in modal logic), it was debated whether the notion of transworld identity is genuinely problematic, or whether, on the contrary, the alleged 'problem of transworld identity' is merely a pseudo-problem. (See Loux 1979, Introduction, Section III; Plantinga 1973 and 1974, Ch. 6; Kripke 1980 (cf. Kripke 1972); Kaplan 1967/1979; Kaplan 1975; Chisholm 1967; for further discussion see, for example, Divers 2002, Ch. 16; Hughes 2004, Ch. 3; van Inwagen 1985; Lewis 1986, Ch. 4.) It is difficult to pin down the alleged problem that is supposed to be at the heart of this dispute. In particular, although the main proponents of the view that the alleged problem is a pseudo-problem clearly intended to attack (inter alia) Lewis's version of modal realism, they did not attempt to rebut the thesis (discussed in Section 1.2 above) that if one is a Lewisian realist about possible worlds, then one should find transworld identity problematic. Matters are complicated by the fact that proponents of the view that the alleged problem of transworld identity is a pseudo-problem were to some extent responding to hypothetical arguments, rather than arguments presented in print by opponents (see Plantinga 1974, 93). However, one central issue was whether the claim that an individual exists in more than one possible world (and hence that there are cases of transworld identity) needs to be backed by the provision of criteria of transworld identity, and, if so, why. The term 'criterion of identity' is ambiguous. In an epistemological sense, a criterion of identity is a way of telling whether an identity statement is true, or a way of recognizing whether an individual A is identical with an individual B. However, the notion of a criterion of identity also has a metaphysical interpretation, according to which it is a set of (non-trivial) necessary and sufficient conditions for the truth of an identity statement. Although a criterion of identity in the second (metaphysical) sense might supply us with a criterion of identity in the first (epistemological) sense, it seems that something could be a criterion of identity in the second sense even if it is unsuited to play the role of a criterion of identity in the first sense. The most influential arguments against the view that there is a genuine problem of transworld identity (or 'problem of transworld identification', to use Kripke's preferred terminology) are probably those presented by Plantinga (1973, 1974) and Kripke (1980). Plantinga and Kripke appear to have, as their target, an alleged problem of transworld identity that rests on one of three assumptions. The first assumption is that we must possess criteria of transworld identity in order to ascertain, on the basis of their properties in other possible worlds, the identities of (perhaps radically disguised) individuals in those worlds. The second assumption is that we must possess criteria of transworld identity if our references to individuals in other possible worlds are not to miss their mark. The third assumption is that we must possess criteria of transworld identity in order to understand transworld identity claims. Anyone who makes one of these assumptions is likely to think that there is a problem of transworld identity--a problem concerning our entitlement to make claims that imply that an individual exists in more than one possible world. For it does not seem that we possess criteria of transworld identity that could fulfil any of these three roles. However, Plantinga and Kripke provide reasons for thinking that none of these three assumptions survives scrutiny. If so, and if these assumptions exhaust the grounds for supposing that there is a problem of transworld identity, the alleged problem may be dismissed as a pseudo-problem. The three assumptions may be illustrated, using our examples of George Eliot and Bertrand Russell, as follows. (The examples are alternated simply for the sake of a little variety.) (1) The 'epistemological' assumption: We must possess a criterion of transworld identity for George Eliot in order to be able to tell, on the basis of knowledge of the properties that an individual has in some other possible world, whether that individual is identical with Eliot. (2) The 'security of reference' assumption: We must possess a criterion of transworld identity for Bertrand Russell in order to know that, when we say such things as 'There is a possible world in which Russell is a playwright', we are talking about Russell rather than someone else. (3) The 'intelligibility' assumption: We must possess a criterion of transworld identity for George Eliot in order to understand the claim that there is a possible world in which she is a scientist. ### 3.1 Against the epistemological assumption The epistemological assumption appears to imply that the point of our having a criterion of transworld identity for George Eliot would be that we could then employ the criterion in order to ascertain which individual in a possible world is Eliot; if, on the other hand, we do not possess such a criterion, we shall be unable to pick her out or identify her in other possible worlds (Plantinga 1973; 1974, Ch. 6; Kripke 1980, 42-53; cf. Loux 1979, Introduction; Kaplan 1967/1979). However, this suggestion, as stated, is vulnerable to the charge that it is the product of confusion. For how could we use a criterion of identity in the way envisaged? We must dismiss as fanciful the idea that if we had a criterion of transworld identity for George Eliot, we could use it to tell, by empirical inspection of the properties of individuals in other possible worlds (perhaps using a powerful telescope (Kripke 1980, 44) or 'Jules Verne-o-scope' (Kaplan 1967/1979, 93; Plantinga 1974, 94)), which, if any, of those individuals is Eliot. For no one (including an extreme realist like Lewis) thinks that our epistemological access to other possible worlds is of this kind. (According to Lewis, other possible worlds are causally isolated from our own, and hence beyond the reach of our telescopes or any other perceptual devices.) But once we face up to the fact that a criterion of transworld identity (if we had one) could have no such empirical use, the argument based on the epistemological assumption appears to collapse. It is tempting to suggest that the argument is the product of the (perhaps surreptitious) influence of a misleading picture of our epistemological access to other possible worlds. As Kripke writes (using President Nixon as his example): One thinks, in this [mistaken] picture, of a possible world as if it were like a foreign country. One looks upon it as an observer. Maybe Nixon has moved to the other country and maybe he hasn't, but one is given only qualities. One can observe all his qualities, but, of course, one doesn't observe that someone is Nixon. One observes that something has red hair (or green or yellow) but not whether something is Nixon. So we had better have a way of telling in terms of properties when we run into the same thing as we saw before; we had better have a way of telling, when we come across one of these other possible worlds, who was Nixon. (1980, 43) (It is possible, though, that in this passage Kripke's principal target is not a mistaken conception of our epistemological access to other possible worlds, but what he takes to be a mistaken ('foreign country') conception of their nature: a conception that (when divorced from the fanciful epistemology) would be entirely appropriate for a Lewisian realist about worlds.) ### 3.2 Against the 'security of reference' assumption It might be suggested that the point of a criterion of transworld identity is that its possession would enable me to tell which individual I am referring to when I say (for example) 'There is a possible world in which Russell is a playwright'. Suppose that I am asked: 'How do you know that the individual you are talking about--this playwright in another possible world--is Bertrand Russell rather than, say, G. E. Moore, or Marlene Dietrich, or perhaps someone who is also a playwright in the actual world, such as Tennessee Williams or Aphra Behn? Don't you need to be able to supply a criterion of transworld identity in order to secure your reference to Russell?' (Cf. Plantinga 1974, 94-97; Kripke 1980, 44-47.) It seems clear that the right answer to this question is 'no'. As Kripke has insisted, it seems spurious to suggest that the question how we know which individual we are referring to when we make such a claim can be answered only by invoking a criterion of transworld identity. For it seems that we can simply stipulate that the individual in question is Bertrand Russell (Kripke 1980, 44). Similarly, perhaps, if I say that there is some past time at which Angela Merkel is a baby, and am asked 'How do you know that it's the infant Angela Merkel that you are talking about, rather than some other infant?', an apparently adequate reply is that I am stipulating that the past state of affairs I am talking about is one that concerns Merkel (and not some other individual). It seems that I can adequately answer the parallel question in the modal case by saying that I am stipulating that, when I say that there is some possible world in which Russell is a playwright, the relevant individual in the possible world (if there is one) is Russell (and not some other potential or actual playwright). ### 3.3 Against the 'intelligibility' assumption A third job for a criterion of transworld identity might be this: in order to understand the claim that there is some possible world in which George Eliot is a scientist, perhaps we must be able to give an informative answer to the question 'What would it take for a scientist in another possible world to be identical with Eliot?' Again, however, it can be argued that this demand is illegitimate, at least if what is demanded is that one be able to specify a set of properties whose possession, in another possible world, by an individual in that world, is non-trivially necessary and sufficient for being George Eliot (cf. Plantinga 1973; Plantinga 1974, 94-97; van Inwagen 1985). For one thing, we may point to the fact that it is doubtful that, in order to understand the claim that there is some past time at which Angela Merkel is a baby, we have to be able to answer the question 'What does it take for an infant at some past time to be identical with Merkel?' in any informative way. Secondly, it may be proposed that in order to understand the claim that there is some possible world in which George Eliot is a scientist, we can rely on our prior understanding of the claim that she might have been a scientist (cf. Kripke 1980, 48, note 15; van Inwagen 1985). ### 3.4 Remaining issues However, even if all three of these assumptions can be dismissed as bad, or at least inadequate, reasons for supposing that transworld identity requires criteria of transworld identity (and hence for supposing that there is a problem of transworld identity), it does not follow that there are no good reasons for this supposition. In particular, even if the three assumptions are discredited, a fourth claim may survive: (4) There must be a criterion of transworld identity for Russell (in the sense of a set of properties whose possession by an object in any possible world is non-trivially necessary and sufficient for being Russell) if the claim that there is a possible world in which Russell is a playwright is to be true. (Such a set of properties would be what is called a non-trivial individual essence of Russell, where an individual essence of an individual A is a property, or set of properties, whose possession by an individual in any possible world is both necessary and sufficient for identity with A.) That this possibility is left open by the arguments so far considered is suggested by at least two points. The first concerns the analogy drawn above between transworld identity and identity through time. Even if we can understand the claim that there is some past time at which Angela Merkel is a baby without being able to specify informative (non-trivial) necessary and sufficient conditions for the identity of the adult Angela Merkel with some previously existing infant, it does not follow that there are no such necessary and sufficient conditions. And many philosophers have supposed that there are such conditions for personal identity over time. Secondly, the fact that one may be able to ensure, by stipulation, that one is talking about a possible world in which Bertrand Russell (and not someone else) is a playwright (if there is such a world) does not imply that, when making this stipulation, one is not implicitly stipulating that this individual satisfies, in that world, conditions non-trivially necessary and sufficient for being Russell, even if one is not in a position to say what these conditions are. This second point is an extension of the observation that, if (as most philosophers believe) Bertrand Russell has some essential properties (properties that he has in all possible worlds in which he exists), to stipulate that one is talking about a possible world in which Russell is a playwright is, at least implicitly, to stipulate that the possible world is one in which someone with Russell's essential properties is a playwright. For example, according to Kripke's 'necessity of origin' thesis, human beings have their parents essentially (Kripke 1980). If this is correct, then, when we say 'There is a possible world in which Russell is a playwright', it seems that, if our stipulation is to be coherent, we must be at least implicitly stipulating that the possible world is one in which someone with Russell's actual parents is a playwright, even if the identity of Russell's parents is unknown to us, and even though we are (obviously) in no position to conduct an empirical investigation into the ancestry, in the possible world, of the individuals who exist there. Thus, it seems, even if Kripke is right in insisting that we need not be able to specify non-trivial necessary and sufficient conditions for being Russell in another possible world if we are legitimately to claim that there are possible worlds in which he is a playwright, it might nevertheless be the case that there are such necessary and sufficient conditions (cf. Kripke 1980, 46-47 and 18, note 17; Lewis 1986, 222). But what positive reasons are there for holding that transworld identities require non-trivial necessary and sufficient conditions (non-trivial individual essences), if arguments that are based on the epistemological, security of reference, and intelligibility assumptions are abandoned? (Similar issues arise for the transworld identities of properties, discussed in the supplement on transworld identity of properties.) ## 4. Individual essences and bare identities The principal argument for this view--that transworld identities require non-trivial individual essences--is that such essences are needed in order to avoid what have been called 'bare identities' across possible worlds. And some regard bare identities as too high a price to pay for the characterization of de re modal statements in terms of transworld identity. If they are right, and if (as many philosophers believe) there are no plausible candidates for non-trivial individual essences (at least for such things as people, cats, trees, and tables) there is, indeed, a serious problem about transworld identity. (The expression 'bare identities' is taken from Forbes 1985. The notion, as used here, is approximately the same as the notion of 'primitive thisness' employed by Adams (1979), although Adams's notion is that of an identity that does not supervene on qualitative facts, rather than an identity that does not supervene on any other facts at all.) Suppose that we combine transworld identity with the claim (without which the introduction of transworld identity seems pointless) that a transworld identity can hold between A in w1 and B in w2 even though the properties that B has in w2 are somewhat different from the properties that A has in w1 (or, to put it more simply, suppose that we combine the claim that there are transworld identities with the claim that not all of a thing's properties are essential to it). Then, it can be argued, unless there are non-trivial individual essences, we are in danger of having to admit the existence of possible worlds that differ from one another only in the identities of some of the individuals that they contain. ### 4.1 Chisholm's Paradox and bare identities One such argument, adapted from Chisholm 1967, goes as follows. Taking Adam and Noah in the actual world as our examples (and pretending, for the sake of the example, that the biblical characters are real people), then, on the plausible assumption that not all of their properties are essential to them, it seems that there is a possible world in which Adam is a little more like the actual Noah than he actually was, and Noah a little more like the actual Adam than he actually was. But if there is such a world, then it seems that there should be a further world in which Adam is yet more like the actual Noah, and Noah yet more like the actual Adam. Proceeding in this way, it looks as if we may arrive ultimately at a possible world that is exactly like the actual world, except that Adam and Noah have 'switched roles' (plus any further differences that follow logically from this, such as the fact that in the 'role-switching' world Eve is the consort of a man who plays the Adam role, but is in fact Noah). But if this can happen with Adam and Noah, then it seems that it could happen with any two actual individuals. For example, it looks as if there will be a possible world that is a duplicate of the actual world except for the fact that in this world you play the role that Queen Victoria plays in the actual world, and she plays the role that you play in the actual world (cf. Chisholm 1967, p. 83 in 1979). But this may seem intolerable. Is it really the case that Queen Victoria could have had all your actual properties (except for identity with you) while you had all of hers (except for identity with her)? However, if one thinks that such conclusions are intolerable, how are they to be avoided? The obvious answer is that what is needed, in the Adam-Noah case, is that the roles played by Adam and Noah in the actual world include some properties that are essential to their bearers' being Adam and Noah respectively: that Adam and Noah differ non-trivially in their essential properties as well as in their accidental properties: more precisely, that Adam has some essential property that Noah essentially lacks, or vice versa. For if 'the Adam role' includes some property that Noah essentially lacks, then, of course, there is no possible world in which Noah has that property, in which case the Adam role (in all its detail) is not a possible role for Noah, and the danger of a role-switching world such as the one described above is avoided. The supposition that Adam and Noah differ in their essential properties in this way, although sufficient to block the generation of this example of a role-switching world, does not by itself imply that each of Adam and Noah has an individual essence: a set of essential properties whose possession is (not only necessary but also) sufficient for being Adam or Noah. Suppose that Adam has, as one of his essential properties, living in the Garden of Eden, whereas Noah essentially lacks this property. This will block the possibility of Noah's playing the Adam role, although it does not, by itself, imply that nothing other than Adam could play that role. However, when we reflect on the potential generality of the argument, it appears that, if we are to block all cases of role-switching concerning actual individuals, we must suppose that every actual individual has some essential property (or set of essential properties) that every other actual individual essentially lacks. For example, to block all cases of role-switching concerning Adam and other actual individuals, there must be some component of 'the Adam role' that is not only essential to being Adam, but also cannot be played, in any possible world, by any actual individual other than Adam. Even if we suppose that all actual individuals are distinguished from one another by such 'distinctive' essential properties, this still does not, strictly speaking, imply that they have individual essences. For example, it does not rule out the existence of a possible world that is exactly like the actual world except that, in this possible world, the Adam role is played, not by Adam, but by some merely possible individual (distinct from all actual individuals). However, if we find intolerable the idea that there are such possible worlds--worlds that, like the role-switching world, differ from the actual world only in the identities of some of the individuals that they contain--then, it seems, we must suppose that individuals like Adam (and Noah and you) have (non-trivial) individual essences, where an individual essence of Adam is (by definition) some property (or set of properties) that is both essential to being Adam and also such that it is not possessed, in any possible world, by any individual other than Adam--i.e., an essential property (or set of properties) that guarantees that its possessor is Adam and no one else. Chisholm (1967) arrives at his role-switching world by a series of steps. Thus his argument appears to rely on the combination of the transitivity of identity (across possible worlds) with the assumption that a succession of small changes can add up to a big change. And 'Chisholm's Paradox' (as it is called) is sometimes regarded as relying crucially on these assumptions, suggesting that it has the form of a sorites paradox (the type of paradox that generates, from apparently impeccable assumptions, such absurd conclusions as that a man with a million hairs on his head is bald). (See, for example, Forbes 1985, Ch. 7.) However, there are versions of the role-switching argument that do not rely on the cumulative effect of a series of small changes. Suppose we assume that Adam and Noah do not differ from one another in their essential properties; in other words, that all the differences between them are accidental (i.e., contingent) differences. It seems immediately to follow that any way that Adam could have been is a way that Noah could have been, and vice versa. But one way that Adam could have been is the way Adam actually is, and one way that Noah could have been is the way Noah actually is. So (if Adam and Noah do not differ in their essential properties) it seems that there is a possible world in which Adam plays the Noah role, and a possible world in which Noah plays the Adam role. But there is no obvious reason why a world in which Adam plays the Noah role and a world in which Noah plays the Adam role shouldn't be the very same world. And in that case there is a possible world in which Adam and Noah have swapped their roles. This argument for the generation of a role-switching world does not rely on a series of small changes: all that it requires is the assumption that there is no essential difference between Noah and Adam: or, to put it another way, that any essential property of Noah is also an essential property of Adam, and vice versa. (See Mackie 2006, Ch. 2; also Adams 1979; cf. Dorr, Hawthorne, and Yli-Vakkuri 2021.) ### 4.2 Forbes on individual essences and bare identities Another type of argument for the conclusion that unless things have non-trivial individual essences there will be 'bare' transworld identities: identities that do not supervene on (are not grounded in) other facts, is presented by Graeme Forbes. (Strictly speaking, Forbes is concerned to avoid identities that are not grounded in what he calls 'intrinsic' properties.) A sketch of a type of argument used by Forbes is this. (What follows is based on Forbes 1985, Ch. 6; see also Mackie 2006, Ch. 3.) Suppose (as is surely plausible) that an actually existing oak tree could have been different in some respects from the way that it is; suppose also that, even if it has some essential properties (perhaps it is essentially an oak tree, for example), it has no non-trivial individual essence consisting in some set of its intrinsic properties. Then there is the danger that there may be three possible worlds (call them 'w2', 'w3', and 'w4'), where in w2 there is an oak tree that is identical with the original tree (w2 representing one way in which the tree could have been different), and in w3 there is an oak tree that is identical with the original tree (w3 representing another way in which the tree could have been different), and in w4 there are two oak trees, one of which is an intrinsic duplicate of the tree as it is in w2, and the other an intrinsic duplicate of the tree as it is in w3. If all of w2, w3, and w4 are possible, then, given that at least one of the trees in w4 is not identical with the original tree (since two things cannot be identical with one thing) there are instances of transworld identity (and transworld distinctness) concerning a tree in one possible world and a tree in another that are not grounded in (do not supervene on) the intrinsic features that those trees possess in those possible worlds. For example, suppose that, of the two trees in w4, the intrinsic duplicate of the w2 tree is not identical with the original tree. Then, obviously, the distinctness (non-identity) between this w4 tree and the tree in w2 is not grounded in the intrinsic features that the trees have in w2 and w4--and nor is the identity between the tree in w2 and the original tree grounded in the intrinsic features that the tree has in w2 and in the actual world. Forbes argues that, in order to avoid this (and similar) consequences, we should suppose that (contrary to the second assumption used in setting up the 'reduplication argument' sketched above) the oak tree does have a non-trivial individual essence consisting in some of its intrinsic properties, and his favoured candidate for its essence is one that includes the tree's coming from the particular acorn from which it actually originated. If the tree does have such an 'intrinsic' individual essence, then, if w2 and w3 are both possible, each of them must contain a tree that has (in that world) intrinsic properties that are guaranteed to be sufficient for identity with the original tree, in which case (as a matter of logic) there can be no world such as w4 that contains intrinsic duplicates of both of them. (See Forbes 1985, Ch. 6, and, for discussion, Mackie 1987; Mackie 2006; Robertson 1998; Yablo 1988; Chihara 1998; Della Rocca 1996; further discussions by Forbes include his 1986, 1994, and 2002.) Finally, it is obvious that the structure of Forbes's argument has nothing to do with the fact that the chosen example is a tree. Forbes's 'reduplication argument' therefore appears to pose a general problem for the characterization of de re modal statements about individuals in terms of transworld identity: either we must admit that their transworld identities can be 'bare', or we must find non-trivial individual essences, based on their intrinsic properties, that can ground their identities across possible worlds. ### 4.3 Transworld identity and conditions for identity over time So far it has been assumed that (non-trivial) necessary and sufficient conditions for transworld identity with a given object would involve the possession, by that object, of an individual essence: a set of properties that it carries with it in every possible world in which it exists. But one might wonder why we should make this assumption. Those who believe that there are (non-trivial) necessary and sufficient conditions for identity over time need not, and almost universally do not, believe that these conditions consist in the possession, by an object, of some 'omnitemporal core' (to use a phrase suggested by Harold Noonan) that it has at every time in its existence. So why should things be different in the modal case? The obvious answer seems to be this. In the case of identity over time, we can appeal to relations (other than mere similarity) between the states of an individual at different times in its existence. For example, it looks as if we can say that the adult Russell is identical with the infant Russell in virtue of the existence of certain spatiotemporal and causal continuities between his infant state in 1873 and his adult state in (say) 1950 that are characteristic of the continued existence of a human being. But no such relations of continuity are available to ground identities across possible worlds (cf. Quine 1976). However, on reflection, it may seem that this is too swift. If we suppose that any possible history for Russell is a possible spatiotemporal and causal extension of the state that he was actually in at some time in his existence, then perhaps we may appeal to the same continuities that ground his identity over time in the actual world in order to ground his identity across possible worlds (cf. Brody 1980, 114-115; 121). For example, perhaps to say that Russell could have been a playwright is to say that there was some time in his actual existence at which he could have become a playwright. If so, then perhaps we can hold that for a playwright in a possible world to be identical with Russell is for that playwright to have, in that world, a life that is, at some early stage, exactly the same as Russell's actual life at some early stage, but which develops from that point, in the spatiotemporally and causally continuous fashion that is characteristic of the continued existence of a human being, into the career of a playwright rather than that of a philosopher. However, although such a conception may seem initially attractive, it runs into difficulties if it is intended to provide conditions that are genuinely both necessary and sufficient for the identity of individuals across possible worlds. These difficulties include the fact that it seems too much to demand that Russell have exactly the same early history (or origin) as his actual early history (or origin) in every possible world in which he exists. Yet if Russell's early history could have been different in certain respects, we face the question: 'In virtue of what is an individual in another possible world with a slightly different early history from Russell's actual early history identical with Russell?'--a question of precisely the type that the provision of necessary and sufficient conditions for transworld identity was intended to answer. (For discussion of this 'branching' conception of possibilities, and its implications for questions of transworld identity and essential properties, see Brody 1980, Ch. 5; Mackie 1998; Mackie 2006, Chs 6-7; Coburn 1986, Section VI; McGinn 1976; Mackie 1974; Prior 1960.) ### 4.4 Responses to the problems The fact that, in the absence of non-trivial individual essences, a transworld identity characterization of de re modal statements appears to generate bare identities (via arguments such as Chisholm's Paradox or Forbes's reduplication argument) may produce a variety of reactions. The moral that Chisholm (1967) drew from his argument was scepticism about transworld identity, based partly on scepticism about whether the non-trivial individual essences that would block the generation of role-switching worlds are available. Others would go further, and conclude that such puzzles provide not only a reason for rejecting transworld identity, but also a reason for adopting counterpart theory. (Note, though, that Lewis's reasons for adopting counterpart theory appear to be largely independent of such puzzles (cf. Lewis 1986, Ch. 4).) A third reaction is to accept bare identities--or, at least, to accept that individuals (including actual individuals) may have qualitative duplicates in other possible worlds, and that transworld identities may involve what have been called 'haecceitistic' differences. (See Adams 1979; Mackie 2006, Chs 2-3; Lewis 1986, Ch. 4, Section 4; also the separate entry on Haecceitism.) A fourth reaction, that of Forbes, is to propose a mixed solution: he holds that for some individuals (including human beings and trees) suitable candidates for non-trivial individual essences can be found (by appeal to distinctive features of their origins), although for others (including most artefacts) it may be that no suitable candidates are available, in which case counterpart theory should be adopted for these cases (see Forbes 1985, Chs 6-7). Koslicki (2020) provides another mixed solution. She accepts (primarily for Quinean reasons) that there is a problem of transworld identity for certain individuals including people: one that can be solved only by attributing to them non-trivial individual essences; she argues that these consist in their individual forms. Such an individual form would provide a substantial answer to questions such as 'what makes an individual Noah in the "role-switching" world rather than Adam?' Thus, it would appear, the problem of transworld identity for individuals such as Noah and Adam is replaced by a problem of transworld identity for their individual forms. But why is this supposed to represent an improvement? As Fine puts it: 'Why should we not simply take the crossworld identity of [non-form] entities as given and not standing in any need of a criterion?' (2020, 430). One reason (considered by Koslicki) is conceptual. It is thought that we need to make sense of de re modal claims in terms of de dicto modal claims. However, if individual forms are to come to the rescue, the question arises how the individual forms are to be distinguished from one another. The problem is acute in cases where the individuals in question are otherwise indiscernible (Fine 2020, 432). Moreover, it is not clear that Koslicki's invocation of the fact that individual forms--unlike, say, unanalysable haecceities--have a 'qualitative' element (2020, Sections 3.3.4-3.5) is supposed to help with this problem. It is perhaps significant, though, that no theorist appears to have argued that a 'non-trivial individual essence' solution can be applied to all the relevant cases. In other words, the consensus appears to be that the price of interpreting all de re modal claims in terms of transworld identity (as opposed to counterpart theory) is the acceptance of (some) bare identities across possible worlds. Salmon (1996) claims 'something like a proof' of this implication, from transworld identities to bare identities. He argues for what he calls Extreme Haecceitism, the view that transworld identities cannot be grounded in general facts about the individuals concerned. The purported proof goes as follows. We consider x in world w and y in world w2 and suppose, for reductio, that x = y and that this fact is reducible to (or grounded in, or entailed by) general facts about x in w and y in w2. But, says Salmon, the fact that x = x is not reducible (grounded, etc.) in this way, since it is a fact of logic. So, says Salmon, 'x differs from y in at least one respect. For x lacks y's feature that its identity with x is grounded in general (cross-world) facts about x and it' (1996, 216), and hence (by Leibniz's Law), x [?] y, contrary to assumption. So if there are any transworld identities, those identities cannot be grounded in general facts about those individuals. They must be 'bare'. Salmon's argument is a variant on Evans's (1978) argument against the possibility of vague objects. It is controversial in general whether arguments of this form succeed, for applications of Leibniz's Law in intensional contexts are questionable. Catterson (2008) argues on independent grounds that Salmon's argument is not sound. We should also note that, even if the argument is successful, it does not directly establish Salmon's Extreme Haecceitism, but only the conditional thesis that if there are any true transworld identity statements x = y, then they are not reducible to general facts about x and y. ### 4.5 Leibniz and hyper-essentialism Finally, it can be noted that the problems concerning transworld identity discussed here arise only because it is assumed that not all of an individual's properties are essential to it (and hence that, if it exists in more than one possible world, it has different properties in different worlds). If, instead, one were to hold that all of an individual's properties are essential properties--and hence, for example, that George Eliot could not have existed with properties in any way different from her actual ones--then no such problems would arise. Moreover, this suggestion, implausible though it may be, is of some historical interest. For, according to a standard interpretation of the views of Gottfried Leibniz, the philosopher who is the father of theories of possible worlds, Leibniz's theory of 'complete individual notions' commits him to the thesis that an individual such as George Eliot does have all her properties essentially (cf. Leibniz, Discourse on Metaphysics (1687), Sections 8 and 13; printed in Leibniz 1973 and elsewhere). According to the 'hyper-essentialist' view to which Leibniz appears to be committed, any individual, in any possible world, whose properties in that world differ from the actual properties of George Eliot is not, strictly speaking, identical with Eliot. However, it also seems clear that this does not represent a way of saving a transworld identity interpretation of de re modality. On the contrary: if there is no possible world in which George Eliot exists with properties different from her actual properties, then it is plausible to conclude that there is no possible world, other than the actual world, in which she exists at all. For unless possible worlds can be exact duplicates (something that Leibniz himself would deny), any merely possible world must differ from the actual world in some respect. If so, then the properties of any individual in another possible world must differ in some respect from the actual properties of Eliot (even if the difference is only a difference in relational properties), in which case, if all Eliot's properties are essential to her, that individual is not Eliot. (Leibniz's views may, however, be seen as a partial anticipation of counterpart theory, which attempts to save the truth of the claim that George Eliot could have been different in some respects (thus denying 'hyper-essentialism') while preserving the metaphysical thesis that no individual who is, strictly speaking, identical with Eliot exists in any other possible world (cf. Kripke 1980, 45, note 13).) ### 4.6 Haecceities and haecceitism The view that an individual's transworld identity is 'bare' is sometimes described as the view that its identity consists in its possession of a 'haecceity' or 'thisness': an unanalysable non-qualitative property that is necessary and sufficient for its being the individual that it is. (The term 'individual essence' is sometimes used to denote such a haecceity. It should be noted that according to the terminology used in this article, although a haecceity would be an individual essence, it would not be a non-trivial individual essence.) However, it is not obvious that the belief in bare identities requires the acceptance of haecceities. One can apparently hold that transworld identities may be 'bare' without holding that they are constituted by any properties at all, even unanalysable haecceities (cf. Lewis 1986, 225; Adams 1979, 6-7). Thus we should distinguish what is standardly known as 'haecceitism' (roughly, the view that there may be bare identities across possible worlds in the sense of identities that do not supervene on qualitative properties) from the belief in haecceities (the belief that individuals have unanalysable non-qualitative properties that constitute their being the individuals that they are). (For more on the use of the term 'haecceitism' see Lewis 1986, Ch. 4, Section 4; Adams 1979; Kaplan 1975, Section IV; also the separate entry on Haecceitism. For the history of the term 'haecceity', see the entry on Medieval Theories of Haecceity.) In addition, it should be noted that to believe in 'bare' transworld identities, in the sense under discussion here, is not to believe in 'bare particulars', if to be a bare particular is to be an entity devoid of (non-trivial) essential properties. As the arguments discussed in Sections 4.1-4.2 above demonstrate, a commitment to a 'bare' (or 'ungrounded') difference in the identities of two individuals A and B in different possible worlds (two human beings, or two trees, for example) does not imply that those individuals have no non-trivial essential properties. All it implies is that A and B do not differ in their non-trivial essential properties--and hence that, although there may well be non-trivial necessary conditions for being A in any possible world, and non-trivial necessary conditions for being B in any possible world, there are no non-trivial necessary conditions for being A that are not also necessary conditions for being B, and vice versa. (Cf. Adams's 'Moderate Haecceitism' (1979, 24-26).) ## 5. Transworld identity and the transitivity of identity It was argued above that the proponent of transworld identity without non-trivial individual essences faces the prospect of bare ('ungrounded') identities across possible worlds. One such argument is Chisholm's Paradox, which relies on the transitivity of identity to produce the result that a series of small changes in the properties of Adam and Noah leads to a world in which Adam and Noah have swapped their roles. However, the transitivity of identity generates additional problems concerning transworld identity, some of which have nothing particularly to do with role-switching possibilities or bare identities. ### 5.1 Chandler's transitivity argument One such argument is given by Chandler (1976). It can be illustrated simply as follows (adapting Chandler's own example). Suppose that there is a bicycle originally composed of three parts: A1, B1, and C1. (We might suppose that A1 is the frame, and B1 and C1 the two wheels.) Suppose we think that any bicycle could have been originally composed of any two thirds of its original parts, with a substitute third component. We may call this (following Forbes 1985) 'the tolerance principle'; it is a development of the intuitively appealing thought that it is too much to demand, of an object such as a bicycle, that it could not have existed unless all of its original parts had been the same. Suppose, further, that we think that no bicycle could have been originally composed of just one third of its original parts, even with substitutes for the other two thirds. Call this 'the restriction principle'. The combination of these assumptions appears to generate a difficulty for the paraphrase of de re modal claims about bicycles in terms of transworld identity. For if there is (as the tolerance principle allows) a possible world w2 in which our bicycle comes into existence composed of parts A1 + B1 + C2, where C1 [?] C2, then, if we apply the tolerance principle to this bicycle we must say that it could have come into existence (in some further possible world w3) with any two thirds of those parts, with a substitute third component: for example, that it could have come into existence (in w3) composed of A1 + B2 + C2, where B1 [?] B2 and C1 [?] C2. The bicycle in w3 is, ex hypothesi, identical with the bicycle in w2, and the bicycle in w2 is, ex hypothesi, identical with the original bicycle; so, by the transitivity of identity, the bicycle in w3 is identical with the original bicycle. Hence our assumptions have generated a contradiction. We have a bicycle in w3, originally composed of A1 + B2 + C2, that both is identical with the original bicycle (by the repeated application of the tolerance principle, together with the transitivity of identity) and is not identical with the original bicycle (by the restriction principle). One might complain that the version of the tolerance principle cited above is too lenient. Perhaps it is not true that the bicycle could have come into existence with just two thirds of its original components: perhaps a threshold of, say, 90% or more is required. However, the simple argument given above can be adapted to generate a contradiction between the restriction principle and any tolerance principle that permits some difference in the bicycle's original composition, simply by introducing a longer chain of possible worlds. Thus the transitivity argument appears to force the proponent of transworld identity to choose between two implausible claims: that an object such as a bicycle has all of its original parts essentially (thus denying any version of the tolerance principle) and that an object such as a bicycle could have come into existence with few (if any) of its original parts (thus denying any (non-trivial) version of the restriction principle). Moreover, it is clear that the problem can be generalized to any object to which versions of the tolerance principle and the restriction principle concerning its original material composition have application, which appears to include all artefacts, if not biological organisms. It seems legitimate to call this puzzle a 'problem of transworld identity', for it turns partly on the transitivity of identity, and can be avoided by interpreting claims about how bicycles could have been different (de re modal claims about bicycles) in terms of a counterpart relation that is not transitive (Chandler 1976). Thus a counterpart theorist may admit that the bicycle could have been originally composed of A1 + B1 + C2 rather than A1 + B1 + C1, on the grounds that (according to the tolerance principle) it has a counterpart (in w2) that is originally so composed. And the counterpart theorist may admit that a bicycle (such as the one in w2) that is originally composed of A1 + B1 + C2 could have been originally composed of A1 + B2 + C2, since (by the tolerance principle) it has a counterpart (in w3) that is originally so composed. However, since the counterpart relation (unlike identity) is not transitive, the counterpart theorist need not say that the bicycle in w3 that is originally composed of A1 + B2 + C2 is a counterpart of the bicycle in the actual world (w1) originally composed of A1 + B1 + C1, for its similarity to the bicycle in w1 may be insufficient to allow it to be that bicycle's counterpart. Thus the non-transitivity of the counterpart relation (a relation based on resemblance) appears neatly to allow the counterpart theorist to respect both the tolerance principle and the restriction principle, without falling into contradiction. ### 5.2 Responses to the transitivity problem One reaction to the transitivity puzzle is to abandon transworld identity in favour of counterpart theory. But how--given the structure of the puzzle--can the theorist who wishes to resist that move, and retain transworld identity, respond? One response would be to give up any non-trivial version of the restriction principle, and hold that an artefact such as a bicycle could have come into existence with an entirely different material composition from its actual original composition. Although this counterintuitive view has been defended (for example, Mackie (2006) argues for it on grounds independent of the transitivity problem), it has few adherents. A second response would be to give up the tolerance principle, and adopt what Roca-Royes (2016) calls an 'inflexible' version of the principle that the material origin of an artefact is essential to it, holding that an artefact such as a bicycle could not have come into existence with a material composition in any way different from its actual original composition. Although this view is admittedly counterintuitive, Roca-Royes argues that it provides the best solution to the 'Four Worlds Paradox' to be discussed in the next section. A third solution to the transitivity problem has been proposed (by Chandler, followed by Salmon) which apparently allows us to reconcile all three of the transitivity of identity, the tolerance principle, and the restriction principle. This is to say that although there are possible worlds (such as w3) in which the bicycle is originally composed of only a small proportion of its actual original parts, such worlds are not possible relative to (not 'accessible to') the initial world w1. From the standpoint of w1, such an original composition for the bicycle is only possibly possible: something that would have been possible, had things been different in some possible way, but is not, as things are, possible (Chandler 1976; Salmon 1979; Salmon 1982, 238-240). Whether this solution is satisfactory is disputed. (See, for example, Dorr, Hawthorne, and Yli-Vakkuri 2021, Chs 7-8.) Admittedly, there are some contexts in which we talk of possibility in a way that may suggest that the 'accessibility relation' between possible worlds is non-transitive: that not everything that would have been possible, had things been different in some possible way, is possible simpliciter. (If Ann had started writing her paper earlier, it would have been possible for her to finish it today. And she could have started writing her paper earlier. But, as things are, it is not possible for her to finish it today.) Nevertheless, the idea that, as regards the type of metaphysical possibility that is involved in puzzles such as that of the bicycle, there might be states of affairs that are possibly possible and yet not possible (and hence that de re metaphysical possibility and necessity do not obey the system of modal logic known as S4) is regarded with suspicion by many philosophers. It should be noted that the 'non-transitivity of accessibility' response is distinct from an even more radical response, which rejects the principle of the transitivity of identity--a principle definitive of the classical notion of identity. For example, Priest (2010) denies the transitivity of identity in the context of his dialetheism about truth and a paraconsistent logic in which the material conditional does not obey the principle of modus ponens. Discussion of this extreme position is, however, beyond the scope of this article. (On dialetheism and paraconsistent logic, see the separate entry on Dialetheism. On the logic of identity, see the entry on Identity.) Finally, Dorr, Hawthorne, and Yli-Vakkuri 2021 propose that, in some cases, the combination of (classical) transworld identity, tolerance, and the restriction principle can all be retained, yet without denying (as do Chandler and Salmon) the transitivity of accessibility between possible worlds. They appeal to two principles: a metaphysical principle of 'plenitude' and a metasemantic principle of 'semantic plasticity' (for the relevant terms). This solution is defended by Dorr et al partly by considering difficulties for rival solutions (including that of Chandler and Salmon). The details are complex, however: interested readers should consult Dorr et al 2021. As with other proposed solutions to the puzzle, the principles to which Dorr et al appeal (plenitude and semantic plasticity for the relevant terms) are controversial. It is fair to say that there is no consensus about how the proponent of transworld identity should respond to the transitivity problem posed by Chandler's example. ### 5.3 The 'Four Worlds Paradox' Chandler's transitivity argument can be adapted to produce a puzzle that is like those discussed in Sections 4.1-4.2 above in that it involves the danger of 'bare identities', a puzzle that Salmon (1982) has called 'The Four Worlds Paradox'. To illustrate the puzzle: suppose that the actual world (w1) contains a bicycle, a, that is (actually) originally composed of A1 + B1 + C1, and suppose that there is a possible world, w5, containing a bicycle, b (not identical with a), that is originally composed (in w5) of A2 + B2 + C1 (where A1 [?] A2 and B1 [?] B2). Then, it seems, the application of the tolerance principle to each of a and b may generate two further possible worlds, in one of which (w6) there is a bicycle with the original composition A1 + B2 + C1 that is identical with a, and in the other of which (w7) there is a bicycle with the original composition A1 + B2 + C1 that is identical with b. Since there need apparently be no further difference between the intrinsic features of w6 and w7 on which this difference in identities could depend, we appear to have a case of bare identities. This 'Four Worlds Paradox' is like Chandler's original transitivity puzzle in that it does not seem that an appeal to individual essences could solve it without conflicting with the tolerance principle. If that is so, the proponent of transworld identity (as opposed to counterpart theory) appears to be left with two options consistent with the transitivity of identity: the denial of the tolerance principle, and the acceptance of bare identities. (But cf. Dorr, Hawthorne, and Yli-Vakkuri 2021.) It may be argued, however, that the acceptance of bare identities can be made more palatable in this case by the adoption of a non-transitive accessibility relation between possible worlds. (See Salmon 1982, 230-252; and, for discussion, Roca-Royes 2016. For a defence of the employment of counterpart theory to solve the Four Worlds Paradox, see Forbes 1985, Ch. 7. For discussion of a radical response that retains the tolerance principle and yet avoids bare identities, but only at the cost of claiming that two bicycles could completely coincide in one possible world, simultaneously sharing all their parts, see Roca-Royes 2016, discussing Williamson 1990. On the relevance to the Four Worlds Paradox to Kripke's principle of the necessity (essentiality) of origin for artefacts, see also Robertson 1998 and Hawthorne and Gendler 2000.) ## 6. Concluding remarks ### 6.1 Transworld identity and counterpart theory One of our initial questions (Section 1 above) was whether a commitment to transworld identity--the view that an individual exists in more than one possible world--is an acceptable commitment for one who believes in possible worlds. The considerations of Sections 4-5 above suggest that this commitment does involve genuine (although perhaps not insuperable) problems, even for one who rejects David Lewis's extreme realism about the nature of possible worlds. The problems do not arise directly from the notion of an individual's existing in more than one possible world with different properties. Rather, they derive principally from the fact that it is hard to accommodate all that we want to say about the modal properties of ordinary individuals (including all the things we want to say about their essential and accidental properties) if de re modal statements about such individuals are characterized in terms of their existence or non-existence in other possible worlds. There is currently no consensus about the appropriate resolution of these problems. In particular, there is no consensus about whether the adoption of counterpart theory is superior to the solutions available to a transworld identity theorist. A full examination of the issue would require a discussion of the objections that have been raised against counterpart theory as an interpretation of de re modality. And a detailed discussion of counterpart theory is beyond the scope of this article. (For David Lewis's presentation of counterpart theory, the reader might start with Lewis 1973, 39-43, followed by (the more technical) Lewis 1968. Early criticisms of Lewis's counterpart theory include those in Kripke 1980; Plantinga 1973; and Plantinga 1974, Ch. 6. Lewis develops the 1968 version of his counterpart theory in Lewis 1971 and 1986, Ch. 4; he responds to criticisms in his "Postscripts to 'Counterpart Theory and Quantified Modal Logic'" (1983, 39-46) and in Lewis 1986, Ch. 4. Other discussions of counterpart theory include Hazen 1979, the relevant sections of Divers 2002, Melia 2003, and the more technical treatment in Forbes 1985. More recent critiques of the theory (postdating Lewis's 1986 response to his critics) include Fara and Williamson 2005, Fara 2009, and Dorr, Hawthorne, and Yli-Vakkuri 2021, Ch. 10.) One way to argue in favour of transworld identity (distinct from the defensive strategies discussed in Sections 4 and 5 above) is what we might call 'the argument from logical simplicity' (Linsky and Zalta 1994, 1996; Williamson 1998, 2000). The argument begins by noting that Quantified Modal Logic--which combines individual quantifiers and modal operators--is greatly simplified when one accepts the validity of the Barcan scheme, [?]x#A - #[?]xA (Marcus 1946). The resulting logic is sound and complete with respect to constant domain semantics, in which each possible world has precisely the same set of individuals in its domain. The simplest philosophical interpretation of this semantics is that one and the same individual exists at every possible world. Several remarks on this argument are in order. First, its conclusion is very strong: it says that any entity that in fact exists or that could have existed exists necessarily. There is no contingent existence. This goes far beyond the claim that there are genuine identities across worlds. (Williamson (2002) defends this conclusion on independent grounds.) Second, the argument does not offer an explanation of how transworld identities are possible; it insists only that there are genuine transworld identities. (Nevertheless, the metaphysical picture that can most naturally be 'read off' the constant-domain semantics treats properties-at-a-world as relations between particulars and worlds, as on McDaniel's modal realism with overlap (McDaniel 2004), discussed in Section 1.2 above.) Third, the argument is not best understood as the claim that, if one does not accept transworld identities, then one is forced into denying the Barcan scheme (and hence forced into uncomfortable logical territory). That claim would be true only if the Barcan scheme were validated only by constant-domain semantics, which is not the case. Counterpart-theoretic semantics can be restricted so as to validate the Barcan scheme, by insisting that the counterpart relation is an equivalence relation which, for each particular x and world w, relates x to a unique particular in w. (One could not then interpret the counterpart relation in terms of similarity, as Lewis does.) Rather, the argument should be understood as the claim that the best way to gain the advantages of a logic containing the Barcan scheme is by adopting constant-domain semantics (and genuine transworld identities along with it). But just which metaphysical view counts as 'best' here will involve a trade-off between many factors. These include the simplicity of the constant-domain semantics, on the one hand, but also arguments of the kind raised by Lewis against modal realism with overlap, on the other. ### 6.2 Lewis on transworld identity and 'existence according to a world' Finally, we can note that Lewis (1986) has presented a challenge to the self-styled champions of 'transworld identity' to explain why the view that they insist on deserves to be called a commitment to transworld identity at all. Throughout this article, it has been assumed that a commitment to transworld identity may be differentiated from a commitment to counterpart theory on the grounds that the transworld identity theorist accepts, while the counterpart theorist denies, that an object exists in more than one possible world (cf. Section 1.2 above). However, as Lewis points out, there is a notion of 'existence according to a (possible) world' that is completely neutral as between a counterpart-theoretic and a 'transworld identity' interpretation. In terms of this neutral conception, as long as the counterpart theorist and transworld identity theorist agree that Bertrand Russell could have been a playwright instead of a philosopher, they must agree that Russell exists according to more than one world. In particular, they must agree that, according to our world, he exists and is a philosopher, and according to some other worlds, he exists and is a non-philosopher playwright (cf. Lewis 1986, 194). The difference between the theorists, then, allegedly consists in their different interpretations of what it is for Russell to exist 'according to' a world. In the view of the counterpart theorist, for Russell to exist according to a possible world in which he is a playwright is for him to have a counterpart in that world who is (in that world) a playwright. According to the transworld identity theorist, it is supposed to be for Russell (himself) to exist in that world as a playwright. If the transworld identity theorist were a Lewisian realist about possible worlds, this notion of existence in a world could be clearly distinguished from the neutral notion of existence according to a world, on the grounds that the existence of Russell in a world would require his complete or partial presence as a part of such a world (cf. Section 1.2 above). But, as Lewis notes, the self-styled champions of 'transworld identity' who oppose his counterpart theory are philosophers who repudiate a Lewisian realist conception of what it takes for Russell to exist in more than one possible world. Hence, he argues, there is a question about their entitlement to claim that, according to their theory, Russell exists in other possible worlds in any sense that goes beyond the neutral thesis (compatible with counterpart theory) that Russell exists according to other worlds. Thus Lewis writes (using the 1968 US presidential candidate Hubert Humphrey as his example): The philosophers' chorus on behalf of 'trans-world identity' is merely insisting that, for instance, it is Humphrey himself who might have existed under other conditions, ... who might have won the presidency, who exists according to many worlds and wins according to some of them. All that is uncontroversial. The controversial question is how he manages to have these modal properties. (1986, 198) A natural reaction to Lewis's challenge is to point out that a proponent of transworld identity who is not a Lewisian realist will typically reject Lewis's counterpart theory on the grounds that his counterpart relation does not have the logic of identity. If so, then (pace Lewis) it is not the case, strictly speaking, that the 'philosophers' chorus on behalf of "trans-world identity"' is merely insisting on the neutral claim that objects exist according to more than one world. However, even if this is correct, it does not answer a further potential challenge. Suppose, as seems plausible, that there could be a counterpart relation that (unlike the one proposed by Lewis himself) is an equivalence relation (transitive, symmetric, and reflexive), and 'one-one between worlds'. What would distinguish, in the case of a theorist who is not a Lewisian realist about possible worlds, between, on the one hand, a commitment to the interpretation of de re modal statements in terms of such an 'identity-resembling' counterpart relation, and, on the other hand, a commitment to genuine transworld identity (and hence to the view that an individual genuinely exists in a number of distinct possible worlds)? The aficionado of transworld identity owes Lewis a reply to this challenge.
persons-means	## 1. Kantian Roots Kant sets forth several formulations of the categorical imperative, that is, the principle he holds to be the supreme principle of morality. One formulation, often called the "Formula of Humanity" states: > > > So act that you treat humanity, whether in your own person or in the > person of any other, always at the same time as an end, never merely > as a means. (Kant 1785: 429, italics > removed)[2] > > > The Formula of Humanity contains the command that we ought never to treat persons merely as means.[3] A few points regarding this command are helpful to keep in view. First, Kant holds that if a person treats someone merely as a means, then she acts wrongly. The Formula of Humanity encompasses an absolute constraint against treating persons merely as means. Second, Kant does not hold that if in acting a person refrains from treating anyone merely as a means, then she acts rightly (Kant 1797: 395). A person can, for example, act wrongly in Kant's view by expressing contempt for another, even if she is not using him at all (Kant 1797: 462-464). She would be acting wrongly by failing to treat the other as an end in himself, rather than by treating him merely as a means. A third related point is that, according to Kant, it is both a necessary and a sufficient condition for one's treating persons in a morally permissible way that one treat them as ends in themselves (Hill 1992: 41-42). Some Kantians, especially those engaged primarily in interpretation, rather than reconstruction, of his views thus understandably hold that far more important than understanding his position on treating persons merely as means is understanding his account of treating persons as ends in themselves (Wood 1999, 143). Fourth, Kant holds that a person can treat herself merely as a means. If a person acts contrary to certain "perfect duties to oneself" (Kant 1797: 421), including her duty not to kill herself (422-423), not to defile herself by lust (424-425), and not to lie (429-430), then she treats herself merely as a means, thereby contravening the Formula of Humanity. It is difficult to discern how, according to Kant, one treats oneself merely as a means in violating these duties (Kerstein 2008; Timmons 2017: Ch. 7). How, for example, does a person's lying to another amount to her treating herself merely as a means? In any case, this article's focus is on treating others merely as means. Doing that is widely discussed as a possible violation of a moral constraint. More specifically, the article explores when someone uses another and either treats or refrains from treating the other merely as a means. It concentrates on concepts that seem to have roots in Kant's work, but that are familiar from ordinary moral discourse.[3] Kant himself devotes little discussion to clarifying the notion of treating others merely as means.[4] Yet as is apparent below, some of his remarks have been a springboard for detailed accounts of the notion. One salient issue outside the article's purview is that of the scope of "others" in the prescription not to treat others merely as means. Does Kant embrace among these others all genetically human beings, including human embryos and individuals in a persistent vegetative state, or does he limit the others to beings who have certain capacities, for example, that to set and rationally pursue ends (Kain 2009; Sussman 2001)? How should Kant or other theorists set the scope of those whom we ought not to treat merely as means? Should we include some non-human animals (e.g., chimpanzees or dolphins) among them? How should we determine to whom to grant this moral status? ## 2. Using Another In order to treat another merely as a means or just use him, an agent must use the other or treat him as a means. But when does someone count as doing that? As noted, using others or treating them as means is often morally permissible. In everyday discourse, expressions such as "She used me" can mean she just used me, or treated me merely as a means and so can imply a negative evaluation of action or attitude. But for our purposes talk of a person using another or treating him as a means implies no such moral judgment. Accounts of treating others merely as means sometimes leave implicit the notions of using another they rely on (Kleingeld 2020). Some points regarding what using another does or does not amount to seem uncontroversial. It does not seem sufficient for us to count as using another as a means that we benefit from what the other has done (Nozick 1974: 31-32). If, on her usual route, a runner derives enjoyment from the singing of a stranger who happens to be walking by, she does not appear to be treating the stranger as a means. Moreover, not all cases of an agent's intentionally doing something in response to another are cases of her using the other as a means. If someone frowns at another approaching, for example, he might not thereby be using the other at all; he might simply be expressing that the other is unwelcome. However, inquiry reveals challenges in specifying what using another amounts to. We might say that an agent uses another or, equivalently, treats her as a means just in case the agent intentionally does something to the other in order to secure or as a part of securing one of his ends (Kerstein 2009: 166). For example, a passenger uses a bus driver if he boards her bus in order to get across town; a wife treats her husband as a means if she lies to him so that his birthday party will be a surprise; and a victim treats a mugger as a means if she punches him in order to escape from his grasp. In contrast, a pilot who drops bombs solely in order to kill enemy combatants might foresee that innocent bystanders will be harmed. Yet if he does not intentionally do anything to the bystanders, then he does not treat them as means, according to this account.[5] But does the account count too much as treating another as a means? Suppose that an usher at a concert is trying to prevent a small child from falling through railing on a balcony. She pushes a spectator out of the way to get to the child. The specification we are considering implies that the usher has used the spectator as a means; for she has intentionally done something to her (i.e., pushed her aside) in order to attain an end (i.e., to get to the child). Some might say that the usher has treated the spectator in some way, namely, as an obstacle to be displaced. Yet she has not used the spectator. In order for an agent to count as using another, it is not enough that she do something to the other in order to realize some end of hers, some have suggested. She must also intend the presence or participation of some aspect of the other to contribute to the end's realization (Scanlon 2008: 106-107; Guerrero 2016: 779). The usher does not intend the spectator's presence or participation to play any role in her preventing the child from falling. She thinks of her simply as "in the way". On one account, an agent uses another (or, equivalently, uses or treats another as a means) if and only if she intentionally does something to or with the other in order to realize her end, and she intends the presence or participation of the other to contribute to the end's realization (Kerstein 2013: 58). On this account, an agent can count as using another when she is striving to benefit him. For example, a physician giving a patient a treatment to save his life is using the patient. Some find this implication of the account implausible (Parfit 2011: 222).[6] Others do not, pointing to cases such as that of a physician using a patient in a study of a new drug in order to ameliorate the patient's condition. In any case, consistent with this account, an agent might use another through using the other's rational, emotional, or physical capacities. A tourist might ask someone for directions, using the other's knowledge to get to his destination; a politician might use his constituents' fear of crime to gain their support for more spending on law enforcement; a doctor might use a vein from a patient's leg to repair her heart. One important question left unanswered by this and other accounts of treating another as a means is one of scope. For example, does an agent use another if he uses biospecimens (e.g., cells) or information (e.g., concerning social media activity) derived from the other? If so, then the scope of a constraint on treating others merely as means would extend to the practices of biobanks and technology companies. ## 3. Sufficient Conditions for Using Others Merely as Means Much debate concerning what it means to treat others merely as means stems from a single passage in the Groundwork of the Metaphysics of Morals. Kant is attempting to demonstrate that the Formula of Humanity generates a duty not to make false promises: > > > He who has it in mind to make a false promise to others sees at once > that he wants to make use of another human being merely as a > means, without the other at the same time containing in himself > the end. For, he whom I want to use for my purposes by such a promise > cannot possibly agree to my way of behaving toward him, and so himself > contain the end of this action. (1785: 429-430) > > > In these brief remarks, Kant hints at various ways in which we might understand conditions for treating another merely as a means. We might understand them in terms of the other's inability to share the agent's end in using him or to consent to her using him, for example. In this section, we discuss elaborations of these ways (and others) of formulating sufficient conditions for someone who is using another to be treating this other merely as a means. ### 3.1 End Sharing On the basis of Kant's remarks, we might claim that if another cannot "contain the end" of an agent's action, that is, share the end the agent is pursuing in using her, then the agent treats the other merely as a means. Two agents presumably share an end just in case they are both trying, or have chosen to try, to realize this end. But what, precisely, does it mean to say that two agents cannot share an end? Returning to the example at hand, what does it mean to say that the promisee cannot share the promisor's end? From the outset, it is important to specify precisely which of the promisor's ends the promisee cannot share. It is presumably the promisor's end of getting money from the promisee without ever paying it back. The promisor's ultimate end might be one that the two can share (e.g., that of curing cancer). What sense of 'cannot' would be plausible to invoke in maintaining that a promisee cannot share a false promisor's end? #### 3.1.1 Logical impossibility of end sharing According to one interpretation of Kant, the promisee cannot share the promisor's end in that it is logically impossible for him to do so (Hill 2002: 69-70). Suppose the promisor, a borrower, has the end of getting money from the promisee, a lender, without ever paying it back. At the time he makes a loan on the basis of this promise, the lender cannot himself share the end of the borrower's getting the money from him without ever paying it back, goes this reading. If the lender shared the borrower's end, then he would not really be making a loan. For according to our practice, it belongs to the very concept of making a loan, as opposed, say, to giving money away, that one believe that what one disburses will be repaid. This interpretation of the false promising case leads naturally to the view that a sufficient condition for an agent's treating another merely as a means is that it is logically impossible for the other to share the end the agent is pursuing in using her in some way. However, this proposed sufficient condition might fail to register as treating others merely as means paradigmatic cases of doing so (Kerstein 2009: 167-168). Take, for example, a loiterer who threatens an innocent passerby with a gun in order to get $100. It would presumably be good if a sufficient condition for treating another merely as a means yielded the conclusion that the loiterer is treating the passerby merely as a means; for he is mugging her, which, intuitively speaking, seems to be a clear case of treating another merely as a means. One might question whether the proposed sufficient condition does this. Even highly unlikely events are logically possible. It is improbable, but still logically possible, that the passerby shares the loiterer's end of his getting $100, one might argue. For example, the passerby might aim to give $100 to the loiterer, but not recognize him when he threatens her and so hands over her money to him as a result of his threat. If this possibility is realized, then the account would not count the loiterer as treating the passerby merely as a means. One might also argue that in the case of the false promise, it is improbable, but still logically possible, that the lender loans money to a borrower (and thereby believes that it will be repaid), all the while sharing the borrower's end that she get money from him (the lender) without paying it back. For example, the lender might believe that the borrower will pay him back, but share her end of her getting money from him without repaying it because he believes that if she does, she will bring about something he covets, namely, the demise of her reputation. Some philosophers insist, however, that this sort of scenario is not logically possible; for in order to be making a loan to another, a person must not only believe that his money will be repaid, but want and hope that it will be (Papadaki 2016: 78), they say. If these philosophers' views are plausible, the proposed sufficient condition in view would deem as instances of treating others merely as means a range of actions many envisage as such. #### 3.1.2 Preventing the other from choosing to pursue one's end According to a different interpretation of Kant, another cannot share the end an agent pursues in using him in some way if how the agent behaves "prevents [the other] from choosing whether to contribute to the realization of that end or not" (Korsgaard 1996: 139). The lender in our example cannot share the borrower's end of getting money without ever repaying it; for the borrower's false promise obscures his end and thus prevents the lender from choosing whether to contribute to it. This reading of possible end sharing might have implausible implications when incorporated into a principle according to which a person who uses another treats the other merely as a means if the other cannot share the end she is pursuing in using him (Kerstein 2013: 63). Consider young men hiking in the Rocky Mountains for the first time who find themselves on a mountain in late afternoon without water and unsure of the way down. To their relief, they spot another hiker, someone whom they saw park his car in the same area below. They follow him, using his knowledge of the terrain to get down the mountain safely. The young men realize that they could, but choose not to, tell the hiker that they are following him. Out of embarrassment for their dependence on him, they ensure that they remain undetected. The way they act prevents the man from choosing whether to contribute to the realization of their end. According to the notion of possible end sharing we are here considering, we might have to embrace a view that some find implausible: Since the hiker cannot share the young men's end, they are treating him merely as a means and thereby acting wrongly. To avoid this implication, one might affirm the following: a person cannot share the end an agent pursues in using him if the agent's behavior prevents the person from choosing whether to contribute to the realization of that end and the person has a right not to be prevented from making this choice (Papadaki 2016: 80). If the hiker fails to have a right not to be prevented from choosing whether to contribute to the young men getting safely down the mountain, then they do not treat him merely as a means according to the amended account. Of course, this amendment invites questions as to when a person has such a right, as well as making the account of treating others merely as means depend on an account of moral rights in a way that Korsgaard (or Kant) might not have intended. #### 3.1.3 Practical irrationality The allusion in the false promising passage to possible end sharing is subject to a third interpretation: the promisee cannot share the promisor's end in that it would be practically irrational for her to do so. In typical cases, it would be irrational for the promisee to try to realize the end of making a loan that is never to be repaid. This end's being brought about would prevent him from realizing other ends he is pursuing, ends such as paying rent, buying groceries, or simply getting his money back. The notion of practical irrationality at work here seems implicit in the Groundwork. Kant there (1785: 413-418) introduces a principle that Thomas Hill, Jr. calls (1992: 17-37) "the hypothetical imperative": If you will an end, then you ought to will the means to it that are necessary and in your power, or give up willing the end. Willing an end presumably involves setting it and attempting to realize it. According to Kant, the hypothetical imperative is a principle of reason: all of us are rationally compelled to conform to this principle. An agent would be violating the hypothetical imperative and thus acting irrationally by willing an end yet, at the same time, willing another end, the attainment of which would, he is aware, make it impossible for him to take the otherwise available and necessary means to his original end. An agent would violate the hypothetical imperative, for example, by willing now to buy a house and yet, at the same time, willing to use the money he knows he needs for the down payment to make a gift to his niece. If he willed to make the gift, he would be failing to will the necessary means in his power to buy the house. The Kantian hypothetical imperative implies that it is irrational to will to be thwarted in attaining ends that one is pursuing. In typical cases, if a promisee willed the end of a false promisor, she would be doing just that. There are two things that an agent who has willed something can do which would bring his action into compliance with the hypothetical imperative. He can either will the means that are necessary and in his power to the end (which, of course, would rule out his willing to be thwarted in attaining the end) or he can give up willing the end. For example, the hypothetical imperative would not imply that it was irrational for the person described above to cease willing now to buy a house and instead use the money that he knows would be required as a down payment on it to make a gift to his niece. A person cannot share an agent's end, according to this third account, if: > > > The person has an end such that his pursuing it at the same time that > he pursues the agent's end would violate the hypothetical > imperative, and the person would be unwilling to give up pursuing this > end, even if he was aware of the likely effects of the agent's > successful pursuit of her end. > > > By way of illustration, suppose that a doctor plans to use a healthy patient to obtain a heart and lungs for transplant, that is, to extract them from him in an operation that would kill him. We can imagine that the patient has many ends, for example, that of attending his daughter's wedding. According to the hypothetical imperative, it would be irrational for him to pursue this end at the same time he was pursuing the doctor's end of getting from him a heart and lungs for transplant. The account implies that the patient cannot share the doctor's end if he would be unwilling to give up his end of attending his daughter's wedding against the background of an awareness of the likely effects of the doctor's successful pursuit of his organs (e.g., his life being lost and other lives being saved). This notion of conditions under which a person cannot share an agent's end might be included in the following account: An agent treats another merely as a means if the other cannot share the proximate end or ends the agent is pursuing in treating him as a means. An agent's proximate end is something she aims to bring about directly from her use of the person. Her proximate end might also be her ultimate end, say, if she uses another to avoid pain. But her proximate end might be far removed from her ultimate end. Someone might, for example, use another to develop her skill as a violinist to earn a good living in an orchestra so she can put her little sisters through college, and so forth. The account invokes proximate ends because they are far more intimately connected to the use that brings them about than ultimate ends need be. Yet, like the other accounts we have considered, this account is subject to criticism. One possible shortcoming stems from cases of competition (Kerstein 2009: 170-171). Sometimes people have the end of being the sole winner in a competition. A competitor pursuing such an end might, according to the account, be treating his competitor merely as a means and thus acting wrongly, even though she abides fully with the competition's rules. To begin, competitors sometimes count as treating one another as means. To invoke one account of doing so (discussed in SS2 above), Player A intentionally does something to her opponent, Player B, for example, tries to defeat him, which requires B's presence or participation. Moreover, A's proximate end in trying to defeat B might be to win top player for the year; and B's proximate end in trying to defeat A might also be to win top player for the year. To focus on A, she is using B to realize an end, namely her (A's) winning top player for the year, but B cannot share this end. In willing that A be top player of the year, B would, in effect, be willing to be thwarted in his attempt to win top player for the year, assuming that there can be no tie for top player. Finally, awareness on B's part of the likely effects of A's successful pursuit of being the top player would presumably not result in B's being willing to give up his (B's) end of being top player. In trying to defeat B to be number one, A would be treating B merely as a means and thereby acting wrongly, the account seems to imply, even if A competed fairly, that is, violated none of the competition's rules. Some might find this implication implausible. Sometimes becoming the best in some endeavor involves defeating (and using) competitors to do so. But defeating (and using) competitors to be the best, especially when they have freely entered a competition, need not amount to acting wrongly, some might insist. ### 3.2 Possible Consent In the passage on false promising, Kant references possible consent. He suggests that the victim of the false promise cannot agree to the use the false promisor is making of him. We might conclude that the victim cannot agree on the grounds that he cannot share the promisor's end; for it would, in the sense invoked above (SS3.1.3), be practically irrational for him to pursue this end. There is, however, another way of interpreting the victim's inability to consent in the context of considering candidates for a plausible sufficient condition for an agent's treating another merely as a means. Another account, prompted by the Groundwork passage, is this: An agent uses another merely as a means if the other cannot consent to her use of him (O'Neill 1989: 113).[7] An agent cannot consent to being treated as a means if he does not have the ability to avert his being treated as such by dissenting, that is, by withholding his agreement to it.[8] If an agent deceives or coerces another, then the other's dissent is "in principle ruled out" (1989: 111) and thus so is his consent. Suppose, for example, that an appliance serviceperson tricks a customer into authorizing an expensive repair. The customer does not really have the opportunity to dissent to the person's action by refusing to give his consent to it. For he does not know what her action is, namely one of lying to him about what is wrong with his refrigerator. (If he did know what her action was, then he would not be deceived.) Or suppose that a mugger approaches you on a dark street, points a gun at you, and tells you that unless you give him all of your money, he will hurt you. He leaves you no opportunity to avert his use of you by withholding your consent. Regardless of what you say, he is presumably going to use you, whether it is through your handing over your wallet or his violently taking it from you. Since you cannot consent to his action, the mugger is treating you merely as a means.[9] The account is subject to objections. It does not suffice for an agent to treat another merely as a means that the other simply be unable to consent to the way he is being used, some argue. If it did suffice, then a passerby giving cardiopulmonary resuscitation (CPR) to a collapsed jogger would be treating the jogger merely as a means and thus acting wrongly. But the passerby does not seem to be doing anything that is morally impermissible. In light of this objection, someone might propose a different account: Suppose an agent uses another. She uses him merely as a means if something she has done or is doing to the other renders him unable to consent to her using him. Of course, although the collapsed jogger has no opportunity to consent to the passerby's giving him CPR, the passerby has not put him in that position. So this account avoids the unwelcome implication that the passerby treats the collapsed jogger merely as a means. However, this account is also open to objection. First, it fails to designate as such some cases that we, intuitively speaking, would surely classify as treating others merely as means (Kerstein 2013: 74). Think for example of a case where one person knocks someone out with a "date rape" drug. Another person, who had no knowledge of or involvement in drugging the victim, sexually assaults him. Since this other person has not rendered the victim unable to consent to his use of him, the account does not yield the conclusion that he treats him merely as a means. The account arguably not only fails to capture some cases of an agent's treating another merely as a means, but also designates as such some cases of deception that, intuitively speaking, are not. For example, in order to make your spouse's birthday party a surprise for her, you need to lie to your sister-in-law about your whereabouts during a certain afternoon. You use her to quell your spouse's suspicions regarding your plans. As you realize, if you told your sister-in-law about the party, she would be unable to keep the secret from your spouse. According to the account, you treat your sister-in-law merely as a means, since your deception leaves her with no opportunity to avert your use of her. This conclusion seems questionable to some, albeit not to others. Here is another case of what some think of as morally permissible deception (Parfit 2011: 178). Suppose that, in order to save the life of an innocent witness to a crime, you use her to pass on a lie you have told her to the perpetrator, Brown. If Brown did not believe the lie, he would kill the witness. You realize that if you let the witness in on what was necessary to save her life and told her to lie to Brown herself, she would not be able to do so effectively. Your treatment of the person renders impossible her consent to your use of her. But it is implausible to conclude that you are treating her merely as a means, some insist. In these two cases, it makes sense to think that the person you are using can share your ends, in the sense specified in SS3.1.3. Your sister-in-law can share the end of your spouse not getting suspicious regarding a surprise party, and, of course, the witness can share the end of Brown's coming to believe some lie. Perhaps that is why in these cases the person's inability to consent to your use of her seems to fall short of plausibly implying that you are using her merely as a means. ### 3.3 Actual Consent A proposal for a sufficient condition for treating another merely as a means might invoke a notion of actual consent. Suppose an agent is using another, the proposal might go; he is using her merely as a means if she has not consented to his use of her (Nozick 1974: 30-31; Scanlon 2008: 107). A more detailed actual consent proposal is the following: "An agent uses another person merely as a means if and only if (1) the agent uses another person as a means in the service of realizing her ends (2) without, as a matter of moral principle, making this use conditional on the other's consent; where (3) by 'consent' is meant the other's genuine actual consent to being used, in a particular manner, as a means to the agent's end" (Kleingeld 2020: 398). Both proposals face challenges. For example, suppose a patient arrives in the emergency room unconscious, with severe injuries. The physician on duty judges that only an experimental treatment could preserve the patient's life. She therefore uses the patient in a study of a new technique to prevent blood loss. Both actual consent proposals imply, seemingly implausibly, that the physician used the patient merely as a means. One might claim that an agent who uses another without the other's actual consent treats him merely as a means unless particular conditions obtain: the agent lacks competency to consent, the other is rationally required to consent (e.g., perhaps to preserve his life), or the agent is rationally forbidden to consent (e.g., perhaps to being someone's slave) (Formosa 2017: 99-100). This proposal would avoid the implication that the physician in the preceding example is using the patient merely as a means; for the patient obviously lacks competency to consent. Yet the proposal seems to suffer from a shortcoming that all the actual consent accounts mentioned have. They all seem to imply, implausibly, that an agent would be treating another merely as a means in cases such as this (Cohen 1995: 243): Someone uses another without the other's consent to prevent the light from shining in her eyes, by positioning herself in an appropriate way. ### 3.4 Invoking Moral Notions (or Not) A distinction worth noting in accounts of using others merely as means is that between ones that do and ones that do not appeal to moral concepts. A clear example of the former type of account is the following: X uses Y merely as a means if and only if (1) X uses Y as a means and (2) X's use of Y violates a duty that X owes to Y (Fahmy 2023: 58). To determine whether an agent uses another merely as a means on this account, it is obviously necessary to rely on views regarding which duties the agent owes to the other. Not surprisingly, philosophers who offer such accounts sometimes invoke Kant's notion that persons have dignity that demands respect as a basis for determining our moral duties (2023: 50) (Formosa 2017: 108). A worry concerning these accounts is that the light they shed on what treating others merely as means amounts to is limited by the clarity, or lack thereof, in the moral notions they invoke. Other philosophers suggest sufficient conditions for a person treating another merely as a means that seem not to depend on any moral concepts (Kerstein 2013: 76) (Audi 2016: 4, 56-57). It is often easier to discern the implications, both plausible and implausible, of these accounts than the implications of accounts that rely on disputed or vague moral concepts such as our dignity or rights as persons. Of course, not only accounts of when agents treat others merely as means, but also accounts of when they do not do so (Section 5) differ in whether they are grounded in moral concepts. ## 4. Treating Another Merely as a Means and Acting Wrongly Kant holds that if someone treats another merely as a means, the person acts wrongly, that is, does something morally impermissible. Some accounts of treating others merely as means seem not to yield the conclusion that if a person treats another in this way, then he acts wrongly. On one "rough definition", we use another merely as a means if we both use the other and regard him > > > as a mere instrument or tool: someone whose well-being and moral > claims we ignore, and whom we would treat in whatever ways would best > achieve our aims. (Parfit 2011: 213 and 227) > > > For example, a kidnapper treats his victim merely as a means if she uses him for profit and thinks of him simply as a tool that she would treat in any way necessary for profit. This account takes quite literally "merely" in "treating others merely as means". According to it, treating another merely as a means amounts roughly to treating the other solely or exclusively as a tool. If this is how we understand treating others merely as means, then doing so does not always amount to acting wrongly, it appears.[10] Suppose that a gangster considers a barista a mere tool to get coffee and that he would treat her in whichever way would best serve his interests. In buying coffee from her, the gangster treats her merely as a means on this account, but does not, it seems, act wrongly (Parfit 2011: 216). On a more detailed, but similar, understanding of treating another merely as a means, it also seems that the gangster might be doing so and yet not be acting wrongly. According to this account: "Treating a person merely as a means is doing something toward that person . . . on the basis of instrumental motivation, with no motivational concern regarding any non-instrumental aspects of the action(s), and with a disposition not to acquire such a concern" (Audi 2016 56-57). These notions of treating others merely as means seem not to coincide with Kant's notion of doing so. Recall Kant's example of making a false promise to another for financial gain. Suppose that a particular false promiser would not do just anything to the other for gain for himself, for example, he would not murder the other's family. And suppose that this false promisor does have non-instrumental motivational concern for the other in that he would not make this false promise if it would likely result in the other needing to declare bankruptcy. According to Kant's notion, but not to these accounts, it appears, the false promisor would be treating the other merely as a means.[11] We might question whether treating another merely as a means amounts to acting wrongly even if we focus on the candidate sufficient conditions examined above. For the sake of simplicity, let us focus on the possible consent account (SS3.2), according to which an agent treats another merely as a means if the other cannot consent to her use of him. (We could just as effectively employ other proposed sufficient conditions we have discussed for just using another.) Suppose that two muggers attack a victim. The victim violently pushes one of the muggers into the other, so that he (the victim) can make his escape. The victim uses the mugger he pushes, and the mugger presumably is unable to avert this use simply by dissenting from it. Yet many would object to the idea that the victim is acting wrongly. One response to this issue would be to build into accounts of treating another merely as a means the specification that one is not doing that if he is using someone in order to prevent himself or someone else from being treated in this way. Building in this specification would, of course, tend to make accounts somewhat unwieldy. Other examples might render more difficult to accept the idea that treating another merely as a means is always morally impermissible. Suppose, for example, that we use one person to save a million people from nuclear conflagration, without giving the one any opportunity to avert the use by dissenting from it. We thereby treat the one merely as a means, according to a possible consent account. But do we act wrongly? Some hold that we do not.[12] They might defend the view that while it is always wrong pro tanto to treat another merely as a means, doing so is sometimes morally permissible, all things considered. In other words, we always have strong moral reasons not to treat others merely as means, but these reasons can be outweighed by other moral considerations, presumably including the good of many lives being preserved. If, contrary to Kant's view, the moral constraint against treating others merely as means is not absolute, then a question arises as to when it gets overridden. ## 5. Using Another, but Not Merely as a Means We have explored sufficient conditions for treating another merely as a means. But just as challenging as pinpointing them is specifying when someone uses another, but not merely as a means. According to one proposal, if an agent uses another, she does not use him merely as a means if he gives his voluntary, informed consent to her use of him. To fix ideas let us say that the consent of the person being used is voluntary only if he is not being coerced into giving it and informed only if he understands how he is being used and to what purpose(s). This proposal seems intuitively attractive. If a person agrees to someone using him and understands her ends in doing so, then how can she be treating him merely as a means? Appealing to reflective common sense, philosophers have tried to illustrate how. We might refer to one range of cases they invoke as exploitation cases because they seem to involve one person taking unfair advantage of another, which is a hallmark of exploitation (Wertheimer 1996). To cite one such case, suppose that a mother of modest means cannot afford to give her children a good education. A rich person proposes to finance her children's enrollment at excellent schools in exchange for her serving as his personal slave (Davis 1984: 392). The mother might understand the use he intends to put her to and for what purposes. Moreover, if we think of coercion as involving an agent threatening to make someone worse off than she would be if she did not interact with the agent, then the rich person would not count as coercing the mother into agreeing to his use of her. The account of using another but not merely as a means that we are considering might, therefore, imply that the rich person is not just using the mother in making her his personal slave. That implication strikes some as implausible. Another type of case that might cause problems for this account invokes unnecessary or otiose threats designed to force another to serve one's purposes. Here is an example of such a case. An elderly salesperson thinks that his company is trying to force him into retirement by keeping its latest sales leads from him. Desperate to make a sale, he intends to use his office manager to obtain the latest leads. The manager has the password to a database housing the leads. He tells her that he really needs to close some deals, and unless she gets the latest leads for him, he is going to reveal to everyone in the office that she is lesbian. He believes reasonably, given his incomplete understanding of her and of the attitudes of his other co-workers, that this revelation would be damaging to her reputation. But the office manager takes the salesperson, whom she thinks of as a friendly colleague, to be making an ill-advised joke. Virtually everyone in the office is already aware of her sexual preference. And, she believes, the salesperson is cognizant that it is company policy that all salespeople are to be granted access to the latest leads upon request. She gives him a puzzled look and agrees to get him the leads right away. The salesperson receives from the office manager her voluntary, informed consent to his use of her to get the leads. She understands that he intends to use her to this end. Granted, he threatens to make her worse off if she does not give him the leads. But it is not the threat, which she does not even register as such, that generates her agreement to his use of her. She agrees voluntarily. Yet, despite obtaining her consent for his use of her, some believe that the salesperson treats the office manager merely as a means and acts (at least pro tanto) wrongly. Others might argue that although the salesperson has revealed a moral deficiency, he has not done anything wrong. Rather, he has simply revealed a morally faulty attitude toward the office manager (Scanlon 2008: 46; Walen 2014: 428-429). If we judge that the salesperson does act wrongly, then we presumably take this case to illustrate that treating another merely as a means does not necessarily amount to harming her. In other words, in treating another merely as a means, an individual can wrong another without harming her. Regardless of whether we judge in this case that the salesperson acts wrongly, the case helps to illustrate a distinction between agent-focused and patient-focused accounts of treating or not treating another merely as a means. According to the account we are considering, let us recall, if an agent uses another, she does not use him merely as a means if he gives his voluntary, informed consent to her use of him. This account focuses on the other, that is, on the individual treated as a means to determine whether the agent is treating him merely as a means. If he (i.e., the other) gives his informed, voluntary consent to being used in some way, then the agent does not treat him merely as a means, according to the account. To make this determination, an agent-focused account would, of course, focus more on the agent. Such an account might hold, for example, that if an agent uses another, she does not use him merely as a means if it is reasonable for her (the agent) to believe that the other gives his voluntary, informed consent to her use of him. According to the notion of reasonable belief invoked here, it is reasonable for someone to believe something roughly if the belief is justifiable given the person's context (e.g., his upbringing, cognitive limitations, and so forth). Contrary to the patient-focused account, the agent-focused account is free from the implication that the salesperson is not treating the office manager merely as a means. It is not reasonable for the salesperson to believe that the office manager has given her voluntary consent to his use of her. It is, rather, reasonable for him to believe that he has coerced her into giving him the sales leads. A parallel patient-based vs. agent-based distinction applies, of course, to accounts of sufficient conditions for treating another merely as a means. One might, for example, hold that an agent just uses another if the other cannot share the end the agent is pursuing in using him (a patient-focused account). Or one might hold that an agent just uses another if it is reasonable for her to believe that the other cannot share the end the agent is pursuing in using him (an agent-focused account). We have been considering actual consent accounts of agents using others, but not treating them merely as means. We can also develop accounts that invoke other concepts familiar from discussion of sufficient conditions for treating others merely as means, including the concepts of possible end-sharing and possible consent. For example, we might suggest that an agent who uses another does not use that other merely as a means if the other can consent to the agent's use of him, that is, if the other can avert the use simply by dissenting from it. This suggestion, as well as others that invoke additional concepts we have considered, such as that of end sharing, might generate questionable verdicts regarding exploitation cases. For example, the mother of modest means discussed above can consent to the rich person's use of her. Her dissent from it alone would stop it from occurring. But some would insist that the rich person is nevertheless treating her merely as a means in making her his personal slave in exchange for educating his children. Another approach to conditions under which an agent uses another, but does not treat him merely as a means takes shape against a literal construal of treating others merely as means. On this construal, discussed above, we use another merely as a means roughly if we both use the other and regard him as a mere tool. According to the approach, we do not treat another merely as a means if our treatment of the other person "is governed or guided in sufficiently important ways by some relevant moral belief or concern" (Parfit 2011: 214). But when is an agent's use of another governed in sufficiently important ways by some relevant moral belief or concern? The agent's use of another is so governed, according to one response, when the agent tries to and succeeds in using the other only in ways to which the other can rationally consent. But when can the other rationally consent? To simplify matters, let us make some background assumptions. Let us suppose that the people who might be used understand what will be done to them and to what purpose, as well as the effects the use will have. Let us also assume that those who might be used have the power to give (or to withhold) consent in the "act-affecting sense" (Parfit 2011: 183-184). When we ask whether they can rationally consent to being used, we are asking whether it would be rational for them to consent (or dissent) supposing that their choice would determine whether or not they were used. Against the background of these assumptions, we can say that, on this account, a person can rationally consent to being treated as a means just in case he has sufficient reasons to consent to it. The account rests on an "objective" view of reasons, according to which > > > there are certain facts that give us reasons both to have certain > desires and aims, and to do whatever might achieve these aims. These > reasons are given by facts about the objects of these desires > or aims, or what we might want or try to achieve. (Parfit 2011: > 45) > > > For example, the fact that a child is in pain as a result of a splinter stuck in his finger gives me reason to want to and to try to get it out.[13] On this account, we have impartial as well as partial reasons for agreeing to be treated in various ways. Our impartial reasons are "person-neutral" (2011: 138). We do not need to invoke ourselves when we describe the facts that yield these reasons. The fact that some event would cause tremendous pain to a particular person, for example, gives us reason (albeit perhaps not sufficient reason) to prevent the event or relieve the pain "whoever this person may be, and whatever this person's relation to us" (2011: 138). Our partial reasons are "person-relative": they "are provided by facts whose description must refer to us" (2011: 138). The fact that the little boy being hurt by the splinter is my son gives me a partial reason to pull it out. We each have partial reasons to be particularly attentive to our own well-being and to the well-being of those in our circle, for example, our family and friends. According to the account, > > > When one of our two possible acts would make things go in some way > that would be impartially better, but the other act would make things > go better either for ourselves or for those to whom we have close > ties, we often have sufficient reasons to act in either of these ways. > (2011: 137) > > > For example, regarding a case in which a person could either save himself from some injury or do something that would save some stranger's life in a distant land, the person presumably has sufficient reasons to do either one. In a similar vein, a person can have sufficient reason to consent to being treated as a means by virtue of some impartial reason, such as the fact that his being so treated will save many lives, even if he also has sufficient reason to dissent from being treated as a means by virtue of some partial reason, such as the fact that his being so treated will result in suffering for him. In sum, this account holds that if an agent uses another, he does not use the other merely as a means if the other has sufficient reasons, as just characterized, for agreeing to be used. The account seems to imply that cases thought to be paradigmatic of treating others merely as means do not involve treating them in this way. Take a case in which a pedestrian is on a bridge above a track where a train is barreling toward five people (Parfit 2011: 218). The only way to save the five would be to open, by remote control, a trap-door on which the pedestrian is standing, so that he would fall in front of the train. The train would kill him, but the impact would trigger its automatic brake. If a bystander opens the trap door, then she uses the pedestrian as a means to save the five. The pedestrian has sufficient reasons to refrain from consenting to being used to stop the train. After all, it will result in his premature death. But, according to the account, he also might have sufficient reasons to consent to being used to stop the train; for his being used in this way would save the lives of five people, contributing to an outcome which is presumably impartially best (2011: 220). Suppose the bystander opens the trap door and uses the person on the bridge to save the five. In so doing, she might be limiting her use of another to ways to which the other can rationally consent. If she is, then she is not treating the person merely as a means, according to the account. Consider another well-known example. Five patients at a hospital are in immediate need of different organs. One patient needs a kidney, another needs a liver, and so forth. If a surgeon used a healthy person undergoing routine tests as a resource for organs, killing him in the process, all of the five would be saved (Harman 1977: 3-4). The healthy person presumably has strong partial reasons to dissent from being used for his organs. But he also arguably has enough impartial reason to consent, namely, that five people will thereby be saved, such that overall he has sufficient reason to consent. So, assuming that the surgeon is trying to treat people only in ways to which they can rationally consent, she might not be treating the healthy person merely as a means, even if before she succeeds in putting him under, he is begging for his life.[14] If the account we are considering of using others, but not merely as means does entail that the bystander and the doctor in these two cases are not treating others merely as means, the account suffers from a significant flaw, according to some. In the words of one philosopher, the idea that it is wrong to treat others merely as means is "both very important and very hard to pin down" (Glover 2006: 65). Our investigation has illustrated challenges in specifying what it means to treat others merely as means. It has not revealed one, univocal concept, grounded in common sense, of what just using another amounts to. In the end, there may be no such concept, but rather a set of overlapping notions, which point to a range of morally problematic actions or attitudes concerning the use of others.
trinity	## 1. One-self Theories One-self theories assert the Trinity, despite initial appearances, to contain exactly one self. ### 1.1 Selves, gods, and modes A self is a being who is in principle capable of knowledge, intentional action, and interpersonal relationships. A deity is commonly understood to be a sort of extraordinary self. In the Bible, the deity Yahweh (a.k.a. "the LORD") commands, forgives, controls history, predicts the future, occasionally appears in humanoid form, enters into covenants with human beings, and sends prophets, whom he even allows to argue with him. More than a common deity in a pantheon of deities, he is portrayed as being the one creator of the cosmos, and as having uniquely great power, knowledge, and goodness. Trinitarians hold this revelation of the one God as a great self to have been either supplemented or superseded by later revelation which shows the one God in some sense to be three "Persons." (Greek: hypostaseis or prosopa, Latin: personae) But if these divine Persons are selves, then the claim is that there are three divine selves, which would seem to be three gods. Some Trinity theories understand the Persons to be selves, and then try to show that the falsity of monotheism does not follow. (See section 2 below.) But a rival approach is to explain that these three divine Persons are really ways the one divine self is, that is to say, modes of the one god. In current terms, one reduces all but one of the three or four apparent divine selves (Father, Son, Spirit, the triune God) to the remaining one. One of these four is the one god, and the others are his modes. Because the New Testament seems to portray the Son and Spirit as somehow subordinate to the one God, one-self Trinity theories always either reduce Father, Son, and Spirit to modes of the one, triune God, or reduce the Son and Spirit to modes of the Father, who is supposed to be numerically identical to the one God. (See section 1.8 for views on which only the Holy Spirit is reduced to a mode of God, that is, the Father.) Because God in the Bible is portrayed as a great self, at the popular level of trinitarian Christianity one-self thinking has a firm hold. Liturgical statements, song lyrics, and sermons frequently use trinitarian names ("Father", "Son", "Jesus", "God", etc.) as if they were interchangeable, co-referring terms, referring directly or indirectly (via a mode) to one and the same divine self. ### 1.2 What is a mode? But, what is a "mode"? It is a "way a thing is", but that might mean several things. A "mode of X" might be * an intrinsic property of X (e.g., a power of X, an action of X) * a relation that X bears to some thing or things (e.g., X's loving itself, X's being greater than Y, X appearing wonderful to Y and to Z) * a state of affairs or event which includes X (e.g., X loving Y, it being the case that X is great) One-self trinitarians often seem to have in mind the last of these. (E.g., The Son is the event of God's relating to us as friend and savior. Or the Son is the event of God's taking on flesh and living and dying to reveal the Father to humankind. Or the Son is the eternal event or state of affairs of God's living in a son-like way.) If an event is (in the simplest case) a substance (thing) having a property (or a relation) at a time, then the Son (etc.) will be identified with God's having a certain property, or being in a certain relation, at a time (or timelessly). By a natural slide of thought and language, the Son (or Spirit) may just be thought of and spoken of as a certain divine property, rather than God's having of it (e.g., God's wisdom). Modes may be essential to the thing or not; a mode may be something a thing could exist without, or something which it must always have so long as it exists. (Or on another way to understand the essential/non-essential distinction, a mode may belong to a thing's definition or not.) There are three ways these modes of an eternal being may be temporally related to one another: maximally overlapping, non-overlapping, or partially overlapping. First, they may be eternally concurrent--such that this being always, or timelessly, has all of them. Second, they may be strictly sequential (non-overlapping): first the being has only one, then only another, then only another. Finally, some of the modes may be had at the same times, partially overlapping in time. ### 1.3 One-self Theories and "Modalism" in Theology Influential 20th century theologians Karl Barth (1886-1968) and Karl Rahner (1904-84) endorse one-self Trinity theories, and suggest replacements for the term "Person". They argue that in modern times "person" has come to mean a self. But three divine selves would be three gods. Hence, even if "Person" should be retained as traditional, its meaning in the context of the Trinity should be expounded using phrases like "modes of being" (Barth) or "manners of subsisting" (Rahner) (Ovey 2008, 203-13; Rahner 1997, 42-5, 103-15). Barth's own summary of his position is: > > As God is in Himself Father from all eternity, He begets Himself as > the Son from all eternity. As He is the Son from all eternity, He is > begotten of Himself as the Father from all eternity. In this eternal > begetting of Himself and being begotten of Himself, He posits Himself > a third time as the Holy Spirit, that is, as the love which unites Him > in Himself. (Barth 1956, 1) > All of Barth's capitalized pronouns here refer to one and the same self, the self-revealing God, eternally existing in three ways. Similarly, Rahner says that God > > ...is - at once and necessarily - the unoriginate who > mediates himself to himself (Father), the one who is in truth uttered > for himself (Son), and the one who is received and accepted in love > for himself (Spirit) - and... as a result of this, > he [i.e. God] is the one who can freely communicate himself. (Rahner > 1997, 101-2) > Similarly, theologian Alastair McGrath writes that > > ...when we talk about God as one person, we mean one person > in the modern sense of the word [i.e. a self], and when we > talk about God as three persons, we mean three persons in the > ancient sense of the word [i.e. a persona or role that is > played]. (McGrath 1988, 131) > All three theologians are assuming that the three modes of God are all essential and maximally overlapping. Mainstream Christian theologians nearly always reject "modalism", meaning a one-self theory like that of Sabellius (fl. 220), an obscure figure who was thought to teach that the Father, Son, and Holy Spirit are sequential, non-essential modes, something like ways God interacts with his creation. Thus, in one epoch, God exists in the mode of Father, during the first century he exists as Son, and then after Christ's resurrection and ascension, he exists as Holy Spirit (Leftow 2004, 327; McGrath 2007, 254-5; Pelikan 1971, 179). Sabellian modalism is usually rejected on the grounds that such modes are strictly sequential, or because they are not intrinsic features of God, or because they are intrinsic but not essential features of God. The first aspect of Sabellian modalism conflicts with episodes in the New Testament where the three appear simultaneously, such as the Baptism of Jesus in Matthew 3:16-7. The last two are widely held to be objectionable because it is held that a doctrine of the Trinity should tell us about how God really is, not merely about how God appears, or because a trinitarian doctrine should express (some of) God's essence. Sabellian and other ancient modalists are sometimes called "monarchians" because they upheld the sole monarchy of the Father, or "patripassians" for their (alleged) acceptance of the view that the Father (and not only the Son) suffered in the life of the man Jesus. While Sabellian one-self theories were rejected for the reasons above, these reasons don't rule out all one-self Trinity theories, such as ones positing the Three as God's modes in the sense of his eternally having certain intrinsic and essential features. Sometimes the Trinity doctrine is expounded by theologians as meaning just this, the creedal formulas being interpreted as asserting that God (non-contingently) acts as Creator, Redeemer, and Comforter, or describing "God as transcendent abyss, God as particular yet unbounded intelligence, and God as the immanent creative energy of being... three distinct ways of being God", with the named modes being intrinsic and essential to God, and not mere ways that God appears (Ward 2002, 236; cf. Ward 2000, 90; Ward 2015). ### 1.4 Trinity as Incoherent The simplest sort of one-self theory affirms that God is, because omniscient, omnipotent, and omnibenevolent, the one divine self, and each Person of the Trinity just is that same self. The "Athanasian" creed (on which see section 5.3) seems to imply that each Person just is God, even while being distinct from the other two Persons. Since the high middle ages trinitarians have used a diagram of this sort to explain the teaching that God is a Trinity. ![The traditional Trinity shield or scutum fidei (shield of faith)](Trinityshield.png) If each occurrence of "is" here expresses numerical identity, commonly expressed in modern logical notation as "=" then the chart illustrates these claims: 1. Father = God 2. Son = God 3. Spirit = God 4. Father [?] Son 5. Son [?] Spirit 6. Spirit [?] Father But the conjunction of these claims, which has been called "popular Latin trinitarianism", is demonstrably incoherent (Tuggy 2003a, 171; Layman 2016, 138-9). Because the numerical identity relation is defined as transitive and symmetrical, claims 1-3 imply the denials of 4-6. If 1-6 are steps in an argument, that argument can continue thus: 7. God = Son (from 2, by the symmetry of =) 8. Father = Son (from 1, 4, by the transitivity of =) 9. God = Spirit (from 3, by the symmetry of =) 10. Son = Spirit (from 2, 6, by the transitivity of =) 11. God = Father (from 1, by the symmetry of =) 12. Spirit = Father (from 3, 7, the transitivity of =) This shows that 1-3 imply the denials of 4-6, namely, 8, 10, and 12. Any Trinity doctrine which implies all of 1-6 is incoherent. To put the matter differently: it is self-evident that things which are numerically identical to the same thing must also be numerically identical to one another. Thus, if each Person just is God, that collapses the Persons into one and the same thing. But then a trinitarian must also say that the Persons are numerically distinct from one another. But none of this is news to the Trinity theorists whose work is surveyed in this entry. Each theory here is built with a view towards undermining the above argument. In other words, each theorist discussed here, with the exception of some mysterians (see section 4.2), denies that "the doctrine of the Trinity", rightly understood, implies all of 1-6. ### 1.5 Divine Life Streams Brian Leftow sets the agenda for his own one-self theory in an attack on "social", that is, multiple-self theories. (See sections 2 and 3.1 below.) In contrast to these, he asserts that > > ...there is just one divine being (or substance), God...[As > Thomas Aquinas says,] God begotten receives numerically the same > nature God begetting has. To make Aquinas' claim perfectly > plain, I introduce a technical term, "trope". Abel and > Cain were both human. So they had the same nature, humanity. Yet each > also had his own nature, and Cain's humanity was not identical > with Abel's... A trope is an individualized case of an > attribute. Their bearers individuate tropes: Cain's humanity is > distinct from Abel's just because it is Cain's, not > Abel's. With this term in hand, I now restate Aquinas' > claim: while Father and Son instance the divine nature (deity), they > have but one trope of deity between them, which is > God's...bearers individuate tropes. If the Father's > deity is God's, this is because the Father just is God. > (1999, 203-4) > Leftow characterizes his one-self Trinity theory as "Latin", following the recent practice of contrasting Western or Latin with Eastern or Greek or "social" Trinity theories. Leftow considers his theory to be in the lineage of some prominent Latin-language theorists. (See the supplementary document on the history of trinitarian doctrines, section 3.3.2, on Augustine, and section 4.1, on Thomas Aquinas.) In a later discussion Leftow adds that this Trinity theory needn't commit to trope theory about properties. Rather, whether or not properties are tropes, "...the Father's having deity = [is numerically identical to] the Son's having deity. For both are at bottom just God's having deity." (Leftow 2007, 358) Leftow makes an extended analogy with time travel; just as a dancer may repeatedly time travel back to the dance stage, resulting in a whole chorus line of dancers, so God may eternally live his life in three "streams" or "strands" (2004, 312-23). Each Person-constituting "strand" of God's life is supposed to (in some sense) count as a "complete" life (although for any one of the three, there's more to God's life than it) (2004, 312). Just as the many stages of the time-traveling dancer's life are united into stages of her by their being causally connected in the right way, so too, analogously, the lives of each of the three Persons count as being the "strands of" the life of God, because of the mysterious but somehow causal inter-trinitarian relations (the Father generating the Son, and the Father and the Son spirating the Spirit) (313-4, cf. 321-2, Leftow 2012a, 313). Time-travel does not require that entities are four-dimensional (2012b, 337). If a single dancer, then, time travels to the past to dance with herself, this does not amount to one temporal part of her dancing with a different temporal part of her. If that were so, neither dancer would be identical to the (whole, temporally extended) woman. But Leftow supposes that both would be identical to her, and so would not be merely her temporal parts. He holds that if time travel is possible, a self may have multiple instances or iterations at a time. His theory is that the Trinity is like this, subtracting out the time dimension. God, in timeless eternity, lives out three lives, or we might say exists in three aspects. In one he's Father, in another Son, and in another the Holy Spirit. But they are all one self, one God, as it were three times repeated or multiplied. Leftow argues that his theory isn't any undesirable form of "modalism" (i.e. a heretical one-self theory) because > > Nothing in my account of the Trinity precludes saying that the > Persons' distinction is an eternal, necessary, non-successive > and intrinsic feature of God's life, one which would be there > even if there were no creatures. (2004, 327) > Leftow wants to show what is wrong with the following argument (2004, 305-6; cf. 2007, 359): 1. the Father = God 2. the Son = God 3. God = God 4. the Father = the Son (from 1-3) 5. the Father generates the Son 6. God generates God (from 1, 2, 5). Creedal orthodoxy requires 1-3 and 5, yet 1-3 imply the unorthodox 4, and 1, 2 and 5 imply the unorthodox (and necessarily false) statement 6. So what to do? Lines 1-4 seem perfectly clear, and the inference from 1-3 to 4 seems valid. So too does the inference from 1, 2, and 5 to 6. Why should 6 be thought impossible? The idea is that whatever its precise meaning, "generation" is some sort of causing or originating, something in principle nothing can do to itself. One would expect Leftow, as a one-self trinitarian, to deny 1 and 2, on the grounds that neither Father nor Son are identical to the one self which is God, but rather, each is a mode of God. But Leftow instead argues that premises 1 and 2 are unclear, and that depending on how they are understood, the argument will either be sound but not heretical, or unsound because it is invalid, 4 not following from 1-3, and 6 not following from 1, 2, and 5. The argument seems straightforward so long as we read "Father" and "Son" as merely singular referring terms. But Leftow asserts that they are also definite descriptions "which may be temporally rigid or non-rigid" (Leftow 2012b, 334-5). A temporally rigid term refers to a being at all parts of its temporal career. Thus, if "the president of the United States" is temporally rigid, then in the year 2013 we may truly say that "The president of the United States lived in Indonesia", not of course, while he was president, but it is true of the man who was president in 2013, that in his past, he lived in Indonesia. If the description "president of the United States" (used in 2013) is not temporally rigid, then it refers to Barack Obama only in the presidential phase of his life, and so the sentence above would be false. "The Father", then, is a disguised description, something like "the God who is in some life unbegotten" (2012b, 335) (For "the Son" we would substitute "begotten" for the last word.) Because the "=" sign can have a temporally non-rigid description on one or both sides of it, then there can be "temporary identities", that is, identity statements which are true only at some times but not others. Leftow gives as an example the sentence "that infant = Lincoln"; this is true when Lincoln is an infant but false when he has grown up. Such identity statements can only be true or false relative to times, or to something time-like (2004, 324). If the terms "Father" and "Son" are temporally rigid, or at least like such a term in that each applies to God at all portions of his life (which isn't temporally ordered), then 4 does follow from 1-3. But 4, Leftow argues, is theologically innocuous, as it means something like "the God who is in some life the Father is also the God who is in some life the Son" (2012b, 335). This is "compatible with the lives, and so the Persons, remaining distinct," seemingly, distinct instances of God (each of which is identical to God), and Leftow accepts 1-4 as sound only if 4 means this (ibid.). If the terms "Father" and "Son" are temporally non-rigid, or at least like such a term in that each applies to God relative to some one portion of his life but not relative to the others, then the argument is unsound. Relative to the Father-strand of God's life, 1 will be true but 2 will be false. Relative to the Son-strand, 2 will be true, but 1 will be false. 3 and 5 will be true relative to any strand, but in any case, we will not be able to establish either 4 or 6. Leftow's theory crucially depends on a concept of modes-intrinsic, essential, eternal ways God is, that is, lives or life-strands. But he does not identify the "persons" of the Trinity with these modes. Rather, he asserts that the modes somehow constitute, cause, or give rise to each Person (2007, 373-5). Like theories that reduce these Persons to mere modes of a self, Leftow's theory has it that what may appear to be three selves actually turn out to be one self, God. But they, all (apparently) three of them, just are (are numerically, absolutely identical to) that one self, that is, God thrice over or thrice repeated. Some philosophers object that Leftow's time-travel analogy is unhelpful because time-travel is impossible (Hasker 2009, 158). Similarly, one may object that Leftow is trying to illuminate the obscure (the Trinity) by the equally or more obscure (the alleged possibility of time travel, and timeless analogues to it). If Leftow's one-self theory is intended as a literal interpretation of trinitarian language, a "rational reconstruction" (Tuggy 2011a), this would be problematic; but if he means it merely as an apologetic defense (i.e. we can't rule out that the Trinity means this, and this can't be proven incoherent) then the fact that some intellectuals believe in the possibility of time travel supports his case. One may wonder whether Leftow's life stream theory is really trinitarian. Do not his Persons really, so to speak, collapse into one, since each is numerically identical to God? Isn't this modalism, rather than trinitarianism? (McCall 2003, 428) Again, one may worry that Leftow's concept of God being "repeated" or having multiple instances or iterations is either incoherent or unintelligible. And how can such Persons be fully God or fully divine when they exist because of something which is more fundamental, God's life-strands? (Byerly 2017, 81) William Hasker objects that assuming Leftow's theory, > > In the Gospels, we have the spectacle of God-as-Son > praying to himself, namely to God-as-Father. > Perhaps most poignant of all... are the words of abandonment on > the cross, "My God, why have you forsaken me?" On the view > we are considering, this comes out as "Why have > I-as-Father forsaken > myself-as-Son?" (Hasker 2009, 166) > In reply, Leftow argues that if we accept the coherence of time travel stories, we should not be bothered by the prospect of "one person at one point in his life begging the same person at another point" (2012a, 321). About the cry of abandonment on the cross, Leftow urges that the New Testament reveals a Christ who (although divine and so omniscient) did not have full access to his knowledge, specifically knowledge of his relation to the Father, and so Christ could not have meant what Hasker said above. Instead, he "would have been using the Son's 'myself' and 'I,' which... pick out only the Son" (2012a, 322). Hasker also objects that Leftow's one-self theory collapses the interpersonal relationships of the members of the Trinity into God's relating to himself, and suggests that in Leftow's view, God would enjoy self-love, but not other-love, and so would not be perfect (2009, 161-2, 2012a, 331). (On this sort of argument see sections 2.3 and 2.5 below.) Leftow replies that the self-love in question would be "relevantly like love of someone else" and so, presumably, of equal value (2012b, 339). Does the theory imply "patripassianism", the traditionally rejected view that the Father suffers? (After all, the Son suffers, and both he and the Father are identical to God.) Leftow argues that nothing heretical follows; if his analysis is right "then claiming that the Father is on the Cross is like claiming that The Newborn [sic] is eligible to join the AARP [an organization for retirees]", that is, true but misleading (2012b, 336). ### 1.6 Analogy to an Extended Simple Recent metaphysicians have discussed the possibility of a simple (partless) object which is nonetheless spatially extended, occupying regions of space, but without having parts that occupy the sub-regions. Martin Pickup (2016) makes an analogy between these and the idea of a triune God, inspiring his own "Latin" account. Motivated by skepticism about any three-self theory, and taking the "Athanasian" creed as a starting point, he understands the claims that each Person "is God" as asserting the numerical identity of God with that Person. While God is like an extended simple, the relationships between the Persons is "analogous to the relationships between the spatial regions that an extended simple occupies" (418). To expound the theory Pickup uses the terminology of a "person space", an imagined realm of all possible persons, which is just "an abstraction of the facts about possible personhood" (422). A point within person space is "a representation of a group of properties that are jointly necessary and sufficient for being a certain possible person" (423). This is supposed to help us "see the conceptual space between being a certain person and being a certain entity" (422, n. 18). Pickup gives some non-theological examples to motivate the idea that one thing may occupy multiple points of person space: the fictional example of Dr. Jekyll and Mr. Hyde, and humans suffering from Dissociative Identity Disorders (420-1). The prima facie trinitarian claims which generate a contradiction are as follows, where each "is" means numerical (absolute, non-relative) identity. 1. The Father is God. 2. The Son is God. 3. The Holy Spirit is God. 4. The Father is not the Son. 5. The Father is not the Holy Spirit. 6. The Son is not the Holy Spirit. If the three Persons are numerically distinct (4-6), then it can't be that all of them are numerically identical to some one God. (1-3) Pickup proposes to understand these using person space concepts. Using p1, p2, and p3 for different points in person space, the points which correspond to being, respectively, the Father, Son, and Spirit, claims 1-3 are read as: 1. The occupant of p1 is God. 2. The occupant of p2 is God. 3. The occupant of p3 is God. These claims entail the 1-3 we started with, understood as claims of numerical identity. But this method of interpretation transforms 4-6, which become: 4. p1 is not p2. 5. p1 is not p3. 6. p2 is not p3. In other words, "4. The Father is not the Son" is not understood as asserting the numerical distinctness of the Father and the Son, but rather, as asserting the distinction of the Father's person space from the Son's person space. (And similarly with 5, 6.) The point of all of this is that the six interpreted claims seem to be coherent, such that possibly, all of them are true. Notably, this account accepts what we can call "Person-collapse", the implication of 1-3 that the Father just is the Son, the Father just is the Spirit, and the Son just is the Spirit. (In other words, those are numerically identical.) Thus, unlike many Trinity theories, this one is arguably compatible with a doctrine of divine simplicity. Pickup defends this account from several objections. Against the objection that this is a heretical modalism, Pickup argues that it can't be, since here God's being three Persons is a fundamental metaphysical fact, not a derivative fact. Further, each Person is a real person, as real as any human person (426-7). Still, one may think the account denies the reality of the Persons. Pickup clarifies that they are not so many distinct realities, but rather each just is God. Nor are they parts of God. When this theory countenances the "distinctness of the Persons" it is not implying the numerical distinctness of the Persons, but only the distinctness of the person spaces they occupy (p1, p2, and p3). But as explained above, person spaces are not understood to be real entities (435). Second, it can be objected that catholic tradition demands the numerical distinctness of the Persons. Pickup denies that it does, since it aims to avoid tritheism and affirms that each just is God (427). Third, Person-collapse entails claims any trinitarian should deny, including that the Father just is the Son, and that the Spirit is three Persons. (Being numerically the same with God, since God is three Persons, the Holy Spirit will be three Persons.) Pickup concedes that at first glance such claims "sound bad", but replies that any "Latin" account of the Trinity will as such have to accept such claims (428). Fourth, it is self-evident that if any x and any y are numerically identical, it follows that x and y can't ever differ. But arguably any account of the Trinity must allow that the Persons differ at least in respect of origin, so that only the Son is begotten, only the Spirit proceeds, and only the Father begets the Son. In reply, Pickup argues that metaphysicians who accept the possibility of extended simples also accept that such "can be heterogeneous: that they can have different properties at the different locations at which they exist" (429). Metaphysicians differ in how they try to show that this is possible (432-4). But according to Pickup, it is plausible that there are "fundamental distributional properties". An example of a distributional property is an object being polka-dotted, which requires that it is one color some places and another color in other places. What is controversial is that such properties can be fundamental, that they can belong to the basic metaphysical structure of reality, rather than being explained as no more than various smaller items having non-distributional properties (430). Thus, we should not think that an extended simple which is heterogeneous both has and lacks a property, being F and not being F. Rather, we should think that it has a single distributional property of being F at some of its locations and not being F at others. If, for instance, an extended simple could be colored, it might be polka-dotted, where this is understood as a fundamental property, rather than, for instance, black in some spots and not-black in others. Thus, we avoid saying that one and the same object at a single time is and is not black (429-31). Applied to the Trinity , Pickup suggests that what at first look like different properties (generating the Son, being generated, and being spirated) really amount to a single distributional property of God, which he calls "generation". This is the property of generating the Son at p1, being generated at p2, and being spirated at p3. This is not a matter of God being different at different person spaces; rather, at each space God has the distributional property called "generation" (431). In this way, the theory does not deny the indiscernibility of identicals, the principle that if any x and any y are numerically identical, then they can't ever differ. The persons of the Trinity, on this account, never do differ, although it may seem that way at first glance. An objection to this is that the Father's generation of the Son implies that the Father is logically or causally prior to the Son, and a distributional property can't account for this. Pickup suggests in reply that perhaps instead it is p1 which is prior to p2, or perhaps God's occupying p1 is prior to his occupying p2 (432). Finally, it can be objected that necessarily, any person just is a certain entity, and that even if this is false, still, it seems that necessarily persons and entities can't be separated, so that if anything is one person it must be one entity as well (436). In reply, Pickup denies that such claims are true, and suggests that conceptually, it seems that one may count persons and beings differently, even in merely human cases like the fictional Jekyll and Hyde, which he takes to be an instance of one entity that is two persons (436, 420). Pickup also argues that it is a virtue of this account that it doesn't specify what it takes to be a person (436). One may object that the suggested paraphrase or interpretation of the statements that no Person is either of the others seems to change the subject from divine Persons to imaginary "person spaces". Again, the cost of denying the numerical distinctness of the Persons may be too high for many trinitarians to accept. And it would seem that trinitarians are committed to many differences between the Persons other than the properties or relations relating to origin. The Son died, but the Father did not. The Holy Spirit descended upon Jesus at his baptism, but the Father and Son did not. The Father at Jesus's baptism said "This is my beloved Son", but the Son and Spirit did not. It is not clear that all of these seeming differences can be understood as really involving fundamental distributional properties of God. ### 1.7 Difficulties for One-self Theories Any one-self theory is hard to square with the New Testament's theme of the interpersonal relationship between Father and Son. (Layman 2016, 129-30; McCall 2010, 87-8, 2014c, 117-27; Plantinga 1989, 23-7) Any one-self theory is also hard to square with the Son's role as mediator between God and humankind (Tuggy and Date 2020, 122-3). These teachings arguably assume the Son to be a self, not a mere mode of a self, and to be a different self than his Father. Theories such as Ward's (section 1.3 above), which make the Son a mere mode, make him something less than a self, whereas others (see section 1.6) make him a self, but the same self as his Father. Either way, the Son seems not to be qualified either to mediate between God and humankind, or to be a friend of the one he calls "Father". Again, some traditional incarnation theories seems to assume that the eternal Son who becomes incarnate (who enters into a hypostatic union with a complete human nature) is the same self as the historical man Jesus of Nazareth. But no mere mode could be the same self as anything, and the New Testament seems to teach that this man was sent by another self, God. Some one-self theories run into trouble about God's relation to the cosmos. If God exists necessarily and is essentially the creator and the redeemer of created beings in need of salvation, this implies it is not possible for there to be no creation, or for there to be no fallen creatures; God could not have avoided creating beings in need of redemption. One-self trinitarians may get around this by more carefully specifying the properties in question: not creator but creator of anything else there might be, and not redeemer but redeemer of any creatures in need of salvation there might be and which he should want to save. ### 1.8 The Holy Spirit as a Mode of God Some ancient Christians, most 17th-19th century unitarians, present-day "biblical unitarians", and some modern subordinationists such as the Jehovah's Witnesses hold the Holy Spirit to be a mode of God--God's power, presence, or action in the world. (See the supplementary document on unitarianism.) Not implying modalism about the Son, this position is harder to refute on New Testament grounds, although mainstream theologians and some subordinationist unitarians reject it as inconsistent with New Testament language from which we should infer that the Holy Spirit is a self (Clarke 1738, 147). Modalists about the Spirit counter with other biblical language which suggests that the "Spirit of God" or "Holy Spirit" refers to either God himself, a mode of God (e.g., his power), or an effect of a mode of God (e.g., supernatural human abilities such as healing). (See Burnap 1845, 226-52; Lardner 1793, 79-174; Wilson 1846, 325-32.) This exegetical dispute is difficult, as all natural languages allow persons to be described in mode-terms ("Hillary is Bill's strength.") and modes to be described in language which literally applies only to persons. ("God's wisdom told him not to create beer-sap trees.") ## 2. Three-self Theories One-self Trinity theories are motivated in part by the concern that if there are three divine selves, this implies that there are three gods. Three-self theories, in various ways, deny this implication. They hold the Persons of the Trinity to be selves (as defined above, section 1.1). A major motivation here is that the New Testament writings seem to assume that the Father and Son (and, some also argue, the Holy Spirit) are different selves (e.g. Layman 2016, 131-2). ### 2.1 Relative Identity Theories Why can't multiple divine selves be one and the same god? It would seem that by being the same god, they must be numerically the same entity; "they" are really one, and so "they" can't differ in any way (that is, this one entity can't differ from itself). But then, they (really: it) can't be different divine selves. Relative identity theorists think there is some mistake in this reasoning, so that things may be different somethings yet the same something else. They hold that the above reasoning falsely assumes something about numerical sameness. They hold that numerical sameness, or identity, either can be or always is relative to a kind or concept. Relative identity theorists are concerned to rebut this sort of argument: 1. The Father is God. 2. The Son is God. 3. Therefore, the Father is the Son. If each occurrence of "is" here is interpreted as identity ("absolute" or non-relative identity), then this argument is indisputably valid. Things identical to the same thing must also be identical to one another. The relative identity trinitarian argues that one should read the "is" in 1 and 2 as meaning "is the same being as" and the "is" in 3 as meaning "is the same divine Person as". Doing this, one may say that the argument is invalid, having true premises but a false conclusion. These theorists reject another response to the above argument, which would be to reject it is invalid because 1 and 2 mean that each is a mode of God (see section 1), and while these claims are true, they don't imply 3, since the Father and the Son are two different modes. Against this, the theories of this section assume that the three Persons of the Trinity are three selves (Rea 2009, 406. 419; van Inwagen 1995, 229-31). Following Rea (2003) we divide relative identity trinitarian theories into the pure and the impure. Pure theories accept (1) either that there is no such relation as absolute identity or that such statements are definable in terms of relative-identity relations, and (2) that trinitarian statements of sameness and difference (e.g. the Father is God, the Father is not the Son) are to be analyzed as involving relative and not absolute identity relations, whereas the impure theories accept only (2), allowing that statements about absolute identity (e.g. a = b) may be both intelligible and true, against (1). (434-8) #### 2.1.1 Pure Relative Identity Theories Peter Geach (1972, 1973, 1980) argues that it is meaningless to ask whether or not some a and b are "the same"; rather, sameness is relative to a sortal concept. Thus, while it is senseless to ask whether or not Paul and Saul are identical, we can ask whether or not Paul and Saul are the same human, same person, same apostle, same animal, etc. The doctrine of the Trinity, then, is construed as the claim that the Father, Son, and Holy Spirit are the same God, but are not the same Person. They are "God-identical but Person-distinct" (Rea 2003, 432). As Joseph Jedwab explains, traditional Trinity language and commitments arguably lead naturally to a relative identity account. > > Prima facie, the doctrine of the Trinity implies the sortal > relativity of identity thesis, which says that where > "R" and "S" are sortals, it > could be that for some x and y, x and y are the same R but > different Ss. The Father and the Son are the same God, else > they are two Gods, which implies polytheism and so is false. But the > Father and the Son are different divine Persons, else they are one > divine Person, which implies the Sabellian heresy and so is false. So > the Father and the Son are the same God but different divine Persons. > (2015, 124) > Geach's approach to the Trinity is developed by Martinich (1978, 1979) and Cain (1989). Jedwab (2015) criticizes Cain's version as implying philosophical, theological and christological difficulties. Cain (2016) defends his more Geachian approach. Pure relative identity trinitarianism depends on the controversial claim that there's no such relation as (non-sortal-relative, absolute) identity. Most philosophers hold, to the contrary, that the identity relation and its logic are well-understood; such are expounded in recent logic text-books, and philosophers frequently argue in ways that assume there is such a relation as identity (Baber 2015, 165; Layman 2016, 141). One might turn to a weaker relative identity doctrine; outside the context of the Trinity, philosopher Nicholas Griffin (1977; cf. Rea 2003, 435-6) has argued that while there are identity relations, they are not basic, but must be understood in terms of relative identity relations. On either view, relative identity relations are fundamental. It has been objected to Geach's claim about the senselessness of asking if a and b are (non-relatively) "the same" that, > > Given that we have succeeded in picking out something by the use of > "a" and in picking out something by the use of > "b" it surely is a complete determinate > proposition that a = b, that is, it is surely either > true or false that the item we have picked out with > "a" is the item we have picked out with > "b". (Alston and Bennett 1984, 558) > Rea objects that relative identity theory presupposes some sort of metaphysical anti-realism, the controversial doctrine that there is no realm of real objects which exists independently of human thought (2003, 435-6). Baber replies that such worries are misguided, as the only aim of relative identity theory should be to show a way in which the Trinity might be coherent (2015, 170). Trenton Merricks objects that if a and b "are the same F", this implies that a is an F, that b is an F, and that a and b are (absolutely, non-relatively) identical. But this widely accepted analysis is precisely what relative identity trinitarians deny. This leads to the objection that relative-identity trinitarian claims are unintelligible (that is, we have no grasp of what they mean). If someone asserts that Fluffy and Spike are "the same dog" and denies that they're both dogs which are one and the same, we have no idea what this person is asserting. Similarly with the claim that Father and Son are "the same God" but are not identical (Merricks 2006, 301-5, 321; cf. Tuggy 2003a, 173-4, Layman 2016, 141-2). Baber (2015) replies that if the sortal dog is "dominant", meaning that for any sortal F, if x and y are the same dog, they will also be the same F, then the claim that Fluffy and Spike are the same dog but not absolutely identical is intelligible. After all, we can understand that the claim implies that Fluffy and Spike are the same animal, the same pet, and so on (167). The relative identity trinitarian, Baber says, must hold that "Being does not dominate [i.e. imply sameness with respect to] Person but rather that Person dominates Being". However, there's no easy way to prove this, and dominance claims are theory-relative (ibid.). But such a claim will just be a part of the relative identity theorist's Trinity theory (169). One may also object to either sort of relative identity account being the historical doctrine on the grounds that only those conversant in the logic of the last 120 years or so have ever had a concept of relative identity. But this may be disputed; Anscombe and Geach (1961, 118) argue that Aquinas should be interpreted along these lines, Richard Cartwright (1987, 193) claims to find the idea of relative identity in the works of Anselm and in the Eleventh Council of Toledo (675 C.E.), and Jeffrey Brower (2006) finds a similar account in the works of Peter Abelard. (On Aquinas, see the supplementary document on the history of trinitarian doctrines section 4.) Christopher Hughes Conn (2019) argues that Anselm was the first to consciously develop a Trinity theory involving relative identity. #### 2.1.2 Impure Relative Identity Theories: A Relative Identity Logic Peter van Inwagen (1995, 2003) tries to show that there is a set of propositions representing a possibly orthodox interpretation of the "Athanasian" creed (see section 5.3) which is demonstrably self-consistent, refuting claims that the Trinity doctrine is obviously self-contradictory. He formulates a trinitarian doctrine using a concept of relative identity, without employing the concept of absolute identity or presupposing that there is or isn't such a thing (1995, 241). Specifically, he proves that the following eight claims (understood as involving relative and never absolute identity, the names being read as descriptions) don't imply a contradiction in his system of relative identity logic. * There is (exactly) one God. * There are (exactly) three divine Persons. * There are three divine Persons in one divine Being. * God is the same being as the Father. * God is a person. * God is the same person as the Father. * God is the same person as the Son. * The Son is not the same person as the Father. * God is the same being as the Father. (249, 254) Van Inwagen neither endorses this Trinity theory, nor presumes to pronounce it orthodox, and he admits that it does little to reduce the mysteriousness of the traditional language. It may be objected, as to the preceding theory, that van Inwagen's relative identity trinitarianism is unintelligible. Merricks argues that this problem is more acute for van Inwagen than for Geach, as the former declines to adopt Geach's claim that all assertions of identity, in all domains of discourse, and in everyday life, are sortal-relative (Merricks 2006, 302-4). Michael Rea (2003) objects that by remaining neutral on the issue of identity, van Inwagen's theory allows that the three Persons are (absolutely) non-identical, in which case "it is hard to see what it could possibly mean to say that they are the same being" (Rea 2003, 441). It seems that any things which are non-identical are not the same being. Thus, van Inwagen must assume that there is absolute identity, and deny that this relation holds between the Persons. Thus, van Inwagen has not demonstrated the consistency of (this version of) trinitarianism. Further, the theory doesn't rule out polytheism, as it doesn't deny that there are non-identical divine beings. In sum, the impure relative identity trinitarian owes us a plausible and orthodox metaphysical story about how non-identical beings may nonetheless be "one God", and van Inwagen hasn't done this, staying as he has in the realm of logic (Rea 2003, 441-2). In a later discussion, van Inwagen goes farther, claiming that trinitarian doctrine is inconsistent "if the standard logic of identity is correct", and denying there is any "relation that is both universally reflexive [i.e., everything bears the relation to itself] and forces indiscernibility [i.e. things standing in the relation can't differ]" (2003, 92). Thus, there's no such relation as classical or absolute identity, but there are instead only various relative identity relations (92-3). In so doing he moves to a "pure" relative identity approach to the Trinity, as described in section 2.1.1. Many philosophers would object that whatever reason there is to believe in the Trinity, it is more obvious that there's such a relation as identity, that the indiscernibility of identicals is true, and that we do successfully use singular referring terms. Vlastimil Vohanka (2013) argues that van Inwagen has done nothing to show the logical possibility of any Trinity theory. Just because a set of claims can't be proven inconsistent in van Inwagen's relative identity logic, it doesn't follow that such claims don't imply a contradiction, or that it is metaphysically possible that all the claims are true. At one point van Inwagen tells a short non-theological story whose claims, when translated into his relative identity logic, have the same forms as the Trinity propositions. The story, he argues, is clearly not self-contradictory; thus, he concludes, neither are the Trinity propositions, since they have the same logical forms. In response, Vohanka concocts a short non-theological story whose claims translate into claims of the same form in relative identity logic, and yet are clearly logically impossible (207-11). He concludes that "there's no ground for thinking that formal consistency in [relative identity logic] guarantees logical possibility", and that "sharing a form in [relative identity logic] with a logically possible proposition does not guarantee logical possibility" (211-2). #### 2.1.3 Impure Relative Identity Theories: Constitution Trinitarianism Another theory claims to possess the sort of metaphysical story van Inwagen's theory lacks. Based on the concept of constitution, Rea and Brower develop a three-self Trinity theory according to which each of the divine Persons is non-identical to the others, as well as to God, but is nonetheless "numerically the same" as all of them (Brower and Rea 2005a; Rea 2009, 2011). They employ an analogy between the Christian God and material objects. When we look at a bronze statue of Athena, we should say that we're viewing one material object. Yet, we can distinguish the lump of bronze from the statue. These cannot be identical, as they differ (e.g., the lump could, but the statue couldn't survive being smashed flat). We should say that the lump and statue stand in a relation of "accidental sameness". This means that they needn't be, but in fact are "numerically the same" without being identical. While they are numerically one physical object, they are two hylomorphic compounds, that is, two compounds of form and matter, sharing their matter. This, they hold, is a plausible solution to the problem of material constitution (Rea 1995). Similarly, the Persons of the Trinity are so many selves constituted by the same stuff (or something analogous to a stuff). These selves, like the lump and statue, are numerically the same without being identical, but they don't stand in a relation of accidental sameness, as they could not fail to be related in this way. Father, Son, and Spirit are three quasi form-matter compounds. The forms are properties like "being the Father, being the Son, and being the Spirit; or perhaps being Unbegotten, being Begotten and Proceeding" (Rea 2009, 419). The single subject of those properties is "something that plays the role of matter," which Rea calls "the divine essence" or "the divine nature" (Brower and Rea 2005a, 68; Rea 2009, 420). Whereas in the earlier discussion "the divine essence [is] not... an individual thing in its own right" (Brower and Rea 2005a, 68; cf. Craig 2005, 79), in a later piece, Rea holds the divine nature to be a substance (i.e. an entity, an individual being), and moreover "numerically the same" substance as each of the three. Thus, it isn't a fourth substance; nor is it a fourth divine Person, as it isn't, like each of the three, a form-(quasi-)matter compound, but only something analogous to a lump of matter, something which constitutes each of the Three (Rea 2009, 420; Rea 2011, Section 6). Rea adds that this divine nature is a fundamental power which is sharable and multiply locatable. He doesn't say whether it is either universal or particular, saying, "I am unsure whether I buy into the universal/particular distinction" (Rea 2011, Section 6). All properties, in his view, are powers, and vice versa. Thus, this divine nature is both a power and a property, and it plays a role like that of matter in the Trinity. This three-self theory may be illustrated as follows (Tuggy 2013a, 134). ![A representation of Constitution Trinitarianism](CT.png) There would seem to be seven realities here, none of which is (absolutely) identical to any of the others. Four of them are properties: the divine nature (d), being unbegotten (u), being begotten (b), and proceeding (p). Three are hylomorphic (form-matter) compounds: Father, Son, and Holy Spirit (f, s, h)-each with the property d playing the role of matter within it, and each having its own additional property (respectively: u, b, and p) playing the role of form within it. Each of these compounds is a divine self. The ovals can be taken to represent the three hylomorphs (form-matter compounds) or the three hylomorphic compounding relations which obtain among the seven realities posited. Three of these seven (f, s, h) are to be counted as one god, because they are hylomorphs with only one divine nature (d) between them. Thus, of the seven items, three are properties (u, b, p), three are substances which are hylomorphic compounds (f, s, h), and one is both a property and a substance, but a simple substance, not a compound one (d). Brower and Rea argue that their theory stands a better chance of being orthodox than its competitors, and point out that a part of their motivation is that leading medieval trinitarians such as Augustine, Anselm, and Aquinas say things which seem to require a concept of numerical sameness without identity. (See Marenbon 2007, Brower 2005, and the supplementary document on the history of Trinity theories, sections 3.3.2, on Augustine, and 4.1, on Thomas Aquinas.) In contrast to other relative-identity theories, this theory seems well-motivated, for its authors can point to something outside trinitarian theology which requires the controversial concept of numerical sameness without identity. This concept, they can argue, was not concocted solely to acquit the trinitarian of inconsistency. But this strength is also its weakness, for on the level of metaphysics, much hostility to the theory is due to the fact that philosophers are heavily divided on the reality, nature, and metaphysical utility of constitution. Thus, some philosophers deny that a metaphysics of material objects should involve constitution, since strictly speaking there are no statues or pillars, for these apparent objects should be understood as mere modes of the particles that compose them. Arguably, truths about statues and pillars supervene on truths about arrangements of particles (Byerly 2019, 82-3). This Constitution theory has been criticized as underdeveloped, unclear in its aims, unintelligible, incompatible with self-evident truths, unorthodox relative to Roman Catholicism, polytheistic and not monotheistic, not truly trinitarian, involving too many divine individuals (primary substances), out of step with the broad historical catholic tradition, implying that the Persons of the Trinity can't simultaneously differ in non-modal and non-temporal properties, not a theological improvement over simpler relative identity approaches, and as wrongly implying that terms like "God" are systematically ambiguous (Craig 2005; Hasker 2010b; Hughes 2009; Layman 2016; Leftow 2018; Pruss 2009, Tuggy 2013a). #### 2.1.4 Impure Relative Identity Theories: Constitution and Indexicals Scott Williams has constructed a similar theory, which he calls a "Latin Social" account. In common with Leftow's "Latin" theory and Hasker's "Social" theory (see sections 1.5, 2.4), Williams says that there is one "concrete instance or trope of the divine nature" which is a constituent of each Person. Each Person is also constituted by an incommunicable attribute, begetting (Father), being begotten (Son), and being spirated (Spirit) (Williams 2017, 324). He understands each Person to be "an incommunicable existence of an intellectual nature" (326). In his view any person is a person "ontologically and explanatorily prior to any cognitive acts of volitions that that person in question has or might have" (Williams 2017, 327; cf. 2013, 2019). And for him, the Persons are persons. Each Person is essentially numerically the same essence as the one divine essence, while being a numerically different Person from the other two Persons. Thus, the account involves irreducible relations of kind-relative numerical sameness. But the divine Persons are not (absolutely) numerically identical to one another, and each is not (absolutely) numerically identical to the divine essence. This divine essence is like an Aristotelian first substance in that it exists on its own (not in another) and in being a concrete particular, but unlike first substances it is communicable, in other words, it can be shared by non-identical things, the divine Persons (Williams 2017, 326). The term "God" can refer to any of the Persons, or to the divine essence. The term "Trinity" is a plural-referring term which refers to the plurality of the divine Persons (Williams 2013, 85). (See section 5.1.) Williams considers it an axiom of trinitarian theorizing that "the divine persons are necessarily unified or necessarily agree regarding all things" (2017, 321). Some rival theories try to account for this "necessary agreement thesis" by showing how, allegedly, the Persons would have to come up with some policy which would prevent disagreement. Williams finds such claims "philosophically unsatisfying", and instead argues that the three Persons can never disagree because they have numerically one will, one power of choosing (322). Unlike any other three persons, the Persons of the Trinity, because they share one divine nature, share one set of powers, and so any exercise of any divine power belongs to each of the three. In this case, Williams analyzes thinking as producing and using a token sentence in what we might call divine mentalese. Building on work by philosopher John Perry on indexical terms like "I", Williams points out that a single token of a sentence in English may be used by different agents, and may thus have multiple meanings. For example, > > Suppose that Peter produces...a sign that reads, "I am > happy," and that Peter > uses this sign by holding it up. > Peter affirms that Peter is happy. Later, > Peter puts the sign on the ground and Paul picks up the > same sign and holds it up such that Paul affirms that > Paul is happy. Paul uses numerically the same token as Peter did, > yet when Paul uses it he affirms > something different than Peter. (Williams 2013, 81) > Similarly, if divine Persons think using a language-like divine mentalese, then one token of this may be used by different Persons and have a different significance for each. The idea is that a person relates to a proposition (the content of his thought) by means of a token sentence which he produces and uses to think. But these mental acts, given that the Persons share one set of powers, must be shared by all three of them. Yet, the thoughts thereby thought will differ. For example, > > ...if the Father uses a mental token of "I am God the > Father" and in so doing affirms a proposition, then the Father > affirms that God the Father is identical to God the Father. If the Son > uses the same mental token of "I am God the > Father"...the Son affirms the proposition that the Son is > essentially numerically the same divine nature as the Father without > being identical to the Father. (Williams 2017, 331) > This account denies what some philosophers assume to be obvious, that "distinct and incommunicable intellectual acts and volitional acts are necessary conditions for being a person" (339). Williams rejects this as an ungrounded modern assumption. While it employs recent thinking about indexical terms and other matters, Williams considers this account to fit well with historical theologians such as Gregory of Nyssa, Henry of Ghent, and John Duns Scotus (345). That the persons share all mental acts does not imply that they share one mind or that there is one consciousness in the Trinity. Rather, the access consciousness, experiential consciousness, and introspective consciousness of each Person may differ (2020, Section 3). A New Testament reader might question the assumption that the Persons of the Trinity can't disagree, given the temptation of the Son (but not of the Father) and an occasion when the Son asked the Father to be excused from a difficult trial (Matthew 4:1-11; James 1:13; Mark 14:36). In a response, William Hasker objects that it seems that sometimes human beings can think without using any language. Why, then, should we suppose divine Persons to think only by means of mental token sentences? Perhaps they can just relate directly to propositions (the contents of their thoughts). Worse, Williams posits that this divine mental language is ambiguous, but Hasker says, "we would naturally expect a divine language of thought to be very precise indeed, perhaps maximally so" (Hasker 2018b, 364). He also objects that the theory wrongly counts mental acts. Hasker imagines that the Holy Spirit intends to become incarnate on some other planet, and that before this Incarnation or the Incarnation of the Son, the three of them together produce the mental token "I shall become incarnate". Hasker urges that this seems to be two uses of the same mental sentence, one by the Son and the other by the Spirit. "To be aware of a proposition is precisely to perform a mental act," and here the Son is aware of one, but the Spirit is aware of another (365). But this clashes with Williams's claim that there is but one mental power shared by the three. Williams replies that divine mental tokens are needed "to explain why a divine person's mental act is directed at (among all possible propositions) the proposition it is directed at" (Williams 2020, 115). Williams denies that ambiguity is always an imperfection of a language, and urges that there is nothing objectionable about divine Persons using mental tokens that can be used to express various propositions (110-1). About Hasker's allegation that the theory mis-counts the mental acts of the Persons, Williams says that to the contrary, we should see but one mental act here, though we should keep in mind the Persons' background knowledge about who will become incarnate, which provides the contexts relative to which the one token mental sentence means different things. Moreover, "Why posit several mental acts here, when one mental act will do the same explanatory work?" Finally, Williams clarifies that the necessary sharing of divine acts does not apply to "internal divine productions", such as the Father's eternal generation of the Son (111-4, 115-6). #### 2.1.5 Impure Relative Identity Theories: Episodic Personhood Another relative identity theory by Justin Mooney (2020) depends on an entirely different metaphysical account to show how multiple persons may each be the same being. Metaphysician Ned Markosian proposes a thought experiment in which a man dies and is mummified, and then a long time later the mummy's parts are re-arranged into a living woman who has an utterly different psychology than the dead man. The point is that the woman is the same object as the man but is not the same person as the man, because the instances of personhood in his life aren't part of the same episode of personhood (3-4). Mooney applies Markosian's ideas about "identity under a sortal" to the Trinity. On this account, "God is a single, divine substance that is simultaneously or atemporally participating in three distinct episodes of personhood-those of the Father, Son, and Spirit" (5). Thus, each Person just is God, but none is the same Person as any other divine Person. The account may be illustrated by modifying the traditional Trinity shield: ![An illustration of Mooney's relative identity theory of the Trinity.](Moonity.png) On this theory, being different Persons doesn't imply being numerically distinct. One may worry that such Persons must be one and the same Person since they have but one substance between them, but Mooney answers that they are individuated by their causal relationships, following Swinburne (1994) (5). In addition, following Effingham, he says that they are not one and the same Person because they aren't linked by immanent causal relations. (5-6) These are "those causal links an entity bears to itself from one time to another whereby the way it is earlier on causes how it is later on" (Effingham 2015, 35). Following Moreland and Craig (2017), Mooney adds that God possesses three mental faculties, each had by one of the Persons (6). Finally, adapting ideas from Swinburne (1994), he says that > > ...the Father's episode of personhood occurs simply because > God is a divine being, and a divine being is essentially a personal > being. By nature, God instantiates whatever psychological properties > are necessary for being a person. The Son's episode of > personhood occurs because the Father wills that there is an > instantiation of personhood by the divine substance which is not > immanent-causally linked to the Father's instantiation of > personhood. And the Spirit's episode occurs because one or both > of these persons will(s) that there is yet another instantiation of > personhood by the divine substance which is not connected by immanent > causal relations to either the Father's or the Son's > instantiation of personhood (6). > He remains neutral on whether this process is either temporal or necessary (ibid.). Unlike other relative identity theories, this account, like some one-self theories, affirms the absolute identity of each Person with God; each is the same thing or being or primary substance, God (7). This generates a concern that the account may count as a heretical modalism. Mooney replies that "if Markosian's episodic view of personal identity is right, the model is not modalist" (ibid.). The reason is that on this account there are three episodes of personhood, which implies that there are three Persons, even though there is one being which is the component thing in each episode, a single subject of the properties that are involved in being a Person. Even though the account has it that these three are different Persons, still, it identifies each with God, which entails their identity with one another; being the same thing as God, they must be the same thing as each other. Given this, it would seem that they can't differ in any way, e.g. the Son becomes incarnate but the Father does not (7). Mooney replies that the Trinity is mysterious, and that probably a sentence like "The Son became incarnate but the Father didn't" might be understood as not requiring a simultaneous or eternal difference between the being that is the Son and the one which is the Father. In Markosian's thought experiment, one would think that person-names would track with the different stages in the career of the one object, so that, say, "Alice" would refer to the thing only in its latest stages, and "Bob" would apply to it only in its pre-mummy career. Thus, names like "Father" and "Son" should refer to God only in one or the other of God's Person-episodes. Mooney suggests, then, that "The Son became incarnate but the Father didn't" will be true if and only if the Son but not the Father is the same person as someone who became incarnate, that is, God becomes incarnate in the Person-episode associated with the name "Son" but not in the one associated with "Father" (8). This, Mooney argues, shows why when counting objects, we should count by (absolute) identity, while when counting persons we should count by the relation same-person. In the Trinity, then, we count one thing but three Persons; the Persons are the same thing but different persons (9). The account also solves his "problem of Triunity" (2018), which is that as normally analyzed, these three statements can't all be true, and yet arguably a trinitarian is committed to all three of them: 1. God is triune. 2. The Son is God. 3. The Son is not triune. (2020, 10) The solution is that even though "strictly speaking, the Son is triune" since the Son just is God and God is triune, the meaning of 3 is that "the Son is not the same person as anyone who is triune", which is both true and consistent with 1 and 2 (10). Mooney adds that the property being triune should not be confused with the property of being the same Person as someone who is triune; only the first, in his view, is an essential divine attribute (10-1). Mooney argues that this account also solves problems relating to divine processions and aseity. The Son, being God, must have the property of aseity. But Mooney suggests that the Father's generation of the Son doesn't explain the Son's existence (which would rule out the Son's aseity), but only the Son's being a person distinct from the Father (12). The viability of this theory rests on a particular metaphysics of personhood. One might think, contra Markosian and Mooney, that the woman in the story is one being, the mummy is a second being (even though composed of many or all of the same parts), and the man who follows is a third being. Similarly, one may wonder whether numerically distinct Persons can each be numerically the same as one god. The theory implies the falsity of the principle that for any x and y, if they are different Fs, then x is an F, y is an F, and x[?]y. One might also question the theory's way of dealing with apparent differences between the (numerically identical) Persons; any differences of the form the "the Father is F but the Son is not F" get analyzed as meaning "the Father is the same Person as someone who is F and the Son is not the same Person as someone who is F". Does the original claim really mean what the analysis says? ### 2.2 20th Century Theologians and "Social" Theories Some influential 20th-century theologians interpreted the Trinity as containing just one self. (See section 1.3 above.) In the second half of the century, many theologians reacted against one-self theories, criticizing them as modalist or as somehow near-modalist. This period also saw the wide and often uncritical adoption of a paradigm for classifying Trinity theories which derives from 19th c. French Catholic theologian Theodore de Regnon (Barnes 1995). On this paradigm, Western or Latin or Augustinian theories are contrasted with Eastern or Greek or Cappadocian theories, and the difference between the camps is said to be merely one of emphases or "starting points". The Western theories, it is said, emphasize or "start with" God's oneness, and try to show how God is also three, whereas the Eastern theories emphasize or "start with" God's threeness, and try to show how God is also one. The two are thought to emphasize, respectively, psychological or social analogies for understanding the Trinity, and so the latter is often called "social" trinitarianism. But this paradigm has been criticized as confused, unhelpful, and simply not accurate to the history of Trinitarian theology (Cross 2002, 2009; Holmes 2012; McCall 2003). Although the language of Latin vs. "social" Trinity theories has been adopted by many analytic philosophers (e.g. Leftow 1999; Hasker 2010c; Tuggy 2003a), these have interpreted the different theories as logically inconsistent (i.e. such that both can't be true), and not merely as differing in style, emphasis, or sequence. Some 20th century theological sources, accepting the de Regnon paradigm, proceed to blame the Western tradition for "overemphasizing the oneness" of God, and recommend that balance may be restored by looking to the Eastern tradition. A number of concerns characterize theologians in this 20th and 21st century movement of "social" trinitarianism: * Preserving genuinely interpersonal relationships between the Persons of the Trinity, particularly the Father and the Son. * Doing justice to the New Testament idea of Christ as a personal mediator between God and humankind. * Suspicion that the "static" categories of Greek philosophy have in previous trinitarian theologies obscured the dynamic and personal nature of the triune God. * Concern that traditional or Western trinitarian theology has made the doctrine irrelevant to practical concerns such as politics, gender relations, and family life. * The idea that to be Love itself, or for God to be perfectly loving, God must contain three subjects or persons (or at any rate, more than one). (See sections 2.3, 2.5, and 2.6.) (For surveys of this literature see Karkkainen 2007; Olson and Hall 2002, 95-115; Peters 1993, 103-45.) These writers are often unclear about what Trinity theory they're endorsing. The views seem to range from tritheism, to the idea that the Trinity is an event, to something that differs only slightly, or only in emphasis, from pro-Nicene or one-self theories (see section 1 and section 3.3 of the supplementary document on the history of trinitarian doctrines). Merricks observes that some views advertised as "social trinitarianism" make it "sound equivalent to the thesis that the Doctrine of the Trinity is true but modalism is false" (Merricks 2006, 306). However, a number of Christian philosophers, and some theologians employing the methods of analytic philosophy, have started with this literature and then proceeded to develop relatively clear three-self Trinity theories, which are surveyed here. They differ in how they attempt to secure monotheism (Leftow 1999). There are many such Trinity theories, and it is not clear that all the options have yet been explored (Davidson 2016). ### 2.3 Ersatz Monotheism A problem for any three-self Trinity theory is that numerically three selves are, it would seem, numerically three things. And according to a theory of essences or natures, a thing which has or which is an instance of an essence or nature is thereby a thing of a certain kind. All Trinity theories include the Nicene claim that the Persons of the Trinity have between them but one essence or nature, the divine one. But it would seem that by definition a thing with the divine essence is a god, and so three such things would be three gods. Some three-self theories in effect concede that they imply tritheism (three things, each of which has properties sufficient for being a god), but argue that surely a correct Trinity theory can't avoid the right type of tritheism, and can avoid any undesirable tritheism, such as ones involving unequal divinity of the Persons, Persons which are in some sense independent, or Persons who are in principle separable (McCall 2010, 2014c; Plantinga 1988, 1989; Yandell 2010, 2015). Richard Swinburne has long developed and defended a type of three-self Trinity theory which in the eyes of most critics seems to be "a fairly straightforward form of tritheism" (Alston 1997, 55. See also Clark 1996; Davidson 2016; Feser 1997; Howard-Snyder 2015a; Moreland and Craig 2017; Rea 2006; van Inwagen and Howard-Snyder 1998; van Inwagen 2003, 88; Vohanka 2014, 56). In a series of articles and books Swinburne's views have changed in significant ways (Swinburne 1988, 1994, 2008, 2018), but this entry focuses on his latest work on the Trinity. Swinburne aims to build his theory on widespread traditional agreements between most catholic theologians since at least the fourth and fifth centuries (Swinburne 2018, Section 1). The Persons of the Trinity are three beings, each a self which satisfies Boethius's definition of a "person" as "an individual substance (substantia) of a rational nature" (421). Each is divine in that each has all the divine attributes. "A divine person is naturally understood as one who is essentially eternally omnipotent and exists (in some sense) 'necessarily'" (427). He argues that omnipotence entails perfect goodness and omniscience. While all three of the Persons exist necessarily (inevitably), the Father does this independently while the Son and Spirit exist of necessity dependently, because necessarily, the Father exists, and his existence implies that he causes them (437, n. 14). These actions of the Father are inevitable and not voluntary, but they are via the Father's will (425, 428). This causing is traditionally described as the eternal generation of the Son by the Father, and the eternal proceeding of the Spirit from the Father, or from the Father and the Son. For Swinburne, the Son is "caused by the Father alone" while the Spirit is "caused by the Father and/or through the Son" (429). The theory then is committed to one of these two models of the divine processions. ![Father causing the Son and the two of the causing the Spirit, or the Father causing the Son and the Father causing the Spirit through the Son](Swincauses.png) Swinburne has constructed a couple versions of an argument which purports to show why, if there is at least one divine being or Person, there must be exactly three, with the second and third being caused ultimately by the first. In other words, given that it is possible that there be a divine Person, it is metaphysically impossible that there be only one, and it is metaphysically necessary that there be exactly three. Most trinitarians have assumed that such an argument is neither possible nor desirable, as the Trinity can be known only by divine revelation. Against this, Swinburne says that "even if you regard the New Testament as an infallible source of doctrine, you cannot derive from it a doctrine of the Trinity", because when it comes to passages about the Spirit, > > ...there are non-Trinitarian ways of interpreting...[these] > which are just as plausible as interpreting them as expressing the > doctrine that the Holy Spirit is a divine person...So unless > Christians today recognize some good a priori argument for a doctrine > of the Trinity (and most of them do not recognize such an argument), > or unless they consider that the fact that the subsequent Church > taught a doctrine of the Trinity is a significant reason for > interpreting the relevant passages in a Trinitarian way, it seems to > me that most Christians today (that is, those not acquainted with any > a priori argument for its truth) would not be justified in believing > the doctrine. (419-20) > One may wonder if there could be two omnipotent beings; there have been arguments from theism (at least one god) to monotheism (exactly one god) based on the idea that it'd be impossible for there to be more than one who is omnipotent. (See the Monotheism entry, section 5.) Suppose that one omnipotent being willed a certain object to move and simultaneously another omnipotent being willed that it should remain in place. It would seem that whether the object moves or stays in place, one of the being's wills is thwarted, so that, contrary to our stipulation, one of them fails to be omnipotent. Swinburne argues that such conflicts of will are impossible given the omniscience, perfect goodness, and causal relations of the omnipotent beings. In his view, in causing the Son and Spirit, the Father must "lay down the rules determining who has the right to do which actions; and the other members of the Trinity would recognize his right, as the source of their being to lay them down" (428). Inspired by similar arguments given by Richard of St. Victor, Swinburne argues that a divine Person must be perfect in love. But > > ...perfect love must be fully mutual love, reciprocated in kind > and quantity, involving total sharing, the kind of love involved in a > perfect marriage; and only a being who could share with him the rule > of the universe could fully reciprocate the love of another such. > ...it would be a unique best action for the Father to cause the > existence of the Son, and so inevitably he would do so. ...at > each moment of everlasting time the Father must always cause the Son > to exist, and so always keep the Son in being. (429-30) > Thus, if there is one divine Person, there must also be another. Further, there must be a third, for > > A twosome can be selfish. ...Perfect love for a > beloved...must involve the wish that the beloved should be loved > by someone else also. Hence it will be a unique best action for the > Father to cause the existence of a third divine being whom Father and > Son could love and by whom each could be loved. Hence the Holy Spirit. > And I suggest that it would be best if the Father included the Son as > co-cause (as he is of all other actions of the Father) in causing the > Spirit. And again they must have caused the Spirit to exist at each > past moment of everlasting time. Hence the Trinity must always have > existed. (430) > What stops this process of deity-proliferation from careening into four, seventy-four, or four million divine Persons? Swinburne replies that it is not better to cause four (or more) divine Persons than it is to cause three, since > > ...when there is an infinite series of incompatible possible good > actions, each better than the previous one, available to some agent, > it is not logically possible that he do the best one-because > there is no best action. An agent is perfectly good in that situation > if he does any one of those good actions. So since to bring about only > three divine persons would be incompatible with an alternative action > of bringing about only four divine persons, and so generally, the > perfect goodness of the Father would be satisfied by his bringing > about only two further divine persons. He does not have to bring about > a fourth divine person in order to fulfil his divine nature. To create > a fourth divine person would therefore be an act of will, not an act > of nature. But then any fourth divine person would not exist > necessarily in the sense in which the second and third divine persons > exist necessarily-his existence would not be a necessary > consequence of the existence of a necessary being; and hence he would > not be divine. So there cannot be a fourth divine person. There must > be and can only be three divine persons. (430-1) > In sum, divinity implies "perfect love", which implies exactly three divine Persons. Lebens and Tuggy (2019) object that such arguments trade on the ambiguity of "perfect love". Divinity, by implying moral perfection, implies the character trait of being perfectly loving. But someone may have this and yet not be in the sort of interpersonal relationship that Swinburne describes as "perfect love". (See also Tuggy 2015.) Using familial analogies, Brian Leftow challenges Swinburne's claim that the three would lack an overriding reason to produce a fourth, noting that "Cooperating with two to love yet another is a greater 'balancing act' than cooperating with one to love yet another" (1999, 241). Tuggy (2004) objects that if a three-self theory like Swinburne's were true, it would seem that one or more members of the Trinity have wrongfully deceived us by leading us to falsely believe that there is only one divine self. He also argues that the New Testament writings assume that "God" and "the Father of Jesus" (in all but a few cases) co-refer, reflecting the assumption that God and the Father are numerically the same. (See also Tuggy 2014, 2019.) Denying this last claim, he argues, amounts to an uncharitable and unreasonable attribution of a serious confusion to the New Testament writers and (if they're to be believed) to Jesus as well. These arguments are rebutted by William Hasker (2009) and the argument is continued in Hasker 2011, Tuggy 2011b, and Tuggy 2014. But as mentioned at the outset, the most common objection to Swinburne's Trinity theory is that it is tritheism and not monotheism. Looking, for instance, at this account of divine processions, a reader wonders why this doesn't amount to one god eternally causing a second god, and with that second god eternally causing a third god. To assuage such concerns, Swinburne argues that on his model of the Trinity, it is natural to say that there is "one God". Swinburne observes that the Greek theos (and equally the Latin deus) may be used either as a name, a singular referring term picking out a certain individual thing, or as a predicate, a descriptive word equivalent in meaning to "divine", which might in principle be applied to more than one thing. Then he observes, > > While no doubt the Fathers of the [381] Council did not have a clear > view of what was the sense in which there is just one > "God" and the sense in which each of the three beings is > "God" the distinction between the two senses of the > crucial words makes available one obvious way of resolving the > apparent contradiction. This is by thinking of these words as having > the former sense [i.e. referring to one thing like a name] when the > Creed says that there is "one God", and as having the > latter sense [i.e. being equivalent the adjective > "divine"] when it claims that each of the beings "is > God." Thus understood, the Creed is saying that there is one > unique thing which it names "God," which consists of three > beings. (420) > The suggestion is that the tradition is somewhat confused, but that charitably, we should think its talk of "one God" should be understood as referring to the Trinity. (However, see the opening line of the Nicene creed.) What sort of thing is this Trinity? It is not a divine Person, and is not a thing (person or not) with the divine essence. Rather, it is a thing of which the three divine beings (selves, persons) are proper parts (425). Despite this complex entity not having the divine essence (and so, not being a god), Swinburne sometimes refers to it as "God himself" (424). He argues that these three beings can't help but cooperate, and so agrees with the traditional claim that apart from the aforementioned eternal causings of one another, any act of one Person of the Trinity is an act of all three Persons (425). In sum, "This common omnipotence, omniscience, and perfect goodness in the community of action makes it that case that in a natural sense there is one God" (428). That is, given the three divine beings described above, it is "natural", when it comes to the name-like use of the word "God," to apply the term to that thing which is the whole consisting of those three Persons. But it remains that there are three things here each of which is divine, and that this whole is not itself divine; it's hard to see why this is monotheism and not tritheism. Brian Leftow objects that in Swinburne's account God is not itself divine. Nor does it makes sense to worship it, as it is not the sort of thing which can be aware of our addressing it. Further, the issue of monotheism isn't the issue of how unified the divine beings are, but rather of how many there are. > > ...it is hardly plausible that Greek paganism would have been a > form of monotheism had Zeus & Co. been more alike, better behaved, > and linked by the right causal connections. (Leftow 1999, 232; cf. Rea > 2006) > Moreover, Swinburne's theory entails serious inequalities of power among the Three, jeopardizes the personhood of each, and carries the serious price of allowing (contrary to most theists) that a divine being may be created, and the possibility of more than one divine being (Leftow 1999, 236-40). Daniel Howard-Snyder (2016) argues that Swinburne is committed to descriptive polytheism, normative polytheism, and cultic polytheism, and so is a "polytheist par excellence". He also argues that Swinburne's account of the Trinity is unorthodox. Daniel Spencer (2019) argues that the several factors which Swinburne and others appeal to in order to lend some sort of unity to the three persons are obviously inadequate to show how they amount to one God and not three gods. At most, we get three divine beings who in some ways resemble a god. Spencer observes that sometimes Swinburne simply accepts tritheism, as when he says that there are three divine individuals or beings (Spencer 2019, 192, 198 n. 2; Swinburne 1994, 170, 179). In his first treatment of the subject Swinburne talks of "three Gods" (Swinburne 1988, 234). In later writings he doesn't use that phrase, but his conception of the Persons is substantially the same. Spencer observes that in principle, making the Persons proper parts of a whole which is the only God might do the trick (195-6), and Swinburne does suggest that there is a part-whole relationship between the Persons and God; however, for Swinburne the whole is not a god. Perhaps the most sympathetic voice in the literature is William Hasker (2013, Chapter 18), but in the end he agrees that Swinburne has not done enough to unify the Persons. (Hasker 2013b, Chapters 25-8, 2018, 5-7; Swinburne 2014) ### 2.4 Trope-Constitution Monotheism William Hasker (2013b) has constructed what is arguably the most developed three-self theory of the Trinity. As with Swinburne, his thoughts have developed over decades (Tuggy 2013b), but this entry will focus on his recent publications. For Hasker, following Plantinga, the Persons of the Trinity are "distinct centers of knowledge, will, love, and action...persons in some full sense of the term" (22; cf. Chapter 24). Hasker argues that such a view is widespread in ancient sources, including Gregory of Nyssa and Augustine (Chapters 4-5, 9). While we can't reasonably retain the ancient doctrine of divine simplicity (Chapter 7, 2016, 2018a, 7-8, 18-9), we ought to uphold as many of the traditional claims as possible, for we should assume divine guidance of theological development, even though "the Church's doctrine of the Trinity is not as such to be found in the New Testament" (8). The "fathers" of the late fourth century should be seen as "the giants on whose shoulders we need to stand" (10). In the second part of his book (Chapters 11-20) Hasker interacts with a number of Trinity theories, attempting to salvage whatever is correct in them for use in his own three-self theory; he incorporates ideas particularly from Leftow, Craig, Rea, and Swinburne. For Hasker, the Persons of the Trinity are three divine selves (Chapters 22-5). Against a modern Protestant trend, Hasker insists that a doctrine of processions must be retained, arguing that it enjoys "significant support" from scripture (217), and he points out the awkwardness accepting "the main results of the [ancient] trinitarian controversy" while thinking that this "developmental process...had at its heart a fundamentally wrong assumption", that is, that the Son and Spirit exist because of the Father (222-3). Hasker spends several chapters (25-8) addressing the question: "in virtue of what do the three persons constitute one God?" (203). The three enjoy some sort of unity of will and fellowship, and they are united in that the second and third exist and have the divine nature because of the first, but such factors don't, by themselves, imply that they somehow amount to a single god. Hasker holds that a crucial factor is the idea of their shared divine nature as a concrete property or trope. Following Craig, sometimes Hasker characterizes this concrete divine nature as a divine mind or soul. He argues that for all we know, it is possible for one such trope of divinity "to support simultaneously three distinct lives" which belong to the Persons (228). He argues that this possibility is indirectly supported by split-brain and multiple-personality phenomena in human psychology. He takes these to show that "It is possible for a single concrete human nature-a single trope of humanness-to support simultaneously two or more centers of consciousness" (236). This supporting or sustaining relation, Hasker says, may optionally be specified to involve the divine nature constituting each Person (Chapter 28). > > We shall say, then, that the one concrete divine nature sustains > eternally the three distinct life-streams of the Father, Son, and Holy > Spirit, and that in virtue of this the nature constitutes > each of the persons although it is not identical with the > persons. (244) > Constitution is defined here as asymmetric, so none of the Persons also constitutes the divine nature (245). In a later discussion, he seems to make constitution central to the theory (Hasker 2018a). Adapting work on the metaphysics of material constitution by Lynne Rudder Baker, Hasker offers this definition: Suppose x has F as its primary kind, and y has G as its primary kind. Then x constitutes y just in case 1. x and y have all their parts in common; 2. x is in "G-favorable circumstances"; 3. necessarily, if an object of primary kind F is in G-favorable circumstances there is an object of primary kind G that has all its parts in common with that object; and 4. it is conceptually possible for x to exist but for there to be no object of primary kind G that has all its parts in common with x. (Hasker 2018a, 16-7, cf. 2013b, 241-3; Howard-Snyder 2015b, 108-9) Applying this doctrine of non-material constitution to the Trinity, to say that the divine nature constitutes the Father is to say that those have all their parts in common and that the nature is in divine-Person-favorable circumstances. For a thing of the type "divine mind/soul or concrete nature" to be in "divine trinitarian Person"-favorable circumstances means that there is a divine trinitarian Person which has all his parts in common with the first thing, and that it is conceivable that the first thing exists even though there is nothing of the type "divine trinitarian Person" that has all its parts in common with it. Hasker clarifies that in his view all the entities mentioned here are simple (lacking in proper parts), so each will be what metaphysicians call an improper part of one another, satisfying condition i. He also clarifies that the conceptual possibility in condition iv does not imply metaphysical possibility; Hasker denies that this is metaphysically possible: the divine nature exists but no divine Person exists (18). He adds that > > The divine nature constitutes the divine trinitarian Persons when it > sustains simultaneously three divine life-streams, each life-stream > including cognitive, affective, and volitional states. Since in fact > the divine nature does sustain three such life-streams simultaneously, > there are exactly three divine Persons. (2018a, 17) > Presumably, the divine-Person-favorable circumstances which the divine nature is in, is support of these life-streams. Hasker argues that the "grammar" of the Trinity forbids a Christian from saying things like "three gods" based on there being three Persons each of which is divine (2013b, 247). Again, although "God" in the New Testament nearly always refers to the Father, one can't infer that the Father and God are numerically one (248). With a nod to a mereological account of the Persons and God, he says that "Each Person is wholly God, but each Person is not the whole of God" (250; cf. 257). Hasker also argues, plausibly, that the "Athanasian" creed can be read as non-paradoxical if we realize that it is laying down rules about what must be said and what must not be said (250-4). In the end, as with Swinburne, the Trinity which is called "God" is not literally a god, as it is not divine. But Hasker suggests that > > ...in virtue of the closeness of their union, the Trinity is at > times referred to as if it were a single person. The Trinity is > divine, exhibiting all the essential divine attributes--not by > possessing knowledge, power, and so on distinct from those of the > divine persons, but rather in view of the fact that the Trinity > consists precisely of those three persons and of nothing else. It is > this Trinity which we are to worship, and obey, and love as our Lord > and our God. (2013b, 258) > In an attack on theories of divine simplicity, in which he sets aside considerations of God as Trinity, Hasker objects that if God is simple, God is "dehumanized" in that God must lack certain qualities which Christians should think God literally shares with human beings, such as caring for and being responsive to his creatures, and being able to either judge or forgive them (2016, Section 5). But while none of those qualities implies being human, each arguably implies selfhood. Yet Hasker denies that God is literally a self. Brian Leftow points out the oddness of ascribing a soul to God the Trinity. > > This [soul] is not God. It is not a Person either. It is some other > sort of concrete divine individual. We had not suspected that a spirit > could have a soul; lo, God does! (Leftow 2018, 10) > Leftow also objects that the sentence "God is the ultimate reality" seems to be true by definition. But on Hasker's theory, this soul (a.k.a. the divine nature), which is not God, would be the ultimate reality, being the source of the Persons and so of God (the Trinity) (12). Again, Leftow objects that this theory is not monotheistic; rather, the theory features three deities which we can't describe as such because there is one object (the divine nature/soul) which constitutes them (15). Daniel Howard-Snyder objects that Hasker's talk of the nature "supporting" or "sustaining" the lives or "life-streams" of the Persons is unintelligible (2015b, 108-10). He also argues that it is unclear quite what constitutes the Persons, as in various places Hasker says that this is the divine mind/soul, the concrete divine nature (a trope of divinity), and a single mental substance-and these would appear to be different claims (110). Also, monotheism uncontroversially implies that there is exactly one god. But Hasker forbids saying that any of the Persons is a god. And by definition being a god implies having the divine nature, and like others Hasker understands divinity to imply perfection in knowledge, power to intentionally act, and moral goodness-thus, divinity implies being a self. This, Howard-Snyder says, is a necessary truth and one with which basically all Christians agree. But Hasker's "God", whether this is a community or a composite object, is not a self, and so is not literally divine. But then, we've run out of candidates for being the only god; if neither the Father, nor the Son, nor the Spirit is a god, then it would seem that for Hasker there is no god! Anticipating monotheism-related objections, Hasker lobs various charges at Howard-Snyder (112-3), but in the end it seems that Hasker's view is just that "God" can be spoken of as if it were a self (114-5). Taking a term from recent philosophy of mind, Howard-Snyder says that for Hasker God is a "zombie", a merely apparent self which in fact lacks any consciousness, any point of view, and any mentality (114). He concludes that Hasker is not aiming for the sober metaphysical truth about the Trinity but is instead settling for some sort of "as-ifery". How, he asks, could it be more accurate to describe God as "omnipotent and omniscient" than it is to describe God as "powerless and ignorant" when on Hasker's account God is straightforwardly the latter? (115) Hasker replies that his claim that the divine nature supports the lives of the Persons is no more unintelligible than is the claim that "my desktop computer supports word processing"; to support is to "maintain in being or in action; to keep up, keep going" (Hasker 2018a, 11). Nor should it worry us that we can't understand how this supporting works (12). ### 2.5 Trinity Monotheism The Trinity monotheist says that even though there are three divine Persons, there is one God because there is one Trinity (Moreland and Craig 2017, 588; Craig 2006; Layman 1988, 2016). William Lane Craig has defended the best known such theory. The aim is to go beyond mere analogies, providing a literal model of how to understand traditional trinitarian claims. Craig and Moreland offer Cerberus, the three-headed dog from Greek mythology as "an image of the Trinity among creatures" (592). The point of this fictional example is that Cerberus would be one dog with three "centers of consciousness". Though only parts of one dog, each head is literally canine. If we were to upgrade the mental capacity of the three here, it would be one dog which is three persons. And if we imagine that Cerberus survives death, in that case we can't say that the three are one dog because they have one body. In fact, we're now imagining one (canine) soul which supports three persons. Change canine to divine, and this is the model of the Trinity. (592-3) > > God is an immaterial substance or soul endowed with three sets of > cognitive faculties each of which is sufficient for personhood, so > that God has three centers of self-consciousness, intentionality, and > will...the persons are [each] divine... since the model > describes a God who is tri-personal. The persons are the minds of God. > (Craig 2006, 101) > Only the Trinity, on this theory, is an instance of the divine nature, as the divine nature includes the property of being triune. (See section 5.1.) Beyond the Trinity "there are no other instances of the divine nature" (2017, 589). So if "being divine" means "being identical with a divinity" (i.e., being a thing which instantiates the nature divinity), then none of the Persons are "divine". But the Father, Son, and Holy Spirit are each "divine" in that they are parts of the one God, somewhat as the bones of a cat are "feline", or as the heads of Cerberus are "canine" (592). But the theory also makes the Persons "divine" in other ways too. In a sense the theory divides the divine attributes between the Persons and the Trinity. > > ...when we ascribe omnipotence and omniscience to God, we are not > making the Trinity a fourth person or agent. Divine attributes like > omniscience, omnipotence and goodness are grounded in the > persons' possessing these properties, while divine attributes > like necessity, aseity and eternity are not so grounded. With respect > to the latter, the persons have these properties because God as a > whole has them. For parts can have some properties in virtue of the > wholes of which they are parts. ...The point is... [the > persons'] deity seems in no way diminished because they are not > instances of the divine nature. (590) > Like Swinburne (see section 2.3) Craig argues that it is impossible for God to be a single Person because "if God is perfectly loving by his very nature, he must [eternally] be giving himself in love to another" (593). And since God is free not to create, but must be loving another, "the other to whom God's love is necessarily directed must be internal to God himself" (ibid.). For Craig this is a plausibility argument rather than a strict proof, in support of the claim that the concept of unipersonal God is incoherent. Unlike Swinburne, he does not seem to think that this argument is important to reasonable belief in the Trinity (Craig believes the Trinity can somehow be derived from the Bible-on which see section 5.4), nor does Craig mount a philosophical argument for why there must be exactly three divine Persons. One may object that this argument depends on an equivocation on the phrase "perfectly loving". One who thinks that God is a perfect being must hold that God has the character trait of being perfectly loving, but this doesn't seem to imply the action of perfectly loving (i.e. engaging in the best kind of loving relationship with another) (Lebens and Tuggy 2019). Daniel Howard-Snyder (2003) offers numerous objections to Craig's theory. First, it can't avoid either polytheism or different levels of divinity, either of which would make it unorthodox. The Cerberus analogy is criticized on the grounds that it would not be one dog with three minds, but rather, three dogs with overlapping bodies. (This seems clear in the parallel case of human conjoined twins; everyone considers them to be siblings, two humans with overlapping bodies, not a human with two heads.) While Craig's theory upholds (with the creeds) one divine substance, by his own criteria each of the three Persons must be a substance as well, and the account says that each Person is divine. Thus, the theory implies polytheism (393-5). Here God is not a personal being, in the sense of being numerically identical with a certain self, even though it (God) has parts which are selves. Craig wants to say, for example, that each of the three is all-knowing, and also that God is all-knowing, in that God has parts which are all-knowing. But Howard-Snyder objects that, > > ...there can be no "lending" of a property [i.e., a > whole "getting" a property from one of its parts] unless > the borrower is antecedently the sort of thing that can have > it....[Therefore,] Unless God is antecedently the sort of thing > that can act intentionally--that is, unless God is a > person--God cannot borrow the property of creating the heavens > and the earth from the Son....All other [statements involving] > acts attributed to God [in the Bible] will likewise turn out to be, > strictly and literally, false. (399-400) > According to Trinity monotheism, a thing can exemplify the divine nature without itself being a self. Nor can divinity include properties which require being a self, e.g., being all-knowing, being perfectly free. This, Howard-Snyder argues, is "an abysmally low" view of the divine nature, since "If God is not a person or agent, then God does not know anything, cannot act, cannot choose, cannot be morally good, cannot be worthy of worship" (401). Craig replies to Howard-Snyder's objection to the Cerberus analogy that the claim that it represents three dogs is "astonishing", as we all speak of two headed snakes, turtles and such (Craig 2003, 102). While on Trinity monotheism God isn't identical to any personal being, it doesn't follow that God isn't "personal". He is personal in the sense of having personal parts. Further, the view that God isn't a self > > ...is part and parcel of Trinitarian > orthodoxy...Howard-Snyder assumes that God cannot have such > properties [i.e., knowledge, freedom, moral goodness, > worship-worthiness] unless He is a person. But it seems to me that God > can have them if God is a soul possessing the rational faculties > sufficient for personhood. If God were a soul endowed with a single > set of rational faculties, then He could do all these things. By being > a more richly endowed soul, is God thereby somehow incapacitated? > (105) > As to the charge of polytheism, Craig accuses Howard-Snyder of confusing monotheism with unitarianism (106), i.e. assuming that the existence of exactly one god entails that there is exactly one divine self. Finally, Craig argues that the issue of whether or not the Three count as parts of God is unimportant (107-13). Tuggy (2013b) presses some of Howard-Snyder's objections, concluding that the theory is either not monotheistic, or turns out to be a one-self theory. Stephen Layman (2016) has constructed a similar and arguably better developed three-self Trinity theory. Motivated by the New Testament, Layman says that the three Persons of the Trinity are three selves (124-31). Each is "divine" in that he is a fitting object of worship, and so is God the Trinity. God the Trinity is literally a social entity, a concrete, primary substance which is strongly analogous to a living thing, and which like a living thing is a self-maintaining event (149-50). "Strictly speaking, only the Trinity, the community of divine persons, is God, that is, ruler of all" (148). Yet the Persons are "of one substance" in that "each belongs to the kind divine being", where this means a person which is a part of a god (150-1, 165-6). One may object that a social entity can't be a god, as such a thing is merely an abstraction. Layman answers that social entities are concrete, not abstract, and can intentionally act (159-60). Intentionally acting requires having intentions, but social entities may have these, even though they are not selves or even subjects of consciousness. Social entities may have intentions because their parts (i.e. various selves) have them. As a fallback, Layman suggests the view that social entities may act even though they're incapable of intentional action (159). Like Craig, Layman argues that the Trinity can be omnipotent, perfectly good, and omniscient because its persons are (160). Why then is the Trinity not a fourth divine person (see section 3.1)? > > In order to count as a person, an entity must be able to refer to > itself rightly with the first-person singular pronoun "I" > (or its equivalent). And the Trinity can not do this. (161) > But doesn't the Bible portray God as a self who speaks in the first-person? Layman concedes that the Old Testament does. But because they believe in progressive divine revelation, Christians should read the Old Testament as corrected by the New Testament. And in the New Testament arguably there are "three divine persons (conscious beings)" (164). Old Testament passages where "God" speaks first-person should be read as the Father speaking on behalf of the Trinity (ibid.). The account is not polytheism because only the Trinity is God, and because of the necessary unity of the three (160, 167). But isn't "Every divine person is a god" true by definition? No, because "divine" can mean relating to a god (without being a god), and in this common meaning the Persons of this theory are "divine". Similarly, a hand can be "human" without itself being a human being (165). How can the Son and Spirit be fully divine if each is caused by the Father and so does not exist a se? Layman answers that "the objector's intuition that divinity requires aseity is not shared by those who drew up the [Nicene] creed" (167). Further, "it seems to me that aseity is clearly not essential to divinity, that is, it is not essential for being worthy of worship" (168). The qualities of omnipotence, omniscience, eternality, perfect goodness, and necessary existence are sufficient to guarantee the worship-worthiness of the Persons (ibid.). Like Swinburne and Craig, Layman argues that a God who is a single self is impossible. While aware than a theist may understand God to be "perfectly loving" in the sense of having a perfect disposition to love which doesn't have to be actualized, Layman nonetheless asserts that > > There is...considerable plausibility in the claim that a truly > solitary person who throughout all eternity never expressed any love > for anyone would not be a perfectly loving person. (153) > Thus, given that God must be perfect independently of creation, "a truly solitary person would not be divine, for it would not be perfectly loving" (154). Additionally, Layman argues that it is "inconceivable" that a divine Person should flourish without loving another, and that surely only the love of finite selves would not be enough (154-5). A solitary divine Person would be "an appropriate object of pity" (155). Again, Layman argues that the Bible suggests that a divine Person must have not only splendor (exalted attributes) but also glory, "something at least akin to fame-a kind of recognition, approval, or appreciation" which is conferred by another (156). A solitary divine Person would be lacking this glory; but presumably a divine Person must have glory. Thus, there couldn't be just one divine person (156-7). Layman is skeptical about philosophical arguments purporting to show why there must be exactly three divine Persons, but he think's he's shown why there can't be only one. The limit of divine Persons to three, in his view, can only come from the Bible (157-8). Tuggy (2015) objects to arguments that there can't be a single divine self based on divine happiness or flourishing, urging instead that a divine self who exists and is perfect of himself would automatically be well-off, happy, or flourishing despite lacking countless important goods. It is too anthropomorphic, he argues, to suppose that a god or a divine self, like a human, is a social animal which can't flourish without interpersonal relationships. ### 2.6 Material Monotheism Christopher Hughes (2009) suggests a theory much like the Constitution theory (section 2.1.2 above) but without its controversial claim that there can be numerical sameness without identity. On this picture, > > ...we have just one (bit of?) divine "matter," three > divine forms, and three ("partially overlapping," > materially indiscernible but formally discernible divine hylomorphs > [compounds of form and matter]. ..."divine person" is > true of the three hylomorphs, but... "God" is true of > the (one and only) (bit of?) "divine matter." (2009, 309) > On this theory, "The Father is God," means that the Father has God for his matter, or that the Father is "materiated by" God, and "The Father is the same God as the Son" means that these two are materiated by the same God (309-10). An objection is that the one God of Christianity is not supposed to be a portion of matter. Hughes replies that perhaps it is orthodox to say that God is a very unusual kind of matter (310). Alternately, Hughes suggests a retreat from matter terminology, and argues that Persons of the Trinity can't bear the same relation they bear to one another that each bears to God. That is, it can't be correct, for example, that Father and Son are consubstantial, and that the Father and God are consubstantial. The reason is that for two things to be consubstantial is for there to be something which both are "substantiated" or "ensubstanced" by. They are consubstantial because they both bear this other relation to a third, substantiating thing. Thus, e.g. "The Father is God" means "The Father is (a person of the substance) God." Thus, even though Father and Son are numerically two, still it can be true that "There is just one (substance) God" (311). On this alternate view, though, what does it mean to say that God is the substance of a divine Person? Hughes suggests that the case is analogous to material objects. A sweater and some wool thread are "co-materiate" in that both are "materiated" or "enmattered" by one portion of matter, though they are numerically distinct (311; cf. 313). Hughes suggests it is an open question whether this is a different theory, or just a restatement of the first "in more traditional theological terminology." It will be the latter "If we can stretch the notion of 'matter' far enough to cover God, and stretch the notion of material substance (aka hylomorph) far enough to cover the divine persons" (312). Hughes ends on a negative mysterian note (see section 4.1 below), claiming that it is an advantage of this last account that ensubstancement is "a (very, though not entirely) mysterious relation" (313). Leftow (2018) objects that this theory features four things which are divine, which is at least one too many. Further, on this account God has but the Persons lack the divine attribute of aseity, which makes the Persons "at best only second-class deities" (10). ### 2.7 Concept-relative Monotheism Einar Bohn (2011) argues that trinitarian problems of self-consistency vanish when one realizes that the Trinity "is just an ordinary case of one-many identity" (363). He takes from Frege the idea that number-properties are concept-relative. Thus, > > > ...conceptualizing the portion of reality that is God as the > Father, the Son, and the Holy Spirit, we have conceptualized it as > being three in number, but it is nonetheless the same portion of > reality as what we might conceptualize as God, and hence as being one > in number. (366) > > > > There is no privileged way of conceptualizing [this portion of > reality] in terms of which we can explain the other way. Both ways are > equally legitimate. (369) > > > A difficulty for this approach is that most philosophers don't think there can be one-many identity relations. Some think identity is of necessity a one-one relation, although others allow there can be many-many identity; for instance, it may be that the three men who committed the robbery are identical to the three men who were convicted of the robbery. Those who believe identity can be one-many typically do so because they accept the controversial thesis that composition (the relation of parts to a whole they compose) should be understood as identity. Although Bohn does accept that thesis (Bohn 2014), he argues that this Trinity theory relies only on our having "a primitive notion of plural identity" (371), that is, a concept we understand without reference to any concept from mereological (parts and wholes) theory. For example, we can recognize a certain human body to be identical to a certain plurality of head, torso, two arms, and two legs. And we can recognize that a pair of shoes is identical to a plurality of shoes (365). Bohn argues that orthodoxy, by the standards of either the New Testament or the "Athanasian" Creed (see section 5.3), requires that the Persons of the Trinity be distinct (i.e. no one is identical to any other) but not that any is identical to the one God. Rather, orthodoxy requires that the one God is identical to the Three considered as a plurality. Thus, e.g. "The Father is God" must be read predicatively, that is, not as identifying the Father with God, but rather as describing the Father as divine (364, 367 n. 13). Does this theory make God's triunity dependent on human thought? And might the divine portion of reality equally well be conceived as seventeen? Bohn replies, > > That numerical properties are relational properties with concepts as > their relational units is compatible with reality having a real and > objective numerical structure. (372) > Thus, it doesn't follow that any conceptualization of this portion of reality is equally correct. While in this context he demurs from saying anything about concepts (372), it seems that Bohn assumes in Fregean fashion that concepts are objective and not mind-dependent (Bohn 2013, Section 1). Joseph Long (2019) objects that the theory is unorthodox because it requires a type of thing which is divine and yet which is neither the Trinity nor any divine Person. Further, Bohn's talk of "portions of reality" is unintelligible. Finally, orthodoxy demands that the Persons of the Trinity "are one God regardless of our conceptual scheme", whereas on this account whether or not the Persons are one god is relative to how we conceptualize them. Sheiva Kleinschmidt argues that theories on which composition is explained in terms of identity are of no use to the trinitarian, for such theories add no significant options to the options the trinitarian already has (Kleinschmidt 2012). ## 3. Four-self, No-self, and Indeterminate Self Theories Some Trinity theories don't fit in to either one-self or three-self categories, because they imply more divine selves than three, less than one, or are unclear about how many selves there might be in the Trinity. ### 3.1 God as a Functional Person Chad McIntosh (2015) formulates a Trinity theory which is similar to three-self theories except that it adds God the Trinity as a fourth divine self. This theory is inspired by recent work by philosophers on group persons. It's a longstanding part of legal tradition to treat various kinds of non-persons, such as corporations, as if they were persons. This is particularly useful, e.g. in holding corporations responsible for damages they cause. But some philosophers have argued for group agency realism, the thesis that some groups of persons are themselves literal persons, with interests, knowledge, freedom, power to intentionally act, and moral responsibility (168-71). McIntosh distinguishes "intrinsicist" persons, persons which are so because of their nature, from "functional" persons, persons which are so because of how they function. On this account the Persons of the Trinity are intrinsicist persons, while God the Trinity is a functional person (171). McIntosh argues that since moral responsibility implies personhood (of some kind), and it is clear that the Trinity must be praiseworthy, e.g. "for having achieved salvation for humankind" or "just for having the character of a loving community", then the Trinity must himself be a person. And it is widely agreed by Christians that the Trinity should be worshiped, but a non-person can't be a fitting recipient of worship (173). One may object that the Christian God is supposed to be a Trinity of Persons, not a Quaternity of Persons. As Leftow objects to another theory, > > ...the very fact that the doctrine the Creed states is known as > that of the Trinity militates against calling a > four-[divine]-individuals view orthodox. Had the Creed-writers > envisioned God plus the Persons as adding up to four divine > individuals, surely the doctrine would have been called Quaternity > from the beginning (Leftow 2018, 10). > McIntosh replies that the tradition demands that there are exactly three Persons (Greek: hypostases) which share the divine nature or essence, which is captured by his claim that there are exactly three intrinsicist persons. This account does not claim that God is a fourth hypostasis, a fourth intrinsicist person. Rather, God is a functional person, a person not by his essence, but rather who exists as a person because of the unified functioning of the Father, Son, and Spirit (174). McIntosh argues that the theory neatly sidesteps a number of common objections to three-self theories: here, God is a self and not merely a group or a composite object which is less than a self. In contrast with three-self theories, as a literal self, personal pronouns may be literally used of him. And this group which is a "he" can have the divine attributes which imply being a self, such as omniscience and moral perfection. (See sections 2.4, 2.5.) It also, McIntosh argues, either falsifies or casts doubt on the key premise in Tuggy's divine deception argument against three-self theories (175-7, 180; see section 2.3.) Following some Old Testament scholars, McIntosh claims that ancient Israelites recognized many groups, including their own nation, as literal (group, functional) persons. He argues that this belief explains a number of oddities in the Old Testament, such as the idea of group guilt, apparent beings which seem in some sense to be extensions of Yahweh's personhood, and texts which switch easily between singular and plural subjects (177-80). ### 3.2 Temporal Parts Monotheism H. E. Baber (2002) argues that a Trinity theory may posit the Persons as "successive, non-overlapping temporal parts of one God" (11). This one God is neither simple nor timeless, but is a temporally extended self with shorter-lived temporally extended selves as his parts. This does not violate the requirement of monotheism, because we should count gods by "tensed identity", which is "not identity but rather the relation that obtains between individuals at a time, t, when they share a stage [i.e. a temporal part] at t" (5). At any given time, only one self bears this relation of temporal-stage sharing with God. How can any of these selves be divine given that they are neither timeless nor everlasting? Following Parfit, she argues that a self may last through time without being identical to any later self at the later times; that is, "identity is not what matters for survival" (6). Each of these non-eternal selves, then, counts as the continuation of the previous one, and is everlasting in the sense that it is a temporal part of an everlasting whole, God. The obscure traditional generation and procession relations are re-interpreted as non-causal relations between God and two of his temporal parts, the Son and Spirit (13-4). In a later paper, she argues that any trinitarian may and should accept this re-interpretation (Baber 2008). Although Baber argues that this is a "minimally decent" Trinity theory, she admits that it is heretical, and names it a "Neo-Sabellian" theory, because on it, the Persons of the Trinity are non-overlapping, temporary modes of the one God (15; on Sabellianism see section 1.3). But the Persons in this theory are not mere modes; they are truly substances and selves, and there are (at least) three of them, though each is counted as the continuation of the one(s) preceding him. It is unclear whether the theory posits only three selves (10-1). But she argues that the theory is preferable to many of its rivals "since it does not commit us to relative identity or require any ad hoc philosophical commitments" (15), and even though its divine selves don't overlap, sense can be made of, e.g. Jesus's interaction with his Father (meaning not the prior divine Person, but God, the temporal whole of whom Jesus is a temporal part) (11-4). This theory is notable in being a case not of rational reconstruction, but of doctrinal revision (Tuggy 2011a). Many of its features are controversial, such as its unorthodoxy, its metaphysical commitments to temporal parts and the lasting of selves without diachronic identity, its denials of divine simplicity and divine timelessness, and its redefinitions of "monotheism", "generation", and "procession". In a later discussion Baber argues that some form or other of Sabellianism about the Trinity is theoretically straightforward and fits well with popular Christian piety. Further, such theories cansurvive the common objections that they imply that God is only contingently trinitarian, and that they characterize God only in relation to the cosmos. While some Sabellian theories do have those implications, Baber argues that a trinitarian may just accept them (2019, 134-8). Alternately, Baber (2019) develops a structuralist approach to the Trinity which doesn't imply anything about how many selves it involves. ### 3.3 Persons as Relational Qua-Objects Rob Koons (2018) constructs an account inspired by Aquinas, Augustine, and recent work on "qua-objects" by Fine and by Asher. Following the latter, Koons understands "qua-modified noun phrases as picking out intentional objects consisting of tropes or accidents, metaphysical parts of the base object" (346). Koons holds that even everyday objects imply the existence of such intentional objects, i.e. things that can be thought. Thus, we can distinguish Trump-qua-husband from Trump-qua-President; while these are real objects of thought, they both amount to being properties of Trump, which Koons thinks of as metaphysical parts of Donald Trump. Unlike most of the other theories in this entry, Koons builds his on the foundation of divine simplicity, traditionally understood. According to this, God is numerically identical with his nature, his one action, and his existence. God has no accidental (non-essential) properties and no proper parts, and he just is any essential property of his. In sum, God has no parts or components in any sense (339). While one might suppose that this would rule out God being a Trinity, Koons argues that to the contrary, we can understand how and why God is a Trinity by "locating God in an extreme and exotic region of logical space" (ibid.) The divine nature just is any divine attribute, e.g. omnipotence. In addition, Koons argues that the divine nature just is "an intentional relation: namely, perfect knowledge and perfect love.". Understanding or knowledge generally should be understood "as an internal relation between the mind and its external object." (340). Following Aquinas, Koons says that God (a.k.a. the divine nature) understands all things through himself; God is essentially omniscient and essentially self-understanding. Thus, the divine nature implies the existence of three relational qua-objects, which are the Persons of the Trinity. These are not merely three ways we can think of God, or three ways God may appear to us, but rather these objects result from God's essential self-understanding (345). Each of these four things-God (the divine nature), the Father, the Son, and the Spirit-has the divine nature as its one metaphysical component, and each has all the divine attributes (346). Each of those four is numerically distinct from each of the three others (347). The theory requires more than the relation of (absolute) numerical sameness or identity. In addition, Koons defines a relation of "real" sameness. Like identity, this relation is reflexive and symmetrical, but unlike identity it is neither transitive nor Euclidian (such that if any x is related to some y and to some z then this implies that y and z are related in that same way). Thus, real sameness is not an equivalence relation (348, 357 n. 5) According to this theory each Person is really the same as the divine nature (God), but not identical to him. But no Person here is really the same as any other Person; all three are really distinct from one another. To summarize: there are four divine realities on this model of the Trinity. The three Persons are so many qua-objects, while God is not. None of these four is really distinct from God; all are really the same as him. Yet none of the four is identical to any of the others (348). One may ask why there should be only three qua-objects here, when objects like a human person or an apple, having many properties, might imply hundreds or thousands of qua-objects. The answer is that not every qua-object of God is a divine Person. Many such, Koons says, are contingent, e.g. God-as-creator, or God-as-friend-of-Abraham; such would not have existed had God not created. And any qua-object of God which involves only an essential property of his, e.g. God-as-omnipotent, is numerically identical to God (345). To be a "hypostatic qua-object" (i.e. a God-as-thing which is a divine Person) something must exist necessarily, be numerically distinct from God, and be such that "It is not wholly grounded in a logical or conceptual way on any other divine qua-object or objects. So, it must be fully determinate (non-general, non-disjunctive, and non-negative) in its definition" (346). This last condition is meant to prevent the proliferation of divine qua-objects (354-6). Koons argues that this account explains why there are exactly three divine Persons. > > My main claim is that, as a matter of metaphysical necessity, there > are exactly three hypostatic qua-objects (namely, Father, > Son, and Spirit, as defined above). This is because there are only two > intrinsic, relational properties of God (knowing and being known), and > these give rise (on purely logical grounds) to only three > non-disjunctive combinations. (346) > Divine love doesn't imply further Persons because it's the same relational property as divine self-knowing. God-as-knower isn't numerically the same as God-as-known because of the essential asymmetry of the knowing relation (ibid.). Divine love, Koons says, is a kind of charity of friendship; thus, lover and beloved can't be numerically identical. So if the Father loves the Son, this implies that they are numerically distinct (non-identical). It also implies that they are really distinct and not really the same. In specifying what he means by real distinctness Koons writes, > > Two qua-objects with the same ultimate base are really > distinct if and only if they are numerically distinct and the > distinction between them is intrinsic to their ultimate base. > (348) > The distinction between these qua-objects Father and Son is intrinsic to their ultimate base, God (the divine nature) because he is the intrinsic yet relational property of love (348-9). This has the consequence that "the divine nature cannot love or be loved by any of the divine Persons" (351). Koons argues that this theory has many advantages over some rivals. Against the constitution based three-self theory of Brower and Rea (see section 2.1.3), it allows for divine simplicity, as the Trinity does not involve any metaphysical components or parts other than the divine nature. And he claims that their account amounts to tritheism, "since each Person is divine in His own unique and incomparable way". In contrast, on Koons's theory, "each of the three Persons is divine in the same way-simply by being a divine qua-object, and the divine nature is complete and fully divine in itself". Again, Koons's theory can, and theirs can't, explain why there are exactly three divine Persons. And their theory requires three different odd and hard to explain personal attributes (352). As contrasted with any "social" theory, this one doesn't have divine Persons which are really distinct from the divine nature (God) (353). Koons recognizes that many will object that this theory is tetratheism; it features four realities, each of which is divine; prima facie, these would be four gods. Koons believes that the real sameness of each of the Persons with God should rule out any polytheism and rule in monotheism. He offers this definition of monotheism: > > There is one and only one thing such that no divine being is really > distinct from it (348). > This would be equivalent to: > > There is one and only one thing such that every divine being is really > the same as it. > But a "divine being" is a god. Thus the meaning of this definition can be restated as: > > There is one and only one thing such that every god is really the same > as it. Or: There is one and only one thing such that no god is really > distinct from it. > This is a controversial definition; one may think that Koons simply redefines "monotheism" as compatible with any number of gods greater than zero. Put differently, one may count things by identity. Why can't one also count gods in this way? Again, if a is a god, and b is a god, and they are non-identical, what is it about "real sameness" that implies that they're really the same god? Why isn't this an ad hoc, theory-saving definition? One may wonder here how the four realities can be equally divine. It would seem that whereas God (the divine nature) would not exist because of any other, and so would exist a se, each of the qua-object persons would exist because of God, their base. Wouldn't this make God greater than each of the Persons? Again, on this account each of these four is intrinsically and essentially divine, yet the Persons can love, while God can not. How then can all four be omnipotent? Some will judge this theory to inherit all the problems of the traditional divine simplicity doctrine it assumes. Others will consider its fit with simplicity to be a feature and not a bug. Koons points out that it also assumes constituent ontology, a Thomistic account of thought, and the claim that the divine nature is an intentional relation (356). ### 3.4 Persons as Improper Parts of God An account of the Trinity by Daniel Molto tries to preserve the idea found especially in Western creeds that "each divine person is, in some sense, all of God" (2018, 395). (Because of this he claims the title "Latin" for the theory.) He employs a non-standard mereology (theory of parts and wholes) to interpret what he considers to be the three requirements for a Trinity theory: that there's only one god, that no Person of the Trinity is identical to any other, and that in some sense it is true that each Person individually "is wholly" God (400). The view is that God, the Father, the Son, and the Spirit are all improper parts of one another, while none is numerically identical to any other. This is shown in the following chart; the lines represent the symmetrical and transitive improper parthood relation. ![A representation of the divine persons and the Trinity as improper parts of one another.](Moltotrin1.png) On more "classical" mereological systems, if A is an improper part of B, this implies that A and B are numerically identical, but in the system suggested by Molto, this is denied. He argues that this change is not merely theologically motivated, but may be applicable to other issues in metaphysics (410-3). Molto discusses a problem for the model which arises from the transitivity of parthood and the axiom that things which are improper parts of one another must have all their proper parts in common. If the body of the incarnate Son is a proper part of him, then given Molto's model of the Trinity, this body would also have to be a proper part of God, of the Father, and of the Holy Spirit - claims which most Christian theologians would reject. In response, he adds three further elements to the model, as shown here: ![A mereological representation of the Trinity as in Molto's mereological one-self Trinity theory.](Moltotrin2.png) Here, D = the divine nature of the Son, N = the human nature of the Son, and B = the body of the Son. As before, the lines with arrows on each end represent the symmetrical improper parthood relation. In this illustration the one-arrow lines represent the asymmetrical properparthood relation. Thus, the divine nature of the Son and the human nature of the Son are proper parts of the composite Son, and the human body is a proper part of the human nature (and thus, also of the composite Son). Molto leaves it up to theologians whether this sort of theory is orthodox (514-7). His suggestion is only that this may be a simpler and less controversial solution to the logical problem of the Trinity, that is, to showing how trinitarian claims do not imply a contradiction. (397-8, 416-7) ## 4. Mysterianism Often "mystery" is used in a merely honorific sense, meaning a great and important truth or thing relating to religion. In this vein it's often said that the doctrine of the Trinity is a mystery to be adored, rather than a problem to be solved. In the Bible a "mystery" (Greek: musterion) is simply a truth or thing which is or has been somehow hidden (i.e., rendered unknowable) by God (Anonymous 1691; Toulmin 1791b). In this sense a "revealed mystery" is a contradiction in terms (Whitby 1841, 101-9). While Paul seems to mainly use "mystery" for what used to be hidden but is now known (Tuggy 2003a, 175), it has been argued that Paul assumes that what has been revealed will continue to be in some sense "mysterious" (Boyer 2007, 98-101). Mysterianism is a meta-theory of the Trinity, that is, a theory about trinitarian theories, to the effect that an acceptable Trinity theory must, given our present epistemic limitations, to some degree lack understandable content. "Understandable content" here means propositions expressed by language which the hearer "grasps" or understands the meaning of, and which seem to her to be consistent. At its extreme, a mysterian may hold that no first-order theory of the Trinity is possible, so we must be content with delineating a consistent "grammar of discourse" about the Trinity, i.e., policies about what should and shouldn't be said about it. In this extreme form, mysterianism may be a sort of sophisticated position by itself--to the effect that one repeats the creedal formulas and refuses on principle to explain how, if at all, one interprets them. More common is a moderate form, where mysterianism supplements a Trinity theory which has some understandable content, but which is vague or otherwise problematic. Thus, mysterianism is commonly held as a supplement to one of the theories of sections 1-3. Again, it may serve as a supplement not to a full-blown theory (i.e., to a literal model of the Trinity) but rather to one or more (admittedly not very helpful) analogies. (See section 3.3.1 in the supplementary document on the history of trinitarian doctrines.) Unitarian views on the Father, Son, and Spirit are typically motivated in part by hostility to mysterianism. (See the supplementary document on unitarianism.) But the same can said of many of the theories of sections 1-3. Mysterians view their stance as an exercise of theological sophistication and epistemic humility. Some mysterians appeal to the medieval tradition of apophatic or negative theology, the view that one can understand and say what God is not, but not what God is, while others simply appeal to the idea that the human mind is ill-equipped to think about transcendent realities. Tuggy (2003a) lists five different meanings of "mystery" in the literature: > > [1]...a truth formerly unknown, and perhaps undiscoverable by > unaided human reason, but which has now been revealed by God and is > known to some... [2] something we don't completely > understand... [3] some fact we can't explain, or > can't fully or adequately explain... [4] an unintelligible > doctrine, the meaning of which can't be grasped....[5] a > truth which one should believe even though it seems, even after > careful reflection, to be impossible and/or contradictory and thus > false. (175-6) > Sophisticated mysterians about the Trinity appeal to "mysteries" in the fourth and fifth senses. The common core of meaning between them is that a "mystery" is a doctrine which is (to some degree) not understood, in the sense explained above. We here call those who call the Trinity a mystery in the fourth sense "negative mysterians" and those who call it a mystery in the fifth sense "positive mysterians". It is most common for theologians to combine the two views, though usually one or the other is emphasized. Sophisticated modern-era mysterians include Leibniz and the theologian Moses Stuart (1780-1852). (Antognazza 2007; Leibniz Theodicy, 73-122; Stuart 1834, 26-50.) ### 4.1 Negative Mysterianism The negative mysterian holds that the true doctrine of the Trinity is not understandable because it is too poor in intelligible content for it to positively seem either consistent or inconsistent to us. In the late fourth-century pro-Nicene consensus this takes the form of refusing to state in literal language what there are three of in God, how they're related to God or to the divine essence, and how they're related to each other. (See section 3.3 in the supplementary document on the history of Trinity theories.) The Persons of the Trinity, in this way of thinking, are somewhat like three men, but also somewhat like a mind, its thought, and its will, and also somewhat like a root, a tree, and a branch. Multiple incongruous analogies are given, the idea being that a minimal content of the doctrine is thereby expressed, though we remain unable to convert the non-literal claims to literal ones, and may even be unable to express in what respects the analogies do and don't fit. Negative mysterianism goes hand in hand with the doctrines of divine incomprehensibility (that God or God's essence can't be understood completely, at all, or adequately) and divine ineffability (that no human concept, or at least none of some subset of these, applies literally to God). Some recent studies have emphasized the centrality of negative mysterianism to the pro-Nicene tradition of trinitarian thought, chastising recent theorists who seem to feel unconstrained by it (Ayres 2004; Coakley 1999; Dixon 2003). The practical upshot of this is being content to merely repeat the approved trinitarian sentences. Thus, after considering and rejecting as inadequate multiple analogies for the Trinity, Gregory of Nazianzus concludes, > > So, in the end, I resolved that it was best to say > "goodbye" to images and shadows, deceptive and utterly > inadequate as they are to express that reality. I resolved to keep > close to the more truly religious view and rest content with some few > words, taking the Spirit as my guide and, in his company and in > partnership with him, safeguarding to the end the genuine illumination > I had received from him, as I strike out a path through this world. To > the best of my powers I will persuade all men to worship Father, Son, > and Holy Spirit as the single Godhead and power, because to him belong > all glory, honor, and might forever and ever. Amen. (Nazianzus, > Oration 31, 143.) > Opponents of this sort of mysterianism object to it as misdirection, special pleading, neglect of common sense, or even deliberate obfuscation. They emphasize that trinitarian theories are human constructs, and a desideratum of any theory is clarity. We literally can't believe what is expressed in trinitarian language, if we don't grasp the meaning of it, and to the extent that we don't understand a doctrine, it can't guide our other theological beliefs, our actions, or our worship (Cartwright 1987; Dixon 2003, 125-31; Nye 1691b, 47; Tuggy 2003a, 176-80). Negative mysterians reply that it is well-grounded in tradition, and that those who are not naively overconfident in human reason expect some unclarity in the content of this doctrine. ### 4.2 Positive Mysterianism In contrast, the positive mysterian holds that the trinitarian doctrine can't be understood because of an abundance of content. That is, the doctrine seems to contain explicit or implicit contradictions. So while we grasp the meaning of its individual claims, taken together they seem inconsistent, and so the conjunction of them is not understandable, in the sense explained above. The positive mysterian holds that the human mind is adequate to understand many truths about God, although it breaks down at a certain stage, when the most profound divinely revealed truths are entertained. Sometimes an analogy with recent physics is offered; if we find mysteries (i.e., apparent contradictions) there, such as light appearing to be both a particle and a wave, why should we be shocked to find them in theology (van Inwagen 1995, 224-7)? The best-developed positive mysterian theory is that of James Anderson (2005, 2007), who develops Alvin Plantinga's epistemology so that beliefs in mysteries (merely apparent contradictions) may be rational, warranted, justified, and known. Orthodox belief about the Trinity, Anderson holds, involves believing, for example, that Jesus is identical to God, the Father is identical to God, and that Jesus and the Father are not identical. Similarly, one must believe that the Son is omniscient, but lacks knowledge about at least one matter. These, he grants, are apparent contradictions, but for the believer they are strongly warranted and justified by the divine testimony of scripture. He argues that numerous attempts by recent theologians and philosophers to interpret one of the apparently contradictory pairs in a way that makes the pair consistent always result in a lapse of orthodoxy (2007, 11-59). He argues that the Christian should take these trinitarian mysteries to be "MACRUEs", merely apparent contradictions resulting from unarticulated equivocations, and he gives plausible non-theological examples of these (220-5). It is plausible that if a claim appears contradictory to someone, she thereby by has a strong epistemic "defeater" for that belief, i.e., a further belief or other mental state which robs the first belief of rational justification and/or warrant. A stock example is a man viewing apparently red objects. The man then learns that a red light is shining on them. In learning this, he acquires a defeater for his belief that the items before him are red. Thus with the Trinity, if the believer discovers an apparent contradiction in her Trinity theory, doesn't that defeat her belief in that theory? Anderson argues that it does not, at least, if she reflects properly on the situation. The above thought, Anderson argues, should be countered with the doctrine of divine incomprehensibility, which says that we don't know all there is to know about God. Given this truth, the believer should not be surprised to find herself in the above epistemic situation, and so, the believer's trinitarian belief is either insulated from defeat, or if it's already been defeated, that defeat is undone by the preceding realization (2007, 209-54). Dale Tuggy (2011a) argues that Anderson's doctrine of divine incomprehensibility is true but trivial, and not obviously relevant to the rationality of belief in apparent contradictions about God. The probability of our being stuck with such beliefs is a function not only of God's greatness in comparison to humans' cognitive powers, but also of what and how much God chooses to reveal about himself. Nor is it clear that God would be motivated to pay the costs of inflicting apparently contradictory divine revelations on us. Moreover, Anderson has not ruled out that the apparent contradictions come not from the texts alone, but also from our theories or pre-existing beliefs. Finally, he argues that due to the comparative strength of "seemings", a believer committed to paradoxes like those cited above will, sooner or later, acquire an epistemic defeater for her beliefs. In a reply, Anderson (2018) denies that divine incomprehensibility is trivial, while agreeing that many things other than God are incomprehensible (297). While Tuggy had attacked his suggestions about why God would want to afflict us with apparent contradictions, Anderson clarifies that > > ...my theory doesn't require me to identify > positive reasons for God permitting or inducing MACRUEs. For even if I > concede Tuggy's point that "the prior probability of God > inducing MACRUEs in us is either low or inscrutable," the > doctrine of [divine] incomprehensibility can still serve as...an > undercutting defeater for the inference from D appears to be > logically inconsistent to D is false. (298-9) > The defense doesn't require, Anderson argues, any more than that MACRUEs are "not very improbable given theism" (299). As to whether these apparent contradictions result from the texts rightly understood, or whether they result from the texts together with mistaken assumptions we bring to them, this is a question only biblical exegesis can decide, not any a priori considerations (300). As to Tuggy's charge that a believer in theological paradoxes will inevitably acquire an undefeated defeater for her beliefs, Anderson argues that this has not been shown, and that Tuggy overlooks how a believer may reasonably add a relevant belief to her seemingly inconsistent set of beliefs, such as that the apparently conflicting claims P and Q are only approximately true, or that "P and Q are the best way for her to conceptualize matters given the information available to her, but they don't represent the whole story" (304). Anderson's central idea is that the alleged contradictions of Christian doctrine will turn out to be merely apparent. In contrast, some theologians have held that doctrines including the Trinity imply not merely apparent but also real contradictions, but are nonetheless true. Such hold that there are exceptions to the law of non-contradiction. While some philosophers have argued on mostly non-religious grounds for dialetheism, the claim that there can be true (genuine, not merely apparent) contradictions, this position has for the most part not been taken seriously by analytic theologians (Anderson 2007, 117-26) (For a recent exception, see Beall 2019.) ## 5. Beyond Coherence Analytic literature on the Trinity has been laser-focused on the logical coherence of "the" doctrine, addressing an imagined critic arguing that the doctrine is clearly incoherent. They do this by suggesting models of the Trinity, intelligible and arguably coherent interpretations of most or all of the traditional language. But in recent work the tools of analytic philosophy have been applied to several closely related issues. ### 5.1 "The Trinity" and Tripersonality The term "Trinity" has been used either as a singular referring term or as a plural referring term (Tuggy 2016, 2020). The first usage goes hand in hand with the claim that the one God just is the tripersonal God, the Trinity. But the earlier use of "Trinity"-(Greek: trias, Latin: trinitas) where that term refers to a "they" and not a "he" or an "it"-still survives, and some Trinty theories imply that the term "Trinity" can refer only in this way. Most statements of faith by trinitarian Christian groups seem to assume or imply that the Trinity just is God (and vice-versa); the only god is the tripersonal God, the Trinity, and "Trinity" is a singular referring term denoting that reality. This God, it is assumed, does not merely happen to be tripersonal, but must be so; on such a view, it looks like tripersonality will be an essential divine attribute. Thus, it can seem axiomatic that "Christians hold that God is Trinitarian in God's essential nature" (Davis and Yang 2017, 226). Some Trinity theories embrace this (section 2.5). However, if this is so, it is hard to see how each of the Persons could be divine in the way the one God is divine, since generally trinitarians don't want to say that each is himself tripersonal. (See section 2.1.5 for an exception.) Thus, some Trinity theories eschew a thing which is tripersonal, while affirming three divine Persons whose divinity does not require tripersonality (sections 2.1.3, 2.1.4). For such theories, "Trinity" is a plural referring term. ### 5.2 Logic Puzzles and Language While many discussions start with claims that are seen as the heart of the "Athanasian" creed, a recent piece by Justin Mooney (2018, 1) starts with this seemingly inconsistent triad of claims: 1. God is triune. 2. The Son is not triune. 3. The Son is God. Dale Tuggy (2014, 186) presents this inconsistent triad. 1. The Christian God is a self. 2. The Christian God is the Trinity. 3. The Trinity is not a self. One-self trinitarians deny 3, and three-self trinitarians deny 1. But Tuggy argues that for scriptural reasons a Christian should deny 2. (See also section 5.4 and the supplementary document on the history of trinitarian doctrines, section 2.) Ryan Byerly (2019) explains "the philosophical challenge of the Trinity" as centering on the key Nicene term "consubstantial" (Greek: homoousios). How can the three Persons be "consubstantial" so that each equally in some sense "is God", where this implies neither their numerical identity, nor that there's more than one god? Jedwab and Keller (2019, 173) see the fundamental challenge for the orthodox trinitarian as showing how this seemingly inconsistent triad of claims is, rightly understood, consistent: 1. There is exactly one God. 2. There are exactly three divine persons. 3. Each divine person is God. They argue that this must involve paraphrases, clearer formulations of 1-3 which can be seen as possibly all true. They compare how the theories of sections 2.1, 2.3, and 1.5 above must do this, and conclude that it is easier for the first of these to provide paraphrases which plausibly express the same claims as the originals, which is a point in favor of such relative identity theories. Beau Branson (2019) explores these claims as constituting "the logical problem of the Trinity": each Person "is God", they are distinct from one another, and yet there is only one thing which "is God". These provide materials for a formidable argument against any doctrine that entails those seven claims. He argues that all possible non-heretical solutions to that problem either equivocate on the predicate "is God" (roughly: what are called "social" theories, discussed in sections 2.2-7) or insist that divine Persons must be counted by some relation other than "absolute" or "classical" identity (i.e. relative identity theories as discussed in section 2.1). Another recent piece compares different approaches to the Trinity by how they respond to an anti-trinitarian argument based on alleged differences between the Father and the Son (Tuggy 2016b). ### 5.3 Foundations A tradition going back at least to Cartwright (1987) is using the language of the so-called "Athanasian" creed to generate contradictions, the task of the philosophical theologian then being to show how these can be avoided by more careful analysis. This Latin document is by an unknown author, and is not the product of any known council. Modern scholarship places it some time in the fifth century, well after the life of Athanasius (d. 373), and sees it as influenced by the writings of Augustine (Kelly 1964). Objecting to making it a standard of trinitarian theology, several authors have pointed out its dubious provenance and coherence, and have observed that it has mainly been accepted in the Western realm and not in the East, and that it seems to stack the deck against three-self theories (Layman 2016 136-7, 169-71; McCall 2003, 427; Tuggy 2003b, 450-55). Tuggy (2016b) objects that starting with this problematic creed causes analytic theologians to neglect the question of if and how the teaching of this creed is the same as various statements from the "ecumenical" councils, pre-Nicene theologies, or the Bible. But William Hasker argues that rightly understood, the claims of this creed may not be paradoxical, as it is largely concerned with what may and may not be said (2013b, 250-4). Apart from the "Athanasian" creed, H.E. Baber describes five different foundations for theorizing about the Trinity, endorsing the fourth. > > This poses the question of what the 'foundations' for the > philosophical investigation of Trinitarian theology should be if it is > not either [1] the declarations of Church councils or [2] the > theological works of the Fathers or [3] Scripture which...does > not include any Trinitarian doctrine. ...what philosophical > theology should be about...[is] [4] the discourse and practice of > the Church, and by that I do not mean [5] the doctrinal claims of the > Church in its pretension to being a teaching institution, but [4] the > liturgy, hymnody, and art, customs and practices, religious objects > and religious devotions which, together, constitute the Christian > religion and its practice. The aim of philosophical theology is to > make sense of the discourse and provide a rationale for the practices > while avoiding logical incoherence. (Baber 2019, 186-7) > Some in the literature fall cleanly into one of Baber's categories, but more commonly, work in analytic theology is done while leaving unclear just what are the foundations of trinitarian theorizing. Following the example of his earlier work on christology, Timothy Pawl (2020) focuses on the teachings of the "ecumenical" councils, which arguably give the central trinitarian language of catholic traditions. In favor of Baber's second approach, Beau Branson (2018) critiques what he calls "the virtue approach"-basically, treating theological issues like metaphysical or logical puzzles calling for creative theorizing-as point-missing, question begging, and unclear. In contrast, he advocates for "the historical approach", which assumes that the content of "the doctrine of the Trinity" should be considered as fixed by the views of "mainly various fourth-century theologians" (Section 4). It is misguided, he argues, to focus merely on the theoretical virtues of various rational reconstructions of what traditional Trinity language is really supposed to be expressing, as most of these will not plausibly be expressing the historical doctrine. New-fangled accounts, Branson argues, have a burden of showing how they, if coherent, imply that the historical doctrine of the Trinity is coherent, and indeed why the former should even count as a version of the latter (Section 5). (Similarly with other theoretical virtues.) At any rate, nothing about the project of analytic theology requires the neglect of the crucial historical definers of the Trinity doctrine (Section 13). Analytic theologians have expended much effort on metaphysical models which if accurate would arguably show that "the doctrine of the Trinity" is coherent (i.e. seemingly not self-contradictory). But only a recent monograph by Vlastimil Vohanka (2014) asks in depth how, if at all, a person might be able to know that the doctrine of the Trinity is logically possible. Vohanka argues for "Weak Modal Scepticism about the Trinity Doctrine" (83), which is the claim that "it is psychologically impossible to see evidently and apart from religious experience that the Trinity doctrine is logically possible" (86). The arguments for this conclusion defy easy summary (but see Chapter 7 and Jaskolla 2015). What is "the doctrine" in question? Vohanka uses a minimal definition, that "there are three persons, each of whom is God but there is just one being which is God" (47). Admittedly, this better fits one-self theories than it does three or four-self theories (52-7) Like most, Vohanka assumes that a Trinity doctrine is essential to Christianity (57-8), so the main thesis implies the impossibility of knowing that Christianity is logically possible. But the overall project is not an attack on the truth or knowability of Christianity (see the author's clarifications about his aims on 244-7, 276-7). Rather, such claims as the Trinity and Christianity seemingly can't be evident to us (i.e. roughly, can't be obviously true to us, as is 1+1=2 and such propositions-see 18), and moreover, we shouldn't expect that we'll have any religious experience in this life which is sufficient to make them evident (277). For all that's been said, such claims "may well be epistemically justified, well-argued, have clearly high non-logical probability, etc." (279). But the Christian philosopher should give up on "fulfilling the classical idea of evidentness in matters viewed by him as of utmost importance: the truth of Christianity and of the Trinity doctrine" (ibid.). ### 5.4 Competing Narratives In some scholarly circles it is taken as obvious that New Testament teaching is not trinitarian-that it neither asserts, nor implies, nor assumes anything about a tripersonal God (see e.g. Baber 2019, 148-56; Kung 1995, 95-6; see also supplement on the history of Trinity doctrines, section 2). But most analytic literature on the Trinity assumes the truth of an orthodox narrative about where Trinity theories come from. According to this, from the beginning Christians were implicitly trinitarian; that is, they held views which imply that God is a Trinity, but typically did not realize this or have adequate language to express it. By at least the late 300s, they had gained enough new language and/or concepts to express what they had been committed to all along. But in recent analytic literature on the Trinity there are two counternarratives, both of which see the idea of a triune God as entering into Christian traditions in the last half of the 300s. Beau Branson argues that "Monarchical Trinitarianism", which in his view correctly understands the theology of the authoritative fourth century Greek "fathers" (Basil of Caesarea, Gregory of Nyssa, and Gregory of Nazianzus) is a trinitarian theology on which "Strictly speaking, The One God just is the Father" (Branson forthcoming, Section 6). He contrasts this with most other Trinity theories, which he calls "egalitarian" or "symmetrical", theories on which "all three persons have an 'equal claim' to being called 'God,' in any and every sense." (ibid.) Branson criticizes Tuggy's (2016a) analyses of the concepts trinitarian and unitarian, and offers rival definitions on which "Monarchical Trinitarianism" is trinitarian and not unitarian. To be trinitarian, a theology needs only to assert that "there are exactly three divine 'persons' (or individuals, etc.). Nevertheless...there is exactly one God" (Section 5). As Branson understands the history of Christian theology, the idea that "the Trinity" is a tripersonal God is a misunderstanding of tradition which is due particularly to "Western" thinkers, such as Augustine. Branson cites some recent Orthodox theologians who hold, like John Behr, that "there is not One God the Trinity, but One God Father Almighty" (Behr 2018, 330). In reply, Tuggy has argued that recent Orthodox theologians seem divided on this point, and that the idea of a triune God (the one God as the Trinity) is found even in some of the Greek writers Branson claims as exemplars of theological orthodoxy (Tuggy 2020). Another recent counternarrative sees ancient mainstream Christian theology as changing from unitarian to trinitarian. Tuggy (2019) argues that in the New Testament the one God is not the Trinity but rather the Father alone. The argument moves from facts about the texts of the New Testament to what the authors probably thought about the one God, using what philosophers of science call the likelihood principle or the prime principle of confirmation. Tuggy sees such identification of the one God with the Father dominating early Christian theologies until around the time of the second ecumenical council in 381 C.E. (Tuggy 2017, Chapter 5). Then, the Son and the Spirit, which in many 2nd to early 4th c. speculations were two lesser deities in addition to God, were taught to, together with the Father, somehow comprise the one God, the Trinity. (2016b, Sections 2-3)
tropes	## 1. Historical Background The father of the contemporary debate on tropes was D. C. Williams (1953a; 1953b; 1963; 1986 [1959]; 2018).[2] Williams defends a one-category theory of tropes (for the first time so labeled), a bundle theory of concrete particulars, and a resemblance class theory of universals. All of which are now elements of the so-called 'standard' view of tropes. Who to count among Williams' trope-theoretical predecessors is unavoidably contentious. It depends on one's views on the nature of the trope itself, as well as on which theses, besides the thesis that tropes exist, one is prepared to accept as part of a trope--or trope-like--theory. According to some philosophers, trope theory has roots going back at least to Aristotle (perhaps to Plato (cf. Buckels 2018), or even the pre-Socratics (cf. Mertz 1996: 83-118)). In the Categories, Aristotle points out that Substance and Quality both come in what we may call a universal and a particular variety (man and this man in the case of substance, and pallor and this pale--to ti leukon--in the case of quality). Not everyone believe that this means that Aristotle accepts the existence of tropes, however. On one interpretation (Owen 1965) this pale names an absolutely determinate, yet perfectly shareable, shade of pallor. But on a more traditional interpretation (cf. Ackrill 1963 and, more recently, Kampa & Wilkins 2018), it picks out a trope, i.e., a particular 'bit' of pallor peculiar to the substance that happens to exemplify it (for a discussion, cf. Cohen 2013). In view of the strong Aristotelian influence on medieval thinkers, it is perhaps not surprising that tropes or trope-like entities are found also here. Often mentioned in this connection is Ockham (cf. also Aquinas, Duns Scotus, and Suarez). Ockham held that there are in total four sorts of individual things: substances, qualities, substantial forms, and matter. Claude Panaccio (2015) argues that all four sorts of individual things are best understood as tropes or, in the case of substances, as bundles of tropes (more precisely as nuclear bundles of tropes similar to those defended by e.g., Simons 1994 and Keinanen 2015). This list of early proponents of trope-like entities could have been made much longer. D. W. Mertz (1996) mentions, besides those already listed, Boethius, Avicenna, and Averroes. And Kevin Mulligan et al. (1984) point out that similar views can be found defended by early modern philosophers, including Leibniz, Locke, Spinoza, Descartes, Berkeley and Hume. Still, it is in the writings of 19th century phenomenological philosophers that the earliest and most systematic pre-Williams 'trope'-theories are found (Mulligan et al. 1984: 293). The clearest example of an early trope theorist of this variety is undoubtedly Edmund Husserl. In the third part of his Logical Investigations (2001 [1900/1913]), Husserl sets out his theory of moments, which is his name for the world's abstract (and essentially dependent) individual parts (Correia 2004; Beyer 2016). Husserl was most likely heavily influenced by Bernard Bolzano, who held that everything real is either a substance or an adherence, i.e., an attribute that cannot be shared (Bolzano 1950 [1851]).[3] Husserl thought of his moments as (one of) the fundamental constituents of phenomenal reality. This was also how fellow phenomenologists G. F. Stout (1921; 1952),[4] Roman Ingarden (1964 [1947-1948]) and Ivar Segelberg (1999 [1945;1947;1953])[5] viewed their fundamental and trope-like posits. Williams' views are not so easily classified. Although he maintained that all our knowledge rests on perceptual experience, he agreed that it should not be limited to the perceptually given and that it could be extended beyond that by legitimate inference (Campbell et al. 2015). That more or less all post-Williams proponents of tropes treat their posits as the fundamental constituents of mind-independent--not phenomenal--reality, is however clear (cf. e.g., Heil 2003). After Williams, the second most influential trope theorist is arguably Keith Campbell (1997 [1981]; 1990). Campbell adopted the basics of Williams' (standard) theory and then further developed and defended it. Later proponents of more or less standard versions of the trope view include Peter Simons, John Bacon, Anna-Sofia Maurin, Douglas Ehring, Jonathan Schaffer, Kris McDaniel, Markku Keinanen, Jani Hakkarainen, Marta Ujvari, Daniel Giberman, Robert K. Garcia, and Anthony Fisher (all of whose most central works on the topic are listed in the Bibliography). A very influential paper also arguing for a version of the theory (inspired more by Husserl than by Williams) is "Truth-Makers" (Mulligan et al. 1984; cf. also Denkel 1996). This paper defends the view that tropes are essentially dependent entities, the objects of perception, and the world's basic truthmakers. Proponents of trope theories which posit tropes as one of several fundamental categories include C. B. Martin (1980), John Heil (2003), George Molnar (2003), and Jonathan Lowe (2006). Molnar and Heil both defend ontologies that include (but are not limited to) tropes understood as powers, and Lowe counts tropes as one of four fundamental categories. Even more unorthodox are the views of Mertz (1996, 2016), whose trope-like entities are categorized as a kind of relation.[6] ## 2. The Nature of Tropes According to several trope theorists--perhaps most notably, according to Williams--what exists when a trope does is an abstract particular. The word 'abstract' is ambiguous. On the one hand, Williams tells us, it means "transcending individual existence, as a universal, essence, or Platonic idea is supposed to transcend it" (1953a: 15). On this meaning, to be abstract is to be non-spatiotemporal. Tropes--which are standardly taken to exist in spacetime--are clearly not (or, not necessarily) abstract in this sense (which explains why some philosophers--e.g., Kung 1967; Giberman 2014, 2022; cf. also Simons 1994: 557--insist on referring to the trope as 'concrete'). But then there is this other--according to Williams "aboriginal"--sense of the term. A sense in which, to be 'abstract' is to be "partial, incomplete, or fragmentary, the trait of what is less than its including whole" (Williams 1953a: 15). It is in this sense that the trope is supposed to be abstract. Is saying of the trope that it is abstract in this sense enlightening? Some have worried that it isn't (cf. e.g., Maurin 2002: 21; cf. also Daly 1997). That this worry is unwarranted has recently been pretty convincingly argued by Fisher (2020). When Williams argues that the trope is abstract in the sense of "partial, incomplete, or fragmentary", Fisher notes, what he means is that it "fails to exhaust the content of the region it occupies or is merely part of the content of that region" (Fisher 2020: 45; cf. also Williams 1986 [1959]: 3). One consequence of this, is that the trope can only be attended to via a process of abstraction. (In Campbell's words (1990: 2), it can only be "brought before the mind ... by a process of selection, of systematic setting aside, of these other qualities of which we are aware".) But this, Fisher points out, does not mean that the trope is abstract because it is a product of abstraction (if it did, then since attending to basically anything (short of everything) involves abstracting away from the surrounding environment, saying of the trope that it is 'abstract' would be saying nothing very informative at all). What it means, rather, is that the trope is something that requires abstraction to be brought before the mind because it is abstract (which, in turn, it is because it is such that it does not exhaust the content of the region it occupies or is merely part of the content of that region). This takes care of this worry (for more reasons to view the abstractness of the trope as integral to its nature, see the discussion in section 3.1 below). ### 2.1 Property or Object? In philosophy, new posits are regularly introduced by being compared with, or likened to, an already familiar item. Tropes are no exception to this rule. In fact, tropes have been introduced by being compared with and likened to not one but two distinct but equally familiar kinds of things: properties and objects (Loux 2015). Up until very recently, that tropes can be introduced in both of these ways was considered a feature of the theory, not a source of concern. Tropes, was the idea, can be compared with and likened to both properties and objects, because tropes are a bit of both. Recently, however, both friends and foes of tropes have started to question whether tropes can be a bit of both. At any rate, this will depend on what being a property and being an object amounts to, an issue on which there is no clear consensus. Looking more closely at existing versions of the trope view, whether one thinks of one's posit primarily as a kind of property or as a kind of object certainly influences what one takes to be true (or not) of it. To see this, compare the tropes defended by Williams with those defended by Mulligan et al. Williams seems to belong to the camp of those who view the trope (primarily) as a kind of object. Tropes are that out of which everything else there is, is constructed. As a consequence, names for tropes should not be understood as abbreviated definite descriptions of the kind 'the Ph-ness of x'. Instead, to name a trope should be likened with baptizing a child or with introducing a man "present in the flesh", i.e., ostensively (Williams 1953a: 5). Mulligan et al. (1984), on the other hand, seem to regard the trope more as a kind of property, and as a consequence of this, argue to the contrary that the correct (in fact the only) way to refer to tropes is precisely by way of expressions such as 'the Ph-ness of x'. This, they claim (again pace Williams), is because tropes are essentially of some object, because they are ways the object is (cf. also Heil 2003: 126f). In general, a theory which models its posit primarily on the property hence thinks of it as a dependent sort of entity, as of something else. And a theory which thinks of its tropes more as objects--as 'the alphabet of being'--thinks of them as independent, either in the sense that they need not make up the things they actually make up, or--more radically--in the sense that they need not make up anything at all (that they could be so-called 'free-floaters'). How one views the nature of properties and objects, respectively, also plays a role for some of the criticisms the trope view has had to face. So, for instance, does Jerrold Levinson think that tropes cannot be a kind of property because having a property--being red--amounts to being in a certain condition, where conditions are not particular (Levinson 1980; 2006). The alternative is that tropes are what he calls 'qualities', by which he means something resembling bits of abstract stuff. However, since a theory accepting the existence of bits of abstract stuff would be both "ontologically extravagant and conceptually outlandish" (Levinson 2006: 564), abstract stuff most likely doesn't exist. Which means that, according to Levinson, tropes are neither a kind of property nor a kind of object. A circumstance that makes him conclude that tropes do not exist. Arkadiusz Chrudzimski, next, has argued that, although tropes can be viewed either as a kind of property or as a kind of object, they cannot be viewed as a bit of both (Chrudzimski 2002). Which means that the theory loses its coveted 'middle-position', and with it any advantage it might have had over rival views. For, he argues, to conceptualize the trope as a property--a way things are--means imputing in it a propositional structure (Levinson 1980: 107 holds a similar view). Not so if the trope is understood as a kind of object. But then, although tropes understood as properties are suitable as semantically efficient truthmakers, the same is not true of tropes understood as a kind of object. Conversely, although tropes understood as a kind of object are suitable candidates for being that from which both concrete particulars and abstract universals are constructed, tropes understood as properties are not. Whichever way we conceive of tropes, therefore, the theory's overall appeal is severely diminished. Both Levinson's and Chrudzimski's pessimistic conclusions can be resisted. One option is simply to refuse to accept that one cannot seriously propose that there are "abstract stuffs". Levinson offers us little more than an incredulous stare in defense of this claim, and incredulous stares are well-known for lacking the force to convince those not similarly incredulous. Another option is to reject the claim that tropes understood as properties must be propositionally structured. Or, more specifically, to reject the claim that complex truths need complex--(again) propositionally structured--truthmakers. Some truthmaker theorists--not surprisingly, Mulligan et al. (1984) are among them--reject this claim. In so doing, they avoid having to draw the sorts of conclusions to which Levinson and (perhaps especially) Chrudzimski gesture. A more radical option, finally, is to simply reject the idea that tropes can be informatively categorized either as a kind of property or as a kind of object. Some of the features we want to attribute to tropes seem to cut across those categories anyhow. So, for instance, if you think 'being shareable' is essential to 'being a property', then, obviously, tropes are not properties. Yet tropes, even if not shareable, can still be ways objects are, and they can still essentially depend on the objects that have them. Likewise, if 'monopolizing one's position in space-time' is understood as a central trait for objects, tropes are not objects. Yet tropes can still be the independent building-blocks out of which everything else there is, is constructed. According to Garcia (2015a, 2016), this is why we ought to frame our discussion of the nature(s) of tropes in terms of another distinction. Rather than distinguishing between tropes understood as a kind of property, and tropes understood as a kind of object--and risk getting caught up in infected debates about the nature of objects and properties generally--Garcia suggests we distinguish between tropes understood as 'modifiers' and tropes understood as 'modules'. The main difference between tropes understood in these two ways--a difference that is the source of a great many further differences--is that tropes understood as modifiers do not have the character they confer (on objects), whereas tropes understood as modules do. With recourse to this way of distinguishing between different versions of the trope-view, Garcia argues, we can now evaluate each version separately, independently of how we view objects and properties, respectively.[7] ### 2.2 Complex or Simple? According to most trope-theorists, tropes are ontologically simple. Here this should be taken to mean that tropes have no constituents, in the sense that they are not 'made up' or 'built' from entities belonging to some other category. Simple tropes, thus understood, can still have parts--even necessarily so--as long as those parts are also tropes (cf. e.g., Giberman 2022; Robb 2005: 469).[8],[9] That tropes are ontologically simple arguably provides the trope theorist with a tie-breaker vis-a-vis states of affairs (if states of affairs are understood as substrates instantiating universals). Prima facie a theory positing states of affairs has the same explanatory power as a theory positing tropes. But a theory of states of affairs posits--apart from the state of affairs itself--at least two (fundamental) sorts of things (universals and substrates), whereas a theory of tropes (at least a theory of tropes according to which tropes are simple and objects are bundles of tropes) posits only one (tropes). From the point of view of ontological parsimony, therefore, the trope view ought to be preferred.[10] Can the trope be simple? According to a number of the theory's critics, it cannot. Here is Herbert Hochberg's argument to that effect (2004: 39; cf. also his 2001: 178-179; for versions of the argument cf. also Brownstein 1973: 47; Moreland 2001; Armstrong 2004; Ehring 2011):[11] > > > Let a basic proposition be one that is either atomic or the negation > of an atomic proposition. Then consider tropes t and > t\* where 't is different from > t\' and 't* is exactly similar to > t\' are both true. Assume you take either > 'diversity' or 'identity' as primitive. Then > both propositions are basic propositions. But they are logically > independent. Hence, they cannot have the same truth makers. Yet, > for...trope theory /.../ they do and must have the same > truth makers. Thus the theory fails. > > > A number of different things can be said in response to this argument. It assumes, first, that if tropes are simple, trope theory must violate what appears to be a truly fundamental principle (call it HP, short for 'Hochberg's principle'): that logically independent basic propositions must have distinct truthmakers. Having to reject HP is hence a cost of having simple tropes. Perhaps this cost is acceptable. That it is, seems to be a view held by Mulligan et al. (a similar view is, according to Armstrong 2005: 310 also held by Robb). For, they claim (1984: 296): > > > ...[w]e conceive it as in principle possible that one and the > same truth-maker may make true sentences with different meanings: this > happens anyway if we take non-atomic sentences into account, and no > arguments occur to us which suggest that this cannot happen for atomic > sentences as well. > > > One reason for rejecting HP has been put forward by Fraser MacBride (2004). HP, he notes, is formulated in terms of 'logically* independent basic propositions'. However, HP is only plausible if it takes more than logical (what MacBride calls 'formal') (in)dependence into account: material independent also matters![12] Only if two propositions are logically and materially independent does it follow that they must have distinct truthmakers. But formal and material independence can--and in this case most likely will--come apart. For (ibid: 190): > > > ...[i]nsofar as truth-makers are conceived as inhabitants of the > world, as creatures that exist independently of language, it is far > from evident that logically independent statements in the formal sense > are compelled to correspond to distinct truth-makers. > > > Another argument against the simplicity of tropes comes from Ehring and was inspired by an argument first delivered by Moreland (2001: 64). J. P. Moreland's argument concludes that trope theory is unintelligible. Ehring thinks this is much too strong. This is therefore the formulation of the objection he prefers (2011: 180): > > > The nature and particularity of a trope are intrinsically grounded in > that trope. If tropes are simple, their nature and their particularity > are hence identical. The natures of a red trope and an orange trope > are inexactly similar. Hence their respective particularities should > be inexactly similar as well. However, these particularities are > exactly similar. Hence, their particularities are not > identical to their natures, and tropes are not simple. > > > Here the trope theorist could probably reply that the objection rests on a kind of category mistake: that 'particularities' are quite simply not things amenable to standing in similarity relations. Alternatively, she might just concede that tropes are complex, yet argue that they are so in the innocent sense of having other tropes as parts (one particularity-trope and a distinct nature-trope). Against this, Ehring (2011: 183f) has argued that if the trope has its particularity reside in one of the tropes that make it up, we can always ask about that trope what grounds its particularity and nature respectively. Which would seem to lead us into an infinite--and most probably vicious--regress. Ehring's own solution to the problem is to adopt a version of the trope view (what he calls Natural Class Trope Nominalism) on which tropes do not resemble each other in virtue of their nature, but rather in virtue of belonging to this or that natural class. On this view, what makes two tropes qualitatively the same (their belonging to the same natural class of tropes) is different from what makes them distinct (the tropes themselves, primitively being what they are). Which means that, that tropes can be distinct yet exactly resemble each other is no longer a reason to think that tropes are complex. A different kind of solution has recently been proposed by Giberman (2022, cf. also his 2014). According to him, because at least some of the tropes there are, are spatiotemporally located (they are what he calls 'concrete')[13], they must have size, shape, and duration. That tropes have these different features means that they are capable of multiple resemblances. Which, claims Giberman, means that they are qualitatively complex. Yet that they are does not threaten the trope view in any way. For, according to Giberman, tropes are (qualitatively complex) ostriches. What this means is that tropes are such that they "primitively account for their own multiple resemblance capacities". Which means that "no (further) ontological machinery is required to explain that an ostrich trope has multiple properties" (even though we are admitting that it does!) (Giberman 2022: 18). In this sense, the trope is like the Quinean concrete particular (Giberman ibid.; cf. also Quine 1948, Devitt 1980): > > > When the Quinean is asked what it is about a given electron that > metaphysically explains its ability to resemble both massive things > and charged things, he will likely answer 'nothing!--it > just is that way'... Similarly, when asked what it is about the > n charge trope that metaphysically explains its ability to > resemble both charged things and things of a certain size, the ostrich > trope theorist answers: 'the trope itself--no more > structure to it than that'. > > > But then, on Giberman's view--and pace Ehring--we are not allowed to ask of the properties of the trope what grounds their particularity and nature (that's primitive!). And if asking this is not allowed, is the idea, no infinite regress can be generated (cf. also fn 20, this entry). What about the advantage the trope view was supposed to have over the state-of-affairs view? Primitive or not, if the complexity is there, doesn't this mean that those views are now on a par parsimony-wise? Giberman doesn't think so, and gives two reasons why not: (1) although the ostrich trope is qualitatively complex, it is not--unlike the state of affairs--categorially complex (no trope has a constituent from outside the category trope) (2022: 16); (2) (more contentiously) even states of affairs, in being spatiotemporal, need to have size and shape etc. But then, apart from having to accept the existence of categorially distinct substrates, the state of affairs theorist must also accept that universals--like tropes--are primitively qualitatively complex (ibid: 18). ### 2.3 Trope Individuation What makes two tropes, existing in the same world at the same time, distinct?[14] A natural suggestion is that we take the way we normally identify and refer to tropes very literally and individuate tropes with reference to the objects that 'have' them: Object Individuation (OI): For any tropes a and b such that a exactly resembles b, a [?] b iff a belongs to an object that is distinct from the object to which b belongs. No trope theorist endorses this natural suggestion, however. The reason why not is that, at least if objects are bundles of tropes, the individuation of objects will depend on the individuation of the tropes that make them up, which means that, on OI, individuation becomes circular (Lowe 1998: 206f.; Schaffer 2001: 249; Ehring 2011: 77). Indeed, matters improve only marginally if objects are understood as substrates in which tropes are instantiated. For, although on this view, OI does not force one to accept a circular account of individuation, this is because it is now the substrate which carries the individuating burden. This leaves the individuation of the substrate still unaccounted for, and so we appear to have gotten nowhere (Mertz 2001). Any trope theorist who accepts the possible existence of 'free-floating' tropes--i.e., tropes that exist unattached to any object--must in any case reject this account of trope individuation (at least as long as she accepts the possibility of there being more than one free-floating trope at any given time). The main-contenders are instead spatiotemporal individuation (SI) and primitivist individuation (PI). According to SI, first, that two tropes belonging to the same world are distinct can be metaphysically explained with reference to a difference in their respective spatiotemporal position (Campbell 1990: 53f.; Lowe 1998: 207; Schaffer 2001: 249; Giberman 2022: 18): Spatiotemporal Individuation (SI): For any tropes a and b such that a exactly resemble b, a [?] b iff a is at non-zero distance from b. This is an account of trope individuation that seems to respect the way tropes are normally picked out, yet which does not--circularly--individuate tropes with reference to the objects they make up and which does not rule out the existence of 'free-floaters'. In spite of this, the majority of the trope theorists (Schaffer 2001 being one important exception) have opted instead for primitivism (cf. e.g., Ehring 2011: 76; Campbell 1990: 69; Keinanen & Hakkarainen 2014). Primitivism is best understood as the denial of the idea that there is any true and informative way of filling out the biconditional 'For any exactly resembling tropes a and b, a [?] b iff ...'. That a and b are distinct--if they are--is hence primitive. It has no further (ontological) analysis or (metaphysical) explanation. According to what is probably the most influential argument in favor of PI over SI (an argument that changed Campbell's mind: cf. his 1990: 55f.; cf. also Moreland 1985: 65), SI should be abandoned because it rules out the (non-empty) possibility that (parts of) reality could be non-spatiotemporal.[15] Against this, proponents of SI have argued that the thesis that reality must be spatiotemporal can be independently justified (primarily because naturalism can be independently justified, cf. Schaffer 2001: 251). And even if it cannot, SI could easily be modified to accommodate the analogue of the locational order of space (Campbell 1997 [1981]: 136; Schaffer ibid.).[16] A common argument in favor of SI is that it allows its proponents to rule out what most agree are empty possibilities: swapping and piling. Swapping: According to the so-called 'swapping argument' (first formulated in Armstrong 1989: 131-132; cf. also Schaffer 2001: 250f; Ehring 2011: 78f.),[17], [18] if properties are tropes, and individuation is primitive, two distinct yet exactly similar tropes might swap places (this redness here might have been there, and vice versa). The result, post-swap, is a situation that is ontologically distinct from that pre-swap. However, empirically/causally the pre- and post-swap situations are the same (cf. LaBossiere 1993: 262 and Denkel 1996: 173f. for reasons to doubt that they are the same). That is, given the natural laws as we know them, that this red-trope here swaps places with that red-trope there makes no difference to the future evolution of things. Which means that, not only would the world look, feel and smell exactly the same to us pre- and post-swap, it would be in principle impossible to construct a device able to distinguish the two situations from one another. For any device able to detect the (primitive) difference between the two situations would also have to be a device able to communicate this difference (by making a sound, by turning a handle, etc...) yet this is precisely what the fact that swapping makes no difference to the future evolution of things prevents (cf. Dasgupta 2009). This makes admitting the possibility of swapping seem unnecessary. If we also accept the (arguably reasonable) Eleatic principle according to which only changes that matter empirically/causally should count as genuine, we can draw the even stronger conclusion that swapping is not genuinely possible, and, hence, that any account of individuation from which it follows that it is, should be abandoned.[19] To accept SI does not immediately block swapping (Schaffer 2001: 250). For, SI (just like OI and PI) is a principle about trope individuation that holds intra-worldly. In this case: within any given world, no two exactly similar tropes are at zero distance from each other. Swapping, on the other hand, concerns what is possibly true (or not) of exactly similar tropes considered inter-worldly. But this means that, although SI does not declare swapping possible, it doesn't rule it out either. According to the proponent of SI, this is actually a good thing. For there is one possibility that it would be unfortunate if one's principle of individuation did block, namely the possibility--called sliding--that this red-trope here could have been there had the wind blown differently (Schaffer 2001: 251). To get the desired result (i.e., to block swapping while allowing for sliding), Schaffer suggests we combine trope theory with SI and a Lewisian counterpart theory of transworld identity (Lewis 1986). The result is an account according to which exactly resembling tropes are intra-worldly identical if they inhabit the same position in space-time. And according to which they are inter-worldly counterparts, if they are distinct, yet stand in sufficiently similar distance- and other types of relations to their respective (intra-worldly) neighbors. With this addition in place, Schaffer claims, a trope theory which individuates its posits with reference to their spatiotemporal position will make room for the possibility of sliding, because (2001: 253): > > > On the counterfactual supposition of a shift in wind, what results is > a redness exactly like the actual one, which is in perfectly > isomorphic resemblance relations to its worldmates as the actual one > is to its worldmates, with just a slight difference in distance with > respect to, e.g., the roundness of the moon. > > > ...yet it won't allow for the possibility of swapping, because: > > > ...the nearest relative of the redness of the rose which is > here at our world would be the redness still here > 'post-swap'. The redness which would be here has exactly > the same inter- and intraworld resemblance relations as the redness > which actually is here, and the same distance relations, and hence is > a better counterpart than the redness which would be > there. > > > This is not necessarily a reason to prefer SI over PI, however. For, PI, just like SI, is an inter-worldly principle of individuation, which means that it, just like SI, could be combined with a Lewisian counterpart theory, thereby preventing swapping yet making room for sliding. It is, in other words, the counterpart theory, and not SI (or PI), which does all the work. In any case, it is not clear that intra-worldly swapping is an empty possibility. According to Ehring, there are circumstances in which a series of slidings constitute one case of swapping, something that he thinks would make swapping more a reason for than against PI (Ehring 2011: 81-85). Piling: Even if swapping does not give us a reason to prefer SI over PI, perhaps its close cousin 'piling' does. Consider a particular red rose. Given trope theory, this rose is red because it is partly constituted by a redness-trope. But what is to prevent more than one--even indefinitely many--exactly similar red-tropes from partly constituting this rose? Given PI: nothing. It is however far from clear how one could empirically detect that the rose has more than one redness trope, just like it is not clear how one could empirically detect how many redness tropes it has, provided it has more than one. This is primarily because it is far from clear how having more than one redness trope could make a causal difference in the world. But if piling makes no empirical/causal difference, then given a (plausible) Eleatic principle, the possibility of piling is empty, which means that PI ought to be rejected (Armstrong 1978: 86; cf. also Simons 1994: 558; Schaffer 2001: 254, fn. 11). In defense of PI, its proponents now point to a special case of piling, called 'pyramiding' (an example being a 5 kg object consisting of five 1 kg tropes). Pyramiding does seem genuinely possible. Yet, if piling is ruled out, so is pyramiding (Ehring 2011: 87ff.; cf. also Armstrong 1997: 64f.; Daly 1997: 155). According to Schaffer, this is fine. For, although admittedly not quite as objectionable as other types of piling (which he calls 'stacking'), pyramiding faces a serious problem with predication: if admitted, it will be true of the 5 kg object that "[i]t has the property of weighing 1 kg" (Schaffer 2001: 254). Against this, Ehring has pointed out that to say of the 5 kg object that "[i]t has the property of weighing 1 kg" is at most pragmatically odd, and that, even if this oddness is regarded as unacceptable, to avoid it would not require the considerable complication of one's theory of predication imagined by Schaffer (Ehring 2011: 88-91). According to Schaffer, the best argument for the possibility of piling--hence the best argument against SI--is rather provided by the existence of so-called bosons (photons being one example). Bosons are entities which do not obey Pauli's Exclusion Principle, and hence such that two or more bosons can occupy the same quantum state. A 'one-high' boson-pile is hence empirically distinguishable from a 'two-high' one, which means that the possibility of piling in general is not ruled out even if we accept an Eleatic principle. Schaffer (2001: 255) suggests we solve this problem for SI by considering the wave--not the particle/boson--as the way the object 'really' is. But this solution comes with complications of its own for the proponent of SI. For, "[t]he wave function lives in configuration space rather than physical space, and the ontology of the wave function, its relation to physical space, and its relation to the relativistic conception of spacetime which SI so naturally fits remain deeply mysterious" (Schaffer 2001: 256).[20] ## 3. Tropes as Building Blocks As we have seen, tropes can be conceptualized, not just as particularized ways things are, but also--and on some versions of trope theory, primarily--as that out of which everything else there is, is constructed. Minimally, this means that tropes must fulfill at least two constructive tasks: that of making up (the equivalent of) the realist's universal, and that of making up (the equivalent of) the nominalist's concrete particular. ### 3.1 Tropes and Universals How can distinct things have things--their properties--in common? This is the problem of 'the One over Many' (cf. e.g., Rodriguez-Pereyra 2000; Maurin 2022: sect. 2). Universals provide a straightforward solution to this problem: distinct things can have things in common, because there is a type of entity--the universal--capable of existing in (and hence characterizing) more than one object at once. The trope theorist--at least if she does not accept the existence of universals in addition to tropes[21]--does not have recourse to entities that can be likewise identical in distinct instances. She must therefore come up with an alternative solution. Here I'll consider three. The Resemblance Class Theory: The most commonly accepted trope-solution to the problem of the One over Many takes two objects a and b to 'share' a property--F-ness-- if at least one of of the tropes that make up a belongs to the same (exact) resemblance class as at least one of the (numerically distinct) tropes that make up b (cf. Williams 1953a: 10; Campbell 1990: 31f.; cf. also Lewis 1986: 64f.). Exact resemblance is an equivalence relation (although cf. Mormann 1995 for an alternative view), which means that it is a symmetrical, reflexive, and transitive relation. Because it is an equivalence relation, exact resemblance partitions the set of tropes into mutually excluding and non-overlapping classes. Exact resemblance classes of tropes, thus understood, function more or less as the traditional universal does. Which is why proponents of this view think the problem can be solved with reference to them. A point of contention among those who hold this view is if accepting it mean having to accept the existence of resemblance relations. Those who think it doesn't, point out that resemblance is an internal relation which supervenes on whatever it relates: that the (degree of) resemblance between distinct tropes is entailed simply given their existence.[22] Assuming that our ontology is 'sparse' (a thought many believe can be independently justified, cf. e.g., Schaffer 2004 and Armstrong 1978), only what is minimally required to make true all truths exists. Which means that, if resemblance is an internal relation, what exists when distinct tropes resemble each other--and so what plays the role of the realist's universal--is nothing but the resembling tropes themselves (Williams 1963: 608; Campbell 1990: 37f.; cf. also Armstrong 1989: 56). Does it follow from the fact that exact resemblance must obtain given the existence of its relata, that it is no ontological addition to them? Some philosophers (Daly 1997: 152 is among them) do not think so. But if it doesn't follow, some have argued, the trope theorist must combat a trope theoretical version of what has become known as 'Russell's regress'. Russell's regress (perhaps better: Russell's regress argument) is an argument to the effect that, if some particulars (Russell was thinking of concrete particulars, not of tropes) (exactly) resemble each other, then either the relation of resemblance exists and is a universal, or we end up in (vicious) infinite regress (Russell 1997 [1912]: 48; cf. also Kung 1967). Chris Daly (1997: 149) provides us with a trope-theoretical version of this argument:[23] > > > Consider three concrete particulars which are the same shade of > red...each of these concrete particulars has a red > trope--call these tropes F, G, and > H--and these concrete particulars exactly resemble each > other in colour because F, G, and H exactly > resemble each other in colour. But it seems that this account is > incomplete. It seems that the account should further claim that > resemblance tropes hold between F, G, and > H. That is, it seems that there are resemblance tropes > holding between the members of the pairs F and G, > G and H, and F and H... Let > us call the resemblance tropes in question R1, > R2, and R3...each of > these resemblance tropes in turn exactly resemble each other. > Therefore, certain resemblance tropes hold between these > tropes...we are launched on a regress. > > > This regress is a problem only if it is vicious. The most convincing reason for thinking that it is not is provided if we consider the 'pattern of dependence' it instantiates.[24] For, as we have seen, even those who do not think that the internality of exact resemblance makes it a mere 'pseudo-addition' to its subvenient base, agree that resemblance, whatever it is, is such that its existence is necessarily incurred simply given the existence of its relata. But then, no matter how many resemblances we regressively generate, ultimately they all depend for their existence on the existence of the resembling tropes, which resemble each other because of their individual nature, which is primitive. This means that the existence of the regress in no way contradicts--it does not function as a reductio against--the resemblance of the original tropes. On the contrary; it is because the tropes resemble each other, that the regress exists. Therefore, the regress is benign (cf. e.g., Campbell 1990: 37; Maurin 2013). This response only works if the nature of individual tropes--their being what they are--is primitive and not further analyzable (i.e., it only works if we assume a standard view of the nature of tropes). To see this, compare the standard view with a view with which it is often confused: resemblance nominalism. On a version of the trope view according to which universals are resemblance classes of tropes, tropes have the same nature if they resemble each other. Yet, importantly, they resemble each other (or not) in virtue of the (primitive) nature they each 'have' (or 'are'). Resemblance nominalism, on the other hand, is the view that two objects have the natures they do in virtue of the resemblance relations which obtain between them. This means that, whether they resemble or not, is not decided given the existence and nature of the objects themselves. Rather, the pattern of dependence is the other way around. And this--arguably--makes the regress vicious. Perhaps for that reason, resemblance nominalism has no explicit proponent among the trope theorists. The Natural Class Theory: Those not convinced that 'Daly's regress' is benign might prefer a view (first defended by Ehring 2011: 175f) according to which the trope is not what it is either primitively or because of whatever resemblance relations it stands in to other tropes, but because of the natural class to which it belongs. Accepting this view provides us with a new solution to the problem of the One over Many, one that does not depend on the existence (or not) of exact resemblance relations. On this view, more precisely, two objects a and b 'share' a property--F-ness-- if at least one of the tropes that make up a belongs to the same natural class as at least one of the tropes that make up b. The problem with this alternative is that it appears to--implausibly--turn explanation on its head. If accepted, tropes do not belong to this or that class because of their nature. Rather, tropes have the natures they do because they belong to this or that class. The Trope-Kind Theory: Although he in his (1953a) briefly 'dallied' (his word) with the view that, to be a universal is to be a resemblance class of tropes, Williams soon thereafter changed his mind (cf. esp. his 1986 [1959] and 1963).[25] According to the view he adopted instead of the resemblance view, universals are neither "made" nor "discovered" but are something we acknowledge "by a relaxation of the identity conditions of thought and language" (1986 [1959]: 8; cf. also Campbell 1990: 44). If the F-ness of a and the F-ness of b is counted in a way that is subject to the rule that anything indiscernible is identical, their sameness is explained with reference to the universal they share. And if the F-ness of a and the F-ness of b is counted in a way that isn't subject to that rule, their sameness is explained with reference to their individual, distinct, yet exactly resembling tropes. Importantly, Williams thinks of this as a kind of realism--an immanent realism--about universals (Fisher 2017: 346; cf. also Fisher 2018 and Heil 2012: 102f.). It takes universals to be "present in, and in fact components of, their instances" (Williams 1986 [1959]: 10). Yet, as noted by Fisher (2017: 346) tropes are still the primitive elements of being. Universals are real in virtue of being mind-independent. Yet, universals aren't fundamental (Fisher: ibid.): > > > Their reality is determined by mind-independent facts about tropes. > Tropes manifest universals in the sense that universals are nothing > over and above property instances as tropes are by their nature of > kinds. > > > The trope-kind theory offers an interesting and so far surprisingly little discussed solution to the problem of the One over Many. So far, it is a view with few if any explicit contemporary proponents (though cf. Paul 2002, 2017 and van Inwagen 2017: 348f.[26]). It is not unlikely that this state of affairs is now about to change. ### 3.2 Tropes and Concrete Particulars The second constructive task facing the trope theorist is that of building something that behaves like a concrete particular, using only tropes. Exactly how a concrete particular behaves is of course a matter that can be debated. This is not a debate to which the trope theorist has had very much--or at least not anything very original--to contribute. Instead, the trope theoretical discussion has been focused on an issue that needs solving before questions concerning what a concrete particular can or cannot do more precisely become relevant: the issue of if and how tropes make up concrete particulars in the first place. Whether this issue is best approached by considering if and how tropes can make up or ground the existence of what me might call 'ordinary' or 'everyday' objects, or if it is better to concentrate instead on the world's simplest, most fundamental, objects--like those you find discussed in e.g., fundamental physics--is another issue on which trope theorists disagree. Campbell thinks we should concentrate on the latter sort of object. In particular, he thinks we should concentrate on objects that have no other objects as parts, as that way we avoid confusing 'substantial' complexity (and unity) with the--here relevant--qualitative one. David Robb (2005) and Kris McDaniel (2001) disagree. This may in part be due to the fact that they both (cf. also Paul 2002, 2017) think that objects are mereologically composed both on the level of their substantial parts and on the level of their qualitative--trope--parts.[27] According to a majority of the trope theorists, objects are bundles of tropes. The alternative is to understand objects as complexes consisting of a substrate in which tropes are instantiated (a view defended by e.g., Martin 1980; Heil 2003; and Lowe 2006). Indeed, according to D. M. Armstrong (1989, 2004)--a staunch but comparatively speaking rather friendly trope-critic--this minority view ought to be accepted by the trope theorist. Armstrong is most likely wrong, however: several reasons exist for why one ought to prefer the bundle view (Maurin 2016). One such reason has to do with parsimony. If you adopt a substrate-attribute view, you accept the existence of substrates on top of the existence of tropes. Accepting this additional category of entity makes at least some sense if properties are universals. For if objects are bundles of universals and universals are entities which are numerically identical across instances, then if object a is qualitatively identical to object b, a is also numerically identical to b. Which means that, if objects are bundles of universals, the Identity of Indiscernibles is not just true, but necessarily true, a consequence few universal realists have been prepared to accept (indeed, a consequence that may turn out to be unacceptable as a matter of empirical fact--cf. e.g., French 1988). If objects are substrates exemplifying universals, on the other hand, although a and b are qualitatively identical, they are nevertheless distinct. They are distinct, that is, in virtue of being partly constituted by (primitively) distinct substrates. This sort of argument in favor of the substrate-attribute view cannot be recreated for a view on which properties are tropes, however. For, if objects are bundles of tropes, because tropes are particulars not universals, even if a and b are qualitatively identical, they are not numerically identical. And they are not numerically identical because they are constituted by numerically distinct tropes. No need for substrates! According to the bundle view, objects consist of, are made up by, or are grounded in, a sufficient number of mutually compresent and/or in some other way mutually dependent tropes (cf. e.g., McDaniel 2001, 2006 and Giberman 2014 for slightly different takes on bundling). What is compresence? When the same question was asked about (exact) resemblance, the trope theorist had the option of treating the relation as a 'pseudo-addition'. This was because resemblance is an internal relation and so holds necessarily simply given the existence of its relata. According to most trope-theorists, however, compresence is an external relation, and hence a real addition to the tropes it relates.[28] But then adding compresence gives rise to an infinite regress (often called 'Bradley's regress' after Bradley 1930 [1893]; cf. also Armstrong 1978; Vallicella 2002 and 2005; Schnieder 2004; Cameron 2008, 2022; Maurin 2012). Unlike what was true in the case of resemblance, this regress is most likely a vicious regress. This is because the 'pattern of dependence' it instantiates is the opposite of that instantiated by the resemblance regress. In the resemblance case, for tropes t1, t2, and t3 to exactly resemble each other, it is enough that they exist. Not so in the compresence case. Tropes t1, t2, and t3 could exist and not be compresent, which means that in order to ensure that they are compresent, a compresence-trope, c1, must be added to the bundle. But c1 could exist without being compresent with those very tropes. Therefore, in order for t1, t2, t3 and c1 to be compresent, there must be something--call it c2--that makes them so. But since c2 could exist and not be compresent with t1, t2, t3 and c1, it too needs something that ensures its compresence with those entities. Enter c3. And so on. The existence of this regress arguably contradicts--and hence functions as a reductio against--the compresence of the original (first-order) tropes and, thereby, the (possible) existence of the concrete particular. Since concrete particulars (possibly) exist, something must be wrong with this argument. One option is to claim that compresence is internal after all, in which case the regress (if there even is one) is benign (Molnar 2003; Heil 2003 and 2012; cf. also Armstrong 2006). This may seem attractive especially to those who think of their tropes as non-transferable and as ways things are. Even given this way of thinking of the nature of the trope, however, to take compresence as internal means having to give up what are arguably some deeply held modal beliefs. For even if you have reason to think that properties must be 'borne' by some object, to be able to solve the regress-problem one would have to accept the much stronger thesis that every trope must be borne by a specific object. If the only reason we have for thinking that compresence is internal in this sense is that this solves the problem with Bradley's regress, therefore, we should opt to go down this route as a last resort only (cf. Cameron 2006; Maurin 2010). As a way of saving at least some of our modal intuitions while still avoiding Bradley's regress, Simons (1994; cf. also Keinanen 2011 and Keinanen and Hakkarainen 2014 for a slightly different version of this view[29]) suggests we view the concrete particular as constituted partly by a 'nucleus' (made up from mutually and specifically dependent tropes) and partly--at least in the normal case--by a 'halo' (made up from tropes that depend specifically on the tropes in the nucleus). The result is a structured bundle such that, although the tropes in the nucleus at most depend for their existence on the existence of tropes of the same kind as those now in its halo, they do not depend specifically on those tropes. In this way, at least some room is made for contingency, yet Bradley's regress is avoided. For, as the tropes in the halo depend specifically for their existence on the tropes that make up the nucleus, their existence is enough to guarantee the existence of the whole to which they belong. This is better but perhaps not good enough. For, although the same object could now have had a slightly different halo, the possibility that the tropes that actually make up the halo could exist and not be joined to this particular nucleus is ruled out with no apparent justification (other than that this helps its proponent solve the problem with the Bradley regress) (cf. also Garcia 2014 for more kinds of criticism of this view). According to several between themselves very different sorts of trope theorists, finally, we should stop bothering with the (nature and dependence of the) related tropes and investigate instead the (special) nature of compresence itself. This seems intuitive enough. After all, is it not the business of a relation to relate? According to one suggestion along these lines (defended in Simons 2010; Maurin 2002, 2010 and 2011; and Wieland and Betti 2008; cf. also Mertz 1996, Robb 2005 and Giberman 2014 for similar views), non-relational tropes have an existence that is independent of the existence of some specific--either non-relational or relational--trope, but relational tropes (including compresence) depend specifically for their existence on the very tropes they relate. This means that if c1 exists, it must relate the tropes it in fact relates, even though those tropes might very well exist and not be compresent (at least not with each other). There is, then, no regress, and except for c1, the tropes involved in constituting the concrete particular could exist without being compresent with each other. And this, in turn, means that our modal intuitions are left more or less intact.[30] According to Mertz, moreover, to be able to do the unifying work for which it is introduced, compresence cannot be a universal. If it were, then if one of the concrete particulars whose constituents it joins ceases to exist, so will every other concrete particular unified by the same (universal) relation of compresence. But, as Mertz points out, "this is absurdly counterfactual!" (Mertz 1996: 190). Nor can it be a state of affairs. For, states of affairs are in themselves complexes, and so could not be used to solve the Bradley problem.[31] It seems, then, that compresence, if understood in a way that blocks the regress, is a trope. Assuming that Bradley's regress threatens any account according to which many things make up one unified thing (i.e., assuming that it does not only threaten the trope-bundle theorist), that there is this threat may therefore turn out to be a reason in favor of positing tropes (Maurin 2011). The suggestion is not without its critics. To these belong MacBride who argues that, "...to call a trope relational is to pack into its essence the relating function it is supposed to perform without explaining what Bradley's regress calls into question, viz. the capacity of relations to relate" (2011: 173). Rather than solve the problem, in other words, MacBride thinks the suggestion "transfers our original puzzlement to that thing [i.e., the compresence-relation]". For, he asks "how can positing the existence of a relational trope explain anything about its capacity to relate when it has been stipulated to be the very essence of R that it relates a and b. It is as though the capacity of relational tropes to relate is explained by mentioning the fact that they have a 'virtus relativa'"(ibid.). Assuming we agree that there is something that needs explaining (i.e., assuming we agree that how several tropes can--contingently--make up one object needs explaining), we can either reject a proposed solution because we prefer what we think is a better solution, or we can reject it because it is in itself bad or unacceptable (irrespective of whether there are any alternative solutions on offer). MacBride appears to suggest we do the latter. More precisely, what MacBride proposes is that the solution fails because it leaves unexplained the special 'power' to relate it attributes to the compresence trope. If this is why the suggestion fails, however, then either this is because no explanation that posits something ('primitively') apt to perform whatever function we need explained, is acceptable, or it is because in this particular case, an explanation of this kind will not do. If the former, the objection risks leading to an overgeneration of explanatory failures. Everyone will at some point need to posit some things as fundamental. And in order for those fundamental posits to be able to contribute somehow to the theory in question, it seems we must be allowed to say something about them. We must, to use the terminology introduced by Schaffer, outfit our fundamental posits with axioms. But then, as Schaffer also points out (2016: 587): "it is a bad question--albeit one that has tempted excellent philosophers from Bradley through van Fraassen and Lewis--to ask how a posit can do what its axioms say, for that work is simply the business of the posit. End of story". If, on the other hand, the problem is isolated to the case at hand, we are owed an explanation of what makes this case so special. MacBride complains that if the 'explanatory task' is that of accounting for the capacity of compresence to relate, being told that compresence has that capacity 'by nature', will not do. Perhaps he is right about this. But, then, the explanatory task is arguably not that one, but rather the task of accounting for the possible existence of concrete objects (contingently) made up from tropes. If this is the explanatory task, it is far from clear why positing a special kind of (relational) trope that is 'by nature' apt to perform its relating function, will not do as an explanation. ## 4. Trope Applications According to the trope proponent, if you accept the existence of tropes, you have the means available to solve or to dissolve a number of serious problems, not just in metaphysics but in philosophy generally. In what follows, the most common trope-applications proposed in the literature are very briefly introduced. ### 4.1 Tropes in Causation and Persistence According to a majority of the trope theorists, an important reason for thinking tropes exist is the role they play in causation. It is after all not the whole stove that burns you, it is its temperature that does the damage. And it is not any temperature, nor temperature in general, which leaves a red mark. That mark is left by the particular temperature had by this particular stove now. It makes sense, therefore, to say that the mark is left by the stove's temperature-trope, which means that tropes are very good candidates for being the world's basic causal relata (Williams 1953a; Campbell 1990; Denkel 1996; Molnar 2003; Heil 2003; ; Garcia-Encinas 2009; Ehring 2011). That tropes can play a role in causation can hardly be doubted. But can this role also provide the trope-proponent with a reason to think that tropes exist? According to the theory's critics, it cannot. The role tropes (can) play in causation does not provide the trope proponent with any special reason to prefer an ontology of tropes over alternative ontologies. More specifically, it does not give her any special reason to prefer an ontology of tropes over one of states of affairs or events. Just like tropes, state of affairs and events are particular. Just like tropes, they are localized. And, just like tropes, they are non-repeatable (although at least the state of affairs contains a repeatable item--the universal--as one of its constituents). Every reason for thinking that tropes are the world's basic causal relata is therefore also a reason to think that this role is played by states of affairs and/or events.[32] Ehring disagrees. To see why, he asks us to consider the following simple scenario: a property-instance at t1 is causally responsible for an instance of the same property at t2. This is a case of causation which is also a case of property persistence. But what does property persistence involve? According to Ehring, property persistence is not just a matter of something not changing its properties. For, even in cases where nothing discernibly changes, the property instantiated at t1 could nevertheless have been replaced by another property of the same type during the period between t1 and t2. To be able to ontologically explain the scenario, therefore, we first need an account of property persistence able to distinguish 'true' property persistence from cases of 'non-salient property change' or what may also be called property type persistence. But, Ehring claims, this is something a theory according to which property instances are states of affairs cannot do (this he demonstrates with the help of a number of thought experiments, which space does not allow me to reproduce here, but cf. Ehring 1997: 91ff). Therefore, causation gives us reason to think that tropes exist. Ehring is not the only one who regards the relationship between (theories of) persistence and tropes as an intimate one. According to McDaniel (2001)--who defends a theory (TOPO) according to which ordinary physical objects are mereological fusions of monadic and polyadic tropes--adopting (his version of) the trope view can be used to argue for one particular theory of persistence: 3-dimensionalism.[33] And according to Benovsky (2013), because (non-presentist) endurantism is incompatible with the view that properties are (immanent) universals, the endurantist must embrace trope theory.[34] According to Garcia (2016), finally, what role tropes can play in causation will depend on how we conceive of the nature of tropes. If tropes are what he calls 'modifiers', they do not have the character they confer, a fact that would seem to make them less suitable as causal relata. Not so if tropes are of the module kind (and so have the character they confer). But if tropes have the character they confer, Garcia points out, we may always ask, e.g.: Is it the couch or is it the couch's couch-shaped mass-trope that causes the indentation in the carpet? Garcia thinks we have reason to think they both do. The couch causes the indentation by courtesy, but the mass trope would have sufficed to cause it even if it had existed alone, unbundled with the couch's other tropes. But this suggests that if tropes are of the module kind, we end up with a world that is (objectionably) systematically causally overdetermined. The role tropes play in causation may therefore be more problematic than what it might initially seem (though cf. Giberman 2022 for an objection to Garcia's argument). ### 4.2 Tropes and Issues in the Philosophy of Mind Suppose Lisa burns herself on the hot stove. One of the causal transactions that then follow can be described thus: Lisa removed her hand from the stove because she felt pain. This is a description which seems to pick out 'being in pain' as one causally relevant property of the cause. That 'being in pain' is a causally relevant property accords well with our intuitions. However, to say it is leads to trouble. The reason for this is that mental properties, like that of 'being in pain', can be realized by physically very different systems. Therefore, mental properties cannot be identified with physical ones. On the other hand, we seem to live in a physically closed and causally non-overdetermined universe. But this means that, contrary to what we have supposed so far, Lisa did not remove her hand because she felt pain. In general, it means that mental properties are not causally relevant, however much they seem to be (cf. Kim 1989 for a famous expression of this problem). If properties are tropes, some trope theorists have proposed, this conclusion can be resisted (cf. Robb 1997; Martin and Heil 1999; Heil and Robb 2003; for a hybrid version cf. Nanay 2009; cf. also Gozzano and Orilia 2008). To see this, we need first to disambiguate our notion of a property. This notion, it is argued, is really two notions, namely: * Property1 = that which imparts on an individual thing its particular nature (property as token), and * Property2 = that which makes distinct things the same (property as type). Once 'property' has been disambiguated, we can see how mental properties can be causally relevant after all. For now, if mental properties1 are tropes, they can be identified with physical properties1. Mental properties2 can still be distinguished from physical properties2, for properties considered as types are--in line with the standard view of tropes--identified with similarity classes of tropes. When Lisa removes her hand from the stove because she feels pain, therefore, she removes her hand in virtue of something that is partly characterized by a trope which is such that it belongs to a class of mentally similar tropes. This trope is identical with a physical trope--it is both mental and physical--because it also belongs to a (distinct) similarity class of physically similar tropes. Therefore, mental properties can be causally relevant in spite of the fact that the mental is multiply realizable by the physical, and in spite of the fact that we live in a physically closed and non-overdetermined universe. This suggestion has been criticized. According to Paul Noordhof (1998: 223) it fails because it does not respect the "bulge in the carpet constraint". For now the question which was ambiguously asked about properties, can be unambiguously asked about tropes: is it in virtue of being mental or in virtue of being physical that the trope is causally relevant for the effect (for a response, cf. Robb 2001 and Ehring 2003)? And Sophie Gibb (2004) has complained that the trope's simple and primitive nature makes it unsuitable for membership in two such radically different classes as that of the mentally and of the physically similar tropes, respectively (for more reasons against the suggestion cf. Macdonald and Macdonald 2006 and Zhang 2021). ### 4.3 Tropes and Perception Another important reason for thinking that tropes exist, it has been proposed, is the role tropes play in perception. That what we perceive are the qualities of the things rather than the things themselves, first, seems plausible (for various claims to this effect, cf. Williams 1953a: 16; Campbell 1997 [1981]: 130; Schaffer 2001: 247; cf. also Nanay 2012 and Almang 2013). And that the qualities we perceive are tropes rather than universals or instantiations of universals (states of affairs) is, according to Lowe, a matter that can be determined with reference to our experience (cf. e.g., Skrzypulec 2021 for an argument to the contrary). Lowe argues (1998: 205; cf. also, Lowe 2008; Mulligan 1999): > > > [W]hen I see the leaf change in colour--perhaps as it > turned brown by a flame--I seem to see something cease to > exist in the location of the leaf, namely, its greenness. But it > could not be the universal greenness which ceases to exist, > at least so long as other green things continue to exist. My opponent > must say that really what I see is not something ceasing to exist, but > merely the leaf's ceasing to instantiate greenness, or greenness > ceasing to be 'wholly present' just here. I can only say > that that suggestion strikes me as being quite false to the > phenomenology of perception. The objects of perception seem, one and > all, to be particulars--and, indeed, a causal theory of > perception (which I myself favour) would appear to require this, since > particulars alone seem capable of entering into causal relations. > > > A similar view is put forth by Mulligan et al. They argue (1984: 306): > > > [W]hoever wishes to reject moments [i.e., tropes] must of course give > an account of those cases where we seem to see and hear them, cases we > report using definite descriptions such as 'the smile that just > appeared on Rupert's face'. This means that he must claim > that in such circumstances we see not just independent things per > se, but also things as falling under certain concepts or > as exemplifying certain universals. On some > accounts...it is even claimed that we see the universal in the > thing. But the friend of moments finds this counterintuitive. When we > see Rupert's smile, we see something just as spatio-temporal as > Rupert himself, and not something as absurd as a spatio-temporal > entity that somehow contains a concept or a universal. > > > These are admittedly not very strong reasons for thinking that it is tropes and not state of affairs that are the objects of perception. For the view that our perception of a trope is not only distinct, but also phenomenologically distinguishable, from our perception of a state of affairs seems grounded in little more than its proponent's introspective intuitions. States of affairs, just like tropes, are particulars (cf. Armstrong 1997: 126 on the "victory of particularity"). And to say, as Mulligan et al. do, that the very idea of something spatiotemporal containing a universal is absurd, clearly begs the question against the view they are opposing. ### 4.4 Tropes and Semantics That language furnishes the trope theorist with solid reasons for thinking that there are tropes has been indicated by several trope theorists and it has also been forcefully argued, especially by Friederike Moltmann (2003, 2007, 2009, 2013a and 2013b; cf. also Mertz 1996: 3-6). Taking (Mulligan et al. 1984) as her point of departure, Moltmann argues that natural language contains several phenomena whose semantic treatment is best spelled out in terms of an ontology that includes tropes. Nominalizations, first, may seem to point in the opposite direction. For, in the classical discussion, the nominalization of predicates such as is wise into nouns fit to refer, has been taken to count in favor of universal realism. A sub-class of nominalizations--such as John's wisdom--can, however, be taken to speak in favor of the existence of tropes. This is a kind of nominalization which, as Moltmann puts it, "introduce 'new' objects, but only partially characterize them" (2007: 363). That these nominalizations refer to tropes rather than to states of affairs, she argues, can be seen once we consider the vast range of adjectival modifiers they allow for, modifiers only tropes and not states of affairs can be the recipients of (2009: 62-63; cf. also her 2003). Bare demonstratives, next, especially as they occur in so-called identificational sentences, provide another reason for thinking that tropes exist (Moltmann 2013a). In combination with the preposition like--as in Turquoise looks like that--they straightforwardly refer to tropes. But even in cases where they do not refer to tropes, tropes nevertheless contribute to the semantics of sentences in which they figure. In particular, tropes contribute to the meaning of sentences like This is Mary or That is a beautiful woman. These are no ordinary identity statements. What makes them stand out, Moltmann points out, is the exceptional neutrality of the demonstratives in subject position. These sentences are best understood in such a way that the bare demonstratives that figure in them do not refer to individuals (like Mary), but rather to perceptual features (which Moltmann thinks of as tropes) in the situation at hand. Identificational sentences, then, involve the identification of a bearer of a trope via the denotation (if not reference) of a (perceptual) trope. Comparatives--like John is happier than Mary--finally, are according to the received view such that they refer to abstract objects that form a total ordering (so-called degrees). According to Moltmann, a better way to understand these sorts of sentences is with reference to tropes. John is happier than Mary should hence be understood as John's happiness exceeds Mary's happiness. Moltmann thinks this way of understanding comparatives is preferable to the standard view, because tropes are easier to live with than "abstract, rarely explicit entities such as degrees or sets of degrees" (Moltmann 2009: 64). Whether nominalizations, bare demonstratives and/or comparatives succeed in providing the trope theorist with strong reasons to think tropes exist will, among other things, depend on whether or not they really do manage to distinguish between tropes and states of affairs. Moltmann thinks they do but, again, this depends on how one understands the nature of the items in question. It will also depend on if and how one thinks goings on at the linguistic level can tell us anything much about what there is at the ontological level. According to quite a few trope theorists (cf. esp. Heil 2003), we should avoid arguing from the way we conceptualize reality to conclusions about the nature of reality itself. Depending on one's take on the relationship between language and world, therefore, semantics might turn out to have precious little to say about the existence (or not) of tropes. ### 4.5 Tropes in Science Discussions of what use can be made of tropes in science can be found scattered in the literature. Examples include Rom Harre's (2009) discussion of the role tropes play (and don't play) in chemistry, and Bence Nanay's (2010) attempt to use tropes to improve on Ernst Mayr's population thinking in biology. Most discussions have however been focused on the relationship between tropes and physics (Kuhlmann et al. 2002). Most influential in this respect is Campbell's field-theory of tropes (defended in his 1990: Ch. 6; cf. also Von Wachter 2000) and Simons' 'nuclear' theory of tropes and the scientific use he tentatively makes of it (Simons 1994; cf. also Morganti 2009 and Wayne 2008). According to Campbell, the world is constituted by a rather limited number of field tropes which, according to our (current) best science, ought to be identified with the fields of gravitation, electromagnetism, and the weak and strong nuclear forces (plus a spacetime field). Standardly, these forces are understood as exerted by bodies that are not themselves fields. Not so on Campbell's view. Instead, matter is thought of as spread out and as present in various strengths across a region without any sharp boundaries to its location. What parts of the mass field we choose to focus on will be to a certain degree arbitrary. A zone in which several fields all sharply increase their intensity will likely be taken as one single entity or particle. But given the overall framework, individuals of this kind are to be viewed as "well-founded appearances" (Campbell 1990: 151). Campbell's views have been criticized by e.g., Christina Schneider (2006). According to Schneider, the field ontology proposed by Campbell (and by Von Wachter) fails, because the notion of a field with which they seem to be working, is not mathematically rigorous.[35] And Matteo Morganti who, just like Campbell, wants to identify tropes with entities described by quantum physics, finds several problems with the identifications actually made by Campbell. He proposes instead that we follow Simons and identify the basic constituents of reality with the fundamental particles, understood as bundles of tropes (Morganti 2009). If we take the basic properties described by the Standard Model as fundamental tropes, is the idea, then the constitution of particles out of more elementary constituents can be readily reconstructed (possibly by using the sheaf-theoretical framework proposed by Mormann 1995, or the algebraic framework suggested by Fuhrmann 1991). ### 4.6 Tropes and Issues in Moral Philosophy Relatively little has so far been written on the topic of tropes in relation to issues in moral philosophy and value theory. Two things have however been argued. First, that tropes (and not, as is more commonly supposed, objects or persons or states of affairs) are the bearers of final value. Second, that moral non-naturalists (who hold that moral facts are fundamentally autonomous from natural facts) must regard properties as tropes to be able to account for the supervenience of the moral on the natural. That tropes are the bearers of (final) value is a view held by several trope theorists. To say that what we value are the particular properties of things and persons is prima facie intuitive (Williams 1953a: 16). And since concrete particulars--but not tropes--are sometimes the subjects of simultaneous yet conflicting evaluations, tropes seem especially suited for the job as (final) value-bearers (Campbell 1997 [1981]: 130-131). That tropes are the only bearers of final value has however been questioned. According to Rabinowicz and Ronnow-Rasmussen (2003), this is because different pro-attitudes are fitting with respect to different kinds of valuable objects. However, according to Olson (2003), even if this is so, it does not show that tropes are not the only bearers of final value. For that conclusion only follows if we assume that, to what we direct our evaluative attitude is indicative of where value is localized. But final value should be understood strictly as the value something has for its own sake, which means that if e.g., a person is valuable because of her courage, then she is not valuable for her own sake but is valuable, rather, for the sake of one of her properties (i.e., her tropes). But this means that, although the evaluative attitude may well be directed at a person or a thing, the person or thing is nevertheless valued because of, or for the sake of, the tropes which characterize it. Non-naturalists, next, are often charged with not being able to explain what appears to be a necessary dependence of moral facts on natural facts. Normally this dependence is explained in terms of supervenience, but in order for such an account to be compatible with the basic tenets of moral non-naturalism, it has been argued, this supervenience must, in turn, be explainable in purely non-naturalistic terms (for an overview of this debate, cf. Ridge 2018). According to Shafer-Landau (2003) (as interpreted by Ridge 2007) this problem is solved if moral and physical properties in the sense of kinds, are distinguished from moral and physical properties in the sense of tokens, or tropes. For then we can say, in analogy with what has been suggested in the debate on the causal relevance of mental properties, that although (necessarily) every moral trope is constituted by some concatenation of natural tropes, it does not follow that every moral type is identical to a natural type. This suggestion is criticized in Ridge (2007).
trust	## 1. The Nature of Trust and Trustworthiness Trust is an attitude we have towards people whom we hope will be trustworthy, where trustworthiness is a property not an attitude. Trust and trustworthiness are therefore distinct although, ideally, those whom we trust will be trustworthy, and those who are trustworthy will be trusted. For trust to be plausible in a relationship, the parties to the relationship must have attitudes toward one another that permit trust. Moreover, for trust to be well-grounded, both parties must be trustworthy. (Note that here and throughout, unless specified otherwise, "trustworthiness" is understood in a thin sense according to which X is trustworthy for me just in case I can trust X.) Trusting requires that we can, (1) be vulnerable to others--vulnerable to betrayal in particular; (2) rely on others to be competent to do what we wish to trust them to do; and (3) rely on them to be willing to do it.[2] Notice that the second two conditions refer to a connection between trust and reliance. For most philosophers, trust is a kind of reliance although it is not mere reliance (Goldberg 2020). Rather, trust involves reliance "plus some extra factor" (Hawley 2014: 5). Controversy surrounds this extra factor, which generally concerns why the trustor (i.e., the one trusting) would rely on the trustee to be willing to do what they are trusted to do. Trustworthiness is likewise a kind of reliability, although it's not obvious what kind. Clear conditions for trustworthiness are that the trustworthy person is competent and willing to do what they are trusted to do. Yet this person may also have to be willing for certain reasons or as a result of having a certain kind of motive for acting (e.g., they care about the trustor). This section explains these various conditions for trust and trustworthiness and highlights the controversy that surrounds the condition about motive and relatedly how trust differs from mere reliance. Included at the end is some discussion about the nature of distrust. Let me begin with the idea that the trustor must accept some level of vulnerability or risk (Becker 1996; Baier 1986). Minimally, what this person risks, or is vulnerable to, is the failure by the trustee to do what the trustor is depending on them to do. The trustor might try to reduce this risk by monitoring or imposing certain constraints on the behavior of the trustee; but after a certain threshold perhaps, the more monitoring and constraining they do, the less they trust this person. Trust is relevant "before one can monitor the actions of ... others" (Dasgupta 1988: 51) or when out of respect for others one refuses to monitor them. One must be content with them having some discretionary power or freedom, and as a result, with being somewhat vulnerable to them (Baier 1986; Dasgupta 1988). One might think that if one is relying while trusting--that is, if trust is a species of reliance--then accepted vulnerability would not be essential for trust. Do we not rely on things only when we believe they will actually happen? And if we believe that, then we don't perceive ourselves as being vulnerable. Many philosophers writing on trust and reliance say otherwise. They endorse the view of Richard Holton, who writes, "When I rely on something happening ... I [only] need to plan on it happening; I need to work around the supposition that it will [happen]" (Holton 1994: 3). I need not be certain of it happening and I could even have doubts that it will happen (Goldberg 2020). I could therefore accept that I am vulnerable. I could do that while trusting if trust is a form of reliance. What does trusting make us vulnerable to, in particular? Annette Baier writes that "trusting can be betrayed, or at least let down, and not just disappointed" (1986: 235). In her view, disappointment is the appropriate response when one merely relied on someone to do something but did not trust them to do it. To elaborate, although people who monitor and constrain others' behavior may rely on them, they do not trust them if their reliance can only be disappointed rather than betrayed. One can rely on inanimate objects, such as alarm clocks, but when they break, one is not betrayed though one might be disappointed. This point reveals that reliance without the possibility of betrayal (or at least "let down") is not trust; people who rely on one another in a way that makes this reaction impossible do not trust one another. But does trust always involve the potential for betrayal? "Therapeutic trust" may be an exception (Nickel 2007: 318; and for further exceptions, see, e.g., Hinchman 2017). To illustrate this type of trust, consider parents who > > > trust their teenagers with the house or the family car, believing that > their [children] may well abuse their trust, but hoping by such trust > to elicit, in the fullness of time, more responsible and responsive > trustworthy behaviour. (McGeer 2008: 241, her emphasis; see also > Horsburgh 1960 and Pettit 1995) > > > Therapeutic trust is not likely to be betrayed rather than merely be disappointed. It is unusual in this respect (arguably) and in other respects that will become evident later on in this entry. The rest of this section deals with usual rather than unusual forms of trust and trustworthiness. Without relying on people to display some competence, we also can't trust them. We usually trust people to do certain things, such as look after our children, give us advice, or be honest with us, which we wouldn't do that if we thought they lacked the relevant skills, including potentially moral skills of knowing what it means to be honest or caring (Jones 1996: 7). Rarely do we trust people completely (i.e., A simply trusts B). Instead, "trust is generally a three-part relation: A trusts B to do X" (Hardin 2002: 9)--or "A trusts B with valued item C" (Baier 1986) or A trusts B in domain D (D'Cruz 2019; Jones 2019).[3] To have trust in a relationship, we do not need to assume that the other person will be competent in every way. Optimism about the person's competence in at least one area is essential, however. When we trust people, we rely on them not only to be competent to do what we trust them to do, but also to be willing or motivated to do it. We could talk about this matter either in terms of what the trustor expects of the trustee or in terms of what the trustee possesses: that is, as a condition for trust or for trustworthiness (and the same is true, of course, of the competence condition). For simplicity's sake and to focus some of this section on trustworthiness rather than trust, the following refers to the motivation of the trustee mostly as a condition for trustworthiness. Although both the competence and motivational elements of trustworthiness are crucial, the exact nature of the latter is unclear. For some philosophers, it matters only that the trustee is motivated, where the central problem of trustworthiness in their view concerns the probability that this motivation will exist or endure (see, e.g., Hardin 2002: 28; Gambetta 1988b). Jones calls these "risk-assessment views" about trust (1999: 68). According to them, we trust people whenever we perceive that the risk of relying on them to act a certain way is low and so we rely on (i.e., "trust") them. They are trustworthy if they are willing, for whatever reason, to do what they are trusted to do. Risk-assessment theories make no attempt to distinguish between trust and mere reliance and have been criticized for this reason (see, e.g., Jones 1999). By contrast, other philosophers say that just being motivated to act in the relevant way is not sufficient for trustworthiness; according to them, the nature of the motivation matters, not just its existence or duration. It matters in particular, they say, for explaining the trust-reliance distinction, which is something they aim to do. The central problem of trustworthiness for them is not simply whether but also how the trustee is motivated to act. Will that person have the kind of motivation that makes trust appropriate? Katherine Hawley identifies theories that respond to this question as "motives-based" theories (2014). To complicate matters, there are "non-motives-based theories", which are also not risk-assessment theories (Hawley 2014). They strive to distinguish between trust and mere reliance, though not by associating a particular kind of motive with trustworthiness. Since most philosophical debate about the nature of trust and trustworthiness centers on theories that are either motives-based or non-motives-based, let me expand on each of these categories. ### 1.1 Motives-based theories Philosophers who endorse this type of theory differ in terms of what kind of motive they associate with trustworthiness. For some, it is self-interest, while for others, it is goodwill or an explicitly moral motive, such as moral integrity or virtue.[4] For example, Russell Hardin defines trustworthiness in terms of self-interest in his "encapsulated interests" account (2002). He says that trustworthy people are motivated by their own interest to maintain the relationship they have with the trustor, which in turn encourages them to encapsulate the interests of that person in their own interests. In addition, trusting people is appropriate when we can reasonably expect them to encapsulate our interests in their own, an expectation which is missing with mere reliance. Hardin's theory may be valuable in explaining many different types of trust relationships, including those between people who can predict little about one another's motives beyond where their self-interest lies. Still, his theory is problematic. To see why, consider how it applies to a sexist employer who has an interest in maintaining relationships with women employees, who treats them reasonably well as a result, but whose interest stems from a desire to keep them around so that he can daydream about having sex with them. This interest conflicts with an interest the women have in not being objectified by their employer. At the same time, if they were not aware of his daydreaming--say they are not--then he can ignore this particular interest of theirs. He can keep his relationships with them going while ignoring this interest and encapsulating enough of their other interests in his own. And this would make him trustworthy on Hardin's account. But is he trustworthy? The answer is "no" or at least the women themselves would say "no" if they knew the main reason for their employment. The point is that being motivated by a desire to maintain a relationship (the central motivation of a trustworthy person on the encapsulated interests view) may not require one to adopt all of the interests of the trustor that would actually make one trustworthy to that person. In the end, the encapsulated interests view seems to describe only reliability, not trustworthiness. The sexist employer may reliably treat the women well, because of his interest in daydreaming about them, but he is not trustworthy because of why he treats them well. A different type of theory is what Jones calls a "will-based" account, which finds trustworthiness only where the trustee is motivated by goodwill (Jones 1999: 68). This view originates in the work of Annette Baier and is influential, even outside of moral philosophy (e.g., in bioethics and law, especially fiduciary law; see, e.g., Pellegrino and Thomasma 1993, O'Neill 2002, and Fox-Decent 2005). According to it, a trustee who is trustworthy will act out of goodwill toward the trustor, to what or to whom the trustee is entrusted with, or both. While many readers might find the goodwill view problematic--surely we can trust people without presuming their goodwill!--it is immune to a criticism that applies to Hardin's theory and also to risk-assessment theories. The criticism is that they fail to require that the trustworthy person care about (i.e., feel goodwill towards) the trustor, or care about what the trustor cares about. As we have seen, such caring appears to be central to a complete account of trustworthiness. The particular reason why care may be central is that it allows us to grasp how trust and reliance differ. The above suggested that they differ because only trust can be betrayed (or at least let down). But why is that true? Why can trust be betrayed, while mere reliance can only be disappointed? The answer Baier gives is that betrayal is the appropriate response to someone on whom one relied to act out of goodwill, as opposed to ill will, selfishness, or habit bred out of indifference (1986: 234-5; see also Baier 1991). Those who say that trusting could involve relying on people to act instead on motives like ill will or selfishness will have trouble distinguishing between trust and mere reliance. While useful in some respects, Baier's will-based account is not perfect. Criticisms have been made that suggest goodwill is neither necessary nor sufficient for trustworthiness. It is not necessary because we can trust other people without presuming that they have goodwill (e.g., O'Neill 2002; Jones 2004), as we arguably do when we put our trust in strangers. As well as being unnecessary, goodwill may not be sufficient for trustworthiness, and that is true for at least three reasons. First, someone trying to manipulate you--a "confidence trickster" (Baier 1986)--could "rely on your goodwill without trusting you", say, to give them money (Holton 1994: 65). You are not trustworthy for them, despite your goodwill, because they are not trusting you but rather are just trying to trick you. Second, basing trustworthiness on goodwill alone cannot explain unwelcome trust. We do not always welcome people's trust, because trust can be burdensome or inappropriate. When that happens, we object not to these people's optimism about our goodwill (who would object to that?), but only to the fact that they are counting on us. Third, we can expect people to be reliably benevolent toward us without trusting them (Jones 1996: 10). We can think that their benevolence is not shaped by the sorts of values that for us are essential to trustworthiness.[5] Criticisms about goodwill not being sufficient for trustworthiness have prompted revisions to Baier's theory and in some cases to the development of new will-based theories. For example, in response to the first criticism--about the confidence trickster--Zac Cogley argues that trust involves the belief not simply that the trustee will display goodwill toward us but that this person owes us goodwill (2012). Since the confidence trickster doesn't believe that their mark owes them goodwill, they don't trust this person, and neither is this person trustworthy for them. In response to the second criticism--the one about unwelcome trust--Jones claims that optimism about the trustee's goodwill must be coupled with the expectation that the trustee will be "favorably moved by the thought that [we are] counting on her" (1996: 9). Jones does that in her early work on trust where she endorses a will-based theory. Finally, in response to the third concern about goodwill not being informed by the sorts of values that would make people trustworthy for us, some maintain that trust involves an expectation about some shared values, norms, or interests (Lahno 2001, 2020; McLeod 2002, 2020; Mullin 2005; Smith 2008). (To be clear, this last expectation tends not to be combined with goodwill to yield a new will-based theory.) One final criticism of will-based accounts concerns how "goodwill" should be interpreted. In much of the discussion above, it is narrowly conceived so that it involves friendly feeling or personal liking. Jones urges us in her early work on trust to understand goodwill more broadly, so that it could amount to benevolence, conscientiousness, or the like, or friendly feeling (1996: 7). But then in her later work, she worries that by defining goodwill so broadly we > > > turn it into a meaningless catchall that merely reports the presence > of some positive motive, and one that may or may not even be directed > toward the truster. (2012a: 67) > > > Jones abandons her own will-based theory upon rejecting both a narrow and a broad construal of goodwill. (The kind of theory she endorses now is a trust responsive one; see below.) If her concerns about defining goodwill are valid, then will-based theories are in serious trouble. To recapitulate about encapsulated-interest and will-based theories, they say that a trustworthy person is motivated by self-interest or goodwill, respectively. Encapsulated-interest theories struggle to explain how trustworthiness differs from mere reliability, while will-based theories are faced with the criticism that goodwill is neither necessary nor sufficient for trustworthiness. Some philosophers who say that goodwill is insufficient develop alternative will-based theories. An example is Cogley's theory according to which trust involves a normative expectation of goodwill (2012). The field of motives-based theories is not exhausted by encapsulated-interest and will-based theories, however. Other motives-based theories include those that describe the motive of trustworthy people in terms of a moral commitment, moral obligation, or virtue. To expand, consider that one could make sense of the trustworthiness of a stranger by presuming that the stranger is motivated not by self-interest or goodwill, but by a commitment to stand by their moral values. In that case, I could trust a stranger to be decent by presuming just that she is committed to common decency. Ultimately, what I am presuming about the stranger is moral integrity, which some say is the relevant motive for trust relations (those that are prototypical; see McLeod 2002). Others identify this motive similarly as moral obligation, and say it is ascribed to the trustee by the very act of trusting them (Nickel 2007; for a similar account, see Cohen and Dienhart 2013). Although compelling in some respects, the worry about these theories is that they moralize trust inappropriately by demanding that the trustworthy person have a moral motive (see below and also Mullin 2005; Jones 2017). Yet one might insist that it is appropriate to moralize trust or at least moralize trustworthiness, which we often think of as a virtuous character trait. Nancy Nyquist Potter refers to the trait as "full trustworthiness", and distinguishes it from "specific trustworthiness", which is trustworthiness that is specific to certain relationships (and equivalent to the thin sense of trustworthiness I have used throughout; 2002: 25). To be fully trustworthy, one must have a disposition to be trustworthy toward everyone, according to Potter. Let us call this the "virtue" account. It may sound odd to insist that trustworthiness is a virtue or, in other words, a moral disposition to be trustworthy (Potter 2002: 25; Hardin 2002: 32). What disposition exactly is it meant to be? A disposition normally to honor people's trust? That would be strange, since trust can be unwanted if the trust is immoral (e.g., being trusted to hide a murder) or if it misinterprets the nature of one's relationship with the trustee (e.g., being trusted to be friends with a mere acquaintance). Perhaps trustworthiness is instead a disposition to respond to trust in appropriate ways, given "who one is in relation" to the trustor and given other virtues that one possesses or ought to possess (e.g., justice, compassion) (Potter 2002: 25). This is essentially Potter's view. Modeling trustworthiness on an Aristotelian conception of virtue, she defines a trustworthy person as "one who can be counted on, as a matter of the sort of person he or she is, to take care of those things that others entrust to one and (following the Doctrine of the Mean) whose ways of caring are neither excessive nor deficient" (her emphasis; 16).[6] A similar account of trustworthiness as a virtue--an epistemic one, specifically--can be found in the literature on testimony (see Frost-Arnold 2014; Daukas 2006, 2011). Criticism of the virtue account comes from Karen Jones (2012a). As she explains, if being trustworthy were a virtue, then being untrustworthy would be a vice, but that can't be right because we can never be required to exhibit a vice, yet we can be required to be untrustworthy (84). An example occurs when we are counted on by two different people to do two incompatible things and being trustworthy to the one demands that we are untrustworthy to the other (83). To defend her virtue theory, Potter would have to insist that in such situations, we are required either to disappoint someone's trust rather than be untrustworthy, or to be untrustworthy in a specific not a full sense.[7] Rather than cling to a virtue theory, however, why not just accept the thin conception of trustworthiness (i.e., "specific trustworthiness"), according to which X is trustworthy for me just in case I can trust X? Two things can be said. First, the thick conception--of trustworthiness as a virtue--is not meant to displace the thin one. We can and do refer to some people as being trustworthy in the specific or thin sense and to others as being trustworthy in the full or thick sense. Second, one could argue that the thick conception explains better than the thin one why fully trustworthy people are as dependable as they are. It is ingrained in their character. They therefore must have an ongoing commitment to being accountable to others, and better still, a commitment that comes from a source that is compatible with trustworthiness (i.e., virtue as opposed to mere self-interest). An account of trustworthiness that includes the idea that trustworthiness is a virtue will seem ideal only if we think that the genesis of the trustworthy person's commitment matters. If we believe, like risk-assessment theorists, that it matters only whether, not how, the trustor will be motivated to act, then we could assume that ill will can do the job as well as a moral disposition. Such controversy explains how and why motives-based and risk-assessment theories diverge from one another. ### 1.2 Non-motives-based theories A final category are theories that base trustworthiness neither on the kind of motivation a trustworthy person has nor on the mere willingness of this person to do what they are relied on to do. These are non-motives-based and also non-risk-assessment theories. The conditions that give rise to trustworthiness according to them reside ultimately in the stance the trustor takes toward the trustee or in what the trustor believes they ought to be able to expect from this person (i.e., in normative expectations of them). These theories share with motives-based theories the goal of describing how trust differs from mere reliance. An example is Richard Holton's theory of trust (1994). Holton argues that trust is unique because of the stance the trustor takes toward the trustee: the "participant stance", which involves treating the trustee as a person--someone who is responsible for their actions--rather than simply as an object (see also Strawson 1962 [1974]). In the case of trust specifically, the stance entails a readiness to feel betrayal (Holton 1994: 4). Holton's claim is that this stance and this readiness are absent when we merely rely on someone or something. Although Holton's theory has garnered positive attention (e.g., by Hieronymi 2008; McGeer 2008), some do find it dissatisfying. For example, some argue that it does not obviously explain what would justify a reaction of betrayal, rather than mere disappointment, when someone fails to do what they are trusted to do (Jones 2004; Nickel 2007). They could fail to do it just by accident, in which case feelings of betrayal would be inappropriate (Jones 2004). Others assert, by contrast, that taking the participant stance toward someone > > > does not always mean trusting that person: some interactions [of this > sort] lie outside the realm of trust and distrust. (Hawley 2014: > 7) > > > To use an example from Hawley, my partner could come to rely on me to make him dinner every night in a way that involves him taking the participant stance toward me. But he needn't trust me to make him dinner and so needn't feel betrayed if I do not. He might know that I am loath for him to trust me in this regard: "to make this [matter of making dinner] a matter of trust" between us (Hawley 2014: 7). Some philosophers have expanded on Holton's theory in a way that might deflect some criticism of it. Margaret Urban Walker emphasizes that in taking a participant stance, we hold people responsible (2006: 79). We expect them to act not simply as we assume they will, but as they should. We have, in other words, normative rather than merely predictive expectations of them. Call this a "normative-expectation" theory, which again is an elaboration on the participant-stance theory. Endorsed by Walker and others (e.g., Jones 2004 and 2012a; Frost-Arnold 2014), this view explains the trust-reliance distinction in terms of the distinction between normative and predictive expectations. It also describes the potential for betrayal in terms of the failure to live up a normative expectation. Walker's theory is non-motives-based because it doesn't specify that trustworthy people must have a certain kind of motive for acting. She says that trustworthiness is compatible with having many different kinds of motives, including, among others, goodwill, "pride in one's role", "fear of penalties for poor performance", and "an impersonal sense of obligation" (2006: 77). What accounts for whether someone is trustworthy in her view is whether they act as they should, not whether they are motivated in a certain way. (By contrast, Cogley's normative-expectation theory says that the trustworthy person both will and ought to act with goodwill. His theory is motives-based.) Prominent in the literature is a kind of normative-expectation theory called a "trust- (or dependence-) responsive" theory (see, e.g., Faulkner and Simpson 2017: 8; Faulkner 2011, 2017; Jones 2012a, 2017, 2019; McGeer and Petit 2017). According to this view, being trustworthy involves being appropriately responsive to the reason you have to do X--what you are being relied on (or "counted on"; Jones 2012a) to do--when it's clear that someone is in fact relying on you. The reason you have to do X exists simply because someone is counting on you; other things being equal, you should do it for this reason. Being appropriately responsive to it, moreover, just means that you find it compelling (Jones 2012a: 70-71). The person trusting you expects you to have this reaction; in other words, they have a normative expectation that the "manifest fact of [their] reliance will weigh on you as a reason for choosing voluntarily to X" (McGeer and Pettit 2017: 16). This expectation is missing in cases of mere reliance. When I merely rely on you, I do not expect my reliance to weigh on you as I do when I trust you. Although trust-responsive theories might seem motives-based, they are not. One might think that to be trustworthy, they require that you to be motivated by the fact that you are being counted on. Instead, they demand only that you be appropriately responsive to the reason you have to do what you are being depended on to do. As Jones explains, you could be responsive in this way and act ultimately out of goodwill, conscientiousness, love, duty, or the like (2012a: 66). The reaction I expect of you, as the trustor, is compatible with you acting on different kinds of motives, although to be clear, not just any motive will do (not like in Walker's theory); some motives are ruled out, including indifference and ill will (Jones 2012a: 68). Being indifferent or hateful towards me means that you are unlikely to view me counting on you as a reason to act. Hence, if I knew you were indifferent or hateful, I would not expect you to be trust responsive. Trust-responsive theories are less restrictive than motives-based theories when it comes to defining what motives people need to be trustworthy. At the same time, they are more restrictive when it comes to stating whether, in order to be trustworthy or trusted, one must be aware that one is being counted on. One couldn't be trust responsive otherwise. In trusting you, I therefore must "make clear to you my assumption that you will prove reliable in doing X" (McGeer and Pettit 2017: 16). I do not have to do that by contrast if, in trusting you, I am relying on you instead to act with a motive like goodwill. Baier herself allows that trust can exist where the trustee is unaware of it (1986: 235; see also Hawley 2014; Lahno 2020). For her, trust is ubiquitous (Jones 2017: 102) in part for this reason; we trust people in a myriad of different ways every single day, often without them knowing it. If she's right about this fact, then trust-responsive theories are incomplete. These theories are also vulnerable to objections raised against normative-expectation theories, because they are again a type of normative-expectation theory. One such concern comes from Hawley. In writing about both trust and distrust, she states that > > > we need a story about when trust, distrust or neither is objectively > appropriate--what is the worldly situation to which (dis)trust] > is an appropriate response? When is it appropriate to have > (dis)trust-related normative expectations of someone? (2014: 11) > > > Normative-expectation theories tend not to provide an answer. And trust-responsive theories suggest only that trust-related normative expectations are appropriate when certain motives are absent (e.g., ill will), which may not to be enough. Hawley responds to the above concern within her "commitment account" of trust (2014, 2019). This theory states that in trusting others, we believe that they have a commitment to doing what we are trusting them to do (2014: 10), a fact which explains why we expect them to act this way, and also why we fail to do so in cases like that of my partner relying on me to make dinner; he knows I have no commitment to making his dinner (or anyone else's) repeatedly. For Hawley, the relevant commitments > > > can be implicit or explicit, weighty or trivial, conferred by roles > and external circumstances, default or acquired, welcome or unwelcome. > (2014: 11) > > > They also needn't actually motivate the trustworthy person. Her theory is non-motives-based because it states that to > > > be trustworthy, in some specific respect, it is enough to behave in > accordance with one's commitment, regardless of motive. (2014: > 16) > > > Similarly, to trust me to do something, it is enough to believe that I > > > have a commitment to do it, and that I will do it, without believing > that I will do it because of my commitment. (2014: 16; her > emphasis) > > > Notice that unlike trust-responsive theories, the commitment account does not require that the trustee be aware of the trust in order to be trustworthy. This person simply needs to have a commitment and to act accordingly. They don't even need to be committed to the trustor, but rather could be committed to anyone and one could trust them to follow through on that commitment (Hawley 2014: 11). So, relying on a promise your daughter's friend makes to her to take her home from the party would count as an instance of trust (Hawley 2014: 11). In this way, the commitment account is less restrictive than trust-responsive theories are. In being non-motives-based, Hawley's theory is also less restrictive than any motives-based theory. Trust could truly be ubiquitous if she's correct about the nature of it. Like the other theories considered here, however, the commitment account is open to criticisms. One might ask whether Hawley gives a satisfactory answer to the question that motivates her theory: when can we reasonably have the normative expectations of someone that go along with trusting them? Hawley's answer is, when this person has the appropriate commitment, where "commitment" is understood very broadly. Yet where the relevant commitment is implicit or unwelcome, it's unclear that we can predict much about the trustee's behavior. In cases like these, the commitment theory may have little to say about whether it is reasonable to trust. A further criticism comes from Andrew Kirton (2020) who claims that we sometimes trust people to act contrary to what they are committed to doing. His central example involves a navy veteran, an enlisted man, whose ship sunk at sea and who trusted those who rescued them (navy men) to ignore a commitment they had to save the officers first, because the officers were relatively safe on lifeboats compared to the enlisted men who were struggling in the water. Instead the rescuers adhered to their military duty, and the enlisted man felt betrayed by them for nearly letting him drown. Assuming it is compelling, this example shows that trust and commitment can come apart and that Hawley's theory is incomplete.[8] The struggle to find a complete theory of trust has led some philosophers to be pluralists about trust--that is, to say, "we must recognise plural forms of trust" (Simpson 2012: 551) or accept that trust is not just one form of reliance, but many forms of it (see also Jacoby 2011; Scheman 2020; McLeod 2020). Readers may be led to this conclusion from the rundown I've given of the many different theories of trust in philosophy and the objections that have been raised to them. Rather than go in the direction of pluralism, however, most philosophers continue to debate what unifies all trust such that it is different from mere reliance. They tend to believe that a unified and suitably developed motives-based theory or non-motives-based theory can explain this difference, although there is little consensus about what this theory should be like. In spite of there being little settled agreement in philosophy about trust, there are thankfully things we can say for certain about it that are relevant to deciding when it is warranted. The trustor must be able to accept that by trusting, they are vulnerable usually to betrayal. Also, the trustee must be competent and willing to do what the trustor expects of them and may have to be willing because of certain attitudes they have. Last, in paradigmatic cases of trust, the trustor must be able to rely on the trustee to exhibit this competence and willingness. ### 1.3 Distrust As suggested above, distrust has been somewhat of an afterthought for philosophers (Hawley 2014),[9] although their attention to it has grown recently. As with trust and trustworthiness, philosophers would agree that distrust has certain features, although the few who have developed theories of distrust disagree ultimately about the nature of it. The following are features of distrust that are relatively uncontroversial (see D'Cruz 2020): 1. Distrust is not just the absence of trust since it is possible to neither distrust nor trust someone (Hawley 2014: 3; Jones 1996: 16; Krishnamurthy 2015). There is gap between the two--"the possibility of being suspended between" them (Ullmann-Margalit 2004 [2017: 184]). (For disagreement, see Faulkner 2017.) 2. Although trust and distrust are not exhaustive, they are exclusive; one cannot at the same time trust and distrust someone about the same matter (Ullmann-Margalit 2004 [2017: 201]). 3. Distrust is "not mere nonreliance" (Hawley 2014: 3). I could choose not to rely on a colleague's assistance because I know she is terribly busy, not because I distrust her. 4. Relatedly, distrust has a normative dimension. If I distrusted a colleague for no good reason and they found out about it, then they would probably be hurt or angry. But the same reaction would not accompany them knowing that I decided not to rely on them (Hawley 2014). Being distrusted is a bad thing (Domenicucci and Holton 2017: 150; D'Cruz 2019: 935), while not being relied on needn't be bad at all. 5. Distrust is normally a kind of nonreliance, just as trust is a kind (or many kinds) of reliance. Distrust involves "action-tendencies" of avoidance or withdrawal (D'Cruz 2019: 935-937), which make it incompatible with reliance--or at least complete reliance. We can be forced to rely on people we distrust, yet even then, we try to keep them at as safe a distance as possible. Given the relationship between trust and distrust and the similarities between them (e.g., one is "richer than [mere] reliance" and the other is "richer than mere nonreliance"; Hawley 2014: 3), one would think that any theory of trust should be able to explain distrust and vice versa. Hawley makes this point and criticizes theories of trust for not being able to make sense of distrust (2014: 6-9). For example, will-based accounts imply that distrust must be nonreliance plus an expectation of ill will, yet the latter is not required for distrust. I could distrust someone because he is careless, not because he harbors ill will toward me (Hawley 2014: 6). Hawley defends her commitment account of trust, in part, because she believes it is immune to the above criticism. It says that distrust is nonreliance plus the belief that the person distrusted is committed to doing what we will not rely on them to do. In spite of them being committed in this way (or so we believe), we do not rely on them (2014: 10). This account does not require that we impute any particular motive or feeling to the one distrusted, like ill will. At the same time, it tells us why distrust is not mere nonreliance and also why it is normative; the suspicion of the one distrusted is that they will fail to meet a commitment they have, which is bad. Some have argued that Hawley's theory of distrust is subject to counterexamples, however (D'Cruz 2020; Tallant 2017). For example, Jason D'Cruz describes a financier who "buys insurance on credit defaults, positioning himself to profit when borrowers default" (2020: 45). The financier believes that the borrowers have a commitment not to default, and he does not rely on them to meet this commitment. The conclusion that Hawley's theory would have us reach is that he distrusts the borrowers, which doesn't seem right. A different kind of theory of distrust can be found in the work of Meena Krishnamurthy (2015), who is interested specifically in the value that distrust has for political democracies, and for political minorities in particular (2015). She offers what she calls a "narrow normative" account of distrust that she derives from the political writings of Martin Luther King Jr. The account is narrow because it serves a specific purpose: of explaining how distrust can motivate people to resist tyranny. It is normative because it concerns what they ought to do (again, resist; 392). The theory states that distrust is the confident belief that others will not act justly. It needn't involve an expectation of ill will; King's own distrust of white moderates was not grounded in such an expectation (Krishnamurthy 2015: 394). To be distrusting, one simply has to believe that others will not act justly, whether out of fear, ignorance, or what have you. D'Cruz complains that Krishnamurthy's theory is too narrow because it requires a belief that the one distrusted will fail to do something (i.e., act justly) (2020); but one can be distrustful of someone--say a salesperson who comes to your door (Jones 1996)--without predicting that they will do anything wrong or threatening. D'Cruz does not explain, however, why Krishnamurthy needs to account for cases like these in her theory, which again is meant to serve a specific purpose. Is it important that distrust can take a form other than "X distrusts Y to [do] Ph" for it to motivate political resistance (D'Cruz 2020: 45)? D'Cruz's objection is sound only if the answer is "yes". Nevertheless, D'Cruz's work is helpful in showing what a descriptive account of distrust should look like--that is, an account that unlike Krishnamurthy's, tracks how we use the concept in many different circumstances. He himself endorses a normative-expectation theory, according to which distrust involves > > > a tendency to withdraw from reliance or vulnerability in contexts of > normative expectation, based on a construal of a person or persons as > malevolent, incompetent, or lacking integrity. (2019: 936) > > > D'Cruz has yet to develop this theory fully, but once he does so, it will almost certainly be a welcome addition to the scant literature in philosophy on distrust. In summary, among the relatively few philosophers who have written on distrust, there is settled agreement about some of its features but not about the nature of distrust in general. The agreed-upon features tell us something about when distrust is warranted (i.e., plausible). For distrust in someone to be plausible, one cannot also trust that person, and normally one will not be reliant on them either. Something else must be true as well, however. For example, one must believe that this person is committed to acting in a certain way but will not follow through on this commitment. The "something else" is crucial because distrust is not the negation of trust and neither is it mere nonreliance. Philosophers have said comparatively little about what distrust is, but a lot about how distrust tends to be influenced by negative social stereotypes that portray whole groups of people as untrustworthy (e.g., Potter 2020; Scheman 2020; D'Cruz 2019; M. Fricker 2007). Trusting attitudes are similar--who we trust can depend significantly on social stereotypes, positive ones--yet there is less discussion about this fact in the literature on trust. This issue concerns the rationality (more precisely, the irrationality) of trust and distrust, which makes it relevant to the next section, which is on the epistemology of trust. ## 2. The Epistemology of Trust Writings on this topic obviously bear on the issue of when trust is warranted (i.e., justified). The central epistemological question about trust is, "Ought I to trust or not?" That is, given the way things seem to me, is it reasonable for me to trust? People tend to ask this sort of question only in situations where they can't take trustworthiness for granted--that is, where they are conscious of the fact that trusting could get them into trouble. Examples are situations similar to those in which they have been betrayed in the past or unlike any they have ever been in before. The question, "Ought I to trust?" is therefore particularly pertinent to a somewhat odd mix of people that includes victims of abuse or the like, as well as immigrants and travelers. The question "Ought I to distrust?" has received comparatively little attention in philosophy despite it arguably being as important as the question of when to trust. People can get into serious trouble by distrusting when they ought not to, rather than just by trusting when they ought not to. The harms of misplaced distrust are both moral and epistemic and include dishonoring people, being out of harmony with them, and being deprived of knowledge via testimony (D'Cruz 2019; M. Fricker 2007). Presumably because they believe that the harms of misplaced trust are greater (D'Cruz 2019), philosophers--and consequently I, in this entry--focus more on the rationality of trusting, as opposed to distrusting. Philosophical work that is relevant to the issue of how to trust well appears either under the general heading of the epistemology or rationality of trust (e.g., Baker 1987; Webb 1992; Wanderer and Townsend 2013) or under the specific heading of testimony--that is, of putting one's trust in the testimony of others. This section focuses on the epistemology of trust generally rather than on trust in testimony specifically. There is a large literature on testimony (see the entry in this encyclopedia) and on the related topic of epistemic injustice, both of which I discuss only insofar as they overlap with the epistemology of trust. Philosophers sometimes ask whether it could ever be rational to trust other people. This question arises for two reasons. First, it appears that trust and rational reflection (e.g., on whether one should be trusting) are in tension with one another. Since trust inherently involves risk, any attempt to eliminate that risk through rational reflection could eliminate one's trust by turning one's stance into mere reliance. Second, trust tends to give us blinkered vision: it makes us resistant to evidence that may contradict our optimism about the trustee (Baker 1987; Jones 1996 and 2019). For example, if I trust my brother not to harm anyone, I will resist the truth of any evidence to the contrary. Here, trust and rationality seem to come apart. Even if some of our trust could be rational, one might insist that not all of it could be rational for various reasons. First, if Baier is right that trust is ubiquitous (1986: 234), then we could not possibly subject all of it to rational reflection. We certainly could not reflect on every bit of knowledge we've acquired through the testimony of others, such as that the earth is round or Antarctica exists (Webb 1993; E. Fricker 1995; Coady 1992). Second, bioethicists point out that some trust is unavoidable and occurs in the absence of rational reflection (e.g., trust in emergency room nurses and physicians; see Zaner 1991). Lastly, some trust--namely the therapeutic variety--purposefully leaps beyond any evidence of trustworthiness in an effort to engender trustworthiness in the trustee. Is this sort of trust rational? Perhaps not, given that there isn't sufficient evidence for it. Many philosophers respond to the skepticism about the rationality of trust by saying that rationality, when applied to trust, needs to be understood differently than it is in each of the skeptical points above. There, "rationality" means something like this: it is rational to believe in something only if one has verified that it will happen or done as much as possible to verify it. For example, it is rational for me to believe that my brother has not harmed anyone only if the evidence points in that direction and I have discovered that to be the case. As we've seen, problems exist with applying this view of rationality to trust, yet it is not the only option; this view is both "truth-directed" and "internalist", while the rationality of trust could instead be "end-directed" or "externalist". Or it could be internalist without requiring that we have done the evidence gathering just discussed. Let me expand on these possibilities, starting with those that concern truth- or end-directed rationality. ### 2.1 Truth- vs. end-directed rationality In discussing the rationality of trust, some authors distinguish between these two types of rationality (also referred to as epistemic vs. strategic rationality; see, e.g., Baker 1987). One could say that we are rational in trusting emergency room physicians, for example, not necessarily because we have good reason to believe that they are trustworthy (our rationality is not truth-directed), but because by trusting them, we can remain calm in a situation over which we have little control (our rationality is therefore end-directed). Similarly, it may be rational for me to trust my brother not because I have good evidence of his trustworthiness but rather because trusting him is essential to our having a loving relationship.[10] Trust can be rational, then, depending on whether one conceives of rationality as truth-directed or end-directed. Notice that it matters also how one conceives of trust, and more specifically, whether one conceives of it as a belief in someone's trustworthiness (see section 4). If trust is a belief, then whether the rationality of trust can be end-directed will depend on whether the rationality of a belief can be end-directed. To put the point more generally, how trust is rationally justified will depend on how beliefs are rationally justified (Jones 1996). Some of the literature on trust and rationality concerns whether the rationality of trust can indeed be end-directed and also what could make therapeutic trust and the like rational. Pamela Hieronymi argues that the ends for which we trust cannot provide reasons for us to trust in the first place (2008). Considerations about how useful or valuable trust is do not bear on the truth of a trusting belief (i.e., a belief in someone's trustworthiness). But Hieronymi claims that trust, in a pure sense at least, always involves a trusting belief. How then does she account for trust that is motivated by how therapeutic (i.e., useful) the trust will be? She believes that trust of this sort is not pure or full-fledged trust. As she explains, people can legitimately complain about not being trusted fully when they are trusted in this way, which occurs when other people lack confidence in them but trust them nonetheless (2008: 230; see also Lahno 2001: 184-185). By contrast, Victoria McGeer believes that trust is more substantial or pure when the available evidence does not support it (2008). She describes how trust of this sort--what she calls "substantial trust"--could be rational and does so without appealing to how important it might be or to the ends it might serve, but instead to whether the trustee will be trustworthy.[11] According to McGeer, what makes "substantial trust" rational is that it involves hope that the trustees will do what they are trusted to do, which "can have a galvanizing effect on how [they] see themselves, as trustors avowedly do, in the fullness of their potential" (2008: 252; see also McGeer and Pettit 2017). Rather than complain (as Hieronymi would assume that trustees might) about trustors being merely hopeful about their trustworthiness, they could respond well to the trustors' attitude toward them. Moreover, if it is likely that they will respond well--in other words, that they will be trust-responsive--then the trust in them must be epistemically rational. That is particularly true if being trustworthy involves being trust-responsive, as it does for McGeer (McGeer and Pettit 2017). McGeer's work suggests that all trust--even therapeutic trust--can be rational in a truth-directed way. As we've seen, there is some dispute about whether trust can be rational in just an end-directed way. What matters here is whether trust is the sort of attitude whose rationality could be end-directed. ### 2.2 Internalism vs. externalism Philosophers who agree that trust can be rational (in a truth- or end-directed way or both) tend to disagree about the extent to which reasons that make it rational must be accessible to the trustor. Some say that these reasons must be available to this person in order for their trust to be rational; in that case, the person is or could be internally justified in trusting as they do. Others say that the reasons need not be internal but can instead be external to the trustor and lie in what caused the trust, or, more specifically, in the epistemic reliability of what caused it. The trustor also needn't have access to or be aware of the reliability of these reasons. The latter's epistemology of trust is externalist, while the former's is internalist. Some epistemologists write as though trust is only rational if the trustor themselves has rationally estimated the likelihood that the trustee is trustworthy. For example, Russell Hardin implies that if my trust in you is rational, then > > > I make a rough estimate of the truth of [the] claim ... that you > will be trustworthy under certain conditions ... and then I > correct my estimate, or "update," as I obtain new evidence > on you. (2002: 112) > > > On this view, I must have reasons for my estimate or for my updates (Hardin 2002: 130), which could come from inductive generalizations I make about my past experience, from my knowledge that social constraints exist that will encourage your trustworthiness or what have you. Such an internalist epistemology of trust is valuable because it coheres with the commonsense idea that one ought to have good reasons for trusting people (i.e., reasons grounded in evidence that they will be trustworthy) particularly when something important is at stake (E. Fricker 1995). One ought, in other words, to be epistemically responsible in one's trusting (see Frost-Arnold 2020). Such an epistemology is also open to criticisms, however. For example, it suggests that rational trust will always be partial rather than complete, given that the rational trustor is open to evidence that contradicts their trust on this theory, while someone who trusts completely in someone else lacks such openness. The theory also implies that the reasons for trusting well (i.e., in a justified way) are accessible to the trustor, at some point or another, which may simply be false. Some reasons for trust may be too "cunning" for this to be the case. Relevant here is the reason for trusting discussed by Philip Pettit (1995): that trust signals to people that they are being held in esteem, which is something they will want to maintain; they will honor the trust because they are naturally "esteem-seeking". (Note that consciously having this as a reason for trusting--of using people's need for esteem to get what you want from them--is incompatible with actually trusting (Wanderer and Townsend 2013: 9), if trust is motives-based and the required motive is something other than self-interest.) Others say that reasons for trust are usually too numerous and varied to be open to the conscious consideration of the trustor (e.g., Baier 1986). There can be very subtle reasons to trust or distrust someone--for example, reasons that have to do with body language, with systematic yet veiled forms of oppression, or with a complicated history of trusting others about which one can't easily generalize. Factors like these can influence trustors without them knowing it, sometimes making their trust irrational (e.g., because it is informed by oppressive biases), and other times making it rational. The concern about there being complex reasons for trusting explain why some philosophers defend externalist epistemologies of trust. Some do so explicitly (e.g., McLeod 2002). They argue for reliabilist theories that make trust rationally justified if and only if it is formed and sustained by reliable processes (i.e., "processes that tend to produce accurate representations of the world", such as drawing on expertise one has rather than simply guessing; Goldman 1992: 113; Goldman and Beddor 2015 [2016]). Others gesture towards externalism (Webb 1993; Baier 1986), as Baier does with what she calls "a moral test for trust". The test is that > > > knowledge of what the other party is relying on for the continuance of > the trust relationship would ... itself destabilize the relation. > (1986: 255) > > > The other party might be relying on a threat advantage or the concealment of their untrustworthiness, in which case the trust would probably fail the test. Because Baier's test focuses on the causal basis for trust, or for what maintains the trust relation, it is externalist. Also, because the trustor often cannot gather the information needed for the test without ceasing to trust the other person (Baier 1986: 260), the test cannot be internalist. Although an externalist theory of trust deals well with some of the worries one might have with an internalist theory, it has problems of its own. One of the most serious issues is the absence of any requirement that trustors themselves have good (motivating) reasons for trusting, especially when their trust makes them seriously vulnerable. Again, it appears that common sense dictates the opposite: that sometimes as trustors, we ought to be able to back up our decisions about when to trust. The same is true about our distrust presumably: that sometimes we ought to be able to defend it. Assuming externalists mean for their epistemology to apply to distrust and not just to trust, their theory violates this bit of common sense as well. Externalism about distrust also seems incompatible with a strategy that some philosophers recommend for dealing with biased distrust. The strategy is to develop what they call "corrective trust" (e.g., Scheman 2020) or "humble trust" (D'Cruz 2019), which demands a humble skepticism toward distrust that aligns with oppressive stereotypes and efforts at correcting the influence of these stereotypes (see also M. Fricker 2007). The concern about an externalist epistemology is that it does not encourage this sort of mental work, since it does not require that we reflect on our reasons for distrusting or trusting. There are alternatives to the kinds of internalist and externalist theories just discussed, especially within the literature on testimony.[12] For example, Paul Faulkner develops an "assurance theory" of testimony that interprets speaker trustworthiness in terms of trust-responsiveness. Recall that on a trust-responsiveness theory of trust, being trusted gives people the reason to be trustworthy that someone is counting on them. They are trustworthy if they are appropriately responsive to this reason, which, in the case of offering testimony, involves giving one's assurance that one is telling the truth (Adler 2006 [2017]). Faulkner uses the trust-responsiveness account of trust, along with a view of trust as an affective attitude (see section 4), to show "how trust can ground reasonable testimonial uptake" (Faulkner and Simpson 2017: 6; Faulkner 2011 and 2020). > > > He proposes that A affectively trust S if and only > if A depends on S Ph-ing, and expects his > dependence on S to motivate S to Ph--for > A's dependence on S to be the reason for which > S Phs .... As a result, affective trust is a > bootstrapping attitude: I can choose to trust someone affectively and > my doing so creates the reasons which justify the attitude. (Faulkner > and Simpson 2017: 6) > > > Most likely, A (the trustor) is aware of the reasons that justify his trust or could be aware of them, making this theory an internalist one. The reasons are also normative and non-evidentiary (Faulkner 2020); they concern what S ought to do because of A's dependence, not what S will do based on evidence that A might gather about S. This view doesn't require that A have evidentiary reasons, and so it is importantly different than the internalist epistemology discussed above. But it is then also subject to the criticisms made of externalist theories that they don't require the kind of scrutiny of our trusting attitudes that we tend to expect and probably ought to expect in societies where some people are stereotyped as more trusting than others. Presumably to avoid having to defend any particular epistemology of trust, some philosophers provide just a list of common justifiers for it (i.e., "facts or states of affairs that determine the justification status of [trust]"; Goldman 1999: 274), which someone could take into account in deciding when to trust (Govier 1998; Jones 1996). Included on these lists are such factors as the social role of the trustee, the domain in which the trust occurs, an "agent-specific" factor that concerns how good a trustor the agent tends to be (Jones 1996: 21), and the social or political climate in which the trust occurs. Philosophers have tended to emphasize this last factor as a justification condition for trust, and so let me elaborate on it briefly. ### 2.3 Social and political climate Although trust is paradigmatically a relation that holds between two individuals, forces larger than those individuals inevitably shape their trust and distrust in one another. Social or political climate contributes to how (un)trustworthy people tend to be and therefore to whether trust and distrust are justified. For example, a climate of virtue is one in which trustworthiness tends to be pervasive, assuming that virtues other than trustworthiness tend to enhance it (Baier 2004).[13] A climate of oppression is one in which untrustworthiness is prevalent, especially between people who are privileged and those who are less privileged (Baier 1986: 259; Potter 2002: 24; D'Cruz 2019). "Social trust", as some call it, is low in these circumstances (Govier 1997; Welch 2013). Social or political climate has a significant influence on the default stance that we ought to take toward people's trustworthiness (see, e.g., Walker 2006). We need such a stance because we can't always stop to reflect carefully on when to trust (i.e., assuming that some rational reflection is required for trusting well). Some philosophers say that the correct stance is trust and do so without referring to the social or political climate; Tony Coady takes this sort of position, for example, on our stance toward others' testimony (Coady 1992). Others disagree that the correct stance could be so universal and claim instead that it is relative to climate, as well as to other factors such as domain (Jones 1999). Our trust or distrust may be prima facie justified if we have the correct default stance, although most philosophers assume that it could only be fully justified (in a truth- or end-directed way) by reasons that are internal to us (evidentiary or non-evidentiary reasons) or by the causal processes that created the attitude in the first place. Whichever epistemology of trust we choose, it ought to be sensitive to the tension that exists between trusting somebody and rationally reflecting on the grounds for that trust. It would be odd, to say the least, if what made an attitude justified destroyed that very attitude. At the same time, our epistemology of trust ought to cohere as much as possible with common sense, which dictates that we should inspect rather than have pure faith in whatever makes us seriously vulnerable to other people, which trust can most definitely do. ## 3. The Value of Trust Someone who asks, "When is trust warranted?" might be interested in knowing what the point of trust is. In other words, what value does it have? Although the value it has for particular people will depend on their circumstances, the value it could have for anyone will depend on why trust is valuable, generally speaking. Trust can have enormous instrumental value and may also have some intrinsic value. In discussing its instrumental value, this section refers to the "goods of trust", which can benefit the trustor, the trustee, or society in general. They are therefore social and/or individual goods. What is more and as emphasized throughout, these goods tend to accompany justified trust, rather than any old trust.[14] Like the other sections of this entry, this one focuses predominantly though not exclusively on trust; it also mentions recent work on the value of distrust. Consider first the possibility that trust has intrinsic value. If trust produced no goods independent of it, would there be any point in trusting? One might say "yes", on the grounds that trust is (or can be; O'Neil 2012: 311) a sign of respect for others. (Similarly, distrust is a sign of disrespect; D'Cruz 2019.) If true, this fact about trust would make it intrinsically worthwhile, at least so long as the trust is justified. Presumably, if it was unjustified, then the respect would be misplaced and the intrinsic value would be lost. But these points are speculative, since philosophers have said comparatively little about trust being worthwhile in itself as opposed to worthwhile because of what it produces, or because of what accompanies it. The discussion going forward centers on the latter, more specifically on the goods of trust. Turning first to the instrumental value of trust to the trustor, some argue that trusting vastly increases our opportunities for cooperating with others and for benefiting from that cooperation, although of course we would only benefit if people we trusted cooperated as well (Gambetta 1988b; Hardin 2002; Dimock 2020). Trust enhances cooperation, while perhaps not being necessary for it (Cook et al. 2005; Skyrms 2008). Because trust removes the incentive to check up on other people, it makes cooperation with trust less complicated than cooperation without it (Luhmann 1973/1975 [1979]). Trust can make cooperation possible, rather than simply easier, if trust is essential to promising. Daniel Friedrich and Nicholas Southwood defend what they call the "Trust View" of promissory obligation (2011), according to which "making a promise involves inviting another individual to trust one to do something" (2011: 277). If this view is correct, then cooperation through promising is impossible without trust. Cooperation of this sort will also not be fruitful unless the trust is justified. Trusting provides us with goods beyond those that come with cooperation, although again, for these goods to materialize, the trust must be justified. Sometimes, trust involves little or no cooperation, so that the trustor is completely dependent on the trustee while the reverse is not true. Examples are the trust of young children in their parents and the trust of severely ill or disabled people in their care providers. Trust is particularly important for these people because they tend to be powerless to exercise their rights or to enforce any kind of contract. The trust they place in their care providers also contributes to them being vulnerable, and so it is essential that they can trust these people (i.e., that their trust is justified). The goods at stake for them are all the goods involved in having a good or decent life. Among the specific goods that philosophers associate with trusting are meaningful relationships or attachments (rather than simply cooperative relationships that further individual self-interests; Harding 2011, Kirton forthcoming) as well as knowledge and autonomy.[15] To expand, trust allows for the kinds of secure attachments that some developmental psychologists ("attachment" theorists) believe are crucial to our well-being and to our ability to be trusting of others (Bowlby 1969-1980; Ainsworth 1969; see Kirton 2020; Wonderly 2016). Particularly important here are parent-child relationships (McLeod et al. 2019). Trust is also crucial for knowledge, given that scientific knowledge (Hardwig 1991), moral knowledge (Jones 1999), and almost all knowledge in fact (Webb 1993) depends for its acquisition on trust in the testimony of others. The basic argument for the need to trust what others say is that no one person has the time, intellect, and experience necessary to independently learn facts about the world that many of us do know. Examples include the scientific fact that the earth is round, the moral fact that the oppression of people from social groups different from our own can be severe (Jones 1999), and the mundane fact that we were born on such-in-such a day (Webb 1993: 261). Of course, trusting the people who testify to these facts could only generate knowledge if the trust was justified. If we were told our date of birth by people who were determined oddly to deceive us about when we were born, then we would not know when we were born. Autonomy is another good that flows from trust insofar as people acquire or exercise autonomy only in social environments where they can trust people (or institutions, etc.) to support their autonomy. Feminists in particular tend to conceive of autonomy this way--that is, as a relational property (Mackenzie and Stoljar 2000). Many feminists emphasize that oppressive social environments can inhibit autonomy, and some say explicitly that conditions necessary for autonomy (e.g., adequate options, knowledge relevant to one's decisions) exist only with the help of people or institutions that are trustworthy (e.g., Oshana 2014; McLeod and Ryman 2020). Justified trust in others to ensure that these conditions exist is essential for our autonomy, if autonomy is indeed relational.[16] Goods of trust that are instrumental to the well-being of the trustee also do not materialize unless the trust is justified. Trust can improve the self-respect and moral maturity of this person. Particularly if it involves reliance on a person's moral character, trust can engender self-respect in the trustee (i.e., through them internalizing the respect signaled by that trust). Being trusted can allow us to be more respectful not only toward ourselves but also toward others, thus enhancing our moral maturity. The explicit goal of therapeutic trust is precisely to bring about this end. The above (section 2) suggests that therapeutic trust can be justified in a truth-directed way over time, provided that the trust has its intended effect of making the trustee more trustworthy (McGeer 2008; Baker 1987: 12). Clearly, for therapeutic trust to benefit the trustee, it would have to be justified in this way, meaning that the therapy would normally have to work. Finally, there are social goods of trust that are linked with the individual goods of cooperation and moral maturity. The former goods include the practice of morality, the very existence of society perhaps, as well as strong social networks. Morality itself is a cooperative activity, which can only get off the ground if people can trust one another to try, at least, to be moral. For this reason, among others, Baier claims that trust is "the very basis of morality" (2004: 180). It could also be the very basis of society, insofar as trust in our fellow citizens to honor social contracts makes those contracts possible. A weaker claim is that trust makes society better or more livable. Some argue that trust is a form of "social capital", meaning roughly that it enables "people to work together for common purposes in groups and organizations" (Fukuyama 1995: 10; quoted in Hardin 2002: 83). As a result, "high-trust" societies have stronger economies and stronger social networks in general than "low-trust" societies (Fukuyama 1995; Inglehart 1999). Of course, this fact about high-trust societies could only be true if, on the whole, the trust within them was justified--that is, if trustees tended not to "defect" and destroy chances for cooperating in the future. The literature on distrust suggests that there are goods associated with it too. For example, there is the social good discussed by Krishnamurthy of "securing democracy by protecting political minorities from tyranny" (2015: 392). Distrust as she understands it (a confident belief that others will not act justly) plays this positive role when it is justified, which is roughly when the threat of tyranny or unjust action is real. Distrust in general is valuable when it is justified--for the distrustors at least, who protect themselves from harm. By contrast, the people distrusted tend to experience negative effects on their reputation or self-respect (D'Cruz 2019). Both trust and distrust are therefore valuable particularly when they are justified. The value of justified trust must be very high if without it, we can't have morality or society and can't be morally mature, autonomous, knowledgeable, or invested with opportunities for collaborating with others. Justified distrust is also essential, for members of minority groups especially. Conversely, trust or distrust that is unjustified can be seriously problematic. Unjustified trust, for example, can leave us open to abuse, terror, and deception. ## 4. Trust and the Will Trust may not be warranted (i.e., plausible) because the agent has lost the ability to trust or simply cannot bring themselves to trust. People can lose trust in almost everyone or everything as a result of trauma (Herman 1991). The trauma of rape, for example, can profoundly reduce one's sense that the world is a safe place with caring people in it (Brison 2002). By contrast, people can lose trust just in particular people or institutions. They can also have no experience trusting in certain people or institutions, making them reluctant to do so. They or others might want them to become more trusting. But the question is, how can that happen? How can trust be restored or generated? The process of building trust is often slow and difficult (Uslaner 1999; Baier 1986; Lahno 2020), and that is true, in part, because of the kind of mental attitude trust is. Many argue that it is not the sort of attitude we can simply will ourselves to have. At the same, it is possible to cultivate trust.[17] This section focuses on these issues, including what kind of mental attitude trust is (e.g., a belief or an emotion). Also discussed briefly is what kind of mental attitude distrust is. Like trust, distrust is an attitude that people may wish to cultivate, particularly when they are too trusting. Consider first why one would think that trust can't be willed. Baier questions whether people are able "to trust simply because of encouragement to trust" (1986: 244; my emphasis). She writes, > > > "Trust me!" is for most of us an invitation which we > cannot accept at will--either we do already trust the one who > says it, in which case it serves at best as reassurance, or it is > properly responded to with, "Why should and how can I, until I > have cause to?". (my emphasis; 1986: 244) > > > Baier is not a voluntarist about trust, just as most people are not voluntarists about belief. In other words, she thinks that we can't simply decide to trust for purely motivational rather than epistemic reasons (i.e., merely because we want to, rather than because we have reason to think that the other person is or could be trustworthy; Mills 1998). That many people feel compelled to say, "I wish I could trust you", suggests that Baier's view is correct; wishing or wanting is not enough. But Holton interprets Baier's view differently. According to him, Baier's point is that we can never decide to trust, not that we can never decide to trust for motivational purposes (1994). This interpretation ignores, however, the attention that Baier gives to situations in which all we have is encouragement (trusting "simply because of encouragement"). The "cause" she refers to ("Why should and how can I, until I have cause to [trust]?"; 1986: 244) is an epistemic cause. Once we have one of those, we can presumably decide whether to trust on the basis of it.[18] But we cannot decide to trust simply because we want to, according to Baier. If trust resembles belief in being non-voluntary, then perhaps trust itself is a belief. Is that right? Many philosophers claim that it is (e.g., Hieronymi 2008; McMyler 2011; Keren 2014), while others disagree (e.g., Jones 1996; Faulkner 2007; D'Cruz 2019). The former contend that trust is a belief that the trustee is trustworthy, at least in the thin sense that the trustee will do what he is trusted to do (Keren 2020). Various reasons exist in favour of such theories, doxastic reasons (see Keren 2020) including that these theories suggest it is impossible to trust a person while holding the belief that this person is not trustworthy, even in the thin sense. Most of us accept this impossibility and would want any theory of trust to explain it. A doxastic account does so by saying that we can't believe a contradiction (not knowingly anyway; Keren 2020: 113). Those who say that trust is not a belief claim that it is possible to trust without believing the trustee is trustworthy.[19] Holton gives the nice example of trusting a friend to be sincere without believing that the friend will be sincere (1994: 75). Arguably, if one already believed that to be the case, then one would have no need to trust the friend. It is also possible to believe that someone is trustworthy without trusting that person, which suggests that trust couldn't just be a belief in someone's trustworthiness (McLeod 2002: 85). I might think that a particular person is trustworthy without trusting them because I have no cause to do so. I might even distrust them despite believing that they are trustworthy (Jones 1996, 2013). As Jones explains, distrust can be recalcitrant in parting "company with belief" (D'Cruz 2019: 940; citing Jones 2013), a fact which makes trouble for doxastic accounts not just of trust but of distrust too (e.g., Krishnamurthy 2015). The latter must explain how distrust could be a belief that someone is untrustworthy that could exist alongside the belief that the person is trustworthy. Among the alternatives to doxasticism are theories stating that trust is an emotion, a kind of stance (i.e., the participant stance; Holton 1994), or a disposition (Kappel 2014; cited in Keren 2020). The most commonly held alternative is the first: that trust is an emotion. Reasons in favour of this view include the fact that trust resembles an emotion in having characteristics that are unique to emotions, at least according to an influential account of them (de Sousa 1987; Calhoun 1984; Rorty 1980; Lahno 2001, 2020). For example, emotions narrow our perception to "fields of evidence" that lend support to the emotions themselves (Jones 1996: 11). When we are in the grip of an emotion, we therefore tend to see facts that affirm its existence and ignore those that negate it. To illustrate, if I am really angry at my mother, then I tend to focus on things that justify my anger while ignoring or refusing to see things that make it unjustified. I can only see those other things once my anger subsides. Similarly with trust: if I genuinely trust my mother, my attention falls on those aspects of her that justify my trust and is averted from evidence that suggests she is untrustworthy (Baker 1987). The same sort of thing happens with distrust, according to Jones (Jones 2019). She refers to this phenomenon as "affective looping", which, in her words, occurs when "a prior emotional state provides grounds for its own continuance" (2019: 956). She also insists that only affective-attitude accounts of trust and distrust can adequately explain it (2019). There may be a kind of doxastic theory, however, that can account for the affective looping of trust, if not of distrust. Arnon Keren, whose work focuses specifically on trust, defends what he calls an "impurely doxastic" theory. He describes trust as believing in someone's trustworthiness and responding to reasons ("preemptive" ones) against taking precautions that this person will not be trustworthy (Keren 2020, 2014). Reasons for trust are themselves reasons of this sort, according to Keren; they oppose actions like those of carefully monitoring the behavior of the trustee or weighing the available evidence that this person is trustworthy. The trustor's response to these preemptive reasons would explain why this person is resistant (or at least not attune) to counter evidence to their trust (Keren 2014, 2020). Deciding in favour of an affective-attitude theory or a purely or impurely doxastic one is important for understanding features of trust like affective looping. Yet it may have little bearing on whether or how trust can be cultivated. For, regardless of whether trust is a belief or an emotion, presumably we can cultivate it by purposefully placing ourselves in a position that allows us to focus on evidence of people's trustworthiness. The goal here could be self-improvement: that is, becoming more trusting, in a good way so that we can reap the benefits of justified trust. Alternatively, we might be striving for the improvement of others: making them more trustworthy by trusting them therapeutically. Alternatively still, we could be engaging in "corrective trust". (See the above discussions of therapeutic and corrective trust.) This section has centered on how to develop trust and how to account for facts about it such as the blinkered vision of the trustor. Similar facts about distrust were also mentioned: those that concern what kind of mental attitude it is. Theorizing about whether trust and distrust are beliefs, emotions or something else allows us to appreciate why they have certain features and also how to build these attitudes. The process for building them, which may be similar regardless of whether they are beliefs or emotions, will be relevant to people who don't trust enough or who trust too much. ## 5. Conclusion This entry as a whole has examined an important practical question about trust: "When is trust warranted?" Also woven into the discussion has been some consideration of when distrust is warranted. Centerstage has been given to trust, however, because philosophers have debated it much more than distrust. Different answers to the question of when trust is warranted give rise to different philosophical puzzles. For example, in response, one could appeal to the nature of trust and trustworthiness and consider whether the conditions are ripe for them (e.g., for the proposed trustor to rely on the trustee's competence). But one would first have to settle the difficult issue of what trust and trustworthiness are, and more specifically, how they differ from mere reliance and reliability, assuming there are these differences. Alternatively, in deciding whether trust is warranted, one could consider whether trust would be rationally justified or valuable. One would consider these things simultaneously when rational justification is understood in an end-directed way, making it dependent on trust's instrumental value. With respect to rational justification alone, puzzles arise when trying to sort out whether reasons for trust must be internal to trustors or could be external to them. In other words, is trust's epistemology internalist or externalist? Because good arguments exist on both sides, it's not clear how trust is rationally justified. Neither is it entirely clear what sort of value trust can have, given the nature of it. For example, trust may or may not have intrinsic moral value depending on whether it signals respect for others. Lastly, one might focus on the fact that trust cannot be warranted when it is impossible, which is the case when the agent does not already exhibit trust and cannot simply will themselves to have it. While trust is arguably not the sort of attitude that one can just will oneself to have, trust can be cultivated. The exact manner or extent to which it can be cultivated, however, may depend again on what sort of mental attitude it is. Since one can respond to the question, "When is trust warranted?" by referring to each of the above dimensions of trust, a complete philosophical answer to this question is complex. The same is true about the question of when to distrust, because the same dimensions (the epistemology of distrust, its value, etc.) are relevant to it. Complete answers to these broad questions about trust and distrust would be philosophically exciting and also socially important. They would be exciting both because of their complexity and because they would draw on a number of different philosophical areas, including epistemology, philosophy of mind, and value theory. The answers would be important because trust and distrust that are warranted contribute to the foundation of a good society, where people thrive through healthy cooperation with others, become morally mature human beings, and are not subject to social ills like tyranny or oppression.
truth	## 1. The neo-classical theories of truth Much of the contemporary literature on truth takes as its starting point some ideas which were prominent in the early part of the 20th century. There were a number of views of truth under discussion at that time, the most significant for the contemporary literature being the correspondence, coherence, and pragmatist theories of truth. These theories all attempt to directly answer the nature question: what is the nature of truth? They take this question at face value: there are truths, and the question to be answered concerns their nature. In answering this question, each theory makes the notion of truth part of a more thoroughgoing metaphysics or epistemology. Explaining the nature of truth becomes an application of some metaphysical system, and truth inherits significant metaphysical presuppositions along the way. The goal of this section is to characterize the ideas of the correspondence, coherence and pragmatist theories which animate the contemporary debate. In some cases, the received forms of these theories depart from the views that were actually defended in the early 20th century. We thus dub them the 'neo-classical theories'. Where appropriate, we pause to indicate how the neo-classical theories emerge from their 'classical' roots in the early 20th century. ### 1.1 The correspondence theory Perhaps the most important of the neo-classical theories for the contemporary literature is the correspondence theory. Ideas that sound strikingly like a correspondence theory are no doubt very old. They might well be found in Aristotle or Aquinas. When we turn to the late 19th and early 20th centuries where we pick up the story of the neo-classical theories of truth, it is clear that ideas about correspondence were central to the discussions of the time. In spite of their importance, however, it is strikingly difficult to find an accurate citation in the early 20th century for the received neo-classical view. Furthermore, the way the correspondence theory actually emerged will provide some valuable reference points for the contemporary debate. For these reasons, we dwell on the origins of the correspondence theory in the late 19th and early 20th centuries at greater length than those of the other neo-classical views, before turning to its contemporary neo-classical form. For an overview of the correspondence theory, see David (2018). ### 1.1.1 The origins of the correspondence theory The basic idea of the correspondence theory is that what we believe or say is true if it corresponds to the way things actually are - to the facts. This idea can be seen in various forms throughout the history of philosophy. Its modern history starts with the beginnings of analytic philosophy at the turn of the 20th century, particularly in the work of G. E. Moore and Bertrand Russell. Let us pick up the thread of this story in the years between 1898 and about 1910. These years are marked by Moore and Russell's rejection of idealism. Yet at this point, they do not hold a correspondence theory of truth. Indeed Moore (1899) sees the correspondence theory as a source of idealism, and rejects it. Russell follows Moore in this regard. (For discussion of Moore's early critique of idealism, where he rejects the correspondence theory of truth, see Baldwin (1991). Hylton (1990) provides an extensive discussion of Russell in the context of British idealism. An overview of these issues is given by Baldwin (2018).) In this period, Moore and Russell hold a version of the identity theory of truth. They say comparatively little about it, but it is stated briefly in Moore (1899; 1902) and Russell (1904). According to the identity theory, a true proposition is identical to a fact. Specifically, in Moore and Russell's hands, the theory begins with propositions, understood as the objects of beliefs and other propositional attitudes. Propositions are what are believed, and give the contents of beliefs. They are also, according to this theory, the primary bearers of truth. When a proposition is true, it is identical to a fact, and a belief in that proposition is correct. (Related ideas about the identity theory and idealism are discussed by McDowell (1994) and further developed by Hornsby (2001).) The identity theory Moore and Russell espoused takes truth to be a property of propositions. Furthermore, taking up an idea familiar to readers of Moore, the property of truth is a simple unanalyzable property. Facts are understood as simply those propositions which are true. There are true propositions and false ones, and facts just are true propositions. There is thus no "difference between truth and the reality to which it is supposed to correspond" (Moore, 1902, p. 21). (For further discussion of the identity theory of truth, see Baldwin (1991), Candlish (1999), Candlish and Damnjanovic (2018), Cartwright (1987), Dodd (2000), and the entry on the identity theory of truth.) Moore and Russell came to reject the identity theory of truth in favor of a correspondence theory, sometime around 1910 (as we see in Moore, 1953, which reports lectures he gave in 1910-1911, and Russell, 1910b). They do so because they came to reject the existence of propositions. Why? Among reasons, they came to doubt that there could be any such things as false propositions, and then concluded that there are no such things as propositions at all. Why did Moore and Russell find false propositions problematic? A full answer to this question is a point of scholarship that would take us too far afield. (Moore himself lamented that he could not "put the objection in a clear and convincing way" (1953, p. 263), but see Cartwright (1987) and David (2001) for careful and clear exploration of the arguments.) But very roughly, the identification of facts with true propositions left them unable to see what a false proposition could be other than something which is just like a fact, though false. If such things existed, we would have fact-like things in the world, which Moore and Russell now see as enough to make false propositions count as true. Hence, they cannot exist, and so there are no false propositions. As Russell (1956, p. 223) later says, propositions seem to be at best "curious shadowy things" in addition to facts. As Cartwright (1987) reminds us, it is useful to think of this argument in the context of Russell's slightly earlier views about propositions. As we see clearly in Russell (1903), for instance, he takes propositions to have constituents. But they are not mere collections of constituents, but a 'unity' which brings the constituents together. (We thus confront the 'problem of the unity of the proposition'.) But what, we might ask, would be the 'unity' of a proposition that Samuel Ramey sings - with constituents Ramey and singing - except Ramey bearing the property of singing? If that is what the unity consists in, then we seem to have nothing other than the fact that Ramey sings. But then we could not have genuine false propositions without having false facts. As Cartwright also reminds us, there is some reason to doubt the cogency of this sort of argument. But let us put the assessment of the arguments aside, and continue the story. From the rejection of propositions a correspondence theory emerges. The primary bearers of truth are no longer propositions, but beliefs themselves. In a slogan: > > A belief is true if and only if it corresponds to a fact. > Views like this are held by Moore (1953) and Russell (1910b; 1912). Of course, to understand such a theory, we need to understand the crucial relation of correspondence, as well as the notion of a fact to which a belief corresponds. We now turn to these questions. In doing so, we will leave the history, and present a somewhat more modern reconstruction of a correspondence theory. (For more on facts and proposition in this period, see Sullivan and Johnston (2018).) ### 1.1.2 The neo-classical correspondence theory The correspondence theory of truth is at its core an ontological thesis: a belief is true if there exists an appropriate entity - a fact - to which it corresponds. If there is no such entity, the belief is false. Facts, for the neo-classical correspondence theory, are entities in their own right. Facts are generally taken to be composed of particulars and properties and relations or universals, at least. The neo-classical correspondence theory thus only makes sense within the setting of a metaphysics that includes such facts. Hence, it is no accident that as Moore and Russell turn away from the identity theory of truth, the metaphysics of facts takes on a much more significant role in their views. This perhaps becomes most vivid in the later Russell (1956, p. 182), where the existence of facts is the "first truism." (The influence of Wittgenstein's ideas to appear in the Tractatus (1922) on Russell in this period was strong, and indeed, the Tractatus remains one of the important sources for the neo-classical correspondence theory. For more recent extensive discussions of facts, see Armstrong (1997) and Neale (2001).) Consider, for example, the belief that Ramey sings. Let us grant that this belief is true. In what does its truth consist, according to the correspondence theory? It consists in there being a fact in the world, built from the individual Ramey, and the property of singing. Let us denote this $\langle$Ramey, Singing$\rangle$. This fact exists. In contrast, the world (we presume) contains no fact $\langle$Ramey, Dancing$\rangle$. The belief that Ramey sings stands in the relation of correspondence to the fact $\langle$Ramey, Singing$\rangle$, and so the belief is true. What is the relation of correspondence? One of the standing objections to the classical correspondence theory is that a fully adequate explanation of correspondence proves elusive. But for a simple belief, like that Ramey sings, we can observe that the structure of the fact $\langle$Ramey, Singing$\rangle$ matches the subject-predicate form of the that-clause which reports the belief, and may well match the structure of the belief itself. So far, we have very much the kind of view that Moore and Russell would have found congenial. But the modern form of the correspondence theory seeks to round out the explanation of correspondence by appeal to propositions. Indeed, it is common to base a correspondence theory of truth upon the notion of a structured proposition. Propositions are again cast as the contents of beliefs and assertions, and propositions have structure which at least roughly corresponds to the structure of sentences. At least, for simple beliefs like that Ramey sings, the proposition has the same subject predicate structure as the sentence. (Proponents of structured propositions, such as Kaplan (1989), often look to Russell (1903) for inspiration, and find unconvincing Russell's reasons for rejecting them.) With facts and structured propositions in hand, an attempt may be made to explain the relation of correspondence. Correspondence holds between a proposition and a fact when the proposition and fact have the same structure, and the same constituents at each structural position. When they correspond, the proposition and fact thus mirror each-other. In our simple example, we might have: \[\begin{matrix} \text{proposition that} & \text{Ramey} & \text{sings} \\ & \downarrow & \downarrow \\ \text{fact} & \langle Ramey, & Singing \rangle \end{matrix}\] Propositions, though structured like facts, can be true or false. In a false case, like the proposition that Ramey dances, we would find no fact at the bottom of the corresponding diagram. Beliefs are true or false depending on whether the propositions which are believed are. We have sketched this view for simple propositions like the proposition that Ramey sings. How to extend it to more complex cases, like general propositions or negative propositions, is an issue we will not delve into here. It requires deciding whether there are complex facts, such as general facts or negative facts, or whether there is a more complex relation of correspondence between complex propositions and simple facts. (The issue of whether there are such complex facts marks a break between Russell (1956) and Wittgenstein (1922) and the earlier views which Moore (1953) and Russell (1912) sketch.) According to the correspondence theory as sketched here, what is key to truth is a relation between propositions and the world, which obtains when the world contains a fact that is structurally similar to the proposition. Though this is not the theory Moore and Russell held, it weaves together ideas of theirs with a more modern take on (structured) propositions. We will thus dub it the neo-classical correspondence theory. This theory offers us a paradigm example of a correspondence theory of truth. The leading idea of the correspondence theory is familiar. It is a form of the older idea that true beliefs show the right kind of resemblance to what is believed. In contrast to earlier empiricist theories, the thesis is not that one's ideas per se resemble what they are about. Rather, the propositions which give the contents of one's true beliefs mirror reality, in virtue of entering into correspondence relations to the right pieces of it. In this theory, it is the way the world provides us with appropriately structured entities that explains truth. Our metaphysics thus explains the nature of truth, by providing the entities needed to enter into correspondence relations. For more on the correspondence theory, see David (1994, 2018) and the entry on the correspondance theory of truth. ### 1.2 The coherence theory Though initially the correspondence theory was seen by its developers as a competitor to the identity theory of truth, it was also understood as opposed to the coherence theory of truth. We will be much briefer with the historical origins of the coherence theory than we were with the correspondence theory. Like the correspondence theory, versions of the coherence theory can be seen throughout the history of philosophy. (See, for instance, Walker (1989) for a discussion of its early modern lineage.) Like the correspondence theory, it was important in the early 20th century British origins of analytic philosophy. Particularly, the coherence theory of truth is associated with the British idealists to whom Moore and Russell were reacting. Many idealists at that time did indeed hold coherence theories. Let us take as an example Joachim (1906). (This is the theory that Russell (1910a) attacks.) Joachim says that: > > Truth in its essential nature is that systematic coherence which is > the character of a significant whole (p. 76). > We will not attempt a full exposition of Joachim's view, which would take us well beyond the discussion of truth into the details of British idealism. But a few remarks about his theory will help to give substance to the quoted passage. Perhaps most importantly, Joachim talks of 'truth' in the singular. This is not merely a turn of phrase, but a reflection of his monistic idealism. Joachim insists that what is true is the "whole complete truth" (p. 90). Individual judgments or beliefs are certainly not the whole complete truth. Such judgments are, according to Joachim, only true to a degree. One aspect of this doctrine is a kind of holism about content, which holds that any individual belief or judgment gets its content only in virtue of being part of a system of judgments. But even these systems are only true to a degree, measuring the extent to which they express the content of the single 'whole complete truth'. Any real judgment we might make will only be partially true. To flesh out Joachim's theory, we would have to explain what a significant whole is. We will not attempt that, as it leads us to some of the more formidable aspects of his view, e.g., that it is a "process of self-fulfillment" (p. 77). But it is clear that Joachim takes 'systematic coherence' to be stronger than consistency. In keeping with his holism about content, he rejects the idea that coherence is a relation between independently identified contents, and so finds it necessary to appeal to 'significant wholes'. As with the correspondence theory, it will be useful to recast the coherence theory in a more modern form, which will abstract away from some of the difficult features of British idealism. As with the correspondence theory, it can be put in a slogan: > > A belief is true if and only if it is part of a coherent system of > beliefs. > To further the contrast with the neo-classical correspondence theory, we may add that a proposition is true if it is the content of a belief in the system, or entailed by a belief in the system. We may assume, with Joachim, that the condition of coherence will be stronger than consistency. With the idealists generally, we might suppose that features of the believing subject will come into play. This theory is offered as an analysis of the nature of truth, and not simply a test or criterion for truth. Put as such, it is clearly not Joachim's theory (it lacks his monism, and he rejects propositions), but it is a standard take on coherence in the contemporary literature. (It is the way the coherence theory is given in Walker (1989), for instance. See also Young (2001) for a recent defense of a coherence theory.) Let us take this as our neo-classical version of the coherence theory. The contrast with the correspondence theory of truth is clear. Far from being a matter of whether the world provides a suitable object to mirror a proposition, truth is a matter of how beliefs are related to each-other. The coherence theory of truth enjoys two sorts of motivations. One is primarily epistemological. Most coherence theorists also hold a coherence theory of knowledge; more specifically, a coherence theory of justification. According to this theory, to be justified is to be part of a coherent system of beliefs. An argument for this is often based on the claim that only another belief could stand in a justification relation to a belief, allowing nothing but properties of systems of belief, including coherence, to be conditions for justification. Combining this with the thesis that a fully justified belief is true forms an argument for the coherence theory of truth. (An argument along these lines is found in Blanshard (1939), who holds a form of the coherence theory closely related to Joachim's.) The steps in this argument may be questioned by a number of contemporary epistemological views. But the coherence theory also goes hand-in-hand with its own metaphysics as well. The coherence theory is typically associated with idealism. As we have already discussed, forms of it were held by British idealists such as Joachim, and later by Blanshard (in America). An idealist should see the last step in the justification argument as quite natural. More generally, an idealist will see little (if any) room between a system of beliefs and the world it is about, leaving the coherence theory of truth as an extremely natural option. It is possible to be an idealist without adopting a coherence theory. (For instance, many scholars read Bradley as holding a version of the identity theory of truth. See Baldwin (1991) for some discussion.) However, it is hard to see much of a way to hold the coherence theory of truth without maintaining some form of idealism. If there is nothing to truth beyond what is to be found in an appropriate system of beliefs, then it would seem one's beliefs constitute the world in a way that amounts to idealism. (Walker (1989) argues that every coherence theorist must be an idealist, but not vice-versa.) The neo-classical correspondence theory seeks to capture the intuition that truth is a content-to-world relation. It captures this in the most straightforward way, by asking for an object in the world to pair up with a true proposition. The neo-classical coherence theory, in contrast, insists that truth is not a content-to-world relation at all; rather, it is a content-to-content, or belief-to-belief, relation. The coherence theory requires some metaphysics which can make the world somehow reflect this, and idealism appears to be it. (A distant descendant of the neo-classical coherence theory that does not require idealism will be discussed in section 6.5 below.) For more on the coherence theory, see Walker (2018) and the entry on the coherence theory of truth. ### 1.3 Pragmatist theories A different perspective on truth was offered by the American pragmatists. As with the neo-classical correspondence and coherence theories, the pragmatist theories go with some typical slogans. For example, Peirce is usually understood as holding the view that: > > Truth is the end of inquiry. > (See, for instance Hartshorne et al., 1931-58, SS3.432.) Both Peirce and James are associated with the slogan that: > > Truth is satisfactory to believe. > James (e.g., 1907) understands this principle as telling us what practical value truth has. True beliefs are guaranteed not to conflict with subsequent experience. Likewise, Peirce's slogan tells us that true beliefs will remain settled at the end of prolonged inquiry. Peirce's slogan is perhaps most typically associated with pragmatist views of truth, so we might take it to be our canonical neo-classical theory. However, the contemporary literature does not seem to have firmly settled upon a received 'neo-classical' pragmatist theory. In her reconstruction (upon which we have relied heavily), Haack (1976) notes that the pragmatists' views on truth also make room for the idea that truth involves a kind of correspondence, insofar as the scientific method of inquiry is answerable to some independent world. Peirce, for instance, does not reject a correspondence theory outright; rather, he complains that it provides merely a 'nominal' or 'transcendental' definition of truth (e.g Hartshorne et al., 1931-58, SS5.553, SS5.572), which is cut off from practical matters of experience, belief, and doubt (SS5.416). (See Misak (2004) for an extended discussion.) This marks an important difference between the pragmatist theories and the coherence theory we just considered. Even so, pragmatist theories also have an affinity with coherence theories, insofar as we expect the end of inquiry to be a coherent system of beliefs. As Haack also notes, James maintains an important verificationist idea: truth is what is verifiable. We will see this idea re-appear in section 4. For more on pragmatist theories of truth, see Misak (2018). James' views are discussed further in the entry on William James. Peirce's views are discussed further in the entry on Charles Sanders Peirce. ## 2. Tarski's theory of truth Modern forms of the classical theories survive. Many of these modern theories, notably correspondence theories, draw on ideas developed by Tarski. In this regard, it is important to bear in mind that his seminal work on truth (1935) is very much of a piece with other works in mathematical logic, such as his (1931), and as much as anything this work lays the ground-work for the modern subject of model theory - a branch of mathematical logic, not the metaphysics of truth. In this respect, Tarski's work provides a set of highly useful tools that may be employed in a wide range of philosophical projects. (See Patterson (2012) for more on Tarski's work in its historical context.) Tarski's work has a number of components, which we will consider in turn. ### 2.1 Sentences as truth-bearers In the classical debate on truth at the beginning of the 20th century we considered in section 1, the issue of truth-bearers was of great significance. For instance, Moore and Russell's turn to the correspondence theory was driven by their views on whether there are propositions to be the bearers of truth. Many theories we reviewed took beliefs to be the bearers of truth. In contrast, Tarski and much of the subsequent work on truth takes sentences to be the primary bearers of truth. This is not an entirely novel development: Russell (1956) also takes truth to apply to sentence (which he calls 'propositions' in that text). But whereas much of the classical debate takes the issue of the primary bearers of truth to be a substantial and important metaphysical one, Tarski is quite casual about it. His primary reason for taking sentences as truth-bearers is convenience, and he explicitly distances himself from any commitment about the philosophically contentious issues surrounding other candidate truth-bearers (e.g., Tarski, 1944). (Russell (1956) makes a similar suggestion that sentences are the appropriate truth-bearers "for the purposes of logic" (p. 184), though he still takes the classical metaphysical issues to be important.) We will return to the issue of the primary bearers of truth in section 6.1. For the moment, it will be useful to simply follow Tarski's lead. But it should be stressed that for this discussion, sentences are fully interpreted sentences, having meanings. We will also assume that the sentences in question do not change their content across occasions of use, i.e., that they display no context-dependence. We are taking sentences to be what Quine (1960) calls 'eternal sentences'. In some places (e.g., Tarski, 1944), Tarski refers to his view as the 'semantic conception of truth'. It is not entirely clear just what Tarski had in mind by this, but it is clear enough that Tarski's theory defines truth for sentences in terms of concepts like reference and satisfaction, which are intimately related to the basic semantic functions of names and predicates (according to many approaches to semantics). For more discussion, see Wolenski (2001). ### 2.2 Convention T Let us suppose we have a fixed language $\mathbf{L}$ whose sentences are fully interpreted. The basic question Tarski poses is what an adequate theory of truth for $\mathbf{L}$ would be. Tarski's answer is embodied in what he calls Convention T: > > An adequate theory of truth for $\mathbf{L}$ must imply, for > each sentence $\phi$ of $\mathbf{L}$ > > > $\ulcorner \phi \urcorner$ is true if and only if $\phi$. > (We have simplified Tarski's presentation somewhat.) This is an adequacy condition for theories, not a theory itself. Given the assumption that $\mathbf{L}$ is fully interpreted, we may assume that each sentence $\phi$ in fact has a truth value. In light of this, Convention T guarantees that the truth predicate given by the theory will be extensionally correct, i.e., have as its extension all and only the true sentences of $\mathbf{L}$. Convention T draws our attention to the biconditionals of the form > > $\ulcorner \ulcorner \phi \urcorner$ > is true if and only if $\phi \urcorner$, > which are usually called the Tarski biconditionals for a language $\mathbf{L}$. ### 2.3 Recursive definition of truth Tarski does not merely propose a condition of adequacy for theories of truth, he also shows how to meet it. One of his insights is that if the language $\mathbf{L}$ displays the right structure, then truth for $\mathbf{L}$ can be defined recursively. For instance, let us suppose that $\mathbf{L}$ is a simple formal language, containing two atomic sentences 'snow is white' and 'grass is green', and the sentential connectives $\vee$ and $\neg$. In spite of its simplicity, $\mathbf{L}$ contains infinitely many distinct sentences. But truth can be defined for all of them by recursion. 1. Base clauses: 1. 'Snow is white' is true if and only if snow is white. 2. 'Grass is green' is true if and only if grass is green. 2. Recursion clauses. For any sentences $\phi$ and $\psi$ of $\mathbf{L}$: 1. $\ulcorner \phi \vee \psi \urcorner$ is true if and only if $\ulcorner \phi \urcorner$ is true or $\ulcorner \psi \urcorner$ is true. 2. $\ulcorner \neg \phi \urcorner$ is true if and only if it is not the case that $\ulcorner \phi \urcorner$ is true. This theory satisfies Convention T. ### 2.4 Reference and satisfaction This may look trivial, but in defining an extensionally correct truth predicate for an infinite language with four clauses, we have made a modest application of a very powerful technique. Tarski's techniques go further, however. They do not stop with atomic sentences. Tarski notes that truth for each atomic sentence can be defined in terms of two closely related notions: reference and satisfaction. Let us consider a language $\mathbf{L}'$, just like $\mathbf{L}$ except that instead of simply having two atomic sentences, $\mathbf{L}'$ breaks atomic sentences into terms and predicates. $\mathbf{L}'$ contains terms 'snow' and 'grass' (let us engage in the idealization that these are simply singular terms), and predicates 'is white' and 'is green'. So $\mathbf{L}'$ is like $\mathbf{L}$, but also contains the sentences 'Snow is green' and 'Grass is white'.) We can define truth for atomic sentences of $\mathbf{L}'$ in the following way. 1. Base clauses: 1. 'Snow' refers to snow. 2. 'Grass' refers to grass. 3. $a$ satisfies 'is white' if and only if $a$ is white. 4. $a$ satisfies 'is green' if and only if $a$ is green. 2. For any atomic sentence $\ulcorner t$ is $P \urcorner$: $\ulcorner t$ is $P \urcorner$ is true if and only if the referent of $\ulcorner t \urcorner$ satisfies $\ulcorner P\urcorner$. One of Tarski's key insights is that the apparatus of satisfaction allows for a recursive definition of truth for sentences with quantifiers, though we will not examine that here. We could repeat the recursion clauses for $\mathbf{L}$ to produce a full theory of truth for $\mathbf{L}'$. Let us say that a Tarskian theory of truth is a recursive theory, built up in ways similar to the theory of truth for $\mathbf{L}'$. Tarski goes on to demonstrate some key applications of such a theory of truth. A Tarskian theory of truth for a language $\mathbf{L}$ can be used to show that theories in $\mathbf{L}$ are consistent. This was especially important to Tarski, who was concerned the Liar paradox would make theories in languages containing a truth predicate inconsistent. For more, see Ray (2018) and the entries on axiomatic theories of truth, the Liar paradox, and Tarski's truth definitions. ## 3. Correspondence revisited The correspondence theory of truth expresses the very natural idea that truth is a content-to-world or word-to-world relation: what we say or think is true or false in virtue of the way the world turns out to be. We suggested that, against a background like the metaphysics of facts, it does so in a straightforward way. But the idea of correspondence is certainly not specific to this framework. Indeed, it is controversial whether a correspondence theory should rely on any particular metaphysics at all. The basic idea of correspondence, as Tarski (1944) and others have suggested, is captured in the slogan from Aristotle's Metaphysics G 7.27, "to say of what is that it is, or of what is not that it is not, is true" (Ross, 1928). 'What is', it is natural enough to say, is a fact, but this natural turn of phrase may well not require a full-blown metaphysics of facts. (For a discussion of Aristotle's views in a historical context, see Szaif (2018).) Yet without the metaphysics of facts, the notion of correspondence as discussed in section 1.1 loses substance. This has led to two distinct strands in contemporary thinking about the correspondence theory. One strand seeks to recast the correspondence theory in a way that does not rely on any particular ontology. Another seeks to find an appropriate ontology for correspondence, either in terms of facts or other entities. We will consider each in turn. ### 3.1 Correspondence without facts Tarski himself sometimes suggested that his theory was a kind of correspondence theory of truth. Whether his own theory is a correspondence theory, and even whether it provides any substantial philosophical account of truth at all, is a matter of controversy. (One rather drastic negative assessment from Putnam (1985-86, p. 333) is that "As a philosophical account of truth, Tarski's theory fails as badly as it is possible for an account to fail.") But a number of philosophers (e.g., Davidson, 1969; Field, 1972) have seen Tarski's theory as providing at least the core of a correspondence theory of truth which dispenses with the metaphysics of facts. Tarski's theory shows how truth for a sentence is determined by certain properties of its constituents; in particular, by properties of reference and satisfaction (as well as by the logical constants). As it is normally understood, reference is the preeminent word-to-world relation. Satisfaction is naturally understood as a word-to-world relation as well, which relates a predicate to the things in the world that bear it. The Tarskian recursive definition shows how truth is determined by reference and satisfaction, and so is in effect determined by the things in the world we refer to and the properties they bear. This, one might propose, is all the correspondence we need. It is not correspondence of sentences or propositions to facts; rather, it is correspondence of our expressions to objects and the properties they bear, and then ways of working out the truth of claims in terms of this. This is certainly not the neo-classical idea of correspondence. In not positing facts, it does not posit any single object to which a true proposition or sentence might correspond. Rather, it shows how truth might be worked out from basic word-to-world relations. However, a number of authors have noted that Tarski's theory cannot by itself provide us with such an account of truth. As we will discuss more fully in section 4.2, Tarski's apparatus is in fact compatible with theories of truth that are certainly not correspondence theories. Field (1972), in an influential discussion and diagnosis of what is lacking in Tarski's account, in effect points out that whether we really have something worthy of the name 'correspondence' depends on our having notions of reference and satisfaction which genuinely establish word-to-world relations. (Field does not use the term 'correspondence', but does talk about e.g., the "connection between words and things" (p. 373).) By itself, Field notes, Tarski's theory does not offer an account of reference and satisfaction at all. Rather, it offers a number of disquotation clauses, such as: 1. 'Snow' refers to snow. 2. $a$ satisfies 'is white' if and only if $a$ is white. These clauses have an air of triviality (though whether they are to be understood as trivial principles or statements of non-trivial semantic facts has been a matter of some debate). With Field, we might propose to supplement clauses like these with an account of reference and satisfaction. Such a theory should tell us what makes it the case that the word 'snow' refer to snow. (In 1972, Field was envisaging a physicalist account, along the lines of the causal theory of reference.) This should inter alia guarantee that truth is really determined by word-to-world relations, so in conjunction with the Tarskian recursive definition, it could provide a correspondence theory of truth. Such a theory clearly does not rely on a metaphysics of facts. Indeed, it is in many ways metaphysically neutral, as it does not take a stand on the nature of particulars, or of the properties or universals that underwrite facts about satisfaction. However, it may not be entirely devoid of metaphysical implications, as we will discuss further in section 4.1. ### 3.2 Representation and Correspondence Much of the subsequent discussion of Field-style approaches to correspondence has focused on the role of representation in these views. Field's own (1972) discussion relies on a causal relation between terms and their referents, and a similar relation for satisfaction. These are instances of representation relations. According to representational views, meaningful items, like perhaps thoughts or sentences or their constituents, have their contents in virtue of standing in the right relation to the things they represent. On many views, including Field's, a name stands in such a relation to its bearer, and the relation is a causal one. The project of developing a naturalist account of the representation relation has been an important one in the philosophy of mind and language. (See the entry on mental representation.) But, it has implications for the theory of truth. Representational views of content lead naturally to correspondence theories of truth. To make this vivid, suppose you hold that sentences or beliefs stand in a representation relation to some objects. It is natural to suppose that for true beliefs or sentences, those objects would be facts. We then have a correspondence theory, with the correspondence relation explicated as a representation relation: a truth bearer is true if it represents a fact. As we have discussed, many contemporary views reject facts, but one can hold a representational view of content without them. One interpretation of Field's theory is just that. The relations of reference and satisfaction are representation relations, and truth for sentences is determined compositionally in terms of those representation relations, and the nature of the objects they represent. If we have such relations, we have the building blocks for a correspondence theory without facts. Field (1972) anticipated a naturalist reduction of the representation via a causal theory, but any view that accepts representation relations for truth bearers or their constituents can provide a similar theory of truth. (See Jackson (2006) and Lynch (2009) for further discussion.) Representational views of content provide a natural way to approach the correspondence theory of truth, and likewise, anti-representational views provide a natural way to avoid the correspondence theory of truth. This is most clear in the work of Davidson, as we will discuss more in section 6.5. ### 3.3 Facts again There have been a number of correspondence theories that do make use of facts. Some are notably different from the neo-classical theory sketched in section 1.1. For instance, Austin (1950) proposes a view in which each statement (understood roughly as an utterance event) corresponds to both a fact or situation, and a type of situation. It is true if the former is of the latter type. This theory, which has been developed by situation theory (e.g., Barwise and Perry, 1986), rejects the idea that correspondence is a kind of mirroring between a fact and a proposition. Rather, correspondence relations to Austin are entirely conventional. (See Vision (2004) for an extended defense of an Austinian correspondence theory.) As an ordinary language philosopher, Austin grounds his notion of fact more in linguistic usage than in an articulated metaphysics, but he defends his use of fact-talk in Austin (1961b). In a somewhat more Tarskian spirit, formal theories of facts or states of affairs have also been developed. For instance, Taylor (1976) provides a recursive definition of a collection of 'states of affairs' for a given language. Taylor's states of affairs seem to reflect the notion of fact at work in the neo-classical theory, though as an exercise in logic, they are officially $n$-tuples of objects and intensions. There are more metaphysically robust notions of fact in the current literature. For instance, Armstrong (1997) defends a metaphysics in which facts (under the name 'states of affairs') are metaphysically fundamental. The view has much in common with the neo-classical one. Like the neo-classical view, Armstrong endorses a version of the correspondence theory. States of affairs are truthmakers for propositions, though Armstrong argues that there may be many such truthmakers for a given proposition, and vice versa. (Armstrong also envisages a naturalistic account of propositions as classes of equivalent belief-tokens.) Armstrong's primary argument is what he calls the 'truthmaker argument'. It begins by advancing a truthmaker principle, which holds that for any given truth, there must be a truthmaker - a "something in the world which makes it the case, that serves as an ontological ground, for this truth" (p. 115). It is then argued that facts are the appropriate truthmakers. In contrast to the approach to correspondence discussed in section 3.1, which offered correspondence with minimal ontological implications, this view returns to the ontological basis of correspondence that was characteristic of the neo-classical theory. For more on facts, see the entry on facts. ### 3.4 Truthmakers The truthmaker principle is often put as the schema: > > If $\phi$, then there is an $x$ such that necessarily, if > $x$ exists, then $\phi$. > (Fox (1987) proposed putting the principle this way, rather than explicitly in terms of truth.) The truthmaker principle expresses the ontological aspect of the neo-classical correspondence theory. Not merely must truth obtain in virtue of word-to-world relations, but there must be a thing that makes each truth true. (For one view on this, see Merricks (2007).) The neo-classical correspondence theory, and Armstrong, cast facts as the appropriate truthmakers. However, it is a non-trivial step from the truthmaker principle to the existence of facts. There are a number of proposals in the literature for how other sorts of objects could be truthmakers; for instance, tropes (called 'moments', in Mulligan et al., 1984). Parsons (1999) argues that the truthmaker principle (presented in a somewhat different form) is compatible with there being only concrete particulars. As we saw in discussing the neo-classical correspondence theory, truthmaker theories, and fact theories in particular, raise a number of issues. One which has been discussed at length, for instance, is whether there are negative facts. Negative facts would be the truthmakers for negated sentences. Russell (1956) notoriously expresses ambivalence about whether there are negative facts. Armstrong (1997) rejects them, while Beall (2000) defends them. (For more discussion of truthmakers, see Cameron (2018) and the papers in Beebee and Dodd (2005).) ## 4. Realism and anti-realism The neo-classical theories we surveyed in section 1 made the theory of truth an application of their background metaphysics (and in some cases epistemology). In section 2 and especially in section 3, we returned to the issue of what sorts of ontological commitments might go with the theory of truth. There we saw a range of options, from relatively ontologically non-committal theories, to theories requiring highly specific ontologies. There is another way in which truth relates to metaphysics. Many ideas about realism and anti-realism are closely related to ideas about truth. Indeed, many approaches to questions about realism and anti-realism simply make them questions about truth. ### 4.1 Realism and truth In discussing the approach to correspondence of section 3.1, we noted that it has few ontological requirements. It relies on there being objects of reference, and something about the world which makes for determinate satisfaction relations; but beyond that, it is ontologically neutral. But as we mentioned there, this is not to say that it has no metaphysical implications. A correspondence theory of truth, of any kind, is often taken to embody a form of realism. The key features of realism, as we will take it, are that: 1. The world exists objectively, independently of the ways we think about it or describe it. 2. Our thoughts and claims are about that world. (Wright (1992) offers a nice statement of this way of thinking about realism.) These theses imply that our claims are objectively true or false, depending on how the world they are about is. The world that we represent in our thoughts or language is an objective world. (Realism may be restricted to some subject-matter, or range of discourse, but for simplicity, we will talk about only its global form.) It is often argued that these theses require some form of the correspondence theory of truth. (Putnam (1978, p. 18) notes, "Whatever else realists say, they typically say that they believe in a 'correspondence theory of truth'.") At least, they are supported by the kind of correspondence theory without facts discussed in section 3.1, such as Field's proposal. Such a theory will provide an account of objective relations of reference and satisfaction, and show how these determine the truth or falsehood of what we say about the world. Field's own approach (1972) to this problem seeks a physicalist explanation of reference. But realism is a more general idea than physicalism. Any theory that provides objective relations of reference and satisfaction, and builds up a theory of truth from them, would give a form of realism. (Making the objectivity of reference the key to realism is characteristic of work of Putnam, e.g., 1978.) Another important mark of realism expressed in terms of truth is the property of bivalence. As Dummett has stressed (e.g., 1959; 1976; 1983; 1991), a realist should see there being a fact of the matter one way or the other about whether any given claim is correct. Hence, one important mark of realism is that it goes together with the principle of bivalence: every truth-bearer (sentence or proposition) is true or false. In much of his work, Dummett has made this the characteristic mark of realism, and often identifies realism about some subject-matter with accepting bivalence for discourse about that subject-matter. At the very least, it captures a great deal of what is more loosely put in the statement of realism above. Both the approaches to realism, through reference and through bivalence, make truth the primary vehicle for an account of realism. A theory of truth which substantiates bivalence, or builds truth from a determinate reference relation, does most of the work of giving a realistic metaphysics. It might even simply be a realistic metaphysics. We have thus turned on its head the relation of truth to metaphysics we saw in our discussion of the neo-classical correspondence theory in section 1.1. There, a correspondence theory of truth was built upon a substantial metaphysics. Here, we have seen how articulating a theory that captures the idea of correspondence can be crucial to providing a realist metaphysics. (For another perspective on realism and truth, see Alston (1996). Devitt (1984) offers an opposing view to the kind we have sketched here, which rejects any characterization of realism in terms of truth or other semantic concepts.) In light of our discussion in section 1.1.1, we should pause to note that the connection between realism and the correspondence theory of truth is not absolute. When Moore and Russell held the identity theory of truth, they were most certainly realists. The right kind of metaphysics of propositions can support a realist view, as can a metaphysics of facts. The modern form of realism we have been discussing here seeks to avoid basing itself on such particular ontological commitments, and so prefers to rely on the kind of correspondence-without-facts approach discussed in section 3.1. This is not to say that realism will be devoid of ontological commitments, but the commitments will flow from whichever specific claims about some subject-matter are taken to be true. For more on realism and truth, see Fumerton (2002) and the entry on realism. ### 4.2 Anti-realism and truth It should come as no surprise that the relation between truth and metaphysics seen by modern realists can also be exploited by anti-realists. Many modern anti-realists see the theory of truth as the key to formulating and defending their views. With Dummett (e.g., 1959; 1976; 1991), we might expect the characteristic mark of anti-realism to be the rejection of bivalence. Indeed, many contemporary forms of anti-realism may be formulated as theories of truth, and they do typically deny bivalence. Anti-realism comes in many forms, but let us take as an example a (somewhat crude) form of verificationism. Such a theory holds that a claim is correct just insofar as it is in principle verifiable, i.e., there is a verification procedure we could in principle carry out which would yield the answer that the claim in question was verified. So understood, verificationism is a theory of truth. The claim is not that verification is the most important epistemic notion, but that truth just is verifiability. As with the kind of realism we considered in section 4.1, this view expresses its metaphysical commitments in its explanation of the nature of truth. Truth is not, to this view, a fully objective matter, independent of us or our thoughts. Instead, truth is constrained by our abilities to verify, and is thus constrained by our epistemic situation. Truth is to a significant degree an epistemic matter, which is typical of many anti-realist positions. As Dummett says, the verificationist notion of truth does not appear to support bivalence. Any statement that reaches beyond what we can in principle verify or refute (verify its negation) will be a counter-example to bivalence. Take, for instance, the claim that there is some substance, say uranium, present in some region of the universe too distant to be inspected by us within the expected lifespan of the universe. Insofar as this really would be in principle unverifiable, we have no reason to maintain it is true or false according to the verificationist theory of truth. Verificationism of this sort is one of a family of anti-realist views. Another example is the view that identifies truth with warranted assertibility. Assertibility, as well as verifiability, has been important in Dummett's work. (See also works of McDowell, e.g., 1976 and Wright, e.g., 1976; 1982; 1992.) Anti-realism of the Dummettian sort is not a descendant of the coherence theory of truth per se. But in some ways, as Dummett himself has noted, it might be construed as a descendant - perhaps very distant - of idealism. If idealism is the most drastic form of rejection of the independence of mind and world, Dummettian anti-realism is a more modest form, which sees epistemology imprinted in the world, rather than the wholesale embedding of world into mind. At the same time, the idea of truth as warranted assertibility or verifiability reiterates a theme from the pragmatist views of truth we surveyed in section 1.3. Anti-realist theories of truth, like the realist ones we discussed in section 4.1, can generally make use of the Tarskian apparatus. Convention T, in particular, does not discriminate between realist and anti-realist notions of truth. Likewise, the base clauses of a Tarskian recursive theory are given as disquotation principles, which are neutral between realist and anti-realist understandings of notions like reference. As we saw with the correspondence theory, giving a full account of the nature of truth will generally require more than the Tarskian apparatus itself. How an anti-realist is to explain the basic concepts that go into a Tarskian theory is a delicate matter. As Dummett and Wright have investigated in great detail, it appears that the background logic in which the theory is developed will have to be non-classical. For more on anti-realism and truth, see Shieh (2018) and the papers in Greenough and Lynch (2006) and the entry on realism. ### 4.3 Anti-realism and pragmatism Many commentators see a close connection between Dummett's anti-realism and the pragmatists' views of truth, in that both put great weight on ideas of verifiability or assertibility. Dummett himself stressed parallels between anti-realism and intuitionism in the philosophy of mathematics. Another view on truth which returns to pragmatist themes is the 'internal realism' of Putnam (1981). There Putnam glosses truth as what would be justified under ideal epistemic conditions. With the pragmatists, Putnam sees the ideal conditions as something which can be approximated, echoing the idea of truth as the end of inquiry. Putnam is cautious about calling his view anti-realism, preferring the label 'internal realism'. But he is clear that he sees his view as opposed to realism ('metaphysical realism', as he calls it). Davidson's views on truth have also been associated with pragmatism, notably by Rorty (1986). Davidson has distanced himself from this interpretation (e.g., 1990), but he does highlight connections between truth and belief and meaning. Insofar as these are human attitudes or relate to human actions, Davidson grants there is some affinity between his views and those of some pragmatists (especially, he says, Dewey). ### 4.4 Truth pluralism Another view that has grown out of the literature on realism and anti-realism, and has become increasingly important in the current literature, is that of pluralism about truth. This view, developed in work of Lynch (e.g. 2001b; 2009) and Wright (e.g. 1992; 1999), proposes that there are multiple ways for truth bearers to be true. Wright, in particular, suggests that in certain domains of discourse what we say is true in virtue of a correspondence-like relation, while in others it is its true in virtue of a kind of assertibility relation that is closer in spirit to the anti-realist views we have just discussed. Such a proposal might suggest there are multiple concepts of truth, or that the term 'true' is itself ambiguous. However, whether or not a pluralist view is committed to such claims has been disputed. In particular, Lynch (2001b; 2009) develops a version of pluralism which takes truth to be a functional role concept. The functional role of truth is characterized by a range of principles that articulate such features of truth as its objectivity, its role in inquiry, and related ideas we have encountered in considering various theories of truth. (A related point about platitudes governing the concept of truth is made by Wright (1992).) But according to Lynch, these display the functional role of truth. Furthermore, Lynch claims that on analogy with analytic functionalism, these principles can be seen as deriving from our pre-theoretic or 'folk' ideas about truth. Like all functional role concepts, truth must be realized, and according to Lynch it may be realized in different ways in different settings. Such multiple realizability has been one of the hallmarks of functional role concepts discussed in the philosophy of mind. For instance, Lynch suggests that for ordinary claims about material objects, truth might be realized by a correspondence property (which he links to representational views), while for moral claims truth might be manifest by an assertibility property along more anti-realist lines. For more on pluralism about truth, see Pedersen and Lynch (2018) and the entry on pluralist theories of truth. ## 5. Deflationism We began in section 1 with the neo-classical theories, which explained the nature of truth within wider metaphysical systems. We then considered some alternatives in sections 2 and 3, some of which had more modest ontological implications. But we still saw in section 4 that substantial theories of truth tend to imply metaphysical theses, or even embody metaphysical positions. One long-standing trend in the discussion of truth is to insist that truth really does not carry metaphysical significance at all. It does not, as it has no significance on its own. A number of different ideas have been advanced along these lines, under the general heading of deflationism. ### 5.1 The redundancy theory Deflationist ideas appear quite early on, including a well-known argument against correspondence in Frege (1918-19). However, many deflationists take their cue from an idea of Ramsey (1927), often called the equivalence thesis: > > $\ulcorner \ulcorner \phi \urcorner$ > is true $\urcorner$ has the same meaning as $\phi$. > (Ramsey himself takes truth-bearers to be propositions rather than sentences. Glanzberg (2003b) questions whether Ramsey's account of propositions really makes him a deflationist.) This can be taken as the core of a theory of truth, often called the redundancy theory. The redundancy theory holds that there is no property of truth at all, and appearances of the expression 'true' in our sentences are redundant, having no effect on what we express. The equivalence thesis can also be understood in terms of speech acts rather than meaning: > > To assert that $\ulcorner \phi \urcorner$ is true is > just to assert that $\phi$. > This view was advanced by Strawson (1949; 1950), though Strawson also argues that there are other important aspects of speech acts involving 'true' beyond what is asserted. For instance, they may be acts of confirming or granting what someone else said. (Strawson would also object to my making sentences the bearers of truth.) In either its speech act or meaning form, the redundancy theory argues there is no property of truth. It is commonly noted that the equivalence thesis itself is not enough to sustain the redundancy theory. It merely holds that when truth occurs in the outermost position in a sentence, and the full sentence to which truth is predicated is quoted, then truth is eliminable. What happens in other environments is left to be seen. Modern developments of the redundancy theory include Grover et al. (1975). ### 5.2 Minimalist theories The equivalence principle looks familiar: it has something like the form of the Tarski biconditionals discussed in section 2.2. However, it is a stronger principle, which identifies the two sides of the biconditional - either their meanings or the speech acts performed with them. The Tarski biconditionals themselves are simply material biconditionals. A number of deflationary theories look to the Tarski biconditionals rather than the full equivalence principle. Their key idea is that even if we do not insist on redundancy, we may still hold the following theses: 1. For a given language $\mathbf{L}$ and every $\phi$ in $\mathbf{L}$, the biconditionals $\ulcorner \ulcorner \phi \urcorner$ is true if and only if $\phi \urcorner$ hold by definition (or analytically, or trivially, or by stipulation ...). 2. This is all there is to say about the concept of truth. We will refer to views which adopt these as minimalist. Officially, this is the name of the view of Horwich (1990), but we will apply it somewhat more widely. (Horwich's view differs in some specific respects from what is presented here, such as predicating truth of propositions, but we believe it is close enough to what is sketched here to justify the name.) The second thesis, that the Tarski biconditionals are all there is to say about truth, captures something similar to the redundancy theory's view. It comes near to saying that truth is not a property at all; to the extent that truth is a property, there is no more to it than the disquotational pattern of the Tarski biconditionals. As Horwich puts it, there is no substantial underlying metaphysics to truth. And as Soames (1984) stresses, certainly nothing that could ground as far-reaching a view as realism or anti-realism. ### 5.3 Other aspects of deflationism If there is no property of truth, or no substantial property of truth, what role does our term 'true' play? Deflationists typically note that the truth predicate provides us with a convenient device of disquotation. Such a device allows us to make some useful claims which we could not formulate otherwise, such as the blind ascription 'The next thing that Bill says will be true'. (For more on blind ascriptions and their relation to deflationism, see Azzouni, 2001.) A predicate obeying the Tarski biconditionals can also be used to express what would otherwise be (potentially) infinite conjunctions or disjunctions, such as the notorious statement of Papal infallibility put 'Everything the Pope says is true'. (Suggestions like this are found in Leeds, 1978 and Quine, 1970.) Recognizing these uses for a truth predicate, we might simply think of it as introduced into a language by stipulation. The Tarski biconditionals themselves might be stipulated, as the minimalists envisage. One could also construe the clauses of a recursive Tarskian theory as stipulated. (There are some significant logical differences between these two options. See Halbach (1999) and Ketland (1999) for discussion.) Other deflationists, such as Beall (2005) or Field (1994), might prefer to focus here on rules of inference or rules of use, rather than the Tarski biconditionals themselves. There are also important connections between deflationist ideas about truth and certain ideas about meaning. These are fundamental to the deflationism of Field (1986; 1994), which will be discussed in section 6.3. For an insightful critique of deflationism, see Gupta (1993). For more on deflationism, see Azzouni (2018) and the entry on the deflationary theory of truth. ## 6. Truth and language One of the important themes in the literature on truth is its connection to meaning, or more generally, to language. This has proved an important application of ideas about truth, and an important issue in the study of truth itself. This section will consider a number of issues relating truth and language. ### 6.1 Truth-bearers There have been many debates in the literature over what the primary bearers of truth are. Candidates typically include beliefs, propositions, sentences, and utterances. We have already seen in section 1 that the classical debates on truth took this issue very seriously, and what sort of theory of truth was viable was often seen to depend on what the bearers of truth are. In spite of the number of options under discussion, and the significance that has sometimes been placed on the choice, there is an important similarity between candidate truth-bearers. Consider the role of truth-bearers in the correspondence theory, for instance. We have seen versions of it which take beliefs, propositions, or interpreted sentences to be the primary bearers of truth. But all of them rely upon the idea that their truth-bearers are meaningful, and are thereby able to say something about what the world is like. (We might say that they are able to represent the world, but that is to use 'represent' in a wider sense than we saw in section 3.2. No assumptions about just what stands in relations to what objects are required to see truth-bearers as meaningful.) It is in virtue of being meaningful that truth-bearers are able to enter into correspondence relations. Truth-bearers are things which meaningfully make claims about what the world is like, and are true or false depending on whether the facts in the world are as described. Exactly the same point can be made for the anti-realist theories of truth we saw in section 4.2, though with different accounts of how truth-bearers are meaningful, and what the world contributes. Though it is somewhat more delicate, something similar can be said for coherence theories, which usually take beliefs, or whole systems of beliefs, as the primary truth-bearers. Though a coherence theory will hardly talk of beliefs representing the facts, it is crucial to the coherence theory that beliefs are contentful beliefs of agents, and that they can enter into coherence relations. Noting the complications in interpreting the genuine classical coherence theories, it appears fair to note that this requires truth-bearers to be meaningful, however the background metaphysics (presumably idealism) understands meaning. Though Tarski works with sentences, the same can be said of his theory. The sentences to which Tarski's theory applies are fully interpreted, and so also are meaningful. They characterize the world as being some way or another, and this in turn determines whether they are true or false. Indeed, Tarski needs there to be a fact of the matter about whether each sentence is true or false (abstracting away from context dependence), to ensure that the Tarski biconditionals do their job of fixing the extension of 'is true'. (But note that just what this fact of the matter consists in is left open by the Tarskian apparatus.) We thus find the usual candidate truth-bearers linked in a tight circle: interpreted sentences, the propositions they express, the belief speakers might hold towards them, and the acts of assertion they might perform with them are all connected by providing something meaningful. This makes them reasonable bearers of truth. For this reason, it seems, contemporary debates on truth have been much less concerned with the issue of truth-bearers than were the classical ones. Some issues remain, of course. Different metaphysical assumptions may place primary weight on some particular node in the circle, and some metaphysical views still challenge the existence of some of the nodes. Perhaps more importantly, different views on the nature of meaning itself might cast doubt on the coherence of some of the nodes. Notoriously for instance, Quineans (e.g., Quine, 1960) deny the existence of intensional entities, including propositions. Even so, it increasingly appears doubtful that attention to truth per se will bias us towards one particular primary bearer of truth. For more on these issues, see King (2018). ### 6.2 Truth and truth conditions There is a related, but somewhat different point, which is important to understanding the theories we have canvassed. The neo-classical theories of truth start with truth-bearers which are already understood to be meaningful, and explain how they get their truth values. But along the way, they often do something more. Take the neo-classical correspondence theory, for instance. This theory, in effect, starts with a view of how propositions are meaningful. They are so in virtue of having constituents in the world, which are brought together in the right way. There are many complications about the nature of meaning, but at a minimum, this tells us what the truth conditions associated with a proposition are. The theory then explains how such truth conditions can lead to the truth value true, by the right fact existing. Many theories of truth are like the neo-classical correspondence theory in being as much theories of how truth-bearers are meaningful as of how their truth values are fixed. Again, abstracting from some complications about meaning, this makes them theories both of truth conditions and truth values. The Tarskian theory of truth can be construed this way too. This can be seen both in the way the Tarski biconditionals are understood, and how a recursive theory of truth is understood. As we explained Convention T in section 2.2, the primary role of a Tarski biconditional of the form $\ulcorner \ulcorner \phi \urcorner$ is true if and only if $\phi \urcorner$ is to fix whether $\phi$ is in the extension of 'is true' or not. But it can also be seen as stating the truth conditions of $\phi$. Both rely on the fact that the unquoted occurrence of $\phi$ is an occurrence of an interpreted sentence, which has a truth value, but also provides its truth conditions upon occasions of use. Likewise, the base clauses of the recursive definition of truth, those for reference and satisfaction, are taken to state the relevant semantic properties of constituents of an interpreted sentence. In discussing Tarski's theory of truth in section 2, we focused on how these determine the truth value of a sentence. But they also show us the truth conditions of a sentence are determined by these semantic properties. For instance, for a simple sentence like 'Snow is white', the theory tells us that the sentence is true if the referent of 'Snow' satisfies 'white'. This can be understood as telling us that the truth conditions of 'Snow is white' are those conditions in which the referent of 'Snow' satisfies the predicate 'is white'. As we saw in sections 3 and 4, the Tarskian apparatus is often seen as needing some kind of supplementation to provide a full theory of truth. A full theory of truth conditions will likewise rest on how the Tarskian apparatus is put to use. In particular, just what kinds of conditions those in which the referent of 'snow' satisfies the predicate 'is white' are will depend on whether we opt for realist or anti-realist theories. The realist option will simply look for the conditions under which the stuff snow bears the property of whiteness; the anti-realist option will look to the conditions under which it can be verified, or asserted with warrant, that snow is white. There is a broad family of theories of truth which are theories of truth conditions as well as truth values. This family includes the correspondence theory in all its forms - classical and modern. Yet this family is much wider than the correspondence theory, and wider than realist theories of truth more generally. Indeed, virtually all the theories of truth that make contributions to the realism/anti-realism debate are theories of truth conditions. In a slogan, for many approaches to truth, a theory of truth is a theory of truth conditions. ### 6.3 Truth conditions and deflationism Any theory that provides a substantial account of truth conditions can offer a simple account of truth values: a truth-bearer provides truth conditions, and it is true if and only if the actual way things are is among them. Because of this, any such theory will imply a strong, but very particular, biconditional, close in form to the Tarski biconditionals. It can be made most vivid if we think of propositions as sets of truth conditions. Let $p$ be a proposition, i.e., a set of truth conditions, and let $a$ be the 'actual world', the condition that actually obtains. Then we can almost trivially see: > > $p$ is true if and only if $a \in p$. > This is presumably necessary. But it is important to observe that it is in one respect crucially different from the genuine Tarski biconditionals. It makes no use of a non-quoted sentence, or in fact any sentence at all. It does not have the disquotational character of the Tarski biconditionals. Though this may look like a principle that deflationists should applaud, it is not. Rather, it shows that deflationists cannot really hold a truth-conditional view of content at all. If they do, then they inter alia have a non-deflationary theory of truth, simply by linking truth value to truth conditions through the above biconditional. It is typical of thoroughgoing deflationist theories to present a non-truth-conditional theory of the contents of sentences: a non-truth-conditional account of what makes truth-bearers meaningful. We take it this is what is offered, for instance, by the use theory of propositions in Horwich (1990). It is certainly one of the leading ideas of Field (1986; 1994), which explore how a conceptual role account of content would ground a deflationist view of truth. Once one has a non-truth-conditional account of content, it is then possible to add a deflationist truth predicate, and use this to give purely deflationist statements of truth conditions. But the starting point must be a non-truth-conditional view of what makes truth-bearers meaningful. Both deflationists and anti-realists start with something other than correspondence truth conditions. But whereas an anti-realist will propose a different theory of truth conditions, a deflationists will start with an account of content which is not a theory of truth conditions at all. The deflationist will then propose that the truth predicate, given by the Tarski biconditionals, is an additional device, not for understanding content, but for disquotation. It is a useful device, as we discussed in section 5.3, but it has nothing to do with content. To a deflationist, the meaningfulness of truth-bearers has nothing to do with truth. ### 6.4 Truth and the theory of meaning It has been an influential idea, since the seminal work of Davidson (e.g., 1967), to see a Tarskian theory of truth as a theory of meaning. At least, as we have seen, a Tarskian theory can be seen as showing how the truth conditions of a sentence are determined by the semantic properties of its parts. More generally, as we see in much of the work of Davidson and of Dummett (e.g., 1959; 1976; 1983; 1991), giving a theory of truth conditions can be understood as a crucial part of giving a theory of meaning. Thus, any theory of truth that falls into the broad category of those which are theories of truth conditions can be seen as part of a theory of meaning. (For more discussion of these issues, see Higginbotham (1986; 1989) and the exchange between Higginbotham (1992) and Soames (1992).) A number of commentators on Tarski (e.g., Etchemendy, 1988; Soames, 1984) have observed that the Tarskian apparatus needs to be understood in a particular way to make it suitable for giving a theory of meaning. Tarski's work is often taken to show how to define a truth predicate. If it is so used, then whether or not a sentence is true becomes, in essence, a truth of mathematics. Presumably what truth conditions sentences of a natural language have is a contingent matter, so a truth predicate defined in this way cannot be used to give a theory of meaning for them. But the Tarskian apparatus need not be used just to explicitly define truth. The recursive characterization of truth can be used to state the semantic properties of sentences and their constituents, as a theory of meaning should. In such an application, truth is not taken to be explicitly defined, but rather the truth conditions of sentences are taken to be described. (See Heck, 1997 for more discussion.) ### 6.5 The coherence theory and meaning Inspired by Quine (e.g., 1960), Davidson himself is well known for taking a different approach to using a theory of truth as a theory of meaning than is implicit in Field (1972). Whereas a Field-inspired representational approach is based on a causal account of reference, Davidson (e.g., 1973) proposes a process of radical interpretation in which an interpreter builds a Tarskian theory to interpret a speaker as holding beliefs which are consistent, coherent, and largely true. This led Davidson (e.g. 1986) to argue that most of our beliefs are true - a conclusion that squares well with the coherence theory of truth. This is a weaker claim than the neo-classical coherence theory would make. It does not insist that all the members of any coherent set of beliefs are true, or that truth simply consists in being a member of such a coherent set. But all the same, the conclusion that most of our beliefs are true, because their contents are to be understood through a process of radical interpretation which will make them a coherent and rational system, has a clear affinity with the neo-classical coherence theory. In Davidson (1986), he thought his view of truth had enough affinity with the neo-classical coherence theory to warrant being called a coherence theory of truth, while at the same time he saw the role of Tarskian apparatus as warranting the claim that his view was also compatible with a kind of correspondence theory of truth. In later work, however, Davidson reconsidered this position. In fact, already in Davidson (1977) he had expressed doubt about any understanding of the role of Tarski's theory in radical interpretation that involves the kind of representational apparatus relied on by Field (1972), as we discussed in sections 3.1 and 3.2. In the "Afterthoughts" to Davidson (1986), he also concluded that his view departs too far from the neo-classical coherence theory to be named one. What is important is rather the role of radical interpretation in the theory of content, and its leading to the idea that belief is veridical. These are indeed points connected to coherence, but not to the coherence theory of truth per se. They also comprise a strong form of anti-representationalism. Thus, though he does not advance a coherence theory of truth, he does advance a theory that stands in opposition to the representational variants of the correspondence theory we discussed in section 3.2. For more on Davidson, see Glanzberg (2013) and the entry on Donald Davidson. ### 6.6 Truth and assertion The relation between truth and meaning is not the only place where truth and language relate closely. Another is the idea, also much-stressed in the writings of Dummett (e.g., 1959), of the relation between truth and assertion. Again, it fits into a platitude: > > Truth is the aim of assertion. > A person making an assertion, the platitude holds, aims to say something true. It is easy to cast this platitude in a way that appears false. Surely, many speakers do not aim to say something true. Any speaker who lies does not. Any speaker whose aim is to flatter, or to deceive, aims at something other than truth. The motivation for the truth-assertion platitude is rather different. It looks at assertion as a practice, in which certain rules are constitutive. As is often noted, the natural parallel here is with games, like chess or baseball, which are defined by certain rules. The platitude holds that it is constitutive of the practice of making assertions that assertions aim at truth. An assertion by its nature presents what it is saying as true, and any assertion which fails to be true is ipso facto liable to criticism, whether or not the person making the assertion themself wished to have said something true or to have lied. Dummett's original discussion of this idea was partially a criticism of deflationism (in particular, of views of Strawson, 1950). The idea that we fully explain the concept of truth by way of the Tarski biconditionals is challenged by the claim that the truth-assertion platitude is fundamental to truth. As Dummett there put it, what is left out by the Tarski biconditionals, and captured by the truth-assertion platitude, is the point of the concept of truth, or what the concept is used for. (For further discussion, see Glanzberg, 2003a and Wright, 1992.) Whether or not assertion has such constitutive rules is, of course, controversial. But among those who accept that it does, the place of truth in the constitutive rules is itself controversial. The leading alternative, defended by Williamson (1996), is that knowledge, not truth, is fundamental to the constitutive rules of assertion. Williamson defends an account of assertion based on the rule that one must assert only what one knows. For more on truth and assertion, see the papers in Brown and Cappelen (2011) and the entry on assertion.
truth-axiomatic	## 1. Motivations There have been many attempts to define truth in terms of correspondence, coherence or other notions. However, it is far from clear that truth is a definable notion. In formal settings satisfying certain natural conditions, Tarski's theorem on the undefinability of the truth predicate shows that a definition of a truth predicate requires resources that go beyond those of the formal language for which truth is going to be defined. In these cases definitional approaches to truth have to fail. By contrast, the axiomatic approach does not presuppose that truth can be defined. Instead, a formal language is expanded by a new primitive predicate for truth or satisfaction, and axioms for that predicate are then laid down. This approach by itself does not preclude the possibility that the truth predicate is definable, although in many cases it can be shown that the truth predicate is not definable. In semantic theories of truth (e.g., Tarski 1935, Kripke 1975), in contrast, a truth predicate is defined for a language, the so-called object language. This definition is carried out in a metalanguage or metatheory, which is typically taken to include set theory or at least another strong theory or expressively rich interpreted language. Tarski's theorem on the undefinability of the truth predicate shows that, given certain general assumptions, the resources of the metalanguage or metatheory must go beyond the resources of the object-language. So semantic approaches usually necessitate the use of a metalanguage that is more powerful than the object-language for which it provides a semantics. As with other formal deductive systems, axiomatic theories of truth can be presented within very weak logical frameworks. These frameworks require very few resources, and in particular, avoid the need for a strong metalanguage and metatheory. Formal work on axiomatic theories of truth has helped to shed some light on semantic theories of truth. For instance, it has yielded information on what is required of a metalanguage that is sufficient for defining a truth predicate. Semantic theories of truth, in turn, provide one with the theoretical tools needed for investigating models of axiomatic theories of truth and with motivations for certain axiomatic theories. Thus axiomatic and semantic approaches to truth are intertwined. This entry outlines the most popular axiomatic theories of truth and mentions some of the formal results that have been obtained concerning them. We give only hints as to their philosophical applications. ### 1.1 Truth, properties and sets Theories of truth and predication are closely related to theories of properties and property attribution. To say that an open formula $\phi(x)$ is true of an individual $a$ seems equivalent (in some sense) to the claim that $a$ has the property of being such that $\phi$ (this property is signified by the open formula). For example, one might say that '$x$ is a poor philosopher' is true of Tom instead of saying that Tom has the property of being a poor philosopher. Quantification over definable properties can then be mimicked in a language with a truth predicate by quantifying over formulas. Instead of saying, for instance, that $a$ and $b$ have exactly the same properties, one says that exactly the same formulas are true of $a$ and $b$. The reduction of properties to truth works also to some extent for sets of individuals. There are also reductions in the other direction: Tarski (1935) has shown that certain second-order existence assumptions (e.g., comprehension axioms) may be utilized to define truth (see the entry on Tarski's definition of truth). The mathematical analysis of axiomatic theories of truth and second-order systems has exhibited many equivalences between these second-order existence assumptions and truth-theoretic assumptions. These results show exactly what is required for defining a truth predicate that satisfies certain axioms, thereby sharpening Tarski's insights into definability of truth. In particular, proof-theoretic equivalences described in Section 3.3 below make explicit to what extent a metalanguage (or rather metatheory) has to be richer than the object language in order to be able to define a truth predicate. The equivalence between second-order theories and truth theories also has bearing on traditional metaphysical topics. The reductions of second-order theories (i.e., theories of properties or sets) to axiomatic theories of truth may be conceived as forms of reductive nominalism, for they replace existence assumptions for sets or properties (e.g., comprehension axioms) by ontologically innocuous assumptions, in the present case by assumptions on the behaviour of the truth predicate. ### 1.2 Truth and reflection According to Godel's incompleteness theorems, the statement that Peano Arithmetic (PA) is consistent, in its guise as a number-theoretic statement (given the technique of Godel numbering), cannot be derived in PA itself. But PA can be strengthened by adding this consistency statement or by stronger axioms. In particular, axioms partially expressing the soundness of PA can be added. These are known as reflection principles. An example of a reflection principle for PA would be the set of sentences $Bew\_{PA}(\ulcorner \phi \urcorner) \rightarrow \phi$ where $\phi$ is a formula of the language of arithmetic, $\ulcorner \phi \urcorner$ a name for $\phi$ and $Bew\_{PA}(x)$ is the standard provability predicate for PA ('$Bew$' was introduced by Godel and is short for the German word 'beweisbar', that is, 'provable'). The process of adding reflection principles can be iterated: one can add, for example, a reflection principle R for PA to PA; this results in a new theory PA+R. Then one adds the reflection principle for the system PA+R to the theory PA+R. This process can be continued into the transfinite (see Feferman 1962 and Franzen 2004). The reflection principles express--at least partially--the soundness of the system. The most natural and full expression of the soundness of a system involves the truth predicate and is known as the Global Reflection Principle (see Kreisel and Levy 1968). The Global Reflection Principle for a formal system S states that all sentences provable in S are true: \[ \forall x(Bew\_{S} (x) \rightarrow Tx) \] $Bew\_{S} (x)$ expresses here provability of sentences in the system S (we omit discussion here of the problems of defining $Bew\_{S} (x))$. The truth predicate has to satisfy certain principles; otherwise the global reflection principle would be vacuous. Thus not only the global reflection principle has to be added, but also axioms for truth. If a natural theory of truth like T(PA) below is added, however, it is no longer necessary to postulate the global reflection principle explicitly, as theories like T(PA) prove already the global reflection principle for PA. One may therefore view truth theories as reflection principles as they prove soundness statements and add the resources to express these statements. Thus instead of iterating reflection principles that are formulated entirely in the language of arithmetic, one can add by iteration new truth predicates and correspondingly new axioms for the new truth predicates. Thereby one might hope to make explicit all the assumptions that are implicit in the acceptance of a theory like PA. The resulting theory is called the reflective closure of the initial theory. Feferman (1991) has proposed the use of a single truth predicate and a single theory (KF), rather than a hierarchy of predicates and theories, in order to explicate the reflective closure of PA and other theories. (KF is discussed further in Section 4.4 below.) The relation of truth theories and (iterated) reflection principles also became prominent in the discussion of truth-theoretic deflationism (see Tennant 2002 and the follow-up discussion). ### 1.3 Truth-theoretic deflationism Many proponents of deflationist theories of truth have chosen to treat truth as a primitive notion and to axiomatize it, often using some version of the $T$-sentences as axioms. $T$-sentences are equivalences of the form $T\ulcorner \phi \urcorner \leftrightarrow \phi$, where $T$ is the truth predicate, $\phi$ is a sentence and $\ulcorner \phi \urcorner$ is a name for the sentence $\phi$. (More refined axioms have also been discussed by deflationists.) At first glance at least, the axiomatic approach seems much less 'deflationary' than those more traditional theories which rely on a definition of truth in terms of correspondence or the like. If truth can be explicitly defined, it can be eliminated, whereas an axiomatized notion of truth may and often does come with commitments that go beyond that of the base theory. If truth does not have any explanatory force, as some deflationists claim, the axioms for truth should not allow us to prove any new theorems that do not involve the truth predicate. Accordingly, Horsten (1995), Shapiro (1998) and Ketland (1999) have suggested that a deflationary axiomatization of truth should be at least conservative. The new axioms for truth are conservative if they do not imply any additional sentences (free of occurrences of the truth-predicate) that aren't already provable without the truth axioms. Thus a non-conservative theory of truth adds new non-semantic content to a theory and has genuine explanatory power, contrary to many deflationist views. Certain natural theories of truth, however, fail to be conservative (see Section 3.3 below, Field 1999 and Shapiro 2002 for further discussion). According to many deflationists, truth serves merely the purpose of expressing infinite conjunctions. It is plain that not all infinite conjunctions can be expressed because there are uncountably many (non-equivalent) infinite conjunctions over a countable language. Since the language with an added truth predicate has only countably many formulas, not every infinite conjunction can be expressed by a different finite formula. The formal work on axiomatic theories of truth has helped to specify exactly which infinite conjunctions can be expressed with a truth predicate. Feferman (1991) provides a proof-theoretic analysis of a fairly strong system. (Again, this will be explained in the discussion about KF in Section 4.4 below.) ## 2. The base theory ### 2.1 The choice of the base theory In most axiomatic theories, truth is conceived as a predicate of objects. There is an extensive philosophical discussion on the category of objects to which truth applies: propositions conceived as objects that are independent of any language, types and tokens of sentences and utterances, thoughts, and many other objects have been proposed. Since the structure of sentences considered as types is relatively clear, sentence types have often been used as the objects that can be true. In many cases there is no need to make very specific metaphysical commitments, because only certain modest assumptions on the structure of these objects are required, independently from whether they are finally taken to be syntactic objects, propositions or still something else. The theory that describes the properties of the objects to which truth can be attributed is called the base theory. The formulation of the base theory does not involve the truth predicate or any specific truth-theoretic assumptions. The base theory could describe the structure of sentences, propositions and the like, so that notions like the negation of such an object can then be used in the formulation of the truth-theoretic axioms. In many axiomatic truth theories, truth is taken as a predicate applying to the Godel numbers of sentences. Peano arithmetic has proved to be a versatile theory of objects to which truth is applied, mainly because adding truth-theoretic axioms to Peano arithmetic yields interesting systems and because Peano arithmetic is equivalent to many straightforward theories of syntax and even theories of propositions. However, other base theories have been considered as well, including formal syntax theories and set theories. Of course, we can also investigate theories which result by adding the truth-theoretic axioms to much stronger theories like set theory. Usually there is no chance of proving the consistency of set theory plus further truth-theoretic axioms because the consistency of set theory itself cannot be established without assumptions transcending set theory. In many cases not even relative consistency proofs are feasible. However, if adding certain truth-theoretic axioms to PA yields a consistent theory, it seems at least plausible that adding analogous axioms to set theory will not lead to an inconsistency. Therefore, the hope is that research on theories of truth over PA will give an some indication of what will happen when we extend stronger theories with axioms for the truth predicate. However, Fujimoto (2012) has shown that some axiomatic truth theories over set theory differ from their counterparts over Peano arithmetic in some aspects. ### 2.2 Notational conventions For the sake of definiteness we assume that the language of arithmetic has exactly $\neg , \wedge$ and $\vee$ as connectives and $\forall$ and $\exists$ as quantifiers. It has as individual constants only the symbol 0 for zero; its only function symbol is the unary successor symbol $S$; addition and multiplication are expressed by predicate symbols. Therefore the only closed terms of the language of arithmetic are the numerals $0, S$(0), $S(S$(0)), $S(S(S$(0))), .... The language of arithmetic does not contain the unary predicate symbol $T$, so let $\mathcal{L}\_T$ be the language of arithmetic augmented by the new unary predicate symbol $T$ for truth. If $\phi$ is a sentence of $\mathcal{L}\_T, \ulcorner \phi \urcorner$ is a name for $\phi$ in the language $\mathcal{L}\_T$; formally speaking, it is the numeral of the Godel number of $\phi$. In general, Greek letters like $\phi$ and $\psi$ are variables of the metalanguage, that is, the language used for talking about theories of truth and the language in which this entry is written (i.e., English enriched by some symbols). $\phi$ and $\psi$ range over formulas of the formal language $\mathcal{L}\_T$. In what follows, we use small, upper case italic letters like ${\scriptsize A}, {\scriptsize B},\ldots$ as variables in $\mathcal{L}\_T$ ranging over sentences (or their Godel numbers, to be precise). Thus $\forall{\scriptsize A}(\ldots{\scriptsize A}\ldots)$ stands for $\forall x(Sent\_T (x) \rightarrow \ldots x\ldots)$, where $Sent\_T (x)$ expresses in the language of arithmetic that $x$ is a sentence of the language of arithmetic extended by the predicate symbol $T$. The syntactical operations of forming a conjunction of two sentences and similar operations can be expressed in the language of arithmetic. Since the language of arithmetic does not contain any function symbol apart from the symbol for successor, these operations must be expressed by sutiable predicate expressions. Thus one can say in the language $\mathcal{L}\_T$ that a negation of a sentence of $\mathcal{L}\_T$ is true if and only if the sentence itself is not true. We would write this as \[ \forall{\scriptsize A}(T[\neg{\scriptsize A}] \leftrightarrow \neg T{\scriptsize A}). \] The square brackets indicate that the operation of forming the negation of ${\scriptsize A}$ is expressed in the language of arithmetic. Since the language of arithmetic does not contain a function symbol representing the function that sends sentences to their negations, appropriate paraphrases involving predicates must be given. Thus, for instance, the expression \[ \forall{\scriptsize A}\forall{\scriptsize B}(T[{\scriptsize A} \wedge{\scriptsize B}] \leftrightarrow(T{\scriptsize A} \wedge T{\scriptsize B})) \] is a single sentence of the language $\mathcal{L}\_T$ saying that a conjunction of sentences of $\mathcal{L}\_T$ is true if and only if both sentences are true. In contrast, \[ T\ulcorner \phi \wedge \psi \urcorner \leftrightarrow (T\ulcorner \phi \urcorner \wedge T\ulcorner \phi \urcorner) \] is only a schema. That is, it stands for the set of all sentences that are obtained from the above expression by substituting sentences of $\mathcal{L}\_T$ for the Greek letters $\phi$ and $\psi$. The single sentence $\forall{\scriptsize A}\forall{\scriptsize B}(T[{\scriptsize A} \wedge{\scriptsize B}] \leftrightarrow (T{\scriptsize A} \wedge T{\scriptsize B}))$ implies all sentences which are instances of the schema, but the instances of the schema do not imply the single universally quantified sentence. In general, the quantified versions are stronger than the corresponding schemata. ## 3. Typed theories of truth In typed theories of truth, only the truth of sentences not containing the same truth predicate is provable, thus avoiding the paradoxes by observing Tarski's distinction between object and metalanguage. ### 3.1 Definable truth predicates Certain truth predicates can be defined within the language of arithmetic. Predicates suitable as truth predicates for sublanguages of the language of arithmetic can be defined within the language of arithmetic, as long as the quantificational complexity of the formulas in the sublanguage is restricted. In particular, there is a formula $Tr\_0 (x)$ that expresses that $x$ is a true atomic sentence of the language of arithmetic, that is, a sentence of the form $n=k$, where $k$ and $n$ are identical numerals. For further information on partial truth predicates see, for instance, Hajek and Pudlak (1993), Kaye (1991) and Takeuti (1987). The definable truth predicates are truly redundant, because they are expressible in PA; therefore there is no need to introduce them axiomatically. All truth predicates in the following are not definable in the language of arithmetic, and therefore not redundant at least in the sense that they are not definable. ### 3.2 The $T$-sentences The typed $T$-sentences are all equivalences of the form $T\ulcorner \phi \urcorner \leftrightarrow \phi$, where $\phi$ is a sentence not containing the truth predicate. Tarski (1935) called any theory proving these equivalences 'materially adequate'. Tarski (1935) criticised an axiomatization of truth relying only on the $T$-sentences, not because he aimed at a definition rather than an axiomatization of truth, but because such a theory seemed too weak. Thus although the theory is materially adequate, Tarski thought that the $T$-sentences are deductively too weak. He observed, in particular, that the $T$-sentences do not prove the principle of completeness, that is, the sentence $\forall{\scriptsize A}(T{\scriptsize A}\vee T[\neg{\scriptsize A}$]) where the quantifier $\forall{\scriptsize A}$ is restricted to sentences not containing T. Theories of truth based on the $T$-sentences, and their formal properties, have also recently been a focus of interest in the context of so-called deflationary theories of truth. The $T$-sentences $T\ulcorner \phi \urcorner \leftrightarrow \phi$ (where $\phi$ does not contain $T)$ are not conservative over first-order logic with identity, that is, they prove a sentence not containing $T$ that is not logically valid. For the $T$-sentences prove that the sentences $0=0$ and $\neg 0=0$ are different and that therefore at least two objects exist. In other words, the $T$-sentences are not conservative over the empty base theory. If the $T$-sentences are added to PA, the resulting theory is conservative over PA. This means that the theory does not prove $T$-free sentences that are not already provable in PA. This result even holds if in addition to the $T$-sentences also all induction axioms containing the truth predicate are added. This may be shown by appealing to the Compactness Theorem. In the form outlined above, T-sentences express the equivalence between $T\ulcorner \phi \urcorner$ and $\phi$ only when $\phi$ is a sentence. In order to capture the equivalence for properties $(x$ has property P iff 'P' is true of $x)$ one must generalise the T-sentences. The result are usually referred to as the uniform T-senences and are formalised by the equivalences $\forall x(T\ulcorner \phi(\underline{x})\urcorner \leftrightarrow \phi(x))$ for each open formula $\phi(v)$ with at most $v$ free in $\phi$. Underlining the variable indicates it is bound from the outside. More precisely, $\ulcorner \phi(\underline{x})\urcorner$ stands for the result of replacing the variable $v$ in $\ulcorner \phi(v)\urcorner$ by the numeral of $x$. ### 3.3 Compositional truth As was observed already by Tarski (1935), certain desirable generalizations don't follow from the T-sentences. For instance, together with reasonable base theories they don't imply that a conjunction is true if both conjuncts are true. In order to obtain systems that also prove universally quantified truth-theoretic principles, one can turn the inductive clauses of Tarski's definition of truth into axioms. In the following axioms, $AtomSent\_{PA}(\ulcorner{\scriptsize A}\urcorner)$ expresses that ${\scriptsize A}$ is an atomic sentence of the language of arithmetic, $Sent\_{PA}(\ulcorner{\scriptsize A}\urcorner)$ expresses that ${\scriptsize A}$ is a sentence of the language of arithmetic. 1. $\forall{\scriptsize A}(AtomSent\_{PA}({\scriptsize A}) \rightarrow(T{\scriptsize A} \leftrightarrow Tr\_0 ({\scriptsize A})))$ 2. $\forall{\scriptsize A}(Sent\_{PA}({\scriptsize A}) \rightarrow(T[\neg{\scriptsize A}] \leftrightarrow \neg T{\scriptsize A}))$ 3. $\forall{\scriptsize A}\forall{\scriptsize B}(Sent\_{PA}({\scriptsize A}) \wedge Sent\_{PA}({\scriptsize B}) \rightarrow (T[{\scriptsize A} \wedge{\scriptsize B}] \leftrightarrow(T{\scriptsize A} \wedge T{\scriptsize B})))$ 4. $\forall{\scriptsize A}\forall{\scriptsize B}(Sent\_{PA}({\scriptsize A}) \wedge Sent\_{PA}({\scriptsize B}) \rightarrow (T[{\scriptsize A} \vee{\scriptsize B}] \leftrightarrow (T{\scriptsize A} \vee T{\scriptsize B})))$ 5. $\forall{\scriptsize A}(v)(Sent\_{PA}(\forall v{\scriptsize A}) \rightarrow(T[\forall v{\scriptsize A}(v)] \leftrightarrow \forall xT[{\scriptsize A}(\underline{x})$])) 6. $\forall{\scriptsize A}(v)(Sent\_{PA}(\forall v{\scriptsize A}) \rightarrow(T[\exists v{\scriptsize A}(v)] \leftrightarrow \exists xT[{\scriptsize A}(\underline{x})$])) Axiom 1 says that an atomic sentence of the language of Peano arithmetic is true if and only if it is true according to the arithmetical truth predicate for this language $(Tr\_0$ was defined in Section 3.1). Axioms 2-6 claim that truth commutes with all connectives and quantifiers. Axiom 5 says that a universally quantified sentence of the language of arithmetic is true if and only if all its numerical instances are true. $Sent\_{PA}(\forall v{\scriptsize A})$ says that ${\scriptsize A}(v)$ is a formula with at most $v$ free (because $\forall v{\scriptsize A}(v)$ is a sentence). If these axioms are to be formulated for a language like set theory that lacks names for all objects, then axioms 5 and 6 require the use of a satisfaction relation rather than a unary truth predicate. Axioms in the style of 1-6 above played a central role in Donald Davidson's theory of meaning and in several deflationist approaches to truth. The theory given by all axioms of PA and Axioms 1-6 but with induction only for $T$-free formulae is conservative over PA, that is, it doesn't prove any new $T$-free theorems that not already provable in PA. However, not all models of PA can be expanded to models of PA + axioms 1-6. This follows from a result due to Lachlan (1981). Kotlarski, Krajewski, and Lachlan (1981) proved the conservativeness very similar to PA + axioms 1-6 by model-theoretic means. Although several authors claimed that this result is also finitarily provable, no such proof was available until Enayat & Visser (2015) and Leigh (2015). Moreover, the theory given by PA + axioms 1-6 is relatively interpretable in PA. However, this result is sensitive to the choice of the base theory: it fails for finitely axiomatized theories (Heck 2015, Nicolai 2016). These proof-theoretic results have been used extensively in the discussion of truth-theoretic deflationism (see Cieslinski 2017). Of course PA + axioms 1-6 is restrictive insofar as it does not contain the induction axioms in the language with the truth predicate. There are various labels for the system that is obtained by adding all induction axioms involving the truth predicate to the system PA + axioms 1-6: T(PA), CT, PA(S) or PA + 'there is a full inductive satisfaction class'. This theory is no longer conservative over its base theory PA. For instance one can formalise the soundness theorem or global reflection principle for PA, that is, the claim that all sentences provable in PA are true. The global reflection principle for PA in turn implies the consistency of PA, which is not provable in pure PA by Godel's Second Incompleteness Theorem. Thus T(PA) is not conservative over PA. T(PA) is much stronger than the mere consistency statement for PA: T(PA) is equivalent to the second-order system ACA of arithmetical comprehension (see Takeuti 1987 and Feferman 1991). More precisely, T(PA) and ACA are intertranslatable in a way that preserves all arithmetical sentences. ACA is given by the axioms of PA with full induction in the second-order language and the following comprehension principle: \[ \exists X\forall y(y\in X \leftrightarrow \phi(x)) \] where $\phi(x)$ is any formula (in which $x$ may or may not be free) that does not contain any second-order quantifiers, but possibly free second-order variables. In T(PA), quantification over sets can be defined as quantification over formulas with one free variable and membership as the truth of the formula as applied to a number. As the global reflection principle entails formal consistency, the conservativeness result for PA + axioms 1-6 implies that the global reflection principle for Peano arithmetic is not derivable in the typed compositional theory without expanding the induction axioms. In fact, this theory proves neither the statement that all logical validities are true (global reflection for pure first-order logic) nor that all the Peano axioms of arithmetic are true. Perhaps surprisingly, of these two unprovable statements it is the former that is the stronger. The latter can be added as an axiom and the theory remains conservative over PA (Enayat and Visser 2015, Leigh 2015). In contrast, over PA + axioms 1-6, the global reflection principle for first-order logic is equivalent to global reflection for Peano arithmetic (Cieslinski 2010), and these two theories have the same arithmetic consequences as adding the axiom of induction for bounded $(\Delta\_0)$ formulas containing the truth predicate (Wcislo and Lelyk 2017). The transition from PA to T(PA) can be imagined as an act of reflection on the truth of $\mathcal{L}$-sentences in PA. Similarly, the step from the typed $T$-sentences to the compositional axioms is also tied to a reflection principle, specifically the uniform reflection principle over the typed uniform $T$-sentences. This is the collection of sentences $\forall x\, Bew\_{S} (\ulcorner \phi(\underline{x})\urcorner) \rightarrow \phi(x) $ where $\phi$ ranges over formulas in $\mathcal{L}\_T$ with one free variable and S is the theory of the uniform typed T-sentences. Uniform reflection exactly captures the difference between the two theories: the reflection principle is both derivable in T(PA) and suffices to derive the six compositional axioms (Halbach 2001). Moreover, the equivalence extends to iterations of uniform reflection, in that for any ordinal $\alpha , 1 + \alpha$ iterations of uniform reflection over the typed $T$-sentences coincides with T(PA) extended by transfinite induction up to the ordinal $\varepsilon\_{\alpha}$, namely the $\alpha$-th ordinal $\kappa$ with the property that $\omega^{\kappa} = \kappa $ (Leigh 2016). Much stronger fragments of second-order arithmetic can be interpreted by type-free truth systems, that is, by theories of truth that prove not only the truth of arithmetical sentences but also the truth of sentences of the language $\mathcal{L}\_T$ with the truth predicate; see Section 4 below. ### 3.4 Hierarchical theories The above mentioned theories of truth can be iterated by introducing indexed truth predicates. One adds to the language of PA truth predicates indexed by ordinals (or ordinal notations) or one adds a binary truth predicate that applies to ordinal notations and sentences. In this respect the hierarchical approach does not fit the framework outlined in Section 2, because the language does not feature a single unary truth predicate applying to sentences but rather many unary truth predicates or a single binary truth predicate (or even a single unary truth predicate applying to pairs of ordinal notations and sentences). In such a language an axiomatization of Tarski's hierarchy of truth predicates can be formulated. On the proof-theoretic side iterating truth theories in the style of T(PA) corresponds to iterating elementary comprehension, that is, to iterating ACA. The system of iterated truth theories corresponds to the system of ramified analysis (see Feferman 1991). Visser (1989) has studied non-wellfounded hierarchies of languages and axiomatizations thereof. If one adds the $T$-sentences $T\_n\ulcorner \phi \urcorner \leftrightarrow \phi$ to the language of arithmetic where $\phi$ contains only truth predicates $T\_k$ with $k\gt n$ to PA, a theory is obtained that does not have a standard $(\omega$-)model. ## 4. Type-free truth The truth predicates in natural languages do not come with any ouvert type restriction. Therefore typed theories of truth (axiomatic as well as semantic theories) have been thought to be inadequate for analysing the truth predicate of natural language, although recently hierarchical theories have been advocated by Glanzberg (2015) and others. This is one motive for investigating type-free theories of truth, that is, systems of truth that allow one to prove the truth of sentences involving the truth predicate. Some type-free theories of truth have much higher expressive power than the typed theories that have been surveyed in the previous section (at least as long as indexed truth predicates are avoided). Therefore type-free theories of truth are much more powerful tools in the reduction of other theories (for instance, second-order ones). ### 4.1 Type-free $T$-sentences The set of all $T$-sentences $T\ulcorner \phi \urcorner \leftrightarrow \phi$, where $\phi$ is any sentence of the language $\mathcal{L}\_T$, that is, where $\phi$ may contain $T$, is inconsistent with PA (or any theory that proves the diagonal lemma) because of the Liar paradox. Therefore one might try to drop from the set of all $T$-sentences only those that lead to an inconsistency. In other words, one may consider maximal consistent sets of $T$-sentences. McGee (1992) showed that there are uncountably many maximal sets of $T$-sentences that are consistent with PA. So the strategy does not lead to a single theory. Even worse, given an arithmetical sentence (i.e., a sentence not containing $T)$ that can neither be proved nor disproved in PA, one can find a consistent $T$-sentence that decides this sentence (McGee 1992). This implies that many consistent sets of $T$-sentences prove false arithmetical statements. Thus the strategy to drop just the $T$-sentences that yield an inconsistency is doomed. A set of $T$-sentences that does not imply any false arithmetical statement may be obtained by allowing only those $\phi$ in $T$-sentences $T\ulcorner \phi \urcorner \leftrightarrow \phi$ that contain $T$ only positively, that is, in the scope of an even number of negation symbols. Like the typed theory in Section 3.2 this theory does not prove certain generalizations but proves the same T-free sentences as the strong type-free compositional Kripke-Feferman theory below (Halbach 2009). Schindler (2015) obtained a deductively very strong truth theory based on stratified disquotational principles. ### 4.2 Compositionality Besides the disquotational feature of truth, one would also like to capture the compositional features of truth and generalize the axioms of typed compositional truth to the type-free case. To this end, axioms or rules concerning the truth of atomic sentences with the truth predicate will have to be added and the restriction to $T$-free sentences in the compositional axioms will have to be lifted. In order to treat truth like other predicates, one will add the axiom $\forall{\scriptsize A}(T[T{\scriptsize A}] \leftrightarrow T{\scriptsize A})$ (where $\forall{\scriptsize A}$ ranges over all sentences). If the type restriction of the typed compositional axiom for negation is removed, the axiom $\forall{\scriptsize A}(T[\neg{\scriptsize A}] \leftrightarrow \neg T{\scriptsize A})$ is obtained. However, the axioms $\forall{\scriptsize A}(T[T{\scriptsize A}] \leftrightarrow T{\scriptsize A})$ and $\forall{\scriptsize A}(T[\neg{\scriptsize A}] \leftrightarrow \neg T{\scriptsize A})$ are inconsistent over weak theories of syntax, so one of them has to be given up. If $\forall{\scriptsize A}(T[\neg{\scriptsize A}] \leftrightarrow \neg T{\scriptsize A})$ is retained, one will have to find weaker axioms or rules for truth iteration, but truth remains a classical concept in the sense that $\forall{\scriptsize A}(T[\neg{\scriptsize A}] \leftrightarrow \neg T{\scriptsize A})$ implies the law of excluded middle (for any sentence either the sentence itself or its negation is true) and the law of noncontradiction (for no sentence the sentence itself and its negation are true). If, in contrast, $\forall{\scriptsize A}(T[\neg{\scriptsize A}] \leftrightarrow \neg T{\scriptsize A})$ is rejected and $\forall{\scriptsize A}(T[T{\scriptsize A}] \leftrightarrow T{\scriptsize A})$ retained, then it will become provable that either some sentences are true together with their negations or that for some sentences neither they nor their negations are true, and thus systems of non-classical truth are obtained, although the systems themselves are still formulated in classical logic. In the next two sections we overview the most prominent system of each kind. ### 4.3 The Friedman-Sheard theory and revision semantics The system FS, named after Friedman and Sheard (1987), retains the negation axiom $\forall{\scriptsize A}(T[\neg{\scriptsize A}] \leftrightarrow \neg T{\scriptsize A})$. The further compositional axioms are obtained by lifting the type restriction to their untyped counterparts: 1. $\forall{\scriptsize A}(AtomSent\_{PA}({\scriptsize A}) \rightarrow(T{\scriptsize A} \leftrightarrow Tr\_0 ({\scriptsize A})))$ 2. $\forall{\scriptsize A}(T[\neg{\scriptsize A}] \leftrightarrow \neg T{\scriptsize A})$ 3. $\forall{\scriptsize A}\forall{\scriptsize B}(T[{\scriptsize A} \wedge{\scriptsize B}] \leftrightarrow(T{\scriptsize A} \wedge T{\scriptsize B}))$ 4. $\forall{\scriptsize A}\forall{\scriptsize B}(T[{\scriptsize A} \vee{\scriptsize B}] \leftrightarrow(T{\scriptsize A} \vee T{\scriptsize B}))$ 5. $\forall{\scriptsize A}(v)(Sent(\forall v{\scriptsize A}) \rightarrow(T[\forall v{\scriptsize A}(v)] \leftrightarrow \forall xT[{\scriptsize A}(\underline{x})$]) 6. $\forall{\scriptsize A}(v)(Sent(\forall v{\scriptsize A}) \rightarrow(T[\exists v{\scriptsize A}(v)] \leftrightarrow \exists xT[{\scriptsize A}(\underline{x})$])) These axioms are added to PA formulated in the language $\mathcal{L}\_T$. As the truth iteration axiom $\forall{\scriptsize A}(T[T{\scriptsize A}] \leftrightarrow T{\scriptsize A})$ is inconsistent, only the following two rules are added: If $\phi$ is a theorem, one may infer $T\ulcorner \phi \urcorner$, and conversely, if $T\ulcorner \phi \urcorner$ is a theorem, one may infer $\phi$. It follows from results due to McGee (1985) that FS is $\omega$-inconsistent, that is, FS proves $\exists x\neg \phi(x)$, but proves also $\phi$(0), $\phi$(1), $\phi$(2), ... for some formula $\phi(x)$ of $\mathcal{L}\_T$. The arithmetical theorems of FS, however, are all correct. In FS one can define all finite levels of the classical Tarskian hierarchy, but FS isn't strong enough to allow one to recover any of its transfinite levels. Indeed, Halbach (1994) determined its proof-theoretic strength to be precisely that of the theory of ramified truth for all finite levels (i.e., finitely iterated T(PA); see Section 3.4) or, equivalently, the theory of ramified analysis for all finite levels. If either direction of the rule is dropped but the other kept, FS retains its proof-theoretic strength (Sheard 2001). It is a virtue of FS that it is thoroughly classical: It is formulated in classical logic; if a sentence is provably true in FS, then the sentence itself is provable in FS; and conversely if a sentence is provable, then it is also provably true. Its drawback is its $\omega$-inconsistency. FS may be seen as an axiomatization of rule-of-revision semantics for all finite levels (see the entry on the revision theory of truth). ### 4.4 The Kripke-Feferman theory The Kripke-Feferman theory retains the truth iteration axiom $\forall{\scriptsize A}(T[T{\scriptsize A}] \leftrightarrow T{\scriptsize A})$, but the notion of truth axiomatized is no longer classical because the negation axiom $\forall{\scriptsize A}(T[\neg{\scriptsize A}] \leftrightarrow \neg T{\scriptsize A})$ is dropped. The semantical construction captured by this theory is a generalization of the Tarskian typed inductive definition of truth captured by T(PA). In the generalized definition one starts with the true atomic sentence of the arithmetical language and then one declares true the complex sentences depending on whether its components are true or not. For instance, as in the typed case, if $\phi$ and $\psi$ are true, their conjunction $\phi \wedge \psi$ will be true as well. In the case of the quantified sentences their truth value is determined by the truth values of their instances (one could render the quantifier clauses purely compositional by using a satisfaction predicate); for instance, a universally quantified sentence will be declared true if and only if all its instances are true. One can now extend this inductive definition of truth to the language $\mathcal{L}\_T$ by declaring a sentence of the form $T\ulcorner \phi \urcorner$ true if $\phi$ is already true. Moreover one will declare $\neg T\ulcorner \phi \urcorner$ true if $\neg \phi$ is true. By making this idea precise, one obtains a variant of Kripke's (1975) theory of truth with the so called Strong Kleene valuation scheme (see the entry on many-valued logic). If axiomatized it leads to the following system, which is known as KF ('Kripke-Feferman'), of which several variants appear in the literature: 1. $\forall{\scriptsize A}(AtomSent\_{PA}({\scriptsize A}) \rightarrow(T{\scriptsize A} \leftrightarrow Tr\_0 ({\scriptsize A})))$ 2. $\forall{\scriptsize A}(AtomSent\_{PA}({\scriptsize A}) \rightarrow(T[\neg{\scriptsize A}] \leftrightarrow \neg Tr\_0 ({\scriptsize A})))$ 3. $\forall{\scriptsize A}(T[T{\scriptsize A}] \leftrightarrow T{\scriptsize A})$ 4. $\forall{\scriptsize A}(T[\neg T{\scriptsize A}] \leftrightarrow T[\neg{\scriptsize A}$]) 5. $\forall{\scriptsize A}(T[\neg \neg{\scriptsize A}] \leftrightarrow T{\scriptsize A})$ 6. $\forall{\scriptsize A}\forall{\scriptsize B}(T[{\scriptsize A} \wedge{\scriptsize B}] \leftrightarrow(T{\scriptsize A} \wedge T{\scriptsize B}))$ 7. $\forall{\scriptsize A}\forall{\scriptsize B}(T[\neg({\scriptsize A} \wedge{\scriptsize B})] \leftrightarrow (T[\neg{\scriptsize A}] \vee T[\neg{\scriptsize B}$])) 8. $\forall{\scriptsize A}\forall{\scriptsize B}(T[{\scriptsize A} \vee{\scriptsize B}] \leftrightarrow(T{\scriptsize A} \vee T{\scriptsize B}))$ 9. $\forall{\scriptsize A}\forall{\scriptsize B}(T[\neg({\scriptsize A} \vee{\scriptsize B})] \leftrightarrow (T[\neg{\scriptsize A}] \wedge T[\neg{\scriptsize B}$])) 10. $\forall{\scriptsize A}(v)(Sent(\forall v{\scriptsize A}) \rightarrow(T[\forall v{\scriptsize A}(v)] \leftrightarrow \forall xT[{\scriptsize A}(\underline{x})$]) 11. $\forall{\scriptsize A}(v)(Sent(\forall v{\scriptsize A}) \rightarrow(T[\neg \forall v{\scriptsize A}(v)] \leftrightarrow \exists xT[\neg{\scriptsize A}(\underline{x})$]) 12. $\forall{\scriptsize A}(v)(Sent(\forall v{\scriptsize A}) \rightarrow(T[\exists v{\scriptsize A}(v)] \leftrightarrow \exists xT[{\scriptsize A}(\underline{x})$])) 13. $\forall{\scriptsize A}(v)(Sent(\forall v{\scriptsize A}) \rightarrow(T[\neg \exists v{\scriptsize A}(v)] \leftrightarrow \forall xT[\neg{\scriptsize A}(\underline{x})$])) Apart from the truth-theoretic axioms, KF comprises all axioms of PA and all induction axioms involving the truth predicate. The system is credited to Feferman on the basis of two lectures for the Association of Symbolic Logic, one in 1979 and the second in 1983, as well as in subsequent manuscripts. Feferman published his version of the system under the label Ref(PA) ('weak reflective closure of PA') only in 1991, after several other versions of KF had already appeared in print (e.g., Reinhardt 1986, Cantini 1989, who both refer to this unpublished work by Feferman). KF itself is formulated in classical logic, but it describes a non-classical notion of truth. For instance, one can prove $T\ulcorner L\urcorner \leftrightarrow T\ulcorner\neg L\urcorner$ if $L$ is the Liar sentence. Thus KF proves that either both the liar sentence and its negation are true or that neither is true. So either is the notion of truth paraconsistent (a sentence is true together with its negation) or paracomplete (neither is true). Some authors have augmented KF with an axiom ruling out truth-value gluts, which makes KF sound for Kripke's model construction, because Kripke had ruled out truth-value gluts. Feferman (1991) showed that KF is proof-theoretically equivalent to the theory of ramified analysis through all levels below $\varepsilon\_0$, the limit of the sequence $\omega , \omega^{\omega}, \omega^{\omega^{ \omega} },\ldots$, or a theory of ramified truth through the same ordinals. This result shows that in KF exactly $\varepsilon\_0$ many levels of the classical Tarskian hierarchy in its axiomatized form can be recovered. Thus KF is far stronger than FS, let alone T(PA). Feferman (1991) devised also a strengthening of KF that is as strong as full predicative analysis, that is ramified analysis or truth up to the ordinal $\Gamma\_0$. Just as with the typed truth predicate, the theory KF (more precisely, a common variant of it) can be obtained via an act of reflection on a system of untyped $T$-sentences. The system of $T$-sentences in question is the extension of the uniform positive untyped $T$-sentences by a primitive falsity predicate, that is, the theory features two unary predicates $T$ and $F$ and axioms \[\begin{align\} &\forall x(T\ulcorner \phi(\underline{x})\urcorner \leftrightarrow \phi(x)) \\ & \forall x(F\ulcorner \phi(\underline{x})\urcorner \leftrightarrow \phi '(x)) \end{align\}\] for every formula $\phi(v)$ positive in both $T$ and $F$, where $\phi '$ represents the De Morgan dual of $\phi$ (exchanging $T$ for $F$ and vice versa). From an application of uniform reflection over this disquotational theory, the truth axioms for the corresponding two predicate version of KF are derivable (Horsten and Leigh, 2016). The converse also holds, as does the generalisation to finite and transfinite iterations of reflection (Leigh, 2017). ### 4.5 Capturing the minimal fixed point As remarked above, if KF proves $T\ulcorner \phi \urcorner$ for some sentence $\phi$ then $\phi$ holds in all Kripke fixed point models. In particular, there are $2^{\aleph\_0}$ fixed points that form a model of the internal theory of KF. Thus from the perspective of KF, the least fixed point (from which Kripke's theory is defined) is not singled out. Burgess (2014) provides an expansion of KF, named $\mu$KF, that attempts to capture the minimal Kripkean fixed point. KF is expanded by additional axioms that express that the internal theory of KF is the smallest class closed under the defining axioms for Kripkean truth. This can be formulated as a single axiom schema that states, for each open formula $\phi$, If $\phi$ satisfies the same axioms of KF as the predicate $T$ then $\phi$ holds of every true sentence. From a proof-theoretic perspective $\mu$KF is significantly stronger than KF. The single axiom schema expressing the minimality of the truth predicate allows one to embed into $\mu$KF the system ID$\_1$ of one arithmetical inductive definition, an impredicative theory. While intuitively plausible, $\mu$KF suffers the same expressive incompleteness as KF: Since the minimal Kripkean fixed point forms a complete $\Pi^{1}\_1$ set and the internal theory of $\mu$KF remains recursively enumerable, there are standard models of the theory in which the interpretation of the truth predicate is not actually the minimal fixed point. At present there lacks a thorough analysis of the models of $\mu$KF. ### 4.6 Axiomatisations of Kripke's theory with supervaluations KF is intended to be an axiomatization of Kripke's (1975) semantical theory. This theory is based on partial logic with the Strong Kleene evaluation scheme. In Strong Kleene logic not every sentence $\phi \vee \neg \phi$ is a theorem; in particular, this disjunction is not true if $\phi$ lacks a truth value. Consequently $T\ulcorner L\vee \neg L\urcorner$ (where $L$ is the Liar sentence) is not a theorem of KF and its negation is even provable. Cantini (1990) has proposed a system VF that is inspired by the supervaluations scheme. In VF all classical tautologies are provably true and $T\ulcorner L \vee \neg L\urcorner$, for instance, is a theorem of VF. VF can be formulated in $\mathcal{L}\_T$ and uses classical logic. It is no longer a compositional theory of truth, for the following is not a theorem of VF: \[ \forall{\scriptsize A}\forall{\scriptsize B}(T[{\scriptsize A} \vee{\scriptsize B}] \leftrightarrow(T{\scriptsize A} \vee T{\scriptsize B})). \] Not only is this principle inconsistent with the other axioms of VF, it does not fit the supervaluationist model for it implies $T\ulcorner L\urcorner \vee T\ulcorner \neg L\urcorner$, which of course is not correct because according to the intended semantics neither the liar sentence nor its negation is true: both lack a truth value. Extending a result due to Friedman and Sheard (1987), Cantini showed that VF is much stronger than KF: VF is proof-theoretically equivalent to the theory ID$\_1$ of non-iterated inductive definitions, which is not predicative. ## 5. Non-classical approaches to self-reference The theories of truth discussed thus far are all axiomatized in classical logic. Some authors have also looked into axiomatic theories of truth based on non-classical logic (see, for example, Field 2008, Halbach and Horsten 2006, Leigh and Rathjen 2012). There are a number of reasons why a logic weaker than classical logic may be preferred. The most obvious is that by weakening the logic, some collections of axioms of truth that were previously inconsistent become consistent. Another common reason is that the axiomatic theory in question intends to capture a particular non-classical semantics of truth, for which a classical background theory may prove unsound. There is also a large number of approaches that employ paraconsistent or substructural logics. In most cases these approaches do not employ an axiomatic base theory such as Peano arithmetic and therefore deviate form the setting considered here, although there is no technical obstacle in applying paraconsistent or substructural logics to truth theories over such base theories. Here we cover only accounts that are close to the setting considered above. For further information on the application of substructural and paraconsistent logics to the truth-theoretic paradoxes see the relevant section in the entry on the liar paradox. ### 5.1 The truth predicate in intuitionistic logic The inconsistency of the $T$-sentences does not rely on classical reasoning. It is also inconsistent over much weaker logics such as minimal logic and partial logic. However, classical logic does play a role in restricting the free use of principles of truth. For instance, over a classical base theory, the compositional axiom for implication $(\rightarrow)$ is equivalent to the principle of completeness, $\forall{\scriptsize A}(T[{\scriptsize A}] \vee T[\neg{\scriptsize A}$]). If the logic under the truth predicate is classical, completeness is equivalent to the compositional axiom for disjunction. Without the law of excluded middle, FS can be formulated as a fully compositional theory while not proving the truth-completeness principle (Leigh & Rathjen 2012). In addition, classical logic has an effect on attempts to combine compositional and self-applicable axioms of truth. If, for example, one drops the axiom of truth-consistency from FS (the left-to-right direction of axiom 2 in Section 4.3) as well as the law of excluded middle for the truth predicate, it is possible to add consistently the truth-iteration axiom $\forall{\scriptsize A}(T[{\scriptsize A}] \rightarrow T[T{\scriptsize A}])$. The resulting theory still bears a strong resemblance to FS in that the constructive version of the rule-of-revision semantics for all finite levels provides a natural model of the theory, and the two theories share the same $\Pi^{0}\_2$ consequences (Leigh & Rathjen 2012; Leigh, 2013). This result should be contrasted with KF which, if formulated without the law of excluded middle, remains maximally consistent with respect to its choice of truth axioms but is a conservative extension of Heyting arithmetic. ### 5.2 Axiomatising Kripke's theory Kripke's (1975) theory in its different guises is based on partial logic. In order to obtain models for a theory in classical logic, the extension of the truth predicate in the partial model is used again as the extension of truth in the classical model. In the classical model false sentences and those without a truth value in the partial model are declared not true. KF is sound with respect to these classical models and thus incorporates two distinct logics. The first is the 'internal' logic of statements under the truth predicate and is formulated with the Strong Kleene valuation schema. The second is the 'external' logic which is full classical logic. An effect of formulating KF in classical logic is that the theory cannot be consistently closed under the truth-introduction rule If $\phi$ is a theorem of KF, so is $T\ulcorner \phi \urcorner$. A second effect of classical logic is the statement of the excluded middle for the liar sentence. Neither the Liar sentence nor its negation obtains a truth value in Kripke's theory, so the disjunction of the two is not valid. The upshot is that KF, if viewed as an axiomatisation of Kripke's theory, is not sound with respect to its intended semantics. For this reason Halbach and Horsten (2006) and Horsten (2011) explore an axiomatization of Kripke's theory with partial logic as inner and outer logic. Their suggestion, a theory labelled PKF ('partial KF'), can be axiomatised as a Gentzen-style two-sided sequent calculus based on Strong Kleene logic (see the entry on many-valued logic). PKF is formed by adding to this calculus the Peano-Dedekind axioms of arithmetic including full induction and the compositional and truth-iteration rules for the truth predicate as proscribed by Kripke's theory. The result is a theory of truth that is sound with respect to Kripke's theory. Halbach and Horsten show that this axiomatization of Kripke's theory is significantly weaker than it's classical cousin KF. The result demonstrates that restricting logic only for sentences with the truth predicate can hamper also the derivation of truth-free theorems. ### 5.3 Adding a conditional Field (2008) and others criticised theories based on partial logic for the absence of a 'proper' conditional and bi-conditional. Various authors have proposed conditionals and bi-conditionals that are not definable in terms of $\neg , \vee$ and $\wedge$. Field (2008) aims at an axiomatic theory of truth not dissimilar to PKF but with a new conditional. Feferman (1984) also introduced a bi-conditional to a theory in non-classical logic. Unlike Field's and his own 1984 theory, Feferman's (2008) theory DT is formulated in classical logic, but it's internal logic is again a partial logic with a strong conditional.
truth-coherence	## 1. Versions of the Coherence Theory of Truth The coherence theory of truth has several versions. These versions differ on two major issues. Different versions of the theory give different accounts of the coherence relation. Different varieties of the theory also give various accounts of the set (or sets) of propositions with which true propositions cohere. (Such a set will be called a specified set.) According to some early versions of the coherence theory, the coherence relation is simply consistency. On this view, to say that a proposition coheres with a specified set of propositions is to say that the proposition is consistent with the set. This account of coherence is unsatisfactory for the following reason. Consider two propositions which do not belong to a specified set. These propositions could both be consistent with a specified set and yet be inconsistent with each other. If coherence is consistency, the coherence theorist would have to claim that both propositions are true, but this is impossible. A more plausible version of the coherence theory states that the coherence relation is some form of entailment. Entailment can be understood here as strict logical entailment, or entailment in some looser sense. According to this version, a proposition coheres with a set of propositions if and only if it is entailed by members of the set. Another more plausible version of the theory, held for example in Bradley (1914), is that coherence is mutual explanatory support between propositions. The second point on which coherence theorists (coherentists, for short) differ is the constitution of the specified set of propositions. Coherentists generally agree that the specified set consists of propositions believed or held to be true. They differ on the questions of who believes the propositions and when. At one extreme, coherence theorists can hold that the specified set of propositions is the largest consistent set of propositions currently believed by actual people. For such a version of the theory, see Young (1995). According to a moderate position, the specified set consists of those propositions which will be believed when people like us (with finite cognitive capacities) have reached some limit of inquiry. For such a coherence theory, see Putnam (1981). At the other extreme, coherence theorists can maintain that the specified set contains the propositions which would be believed by an omniscient being. Some idealists seem to accept this account of the specified set. If the specified set is a set actually believed, or even a set which would be believed by people like us at some limit of inquiry, coherentism involves the rejection of realism about truth. Realism about truth involves acceptance of the principle of bivalence (according to which every proposition is either true or false) and the principle of transcendence (which says that a proposition may be true even though it cannot be known to be true). Coherentists who do not believe that the specified set is the set of propositions believed by an omniscient being are committed to rejection of the principle of bivalence since it is not the case that for every proposition either it or a contrary proposition coheres with the specified set. They reject the principle of transcendence since, if a proposition coheres with a set of beliefs, it can be known to cohere with the set. ## 2. Arguments for Coherence Theories of Truth Two principal lines of argument have led philosophers to adopt a coherence theory of truth. Early advocates of coherence theories were persuaded by reflection on metaphysical questions. More recently, epistemological and semantic considerations have been the basis for coherence theories. ### 2.1 The Metaphysical Route to Coherentism Early versions of the coherence theory were associated with idealism. Walker (1989) attributes coherentism to Spinoza, Kant, Fichte and Hegel. Certainly a coherence theory was adopted by a number of British Idealists in the last years of the nineteenth century and the first decades of the twentieth. See, for example, Bradley (1914). Idealists are led to a coherence theory of truth by their metaphysical position. Advocates of the correspondence theory believe that a belief is (at least most of the time) ontologically distinct from the objective conditions which make the belief true. Idealists do not believe that there is an ontological distinction between beliefs and what makes beliefs true. From the idealists' perspective, reality is something like a collection of beliefs. Consequently, a belief cannot be true because it corresponds to something which is not a belief. Instead, the truth of a belief can only consist in its coherence with other beliefs. A coherence theory of truth which results from idealism usually leads to the view that truth comes in degrees. A belief is true to the degree that it coheres with other beliefs. Since idealists do not recognize an ontological distinction between beliefs and what makes them true, distinguishing between versions of the coherence theory of truth adopted by idealists and an identity theory of truth can be difficult. The article on Bradley in this Encyclopedia (Candlish 2006) argues that Bradley had an identity theory, not a coherence theory. In recent years metaphysical arguments for coherentism have found few advocates. This is due to the fact that idealism is not widely held. ### 2.2 Epistemological Routes to Coherentism Blanshard (1939, ch. XXVI) argues that a coherence theory of justification leads to a coherence theory of truth. His argument runs as follows. Someone might hold that coherence with a set of beliefs is the test of truth but that truth consists in correspondence to objective facts. If, however, truth consists in correspondence to objective facts, coherence with a set of beliefs will not be a test of truth. This is the case since there is no guarantee that a perfectly coherent set of beliefs matches objective reality. Since coherence with a set of beliefs is a test of truth, truth cannot consist in correspondence. Blanshard's argument has been criticised by, for example, Rescher (1973). Blanshard's argument depends on the claim that coherence with a set of beliefs is the test of truth. Understood in one sense, this claim is plausible enough. Blanshard, however, has to understand this claim in a very strong sense: coherence with a set of beliefs is an infallible test of truth. If coherence with a set of beliefs is simply a good but fallible test of truth, as Rescher suggests, the argument fails. The "falling apart" of truth and justification to which Blanshard refers is to be expected if truth is only a fallible test of truth. Another epistemological argument for coherentism is based on the view that we cannot "get outside" our set of beliefs and compare propositions to objective facts. A version of this argument was advanced by some logical positivists including Hempel (1935) and Neurath (1983). This argument, like Blanshard's, depends on a coherence theory of justification. The argument infers from such a theory that we can only know that a proposition coheres with a set of beliefs. We can never know that a proposition corresponds to reality. This argument is subject to at least two criticisms. For a start, it depends on a coherence theory of justification, and is vulnerable to any objections to this theory. More importantly, a coherence theory of truth does not follow from the premisses. We cannot infer from the fact that a proposition cannot be known to correspond to reality that it does not correspond to reality. Even if correspondence theorists admit that we can only know which propositions cohere with our beliefs, they can still hold that truth consists in correspondence. If correspondence theorists adopt this position, they accept that there may be truths which cannot be known. Alternatively, they can argue, as does Davidson (1986), that the coherence of a proposition with a set of beliefs is a good indication that the proposition corresponds to objective facts and that we can know that propositions correspond. Coherence theorists need to argue that propositions cannot correspond to objective facts, not merely that they cannot be known to correspond. In order to do this, the foregoing argument for coherentism must be supplemented. One way to supplement the argument would be to argue as follows. As noted above, the correspondence and coherence theories have differing views about the nature of truth conditions. One way to decide which account of truth conditions is correct is to pay attention to the process by which propositions are assigned truth conditions. Coherence theorists can argue that the truth conditions of a proposition are the conditions under which speakers make a practice of asserting it. Coherentists can then maintain that speakers can only make a practice of asserting a proposition under conditions the speakers are able to recognise as justifying the proposition. Now the (supposed) inability of speakers to "get outside" of their beliefs is significant. Coherentists can argue that the only conditions speakers can recognise as justifying a proposition are the conditions under which it coheres with their beliefs. When the speakers make a practice of asserting the proposition under these conditions, they become the proposition's truth conditions. For an argument of this sort see Young (1995). ## 3. Criticisms of Coherence Theories of Truth Any coherence theory of truth faces two principal challenges. The first may be called the specification objection. The second is the transcendence objection. ### 3.1 The Specification Objection According to the specification objection, coherence theorists have no way to identify the specified set of propositions without contradicting their position. This objection originates in Russell (1907). Opponents of the coherence theory can argue as follows. The proposition (1) "Jane Austen was hanged for murder" coheres with some set of propositions. (2) "Jane Austen died in her bed" coheres with another set of propositions. No one supposes that the first of these propositions is true, in spite of the fact that it coheres with a set of propositions. The specification objection charges that coherence theorists have no grounds for saying that (1) is false and (2) true. Some responses to the specification problem are unsuccessful. One could say that we have grounds for saying that (1) is false and (2) is true because the latter coheres with propositions which correspond to the facts. Coherentists cannot, however, adopt this response without contradicting their position. Sometimes coherence theorists maintain that the specified system is the most comprehensive system, but this is not the basis of a successful response to the specification problem. Coherentists can only, unless they are to compromise their position, define comprehensiveness in terms of the size of a system. Coherentists cannot, for example, talk about the most comprehensive system composed of propositions which correspond to reality. There is no reason, however, why two or more systems cannot be equally large. Other criteria of the specified system, to which coherentists frequently appeal, are similarly unable to solve the specification problem. These criteria include simplicity, empirical adequacy and others. Again, there seems to be no reason why two or more systems cannot equally meet these criteria. Although some responses to the Russell's version of the specification objection are unsuccessful, it is unable to refute the coherence theory. Coherentists do not believe that the truth of a proposition consists in coherence with any arbitrarily chosen set of propositions. Rather, they hold that truth consists in coherence with a set of beliefs, or with a set of propositions held to be true. No one actually believes the set of propositions with which (1) coheres. Coherence theorists conclude that they can hold that (1) is false without contradicting themselves. A more sophisticated version of the specification objection has been advanced by Walker (1989); for a discussion, see Wright (1995). Walker argues as follows. In responding to Russell's version of the specification objection, coherentists claim that some set of propositions, call it S, is believed. They are committed to the truth of (3) "S is believed." The question of what it is for (3) to be true then arises. Coherence theorists might answer this question by saying that "'S is believed' is believed" is true. If they give this answer, they are apparently off on an infinite regress, and they will never say what it is for a proposition to be true. Their plight is worsened by the fact that arbitrarily chosen sets of propositions can include propositions about what is believed. So, for example, there will be a set which contains "Jane Austen was hanged for murder," "'Jane Austen was hanged for murder' is believed," and so on. The only way to stop the regress seems to be to say that the truth conditions of (3) consist in the objective fact S is believed. If, however, coherence theorists adopt this position, they seem to contradict their own position by accepting that the truth conditions of some proposition consist in facts, not in propositions in a set of beliefs. There is some doubt about whether Walker's version of the specification objection succeeds. Coherence theorists can reply to Walker by saying that nothing in their position is inconsistent with the view that there is a set of propositions which is believed. Even though this objective fact obtains, the truth conditions of propositions, including propositions about which sets of propositions are believed, are the conditions under which they cohere with a set of propositions. For a defence of the coherence theory against Walker's version of the specification objection, see Young (2001). A coherence theory of truth gives rise to a regress, but it is not a vicious regress and the correspondence theory faces a similar regress. If we say that p is true if and only if it coheres with a specified set of propositions, we may be asked about the truth conditions of "p coheres with a specified set." Plainly, this is the start of a regress, but not one to worry about. It is just what one would expect, given that the coherence theory states that it gives an account of the truth conditions of all propositions. The correspondence theory faces a similar benign regress. The correspondence theory states that a proposition is true if and only if it corresponds to certain objective conditions. The proposition "p corresponds to certain objective conditions" is also true if and only if it corresponds to certain objective conditions, and so on. ### 3.2 The Transcendence Objection The transcendence objection charges that a coherence theory of truth is unable to account for the fact that some propositions are true which cohere with no set of beliefs. According to this objection, truth transcends any set of beliefs. Someone might argue, for example, that the proposition "Jane Austen wrote ten sentences on November 17th, 1807" is either true or false. If it is false, some other proposition about how many sentences Austen wrote that day is true. No proposition, however, about precisely how many sentences Austen wrote coheres with any set of beliefs and we may safely assume that none will ever cohere with a set of beliefs. Opponents of the coherence theory will conclude that there is at least one true proposition which does not cohere with any set of beliefs. Some versions of the coherence theory are immune to the transcendence objection. A version which holds that truth is coherence with the beliefs of an omniscient being is proof against the objection. Every truth coheres with the set of beliefs of an omniscient being. All other versions of the theory, however, have to cope with the objection, including the view that truth is coherence with a set of propositions believed at the limit of inquiry. Even at the limit of inquiry, finite creatures will not be able to decide every question, and truth may transcend what coheres with their beliefs. Coherence theorists can defend their position against the transcendence objection by maintaining that the objection begs the question. Those who present the objection assume, generally without argument, that it is possible that some proposition be true even though it does not cohere with any set of beliefs. This is precisely what coherence theorists deny. Coherence theorists have arguments for believing that truth cannot transcend what coheres with some set of beliefs. Their opponents need to take issue with these arguments rather than simply assert that truth can transcend what coheres with a specified system. ### 3.3 The Logic Objection Russell (1912) presented a third classic objection to the coherence theory of truth. According to this objection, any talk about coherence presupposes the truth of the laws of logic. For example, Russell argues, to say that two propositions cohere with each other is to presuppose the truth of the law of non-contradiction. In this case, coherentism has no account of the truth of law of non-contradiction. If, however, the coherence theorist holds that the truth of the law of non-contradiction depends on its coherence with a system of beliefs, and it were supposed to be false, then propositions cannot cohere or fail to cohere. In this case, the coherence theory of truth completely breaks down since propositions cannot cohere with each other. Coherentists have a plausible response to this objection. They may hold that the law of non-contradiction, like any other truth, is true because it coheres with a system of beliefs. In particular, the law of non-contradiction is supported by the belief that, for example, communication and reasoning would be impossible unless every system of beliefs contains something like law of non-contradiction (and the belief that communication and reasoning are possible). It is true that, as Russell says, if the law is supposed not to cohere with a system of beliefs, then propositions can neither cohere nor fail to cohere. However, coherence theorists may hold, they do not suppose the law of non-contradiction to be false. On the contrary, they are likely to hold that any coherent set of beliefs must include the law of non-contradiction or a similar law. ## 4. New Objections to Coherentism Paul Thagard is the author of the first of two recent new arguments against the coherence theory. Thagard states his argument as follows: > if there is a world independent of representations of it, > as historical evidence suggests, then the aim of representation should > be to describe the world, not just to relate to other > representations. My argument does not refute the coherence theory, but > shows that it implausibly gives minds too large a place in > constituting truth. (Thagard 2007: 29-30) Thagard's argument seems to be that if there is a mind-independent world, then our representations are representations of the world. (He says representations "should be" of the world, but the argument is invalid with the addition of the auxiliary verb.) The world existed before humans and our representations, including our propositional representations. (So history and, Thagard would likely say, our best science tells us.) Therefore, representations, including propositional representations, are representations of a mind-independent world. The second sentence of the passage just quoted suggests that the only way that coherentists can reject this argument is to adopt some sort of idealism. That is, they can only reject the minor premiss of the argument as reconstructed. Otherwise they are committed to saying that propositions represent the world and, Thagard seems to suggest, this is to say that propositions have the sort of truth-conditions posited by a correspondence theory. So the coherence theory is false. In reply to this argument, coherentists can deny that propositions are representations of a mind-independent world. To say that a proposition is true is to say that it is supported by a specified system of propositions. So, the coherentist can say, propositions are representations of systems of beliefs, not representations of a mind-independent world. To assert a proposition is to assert that it is entailed by a system of beliefs. The coherentist holds that even if there is a mind-independent world, it does not follow that the "the point" of representations is to represent this world. If coherentists have been led to their position by an epistemological route, they believe that we cannot "get outside" our system of beliefs. If we cannot get outside of our system of beliefs, then it is hard to see how we can be said to represent a mind-independent reality. Colin McGinn has proposed the other new objection to coherentism. He argues (McGinn 2002: 195) that coherence theorists are committed to idealism. Like Thagard, he takes idealism to be obviously false, so the argument is a reductio. McGinn's argument runs as follows. Coherentists are committed to the view that, for example, 'Snow falls from the sky' is true iff the belief that snow falls from the sky coheres with other beliefs. Now it follows from this and the redundancy biconditional (p is true iff p) that snow falls from the sky iff the belief that snow falls from the sky coheres with other beliefs. It appears then that the coherence theorist is committed to the view that snow could not fall from the sky unless the belief that snow falls from the sky coheres with other beliefs. From this it follows that how things are depends on what is believed about them. This seems strange to McGinn since he thinks, reasonably, that snow could fall from the sky even if there were no beliefs about snow, or anything else. The linking of how things are and how they are believed to be leads McGinn to say that coherentists are committed to idealism, this being the view that how things are is mind-dependent. Coherentists have a response to this objection. McGinn's argument works because he takes it that the redundancy biconditional means something like "p is true because p". Only if redundancy biconditionals are understood in this way does McGinn's argument go through. McGinn needs to be talking about what makes "Snow falls from the sky" true for his reductio to work. Otherwise, coherentists who reject his argument cannot be charged with idealism. He assumes, in a way that a coherent theorist can regard as question-begging, that the truth-maker of the sentence in question is an objective way the world is. Coherentists deny that any sentences are made true by objective conditions. In particular, they hold that the falling of snow from the sky does not make "Snow falls from the sky" true. Coherentists hold that it, like any other sentence, is true because it coheres with a system of beliefs. So coherentists appear to have a plausible defence against McGinn's objection.
truth-correspondence	## 1. History of the Correspondence Theory The correspondence theory is often traced back to Aristotle's well-known definition of truth (Metaphysics 1011b25): "To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true"--but virtually identical formulations can be found in Plato (Cratylus 385b2, Sophist 263b). It is noteworthy that this definition does not highlight the basic correspondence intuition. Although it does allude to a relation (saying something of something) to reality (what is), the relation is not made very explicit, and there is no specification of what on the part of reality is responsible for the truth of a saying. As such, the definition offers a muted, relatively minimal version of a correspondence theory. (For this reason it has also been claimed as a precursor of deflationary theories of truth.) Aristotle sounds much more like a genuine correspondence theorist in the Categories (12b11, 14b14), where he talks of underlying things that make statements true and implies that these things (pragmata) are logically structured situations or facts (viz., his sitting and his not sitting are said to underlie the statements "He is sitting" and "He is not sitting", respectively). Most influential is Aristotle's claim in De Interpretatione (16a3) that thoughts are "likenesses" (homoiomata) of things. Although he nowhere defines truth in terms of a thought's likeness to a thing or fact, it is clear that such a definition would fit well into his overall philosophy of mind. (Cf. Crivelli 2004; Szaif 2006.) ### 1.1 Metaphysical and Semantic Versions In medieval authors we find a division between "metaphysical" and "semantic" versions of the correspondence theory. The former are indebted to the truth-as-likeness theme suggested by Aristotle's overall views, the latter are modeled on Aristotle's more austere definition from Metaphysics 1011b25. The metaphysical version presented by Thomas Aquinas is the best known: "Veritas est adaequatio rei et intellectus" (Truth is the equation of thing and intellect), which he restates as: "A judgment is said to be true when it conforms to the external reality". He tends to use "conformitas" and "adaequatio", but also uses "correspondentia", giving the latter a more generic sense (De Veritate, Q.1, A.1-3; cf. Summa Theologiae, Q.16). Aquinas credits the Neoplatonist Isaac Israeli with this definition, but there is no such definition in Isaac. Correspondence formulations can be traced back to the Academic skeptic Carneades, 2nd century B.C., whom Sextus Empiricus (Adversos Mathematicos, vii, 168) reports as having taught that a presentation "is true when it is in accord (symphonos) with the object presented, and false when it is in discord with it". Similar accounts can be found in various early commentators on Plato and Aristotle (cf. Kunne 2003, chap. 3.1), including some Neoplatonists: Proklos (In Tim., II 287, 1) speaks of truth as the agreement or adjustment (epharmoge) between knower and the known. Philoponus (In Cat., 81, 25-34) emphasizes that truth is neither in the things or states of affairs (pragmata) themselves, nor in the statement itself, but lies in the agreement between the two. He gives the simile of the fitting shoe, the fit consisting in a relation between shoe and foot, not to be found in either one by itself. Note that his emphasis on the relation as opposed to its relata is laudable but potentially misleading, because x's truth (its being true) is not to be identified with a relation, R, between x and y, but with a general relational property of x, taking the form ([?]y)(xRy & Fy). Further early correspondence formulations can be found in Avicenna (Metaphysica, 1.8-9) and Averroes (Tahafut, 103, 302). They were introduced to the scholastics by William of Auxerre, who may have been the intended recipient of Aquinas' mistaken attribution (cf. Boehner 1958; Wolenski 1994). Aquinas' balanced formula "equation of thing and intellect" is intended to leave room for the idea that "true" can be applied not only to thoughts and judgments but also to things or persons (e.g. a true friend). Aquinas explains that a thought is said to be true because it conforms to reality, whereas a thing or person is said to be true because it conforms to a thought (a friend is true insofar as, and because, she conforms to our, or God's, conception of what a friend ought to be). Medieval theologians regarded both, judgment-truth as well as thing/person-truth, as somehow flowing from, or grounded in, the deepest truth which, according to the Bible, is God: "I am the way and the truth and the life" (John 14, 6). Their attempts to integrate this Biblical passage with more ordinary thinking involving truth gave rise to deep metaphysico-theological reflections. The notion of thing/person-truth, which thus played a very important role in medieval thinking, is disregarded by modern and contemporary analytic philosophers but survives to some extent in existentialist and continental philosophy. Medieval authors who prefer a semantic version of the correspondence theory often use a peculiarly truncated formula to render Aristotle's definition: A (mental) sentence is true if and only if, as it signifies, so it is (sicut significat, ita est). This emphasizes the semantic relation of signification while remaining maximally elusive about what the "it" is that is signified by a true sentence and de-emphasizing the correspondence relation (putting it into the little words "as" and "so"). Foreshadowing a favorite approach of the 20th century, medieval semanticists like Ockham (Summa Logicae, II) and Buridan (Sophismata, II) give exhaustive lists of different truth-conditional clauses for sentences of different grammatical categories. They refrain from associating true sentences in general with items from a single ontological category. (Cf. Moody 1953; Adams McCord 1987; Perler 2006.) Authors of the modern period generally convey the impression that the correspondence theory of truth is far too obvious to merit much, or any, discussion. Brief statements of some version or other can be found in almost all major writers; see e.g.: Descartes 1639, ATII 597; Spinoza, Ethics, axiom vi; Locke, Essay, 4.5.1; Leibniz, New Essays, 4.5.2; Hume, Treatise, 3.1.1; and Kant 1787, B82. Berkeley, who does not seem to offer any account of truth, is a potentially significant exception. Due to the influence of Thomism, metaphysical versions of the theory are much more popular with the moderns than semantic versions. But since the moderns generally subscribe to a representational theory of the mind (the theory of ideas), they would seem to be ultimately committed to spelling out relations like correspondence or conformity in terms of a psycho-semantic representation relation holding between ideas, or sentential sequences of ideas (Locke's "mental propositions"), and appropriate portions of reality, thereby effecting a merger between metaphysical and semantic versions of the correspondence theory. ### 1.2 Object-Based and Fact-Based Versions It is helpful to distinguish between "object-based" and "fact-based" versions of correspondence theories, depending on whether the corresponding portion of reality is said to be an object or a fact (cf. Kunne 2003, chap. 3). Traditional versions of object-based theories assumed that the truth-bearing items (usually taken to be judgments) have subject-predicate structure. An object-based definition of truth might look like this: > A judgment is true if and only if its predicate > corresponds to its object (i.e., to the object referred to by the > subject term of the judgment). Note that this actually involves two relations to an object: (i) a reference relation, holding between the subject term of the judgment and the object the judgment is about (its object); and (ii) a correspondence relation, holding between the predicate term of the judgment and a property of the object. Owing to its reliance on the subject-predicate structure of truth-bearing items, the account suffers from an inherent limitation: it does not cover truthbearers that lack subject-predicate structure (e.g. conditionals, disjunctions), and it is not clear how the account might be extended to cover them. The problem is obvious and serious; it was nevertheless simply ignored in most writings. Object-based correspondence was the norm until relatively recently. Object-based correspondence became the norm through Plato's pivotal engagement with the problem of falsehood, which was apparently notorious at its time. In a number of dialogues, Plato comes up against an argument, advanced by various Sophists, to the effect that false judgment is impossible--roughly: To judge falsely is to judge what is not. But one cannot judge what is not, for it is not there to be judged. To judge something that is not is to judge nothing, hence, not to judge at all. Therefore, false judgment is impossible. (Cf. Euthydemus 283e-288a; Cratylus 429c-e; Republic 478a-c; Theaetetus 188d-190e.) Plato has no good answer to this patent absurdity until the Sophist (236d-264b), where he finally confronts the issue at length. The key step in his solution is the analysis of truthbearers as structured complexes. A simple sentence, such as "Theaetetus sits.", though simple as a sentence, is still a complex whole consisting of words of different kinds--a name (onoma) and a verb (rhema)--having different functions. By weaving together verbs with names the speaker does not just name a number of things, but accomplishes something: meaningful speech (logos) expressive of the interweaving of ideas (eidon symploken). The simple sentence is true when Theaetetus, the person named by the name, is in the state of sitting, ascribed to him through the verb, and false, when Theaetetus is not in that state but in another one (cf. 261c-263d; see Denyer 1991; Szaif 1998). Only things that are show up in this account: in the case of falsehood, the ascribed state still is, but it is a state different from the one Theaetetus is in. The account is extended from speech to thought and belief via Plato's well known thesis that "thought is speech that occurs without voice, inside the soul in conversation with itself" (263e)--the historical origin of the language-of-thought hypothesis. The account does not take into consideration sentences that contain a name of something that is not ("Pegasus flies"), thus bequeathing to posterity a residual problem that would become more notorious than the problem of falsehood. Aristotle, in De Interpretatione, adopts Plato's account without much ado--indeed, the beginning of De Interpretatione reads like a direct continuation of the passages from the Sophist mentioned above. He emphasizes that truth and falsehood have to do with combination and separation (cf. De Int. 16a10; in De Anima 430a25, he says: "where the alternative of true and false applies, there we always find a sort of combining of objects of thought in a quasi-unity"). Unlike Plato, Aristotle feels the need to characterize simple affirmative and negative statements (predications) separately--translating rather more literally than is usual: "An affirmation is a predication of something toward something, a negation is a predication of something away from something" (De Int. 17a25). This characterization reappears early in the Prior Analytics (24a). It thus seems fair to say that the subject-predicate analysis of simple declarative sentences--the most basic feature of Aristotelian term logic which was to reign supreme for many centuries--had its origin in Plato's response to a sophistical argument against the possibility of falsehood. One may note that Aristotle's famous definition of truth (see Section 1) actually begins with the definition of falsehood. Fact-based correspondence theories became prominent only in the 20th century, though one can find remarks in Aristotle that fit this approach (see Section 1)--somewhat surprisingly in light of his repeated emphasis on subject-predicate structure wherever truth and falsehood are concerned. Fact-based theories do not presuppose that the truth-bearing items have subject-predicate structure; indeed, they can be stated without any explicit reference to the structure of truth-bearing items. The approach thus embodies an alternative response to the problem of falsehood, a response that may claim to extricate the theory of truth from the limitations imposed on it through the presupposition of subject-predicate structure inherited from the response to the problem of falsehood favored by Plato, Aristotle, and the medieval and modern tradition. The now classical formulation of a fact-based correspondence theory was foreshadowed by Hume (Treatise, 3.1.1) and Mill (Logic, 1.5.1). It appears in its canonical form early in the 20th century in Moore (1910-11, chap. 15) and Russell: "Thus a belief is true when there is a corresponding fact, and is false when there is no corresponding fact" (1912, p. 129; cf. also his 1905, 1906, 1910, and 1913). The self-conscious emphasis on facts as the corresponding portions of reality--and a more serious concern with problems raised by falsehood--distinguishes this version from its foreshadowings. Russell and Moore's forceful advocacy of truth as correspondence to a fact was, at the time, an integral part of their defense of metaphysical realism. Somewhat ironically, their formulations are indebted to their idealist opponents, F. H. Bradley (1883, chaps. 1&2), and H. H. Joachim (1906), the latter was an early advocate of the competing coherence theory, who had set up a correspondence-to-fact account of truth as the main target of his attack on realism. Later, Wittgenstein (1921) and Russell (1918) developed "logical atomism", which introduces an important modification of the fact-based correspondence approach (see below, Section 7.1). Further modifications of the correspondence theory, bringing a return to more overtly semantic and broadly object-based versions, were influenced by Tarski's (1935) technical work on truth (cf. Field 1972, Popper 1972). ## 2. Truthbearers, Truthmakers, Truth ### 2.1 Truthbearers Correspondence theories of truth have been given for beliefs, thoughts, ideas, judgments, statements, assertions, utterances, sentences, and propositions. It has become customary to talk of truthbearers whenever one wants to stay neutral between these choices. Five points should be kept in mind: 1. The term "truthbearer" is somewhat misleading. It is intended to refer to bearers of truth or falsehood (truth-value-bearers), or alternatively, to things of which it makes sense to ask whether they are true or false, thus allowing for the possibility that some of them might be neither. 2. One distinguishes between secondary and primary truthbearers. Secondary truthbearers are those whose truth-values (truth or falsehood) are derived from the truth-values of primary truthbearers, whose truth-values are not derived from any other truthbearers. Consequently, the term "true" is usually regarded as ambiguous, taking its primary meaning when applied to primary truthbearers and various secondary meanings when applied to other truthbearers. This is, however, not a brute ambiguity, since the secondary meanings are supposed to be derived, i.e. definable from, the primary meaning together with additional relations. For example, one might hold that propositions are true or false in the primary sense, whereas sentences are true or false in a secondary sense, insofar as they express propositions that are true or false (in the primary sense). The meanings of "true", when applied to truthbearers of different kinds, are thus connected in a manner familiar from what Aristotelians called "analogical" uses of a term--nowadays one would call this "focal meaning"; e.g., "healthy" in "healthy organism" and "healthy food", the latter being defined as healthy in the secondary sense of contributing to the healthiness (primary sense) of an organism. 3. It is often unproblematic to advocate one theory of truth for bearers of one kind and another theory for bearers of a different kind (e.g., a deflationary theory of truth, or an identity theory, applied to propositions, could be a component of some form of correspondence theory of truth for sentences). Different theories of truth applied to bearers of different kinds do not automatically compete. The standard segregation of truth theories into competing camps (found in textbooks, handbooks, and dictionaries) proceeds under the assumption--really a pretense--that they are intended for primary truthbearers of the same kind. 4. Confusingly, there is little agreement as to which entities are properly taken to be primary truthbearers. Nowadays, the main contenders are public language sentences, sentences of the language of thought (sentential mental representations), and propositions. Popular earlier contenders--beliefs, judgments, statements, and assertions--have fallen out of favor, mainly for two reasons: 1. The problem of logically complex truthbearers. A subject, S, may hold a disjunctive belief (the baby will be a boy or the baby will be a girl), while believing only one, or neither, of the disjuncts. Also, S may hold a conditional belief (if whales are fish, then some fish are mammals) without believing the antecedent or the consequent. Also, S will usually hold a negative belief (not everyone is lucky) without believing what is negated. In such cases, the truth-values of S's complex beliefs depend on the truth-values of their constituents, although the constituents may not be believed by S or by anyone. This means that a view according to which beliefs are primary truthbearers seems unable to account for how the truth-values of complex beliefs are connected to the truth-values of their simpler constituents--to do this one needs to be able to apply truth and falsehood to belief-constituents even when they are not believed. This point, which is equally fundamental for a proper understanding of logic, was made by all early advocates of propositions (cf. Bolzano 1837, I.SSSS22, 34; Frege 1879, SSSS2-5; Husserl 1900, I.SS11; Meinong 1902, SS6). The problem arises in much the same form for views that would take judgments, statements, or assertions as primary truthbearers. The problem is not easily evaded. Talk of unbelieved beliefs (unjudged judgments, unstated statements, unasserted assertions) is either absurd or simply amounts to talk of unbelieved (unjudged, unstated, unasserted) propositions or sentences. It is noteworthy, incidentally, that quite a few philosophical proposals (concerning truth as well as other matters) run afoul of the simple observation that there are unasserted and unbelieved truthbearers (cf. Geach 1960 & 1965). 2. The duality of state/content a.k.a. act/object. The noun "belief" can refer to the state of believing or to its content, i.e., to what is believed. If the former, the state of believing, can be said to be true or false at all, which is highly questionable, then only insofar as the latter, what is believed, is true or false. Similarly for nouns referring to mental acts or their objects (contents), such as "judgment", "statement", and "assertion". 5. Mental sentences were the preferred primary truthbearers throughout the medieval period. They were neglected in the first half of the 20th century, but made a comeback in the second half through the revival of the representational theory of the mind (especially in the form of the language-of-thought hypothesis, cf. Fodor 1975). Somewhat confusingly (to us now), for many centuries the term "proposition" (propositio) was reserved exclusively for sentences, written, spoken or mental. This use was made official by Boethius in the 6th century, and is still found in Locke's Essay in 1705 and in Mill's Logic in 1843. Some time after that, e.g., in Moore's 1901-01, "proposition" switched sides, the term now being used for what is said by uttering a sentence, for what is believed, judged, stated, assumed (etc.)--with occasional reversions to medieval usage, e.g. in Russell (1918, 1919). ### 2.2 Truthmakers Talk of truthmakers serves a function similar, but correlative, to talk of truthbearers. A truthmaker is anything that makes some truthbearer true. Different versions of the correspondence theory will have different, and often competing, views about what sort of items true truthbearers correspond to (facts, states of affairs, events, things, tropes, properties). It is convenient to talk of truthmakers whenever one wants to stay neutral between these choices. Four points should be kept in mind: 1. The notion of a truthmaker is tightly connected with, and dependent on, the relational notion of truthmaking: a truthmaker is whatever stands in the truthmaking relation to some truthbearer. Despite the causal overtones of "maker" and "making", this relation is usually not supposed to be a causal relation. 2. The terms "truthmaking" and "truthmaker" are ambiguous. For illustration, consider a classical correspondence theory on which x is true if and only if x corresponds to some fact. One can say (a) that x is made true by a fact, namely the fact (or a fact) that x corresponds to. One can also say (b) that x is made true by x's correspondence to a fact. Both uses of "is made true by" are correct and both occur in discussions of truth. But they are importantly different and must be distinguished. The (a)-use is usually the intended one; it expresses a relation peculiar to truth and leads to a use of "truthmaker" that actually picks out the items that would normally be intended by those using the term. The (b)-use does not express a relation peculiar to truth; it is just an instance (for "F" = "true") of the generic formula "what makes an F-thing an F" that can be employed to elicit the definiens of a proposed definition of F. Compare: what makes an even number even is its divisibility by 2; what makes a right action right is its having better consequences than available alternative actions. Note that anyone proposing a definition or account of truth can avail themselves of the notion of truthmaking in the (b)-sense; e.g., a coherence theorist, advocating that a belief is true if and only if it coheres with other beliefs, can say: what makes a true belief true is its coherence with other beliefs. So, on the (b)-use, "truthmaking" and "truthmaker" do not signal any affinity with the basic idea underlying the correspondence theory of truth, whereas on the (a)-use these terms do signal such an affinity. 3. Talk of truthmaking and truthmakers goes well with the basic idea underlying the correspondence theory; hence, it might seem natural to describe a traditional fact-based correspondence theory as maintaining that the truthmakers are facts and that the correspondence relation is the truthmaking relation. However, the assumption that the correspondence relation can be regarded as (a species of) the truthmaking relation is dubious. Correspondence appears to be a symmetric relation (if x corresponds to y, then y corresponds to x), whereas it is usually taken for granted that truthmaking is an asymmetric relation, or at least not a symmetric one. It is hard to see how a symmetric relation could be (a species of) an asymmetric or non-symmetric relation (cf. David 2009.) 4. Talk of truthmaking and truthmakers is frequently employed during informal discussions involving truth but tends to be dropped when a more formal or official formulation of a theory of truth is produced (one reason being that it seems circular to define or explain truth in terms of truthmakers or truthmaking). However, in recent years, the informal talk has been turned into an official doctrine: "truthmaker theory". This theory should be distinguished from informal truthmaker talk: not everyone employing the latter would subscribe to the former. Moreover, truthmaker theory should not simply be assumed to be a version of the correspondence theory; indeed, some advocates present it as a competitor to the correspondence theory (see below, Section 8.5). ### 2.3 Truth The abstract noun "truth" has various uses. (a) It can be used to refer to the general relational property otherwise referred to as being true; though the latter label would be more perspicuous, it is rarely used, even in philosophical discussions. (b) The noun "truth" can be used to refer to the concept that "picks out" the property and is expressed in English by the adjective "true". Some authors do not distinguish between concept and property; others do, or should: an account of the concept might differ significantly from an account of the property. To mention just one example, one might maintain, with some plausibility, that an account of the concept ought to succumb to the liar paradox (see the entry on the liar paradox), otherwise it wouldn't be an adequate account of our concept of truth; this idea is considerably less plausible in the case of the property. Any proposed "definition of truth" might be intend as a definition of the property or of the concept or both; its author may or may not be alive to the difference. (c) The noun "truth" can be used, finally, to refer to some set of true truthbarers (possibly unknown), as in: "The truth is out there", and: "The truth about this matter will never be known". ## 3. Simple Versions of the Correspondence Theory The traditional centerpiece of any correspondence theory is a definition of truth. Nowadays, a correspondence definition is most likely intended as a "real definition", i.e., as a definition of the property, which does not commit its advocate to the claim that the definition provides a synonym for the term "true". Most correspondence theorists would consider it implausible and unnecessarily bold to maintain that "true" means the same as "corresponds with a fact". Some simple forms of correspondence definitions of truth should be distinguished ("iff" means "if and only if"; the variable, "x", ranges over whatever truthbearers are taken as primary; the notion of correspondence might be replaced by various related notions): > > > > \| \| \| > \| --- \| --- \| > \| (1) \| x is true iff x corresponds to some fact; > > x is false iff x does not correspond to any fact. > \| > > > > > > > \| \| \| > \| --- \| --- \| > \| (2) \| x is true iff x corresponds to some state of > affairs that obtains; > x is false iff x corresponds to some state of > affairs that does not obtain. \| > > > Both forms invoke portions of reality--facts/states of affairs--that are typically denoted by that-clauses or by sentential gerundives, viz. the fact/state of affairs that snow is white, or the fact/state of affairs of snow's being white. (2)'s definition of falsehood is committed to there being (existing) entities of this sort that nevertheless fail to obtain, such as snow's being green. (1)'s definition of falsehood is not so committed: to say that a fact does not obtain means, at best, that there is no such fact, that no such fact exists. It should be noted that this terminology is not standardized: some authors use "state of affairs" much like "fact" is used here (e.g. Armstrong 1997). The question whether non-obtaining beings of the relevant sort are to be accepted is the substantive issue behind such terminological variations. The difference between (2) and (1) is akin to the difference between Platonism about properties (embraces uninstantiated properties) and Aristotelianism about properties (rejects uninstantiated properties). Advocates of (2) hold that facts are states of affairs that obtain, i.e., they hold that their account of truth is in effect an analysis of (1)'s account of truth. So disagreement turns largely on the treatment of falsehood, which (1) simply identifies with the absence of truth. The following points might be made for preferring (2) over (1): (a) Form (2) does not imply that things outside the category of truthbearers (tables, dogs) are false just because they don't correspond to any facts. One might think this "flaw" of (1) is easily repaired: just put an explicit specification of the desired category of truthbearers into both sides of (1). However, some worry that truthbearer categories, e.g. declarative sentences or propositions, cannot be defined without invoking truth and falsehood, which would make the resultant definition implicitly circular. (b) Form (2) allows for items within the category of truthbearers that are neither true nor false, i.e., it allows for the failure of bivalence. Some, though not all, will regard this as a significant advantage. (c) If the primary truthbearers are sentences or mental states, then states of affairs could be their meanings or contents, and the correspondence relation in (2) could be understood accordingly, as the relation of representation, signification, meaning, or having-as-content. Facts, on the other hand, cannot be identified with the meanings or contents of sentences or mental states, on pain of the absurd consequence that false sentences and beliefs have no meaning or content. (d) Take a truth of the form 'p or q', where 'p' is true and 'q' false. What are the constituents of the corresponding fact? Since 'q' is false, they cannot both be facts (cf. Russell 1906-07, p. 47f.). Form (2) allows that the fact corresponding to 'p or q' is an obtaining disjunctive state of affairs composed of a state of affairs that obtains and a state of affairs that does not obtain. The main point in favor of (1) over (2) is that (1) is not committed to counting non-obtaining states of affairs, like the state of affairs that snow is green, as constituents of reality. (One might observe that, strictly speaking, (1) and (2), being biconditionals, are not ontologically committed to anything. Their respective commitments to facts and states of affairs arise only when they are combined with claims to the effect that there is something that is true and something that is false. The discussion assumes some such claims as given.) Both forms, (1) and (2), should be distinguished from: > > > > \| \| \| > \| --- \| --- \| > \| (3) \| x is true iff x corresponds to some fact that exists; > x is false iff x corresponds to some fact that does not exist, > \| > > > which is a confused version of (1), or a confused version of (2), or, if unconfused, signals commitment to Meinongianism, i.e., the thesis that there are things/facts that do not exist. The lure of (3) stems from the desire to offer more than a purely negative correspondence account of falsehood while avoiding commitment to non-obtaining states of affairs. Moore at times succumbs to (3)'s temptations (1910-11, pp. 267 & 269, but see p. 277). It can also be found in the 1961 translation of Wittgenstein (1921, 4.25), who uses "state of affairs" (Sachverhalt) to refer to (atomic) facts. The translation has Wittgenstein saying that an elementary proposition is false, when the corresponding state of affairs (atomic fact) does not exist--but the German original of the same passage looks rather like a version of (2). Somewhat ironically, a definition of form (3) reintroduces Plato's problem of falsehood into a fact-based correspondence theory, i.e., into a theory of the sort that was supposed to provide an alternative solution to that very problem (see Section 1.2). A fourth simple form of correspondence definition was popular for a time (cf. Russell 1918, secs. 1 & 3; Broad 1933, IV.2.23; Austin 1950, fn. 23), but seems to have fallen out of favor: > > > > \| \| \| > \| --- \| --- \| > \| (4) \| x is true iff x > corresponds (agrees) with some fact; > x is false iff x mis-corresponds (disagrees) with some fact. \| > > > This formulation attempts to avoid (2)'s commitment to non-obtaining states of affairs and (3)'s commitment to non-existent facts by invoking the relation of mis-correspondence, or disagreement, to account for falsehood. It differs from (1) in that it attempts to keep items outside the intended category of x's from being false: supposedly, tables and dogs cannot mis-correspond with a fact. Main worries about (4) are: (a) its invocation of an additional, potentially mysterious, relation, which (b) seems difficult to tame: Which fact is the one that mis-corresponds with a given falsehood? and: What keeps a truth, which by definition corresponds with some fact, from also mis-corresponding with some other fact, i.e., from being a falsehood as well? In the following, I will treat definitions (1) and (2) as paradigmatic; moreover, since advocates of (2) agree that obtaining states of affairs are facts, it is often convenient to condense the correspondence theory into the simpler formula provided by (1), "truth is correspondence to a fact", at least as long as one is not particularly concerned with issues raised by falsehood. ## 4. Arguments for the Correspondence Theory The main positive argument given by advocates of the correspondence theory of truth is its obviousness. Descartes: "I have never had any doubts about truth, because it seems a notion so transcendentally clear that nobody can be ignorant of it...the word 'truth', in the strict sense, denotes the conformity of thought with its object" (1639, AT II 597). Even philosophers whose overall views may well lead one to expect otherwise tend to agree. Kant: "The nominal definition of truth, that it is the agreement of [a cognition] with its object, is assumed as granted" (1787, B82). William James: "Truth, as any dictionary will tell you, is a property of certain of our ideas. It means their 'agreement', as falsity means their disagreement, with 'reality'" (1907, p. 96). Indeed, The Oxford English Dictionary tells us: "Truth, n. Conformity with fact; agreement with reality". In view of its claimed obviousness, it would seem interesting to learn how popular the correspondence theory actually is. There are some empirical data. The PhilPapers Survey (conducted in 2009; cf. Bourget and Chalmers 2014), more specifically, the part of the survey targeting all regular faculty members in 99 leading departments of philosophy, reports the following responses to the question: "Truth: correspondence, deflationary, or epistemic?" Accept or lean toward: correspondence 50.8%; deflationary 24.8%; other 17.5%; epistemic 6.9%. The data suggest that correspondence-type theories may enjoy a weak majority among professional philosophers and that the opposition is divided. This fits with the observation that typically, discussions of the nature of truth take some version of the correspondence theory as the default view, the view to be criticized or to be defended against criticism. Historically, the correspondence theory, usually in an object-based version, was taken for granted, so much so that it did not acquire this name until comparatively recently, and explicit arguments for the view are very hard to find. Since the (comparatively recent) arrival of apparently competing approaches, correspondence theorists have developed negative arguments, defending their view against objections and attacking (sometimes ridiculing) competing views. ## 5. Objections to the Correspondence Theory **Objection 1:** Definitions like (1) or (2) are too narrow. Although they apply to truths from some domains of discourse, e.g., the domain of science, they fail for others, e.g. the domain of morality: there are no moral facts. The objection recognizes moral truths, but rejects the idea that reality contains moral facts for moral truths to correspond to. Logic provides another example of a domain that has been "flagged" in this way. The logical positivists recognized logical truths but rejected logical facts. Their intellectual ancestor, Hume, had already given two definitions of "true", one for logical truths, broadly conceived, the other for non-logical truths: "Truth or falsehood consists in an agreement or disagreement either to the real relations of ideas, or to real existence and matter of fact" (Hume, Treatise, 3.1.1, cf. 2.3.10; see also Locke, Essay, 4.5.6, for a similarly two-pronged account but in terms of object-based correspondence). There are four possible responses to objections of this sort: (a) Noncognitivism, which says that, despite appearances to the contrary, claims from the flagged domain are not truth-evaluable to begin with, e.g., moral claims are commands or expressions of emotions disguised as truthbearers; (b) Error theory, which says that all claims from the flagged domain are false; (c) Reductionism, which says that truths from the flagged domain correspond to facts of a different domain regarded as unproblematic, e.g., moral truths correspond to social-behavioral facts, logical truths correspond to facts about linguistic conventions; and (d) Standing firm, i.e., embracing facts of the flagged domain. The objection in effect maintains that there are different brands of truth (of the property being true, not just different brands of truths) for different domains. On the face of it, this conflicts with the observation that there are many obviously valid arguments combining premises from flagged and unflagged domains. The observation is widely regarded as refuting non-cognitivism, once the most popular (concessive) response to the objection. In connection with this objection, one should take note of the recently developed "multiple realizability" view of truth, according to which truth is not to be identified with correspondence to fact but can be realized by correspondence to fact for truthbearers of some domains of discourse and by other properties for truthbearers of other domains of discourse, including "flagged" domains. Though it retains important elements of the correspondence theory, this view does not, strictly speaking, offer a response to the objection on behalf of the correspondence theory and should be regarded as one of its competitors (see below, Section 8.2). **Objection 2:** Correspondence theories are too obvious. They are trivial, vacuous, trading in mere platitudes. Locutions from the "corresponds to the facts"-family are used regularly in everyday language as idiomatic substitutes for "true". Such common turns of phrase should not be taken to indicate commitment to a correspondence theory in any serious sense. Definitions like (1) or (2) merely condense some trivial idioms into handy formulas; they don't deserve the grand label "theory": there is no theoretical weight behind them (cf. Woozley 1949, chap. 6; Davidson 1969; Blackburn 1984, chap. 7.1). In response, one could point out: (a) Definitions like (1) or (2) are "mini-theories"--mini-theories are quite common in philosophy--and it is not at all obvious that they are vacuous merely because they are modeled on common usage. (b) There are correspondence theories that go beyond these definitions. (c) The complaint implies that definitions like (1) and/or (2) are generally accepted and are, moreover, so shallow that they are compatible with any deeper theory of truth. This makes it rather difficult to explain why some thinkers emphatically reject all correspondence formulations. (d) The objection implies that the correspondence of S's belief with a fact could be said to consist in, e.g., the belief's coherence with S's overall belief system. This is wildly implausible, even on the most shallow understanding of "correspondence" and "fact". **Objection 3:** Correspondence theories are too obscure. Objections of this sort, which are the most common, protest that the central notions of a correspondence theory carry unacceptable commitments and/or cannot be accounted for in any respectable manner. The objections can be divided into objections primarily aimed at the correspondence relation and its relatives (3.C1, 3.C2), and objections primarily aimed at the notions of fact or state of affairs (3.F1, 3.F2): *3.C1: The correspondence relation must be some sort of resemblance relation. But truthbearers do not resemble anything in the world except other truthbearers--echoing Berkeley's "an idea can be like nothing but an idea". 3.C2: The correspondence relation is very mysterious: it seems to reach into the most distant regions of space (faster than light?) and time (past and future). How could such a relation possibly be accounted for within a naturalistic framework? What physical relation could it possibly be? 3.F1: Given the great variety of complex truthbearers, a correspondence theory will be committed to all sorts of complex "funny facts" that are ontologically disreputable. Negative, disjunctive, conditional, universal, probabilistic, subjunctive, and counterfactual facts have all given cause for complaint on this score. 3.F2: All facts, even the most simple ones, are disreputable. Fact-talk, being wedded to that-clauses, is entirely parasitic on truth-talk. Facts are too much like truthbearers. Facts are fictions, spurious sentence-like slices of reality, "projected from true sentences for the sake of correspondence" (Quine 1987, p. 213; cf. Strawson 1950). ## 6. Correspondence as Isomorphism Some correspondence theories of truth are two-liner mini-theories, consisting of little more than a specific version of (1) or (2). Normally, one would expect a bit more, even from a philosophical theory (though mini-theories are quite common in philosophy). One would expect a correspondence theory to go beyond a mere definition like (1) or (2) and discharge a triple task: it should tell us about the workings of the correspondence relation, about the nature of facts, and about the conditions that determine which truthbearers correspond to which facts. One can approach this by considering some general principles a correspondence theory might want to add to its central principle to flesh out her theory. The first such principle says that the correspondence relation must not collapse into identity--"It takes two to make a truth" (Austin 1950, p. 118): > > Nonidentity:* > No truth is identical with a fact > correspondence to which is sufficient for its being a truth. > It would be much simpler to say that no truth is identical with a fact. However, some authors, e.g. Wittgenstein 1921, hold that a proposition (Satz, his truthbearer) is itself a fact, though not the same fact as the one that makes the proposition true (see also King 2007). Nonidentity is usually taken for granted by correspondence theorists as constitutive of the very idea of a correspondence theory--authors who advance contrary arguments to the effect that correspondence must collapse into identity regard their arguments as objections to any form of correspondence theory (cf. Moore 1901/02, Frege 1918-19, p. 60). Concerning the correspondence relation, two aspects can be distinguished: correspondence as correlation and correspondence as isomorphism (cf. Pitcher 1964; Kirkham 1992, chap. 4). Pertaining to the first aspect, familiar from mathematical contexts, a correspondence theorist is likely to adopt claim (a), and some may in addition adopt claim (b), of: > > Correlation: > > (a) Every truth corresponds to exactly one fact; > > (b) Different truths correspond to different facts. > Together, (a) and (b) say that correspondence is a one-one relation. This seems needlessly strong, and it is not easy to find real-life correspondence theorists who explicitly embrace part (b): Why shouldn't different truths correspond to the same fact, as long as they are not too different? Explicit commitment to (a) is also quite rare. However, correspondence theorists tend to move comfortably from talk about a given truth to talk about the fact it corresponds to--a move that signals commitment to (a). Correlation does not imply anything about the inner nature of the corresponding items. Contrast this with correspondence as isomorphism, which requires the corresponding items to have the same, or sufficiently similar, constituent structure. This aspect of correspondence, which is more prominent (and more notorious) than the previous one, is also much more difficult to make precise. Let us say, roughly, that a correspondence theorist may want to add a claim to her theory committing her to something like the following: > > Structure: > If an item of kind K corresponds to a certain > fact, then they have the same or sufficiently similar structure: the > overall correspondence between a true K and a fact is a matter of > part-wise correspondences, i.e. of their having corresponding > constituents in corresponding places in the same structure, or in > sufficiently similar structures. > The basic idea is that truthbearers and facts are both complex structured entities: truthbearers are composed of (other truthbearers and ultimately of) words, or concepts; facts are composed of (other facts or states of affairs and ultimately of) things, properties, and relations. The aim is to show how the correspondence relation is generated from underlying relations between the ultimate constituents of truthbearers, on the one hand, and the ultimate constituents of their corresponding facts, on the other. One part of the project will be concerned with these correspondence-generating relations: it will lead into a theory that addresses the question how simple words, or concepts, can be about things, properties, and relations; i.e., it will merge with semantics or psycho-semantics (depending on what the truthbearers are taken to be). The other part of the project, the specifically ontological part, will have to provide identity criteria for facts and explain how their simple constituents combine into complex wholes. Putting all this together should yield an account of the conditions determining which truthbearers correspond to which facts. Correlation and Structure reflect distinct aspects of correspondence. One might want to endorse the former without the latter, though it is hard to see how one could endorse the latter without embracing at least part (a) of the former. The isomorphism approach offers an answer to objection 3.C1. Although the truth that the cat is on the mat does not resemble the cat or the mat (the truth doesn't meow or smell, etc.), it does resemble the fact that the cat is on the mat. This is not a qualitative resemblance; it is a more abstract, structural resemblance. The approach also puts objection 3.C2 in some perspective. The correspondence relation is supposed to reduce to underlying relations between words, or concepts, and reality. Consequently, a correspondence theory is little more than a spin-off from semantics and/or psycho-semantics, i.e. the theory of intentionality construed as incorporating a representational theory of the mind (cf. Fodor 1989). This reminds us that, as a relation, correspondence is no more--but also no less--mysterious than semantic relations in general. Such relations have some curious features, and they raise a host of puzzles and difficult questions--most notoriously: Can they be explained in terms of natural (causal) relations, or do they have to be regarded as irreducibly non-natural aspects of reality? Some philosophers have claimed that semantic relations are too mysterious to be taken seriously, usually on the grounds that they are not explainable in naturalistic terms. But one should bear in mind that this is a very general and extremely radical attack on semantics as a whole, on the very idea that words and concepts can be about things. The common practice to aim this attack specifically at the correspondence theory seems misleading. As far as the intelligibility of the correspondence relation is concerned, the correspondence theory will stand, or fall, with the general theory of reference and intentionality. It should be noted, though, that these points concerning objections 3.C1 and 3.C2 are not independent of one's views about the nature of the primary truthbearers. If truthbearers are taken to be sentences of an ordinary language (or an idealized version thereof), or if they are taken to be mental representations (sentences of the language of thought), the above points hold without qualification: correspondence will be a semantic or psycho-semantic relation. If, on the other hand, the primary truthbearers are taken to be propositions, there is a complication: 1. On a broadly Fregean view of propositions, propositions are constituted by concepts of objects and properties (in the logical, not the psychological, sense of "concept"). On this view, the above points still hold, since the relation between concepts, on the one hand, and the objects and properties they are concepts of, on the other, appears to be a semantic relation, a concept-semantic relation. 2. On the so-called Russellian view of propositions (which the early Russell inherited mostly from early Moore), propositions are constituted, not of concepts of objects and properties, but of the objects and properties themselves (cf. Russell 1903). On this view, the points above will most likely fail, since the correspondence relation would appear to collapse into the identity relation when applied to true Russellian propositions. It is hard to see how a true Russellian proposition could be anything but a fact: What would a fact be, if not this sort of thing? So the principle of Nonidentity is rejected, and with it goes the correspondence theory of truth: "Once it is definitely recognized that the proposition is to denote, not a belief or form of words, but an object of belief, it seems plain that a truth differs in no respect from the reality with which it was supposed merely to correspond" (Moore 1901-02, p. 717). A simple, fact-based correspondence theory, applied to propositions understood in the Russellian way, thus reduces to an identity theory of truth, on which a proposition is true iff it is a fact, and false, iff it is not a fact. See below, Section 8.3; and the entries on propositions, singular propositions, and structured propositions in this encyclopedia. But Russellians don't usually renounce the correspondence theory entirely. Though they have no room for (1) from Section 3, when applied to propositions as truthbearers, correspondence will enter into their account of truth for sentences, public or mental. The account will take the form of Section 3's (2), applied to categories of truthbearers other than propositions, where Russellian propositions show up on the right-hand side in the guise of states of affairs that obtain or fail to obtain. Commitment to states of affairs in addition to propositions is sometimes regarded with scorn, as a gratuitous ontological duplication. But Russellians are not committed to states of affairs in addition to propositions, for propositions, on their view, must already be states of affairs. This conclusion is well nigh inevitable, once true propositions have been identified with facts. If a true proposition is a fact, then a false proposition that might have been true would have been a fact, if it had been true. So, a (contingent) false proposition must be the same kind of being as a fact, only not a fact--an unfact; but that just is a non-obtaining state of affairs under a different name. Russellian propositions are states of affairs: the false ones are states of affairs that do not obtain, and the true ones are states of affairs that do obtain. The Russellian view of propositions is popular nowadays. Somewhat curiously, contemporary Russellians hardly ever refer to propositions as facts or states of affairs. This is because they are much concerned with understanding belief, belief attributions, and the semantics of sentences. In such contexts, it is more natural to talk proposition-language than state-of-affairs-language. It feels odd (wrong) to say that someone believes a state of affairs, or that states of affairs are true or false. For that matter, it also feels odd (wrong) to say that some propositions are facts, that facts are true, and that propositions obtain or fail to obtain. Nevertheless, all of this must be the literal truth, according to the Russellians. They have to claim that "proposition" and "state of affairs", much like "evening star" and "morning star", are different names for the same things--they come with different associations and are at home in somewhat different linguistic environments, which accounts for the felt oddness when one name is transported to the other's environment. Returning to the isomorphism approach in general, on a strict or naive implementation of this approach, correspondence will be a one-one relation between truths and corresponding facts, which leaves the approach vulnerable to objections against funny facts (3.F1): each true truthbearer, no matter how complex, will be assigned a matching fact. Moreover, since a strict implementation of isomorphism assigns corresponding entities to all (relevant) constituents of truthbearers, complex facts will contain objects corresponding to the logical constants ("not", "or", "if-then", etc.), and these "logical objects" will have to be regarded as constituents of the world. Many philosophers have found it hard to believe in the existence of all these funny facts and funny quasi-logical objects. The isomorphism approach has never been advocated in a fully naive form, assigning corresponding objects to each and every wrinkle of our verbal or mental utterings. Instead, proponents try to isolate the "relevant" constituents of truthbearers through meaning analysis, aiming to uncover the logical form, or deep structure, behind ordinary language and thought. This deep structure might then be expressed in an ideal-language (typically, the language of predicate logic), whose syntactic structure is designed to mirror perfectly the ontological structure of reality. The resulting view--correspondence as isomorphism between properly analyzed truthbearers and facts--avoids assigning strange objects to such phrases as "the average husband", "the sake of", and "the present king of France"; but the view remains committed to logically complex facts and to logical objects corresponding to the logical constants. Austin (1950) rejects the isomorphism approach on the grounds that it projects the structure of our language onto the world. On his version of the correspondence theory (a more elaborated variant of (4) applied to statements), a statement as a whole is correlated to a state of affairs by arbitrary linguistic conventions without mirroring the inner structure of its correlate (cf. also Vision 2004). This approach appears vulnerable to the objection that it avoids funny facts at the price of neglecting systematicity. Language does not provide separate linguistic conventions for each statement: that would require too vast a number of conventions. Rather, it seems that the truth-values of statements are systematically determined, via a relatively small set of conventions, by the semantic values (relations to reality) of their simpler constituents. Recognition of this systematicity is built right into the isomorphism approach. Critics frequently echo Austin's "projection"-complaint, 3.F2, that a traditional correspondence theory commits "the error of reading back into the world the features of language" (Austin 1950, p. 155; cf. also, e.g., Rorty 1981). At bottom, this is a pessimistic stance: if there is a prima facie structural resemblance between a mode of speech or thought and some ontological category, it is inferred, pessimistically, that the ontological category is an illusion, a matter of us projecting the structure of our language or thought into the world. Advocates of traditional correspondence theories can be seen as taking the opposite stance: unless there are specific reasons to the contrary, they are prepared to assume, optimistically, that the structure of our language and/or thought reflects genuine ontological categories, that the structure of our language and/or thought is, at least to a significant extent, the way it is because of the structure of the world. ## 7. Modified Versions of the Correspondence Theory ### 7.1 Logical Atomism Wittgenstein (1921) and Russell (1918) propose modified fact-based correspondence accounts of truth as part of their program of logical atomism. Such accounts proceed in two stages. At the first stage, the basic truth-definition, say (1) from Section 3, is restricted to a special subclass of truthbearers, the so-called elementary or atomic truthbearers, whose truth is said to consist in their correspondence to (atomic) facts: if x is elementary, then x is true iff x corresponds to some (atomic) fact. This restricted definition serves as the base-clause for truth-conditional recursion-clauses given at the second stage, at which the truth-values of non-elementary, or molecular, truthbearers are explained recursively in terms of their logical structure and the truth-values of their simpler constituents. For example: a sentence of the form 'not-p' is true iff 'p' is false; a sentence of the form 'p and q' is true iff 'p' is true and 'q' is true; a sentence of the form 'p or q' is true iff 'p' is true or 'q' is true, etc. These recursive clauses (called "truth conditions") can be reapplied until the truth of a non-elementary, molecular sentence of arbitrary complexity is reduced to the truth or falsehood of its elementary, atomic constituents. Logical atomism exploits the familiar rules, enshrined in the truth-tables, for evaluating complex formulas on the basis of their simpler constituents. These rules can be understood in two different ways: (a) as tracing the ontological relations between complex facts and constituent simpler facts, or (b) as tracing logico-semantic relations, exhibiting how the truth-values of complex sentences can be explained in terms of their logical relations to simpler constituent sentences together with the correspondence and non-correspondence of simple, elementary sentences to atomic facts. Logical atomism takes option (b). Logical atomism is designed to go with the ontological view that the world is the totality of atomic facts (cf. Wittgenstein 1921, 2.04); thus accommodating objection 3.F2 by doing without funny facts: atomic facts are all the facts there are--although real-life atomists tend to allow conjunctive facts, regarding them as mere aggregates of atomic facts. An elementary truth is true because it corresponds to an atomic fact: correspondence is still isomorphism, but it holds exclusively between elementary truths and atomic facts. There is no match between truths and facts at the level of non-elementary, molecular truths; e.g., 'p', 'p or q', and 'p or r' might all be true merely because 'p' corresponds to a fact). The trick for avoiding logically complex facts lies in not assigning any entities to the logical constants. Logical complexity, so the idea goes, belongs to the structure of language and/or thought; it is not a feature of the world. This is expressed by Wittgenstein in an often quoted passage (1921, 4.0312): "My fundamental idea is that the 'logical constants' are not representatives; that there can be no representatives of the logic of facts"; and also by Russell (1918, p. 209f.): "You must not look about the real world for an object which you can call 'or', and say 'Now look at this. This is 'or''". Though accounts of this sort are naturally classified as versions of the correspondence theory, it should be noted that they are strictly speaking in conflict with the basic forms presented in Section 3. According to logical atomism, it is not the case that for every truth there is a corresponding fact. It is, however, still the case that the being true of every truth is explained in terms of correspondence to a fact (or non-correspondence to any fact) together with (in the case of molecular truths) logical notions detailing the logical structure of complex truthbearers. Logical atomism attempts to avoid commitment to logically complex, funny facts via structural analysis of truthbearers. It should not be confused with a superficially similar account maintaining that molecular facts are ultimately constituted by atomic facts. The latter account would admit complex facts, offering an ontological analysis of their structure, and would thus be compatible with the basic forms presented in Section 3, because it would be compatible with the claim that for every truth there is a corresponding fact. (For more on classical logical atomism, see Wisdom 1931-1933, Urmson 1953, and the entries on Russell's logical atomism and Wittgenstein's logical atomism in this encyclopedia.) While Wittgenstein and Russell seem to have held that the constituents of atomic facts are to be determined on the basis of a priori considerations, Armstrong (1997, 2004) advocates an a posteriori form of logical atomism. On his view, atomic facts are composed of particulars and simple universals (properties and relations). The latter are objective features of the world that ground the objective resemblances between particulars and explain their causal powers. Accordingly, what particulars and universals there are will have to be determined on the basis of total science. Problems: Logical atomism is not easy to sustain and has rarely been held in a pure form. Among its difficulties are the following: (a) What, exactly, are the elementary truthbearers? How are they determined? (b) There are molecular truthbearers, such as subjunctives and counterfactuals, that tend to provoke the funny-fact objection but cannot be handled by simple truth-conditional clauses, because their truth-values do not seem to be determined by the truth-values of their elementary constituents. (c) Are there universal facts corresponding to true universal generalizations? Wittgenstein (1921) disapproves of universal facts; apparently, he wants to re-analyze universal generalizations as infinite conjunctions of their instances. Russell (1918) and Armstrong (1997, 2004) reject this analysis; they admit universal facts. (d) Negative truths are the most notorious problem case, because they clash with an appealing principle, the "truthmaker principle" (cf. Section 8.5), which says that for every truth there must be something in the world that makes it true, i.e., every true truthbearer must have a truthmaker. Suppose 'p' is elementary. On the account given above, 'not-p' is true iff 'p' is false iff 'p' does not correspond to any fact; hence, 'not-p', if true, is not made true by any fact: it does not seem to have a truthmaker. Russell finds himself driven to admit negative facts, regarded by many as paradigmatically disreputable portions of reality. Wittgenstein sometimes talks of atomic facts that do not exist and calls their very nonexistence a negative fact (cf. 1921, 2.06)--but this is hardly an atomic fact itself. Armstrong (1997, chap. 8.7; 2004, chaps. 5-6) holds that negative truths are made true by a second-order "totality fact" which says of all the (positive) first-order facts that they are all the first-order facts. Atomism and the Russellian view of propositions (see Section 6). By the time Russell advocated logical atomism (around 1918), he had given up on what is now referred to as the Russellian conception of propositions (which he and G. E. Moore held around 1903). But Russellian propositons are popular nowadays. Note that logical atomism is not for the friends of Russellian propositions. The argument is straightforward. We have logically complex beliefs some of which are true. According to the friends of Russellian propositions, the contents of our beliefs are Russellian propositions, and the contents of our true beliefs are true Russellian propositions. Since true Russellian propositions are facts, there must be at least as many complex facts as there are true beliefs with complex contents (and at least as many complex states of affairs as there are true or false beliefs with complex contents). Atomism may work for sentences, public or mental, and for Fregean propositions; but not for Russellian propositions. Logical atomism is designed to address objections to funny facts (3.F1). It is not designed to address objections to facts in general (3.F2). Here logical atomists will respond by defending (atomic) facts. According to one defense, facts are needed because mere objects are not sufficiently articulated to serve as truthmakers. If a were the sole truthmaker of 'a is F', then the latter should imply 'a is G', for any 'G'. So the truthmaker for 'a is F' needs at least to involve a and Fness. But since Fness is a universal, it could be instantiated in another object, b, hence the mere existence of a and Fness is not sufficient for making true the claim 'a is F': a and Fness need to be tied together in the fact of a's being F. Armstrong (1997) and Olson (1987) also maintain that facts are needed to make sense of the tie that binds particular objects to universals. In this context it is usually emphasized that facts do not supervene on, hence, are not reducible to, their constituents. Facts are entities over and above the particulars and universals of which they are composed: a's loving b and b's loving a are not the same fact even though they have the very same constituents. Another defense of facts, surprisingly rare, would point out that many facts are observable: one can see that the cat is on the mat; and this is different from seeing the cat, or the mat, or both. The objection that many facts are not observable would invite the rejoinder that many objects are not observable either. (See Austin 1961, Vendler 1967, chap. 5, and Vision 2004, chap. 3, for more discussion of anti-fact arguments; see also the entry facts in this encyclopedia.) Some atomists propose an atomistic version of definition (1), but without facts, because they regard facts as slices of reality too suspiciously sentence-like to be taken with full ontological seriousness. Instead, they propose events and/or objects-plus-tropes (a.k.a. modes, particularized qualities, moments) as the corresponding portions of reality. It is claimed that these items are more "thingy" than facts but still sufficiently articulated--and sufficiently abundant--to serve as adequate truthmakers (cf. Mulligan, Simons, and Smith 1984). ### 7.2 Logical "Subatomism" Logical atomism aims at getting by without logically complex truthmakers by restricting definitions like (1) or (2) from Section 3 to elementary truthbearers and accounting for the truth-values of molecular truthbearers recursively in terms of their logical structure and atomic truthmakers (atomic facts, events, objects-plus-tropes). More radical modifications of the correspondence theory push the recursive strategy even further, entirely discarding definitions like (1) or (2), and hence the need for atomic truthmakers, by going, as it were, "subatomic". Such accounts analyze truthbearers, e.g., sentences, into their subsentential constituents and dissolve the relation of correspondence into appropriate semantic subrelations: names refer to, or denote, objects; predicates (open sentences) apply to, or are satisfied by objects. Satisfaction of complex predicates can be handled recursively in terms of logical structure and satisfaction of simpler constituent predicates: an object o satisfies 'x is not F' iff o does not satisfy 'x is F'; o satisfies 'x is F or x is G' iff o satisfies 'x is F' or o satisfies 'x is G'; and so on. These recursions are anchored in a base-clause addressing the satisfaction of primitive predicates: an object o satisfies 'x is F' iff o instantiates the property expressed by 'F'. Some would prefer a more nominalistic base-clause for satisfaction, hoping to get by without seriously invoking properties. Truth for singular sentences, consisting of a name and an arbitrarily complex predicate, is defined thus: A singular sentence is true iff the object denoted by the name satisfies the predicate. Logical machinery provided by Tarski (1935) can be used to turn this simplified sketch into a more general definition of truth--a definition that handles sentences containing relational predicates and quantifiers and covers molecular sentences as well. Whether Tarski's own definition of truth can be regarded as a correspondence definition, even in this modified sense, is under debate (cf. Popper 1972; Field 1972, 1986; Kirkham 1992, chaps. 5-6; Soames 1999; Kunne 2003, chap. 4; Patterson 2008.) Subatomism constitutes a return to (broadly) object-based correspondence. Since it promises to avoid facts and all similarly articulated, sentence-like slices of reality, correspondence theorists who take seriously objection 3.F2 favor this approach: not even elementary truthbearers are assigned any matching truthmakers. The correspondence relation itself has given way to two semantic relations between constituents of truthbearers and objects: reference (or denotation) and satisfaction--relations central to any semantic theory. Some advocates envision causal accounts of reference and satisfaction (cf. Field 1972; Devitt 1982, 1984; Schmitt 1995; Kirkham 1992, chaps. 5-6). It turns out that relational predicates require talk of satisfaction by ordered sequences of objects. Davidson (1969, 1977) maintains that satisfaction by sequences is all that remains of the traditional idea of correspondence to facts; he regards reference and satisfaction as "theoretical constructs" not in need of causal, or any, explanation. Problems: (a) The subatomistic approach accounts for the truth-values of molecular truthbearers in the same way as the atomistic approach; consequently, molecular truthbearers that are not truth-functional still pose the same problems as in atomism. (b) Belief attributions and modal claims pose special problems; e.g., it seems that "believes" is a relational predicate, so that "John believes that snow is white" is true iff "believes" is satisfied by John and the object denoted by "that snow is white"; but the latter appears to be a proposition or state of affairs, which threatens to let in through the back-door the very sentence-like slices of reality the subatomic approach was supposed to avoid, thus undermining the motivation for going subatomic. (c) The phenomenon of referential indeterminacy threatens to undermine the idea that the truth-values of elementary truthbearers are always determined by the denotation and/or satisfaction of their constituents; e.g., pre-relativistic uses of the term "mass" are plausibly taken to lack determinate reference (referring determinately neither to relativistic mass nor to rest mass); yet a claim like "The mass of the earth is greater than the mass of the moon" seems to be determinately true even when made by Newton (cf. Field 1973). Problems for both versions of modified correspondence theories: (a) It is not known whether an entirely general recursive definition of truth, one that covers all truthbearers, can be made available. This depends on unresolved issues concerning the extent to which truthbearers are amenable to the kind of structural analyses that are presupposed by the recursive clauses. The more an account of truth wants to exploit the internal structure of truthbearers, the more it will be hostage to the (limited) availability of appropriate structural analyses of the relevant truthbearers. (b) Any account of truth employing a recursive framework may be virtually committed to taking sentences (maybe sentences of the language of thought) as primary truthbearers. After all, the recursive clauses rely heavily on what appears to be the logico-syntactic structure of truthbearers, and it is unclear whether anything but sentences can plausibly be said to possess that kind of structure. But the thesis that sentences of any sort are to be regarded as the primary truthbearers is contentious. Whether propositions can meaningfully be said to have an analogous (albeit non-linguistic) structure is under debate (cf. Russell 1913, King 2007). (c) If clauses like "'p or q' is true iff 'p' is true or 'q' is true" are to be used in a recursive account of our notion of truth, as opposed to some other notion, it has to be presupposed that 'or' expresses disjunction: one cannot define "or" and "true" at the same time. To avoid circularity, a modified correspondence theory (be it atomic or subatomic) must hold that the logical connectives can be understood without reference to correspondence truth. ### 7.3 Relocating Correspondence Definitions like (1) and (2) from Section 3 assume, naturally, that truthbearers are true because they, the truthbearers themselves, correspond to facts. There are however views that reject this natural assumption. They propose to account for the truth of truthbearers of certain kinds, propositions, not by way of their correspondence to facts, but by way of the correspondence to facts of other items, the ones that have propositions as their contents. Consider the state of believing that p (or the activity of judging that p). The state (the activity) is not, strictly speaking, true or false; rather, what is true or false is its content, the proposition that p. Nevertheless, on the present view, it is the state of believing that p that corresponds or fails to correspond to a fact. So truth/falsehood for propositions can be defined in the following manner: x is a true/false proposition iff there is a belief state B such that x is the content of B and B corresponds/fails to correspond to a fact. Such a modification of fact-based correspondence can be found in Moore (1927, p. 83) and Armstrong (1973, 4.iv & 9). It can be adapted to atomistic (Armstrong) and subatomistic views, and to views on which sentences (of the language of thought) are the primary bearers of truth and falsehood. However, by taking the content-carrying states as the primary corresponders, it entails that there are no truths/falsehoods that are not believed by someone. Most advocates of propositions as primary bearers of truth and falsehood will regard this as a serious weakness, holding that there are very many true and false propositions that are not believed, or even entertained, by anyone. Armstrong (1973) combines the view with an instrumentalist attitude towards propositions, on which propositions are mere abstractions from mental states and should not be taken seriously, ontologically speaking. ## 8. The Correspondence Theory and Its Competitors ### 8.1 Traditional Competitors Against the traditional competitors--coherentist, pragmatist, and verificationist and other epistemic theories of truth--correspondence theorists raise two main sorts of objections. First, such accounts tend to lead into relativism. Take, e.g., a coherentist account of truth. Since it is possible that 'p' coheres with the belief system of S while 'not-p' coheres with the belief system of S\, the coherentist account seems to imply, absurdly, that contradictories, 'p' and 'not-p', could both be true. To avoid embracing contradictions, coherentists often commit themselves (if only covertly) to the objectionable relativistic view that 'p' is true-for-S* and 'not-p' is true-for-S\. Second, the accounts tend to lead into some form of idealism or anti-realism, e.g., it is possible for the belief that p* to cohere with someone's belief system, even though it is not a fact that p; also, it is possible for it to be a fact that p, even if no one believes that p at all or if the belief does not cohere with anyone's belief system. Cases of this sort are frequently cited as counterexamples to coherentist accounts of truth. Dedicated coherentists tend to reject such counterexamples, insisting that they are not possible after all. Since it is hard to see why they would not be possible, unless its being a fact that p were determined by the belief's coherence with other beliefs, this reaction commits them to the anti-realist view that the facts are (largely) determined by what we believe. This offers a bare outline of the overall shape the debates tend to take. For more on the correspondence theory vs. its traditional competitors see, e.g., Vision 1988; Kirkham 1992, chaps. 3, 7-8; Schmitt 1995; Kunne 2003, chap. 7; and essays in Lynch 2001. Walker 1989 is a book-lenght discussion of coherence theories of truth. See also the entries on pragmatism, relativism, the coherence theory of truth, in this encyclopedia. ### 8.2 Pluralism The correspondence theory is sometimes accused of overreaching itself: it does apply, so the objection goes, to truths from some domains of discourse, e.g., scientific discourse and/or discourse about everyday midsized physical things, but not to truths from various other domains of discourse, e.g., ethical and/or aesthetic discourse (see the first objection in Section 5 above). Alethic pluralism grows out of this objection, maintaining that truth is constituted by different properties for true propositions from different domains of discourse: by correspondence to fact for true propositions from the domain of scientific or everyday discourse about physical things; by some epistemic property, such as coherence or superassertibility, for true propositions from the domain of ethical and aesthetic discourse, and maybe by still other properties for other domains of discourse. This suggests a position on which the term "true" is multiply ambiguous, expressing different properties when applied to propositions from different domains. However, contemporary pluralists reject this problematic idea, maintaining instead that truth is "multiply realizable". That is, the term "true" is univocal, it expresses one concept or property, truth (being true), but one that can be realized by or manifested in different properties (correspondence to fact, coherence or superassertibility, and maybe others) for true propositions from different domains of discourse. Truth itself is not to be identified with any of its realizing properties. Instead, it is characterized, quasi axiomatically, by a set of alleged "platitudes", including, according to Crispin Wright's (1999) version, "transparency" (to assert is to present as true), "contrast" (a proposition may be true without being justified, and v.v.), "timelesness" (if a proposition is ever true, then it always is), "absoluteness" (there is no such thing as a proposition being more or less true), and others. Though it contains the correspondence theory as one ingredient, alethic pluralism is nevertheless a genuine competitor, for it rejects the thesis that truth is correspondence to reality. Moreover, it equally contains competitors of the correspondence theory as further ingredients. Alethic pluralism in its contemporary form is a relatively young position. It was inaugurated by Crispin Wright (1992; see also 1999) and was later developed into a somewhat different form by Lynch (2009). Critical discussion is still at a relatively nascent stage (but see Vision 2004, chap. 4, for extended discussion of Wright). It will likely focus on two main problem areas. First, it seems difficult to sort propositions into distinct kinds according to the subject matter they are about. Take, e.g., the proposition that killing is morally wrong, or the proposition that immoral acts happen in space-time. What are they about? Intuitively, their subject matter is mixed, belonging to the physical domain, the biological domain, and the domain of ethical discourse. It is hard to see how pluralism can account for the truth of such mixed propositions, belonging to more than one domain of discourse: What will be the realizing property? Second, pluralists are expected to explain how the platitudes can be "converted" into an account of truth itself. Lynch (2009) proposes to construe truth as a functional property, defined in terms of a complex functional role which is given by the conjunction of the platitudes (somewhat analogous to the way in which functionalists in the philosophy of mind construe mental states as functional states, specified in terms of their functional roles--though in their case the relevant functional roles are causal roles, which is not a feasible option when it comes to the truth-role). Here the main issue will be to determine (a) whether such an account really works, when the technical details are laid out, and (b) whether it is plausible to claim that properties as different as correspondence to a fact, on the one hand, and coherence or superassertibilty, on the other, can be said to play one and the same role--a claim that seems required by the thesis that these different properties all realize the same property, being true. For more on pluralism, see e.g. the essays in Monnoyer (2007) and in Pedersen & Wright (2013); and the entry on pluralist theories of truth in this encyclopedia. ### 8.3 The Identity Theory of Truth According to the identity theory of truth, true propositions do not correspond to facts, they are facts: the true proposition that snow is white = the fact that snow is white. This non-traditional competitor of the correspondence theory threatens to collapse the correspondence relation into identity. (See Moore 1901-02; and Dodd 2000 for a book-length defense of this theory and discussion contrasting it with the correspondence theory; and see the entry the identity theory of truth: in this encyclopedia.) In response, a correspondence theorist will point out: (a) The identity theory is defensible only for propositions as truthbearers, and only for propositions construed in a certain way, namely as having objects and properties as constituents rather than ideas or concepts of objects and properties; that is, for Russellian propositions. Hence, there will be ample room (and need) for correspondence accounts of truth for other types of truthbearers, including propositions, if they are construed as constituted, partly or wholly, of concepts of objects and properties. (b) The identity theory is committed to the unacceptable consequence that facts are true. (c) The identity theory rests on the assumption that that-clauses always denote propositions, so that the that-clause in "the fact that snow is white" denotes the proposition that snow is white. The assumption can be questioned. That-clauses can be understood as ambiguous names, sometimes denoting propositions and sometimes denoting facts. The descriptive phrases "the proposition..." and "the fact..." can be regarded as serving to disambiguate the succeeding ambiguous that-clauses--much like the descriptive phrases in "the philosopher Socrates" and "the soccer-player Socrates" serve to disambiguate the ambiguous name "Socrates" (cf. David 2002). ### 8.4 Deflationism About Truth At present the most noticeable competitors to correspondence theories are deflationary accounts of truth (or 'true'). Deflationists maintain that correspondence theories need to be deflated; that their central notions, correspondence and fact (and their relatives), play no legitimate role in an adequate account of truth and can be excised without loss. A correspondence-type formulation like > (5) "Snow is white" is true iff it corresponds > to the fact that snow is white, is to be deflated to > (6) "Snow is white" is true iff snow is > white, which, according to deflationists, says all there is to be said about the truth of "Snow is white", without superfluous embellishments (cf. Quine 1987, p. 213). Correspondence theorists protest that (6) cannot lead to anything deserving to be regarded as an account of truth. It is concerned with only one particular sentence ("Snow is white"), and it resists generalization. (6) is a substitution instance of the schema > (7) "p" is true iff > p, which does not actually say anything itself (it is not truth-evaluable) and cannot be turned into a genuine generalization about truth, because of its essential reliance on the schematic letter "p", a mere placeholder. The attempt to turn (7) into a generalization produces nonsense along the lines of "For every x, "x" is true iff x", or requires invocation of truth: "Every substitution instance of the schema ""p" is true iff p" is true". Moreover, no genuine generalizations about truth can be accounted for on the basis of (7). Correspondence definitions, on the other hand, do yield genuine generalizations about truth. Note that definitions like (1) and (2) in Section 3 employ ordinary objectual variables (not mere schematic placeholders); the definitions are easily turned into genuine generalizations by prefixing the quantifier phrase "For every x", which is customarily omitted in formulations intended as definitions. It should be noted that the deflationist's starting point, (5), which lends itself to deflating excisions, actually misrepresents the correspondence theory. According to (5), corresponding to the fact that snow is white is sufficient and necessary for "Snow is white" to be true. Yet, according to (1) and (2), it is sufficient but not necessary: "Snow is white" will be true as long as it corresponds to some fact or other. The genuine article, (1) or (2), is not as easily deflated as the impostor (5). The debate turns crucially on the question whether anything deserving to be called an "account" or "theory" of truth ought to take the form of a genuine generalization (and ought to be able to account for genuine generalizations involving truth). Correspondence theorists tend to regard this as a (minimal) requirement. Deflationists argue that truth is a shallow (sometimes "logical") notion--a notion that has no serious explanatory role to play: as such it does not require a full-fledged account, a real theory, that would have to take the form of a genuine generalization. There is now a substantial body of literature on truth-deflationism in general and its relation to the correspondence theory in particular; the following is a small selection: Quine 1970, 1987; Devitt 1984; Field 1986; Horwich 1990 & 19982; Kirkham 1992; Gupta 1993; David 1994, 2008; Schmitt 1995; Kunne 2003, chap. 4; Rami 2009. Relevant essays are contained in Blackburn and Simmons 1999; Schantz 2002; Armour-Garb and Beall 2005; and Wright and Pedersen 2010. See also the entry the deflationary theory of truth in this encyclopedia. ### 8.5 Truthmaker Theory This approach centers on the truthmaker or truthmaking principle: Every truth has a truthmaker; or alternatively: For every truth there is something that makes it true. The principle is usually understood as an expression of a realist attitude, emphasizing the crucial contribution the world makes to the truth of a proposition. Advocates tend to treat truthmaker theory primarily as a guide to ontology, asking: To entities of what ontological categories are we committed as truthmakers of the propositions we accept as true? Most advocates maintain that propositions of different logical types can be made true by items from different ontological categories: e.g., propositions of some types are made true by facts, others just by individual things, others by events, others by tropes (cf., e.g. Armstrong 1997). This is claimed as a significant improvement over traditional correspondence theories which are understood--correctly in most but by no means all cases--to be committed to all truthmakers belonging to a single ontological category (albeit disagreeing about which category that is). All advocates of truthmaker theory maintain that the truthmaking relation is not one-one but many-many: some truths are made true by more than one truthmaker; some truthmakers make true more than one truth. This is also claimed as a significant improvement over traditional correspondence theories which are often portrayed as committed to correspondence being a one-one relation. This portrayal is only partly justified. While it is fairly easy to find real-life correspondence theorists committing themselves to the view that each truth corresponds to exactly one fact (at least by implication, talking about the corresponding fact), it is difficult to find real-life correspondence theorists committing themselves to the view that only one truth can correspond to a given fact (but see Moore 1910-11, p. 256). A truthmaker theory may be presented as a competitor to the correspondence theory or as a version of the correspondence theory. This depends considerably on how narrowly or broadly one construes "correspondence theory", i.e. on terminological issues. Some advocates would agree with Dummett (1959, p. 14) who said that, although "we have nowadays abandoned the correspondence theory of truth", it nevertheless "expresses one important feature of the concept of truth...: that a statement is true only if there is something in the world in virtue of which it is true". Other advocates would follow Armstrong who tends to present his truthmaker theory as a liberal form of correspondence theory; indeed, he seems committed to the view that the truth of a (contingent) elementary proposition consists in its correspondence with some (atomic) fact (cf. Armstrong 1997; 2004, pp. 22-3, 48-50). It is not easy to find a substantive difference between truthmaker theory and various brands of the sort of modified correspondence theory treated above under the heading "Logical Atomism" (see Section 7.1). Logical atomists, such as Russell (1918) and Wittgenstein (1921), will hold that the truth or falsehood of every truth-value bearer can be explained in terms of (can be derived from) logical relations between truth-value bearers, by way of the recursive clauses, together with the base clauses, i.e., the correspondence and non-correspondence of elementary truth-value bearers with facts. This recursive strategy could be pursued with the aim to reject the truthmaker principle: not all truths have truthmakers, only elementary truths have truthmakers (here understood as corresponding atomic facts). But it could also be pursued--and this seems to have been Russell's intention at the time--with the aim to secure the truthmaker principle, even though the simple correspondence definition has been abandoned: not every truth corresponds to a fact, only elementary truths do, but every truth has a truthmaker; where the recursive clauses are supposed to show how truthmaking without correspondence, but grounded in correspondence, comes about. There is one straightforward difference between truthmaker theory and most correspondence theories. The latter are designed to answer the question "What is truth?". Simple (unmodified) correspondence theories center on a biconditional, such as "x is true iff x corresponds to a fact", intended to convey a definition of truth (at least a "real definition" which does not commit them to the claim that the term "true" is synonymous with "corresponds to a fact"--especially nowadays most correspondence theorists would consider such a claim to be implausibly and unnecessarily bold). Modified correspondence theories also aim at providing a definition of truth, though in their case the definition will be considerably more complex, owing to the recursive character of the account. Truthmaker theory, on the other hand, centers on the truthmaker principle: For every truth there is something that makes it true. Though this principle will deliver the biconditional "x is true iff something makes x true" (since "something makes x true" trivially implies "x is true"), this does not yield a promising candidate for a definition of truth: defining truth in terms of truthmaking would appear to be circular. Unlike most correspondence theories, truthmaker theory is not equipped, and usually not designed, to answer the question "What is truth?"--at least not if one expects the answer to take the form of a feasible candidate for a definition of truth. There is a growing body of literature on truthmaker theory; see for example: Russell 1918; Mullligan, Simons, and Smith 1984; Fox 1987; Armstrong 1997, 2004; Merricks 2007; and the essays in Beebe and Dodd 2005; Monnoyer 2007; and in Lowe and Rami 2009. See also the entry on truthmakers in this encyclopedia. ## 9. More Objections to the Correspondence Theory Two final objections to the correspondence theory deserve separate mention. ### 9.1 The Big Fact Inspired by an allegedly similar argument of Frege's, Davidson (1969) argues that the correspondence theory is bankrupt because it cannot avoid the consequence that all true sentences correspond to the same fact: the Big Fact. The argument is based on two crucial assumptions: (i) Logically equivalent sentences can be substituted salva veritate in the context 'the fact that...'; and (ii) If two singular terms denoting the same thing can be substituted for each other in a given sentence salva veritate, they can still be so substituted if that sentence is embedded within the context 'the fact that...'. In the version below, the relevant singular terms will be the following: '(the x such that x = Diogenes & p)' and '(the x such that x = Diogenes & q)'. Now, assume that a given sentence, s, corresponds to the fact that p; and assume that 'p' and 'q' are sentences with the same truth-value. We have: * s corresponds to the fact that p which, by (i), implies * s corresponds to the fact that [(the x such that x = Diogenes & p) = (the x such that x = Diogenes)], which, by (ii), implies * s corresponds to the fact that [(the x such that x = Diogenes & q) = (the x such that x = Diogenes)], which, by (i), implies * s corresponds to the fact that q. Since the only restriction on 'q' was that it have the same truth-value as 'p', it would follow that any sentence s that corresponds to any fact corresponds to every fact; so that all true sentences correspond to the same facts, thereby proving the emptiness of the correspondence theory--the conclusion of the argument is taken as tantamount to the conclusion that every true sentence corresponds to the totality of all the facts, i.e, the Big Fact, i.e., the world as a whole. This argument belongs to a type now called "slingshot arguments" (because a giant opponent is brought down by a single small weapon, allegedly). The first versions of this type of argument were given by Church (1943) and Godel (1944); it was later adapted by Quine (1953, 1960) in his crusade against quantified modal logic. Davidson is offering yet another adaption, this time involving the expression "corresponds to the fact that". The argument has been criticized repeatedly. Critics point to the two questionable assumptions on which it relies, (i) and (ii). It is far from obvious why a correspondence theorist should be tempted by either one of them. Opposition to assumption (i) rests on the view that expressibility by logically equivalent sentences may be a necessary, but is not a sufficient condition for fact identity. Opposition to assumption (ii) rests on the observation that the (alleged) singular terms used in the argument are definite descriptions: their status as genuine singular terms is in doubt, and it is well-known that they behave rather differently than proper names for which assumption (ii) is probably valid (cf. Follesdal 1966/2004; Olson 1987; Kunne 2003; and especially the extended discussion and criticism in Neale 2001.) ### 9.2 No Independent Access to Reality The objection that may well have been the most effective in causing discontent with the correspondence theory is based on an epistemological concern. In a nutshell, the objection is that a correspondence theory of truth must inevitably lead into skepticism about the external world, because the required correspondence between our thoughts and reality is not ascertainable. Ever since Berkeley's attack on the representational theory of the mind, objections of this sort have enjoyed considerable popularity. It is typically pointed out that we cannot step outside our own minds to compare our thoughts with mind-independent reality. Yet--so the objection continues--on the correspondence theory of truth, this is precisely what we would have to do to gain knowledge. We would have to access reality as it is in itself, independently of our cognition, and determine whether our thoughts correspond to it. Since this is impossible, since all our access to the world is mediated by our cognition, the correspondence theory makes knowledge impossible (cf. Kant 1800, intro vii). Assuming that the resulting skepticism is unacceptable, the correspondence theory has to be rejected, and some other account of truth, an epistemic (anti-realist) account of some sort, has to be put in its place (cf., e.g., Blanshard 1941.) This type of objection brings up a host of issues in epistemology, the philosophy of mind, and general metaphysics. All that can be done here is to hint at a few pertinent points (cf. Searle 1995, chap. 7; David 2004, 6.7). The objection makes use of the following line of reasoning: "If truth is correspondence, then, since knowledge requires truth, we have to know that our beliefs correspond to reality, if we are to know anything about reality". There are two assumptions implicit in this line of reasoning, both of them debatable. (i) It is assumed that S knows x, only if S knows that x is true--a requirement not underwritten by standard definitions of knowledge, which tell us that S knows x, only if x is true and S is justified in believing x. The assumption may rest on confusing requirements for knowing x with requirements for knowing that one knows x. (ii) It is assumed that, if truth = F, then S knows that x is true, only if S knows that x has F. This is highly implausible. By the same standard it would follow that no one who does not know that water is H2O can know that the Nile contains water--which would mean, of course, that until fairly recently nobody knew that the Nile contained water (and that, until fairly recently, nobody knew that there were stars in the sky, whales in the sea, or that the sun gives light). Moreover, even if one does know that water is H2O, one's strategy for finding out whether the liquid in one's glass is water does not have to involve chemical analysis: one could simply taste it, or ask a reliable informant. Similarly, as far as knowing that x is true is concerned, the correspondence theory does not entail that we have to know that a belief corresponds to a fact in order to know that it is true, or that our method of finding out whether a belief is true has to involve a strategy of actually comparing a belief with a fact--although the theory does of course entail that one obtains knowledge only if one obtains a belief that corresponds to a fact. More generally, one might question whether the objection still has much bite once the metaphors of "accessing" and "comparing" are spelled out with more attention to the psychological details of belief formation and to epistemological issues concerning the conditions under which beliefs are justified or warranted. For example, it is quite unclear how the metaphor of "comparing" applies to knowledge gained through perceptual belief-formation. A perceptual belief that p may be true, and by having acquired that belief, one may have come to know that p, without having "compared" (the content of) one's belief with anything. One might also wonder whether its competitors actually enjoy any significant advantage over the correspondence theory, once they are held to the standards set up by this sort of objection. For example, why should it be easier to find out whether one particular belief coheres with all of one's other beliefs than it is to find out whether a belief corresponds with a fact? In one form or other, the "No independent access to reality"-objection against correspondence theoretic approaches has been one of the, if not the, main source and motivation for idealist and anti-realist stances in philosophy (cf. Stove 1991). However, the connection between correspondence theories of truth and the metaphysical realism vs. anti-realism (or idealism) debate is less immediate than is often assumed. On the one hand, deflationists and identity theorists can be, and typically are, metaphysical realists while rejecting the correspondence theory. On the other hand, advocates of a correspondence theory can, in principle, be metaphysical idealists (e.g. McTaggart 1921) or anti-realists, for one might advocate a correspondence theory while maintaining, at the same time, (a) that all facts are constituted by mind or (b) that what facts there are depends somehow on what we believe or are capable of believing, or (c) that the correspondence relation between true propositions and facts depends somehow on what we believe or are capable of believing (claiming that the correspondence relation between true beliefs or true sentences and facts depends on what we believe can hardly count as a commitment to anti-realism). Keeping this point in mind, one can nevertheless acknowledge that advocacy of a correspondence theory of truth comes much more naturally when combined with a metaphysically realist stance and usually signals commitment to such a stance.
truth-deflationary	## 1. Central Themes in Deflationism ### 1.1 The Equivalence Schema While deflationism can be developed in different ways, it is possible to isolate some central themes emphasized by most philosophers who think of themselves as deflationists. These shared themes pertain to endorsing a kind of metaphysical parsimony and positing a "deflated" role for what we can call the alethic locutions (most centrally, the expressions 'true' and 'false') in the instances of what is often called truth-talk. In this section, we will isolate three of these themes. The first, and perhaps most overarching, one has already been mentioned: According to deflationists, there is some strong equivalence between a statement like 'snow is white' and a statement like "'snow is white' is true," and this is all that can significantly be said about that application of the notion of truth. We may capture this idea more generally with the help of a schema, what is sometimes called the equivalence schema: (ES) $\langle p\rangle$ is true if, and only if, $p$. In this schema, the angle brackets indicate an appropriate name-forming or nominalizing device, e.g., quotation marks or 'the proposition that ...', and the occurrences of '$p$' are replaced with matching declarative sentences to yield instances of the schema. The equivalence schema is often associated with the formal work of Alfred Tarski (1935 [1956], 1944), which introduced the schema, (T) $X$ is true if, and only if, $p$. In the instances of schema (T) (sometimes called "Convention (T)"), the '$X$' gets filled in with a name of the sentence that goes in for the '$p$', making (T) a version of (ES). Tarski considered (T) to provide a criterion of adequacy for any theory of truth, thereby allowing that there could be more to say about truth than what the instances of the schema cover. Given that, together with the fact that he took the instances of (T) to be contingent, his theory does not qualify as deflationary. By contrast with the Tarskian perspective on (T)/(ES), we can formulate the central theme of deflationism under consideration as the view, roughly, that the instances of (some version of) this schema do capture everything significant that can be said about applications of the notion of truth; in a slogan, the instances of the schema exhaust the notion of truth. Approaches which depart from deflationism don't disagree that (ES) tells us something about truth; what they (with Tarski) deny is that it is exhaustive, that it tells us the whole truth about truth. Since such approaches add substantive explanations of why the instances of the equivalence schema hold, they are now often called inflationary approaches to truth. Inflationism is the general approach shared by such traditional views as the correspondence theory of truth, coherence theory of truth, pragmatic theory of truth, identity theory of truth, and primitivist theory of truth, These theories all share a collection of connected assumptions about the alethic locutions, the concept of truth, and the property of truth. Inflationary theories all assume that the expression 'is true' is a descriptive predicate, expressing an explanatory concept of truth, which determines a substantive property of truth. From that shared set of presuppositions, the various traditional inflationary theories then diverge from one another by providing different accounts of the assumed truth property. On inflationary views, the nature of the truth property explains why the instances of (ES) hold. Deflationary views, by contrast, reject some if not all of the standard assumptions that lead to inflationary theories, resisting at least their move to positing any substantive truth property. Instead, deflationists offer a different understanding of both the concept of truth and the functioning of the alethic locutions. A deflationist will take the instances of (ES) to be "conceptually basic and explanatorily fundamental" (Horwich 1998a, 21, n. 4; 50), or to be direct consequences of how the expression 'true' operates (cf. Quine 1970 [1986], Brandom 1988, and Field 1994a). It is important to notice that even among deflationists the equivalence schema may be interpreted in different ways, and this is one way to distinguish different versions of deflationism from one another. One question about (ES) concerns the issue of what instances of the schema are assumed to be about (equivalently: to what the names in instances of (ES) are assumed to refer). According to one view, the instances of this schema are about sentences, where a name for a sentence can be formulated simply by putting quotation marks around it. In other words, for those who hold what might be called a sententialist version of deflationism, the equivalence schema has instances like (1): (1) 'Brutus killed Caesar' is true if, and only if, Brutus killed Caesar. To make this explicit, we might say that, according to sententialist deflationism, the equivalence schema is: (ES-sent) The sentence '$p$' is true if, and only if, $p$. Notice that in this schema, the angle-brackets of (ES) have been replaced by quotation marks. According to those who hold what might be called a propositionalist version of deflationism, by contrast, instances of the equivalence schema are about propositions, where names of propositions are, or can be taken to be, expressions of the form 'the proposition that $p$', where '$p$' is filled in with a declarative sentence. For the propositionalist, in other words, instances of the equivalence schema are properly interpreted not as being about sentences but instead as being about propositions, i.e., as biconditionals like (2) rather than (1): (2) The proposition that Brutus killed Caesar is true if, and only if, Brutus killed Caesar. To make this explicit, we might say that, according to propositionalist deflationism, the equivalence schema is: (ES-prop) The proposition that $p$ is true if, and only if, $p$. Interpreting the equivalence schema as (ES-sent) rather than as (ES-prop), or vice versa, thus yields different versions of deflationism, sententialist and propositionalist versions, respectively. Another aspect that different readings of (ES) can vary across concerns the nature of the equivalence that its instances assert. On one view, the right-hand side and the left-hand side of such instances are synonymous or analytically equivalent. Thus, for sententialists who endorse this level of equivalence, (1) asserts that, "'Brutus killed Caesar' is true" means just what 'Brutus killed Caesar' means; while for propositionalists who endorse analytic equivalence, (2) asserts that 'the proposition that Brutus killed Caesar is true' means the same as 'Brutus killed Caesar'. A second view is that the right-hand and left-hand sides of claims such as (1) and (2) are not synonymous but are nonetheless necessarily equivalent; this view maintains that the two sides of each equivalence stand or fall together in every possible world, despite having different meanings. And a third possible view is that claims such as (1) and (2) assert only a material equivalence; this view interprets the 'if and only if' in both (1) and (2) as simply the biconditional of classical logic. This tripartite distinction between analytic, necessary, and material equivalence, when combined with the distinction between sententialism and propositionalism, yields six different possible (although not exhaustive) readings of the instances of (ES): \| \| Sentential \| Propositional \| \| --- \| --- \| --- \| \| Analytic \| $\mathbf{A}$ \| $\mathbf{B}$ \| \| Necessary \| $\mathbf{C}$ \| $\mathbf{D}$ \| \| Material \| $\mathbf{E}$ \| $\mathbf{F}$ \| While different versions of deflationism can be correlated to some extent with different positions in this chart, some chart positions have also been occupied by more than one version of deflationism. The labels 'redundancy theory', 'disappearance theory' and 'no-truth theory' have been used to apply to analytic versions of deflationism: positions $\mathbf{A}$ or $\mathbf{B}$. But there is a sense in which position $\mathbf{A}$ is also occupied by versions of what is called "disquotationalism" (although the most prominent disquotationalists tend to be leary of the notions of analyticity or synonymy), and what is called "prosententialism" also posits an equivalence of what is said with the left- and right-hand sides of the instances of (ES). The latter version of deflationism, however, does this without making the left-hand sides about sentences named via quotation or about propositions understood as abstract entities. No deflationist has offered an account occupying position $\mathbf{C}$, $\mathbf{E}$, or $\mathbf{F}$ (although the explicit inspiration some disquotationalists have found in Tarski's work and his deployment of material equivalence might misleadingly suggest position $\mathbf{E})$. Paul Horwich (1998a) uses the label 'minimalism' for a version of propositionalist deflationism that takes the instances of (ES-prop) to involve a necessary equivalence, thereby occupying position $\mathbf{D}$. To a large extent, philosophers prefer one or another (or none) of the positions in the chart on the basis of their views from other parts of philosophy, typically their views about the philosophy of language and metaphysics. ### 1.2 The Property of Truth The second theme we will discuss focuses on the fact that when we say, for example, that the proposition that Brutus killed Caesar is true, we seem to be attributing a property to that proposition, namely, the property of being true. Deflationists are typically wary of that claim, insisting either that there is no property of being true at all, or, if there is one, it is of a certain kind, often called "thin" or "insubstantial". The suggestion that there is no truth property at all is advanced by some philosophers in the deflationary camp; we will look at some examples below. What makes this position difficult to sustain is that 'is true' is grammatically speaking a predicate much like 'is metal'. If one assumes that grammatical predicates such as 'is metal' express properties, then, prima facie, the same would seem to go for 'is true.' This point is not decisive, however. For one thing, it might be possible to distinguish the grammatical form of claims containing 'is true' from their logical form; at the level of logical form, it might be, as prosententialists maintain, that 'is true' is not a predicate. For another, nominalists about properties have developed ways of thinking about grammatical predicates according to which these expressions don't express properties at all. A deflationist might appeal, perhaps selectively, to such proposals, in order to say that 'is true', while a predicate, does not express a property. Whatever the ultimate fate of these attempts to say that there is no property of truth may be, a suggestion among certain deflationists has been to concede that there is a truth property but to deny it is a property of a certain kind; in particular to deny that it is (as we will say) a substantive property. To illustrate the general idea, consider (3) and (4): (3) Caracas is the capital of Venezuela. (4) The earth revolves around the sun. Do the propositions that these sentences express share a property of being true? Well, in one intuitive sense they do: Since they both are true, we might infer that they both have the property of being true. From this point of view, there is a truth property: It is simply the property that all true propositions have. On the other hand, when we say that two things share a property of Fness, we often mean more than simply that they are both $F$. We often mean that two things that are $F$ have some underlying nature in common, for example, that there is a common explanation as to why they are both $F$. It is in this second claim that deflationists have in mind when they say that truth is not a substantive property. Thus, in the case of our example, what, if anything, explains the truth of (3) is that Caracas is the capital of Venezuela, and what explains this is the political history of Venezuela. On the other hand, what, if anything, explains the truth of (4) is that the earth revolves around the sun, and what explains this is the physical nature of the solar system. The physical nature of the solar system, however, has nothing to do with the political history of Venezuela (or if it does, the connections are completely accidental!) and to that extent there is no shared explanation as to why (3) and (4) are both true. Therefore, in this substantive sense, they have no property in common. It will help to bring out the contrast being invoked here if we consider two properties distinct from a supposed property of being true: the property of being a game and the property of being a mammal. Consider the games of chess and catch. Do both of these have the property of being a game? Well, in one sense, they do: they are both games that people can play. On the other hand, however, there is no common explanation as to why each counts as a game (cf. Wittgenstein 1953, SS66). We might then say that being a game is not a substantive property and mean just this. But now compare the property of being a mammal. If two things are mammals, they have the property of being a mammal, but in addition there is some common explanation as to why they are both mammals - both are descended from the same family of creatures, say. According to one development of deflationism, the property of being true is more like the property of being a game than it is like the property of being a mammal. The comparisons between being true, being a game, and being a mammal are suggestive, but they still do not nail down exactly what it means to say that truth is not a substantive property. The contemporary literature on deflationism contains several different approaches to the idea. One such approach, which we will consider in detail in Section 4.1, involves denying that truth plays an explanatory role. Another approach, pursuing an analogy between being true and existing, describes truth as a "logical property" (for example, Field 1992, 322; Horwich 1998a, 37; Kunne 2003, 91). A further approach appeals to David Lewis's (1983, 1986) view that, while every set of entities underwrites a property, there is a distinction between sparse, or natural, properties and more motley or disjointed abundant properties. On this approach, a deflationist might say that there is an abundant property of being true rather than a sparse one (cf. Edwards 2013, Asay 2014, Kukla and Winsberg 2015, and Armour-Garb forthcoming). A different metaphysical idea may be to appeal to the contemporary discussion of grounding and the distinction between groundable and ungroundable properties. In this context, a groundable property is one that is capable of being grounded in some other property, whether or not it is in fact grounded; an ungroundable property is a property that is not groundable (see Dasgupta 2015, 2016 and Rabin 2020). From this point of view, a deflationist might say that being true is an ungroundable property. Hence it is unlike ordinary, sparse/natural properties, such as being iron, which are both capable of being grounded and are grounded, and it is also unlike fundamental physical properties, such as being a lepton, which are capable of being grounded (in some other possible world) but are not (actually) grounded. We will not try to decide here which of these different views of properties is correct but simply note that deflationists who want to claim that there is a truth property, just not a substantive one, have options for explaining what this means. ### 1.3 The Utility of the Concept of Truth In light of the two central ideas discussed so far - the idea that the equivalence schema is exhaustive of the notion of truth and the idea that there is no substantive truth property - you might wonder why we have a concept of truth in the first place. After all, contrast this question with the explanation of why we have the concept of mammals. A natural suggestion is that it allows us to think and talk about mammals and to develop theories of them. For deflationism, however, as we have just seen, being true is completely different from being a mammal; why then do we have a concept of truth? (An analogous question might be asked about the word 'true', i.e., why we have the word 'true' and related words in our language at all. In the following discussion we will not discriminate between questions about the concept of truth and questions about the word 'true' and will move back and forth between them.) The question of why we have the concept of truth allows us to introduce a third central theme in deflationism, which is an emphasis not merely on the property of truth but on the concept of truth, or, equivalently for present purposes, on the word 'true' (cf. Leeds 1978). Far from supposing that there is no point having the concept of truth, deflationists are usually at pains to point out that anyone who has the concept of truth is in possession of a very useful concept indeed; in particular, anyone who has this concept is in a position to express generalizations that would otherwise require non-standard logical devices, such as sentential variables and quantifiers for them. Suppose, for example, that Jones for whatever reason decides that Smith is an infallible guide to the nature of reality. We might then say that Jones believes everything that Smith says. To say this much, however, is not to capture the content of Jones's belief. In order to do that we need some way of generalizing on the embedded sentence positions in a claim like: (5) If Smith says that birds are dinosaurs, then birds are dinosaurs. To generalize on the relationship indicated in (5), beyond just what Smith says about birds to anything she might say, what we want to do is generalize on the embedded occurrences of 'birds are dinosaurs'. So, we need a (declarative) sentential variable, '$p$', and a universal quantifier governing it. What we want is a way of capturing something along the lines of (6) For all $p$, if Smith says that $p$, then $p.$ The problem is that we cannot formulate this in English with our most familiar way of generalizing because the '$p$' in the consequent is in a sentence-in-use position, rather than mentioned or nominalized context (as it is in the antecedent), meaning that this formal variable cannot be replaced with a familiar English object-variable expression, e.g., 'it'. This is where the concept of truth comes in. What we do in order to generalize in the way under consideration is employ the truth predicate with an object variable to produce the sentence, (7) For all $x$, if what Smith said $= x$, then $x$ is true. Re-rendering the quasi-formal (7) into natural language yields, (8) Everything is such that, if it is what Smith said, then it is true. Or, to put the same thing more colloquially: (9) Everything Smith says is true. The equivalence schema (ES-prop) allows us to use (7) (and therefore (9)) to express what it would otherwise require the unstatable (6) to express. For, on the basis of the schema, there is always an equivalence between whatever goes in for a sentence-in-use occurrence of the variable '$p$' and a context in which that filling of the sentential variable is nominalized. This reveals how the truth predicate can be used to provide a surrogate for sentential variables, simulating this non-standard logical device while still deploying the standard object variables already available in ordinary language ('it') and the usual object quantifiers ('everything') that govern them. This is how the use of the truth predicate in (9) gives us the content of Jones's belief. And the important point for deflationists is that we could not have stated the content of this belief unless we had the concept of truth (the expression 'true'). In fact, for most deflationists, it is this feature of the concept of truth - its role in the formation of these sorts of generalizations - that explains why we have a concept of truth at all. This is, as it is often put, the raison d'etre of the concept of truth (cf. Field 1994a and Horwich 1998a). ## 2. History of Deflationism According to Michael Dummett (1959 [1978]), deflationism originates with Gottlob Frege, as expressed in this famous quote by the latter: > > It is ... worthy of notice that the sentence 'I smell the > scent of violets' has just the same content as the sentence > 'It is true that I smell the scent of violets'. So it > seems, then, that nothing is added to the thought by my ascribing to > it the property of truth. (Frege 1918, 6) > This passage suggests that Frege embraces a deflationary view in position $\mathbf{B}$ (in the chart above), namely, an analytic propositionalist version of deflationism. But this interpretation of his view is not so clear. As Scott Soames (1999, 21ff) points out, Frege (ibid.) distinguishes what we will call "opaque" truth ascriptions, like 'My conjecture is true', from transparent truth-ascriptions, like the one mentioned in the quote from Frege. Unlike with transparent cases, in opaque instances, one cannot simply strip 'is true' away and obtain an equivalent sentence, since the result is not even a sentence at all. Frank Ramsey is the first philosopher to have suggested a position like $\mathbf{B}$ (although he does not really accept propositions as abstract entities (see Ramsey 1927 (34-5) and 1929 (7)), despite sometimes talking in terms of propositions): > > Truth and falsity are ascribed primarily to propositions. The > proposition to which they are ascribed may be either explicitly given > or described. Suppose first that it is explicitly given; then it is > evident that 'It is true that Caesar was murdered' means > no more than that Caesar was murdered, and 'It is false that > Caesar was murdered' means no more than Caesar was not murdered. > .... In the second case in which the proposition is described and > not given explicitly we have perhaps more of a problem, for we get > statements from which we cannot in ordinary language eliminate the > words 'true' or 'false'. Thus if I say > 'He is always right', I mean that the propositions he > asserts are always true, and there does not seem to be any way of > expressing this without using the word 'true'. But suppose > we put it thus 'For all $p$, if he asserts $p$, $p$ is > true', then we see that the propositional function $p$ is true > is simply the same as $p$, as e.g. its value 'Caesar was > murdered is true' is the same as 'Caesar was > murdered'. (Ramsey 1927, 38-9) > On Ramsey's redundancy theory (as it is often called), the truth operator, 'it is true that' adds no content when prefixed to a sentence, meaning that in the instances of what we can think of as the truth-operator version of (ES), (ES-op) It is true that $p$ iff $p$, the left- and right-hand sides are meaning-equivalent. But Ramsey extends his redundancy theory beyond just the transparent instances of truth-talk, maintaining that the truth predicate is, in principle, eliminable even in opaque ascriptions of the form '$B$ is true' (which he (1929, 15, n. 7) explains in terms of sentential variables via a formula along the lines of '$\exists p$ ($p \amp B$ is a belief that $p$)') and in explicitly quantificational instances, like 'Everything Einstein said is true' (explained as above). As the above quote illustrates, Ramsey recognizes that in truth ascriptions like these the truth predicate fills a grammatical need, which keeps us from eliminating it altogether, but he held that even in these cases it contributes no content to anything said using it. A.J. Ayer endorses a view similar to Ramsey's. The following quote shows that he embraces a meaning equivalence between the two sides of the instances of both the sentential (position $\mathbf{A})$ and something like (since, despite his use of the expression 'proposition' to mean sentence, he also considers instances of truth-talk involving the prefix 'it is true that', which could be read as employing 'that'-clauses) the propositional (position $\mathbf{B})$ version of (ES). > > [I]t is evident that a sentence of the form "$p$ is > true" or "it is true that $p$" the reference to > truth never adds anything to the sense. If I say that it is true that > Shakespeare wrote Hamlet, or that the proposition > "Shakespeare wrote Hamlet" is true, I am saying > no more than that Shakespeare wrote Hamlet. Similarly, if I > say that it is false that Shakespeare wrote the Iliad, I am > saying no more than that Shakespeare did not write the Iliad. > And this shows that the words 'true' and > 'false' are not used to stand for anything, but function > in the sentence merely as assertion and negation signs. That is to > say, truth and falsehood are not genuine concepts. Consequently, there > can be no logical problem concerning the nature of truth. (Ayer 1935, > 28. Cf. Ayer 1936 [1952, 89]) > Ludwig Wittgenstein, under Ramsey's influence, makes claims with strong affinities to deflationism in his later work. We can see a suggestion of an endorsement of deflationary positions $\mathbf{A}$ or $\mathbf{B}$ in his (1953, SS136) statement that "$p$ is true $= p$" and "$p$ is false = not-$p$", indicating that ascribing truth (or falsity) to a statement just amounts to asserting that very proposition (or its negation). Wittgenstein also expresses this kind of view in manuscripts from the 1930s, where he claims, "What he says is true = Things are as he says" and "[t]he word 'true' is used in contexts such as 'What he says is true', but that says the same thing as 'He says $\ldquo p\rdquo,$ and $p$ is the case'". (Wittgenstein 1934 [1974, 123]) and 1937 [2005, 61]), respectively) Peter Strawson's views on truth emerge most fully in his 1950 debate with J.L. Austin. In keeping with deflationary position $\mathbf{B}$, Strawson (1950, 145-7) maintains that an utterance of 'It is true that $p$' just makes the same statement as an utterance of '$p$'. However, in Strawson 1949 and 1950, he further endorses a performative view, according to which an utterance of a sentence like 'That is true' mainly functions to do something beyond mere re-assertion. This represents a shift to an account of what the expression 'true' does, from traditional accounts of what truth is, or even accounts of what 'true' means. Another figure briefly mentioned above who looms large in the development of deflationism is Alfred Tarski, with his (1935 [1956] and 1944) identification of a precise criterion of adequacy for any formal definition of truth: its implying all of the instances of what is sometimes called "Convention (T)" or "the (T)-schema", (T) $X$ is true if, and only if, $p$. To explain this schema a bit more precisely, in its instances the '$X$' gets replaced by a name of a sentence from the object-language for which the truth predicate is being defined, and the '$p$' gets replaced by a sentence that is a translation of that sentence into the meta-language in which the truth predicate is being defined. For Tarski, the 'if and only if' deployed in any instance of (T) expresses just a material equivalence, putting his view at position $\mathbf{E}$ in the chart from Section 1.1. Although this means that Tarski is not a deflationist himself (cf. Field 1994a, Ketland 1999, and Patterson 2012), there is no denying the influence that his work and its promotion of the (T)-schema have had on deflationism. Indeed, some early deflationists, such as W.V.O. Quine and Stephen Leeds, are quite explicit about taking inspiration from Tarski's work in developing their "disquotational" views, as is Horwich in his initial discussion of deflationism. Even critics of deflationism have linked it with Tarski: Hilary Putnam (1983b, 1985) identifies deflationists as theorists who "refer to the work of Alfred Tarski and to the semantical conception of truth" and who take Tarski's work "as a solution to the philosophical problem of truth". The first fully developed deflationary view is the one that Quine (1970 [1986, 10-2]) presents. Given his skepticism about the existence of propositions, Quine takes sentences to be the primary entities to which 'is true' may be applied, making the instances of (ES-sent) the equivalences that he accepts. He defines a category of sentence that he dubs "eternal", viz., sentence types that have all their indexical/contextual factors specified, the tokens of which always have the same truth-values. It is for these sentences that Quine offers his disquotational view. As he (ibid., 12) puts it, > > > This cancellatory force of the truth predicate is explicit in > Tarski's paradigm: > > > > > > > 'Snow is white' is true if and only if snow is white. > > > > > > Quotation marks make all the difference between talking about words > and talking about snow. The quotation is a name of a sentence that > contains the name, namely 'snow', of snow. By calling the > sentence true, we call snow white. The truth predicate is a device of > disquotation. > > > As this quote suggests, Quine sees Tarski's formal work on defining truth predicates for formalized languages and his criterion of adequacy for doing so as underwriting a disquotational analysis of the truth predicate. This makes Quine's view a different kind of position-$\mathbf{A}$ account, since he takes the left-hand side of each instance of (ES-sent) to be, as we will put it (since Quine rejects the whole idea of meaning and meaning equivalence), something like a mere syntactic variant of the right-hand side. This also means that Quine's version of deflationism departs from inflationism by rejecting the latter's presupposition that truth predicates function to describe the entities they get applied to, the way that other predicates, such as 'is metal', do. Quine also emphasizes the importance of the truth predicate's role as a means for expressing the kinds of otherwise inexpressible generalizations discussed in Section 1.3. As he (1992, 80-1) explains it, > > The truth predicate proves invaluable when we want to generalize along > a dimension that cannot be swept out by a general term ... The > harder sort of generalization is illustrated by generalization on the > clause 'time flies' in 'If time flies then time > flies'.... We could not generalize as in 'All men are > mortal' because 'time flies' is not, like > 'Socrates', a name of one of a range of objects (men) over > which to generalize. We cleared this obstacle by semantic > ascent: by ascending to a level where there were indeed objects > over which to generalize, namely linguistic objects, sentences. > So, if we want to generalize on embedded sentence-positions within some sentences, "we ascend to talk of truth and sentences" (Quine 1970 [1986, 11]). This maneuver allows us to "affirm some infinite lot of sentences that we can demarcate only by talking about the sentences" (ibid., 12). Leeds (1978) (following Quine) makes it clear how the truth predicate is crucial for extending the expressive power of a language, despite the triviality that disquotationalism suggests for the transparent instances of truth-talk. He (ibid., 121) emphasizes the logical role of the truth predicate in the expression of certain kinds of generalizations that would otherwise be inexpressible in natural language. Leeds, like Quine, notes that a central utility of the truth predicate, in virtue of its yielding every instance of (ES-sent), is the simulation of quantification into sentence-positions. But, unlike Quine, Leeds glosses this logical role in terms of expressing potentially infinite conjunctions (for universal generalization) or potentially infinite disjunctions (for existential generalization). The truth predicate allows us to use the ordinary devices of first-order logic in ways that provide surrogates for the non-standard logical devices this would otherwise require. Leeds is also clear about accepting the consequences of deflationism, that is, of taking the logically expressive role of the truth predicate to exhaust its function. In particular, he points out that there is no need to think that truth plays any sort of explanatory role. We will return to this point in Section 4.1. Dorothy Grover, Joseph Camp, and Nuel Belnap (1975) develop a different variety of deflationism that they call a "prosentential theory". This theory descends principally from Ramsey's views. In fact, Ramsey (1929, 10) made what is probably the earliest use of the term 'pro-sentence' in his account of the purpose of truth-talk. Prosentences are explained as the sentence-level analog of pronouns. As in the case of pronouns, prosentences inherit their content anaphorically from other linguistic items, in this case from some sentence typically called the prosentence's "anaphoric antecedent" (although it need not actually occur before the prosentence). As Grover, et al. develop this idea, this content inheritance can happen in two ways. The most basic one is called "lazy" anaphora. Here the prosentence could simply be replaced with a repetition of its antecedent, as in the sort of case that Strawson emphasized, where one says "That is true" after someone else has made an assertion. According to Grover, et al., this instance of truth-talk is a prosentence that inherits its content anaphorically from the other speaker's utterance, so that the two speakers assert the same thing. As a result, Grover, et al. would take the instances of (ES) to express meaning equivalences, but since they (ibid., 113-5) do not take the instances of truth-talk on the left-hand sides of these equivalences to say anything about any named entities, they would not read (ES) as either (ES-sent) or (ES-prop) on their standard interpretations. So, while their prosententialism is similar to views in position $\mathbf{A}$ or in position $\mathbf{B}$ in the chart above, it is also somewhat different from both. Grover, et al.'s project is to develop the theory "that 'true' can be thought of always as part of a prosentence" (ibid., 83). They explain that 'it is true' and 'that is true' are generally available prosentences that can go into any sentence-position. They consider these expressions to be "atomic" in the sense of not being susceptible to a subject-predicate analysis giving the 'that' or 'it' separate references (ibid., 91). Both of these prosentences can function in the "lazy" way, and Grover, et al. claim (ibid., 91-2, 114) that 'it is true' can also operate as a quantificational prosentence (i.e., a sentential variable), for example, in a re-rendering of a sentence like, (9) Everything Smith says is true. in terms of a "long-form" equivalent claim, such as (8') Everything is such that, if Smith says that it is true, then it is true. One immediate concern that this version of prosententialism faces pertains to what one might call the "paraphrastic gymnastics" that it requires. For example, a sentence like 'It is true that humans are causing climate change' is said to have for its underlying logical form the same form as 'Humans are causing climate change. That is true' (ibid., 94). As a result, when one utters an instance of truth-talk of the form 'It is true that $p$', one states the content of the sentence that goes in for '$p$' twice. In cases of quotation, like "'Birds are dinosaurs' is true", Grover, et al. offer the following rendering, 'Consider: Birds are dinosaurs. That is true' (ibid., 103). But taking this as the underlying form of quotational instance of truth-talk requires rejecting the standard view that putting quotation marks around linguistic items forms names of those items. These issues raise concerns regarding the adequacy of this version of prosententialism. ## 3. The Varieties of Contemporary Deflationism In this section, we explain the details of three prominent, contemporary accounts and indicate some concerns peculiar to each. ### 3.1 Minimalism Minimalism is the version of deflationism that diverges the least from inflationism because it accepts many of the standard inflationary presuppositions, including that 'is true' is a predicate used to describe entities as having (or lacking) a truth property. What makes minimalism a version of deflationism is its denial of inflationism's final assumption, namely, that the property expressed by the truth predicate has a substantive nature. Drawing inspiration from Leeds (1978), Horwich (1982, 182) actually coins the term 'deflationism' while describing "the deflationary redundancy theory which denies the existence of surplus meaning and contends that Tarski's schema ["$p$" is true iff $p$] is quite sufficient to capture the concept." Minimalism, Horwich's mature deflationary position (1998a [First Edition, 1990]), adds to this earlier view. In particular, Horwich (ibid., 37, 125, 142) comes to embrace the idea that 'is true' does express a property, but it is merely a "logical property" (cf. Field 1992), rather than any substantive or naturalistic property of truth with an analyzable underlying nature (Horwich 1998a, 2, 38, 120-1). On the basis of natural language considerations, Horwich (ibid., 2-3, 39-40) holds that propositions are what the alethic locutions describe directly. Any other entities that we can properly call true are so only derivatively, on the basis of having some relation to true propositions (ibid., 100-1 and Horwich 1998b, 82-5). This seems to position Horwich well with respect to explaining the instances of truth-talk that cause problems for Quine and Leeds, e.g., those about beliefs and theories. Regarding truth applied directly to propositions, however, Horwich (1998a, 2-3) still explicitly endorses the thesis that Leeds emphasizes about the utility of the truth predicate (and, Horwich adds, the concept it expresses), namely, that it "exists solely for the sake of a certain logical need". While Horwich (ibid., 138-9) goes so far as to claim that the concept of truth has a "non-descriptive" function, he does not follow Quine and Leeds all the way to their rejection of the assumption that the alethic predicates function to describe truth-bearers. Rather, his (ibid., 31-3, 37) point of agreement with them is that the main function of the truth predicate is its role in providing a means for generalizing on embedded sentence positions, rather than some role in the indication of specifically truth-involving states of affairs. Even so, Horwich (ibid., 38-40) still contends that the instances of truth-talk do describe propositions, in the sense that they make statements about them, and they do so by attributing a property to those propositions. The version of (ES) that Horwich (1998a, 6) makes the basis of his theory is what he also calls "the equivalence schema", (E) It is true that $p$ if and only if $p$. Since he takes truth-talk to involve describing propositions with a predicate, Horwich considers 'it is true that $p$' to be just a trivial variant of 'The proposition that $p$ is true', meaning that his (E) is a version of (ES-prop) rather than of Ramsey's (ES-op). He also employs the notation '$\langle p\rangle$' as shorthand specifically for 'the proposition that $p$', generating a further rendering of his equivalence schema (ibid., 10) that we can clearly recognize as a version of (ES-prop), namely (E) $\langle p\rangle$ is true iff $p$. Horwich considers the instances of (E) to constitute the axioms of both an account of the property of truth and an account of the concept of truth, i.e., what is meant by the word 'true' (ibid., 136). According to minimalism, the instances of (E) are explanatorily fundamental, which Horwich suggests is a reason for taking them to be necessary (ibid., 21, n. 4). This, combined with his view that the equivalence schema applies to propositions, places his minimalism in position $\mathbf{D}$ in the chart given in Section 1.1. The instances of (ES-prop) are thus explanatory of the functioning of the truth predicate (of its role as a de-nominalizer of 'that'-clauses (ibid., 5)), rather than being explained by that functioning (as the analogous equivalences are for both disquotationalism and prosententialism). Moreover, Horwich (ibid., 50, 138) claims that they are also conceptually basic and a priori. He (ibid., 27-30, 33, 112) denies that truth admits of any sort of explicit definition or reductive analysis in terms of other concepts, such as reference or predicate-satisfaction. In fact, Horwich (ibid., 10-1, 111-2, 115-6) holds that these other semantic notions should both be given their own, infinitely axiomatized, minimalist accounts, which would then clarify the non-reductive nature of the intuitive connections between them and the notion of truth. Horwich (ibid., 27-30) maintains that the infinite axiomatic nature of minimalism is unavoidable. He (ibid., 25) rejects the possibility of a finite formulation of minimalism via the use of substitutional quantification. On the usual understanding of this non-standard type of quantification, the quantifiers govern variables that serve to mark places in linguistic strings, indicating that either all or some of the elements of an associated substitution class of linguistic items of a particular category can be substituted in for the variables. Since it is possible for the variables so governed to take sentences as their substitution items, this allows for a type of quantification governing sentence positions in complex sentences. Using this sort of sentential substitutional quantification, the thought is, one can formulate a finite general principle that expresses Horwich's account of truth as follows: (GT) $\forall x\, (x$ is true iff $\Sigma p (x = \langle p\rangle \amp p)),$ where '$\Sigma$' is the existential substitutional quantifier. (GT) is formally equivalent to the formulation that Marian David (1994, 100) presents as disquotationalism's definition of 'true sentence', here formulated for propositions instead. Horwich's main reason for rejecting the proposed finite formulation of minimalism, (GT), is that an account of substitutional quantifiers seems (contra David 1994, 98-9) to require an appeal to truth (since the quantifiers are explained as expressing that at least one or that every item in the associated substitution class yields a true sentence when substituted in for the governed variables), generating circularity concerns (Horwich 1998a, 25-6). Moreover, on Horwich's (ibid., 4, n. 1; Cf. 25, 32-3) understanding, the point of the truth predicate is to provide a surrogate for substitutional quantification and sentence-variables in natural language, so as "to achieve the effect of generalizing substitutionally over sentences ... but by means of ordinary [quantifiers and] variables (i.e., pronouns), which range over objects" (italics original). Horwich maintains that the infinite "list-like" nature of minimalism poses no problem for the view's adequacy with respect to explaining all of our uses of the truth predicate, and the bulk of Horwich 1998a attempts to establish just that. However, Anil Gupta (1993a, 365) has pointed out that minimalism's infinite axiomatization in terms of the instances of (E) for every (non-paradox-inducing) proposition makes it maximally ideologically complex, in virtue of involving every other concept. (Moreover, the overtly "fragmented" nature of the theory also makes it particularly vulnerable to the Generalization Problem that Gupta has raised, which we discuss in Section 4.5, below.) Christopher Hill (2002) attempts to deal with some of the problems that Horwich's view faces, by presenting a view that he takes to be a newer version of minimalism, replacing Horwich's equivalence schema with a universally quantified formula, employing a kind of substitutional quantification to provide a finite definition of 'true thought (proposition)'. Hill's (ibid., 22) formulation of his account, (S) For any object $x$, $x$ is true if and only if $(\Sigma p)((x =$ the thought that $p)$ and $p)$, is formally similar to the formulation of minimalism in terms of (GT) that Horwich rejects, but to avoid the circularity concerns driving that rejection, Hill's (ibid., 18-22) idea is to offer introduction and elimination rules in the style of Gerhard Gentzen (1935 [1969]) as a means for defining the substitutional quantifiers. Horwich (1998a, 26) rejects even this inference-rule sort of approach, but he directs his critique against defining linguistic substitutional quantification this way. Hill takes his substitutional quantifiers to apply to thoughts (propositions) instead of sentences. But serious concerns have been raised regarding the coherence of this non-linguistic notion of substitutional quantification (cf. David 2006, Gupta 2006b, Simmons 2006). As a result, it is unclear that Hill's account is an improvement on Horwich's version of minimalism. ### 3.2 Disquotationalism Like minimalism, disquotationalism agrees with inflationary accounts of truth that the alethic locutions function as predicates, at least logically speaking. However, as we explained in discussing Quine's view in Section 2, disquotationalism diverges from inflationary views (and minimalism) at their shared assumption that these (alethic) predicates serve to describe the entities picked out by the expressions with which they are combined, specifically as having or lacking a certain property. Although Quine's disquotationalism is inspired by Tarski's recursive method for defining a truth predicate, that method is not what Quine's view emphasizes. Field's contemporary disquotationism further departs from that aspect of Tarski's work by looking directly to the instances of the (T)-schema that the recursive method must generate in order to satisfy Tarski's criterion of material adequacy. Tarski himself (1944, 344-5) suggests at one point that each instance of (T) could be considered a "partial definition" of truth and considers (but ultimately rejects; see Section 4.5) the thesis that a logical conjunction of all of these partial definitions amounts to a general definition of truth (for the language that the sentences belonged to). Generalizing slightly from Tarski, we can call this alternative approach "(T)-schema disquotationalism", in contrast with the Tarski-inspired approach that David (1994, 110-1) calls "recursive disquotationalism". Field (1987, 1994a) develops a version of (T)-schema disquotationalism that he calls "pure disquotational truth", focusing specifically on the instances of his preferred version of (ES), the "disquotational schema" (Field 1994a, 258), (T[/ES-sent]) "$p$" is true if and only if $p$. Similar to the "single principle" formulation, (GT), rejected by Horwich (but endorsed by Hill), Field (ibid., 267) allows that one could take a "generalized" version of (T/ES-sent), prefixed with a universal substitutional quantifier, '$\Pi$', as having axiomatic status, or one could incorporate schematic sentence variables directly into one's theorizing language and reason directly with (T/ES-sent) as a schema (cf. ibid., 259). Either way, in setting out his version of deflationism, Field (ibid., 250), in contrast with Horwich, does not take the instances of his version of (ES) as fundamental but instead as following from the functioning of the truth predicate. On Field's reading of (T/ES-sent), the use of the truth predicate on the left-hand side of an instance does not add any cognitive content beyond that which the mentioned utterance has (for the speaker) on its own when used (as on the right-hand-side of (T/ES-sent)). As a result, each instance of (T/ES-sent) "holds of conceptual necessity, that is, by virtue of the cognitive equivalence of the left and right hand sides" (ibid., 258). This places Field's deflationism also in position $\mathbf{A}$ in the chart from Section 1.1. Following Leeds and Quine, Field (1999, 533-4) sees the central utility of a purely disquotational truth predicate to be providing for the expression of certain "fertile generalizations" that cannot be made without using the truth predicate but which do not really involve the notion of truth. Field (1994a, 264) notes that the truth predicate plays "an important logical role: it allows us to formulate certain infinite conjunctions and disjunctions that can't be formulated otherwise [n. 17: at least in a language that does not contain substitutional quantifiers]". Field's disquotationalism addresses some of the worries that arose for earlier versions of this variety of deflationism, due to their connections with Tarski's method of defining truth predicates. It also explains how to apply a disquotational truth predicate to ambiguous and indexical utterances, thereby going beyond Quine's (1970 [1986]) insistence on taking eternal sentences as the subjects of the instances of (ES-sent) (cf. Field 1994a, 278-81). So, Field's view addresses some of the concerns that David (1994, 130-66) raises for disquotationalism. However, an abiding concern about this variety of deflationism is that it is an account of truth as applied specifically to sentences. This opens the door to a version of the complaint that Strawson (1950) makes against Austin's account of truth, that it is not one's act of stating [here: the sentence one utters] but what thereby gets stated that is the target of a truth ascription. William Alston (1996, 14) makes a similar point. While disquotationalists do not worry much about this, this scope restriction might strike others as problematic because it raises questions about how we are to understand truth applied to beliefs or judgments, something that Hill (2002) worries about. Field (1978) treats beliefs as mental states relating thinkers to sentences (of a language of thought). But David (1994, 172-7) raises worries for applying disquotationalism to beliefs, even in the context of an account like Field's. The view that we believe sentences remains highly controversial, but it is one that, it seems, a Field-style disquotationalist must endorse. Similarly, such disquotationalists must take scientific theories to consist of sets of sentences, in order for truth to be applicable to them. This too runs up against Strawson's complaint because it suggests that one could not state the same theory in a different language. These sorts of concerns continue to press for disquotationalists. ### 3.3 Prosententialism As emerges from the discussion of Grover, et al. (1975) in Section 2, prosententialism is the form of deflationism that contrasts the most with inflationism, rejecting even the latter's initial assumption that the alethic locutions function as predicates. Partly in response to the difficulties confronting Grover, et al.'s prosentential account, Robert Brandom (1988 and 1994) has developed a variation on their view with an important modification. In place of taking the underlying logic of 'true' as having this expression occur only as a non-separable component of the semantically atomic prosentential expressions, 'that is true' and 'it is true', Brandom treats 'is true' as a separable prosentence-forming operator. "It applies to a term that is a sentence nominalization or that refers to or picks out a sentence tokening. It yields a prosentence that has that tokening as its anaphoric antecedent" (Brandom 1994, 305). In this way, Brandom's account avoids most of the paraphrase concerns that Grover, et al.'s prosententialism faces, while still maintaining prosententialism's rejection of the contention that the alethic locutions function predicatively. As a consequence of his operator approach, Brandom gives quantificational uses of prosentences a slightly different analysis. He (re)expands instances of truth-talk like the following, (9) Everything Smith says is true. (10) Something Jones said is true. "back" into longer forms, such as (8\) For anything one can say, if Smith says it, then it is true. (11) For something one can say, Jones said it, and it is true. and explains only the second 'it' as involved in a prosentence. The first 'it' in (8\) and (11) still functions as a pronoun, anaphorically linked to a set of noun phrases (sentence nominalizations) supplying objects (sentence tokenings) as a domain being quantified over with standard (as opposed to sentential or "propositional") quantifiers (ibid., 302). Brandom presents a highly flexible view that takes 'is true' as a general "denominalizing" device that applies to singular terms formed from the nominalization of sentences broadly, not just to pronouns that indicate them. A sentence like 'It is true that humans are causing climate change', considered via a re-rendering as 'That humans are causing climate change is true', is already a prosentence on his view, as is a quote-name case like "'Birds are dinosaurs' is true", and an opaque instance of truth-talk like 'Goldbach's Conjecture is true'. In this way, Brandom offers a univocal and broader prosentential account, according to which, "[i]n each use, a prosentence will have an anaphoric antecedent that determines a class of admissible substituends for the prosentence (in the lazy case, a singleton). This class of substituends determines the significance of the prosentence associated with it" (ibid.). As a result, Brandom can accept both (ES-sent) and (ES-prop) - the latter understood as involving no commitment to propositions as entities - on readings closer to their standard interpretations, taking the instances of both to express meaning equivalences. Brandom's account thus seems to be located in both position $\mathbf{A}$ and position $\mathbf{B}$ in the chart from Section 1.1, although, as with any prosententialist view, it still denies that the instances of (ES) say anything about either sentences or propositions. Despite its greater flexibility, however, Brandom's account still faces the central worry confronting prosentential views, namely that truth-talk really does seem predicative, and not just in its surface grammatical form but in our inferential practices with it as well. In arguing for the superiority of his view over that of Grover, et al., Brandom states that "[t]he account of truth talk should bear the weight of ... divergence of logical from grammatical form only if no similarly adequate account can be constructed that lacks this feature" (ibid., 304). One might find it plausible to extend this principle beyond grammatical form, to behavior in inferences as well. This is an abiding concern for attempts to resist inflationism by rejecting its initial assumption, namely, that the alethic locutions function as predicates. ## 4. Objections to Deflationism In the remainder of this article, we consider a number of objections to deflationism. These are by no means the only objections that have been advanced against the approach, but they seem to be particularly obvious and important ones. ### 4.1 The Explanatory Role of Truth The first objection starts from the observation that (a) in certain contexts an appeal to the notion of truth appears to have an explanatory role and (b) deflationism seems to be inconsistent with that appearance. Some of the contexts in which truth seems to have an explanatory role involve philosophical projects, such as the theory of meaning (which we will consider below) or explaining the nature of knowledge. In these cases, the notion of explanation at issue is not so much causal as it is conceptual (see Armour-Garb and Woodbridge forthcoming, for more on this). But the notion of truth seems also sometimes to play a causal explanatory role, especially with regard to explaining various kinds of success - mainly the success of scientific theories/method (cf. Putnam 1978 and Boyd 1983) and of people's behavior (cf. Putnam 1978 and Field 1987), but also the kind of success involved in learning from others (Field 1972). The causal-explanatory role that the notion of truth appears to play in accounts of these various kinds of success has seemed to many philosophers to constitute a major problem for deflationism. For example, Putnam (1978, 20-1, 38) claims, "the notions of 'truth' and 'reference' have a causal-explanatory role in ... an explanation of the behavior of scientists and the success of science", and "the notion of truth can be used in causal explanations - the success of a man's behavior may, after all, depend on the fact that certain of his beliefs are true - and the formal logic of 'true' [the feature emphasized by deflationism] is not all there is to the notion of truth". While a few early arguments against deflationism focus on the role of truth in explanations of the success of science (see Williams 1986 and Fine 1984a, 1984b for deflationary responses to Putnam and Boyd on this), according to Field (1994a, 271), "the most serious worry about deflationism is that it can't make sense of the explanatory role of truth conditions: e.g., their role in explaining behavior, or their role in explaining the extent to which behavior is successful". While few theorists endorse the thesis that explanations of behavior in general need to appeal to the notion of truth (even a pre-deflationary Field (1987, 84-5) rejects this, but see Devitt 1997, 325-330, for an opposing position), explanations of the latter, i.e., of behavioral success, still typically proceed in terms of an appeal to truth. This poses a prima facie challenge to deflationary views. To illustrate the problem, consider the role of the truth-value of an individual's belief in whether that person succeeds in satisfying her desires. Let us suppose that Mary wants to get to a party, and she believes that it is being held at 1001 Northside Avenue. If her belief is true, then, other things being equal, she is likely to get to the party and get what she wants. But suppose that her belief is false, and the party is in fact being held at 1001 Southside Avenue. Then it would be more likely, other things being equal, that she won't get what she wants. In an example of this sort, the truth of her belief seems to be playing a particular role in explaining why she gets what she wants. Assuming that Mary's belief is true, and she gets to the party, it might seem natural to say that the latter success occurs because her belief is true, which might seem to pose a problem for deflationists. However, truth-involving explanations of particular instances of success like this don't really pose a genuine problem. This is because if we are told the specific content of the relevant belief, it is possible to replace the apparently explanatory claim that the belief is true with an equivalent claim that does not appeal to truth. In Mary's particular case, we could replace i) the claim that she believes that the party is being held at 1001 Northside Avenue, and her belief is true, with ii) the claim that she believes that the party is being held at 1001 Northside Avenue, and the party $is$ being held at 1001 Northside Avenue. A deflationist can claim that the appeal to truth in the explanation of Mary's success just provides an expressive convenience (including, perhaps, the convenience of expressing what would otherwise require an infinite disjunction (of conjunctions like ii)), by saying just that what Mary believed was true, if one did not know exactly which belief Mary acted on) (cf. Horwich 1998a, 22-3, 44-6). While deflationists seem to be able to account for appeals to truth in explanations of particular instances of success, the explanatory-role challenge to deflationism also cites the explanatory role that an appeal to truth appears to play in explaining the phenomenon of behavioral success more generally. An explanation of this sort might take the following form: > > > [1] > People act (in general) in such a way that their goals will be > obtained (as well as possible in the given situation), or in such a > way that their expectations will not be frustrated, ... > if their beliefs are true. > [2] > Many beliefs [people have about how to attain their goals] > are true. > [3] > So, as a consequence of [1] and [2], people have a tendency to > attain certain kinds of goals. (Putnam 1978, 101) > > The generality of [1] in this explanation seems to cover more cases than any definite list of actual beliefs that someone has could include. Moreover, the fact that [1] supports counterfactuals by applying to whatever one might possibly believe (about attaining goals) suggests that it is a law-like generalization. If the truth predicate played a fundamental role in the expression of an explanatory law, then deflationism would seem to be unsustainable. A standard deflationary response to this line of reasoning involves rejecting the thesis that [1] is a law, seeing it (and truth-involving claims like it) instead as functioning similarly to how the claim 'What Mary believes is true' functions in an explanation of her particular instance of behavioral success, just expressing an even more indefinite, and thus potentially infinite claim. The latter is what makes a claim like [1] seem like an explanatory law, but even considering this indefiniteness, the standard deflationary account of [1] claims that the function of the appeal to the notion of truth there is still just to express a kind of generalization. One way to bring out this response is to note that, similar to the deflationary "infinite disjunction" account of the claim 'What Mary believes is true', generalizations of the kind offered in [1] entail infinite conjunctions of their instances, which are claims that can be formulated without appeal to truth. For example, in the case of explaining someone, $A$, accomplishing their goal of getting to a party, deflationsts typically claim that the role of citing possession of a true belief is really just to express an infinite conjunction with something like the following form: > > If $A$ believes that the party is 1001 Northside Avenue, and the > party is at 1001 Northside Avenue, then $A$ will get what they want; > and if $A$ believes that the party is at 1001 Southside Avenue, and > the party is at 1001 Southside Ave, then $A$ will get what they > want; and if $A$ believes that party is at 17 Elm St, and the party > is at 17 Elm St, then $A$ will get what they want; ... and so > on. > The equivalence schema (ES) allows one to capture this infinite conjunction (of conditionals) in a finite way. For, on the basis of the schema, one can reformulate the infinite conjunction as: > > If $A$ believes that the party is 1001 Northside Avenue, and that > the party is 1001 Northside Avenue is true, then $A$ will get what > they want; and if $A$ believes that the party is at 1001 Southside > Avenue, and that the party is at 1001 Southside Avenue is true, then > $A$ will get what they want, and if $A$ believes that the party is > at 17 Elm Street, and that the party is at 17 Elm Street is true, then > $A$ will get what they want; ... and so on. > In turn, this (ES)-reformulated infinite conjunction can be expressed as a finite statement with a universal quantifier ranging over propositions: > > For every proposition $x$, if what $A$ believes $= x$, and $x$ > is true, then $A$ will get what they want, other things being equal. > The important point for a deflationist is that one could not express the infinite conjunction regarding the agent's beliefs and behavioral success unless one had the concept of truth. But deflationists also claim that this is all that the notion of truth is doing here and in similar explanations (cf. Leeds 1978, 1995; Williams 1986, Horwich 1998a). How successful is this standard deflationary response? There are several critiques in the literature. Some (e.g., Damnjanovic 2005) argue that there is no distinction in the first place between appearing in a causal-explanatory generalization and being a causal-explanatory property. After all, suppose it is a true generalization that metal objects conduct electricity. That would normally be taken as sufficient to show that being metal is a causal-explanatory property that one can cite in explaining why something conducts electricity. But isn't this a counter, then, to deflationism's thesis that, assuming there is a property of truth at all, it is at most an insubstantial one? If a property is a causal or explanatory property, after all, it is hard to view it as insubstantial. The reasoning at issue here may be presented conveniently by expanding on the general argument considered above and proceeding from an apparently true causal generalization to the falsity of deflationism (ibid.): > > > P1. > If a person $A$ has true beliefs, they will get what they want, > other things being equal. > C1. > Therefore, if $A$ has beliefs with the property of being true, > $A$ will get what they want other things being equal. > C2. > Therefore, the property of being true appears in a > causal-explanatory generalization. > C3. > Therefore, the property of being true is a causal-explanatory > property. > C4. > Therefore, deflationism is false. > > Can a deflationist apply the standard deflationary response to this argument? Doing so would seem to involve rejecting the inference from C2 to C3. After all, the standard reply would say that the role that the appeal to truth plays in P1, the apparent causal generalization, is simply its generalizing role of expressing a potentially infinite, disjointed conjunction of unrelated causal connections (cf. Leeds 1995). So, applying this deflationary response basically hinges on the plausibility of rejecting the initial assumption that there is no distinction between appearing in a causal-explanatory generalization and being a causal-explanatory property. It is worth noting two other responses beyond the standard one that a deflationist might make to the reasoning just set out. The first option is to deny the step from P1 to C1. This inference involves the explicit introduction of the property of being true, and, as we have seen, some deflationists deny that there is a truth property at all (cf. Quine 1970 [1986], Grover, et al. 1975, Leeds 1978, Brandom 1994). But, as we noted above, the idea that there is no truth property may be difficult to sustain given the apparent fact that 'is true' functions grammatically as a predicate. The second option is to deny the final step from C3 to C4 and concede that there is a sense in which truth is a causal-explanatory property and yet say that it is still not a substantive property (cf. Damnjanovic 2005). For example, some philosophers (e.g., Friedman 1974, van Fraassen 1980, Kitcher 1989, Jackson and Pettit 1990) have offered different understandings of scientific explanation and causal explanation, according to which being a causal and explanatory property might not conflict with being insubstantial (perhaps by being an abundant or ungroundable property). This might be enough to sustain a deflationary position. The standard deflationary response to the explanatory-role challenge has also met with criticisms focused on providing explanations of certain "higher-level" phenomena. Philip Kitcher (2002, 355-60) concludes that Horwich's (1998a, 22-3) application of the standard response, in his account of how the notion of truth functions in explanations of behavioral success, misses the more systematic role that truth plays in explaining patterns of successful behavior, such as when mean-ends beliefs flow from a representational device, like a map. Chase Wrenn (2011) agrees with Kitcher that deflationists need to explain systematic as opposed to just singular success, but against Kitcher he argues that deflationists are actually better off than inflationists on this front. Will Gamester (2018, 1252-5) raises a different "higher-level factor" challenge, one based on the putative inability of the standard deflationary account of the role of truth in explanations of behavioral success to distinguish between coincidental and non-coincidental success. Gamester (ibid., 1256-7) claims that an inflationist could mark and account for the difference between the two kinds of success with an explanation that appeals to the notion of truth. But it is not clear that a deflationist cannot also avail herself of a version of this truth-involving explanation, taking it just as the way of expressing in natural language what one might formally express with sentential variables and quantifiers (cf. Ramsey 1927, 1929; Prior 1971, Wrenn 2021, and Armour-Garb and Woodbridge forthcoming). ### 4.2 Propositions Versus Sentences We noted earlier that deflationism can be presented in either a sententialist version or a propositionalist version. Some philosophers have suggested, however, that the choice between these two versions constitutes a dilemma for deflationism (Jackson, Oppy, and Smith 1994). The objection is that if deflationism is construed in accordance with propositionalism, then it is trivial, but if it is construed in accordance with sententialism, it is false. To illustrate the dilemma, consider the following claim: (12) Snow is white is true if and only if snow is white. Now, does 'snow is white' in (12) refer to a sentence or a proposition? If, on the one hand, we take (12) to be about a sentence, then, assuming (12) can be interpreted as making a necessary claim, it is false. On the face of it, after all, it takes a lot more than snow's being white for it to be the case that 'snow is white' is true. In order for 'snow is white' to be true, it must be the case not only that snow is white, it must, in addition, be the case that 'snow is white' means that snow is white. But this is a fact about language that (12) ignores. On the other hand, suppose we take 'snow is white' in (12) to denote the proposition that snow is white. Then the approach looks to be trivial, since the proposition that snow is white is defined as being the one that is true just in case snow is white. Thus, deflationism faces the dilemma of being false or trivial. One response for the deflationist is to remain with the propositionalist version of their doctrine and accept its triviality. A trivial doctrine, after all, at least has the advantage of being true. A second response is to resist the suggestion that propositionist deflationism is trivial. For one thing, the triviality here does not have its source in the concept of truth, but rather in the concept of a proposition. Moreover, even if we agree that the proposition that snow is white is defined as the one that is true if and only if snow is white, this still leaves open whether truth is a substantive property of that proposition; as such it leaves open whether deflationism or inflationism is correct. A third response to this dilemma is to accept that deflationism applies inter alia to sentences, but to argue (following Field 1994a) that the sentences to which it applies must be interpreted sentences, i.e., sentences which already have meaning attached to them. While it takes more than snow being white to make the sentence 'snow is white' true, when we think of it as divorced from its meaning, that is not so clear when we treat it as having the meaning it in fact has. ### 4.3 Correspondence It is often said to be a platitude that true statements correspond to the facts. The so-called "correspondence theory of truth" is built around this intuition and tries to explain the notion of truth by appealing to the notions of correspondence and fact. But even if one does not build one's approach to truth around this intuition, many philosophers regard it as a condition of adequacy on any approach that it accommodate this correspondence intuition. It is often claimed, however, that deflationism has trouble meeting this adequacy condition. One way to bring out the problem here is by focusing on a particular articulation of the correspondence intuition, one favored by deflationists themselves (e.g., Horwich 1998a). According to this way of spelling it out, the intuition that a certain sentence or proposition "corresponds to the facts" is the intuition that the sentence or proposition is true because of how the world is; that is, the truth of the proposition is explained by some fact, which is usually external to the proposition itself. We might express this by saying that someone who endorses the correspondence intuition so understood would endorse: (6) The proposition that snow is white is true because snow is white. The problem with (6) is that, when we combine it with deflationism - or at least with a necessary version of that approach - we can derive something that is plainly false. Anyone who assumes that the instances of the equivalence schema are necessary would clearly be committed to the necessary truth of: (7) The proposition that snow is white is true iff snow is white. And, since (7) is a necessary truth, under that assumption, it is very plausible to suppose that (6) and (7) together entail: (8) Snow is white because snow is white. But (8) is clearly false. The reason is that the 'because' in (6) and (8) is a causal or explanatory relation, and plausibly such relations must obtain between distinct relata. But the relata in (8) are (obviously) not distinct. Hence, (8) is false, and this means that the conjunction of (6) and (7) must be false, and that deflationism is inconsistent with the correspondence intuition. To borrow a phrase of Mark Johnston's (1989) - who mounts a similar argument in a different context - we might say that if deflationism is true, then what seems to be a perfectly good explanation in (6) goes missing; if deflationism is true, after all, then (6) is equivalent to (8), and (8) is not an explanation of anything. One way a deflationist might attempt to respond to this objection is by providing a different articulation of the correspondence intuition. For example, one might point out that the connection between the proposition that snow is white being true and snow's being white is not a contingent connection and suggest that this rules out (6) as a successful articulation of the correspondence intuition. That intuition (one might continue) is more plausibly given voice by (6\) 'Snow is white' is true because snow is white. However, when (6\) is conjoined with (7), one cannot derive the problematic (8), and thus, one might think, the objection from correspondence might be avoided. Now, certainly this is a possible suggestion; the problem with it, however, is that a deflationist who thinks that (6\) is true is most plausibly construed as holding a sententialist, rather than a propositionalist, version of deflationism. A sententialist version of deflationism will supply a version of (7), viz.: (7\) 'Snow is white' is true iff snow is white. which, at least if it is interpreted as a necessary (or analytic) truth, will conspire with (6\) to yield (8). And we are back where we started. Another response would be to object that 'because' creates an opaque context* - that is, the kind of context within which one cannot substitute co-referring expressions and preserve truth. However, for this to work, 'because' must create an opaque context of the right kind. In general, we can distinguish two kinds of opaque context: intensional contexts, which allow the substitution of necessarily co-referring expressions but not contingently co-referring expressions; and hyperintensional contexts, which do not even allow the substitution of necessarily co-referring expressions. If the inference from (6) and (7) to (8) is to be successfully blocked, it is necessary that 'because' creates a hyperintensional context. A proponent of the correspondence objection might try to argue that while 'because' creates an intensional context, it does not create a hyperintensional context. But since a hyperintensional reading of 'because' has become standard fare, this approach remains open to a deflationist and is not an ad hoc fix. A final, and most radical, response would be to reject the correspondence intuition outright. This response is not as drastic as it sounds. In particular, deflationists do not have to say that someone who says 'the proposition that snow is white corresponds to the facts' is speaking falsely. Deflationists might do better by saying that such a person is simply using a picturesque or ornate way of saying that the proposition is true, where truth is understood in accordance with deflationism. Indeed, a deflationist can even agree that, for certain rhetorical or conversational purposes, it might be more effective to use talk of "correspondence to the facts". Nevertheless, it is important to see that this response does involve a burden, since it involves rejecting a condition of adequacy that many regard as binding. ### 4.4 Truth-Value Gaps According to some metaethicists (moral non-cognitivists or expressivists), moral claims - such as the injunction that one ought to return people's phone calls - are neither true nor false. The same situation holds, according to some philosophers of language, for claims that presuppose the existence of something which does not in fact exist, such as the claim that the present King of France is bald; for sentences that are vague, such as 'These grains of sand constitute a heap'; and for sentences that are paradoxical, such as those that arise in connection with the Liar Paradox. Let us call this thesis the gap, since it finds a gap in the class of sentences between those that are true and those that are false. The deflationary approach to truth has seemed to be inconsistent with the gap, and this has been thought by some (e.g., Dummett 1959 [1978, 4] and Holton 2000) to be an objection. The reason for the apparent inconsistency flows from a natural way to extend the deflationary approach from truth to falsity. The most natural thing for a deflationist to do is to introduce a falsity schema like: (F-sent) '$p$' is false iff ${\sim}p.$ Following Holton (1993, 2000), we consider (F-sent) to be the relevant schema for falsity, rather than some propositional schema, since the standard understanding of a gappy sentence is as one that does not express a proposition (cf. Jackson, et al. 1994). With a schema like (F-sent) in hand, deflationists could say things about falsity similar to what they say about truth: (F-sent) exhausts the notion of falsity, there is no substantive property of falsity, the utility of the concept of falsity is just a matter of facilitating the expression of certain generalizations, etc. However, there is a seeming incompatibility between (F-sent) and the gap. Suppose, for reductio, that 'S' is a sentence that is neither true nor false. In that case, it is not the case that 'S' is true, and it is not the case that 'S' is false. But then, by (ES-sent) and (F-sent), we can infer that it is not the case that S, and it is not the case that not-S; in short: ${\sim}$S and ${\sim}{\sim}$S, which is a classical contradiction. Clearly, then, we must give up one of these things. But which one can we give up consistently with deflationism? In the context of ethical non-cognitivism, one possible response to the apparent dilemma is to distinguish between a deflationary account of truth and a deflationary account of truth-aptitude (cf. Jackson, et al. 1994). By accepting an inflationary account of the latter, one can claim that ethical statements fail the robust criteria of "truth-aptitude" (reidentified in terms of expression of belief), even if a deflationary view of truth still allows the application of the truth predicate to them, via instances of (ES). In the case of vagueness, one might adopt epistemicism about it and claim that vague sentences actually have truth-values, we just can't know them (cf. Williamson 1994. For an alternative, see Field 1994b). With respect to the Liar Paradox, the apparent conflict between deflationism and the gap has led some (e.g., Simmons 1999) to conclude that deflationism is hobbled with respect to dealing with the problem, since most prominent approaches to doing so, stemming from the work of Saul Kripke (1975), involve an appeal to truth-value gaps. One alternative strategy a deflationist might pursue in attempting to resolve the Liar is to offer a non-classical logic. Field 2008 adopts this approach and restricts the law of the excluded middle. JC Beall (2002) combines truth-value gaps with Kleene logic (see the entry on many-valued Logic) and makes use of both weak and strong negation. Armour-Garb and Beall (2001, 2003) argue that deflationists can and should be dialetheists and accept that some truthbearers are both true and not true (see also, Woodbridge 2005, 152-3, on adopting a paraconsistent logic that remains "quasi-classical"). By contrast, Armour-Garb and Woodbridge (2013, 2015) develop a version of the "meaningless strategy" with respect to the Liar (based on Grover 1977), which they claim a deflationist can use to dissolve that paradox and semantic pathology more generally, without accepting genuine truth-value gaps or giving up classical logic. ### 4.5 The Generalization Problem Since deflationists place such heavy emphasis on the role of the concept of truth in expressing generalizations, it seems somewhat ironic that certain versions of deflationism have been criticized for being incapable of accounting for generalizations involving truth (Gupta 1993a, 1993b; Field 1994a, 2008; Horwich 1998a (137-8), 2001; Halbach 1999 and 2011 (57-9); Soames 1999, Armour-Garb 2004, 2010, 2011). The "Generalization Problem" (henceforth, $GP)$ captures the worry that a deflationary account of truth is inadequate for explaining our commitments to general facts we express with certain uses of 'true'. This raises the question of whether and, if so, how, deflationary accounts earn the right to endorse such generalizations. Although Tarski (1935 [1956]) places great importance on the instances of his (T)-schema, he comes to recognize that those instances do not provide a fully adequate way of characterizing truth. Moreover, even when the instances of (T) are taken as theorems, Tarski (ibid.) points out that taken all together they are insufficient for proving a 'true'-involving generalization like (A) All sentences of the form 'If $p$, then $p$' are true. since the collection of the instances of (T) is $\omega$-incomplete (where a theory, $\theta$, is $\omega$-incomplete if $\theta$ can prove every instance of an open formula '$Fx$' but cannot prove the universal generalization, '$\forall xFx$'). We arrive at a related problem when we combine a reliance on the instances of some version of (ES) with Quine's view about the functioning and utility of the truth predicate. He (1992, 81) considers the purpose of (A) to be to express a generalization over sentences like the following: (B) If it is raining, then it is raining. (C) If snow is white, then snow is white. Quine points out that we want to be able to generalize on the embedded sentences in those conditionals, by semantically ascending, abstracting logical form, and deriving (A). But, as Tarski (ibid.) notes, this feat cannot be achieved, given only a commitment to (the instances of) (T). From (T) and (A), we can prove (B) and (C) but, given the finitude of deduction, when equipped only with the instances of (T), we cannot prove (A). As a consequence of the Compactness Theorem of first-order logic, anything provable from the totality of the instances of (T) is provable from just finitely many of them, so any theory that takes the totality of the instances of (T) to characterize truth will be unable to prove any generalization like (A). To address the question of why we need to be able to prove these truth-involving generalizations, suppose that we accept a proposition like $\langle$Every proposition of the form $\langle$if $p$, then $p\rangle$ is true$\rangle$. Call this proposition "$\beta$". Now take '$\Gamma$' to stand for the collection of propositions that are the instances of $\beta$. Horwich (2001) maintains that an account of the meaning of 'true' will be adequate only if it aids in explaining why we accept the members of $\Gamma$, where such explanations amount to proofs of those propositions by, among other things, employing an explanatory premise that does not explicitly concern the truth predicate. So, one reason it is important to be able to prove a 'true'-involving generalization is because this is a condition of adequacy for an account of the meaning of that term. One might argue that anyone who grasps the concept of truth, and that of the relevant conditional, should be said to know $\beta$. But if a given account of truth, together with an account of the conditional (along, perhaps, with an account of other logical notions), does not entail $\beta$, then it does not provide an acceptable account of truth. Here is another reason for thinking that generalizations like $\beta$ must be proved. A theory of the meaning of 'true' should explain our acceptance of propositions like $\beta$, which, as Gupta (1993a) and Hill (2002) emphasize, should be knowable a priori by anyone who possesses the concept of truth (and who grasps the relevant logical concepts). But if such a proposition can be known a priori on the basis of a grasp of the concept of truth (and of the relevant logical concepts), then a theory that purports to specify the meaning of 'true' should be able to explain our acceptance of that proposition. But if an account of the meaning of 'true' is going to do this, it must be possible to derive the proposition from one or more of the clauses that constitute our grasp of the concept of truth. This creates a problem for a Horwichian minimalist. Let us suppose that $\beta$ is one of the general propositions that must be provable. Restricted to the resources available through Horwich's minimalism, we can show that $\beta$ cannot be derived. If a Horwichian minimalist could derive $\beta$, it would have to be derived from the instances of (E) $\langle p\rangle$ is true iff $p.$ But there cannot be a valid derivation of a universal generalization from a set of particular propositions unless that set is inconsistent. Since, according to Horwich (1998a), every instance of (E) that is part of his theory of truth is consistent, it follows that there cannot be a derivation of $\beta$ from the instances of (E). This is a purely logical point. As such, considerations of pure logic dictate that our acceptance of $\beta$ cannot be explained by Horwich's account of truth. Since Horwich takes all instances of the propositional version of (T) (i.e., (ES-prop)) as axioms, he can prove each of those instances. But, as we have seen, restricted to the instances of the equivalence schema, he cannot prove the generalization, $\beta$, i.e., $\langle$Every proposition of the form $\langle$if $p$ then $p\rangle$ is true$\rangle$. Some deflationists respond to the GP by using a version of (GT) to formulate their approach: (GT) $\forall x$ $(x$ is true iff $\Sigma p(x = \langle p\rangle \amp p)).$ In this context, there are two things to notice about (GT). First, it is not a schema but a universally quantified formula. For this reason, it is possible to derive a generalization like $\beta$ from it. Second, the existential quantifier, '$\Sigma$', in (GT) must be a higher-order quantifier (see the entry on second-order and higher-order logic) that quantifies into sentential positions. We mentioned above an approach that takes this quantifier as a substitutional one, where the substitution class consists of sentences. We also mentioned Hill's (2002) alternative version that takes the substitution class to be the set of all propositions. Kunne (2003) suggests a different approach that takes '$\Sigma$' to be an objectual (domain and values) quantifier ranging over propositions. However, parallel to Horwich's rejection of (GT) discussed in Section 3.1, all of these approaches have drawn criticism on the grounds that the use of higher-order quantifiers to define truth is circular (cf. Platts 1980, McGrath 2000), and may get the extension of the concept of truth wrong (cf. Sosa 1993). An alternative deflationist approach to the GP attempts to show that, despite appearances, certain deflationary theories do have the resources to derive the relevant generalizations. Field (1994a, 2001a), for example, suggests that we allow reasoning with schemas directly and proposes rules that would allow the derivation of generalizations. Horwich (1998a, 2001) suggests a more informal approach according to which we are justified in deriving $\beta$ since an informal inspection of a derivation of some instance of $\beta$ shows us that we could derive any instance of it. For replies to Horwich, see Armour-Garb 2004, 2010, 2011; Gupta 1993a, 1993b; and Soames 1999. For responses to Armour-Garb's attack on Horwich 2001, see Oms 2019 and Cieslinski 2018. ### 4.6 Conservativeness An ideal theory of truth will be both consistent (e.g., avoid the Liar Paradox) and adequate (e.g., allow us to derive all the essential laws of truth, such as those at issue in the Generalization Problem). Yet it has recently been argued that even if deflationists can provide a consistent theory of truth and avoid the GP, they still cannot provide an adequate theory. This argument turns on the notion of a conservative extension of a theory. Informally, a conservative extension of a theory is one that does not allow us to prove anything that could not be proved from the original, unextended theory. More formally, and applied to theories of truth, a truth theory, $Tr$ is conservative over some theory $T$ formulated in language $L$ if and only if for every sentence $\phi$ of $L$ in which the truth predicate does not occur, if $Tr \cup L \vdash \phi$, then $L \vdash \phi$ (where '$\vdash$' represents provability). Certain truth theories are conservative over arithmetic - e.g., theories that implicitly define truth using only the instances of some version of (ES) - and certain truth theories are not - e.g., Tarski's (1935 [1956], 1944) compositional theory. Specifically, the addition of certain truth theories allows us to prove that arithmetic is consistent, something that we cannot do if we are confined to arithmetic itself. It has been argued (a) that conservative truth theories are inadequate and (b) that deflationists are committed to conservative truth theories. (See Shapiro 1998 and Ketland 1999; Horsten 1995 provides an earlier version of this argument.) We will explain the arguments for (a) below but to get a flavor of the arguments for (b), consider Shapiro's rhetorical question: "How thin can the notion of arithmetic truth be, if by invoking it we can learn more about the natural numbers?" Shapiro is surely right to press deflationists on their frequent claims that truth is "thin" or "insubstantial". It might also be a worry for deflationists if any adequate truth theory allowed us to derive non-logical truths, if they endorse the thesis that truth is merely a "logical property". On the other hand, deflationists themselves insist that truth is an expressively useful device, and so they cannot be faulted for promoting a theory of truth that allows us to say more about matters not involving truth. To see an argument for (a), consider a Godel sentence, $G$, formulated within the language of Peano Arithmetic (henceforth, $PA)$. $G$ is not a theorem of PA if PA is consistent (cf. the entry on Godel's incompleteness theorems). But $G$ becomes a theorem when PA is expanded by adding certain plausible principles that appear to govern a truth predicate. Thus, the resultant theory of arithmetical truth is strong enough to prove G and appears therefore to be non-conservative over arithmetic. If, as has been argued by a number of theorists, any adequate account of truth will be non-conservative over a base theory, then deflationists appear to be in trouble. Understood in this way, the "Conservativeness Argument" (henceforth, $CA)$ is a variant of the objection considered in Section 4.1, claiming that truth plays an explanatory role that deflationism cannot accommodate. There are several deflationary responses to the CA. Field (1999) argues that the worries that arise from the claim that deflationists are in violation of explanatory conservativeness is unfounded. He (ibid., 537) appeals to the expressive role of the truth predicate and maintains that deflationists are committed to a form of "explanatory conservativeness" only insofar as there are no explanations in which the truth predicate is not playing its generalizing role. As a result, he (ibid.) notes that "any use of 'true' in explanations which derives solely from its role as a device of generalization should be perfectly acceptable". For responses to Field, see Horsten 2011 (61) and Halbach 2011 (315-6). Responding to the CA, Daniel Waxman (2017) identifies two readings of 'conservativeness', one semantic and the other syntactic, which correspond to two conceptions of arithmetic. On the first conception, arithmetic is understood categorically as given by the standard model. On the second conception, arithmetic is understood axiomatically and is captured by the acceptance of some first-order theory, such as PA. Waxman argues that deflationism can be conservative given either conception, so that the CA does not go through. Julien Murzi and Lorenzo Rossi (2018) argue that Waxman's attempt at marrying deflationism with conservativeness - his "conservative deflationism" - is unsuccessful. They (ibid.) reject the adoption of this view on the assumption that one's conception of arithmetic is axiomatic, claiming, in effect, that a deflationist's commitment to a conservative conception of truth is misguided (cf. Halbach 2011, Horsten 2011, Cieslinski 2015, and Galinon 2015). Jody Azzouni (1999) defends the "first-order deflationist", viz., a deflationist who endorses what Waxman (ibid.) calls "the axiomatic conception of arithmetic" and whose subsequent understanding cannot rule out the eligibility of non-standard models. Azzouni accepts the need to prove certain 'true'-involving generalizations, but he maintains that there are some generalizations that are about truths that a first-order deflationist need not prove. He further contends that if one does extend her theory of truth in a way that allows her to establish these generalizations, she should not expect her theory to be conservative, nor should she continue describing it as a deflationary view of truth. For a response to Azzouni (ibid.), see Waxman (2017, 453). In line with Field's response to the CA, Lavinia Picollo and Thomas Schindler (2020) argue that the conservativeness constraint imposed by Horsten 1995, Shapiro 1998, Ketland 1999, and others is not a reasonable requirement to impose on deflationary accounts. They contend that the insistence on conservativeness arises from making too much of the metaphor of "insubstantiality" and that it fails to see what the function of the truth predicate really amounts to. Their leading idea is that, from a deflationist's perspective, the logico-linguistic function of the truth predicate is to simulate sentential and predicate quantification in a first-order setting (cf. Horwich 1998a, 4, n. 1). They maintain that, for a deflationist, in conjunction with first-order quantifiers, the truth predicate has the same function as sentential and predicate quantifiers. So, we should not expect the deflationist's truth theory to conservatively extend its base theory. ### 4.7 Normativity It is commonly said that our beliefs and assertions aim at truth, or present things as being true, and that truth is therefore a norm of assertion and belief. This putative fact about truth and assertion in particular has been seen to suggest that deflationism must be false (cf. Wright 1992 and Bar-On and Simmons 2007). However, the felt incompatibility between normativity and deflationism is difficult to make precise. The first thing to note is that there is certainly a sense in which deflationism is consistent with the idea that truth is a norm of assertion. To illustrate this, notice (as we saw in examining truth's putative explanatory role) that we can obtain an intuitive understanding of this idea without mentioning truth at all, so long as we focus on a particular case. Suppose that for whatever reason Mary sincerely believes that snow is green, has good evidence for this belief, and on the basis of this belief and evidence asserts that snow is green. We might say that there is a norm of assertion that implies that Mary is still open to criticism in this case. After all, since snow is not green, there must be something incorrect or defective about Mary's assertion (and similarly for her belief). It is this incorrectness or defectiveness that the idea that truth is a norm of assertion (and belief) is trying to capture. To arrive at a general statement of the norm that lies behind this particular case, consider that here, what we recognize is (13) If someone asserts that snow is green when snow is not green, then that assertion is open to criticism. To generalize on this, what we want to do is generalize on the positions occupied by 'snow is green' and express something along the lines of (14) For all $p$, if someone asserts that $p$ when ${\sim}p$, then that assertion is open to criticism. The problem of providing a general statement like (14) is the same issue first raised in Section 1.3, and the solution by now should be familiar. To state the norm in general we would need to be able to do something we seem unable to do in ordinary language, namely, employ sentential variables and quantifiers for them. But this is where the notion of truth comes in. Because (ES) gives us its contraposition, (ES-con) ${\sim}p$ iff $\langle p\rangle$ is not true. Reading '$\langle p\rangle$' as 'that $p$', we can reformulate (14) as (15) For all $p$, if someone asserts that $p$ when that $p$ is not true, then that assertion is open to criticism. But since the variable '$p$' occurs only in nominalized contexts in (15), we can replace it with an object variable, '$x$', and bind this with an ordinary objectual quantifier, to get (16) For all $x$, if someone asserts $x$, and $x$ is not true, then that assertion is open to criticism. Or, to put it as some philosophers might: (N) Truth is a norm of assertion. In short, then, deflationists need not deny that we appeal to the notion of truth to express a norm of assertion; on the contrary, the concept of truth seems required to state that very generalization. If deflationists can account for the fact that we must apply the notion of truth to express a norm of assertion, then does normativity pose any problem for deflationism? Crispin Wright (1992, 15-23) argues that it does, claiming that deflationism is inherently unstable because there is a distinctive norm for assertoric practice that goes beyond the norms for warranted assertibility - that the norms of truth and warranted assertibility are potentially extensionally divergent. This separate norm of truth, he claims, is already implicit just in acceptance of the instances of (ES). He points out that not having warrant to assert some sentence does not yield having warrant to assert its negation. However, because (ES) gives us (ES-con), we have, in each instance, an inference (going from right to left) from the sentence mentioned not being true to the negation of the sentence. But the instance of (ES) for the negation of any sentence, (ES-neg) $\langle{\sim} p \rangle$ is true iff ${\sim}p,$ takes us (again, going from right to left) from the negated sentence to an ascription of truth to that negated sentence. Thus, some sentence not being true does yield that the negation of the sentence is true, in contrast with warranted assertibility. This difference, Wright (ibid., 18) claims, reveals that, by deflationism's own lights, the truth predicate expresses a distinct norm governing assertion, which is incompatible with the deflationary contention "that 'true' is only grammatically a predicate whose role is not to attribute a substantial characteristic". Rejecting Wright's argument for the instability of deflationism, Ian Rumfitt (1995, 103) notes that if we add the ideas of denying something and of having warrant for doing so ("anti-warrant") to Wright's characterization of deflationism, this would make 'is not true' simply a device of rejection governed by the norm that "[t]he predicate 'is not true' may be applied to any sentence for which one has an anti-warrant". But then truth-talk's behavior with negation would not have to be seen as indicating that it marks a distinct norm beyond justified assertibility and justifiable deniability, which would be perfectly compatible with deflationism. Field (1994a, 264-5) offers a deflationary response to Wright's challenge (as well as to a similar objection regarding normativity from Putnam (1983a, 279-80)), pointing again to the generalizing role of the truth predicate in such normative desires as one to utter only true sentences or one to have only true beliefs. Field agrees with Wright that truth-talk expresses a norm beyond warranted assertibility, but he (1994a, 265) also maintains that "there is no difficulty in desiring that all one's beliefs be disquotationally true; and not only can each of us desire such things, there can be a general practice of badgering other to into having such desires". Horwich (1996, 879-80) argues that Wright's rejection of deflationism does not follow from showing that one can use the truth predicate to express a norm beyond warranted assertibility. Like Field, Horwich claims that Wright missed the point that, in the expression of such a norm, the truth predicate is just playing its generalizing role. For other objections to deflationism based on truth's normative role, see Price 1998, 2003 and McGrath 2003. ### 4.8 Inflationist Deflationism? Another objection to deflationism begins by drawing attention to a little-known doctrine about truth that G.E. Moore held at the beginning of the 20th Century. Richard Cartwright (1987, 73) describes the view as follows: "a true proposition is one that has a certain simple unanalyzable property, and a false proposition is one that lacks the property". This doctrine about truth is to be understood as the analogue for the doctrine that Moore held about goodness, namely that goodness is a simple, unanalyzable quality. The potential problem that this Moorean view about truth presents for deflationism might best be expressed in the form of a question: What is the difference between the Moorean view and deflationism? One might reply that, according to deflationary theories, the concept of truth has an important logical role, i.e., expressing certain generalizations, whereas the concept of goodness does not. However, this doesn't really answer our question. For one thing, it isn't clear that Moore's notion of truth does not also capture generalizations, since it too will yield all of the instances of (ES). For another, the idea that the concept of truth plays an important logical role doesn't distinguish the metaphysics of deflationary conceptions from the metaphysics of the Moorean view, and it is the metaphysics of the matter that the present objection really brings into focus. Alternatively, one might suggest that the distinction between truth according to Moore's view and deflationary conceptions of truth is the distinction between having a simple unanalyzable nature, and not having any underlying nature at all. But what is that distinction? It is certainly not obvious that there is any distinction between having a nature about which nothing can be said and having no nature at all. How might a deflationist respond to this alleged problem? The key move will be to focus on the property of being true. For the Moorean, this property is a simple unanalyzable one. But deflationists need not be committed to this. As we have seen, some deflationists think that there is no truth property at all. And even among deflationists who accept that there is some insubstantial truth property, it is not clear that this is the sort of property that the Moorean has in mind. To say that a property is unanalyzable suggests that the property is a fundamental property. One might understand this in something like the sense that Lewis proposes, i.e., as a property that is sparse and perfectly natural. Or one might understand a fundamental property as one that is groundable but not grounded in anything. But deflationists need not understand a purported property of being true in either of these ways. As noted in Section 1.2, they may think of it as an abundant property rather than a sparse one, or as one that is ungroundable. In this way, there are options available for deflationists who want to distinguish themselves from the Moorean view of truth. ### 4.9 Truth and Meaning The final objection that we will consider concerns the relation between deflationism and theories of meaning, i.e., theories about how sentences get their meanings. The orthodox approach to this last question appeals to the notion of truth, suggesting, roughly, that a sentence $S$ means that $p$ just in case $S$ is true if and only if $p$. This approach to meaning, known widely as "truth-conditional semantics", is historically associated with Donald Davidson's (1967) thesis that the "(T)-sentences" (i.e., the instances of the (T)-schema) generated by a Tarski-truth-definition for a language give the meanings of the sentences they mention, by specifying their truth conditions. In contemporary linguistics, a prominent approach to sentence meaning explains it in terms of propositions, understood (following Lewis 1970 and Stalnaker 1970) as sets of possible worlds, which amount to encapsulations of truth conditions (Elbourne 2011, 50-51). This has led a number of philosophers to argue that, on pain of circularity, deflationism cannot be combined with theories of meaning that make use of the notion of truth to explain meaning - in other words, that deflationism is incompatible with truth-conditional theories of meaning. This assessment of deflationism stems from Dummett's (1959 [1978, 7]) claim that the (T)-sentences (or the instances of any other version of (ES)) cannot both tell us what the sentences they nominalize mean and give us an account of 'true'. As Horwich (1998a, 68) puts it, "we would be faced with something like a single equation and two unknowns" (cf. Davidson 1990, 1996; Horwich 1998b, Kalderon 1999, Collins 2002). The first thing to say regarding this objection is that, even if deflationism were inconsistent with truth-conditional theories of meaning, this would not automatically constitute an objection to deflationism. After all, there are alternative theories of meaning available, and most deflationists reject truth-conditional semantics in favor of some "truth-free" alternative. The main alternatives available include Brandom's (1994) inferentialism, developed following Wilfrid Sellars (1974, 1979); Horwich's (1998b) use-theory of meaning, inspired by Wittgenstein 1953; and Field's (1994a, 2001b) computational-role + indication-relations account. There is, however, a lot of work to be done before any of these theories can be regarded as adequate. Devitt 2001 makes this point in rejecting all of these alternative approaches to meaning, claiming that the only viable approach is (referential/)truth-conditional semantics. While the orthodoxy of truth-conditional accounts of meaning adds weight to this challenge to deflationism, it is still, contra Devitt, an open question whether any non-truth-conditional account of meaning will turn out to be adequate. Moreover, even if there is no viable "truth-free" account, several philosophers (e.g., Bar-on, et al. 2005) have argued that deflationism has no more problem with a truth-conditional theory of meaning than any other approach to truth does. Others have gone further, arguing positively that there is no incompatibility between deflationism and truth-conditional theories. Alexis Burgess (2011, 407-410) argues that at least some versions of deflationism are compatible with mainstream model-theoretic semantics in linguistics (understood as providing explanations of truth conditions) and the recognition of "the manifest power and progress of truth-conditional semantics". Claire Horisk (2008) argues that "circularity" arguments for incompatibility (or, as she prefers, "immiscibility") fail. The only circularity involved here, she claims, is a harmless kind, so long as one is (like some proponents of truth-conditional semantics) offering a reciprocal, rather than a reductive, analysis of meaning. Mark Lance (1997, 186-7) claims that any version of deflationism based on an anaphoric reading of 'is true' (as in Brandom 1988, 1994) is independent of, and thus compatible with, any underlying account of meaning, including a truth-conditional one. Williams 1999 likewise claims that the role of truth in meaning theories for particular languages is just the expressive role that deflationists claim exhausts the notion of truth's function. Of course, all of these claims have been contested, but they seem to show that the thesis that deflationism is inconsistent with a truth-conditional theory of meaning is neither a forgone conclusion nor necessarily an objection to deflationism, even if the thesis is correct. (For further discussion of this issue, see Gupta 1993b, Field 2005, Gupta and Martinez-Fernandez 2005, Patterson 2005, and Horisk 2007.) That said, one outstanding question that remains regarding the viability of any deflationary approach to truth is whether that approach can be squared with an adequate account of meaning.
truth-identity	## 1. Definition and Preliminary Exposition Declarative sentences seem to take truth-values, for we say things like (1) "Socrates is wise" is true. But sentences are apparently not the only bearers of truth-values: for we also seem to allow that what such sentences express, or mean, may be true or false, saying such things as (2) "Socrates is wise" means that Socrates is wise,[1] and (3) That Socrates is wise is true or (4) It is true that Socrates is wise. If, provisionally, we call the things that declarative sentences express, or mean, their contents--again provisionally, these will be such things as that Socrates is wise--then the identity theory of truth, in its most general form, states that (cf. Baldwin 1991: 35): (5) A declarative sentence's content is true just if that content is (identical with) a fact. A fact is here to be thought of as, very generally, a way things are, or a way the world is. On this approach, the identity theory secures an intimate connection between language (what language expresses) and world. Of course there would in principle be theoretical room for a view that identified not the content of, say, the true declarative sentence "Socrates is wise"--let us assume from now on that this sentence is true--with the fact that Socrates is wise, but rather that sentence itself. But this is not a version of the theory that anyone has ever advanced, nor does it appear that it would be plausible to do so (see Candlish 1999b: 200-2; Kunne 2003: 6). The early Wittgenstein does regard sentences as being themselves facts, but they are not identical with the facts that make them true. Alternatively, and using a different locution, one might say that, to continue with the same example, (6) That Socrates is wise is true just if that Socrates is wise is the case. The idea here is that (6) makes a connection between language and reality: on the left-hand side we have something expressed by a piece of language, and on the right-hand side we allude to a bit of reality. Now (6) might look truistic, and that status has indeed been claimed for the identity theory, at least in one of its manifestations. John McDowell has argued that what he calls true "thinkables" are identical with facts (1996: 27-8, 179-80). Thinkables are things like that Socrates is wise regarded as possible objects of thought. For we can think that Socrates is wise; and it can also be the case that Socrates is wise. So the idea is that what we can think can also be (identical with) what is the case. That identity, McDowell claims, is truistic. On this approach, one might prefer one's identity theory to take the form (cf. Hornsby 1997: 2): (7) All true thinkables are (identical with) facts. On this approach the identity theory explicitly aims to secure an intimate connection between mind (what we think) and world. A point which has perhaps been obscured in the literature on this topic, but which should be noticed, is that (7) asserts a relation of subordination: it says that true thinkables are a (proper or improper) subset of facts; it implicitly allows that there might be facts that are not identical with true thinkables. So (7) is not to be confounded with its converse, (8) All facts are (identical with) true thinkables, which asserts the opposite subordination, and says that facts are a (proper or improper) subset of true thinkables, implicitly allowing, this time, that there might be true thinkables that are not identical with facts. (8) is therefore distinct from (7), and if (7) is controversial, (8) is equally or more so, but for reasons that are at least in part different. (8) denies the existence of facts that cannot be grasped in thought. But many philosophers hold it to be evident that there are, or at least could be, such facts--perhaps certain facts involving indefinable real numbers, for example, or in some other way going beyond the powers of human thought. So (8) could be false; its status remains to be established; it can hardly be regarded as truistic. Accordingly, one might expect that an identity theorist who wished to affirm (7), and certainly anyone who wanted to say that (7) (or (6)) was truistic, would--at least qua identity theorist--steer clear of (8), and leave its status sub judice. In fact, however, a good number of identity theorists, both historical and contemporary, incorporate (8) as well as--or even instead of--(7) into their statement of the theory. Richard Cartwright, who published the first modern discussion of the theory in 1987, wrote that if one were formulating the theory, it would say "that every true proposition is a fact and every fact a true proposition" (1987: 74). McDowell states that > > > true thinkables already belong just as much to the world as to minds > [i.e., (7)], and things that are the case already belong just as much > to minds as to the world [i.e., (8)]. It should not even seem that we > need to choose a direction in which to read the claim of identity. > (2005: 84) > > > Jennifer Hornsby takes the theory to state that true thinkables and facts coincide (1997: 2, 9, 17, 20)--they are the same set--so that she in effect identifies that theory with the conjunction of (7) and (8), as also, in effect, does Julian Dodd (2008a: passim). Now, (8) is certainly an interesting thesis that merits much more consideration than it has hitherto received (at least in the recent philosophical literature), and, as indicated, some expositions of the identity theory have as much invested in (8) as in (5) or (7): on this point see further SS2 below. Nevertheless, it will make for clarity of discussion if we associate the identity theory of truth, more narrowly, with something along the lines of (5) or (7), and omit (8) from this particular discussion.[2] That will be the policy here. Whether or not (6) is truistic, both (5) and (7) involve technical or semi-technical vocabulary; moreover, they have been advanced as moves in a technical debate, namely one concerning the viability of the correspondence theory of truth. For these reasons it seems difficult to regard them as truisms (see Dodd 2008a: 179). What (5) and (7) mean, and which of them one will prefer as one's statement of the identity theory of truth, if one is favorably disposed to that theory--one may of course be happy with both--will depend, among other things, on what exactly one thinks about the nature of such entities as that Socrates is wise. In order to get clear on this point, discussion of the identity theory has naturally been conducted in the context of the Fregean semantical hierarchy, which distinguishes between levels of language, sense, and reference. Frege recognized what he called "thoughts" (Gedanken) at the level of sense corresponding to (presented by) declarative sentences at the level of language. McDowell's thinkables are meant to be Fregean thoughts: the change of terminology is intended to stress the fact that these entities are not thoughts in the sense of dated and perhaps spatially located individual occurrences (thinking events), but are abstract contents that are at least in principle available to be grasped by different thinkers at different times and places. So a Fregean identity theory of truth would regard both such entities as that Socrates is wise and, correlatively, facts as sense-level entities: this kind of identity theory will then state that true such entities are identical with facts. This approach will naturally favor (7) as its expression of the identity theory. By contrast with Frege, Russell abjured the level of sense and (at least around 1903-4) recognized what, following Moore, he called "propositions" as worldly entities composed of objects and properties. A modern Russellian approach might adopt these propositions--or something like them: the details of Russell's own conception are quite vague--as the referents of declarative sentences, and identity theorists who followed this line might prefer to take a particular reading of (5) as their slogan. So these Russellians would affirm something along the lines of: (9) All true Russellian propositions are identical with facts (at the level of reference), by contrast with the Fregean (10) All true Fregean thoughts are identical with facts (at the level of sense). This way of formulating the relevant identity claims has the advantage of suggesting that it would, at least in principle, be open to a theorist to combine (9) and (10) in a hybrid position that (i) departed from Russell and followed Frege by admitting both a level of Fregean sense and one of reference, and also, having admitted both levels to the semantic hierarchy, (ii) both located Fregean thoughts at the level of sense and located Russellian propositions at the level of reference. Sense being mode of presentation of reference, the idea would be that declarative sentences refer, via Fregean thoughts, to Russellian propositions (for this disposition, see Gaskin 2006: 203-20; 2008: 56-127). So someone adopting this hybrid approach would affirm both (9) and (10). Of course, the facts mentioned in (9) would be categorially different from the facts mentioned in (10), and one might choose to avoid confusion by distinguishing them terminologically, and perhaps also by privileging one set of facts, ontologically, over the other. If one wanted to follow this privileging strategy, one might say, for instance, that only reference-level facts were genuine facts, the relata of the identity relation at the level of sense being merely fact-like entities, not bona fide facts. That would be to give the combination of (9) and (10) a Russellian spin. Alternatively, someone who took the hybrid line might prefer to give it a Fregean spin, saying that the entities with which true Fregean thoughts were identical were the genuine facts, and that the corresponding entities at the level of reference that true Russellian propositions were identical with were not facts as such, but fact-like correlates of the genuine facts. Without more detail, of course, these privileging strategies leave the status of the entities they are treating as merely fact-like unclear; and, as far as the Fregean version of the identity theory goes, commentators who identify facts with sense-level Fregean thoughts usually, as we shall see, repudiate reference-level Russellian propositions altogether, rather than merely downgrading their ontological status, and so affirm (10) but reject (9). We shall return to these issues in SS4 below. ## 2. Historical Background The expression "the identity theory of truth" was first used--or, at any rate, first used in the relevant sense--by Stewart Candlish in an article on F. H. Bradley published in 1989. But the general idea of the theory had been in the air during the 1980s: for example, in a discussion first published in 1985, concerning John Mackie's theory of truth, McDowell criticized that theory for making > > > truth consist in a relation of correspondence (rather than identity) > between how things are and how things are represented as being. (1985 > [1998: 137 n. 21]) > > > The implication is that identity would be the right way to conceive the given relation. And versions of the identity theory go back at least to Bradley (see, e.g., Bradley 1914: 112-13; for further discussion and references, see Candlish 1989; 1995; 1999b: 209-12; T. Baldwin 1991: 36-40), and to the founding fathers of the analytic tradition (Sullivan 2005: 56-7 n. 4). The theory can be found in G. E. Moore's "The Nature of Judgment" (1899), and in the entry he wrote on "Truth" for J. Baldwin's Dictionary of Philosophy and Psychology (1902-3; reprinted Moore 1993: 4-8, 20-1; see T. Baldwin 1991: 40-3). Russell embraced the identity theory at least during the period of his 1904 discussions of Meinong (see, e.g., 1973: 75), possibly also in his The Principles of Mathematics of 1903, and for a few years after these publications as well (see T. Baldwin 1991: 44-8; Candlish 1999a: 234; 1999b: 206-9). Frege has a statement of the theory in his 1919 essay "The Thought", and may have held it earlier (Frege 1918-19: 74 [1977: 25]; see Hornsby 1997: 4-6; Milne 2010: 467-8). Wittgenstein's Tractatus (1922) is usually held to propound a correspondence rather than an identity theory of truth; however this is questionable. In the Tractatus, declarative sentences (Satze) are said to be facts (arrangements of names), and states of affairs (Sachlagen, Sachverhalte, Tatsachen) are also said to be facts (arrangements of objects). If the Tractatus is taken to put forward a correspondence theory of truth, then presumably the idea is that a sentence will be true just if there is an appropriate relation of correspondence (an isomorphism) between sentence and state of affairs. However, the problem with this interpretation is that, in the Tractatus, a relation of isomorphism between a sentence and reality is generally conceived as a condition of the meaningfulness of that sentence, not specifically of its truth. False sentences, as well as true, are isomorphic with states of affairs--only, in their case the states of affairs do not obtain. For Wittgenstein, states of affairs may either obtain or fail to obtain--both possibilities are, in general, available to them.[3] Correlatively, it has been suggested that the Tractatus contains two different conceptions of fact, a factive and a non-factive one. According to the former conception, facts necessarily obtain or are the case; according to the latter, facts may fail to obtain or not be the case. This non-factive conception has been discerned at Tractatus 1.2-1.21, and at 2.1 (see Johnston 2013: 382). Given that, in the Tractatus, states of affairs (and perhaps facts) have two poles--obtaining or being the case, and non-obtaining or not being the case--it seems to follow that, while Wittgenstein is committed to a correspondence theory of meaning, his theory of truth must be (some version of) an identity theory, along the lines of > > > A declarative sentence is true just if what it is semantically > correlated with is identical with an obtaining state of affairs (a > factive fact). > > > (Identity theorists normally presuppose the factive conception of facts, so that "factive" is redundant in the phrase "factive facts", and that is the policy which will be followed here.) Though a bipolar conception of facts (if indeed Wittgenstein has it) may seem odd, the bipolar conception of states of affairs (which, it is generally agreed, he does have) seems quite natural: here the identity theorist says that a true proposition is identical with an obtaining state of affairs (see Candlish & Damnjanovic 2018: 271-2). Peter Sullivan has suggested a different way of imputing an identity theory to the Tractarian Wittgenstein (2005: 58-9). His idea is that Wittgenstein's simple objects are to be identified with Fregean senses, and that in effect the Tractatus contains an identity theory along the lines of (7) or (10). Sullivan's ground for treating Tractarian objects as senses is that, like bona fide Fregean senses, they are transparent: they cannot be grasped in different ways. An apparent difficulty with this view is that there is plausibly more to Fregean sense than just the property of transparency: after all, Russell also attached the property of transparency to his basic objects, but it has not been suggested that Russellian basic objects are really senses, and the suggestion would seem to have little going for it (partly, though not only, because Russell himself disavowed the whole idea of Fregean sense).The orthodox position, which will be presupposed here, is that the Tractarian Wittgenstein like Russell, finds no use for a level of Fregean sense, so that his semantical hierarchy consists exclusively of levels of language and reference, with nothing of a mediatory or similar nature located between these levels. (Wittgenstein does appeal to the concepts of sense and reference in the Tractatus, but it is generally agreed that they do not figure in a Fregean way, according to which both names and sentences, for example, have both sense and reference; for Wittgenstein, by contrast, sentences have sense but not reference, whereas names have reference but not sense.) ## 3. Motivation What motivates the identity theory of truth? It can be viewed as a response to difficulties that seem to accrue to at least some versions of the correspondence theory (cf. Dodd 2008a: 120, 124). The correspondence theory of truth holds that truth consists in a relation of correspondence between something linguistic or quasi-linguistic, on the one hand, and something worldly on the other. Generally, the items on the worldly end of the relation are taken to be facts or (obtaining) states of affairs. For many purposes these two latter kinds of entity (facts, obtaining states of affairs) are assimilated to one another, and that strategy will be followed here. The exact nature of the correspondence theory will then depend on what the other relatum is taken to be. The items mentioned so far make available three distinct versions of the correspondence theory, depending on whether this relatum is taken to consist of declarative sentences, Fregean thoughts, or Russellian propositions. Modern correspondence theorists make a distinction between truth-bearers, which would typically fall under one of these three classifications, and truth-makers,[4] the worldly entities making truth-bearers true, when they are true. If these latter entities are facts, then true declarative sentences, Fregean thoughts, or Russellian propositions--whichever of these one selects as the relata of the correspondence relation on the language side of the language-world divide--correspond to facts in the sense that facts are what make those sentences, thoughts, or propositions true, when they are true. (Henceforth we shall normally speak simply of thoughts and propositions, understanding these to be Fregean thoughts and Russellian propositions respectively, unless otherwise specified.) That, according to the correspondence theorist (and the identity theorist can agree so far), immediately gives us a constraint on the shape of worldly facts. Take our sample sentence "Socrates is wise", and recall that this sentence is here assumed to be true. At the level of reference we encounter the object Socrates and (assuming realism about properties)[5] the property of wisdom. Both of these may be taken to be entities in the world, but it is plausible that neither amounts to a fact: neither amounts to a plausible truth-maker for the sentence "Socrates is wise", or for its expressed thought, or for its expressed proposition. That is because the man Socrates, just as such, and the property of wisdom, just as such, are not, so the argument goes, propositionally structured, either jointly or severally, and so do not amount to enough to make it true that Socrates is wise (cf. D. Armstrong 1997: 115-16; Dodd 2008a: 7; Hofweber 2016: 288). Even if we add in further universals, such as the relation of instantiation, and indeed the instantiation of instantiation to any degree, the basic point seems to be unaffected. In fact it can plausibly be maintained (although some commentators disagree; Merricks 2007: ch. 1, passim, and pp. 82, 117, 168; Asay 2013: 63-4; Jago 2018: passim, e.g., pp. 73, 84, 185, 218, 250, though cf. p. 161) that the man Socrates, just as such, is not even competent to make it true that Socrates exists; for that we need the existence of the man Socrates. Hence, it would appear that, if there are to be truth-makers in the world, they will have to be structured, syntactically or quasi-syntactically, in the same general way as declarative sentences, thoughts, and propositions. For convenience we can refer to structure in this general sense as "propositional structure": the point then is that neither Socrates, nor the property of wisdom, nor (if we want to adduce it) the relation of instantiation is, just as such, propositionally structured. Following this line of argument through, we reach the conclusion that nothing short of full-blown, propositionally structured entities like the fact that Socrates is wise will be competent to make the sentence "Socrates is wise", or the thought or proposition expressed by that sentence, true. (A question that arises here is whether tropes might be able to provide a "thinner" alternative to such ontologically "rich" entities as the fact that Socrates is wise. One problem that seems to confront any such strategy is that of making the proposed alternative a genuine one, that is, of construing the relevant tropes in such a way that they do not simply collapse into, or ontologically depend on, entities of the relatively rich form that Socrates is wise. For discussion see Dodd 2008a: 7-9.) The question facing the correspondence theorist is now: if such propositionally structured entities are truth-makers, are they truth-makers for sentences, thoughts, or propositions? It is at this point that the identity theorist finds the correspondence theory unsatisfactory. Consider first the suggestion that the worldly fact that Socrates is wise is the truth-maker for the reference-level proposition that Socrates is wise (see, e.g., Jago 2018: 72-3, and passim). There surely are such facts as the fact that Socrates is wise: we talk about such things all the time. The problem would seem to be not with the existence of such facts, but rather with the relation of correspondence which is said by the version of the correspondence theory that we are currently considering to obtain between the fact that Socrates is wise and the proposition that Socrates is wise. As emerges from this way of expressing the difficulty, there seems to be no linguistic difference between the way we talk about propositions and the way we talk about facts, when these entities are specified by "that" clauses. That suggests that facts just are true propositions. If that is right, then the relation between facts and true propositions is not one of correspondence--which, as Frege famously observed (Frege 1918-19: 60 [1977: 3]; cf. Kunne 2003: 8; Milne 2010: 467-8), implies the distinctness of the relata--but identity. This line of argument can be strengthened by noting the following point about explanation. Correspondence theorists have typically wanted the relation of correspondence to explain truth: they have usually wanted to say that it is because the proposition that Socrates is wise corresponds to a fact that it is true, and because the proposition that Socrates is foolish--or rather: It is not the case that Socrates is wise (after all, his merely being foolish is not enough to guarantee that he is not wise, for he might, like James I and VI, be both wise and foolish)--does not correspond to a fact that it is false. But the distance between the true proposition that Socrates is wise and the fact that Socrates is wise seems to be too small to provide for explanatory leverage. Indeed the identity theorist's claim is that there is no distance at all. Suppose we ask: Why is the proposition that Socrates is wise true? If we reply by saying that it is true because it is a fact that Socrates is wise, we seem to have explained nothing, but merely repeated ourselves (cf. Strawson 1971: 197; Anscombe 2000: 8; Rasmussen 2014: 39-43). So correspondence apparently gives way to identity as the relation which must hold or fail to hold between a proposition and a state of affairs if the proposition is to be true or false: the proposition is true just if it is identical with an obtaining state of affairs and false if it is not (cf. Horwich 1998: 106). And it would seem that, if the identity theorist is right about this disposition, explanatory pretensions will have to be abandoned: for while it will be correct to say that a proposition is true just if it is identical with a fact, false otherwise, it is hard to see that much of substance has thereby been said about truth (cf. Hornsby 1997; 2; Dodd 2008a; 135). It might be replied here that there are circumstances in which we tolerate statements of the form "A because B" when an appropriate identity--perhaps even identity of sense, or reference, or both--obtains between "A" and "B". For example, we say things like "He is your first cousin because he is a child of a sibling of one of your parents" (Kunne 2003: 155). But here it is plausible that there is a definitional connection between left-hand side and right-hand side, which seems not to hold of > > > The proposition that Socrates is wise is true because it is a > fact that Socrates is wise. > > > In the latter case there is surely no question of definition; rather, we are supposed, according to the correspondence theorist, to have an example of metaphysical explanation, and that is just what, according to the identity theorist, we do not have. After all, the identity theorist will insist, it seems obvious that the relation, whatever it is, between the proposition that Socrates is wise and the fact that Socrates is wise must, given that the proposition is true, be an extremely close one: what could this relation be? If the identity theorist is right that the relation cannot be one of metaphysical explanation (in either direction), then it looks as though it will be hard to resist the insinuation of the linguistic data that the relation is one of identity. It is for this reason that identity theorists sometimes insist that their position should not be defined in terms of an identity between truth-bearer and truth-maker: that way of expressing the theory looks too much in thrall to correspondence theorists' talk (cf. Candlish 1999b: 200-1, 213). For the identity theorist, to speak of both truth-makers and truth-bearers would imply that the things allegedly doing the truth-making were distinct from the things that were made true. But, since in the identity theorist's view there are no truth-makers distinct from truth-bearers, if the latter are conceived as propositions, and since nothing can make itself true, it follows that there are no truth-makers simpliciter, only truth-bearers. It seems to follow, too, that it would be ill-advised to attack the identity theory by pointing out that some (or all) truths lack truth-makers (so Merricks 2007: 181): so long as truths are taken to be propositions, that is exactly what identity theorists themselves say. From the identity theorist's point of view, truth-maker theory looks very much like an exercise in splitting the level of reference in half and then finding a bogus match between the two halves (see McDowell 1998: 137 n. 21; Gaskin 2006: 203; 2008: 119-27). For example, when David Armstrong remarks that > > > What is needed is something in the world which ensures that a > is F, some truth-maker or ontological ground for > a's being F. What can this be except the state of > affairs of a's being F? (1991: 190) > > > the identity theorist is likely to retort that a's being F, which according to Armstrong "ensures" that a is F,just is the entity (whatever it is) that a is F. The identity theorist maps conceptual connections that we draw between the notions of proposition, truth, falsity, state of affairs, and fact. These connections look trivial, when spelt out--of course, an identity theorist will counter that to go further would be to fall into error--so that to speak of an identity theory can readily appear too grand (McDowell 2005: 83; 2007: 352. But cf. David 2002: 126). So much for the thesis that facts are truth-makers and propositions truth-bearers; an exactly parallel argument applies to the version of the correspondence theory that treats facts as truth-makers and thoughts as truth-bearers. Consider now the suggestion that obtaining states of affairs, as the correspondence theorist conceives them, make declarative sentences (as opposed to propositions) true (cf. Horwich 1998: 106-7). In this case there appears to be no threat of triviality of the sort that apparently plagued the previous version of the correspondence theory, because states of affairs like that Socrates is wise are genuinely distinct from linguistic items such as the sentence "Socrates is wise". To that extent friends of the identity theory need not jib at the suggestion that such sentences have worldly truth-makers, if that is how the relation of correspondence is being glossed. But they might question the appropriateness of the gloss. For, they might point out, it does not seem possible, without falsification, to draw detailed links between sentences and bits of the world. After all, different sentences in the same or different languages can "correspond" to the same bit of the world, and these different sentences might have very different (numbers of) components. The English sentence "There are cows" contains three words: are there then three bits in the world corresponding to this sentence, and making it true? (cf. Neale 2001: 177). The sentence "Cows exist" contains only two words, but would not the correspondence theorist want to say that it was made true by the same chunk of reality? And when we take other languages into account, there seems in principle to be no reason to privilege any particular number and say that a sentence corresponding to the relevant segment of reality must contain that number of words: why might there not, in principle, be sentences of actual or possible languages such that, for any n [?] 1, there existed a sentence comprising n words and meaning the same as the English "There are cows"? (In fact, is English not already such a language? Just prefix and then iterate ad lib. a vacuous operator like "Really".) In a nutshell, then, the identity theorist's case against the correspondence theory is that, when the truth-making relation is conceived as originating in a worldly fact (or similar) and having as its other relatum a true sentence, the claim that this relation is one of correspondence cannot be made out; if, on the other hand, the relevant relation targets a proposition (or thought), then that relation must be held to be one of identity, not correspondence. ## 4. Identity, Sense, and Reference Identity theorists are agreed that, in the case of any particular relevant identity, a fact will constitute the worldly relatum of the relation, but there is significant disagreement among them on the question what the item on the other end of the relation is--whether a thought or a proposition (or both). As we have seen, there are three possible positions here: (i) one which places the identity relation exclusively between true thoughts and facts, (ii) one which places it exclusively between true propositions and facts, and (iii) a hybrid position which allows identities of both sorts (identities obtaining at the level of sense will of course be quite distinct from identities obtaining at the level of reference). Which of these positions an identity theorist adopts will depend on wider metaphysical and linguistic considerations that are strictly extraneous to the identity theory as such. Identity theorists who favor (i) generally do so because they want to have nothing to do with propositions as such. That is to say, such theorists eschew propositions as reference-level entities: of course the word "proposition" may be, and sometimes is, applied to Fregean thoughts at the level of sense, rather than to Russellian propositions at the level of reference. For example, Hornsby (1997: 2-3) uses "proposition" and "thinkable" interchangeably. So far, this terminological policy might be considered neutral with respect to the location of propositions and thinkables in the Fregean semantic hierarchy: that is to say, if one encounters a theorist who talks about "thinkables" and "propositions", even identifying them, one does not, just so far, know where in the semantic hierarchy this theorist places these entities. In particular, we cannot assume, unless we are specifically told so, that our theorist locates either propositions or thinkables at the level of sense. After all, someone who houses propositions at the level of reference holds that these reference-level entities are thinkable, in the sense that they are graspable in thought (perhaps via thoughts at the level of sense). But they are not thinkables if this latter word is taken, as it is by McDowell and Hornsby, to be a technical term referring to entities at the level of sense. For clarity the policy here will be to continue to apply the word "proposition" exclusively to Russellian propositions at the level of reference. Such propositions, it is plausible to suppose, can be grasped in thought, but by definition they are not thoughts or thinkables, where these two latter terms have, respectively, their Fregean and McDowellian meanings. It is worth noting that this point, though superficially a merely terminological one, engages significantly with the interface between the philosophies of language and mind that was touched on in the opening paragraph. Anyone who holds that reference-level propositions can, in the ordinary sense, be thought--are thinkable--is likely to be unsatisfied with any terminology that seems to limit the domain of the thinkable and of what is thought to the level of sense (On this point see further below in this section, and Gaskin 2020: 101-2). Usually, as has been noted, identity theorists who favor (i) above have this preference because they repudiate propositions as that term is being employed here: that is, they repudiate propositionally structured reference-level entities. There are several reasons why such identity theorists feel uncomfortable with propositions when these are understood to be reference-level entities. There is a fear that such propositions, if they existed, would have to be construed as truth-makers; and identity theorists, as we have seen, want to have nothing to do with truth-makers (Dodd 2008a: 112). That fear could perhaps be defused if facts were also located at the level of reference for true propositions to be identical with. This move would take us to an identity theory in the style of (ii) or (iii) above. Another reason for suspicion of reference-level propositions is that commentators often follow Russell in his post-1904 aversion specifically to false objectives, that is, to false propositions in re (Russell 1966: 152; Cartwright 1987: 79-84). Such entities are often regarded as too absurd to take seriously as components of reality (so T. Baldwin 1991: 46; Dodd 1995: 163; 1996; 2008a: 66-70, 113-14, 162-6). More especially, it has been argued that false propositions in re could not be unities, that the price of unifying a proposition at the level of reference would be to make it true: if this point were correct it would arguably constitute a reductio ad absurdum of the whole idea of reference-level propositions, since it is plausible to suppose that if there cannot be false reference-level propositions, there cannot be true ones either (see Dodd 2008a: 165). If, on the other hand, one is happy with the existence of propositions in re or reference-level propositions, both true and false,[6] one is likely to favor an identity theory in the style of (ii) or (iii). And, once one has got as far as jettisoning (i) and deciding between (ii) and (iii), there must surely be a good case for adopting (iii): for if one has admitted propositionally structured entities both at the level of sense (as senses of declarative sentences) and at the level of reference (propositions), there seems no good reason not to be maximally liberal in allowing identities between entities of these two types and, respectively, sense- and reference-level kinds of fact (or fact-like entities). Against what was suggested above about Frege (SS2), it has been objected that Frege could not have held an identity theory of truth (Baldwin 1991: 43); the idea here is that, even if he had acknowledged states of affairs as bona fide elements of reality, Frege could not have identified true thoughts with them on pain of confusing the levels of sense and reference. As far as the exegetical issue is concerned, the objection might be said to overlook the possibility that Frege identified true thoughts with facts construed as sense-level entities, rather than with states of affairs taken as reference-level entities; and, as we have noted, Frege does indeed appear to have done just this (see Dodd & Hornsby 1992). Still, the objection raises an important theoretical issue. It would surely be a serious confusion to try to construct an identity across the categorial division separating sense and reference, in particular to attempt to identify true Fregean thoughts with reference-level facts or states of affairs.[7] It has been suggested that McDowell and Hornsby are guilty of this confusion;[8] they have each rejected the charge,[9] insisting that, for them, facts are not reference-level entities, but are, like Fregean thoughts, sense-level entities.[10] But, if one adheres to the Fregean version of the identity theory ((i) above), which identifies true thoughts with facts located at the level of sense, and admits no correlative identity, in addition, connecting true propositions located at the level of reference with facts or fact-like entities also located at that level, it looks as though one faces a difficult dilemma. At what level in the semantical hierarchy is the world to be placed? Suppose first one puts it at the level of reference (this appears to be Dodd's favored view: see 2008a: 180-1, and passim). In that case the world will contain no facts or propositions, but just objects and properties hanging loose in splendid isolation from one another, a dispensation which looks like a version of Kantian transcendental idealism. (Simply insisting that the properties include not merely monadic but also polyadic ones, such as the relation of instantiation, will not in itself solve the problem: we will still just have a bunch of separate objects, properties, and relations.) If there are no true propositions--no facts--or even false propositions to be found at the level of reference, but if also, notwithstanding that absence, the world is located there, the objects it contains will, it seems, have to be conceived as bare objects, not as things of certain sorts. Some philosophers of a nominalistic bias might be happy with this upshot; but the problem is how to make sense of the idea of a bare object--that is, an object not characterized by any properties. (Properties not instantiated by any objects, by contrast, will not be problematic, at least not for a realist.) So suppose, on the other hand, that one places the world at the level of sense, on the grounds that the world is composed of facts, and that that is where facts are located. This ontological dispensation is explicitly embraced by McDowell (1996: 179). The problem with this way out of the dilemma would seem to be that, since Fregean senses are constitutively modes of presentation of referents, the strategy under current consideration would take the world to be made up of modes of presentation--but of what? Of objects and properties? These are certainly reference-level entities, but if they are presented by items in the realm of sense, which is being identified on this approach with the world, then again, as on the first horn of the dilemma, they would appear to be condemned to an existence at the level of reference in splendid isolation from one another, rather than in propositionally structured combinations, so that once more we would seem to be committed to a form of Kantian transcendental idealism (see Suhm, Wagemann, & Wessels 2000: 32; Sullivan 2005: 59-61; Gaskin 2006:199-203). Both ways out of the dilemma appear to have this unattractive consequence. The only difference between those ways concerns where exactly in the semantic hierarchy we locate the world; but it is plausible that that issue, in itself, is or ought to be of less concern to metaphysicians than the requirement to avoid divorcing objects from the properties that make those objects things of certain sorts; and both ways out of the dilemma appear to flout this requirement. To respect the requirement, we need to nest reference-level objects and properties in propositions, or proposition-like structures, also located at the level of reference. And then some of these structured reference-level entities--the true or obtaining ones--will, it seems, be facts, or at least fact-like. Furthermore, once one acknowledges the existence of facts, or fact-like entities, existing at the level of sense, it seems in any case impossible to prevent the automatic generation of facts, or fact-like entities, residing at the level of reference. For sense is mode of presentation of reference. So we need reference-level facts or fact-like entities to be what sense-level facts or fact-like entities present. One has to decide how to treat these variously housed fact-like entities theoretically. If one were to insist that the sense-level fact-like entities were the genuine and only facts, the corresponding reference-level entities would be no better than fact-like, and contrariwise. But, regardless whether the propositionally structured entities automatically generated in this way by sense-level propositionally structured entities are to be thought of as proper facts or merely as fact-like entities, it would seem perverse not to identify the world with these entities.[11] For to insist on continuing to identify the world with sense-level rather than reference-level propositionally structured entities would seem to fly in the face of a requirement to regard the world as maximally objective and maximally non-perspectival. McDowell himself hopes to avert any charge of embracing an unacceptable idealism consequent on his location of the world at the level of sense by relying on the point that senses present their references directly, not descriptively, so that reference is, as it were, contained in sense (1996: 179-80). To this it might be objected that the requirement of maximal objectivity forces an identification of the world with the contained, not the containing, entities in this scenario, which in turn seems to force the upshot--if the threat of Kantian transcendental idealism is really to be obviated--that the contained entities be propositionally structured as such, that is, as contained entities, and not simply in virtue of being contained in propositionally structured containing entities. (For a different objection to McDowell, see Sullivan 2005: 60 n. 6.) ## 5. Difficulties with the Theory and Possible Solutions ### 5.1 The modal problem G. E. Moore drew attention to a point that might look (and has been held to be) problematic for the identity theory (Moore 1953: 308; Fine 1982: 46-7; Kunne 2003: 9-10). The proposition that Socrates is wise exists in all possible worlds where Socrates and the property of wisdom exist, but in some of those worlds this proposition is true and in others it is false. The fact that Socrates is wise, by contrast, only exists in those worlds where the proposition both exists and is true. So it would seem that the proposition that Socrates is wise cannot be identical with the fact that Socrates is wise. They have different modal properties, and so by the principle of the indiscernibility of identicals they cannot be identical. Note, first, that this problem, if it is a problem, has nothing especially to do with the identity theory of truth or with facts. It seems to arise already for true propositions and propositions taken simpliciter before ever we get to the topic of facts. That is, one might think that the proposition that Socrates is wise is identical with the true proposition that Socrates is wise (assuming, as we are doing, that this proposition is true); but we then face the objection that the proposition taken simpliciter and the true proposition differ in their modal properties, since (as one might suppose) the true proposition that Socrates is wise does not exist at worlds where the proposition that Socrates is wise is false, but the proposition taken simpliciter does. Indeed the problem, if it is a problem, is still more general, and purported solutions to it go back at least to the Middle Ages (when it was discussed in connection with Duns Scotus' formal distinction; see Gaskin 2002 [with references to further relevant literature]). Suppose that Socrates is a cantankerous old curmudgeon. Now grumpy Socrates, one would think, is identical with Socrates. But in some other possible worlds Socrates is of a sunny and genial disposition. So it would seem that Socrates cannot be identical with grumpy Socrates after all, because in these other possible worlds, while Socrates goes on existing, grumpy Socrates does not exist--or so one might argue. Can the identity theorist deal with this problem, and if so how? Here is one suggestion. Suppose we hold, staying with grumpy Socrates for a moment, that, against the assumption made at the end of the last paragraph, grumpy Socrates does in fact exist in worlds where Socrates has a sunny disposition. The basis for this move would be the thought that, after all, grumpy Socrates is identical with Socrates, and Socrates exists in these other worlds. So grumpy Socrates exists in those worlds too; it is just that he is not grumpy in those worlds. (Suppose Socrates is very grumpy; suppose in fact that grumpiness is so deeply ingrained in his character that worlds in which he is genial are quite far away. Someone surveying the array of possible worlds, starting from the actual world and moving out in circles, and stumbling at long last upon a world with a pleasant Socrates in it, might register the discovery by exclaiming, with relief, "Oh look! Grumpy Socrates is not grumpy over here!".) Similarly, one might contend, the true proposition, and fact, that Socrates is wise goes on existing in the worlds where Socrates is not wise, because the true proposition, and fact, that Socrates is wise just is the proposition that Socrates is wise, and that proposition goes on existing in these other worlds, but in those worlds that true proposition, and fact, is not a true proposition, or a fact. (In Scotist terms one might say that the proposition that Socrates is wise and the fact that Socrates is wise are really identical but formally distinct.) This solution was, in outline, proposed by Richard Cartwright in his 1987 discussion of the identity theory (Cartwright 1987: 76-8; cf. David 2002: 128-9; Dodd 2008a: 86-8; Candlish & Damnjanovic 2018: 265-6). According to Cartwright, the true proposition, and fact, that there are subways in Boston exists in other possible worlds where Boston does not have subways, even though in those worlds that fact would be not be a fact. (Compare: grumpy Socrates exists in worlds where Socrates is genial and sunny, but he is not grumpy there.) So even in worlds where it is not a fact that Boston has subways, that fact, namely the fact that Boston has subways, continues to exist. Cartwright embellishes his solution with two controversial points. First, he draws on Kripke's distinction between rigid and non-rigid designation, suggesting that his solution can be described by saying that the expression "The fact that Boston has subways" is a non-rigid designator. But it is plausible that that expression goes on referring to, or being satisfied by (depending on how exactly one wants to set up the semantics of definite descriptions: see Gaskin 2008: 56-81), the fact that Boston has subways in possible worlds where Boston does not have subways; it is just that, though that fact exists in those worlds, it is not a fact there. But that upshot does not appear to derogate from the rigidity of the expression in question. Secondly, Cartwright allows for a true reading of "The fact that there are subways in Boston might not have been the fact that there are subways in Boston". But it is arguable that we should say that this sentence is just false (David 2002: 129). The fact that there are subways in Boston would still have gone on being the same fact in worlds where Boston has no subways, namely the fact that there are subways in Boston; it is just that in those worlds this fact would not have been a fact. You might say: in that world the fact that there are subways in Boston would not be correctly described as a fact, but in talking about that world we are talking about it from the point of view of our world, and in our world it is a fact. (Similarly with grumpy Socrates.) Now, an objector may want to press the following point against the above purported solution to the difficulty. Consider again the fact that Socrates is wise. Surely, it might be said, it is more natural to maintain that that fact does not exist in a possible world where Socrates is not wise, rather than that it exists there all right, but is not a fact. After all, imagine a conversation about a world in which Socrates is not wise and suppose that Speaker A claims that Socrates is indeed wise in that world. Speaker B might counter with > > > No, sorry, you're wrong: there is no such fact in that world; > the purported fact that Socrates is wise simply does not > exist in that world. > > > It might seem odd to insist that B is not allowed to say this and must say instead > > > Yes, you're right that there is such a fact in that world, > namely the fact that Socrates is wise, but in that world > that fact is not a fact;. > > > How might the identity theorist respond to this objection? One possible strategy would be to make a distinction between fact and factuality, as follows. Factuality, one might say, is a reification of facts. Once you have a fact, you also get, as an ontological spin-off, the factuality of that fact. The fact, being a proposition, exists at all possible worlds where the proposition exists, though in some of these worlds it may not be a fact: it will not be a fact in worlds where the proposition is false. The factuality of that fact, by contrast, only exists at those worlds where the fact is a fact--where the proposition is true. So factuality is a bit like a trope. Compare grumpy Socrates again. Grumpy Socrates, the identity theorist might contend, exists at all worlds where Socrates exists, though at some of those worlds he is not grumpy. But Socrates' grumpiness--that particular trope--exists only at worlds where Socrates is grumpy. That seems to obviate the problem, because the suggestion being canvassed here is that grumpy Socrates is identical not with Socrates' grumpiness--so that the fact that these two entities have different modal properties need embarrass no one--but rather with Socrates. Similarly, the suggestion is that the proposition that Socrates is wise is identical not with the factuality of the fact that Socrates is wise, but just with that fact. So the identity theorist would accommodate the objector's point by insisting that facts exist at possible worlds where their factualities do not exist. The reader may be wondering why this problem was ever raised against the identity theory of truth in the first place. After all, the identity theorist does not say that propositions simpliciter are identical with facts, but that true propositions are identical with facts, and now true propositions and facts surely have exactly the same modal properties: for regardless how things are with the sheer proposition that Socrates is wise, at any rate the true proposition that Socrates is wise must surely be thought to exist at the same worlds as the fact that Socrates is wise, whatever those worlds are. However, as against this quick way with the purported problem, there stands the intuition, mentioned and exploited above, that the true proposition that Socrates is wise is identical with the proposition that Socrates is wise. So long as that intuition is in play, the problem does indeed seem to arise--for true propositions, in the first instance, and then for facts by transitivity of identity. But the identity theorist will maintain that, as explained, the problem has a satisfactory solution. ### 5.2 The "right fact" problem Candlish, following Cartwright, has urged that the identity theory of truth is faced with the difficulty of getting hold of the "right fact" (Cartwright 1987: 74-5; Candlish 1999a: 238-9; 1999b: 202-4). Consider a version of the identity theory that states: (11) The proposition that p** is true just if it is identical with a fact.[12] Candlish's objection is now that (11) > > > does not specify which fact has to be identical with the > proposition for the proposition to be true. But what the identity > theory requires is not that a true proposition be identical with > some fact or other, it is that it be identical with the > right fact. (1999b: 203) > > > In another paper Candlish puts the matter like this: > > > But after all, any proposition might be identical with some > fact or other (and there are reasons identified in the > Tractatus for supposing that all propositions are themselves > facts), and so all might be true. What the identity theory needs to > capture is the idea that it is by virtue of being identical > with the appropriate fact that a proposition is true. (1999a: > 239) > > > The reference to the Tractatus is suggestive. Of course, it might be objected that the Tractatus does not have propositions in the sense of that word figuring here: that is, it does not recognize Russellian propositions (propositions at the level of reference). Nor indeed does it appear to recognize Fregean thoughts. In the Tractatus, as we have noted (SS2), declarative sentences (Satze) are facts (arrangements of names), and states of affairs (Sachlagen, Sachverhalte, Tatsachen) are also facts (arrangements of objects). Even so, Candlish's allusion to the Tractatus reminds us that propositions (in our sense) are Tractarian inasmuch as they are structured arrangements of entities, namely objects and properties. (Correlatively, thoughts are structured arrangements of senses.) False propositions (and false thoughts) will equally be arrangements of objects and properties (respectively, senses). So the difficulty that Cartwright and Candlish have identified can be put like this. Plausibly any proposition, whether or not it is true, is identical with some fact or other given that a proposition is an arrangement of entities of the appropriate sort. But if propositions just are facts, then every proposition is identical with some fact--at the very least, with itself--whether it is true or false. So the right-to-left direction of (11) looks incorrect. J. C. Beall (2000) attempts to dissolve this problem on the identity theorist's behalf by invoking the principle of the indiscernibility of identicals. His proposal works as follows. If we ask, in respect of (11), what the "right" fact is, it seems that we can answer that the "right" fact must at least have the property of being identical with the proposition that p*, and the indiscernibility principle then guarantees that there is only one such fact. This proposal is open to an obvious retort. Suppose that the proposition that p* is false. That proposition will still be identical with itself, and if we are saying (in Wittgensteinian spirit) that propositions are facts, then that proposition will be identical with at least one fact, namely itself. So it will satisfy the right-hand side of (11), its falsity notwithstanding. But reflection on this retort suggests a patch-up to Beall's proposal: why not say that the right fact is the fact that p*? We would then be able to gloss (11) with (12) The proposition that p* is true just if (a) it is a fact that p*, and (b) the proposition that p* is identical with that fact. Falsity, it seems, now no longer presents a difficulty, because if it is false that p** then it is not a fact that p*, so that (a) fails, and there is no appropriate candidate for the proposition that p* to be identical with.[13] Notice that, in view of the considerations already aired in connection with the modal problem ((i) of this section), caution is here required. Suppose that it is true that p** in the actual world, but false in some other possible world. According to the strategy that we have been considering on the identity theorist's behalf, it would be wrong to say that, in the possible world where it is false that p*, there is no such fact as the fact that p. The strategy has it that there is indeed such a fact, because it is (in the actual world) a fact that p, and that fact, and the true proposition, that p, go on existing in the possible world where it is false that p; it is just that that* fact is not a fact in that possible world. But (12), the identity theorist will maintain, deals with this subtlety. In the possible world we are considering, where it is false that p*, though the fact that p* exists, it is not a fact that p*, so (a) fails, and there is accordingly no risk of our getting hold of the "wrong" fact. Note also that if a Wittgensteinian line is adopted, while the (false) proposition that p* will admittedly be identical with a fact--at the very least with itself--it will be possible, given the failure of (a), for the identity theorist to contend with a clear conscience that that fact is the wrong fact, which does not suffice to render the proposition true. ### 5.3 The "slingshot" problem If the notorious "slingshot" argument worked, it would pose a problem for the identity theory of truth. The argument exists in a number of different, though related, forms, and this is not the place to explore all of these in detail.[14] Here we shall look briefly at what is one of the simplest and most familiar versions of the argument, namely Davidson's. This version of the argument aims to show that if true declarative sentences refer to anything (for example to propositions or facts), then they all refer to the same thing (to the "Great Proposition", or to the "Great Fact"). This upshot would be unacceptable to an identity theorist of a Russellian cast, who thinks that declarative sentences refer to propositions, and that true such propositions are identical with facts: any such theorist is naturally going to want to insist that the propositions referred to by different declarative sentences are, at least in general, distinct from one another, and likewise that the facts with which distinct true propositions are identical are also distinct from one another. Davidson expresses the problem that the slingshot argument purportedly throws up as follows: > > > The difficulty follows upon making two reasonable assumptions: that > logically equivalent singular terms have the same reference; and that > a singular term does not change its reference if a contained singular > term is replaced by another with the same reference. But now suppose > that "R" and "S" abbreviate any > two sentences alike in truth value. (1984: 19) > > > He then argues that the following four sentences have the same reference: (13) $R$ (14) $\hat{z}(z\! =\! z \amp R) = \hat{z}(z\! =\! z)$ (15) $\hat{z}(z\! =\! z \amp S) = \hat{z}(z\! =\! z $) (16) $S$ (The hat over a variable symbolizes the description operator: so "$\hat{z}$" means the $z$ such that ...) This is because (13) and (14) are logically equivalent, as are (15) and (16), while the only difference between (14) and (15) is that (14) contains the expression (Davidson calls it a "singular term") "$\hat{z} (z\! =\! z \amp R)$" whereas (15) contains "$\hat{z} (z\! =\! z \amp S)$", > > > and these refer to the same thing if S and R are alike > in truth value. Hence any two sentences have the same reference if > they have the same truth value. (1984: 19) > > > The difficulty with this argument, as a number of writers have pointed out (see, e.g., Yourgrau 1987; Gaskin 1997: 153 n. 17; Kunne 2003: 133-41), and the place where the identity theorist is likely to raise a cavil, lies in the first assumption on which it depends. Davidson calls this assumption "reasonable", but it has been widely questioned. It states "that logically equivalent singular terms have the same reference". But intuitively, the ideas of logical equivalence and reference seem to be quite distinct, indeed to have, as such, little to do with one another, so that it would be odd if there were some a priori reason why the assumption had to hold. And it is not difficult to think of apparent counterexamples: the sentence "It is raining" is logically equivalent to the sentence "It is raining and (either Pluto is larger than Mercury or it is not the case that Pluto is larger than Mercury)", but the latter sentence seems to carry a referential payload that the former does not. Of course, if declarative sentences refer to truth-values, as Frege thought, then the two sentences will indeed be co-referential, but to assume that sentences refer to truth-values would be question-begging in the context of an argument designed to establish that all true sentences refer to the same thing. ### 5.4 The congruence problem A further objection to the identity theory, going back to an observation of Strawson's, takes its cue from the point that canonical names of propositions and of facts are often not straightforwardly congruent with one another: they are often not intersubstitutable salva congruitate (or, if they are, they may not be intersubstitutable salva veritate) (Strawson 1971: 196; cf. Kunne 2003: 10-12). For example, we say that propositions are true, not that they obtain, whereas we say that facts obtain, not that they are true. How serious is this point? The objection in effect presupposes that for two expressions to be co-referential, or satisfied by one and the same thing, they must be syntactically congruent, have the same truth-value potential, and match in terms of general contextual suitability. The assumption of the syntactic congruence of co-referential expressions is controversial, and it may be possible for the identity theorist simply to deny it (see Gaskin 2008: 106-10, for argument on the point, with references to further literature; cf. Dodd 2008a: 83-6.). Whether co-referential expressions must be syntactically congruent depends on one's conception of reference, a matter that cannot be further pursued here (for discussion see Gaskin 2008: ch. 2; 2020: chs. 3-5). There has been a good deal of discussion in the literature concerning the question whether an identification of facts with true propositions is undermined not specifically by phenomena of syntactic incongruence but rather by failure of relevant intersubstitutions to preserve truth-values (see, e.g., King 2007: ch. 5; King in King, Soames, & Speaks 2014: 64-70, 201-8; Hofweber 2016: 215-23; Candlish & Damnjanovic 2018: 264). The discussion has focused on examples like the following: (17) Daniel remembers the fact that this is a leap year; (18) Daniel remembers the true proposition that this is a leap year; (19) The fact that my local baker has shut down is just appalling; (20) The true proposition that my local baker has shut down is just appalling. The problem here is said to be that the substitution of "true proposition" for "fact" or vice versa generates different readings (in particular, readings with different truth-values). Suppose Daniel has to memorize a list of true propositions, of which one is the proposition that this is a leap year. Then it is contended that we can easily imagine a scenario in which (17) and (18) differ in truth-value. Another way of putting the same point might be to say that (17) is equivalent to (21) Daniel remembers that this is a leap year, but that (18) is not equivalent to (21), because--so the argument goes--(18) but not (21) would be true if Daniel had memorized his list of true propositions without realizing that they were true. Similar differences can be argued to apply, mutatis mutandis, to (19) and (20). Can the identity theorist deal with this difficulty? In the first place one might suggest that the alleged mismatch between (17) and (18) is less clear than the objector claims. (17) surely does have a reading like the one that is said to be appropriate for (18). Suppose Daniel has to memorize a list of facts. (17) could then diverge in truth-value from (22) Daniel remembers the fact that this is a leap year as a fact. For there is a reading of (17) on which, notwithstanding (17)'s truth, (22) is false: this is the reading on which Daniel has indeed memorized a list of facts, but without necessarily realizing that the things he is memorizing are facts. He has memorized the relevant fact (that this is a leap year), we might say, but not as a fact. That is parallel to the reading of (18) according to which Daniel has memorized the true proposition that this is a leap year, but not as a true proposition. The identity theorist might then aver that, perhaps surprisingly, the same point actually applies to the simple (21), on the grounds that this sentence can mean that Daniel remembers the propositional object that this is a leap year (from a list of such objects, say, that he has been asked to memorize), with no implication that he remembers it either as a proposition or as a fact. So, according to this response, the transparent reading of (18)--which has Daniel remember the propositional object, namely that this is a leap year, but not necessarily remember it as a fact, or even as the propositional object that this is a leap year (he remembers it under some other mode of presentation)--is also available for (17) and for (21). What about the opaque reading of either (17) or (21), which implies that Daniel knows for a fact that this is a leap year--is that reading available for (18) too? The identity theorist might maintain that this reading is indeed available, and then explain why we tend not to use sentences like (18) in the relevant sense, preferring sentences of the form of (17) or (21), on the basis of the relative technicality of the vocabulary of (18). The idea would be that it is just an accident of language that we prefer either (17) or (21) to (18) where what is in question is the sense that implies that Daniel has propositional knowledge that this is a leap year (is acquainted with that fact as a fact), as opposed to having mere acquaintance, under some mode of presentation or other, with the propositional object which happens to be (the fact) that this is a leap year. And if we ask why we prefer (17) or (21) to (23) Daniel remembers the proposition that this is a leap year, then the answer will be the Gricean one that (23) conveys less information than (17) or (21), under the reading of these two sentences that we are usually interested in, according to which Daniel remembers the relevant fact as a fact, for (23) is compatible with the falsity of "This is a leap year". Hence to use (23) in a situation where one was in a position to use (17) or (21) would carry a misleading conversational implicature. That, at any rate, is one possible line for the identity theorist to take. (It is worth noting here that, if the identity theorist is right about this, it will follow that the "know that" construction will be subject to a similar ambiguity as the "remember that" construction, given that remembering is a special case of knowing. That is: "A knows that p*" will mean either "A* is acquainted with the fact that p*, and is acquainted with it as a fact" or merely "A* is acquainted with the fact that p*, but not necessarily with it as such--either as a fact or even as a propositional object".) ### 5.5 The individuation problem It might appear that we individuate propositions more finely than facts: for example, one might argue that the fact that Hesperus is bright* is the same fact as the fact that Phosphorus is bright, but that the propositions in question are different (see on this point Kunne 2003: 10-12; Candlish & Damnjanovic 2018: 266-7). The identity theorist has a number of strategies in response to this objection. One would be simply to deny it, and maintain that facts are individuated as finely as propositions: if one is a supporter of the Fregean version of the identity theory, this is likely to be one's response (see, e.g., Dodd 2008a: 90-3). Alternatively, one might respond by saying that, if there is a good point hereabouts, at best it tells only against the Fregean and Russellian versions of the identity theory, not against the hybrid version. The identity theory in the hybrid version can agree that we sometimes think of facts as extensional, reference-level entities and sometimes also individuate propositions or proposition-like entities intensionally. Arguably, these twin points do indeed tell against either a strict Fregean or a strict Russellian version of the identity theory: they tell against the strict Fregean position because, as well as individuating facts intensionally, we also, sometimes, individuate facts extensionally; and they tell against the strict Russellian position because, as well as individuating facts extensionally, we also, sometimes, individuate facts intensionally. But it is plausible that the hybrid version of the identity theory is not touched by the objection, because that version of the theory accommodates propositionally structured and factual entities at both levels of sense and reference, though different sorts of these entities at these different levels--either propositions at the level of sense and correlative proposition-like entities at the level of reference or vice versa, and similarly, mutatis mutandis, for facts and fact-like entities. It will follow, then, for this version of the identity theory, that Fregean thoughts and Russellian propositions are available, if true, to be identical with the factual entities of the appropriate level (sense and reference, respectively), and the individuation problem will not then, it seems, arise. Propositions or propositionally structured entities will be individuated just as finely as we want them to be individuated, and at each level of resolution there will be facts or fact-like entities, individuated to the same resolution, for them to be identical with, if true.[15] ### 5.6 Truth and Intrinsicism The solutions to these problems, if judged satisfactory, seem to direct us to a conception of truth that has been called "intrinsicist" (Wright 1999: 207-9), and "primitivist" (Candlish 1999b: 207). This was a conception recognized by Moore and Russell who, in the period when they were sympathetic to the identity theory, spoke of truth as a simple and unanalysable property (Moore 1953: 261; 1993: 20-1; Russell 1973: 75; Cartwright 1987: 72-5; Johnston 2013: 384). The point here would be as follows. There are particular, individual explanations of the truth of many propositions. For example, the true proposition that there was a fire in the building will be explained by alluding to the presence of combustible material, enough oxygen, a spark caused by a short-circuit, etc. So, case by case, we will (at least often) be able to provide explanations why given propositions are true, and science is expanding the field of such explanations all the time. But according to the intrinsicist, there is no prospect of providing a general explanation of truth, in the sense of an account that would explain, in utterly general terms, why any true proposition was true. At that general level, according to the intrinsicist, there is nothing interesting to be said about what makes true propositions true: there are only the detailed case-histories. An intrinsicist may embrace one or another version of the identity theory of truth: what has to be rejected is the idea that the truth of a true proposition might consist in a relation to a distinct fact--that the truth of the true proposition that Socrates is wise, for example, might consist in anything other than identity with the fact that Socrates is wise. On this approach, truth is held to be both intrinsic to propositions, and primitive.[16] Intrinsicism was not a popular position, at least until recently: Candlish described it as "so implausible that almost no one else [apart from Russell, in 1903-4] has been able to take it seriously" (1999b: 208); but it may be gaining in popularity now (see, e.g., Asay 2013). Candlish (1999b: 208) thinks that intrinsicism and the identity theory are competitors, but perhaps that view is not mandatory. Intrinsicism says that truth is a simple and unanalysable property of propositions; the identity theory says that a true proposition is identical with a fact (and with the right fact). These statements will seem to clash only if the identity theory is taken to propound a heavy-duty analysis of truth. But if, following recent exponents of the theory, we take it rather to be merely spelling out a connection between two entities that we have in our ontology anyway, namely true propositions and facts, and which turn out (like Hesperus and Phosphorus) to be identical, on the basis of a realization that an entity like that Socrates is wise is both a true proposition and a fact, then any incipient clash between intrinsicism and the identity theory will, it seems, be averted. On this approach, a natural thing to say will be that the identity theory describes the way in which truth is a simple and unanalysable property.
truth-pluralist	## 1. Alethic pluralism about truth: a plurality of properties ### 1.1 Strength The pluralist's thesis that there are many ways of being true is typically construed as being tantamount to the claim that the number of truth properties is greater than one. However, this basic interpretation, * (1) there is more than one truth property. is compatible with both moderate as well as more radical precisifications. According to moderate pluralism, at least one way of being true among the multitude of others is universally shared: * (2) there is more than one truth property, some of which are had by all true sentences. According to strong pluralism, however, there is no such universal or common way of being true: * (3) there is more than one truth property, none of which is had by all true sentences. Precisifying pluralism about truth in these two ways brings several consequences to the fore. Firstly, both versions of pluralism conflict with strong monism about truth: * (4) there is exactly one truth property, which is had by all true sentences. Secondly, moderate--but not strong--pluralism is compatible with a moderate version of monism about truth: * (5) there is one truth property, which is had by all true sentences. (2) and (5) are compatible because (5) does not rule out the possibility that the truth property had by all true sentences might be one among the multitude of truth properties endorsed by the moderate pluralist (i.e., by someone who endorses (2)). Only strong pluralism in (3) entails the denial of the claim that all true sentences are true in the same way. Thus, moderate pluralists and moderate monists can in principle find common ground. ### 1.2 Related kinds of pluralism and neighboring views Not infrequently, pluralism about truth fails to be distinguished from various other theses about associated conceptual, pragmatic, linguistic, semantic, and normative phenomena. Each of these other theses involves attributing plurality to a different aspect of the analysandum (explanandum, definiendum, etc.). For instance, linguistically, one may maintain that there is a plurality of truth predicates (Wright 1992; Tappolet 1997; Lynch 2000; Pedersen 2006, 2010). Semantically, one may maintain that alethic terms like 'true' have multiple meanings (Pratt 1908; Tarski 1944; Kolbel 2008, 2013; Wright 2010). Cognitively or conceptually, one may maintain that there is a multiplicity of truth concepts or regimented ways of conceptualizing truth (Kunne 2003; cf. Lynch 2006). Normatively, one might think that truth has a plurality of profiles (Ferrari 2016, 2018). These parameters or dimensions suggest that pluralism is itself not just a single, monolithic theory (see also Sher 1998; Wright 2013). Any fully developed version of pluralism about truth is likely to make definitive commitments about at least some of these other phenomena. (However, it hardly entails them; one can consistently be an alethic pluralist about truth, for instance, without necessarily having commitments to linguistic pluralism about truth predicates, or about concepts like fact or actuality.) Nonetheless, theses about these other phenomena should be distinguished from pluralism about truth, as understood here. Likewise, pluralism about truth must be distinguished from several neighbouring views, such as subjectivism, contextualism, relativism, or even nihilism about truth. For example, one can maintain some form of subjectivism about truth while remaining agnostic about how many ways of being true there are. Or again, one can consistently maintain that there is exactly one way of being true, which is always and everywhere dependent on context. Nor is it inconsistent to be both a pluralist and an absolutist or other anti-relativist about truth. For example, one might argue that each of the different ways of being true holds absolutely if it holds at all (Wright 1992). Alternatively, one might explicate a compatibilist view, in which there are at least two kinds of truth, absolute and relative truth (Joachim 1905), or deflationist and substantivist (Kolbel 2013). Such views would be, necessarily, pluralistic. Occasionally, pluralists have also been lumped together with various groups of so-called 'nihilists', 'deniers', and 'cynics', and even associated with an 'anything goes' approach to truth (Williams 2002). However, any version of pluralism is prima facie inconsistent with any view that denies truth properties, such as nihilism and certain forms of nominalism. ### 1.3 Alethic pluralism, inflationism, and deflationism The foregoing varieties of pluralism are consistent with various further analyses of pluralists' ideas about truth. For instance, pluralists may--but need not--hold that truth properties are simply one-place properties, since commitments to truth's being monadic are orthogonal to commitments to its being monistic. However, most pluralists converge on the idea that truth is a substantive property and take this idea as the point of departure for articulating their view. A property is substantive just in case there is more to its nature than what is given in our concept of the property. A paradigmatic example of a substantive property is the property of being water. There is more to the nature of water--being composed of H$\_2$O, e.g.--than what is revealed in our concept of water (the colourless, odourless liquid that comes out of taps, fills lakes, etc.) The issue of substantiveness connects with one of the major issues in the truth debate: the rift between deflationary theories of truth and their inflationary counterparts (Horwich 1990; Edwards 2013b; Kunne 2003; Sher 2016b; Wyatt 2016; Wyatt & Lynch 2016). A common way to understand the divide between deflationists and inflationists is in terms of the question whether or not truth is a substantive property. Inflationists endorse this idea, while deflationists reject it. More specifically, deflationists and inflationists can be seen as disagreeing over the following claim: * (6) there exists some property $F$ (coherence, correspondence, etc.) such that any sentence, if true, is so in virtue of being $F$--and this is a fact that is not transparent in the concept of truth. The inflationist accepts (6). According to her, it is not transparent in the concept of truth that being true is a matter of possessing some further property (cohering, corresponding, etc.). This makes truth a substantive property. The deflationist, on the other hand, rejects (6) because she is committed to the idea that everything there is to know about truth is transparent in the concept--which, on the deflationist's view, is exhausted by the disquotational schema ('$p$' is true if, and only if, $p)$, or some principle like it. Deflationists also tend to reject a further claim about truth's explanatory role: * (7) $F$ is necessary and sufficient for explaining the truth of any true sentence $p$. Inflationists, on the other hand, typically accept both (6) and (7). Strong and moderate versions of pluralism are perhaps best understood as versions of a non-traditional inflationary theory (for an exception, see Beall 2013; for refinements, see Edwards 2012b and Ferrari & Moruzzi forthcoming). Pluralists side with inflationists on (6) and (7), and so, their views count as inflationary. Yet, traditional inflationary theories are also predominantly monistic. They differ about which property $F$--coherence, identity, superwarrant, correspondence, etc.--truth consists in, but concur that there is precisely one such property: * (8) there is a single property $F$ and truth consists in a single property $F$. The monistic supposition in (8) is tantamount to the claim that there is but one way of being true. In opposing that claim, pluralism counts as non-traditional. ## 2. Motivating pluralism: the scope problem Pluralists' rejection of (8) typically begins by rendering it as a claim about the invariant nature of truth across all regions of discourse (Acton 1935; Wright 1992, 1996; Lynch 2000, 2001; for more on domains see Edwards 2018b; Kim & Pedersen 2018, Wyatt 2013; Yu 2017). Thus rendered, the claim appears to be at odds with the following observation: * (9) the plausibility of each inflationist's candidate for the property $F$ differs across different regions of discourse. For example, some theories--such as correspondence theories--seem intuitively plausible when applied to truths about ladders, ladles, and other ordinary objects. However, those theories seem much less convincing when applied to truths about comedy, fashion, ethical mores, numbers, jurisprudential dictates, etc. Conversely, theories that seem intuitively plausible when applied to legal, comic, or mathematical truths--such as those suggesting that the nature of truth is coherence--seem less convincing when applied to truths about the empirical world. Pluralists typically take traditional inflationary theories of truth to be correct in analyzing truth in terms of some substantive property $F$. Yet, the problem with their monistic suppositions lies with generalization: a given property $F$ might be necessary and sufficient for explaining why sentences about a certain subject matter are true, but no single property is necessary and sufficient for explaining why $p$ is true for all sentences $p$, whatever its subject matter. Subsequently, those theories' inability to generalize their explanatory scope beyond the select few regions of discourse for which they are intuitively plausible casts aspersion on their candidate for $F$. This problem has gone by various names, but has come to be known as 'the scope problem' (Lynch 2004b, 2009; cf. Sher 1998). Pluralists respond to the scope problem by first rejecting (8) and replacing it with: * (10) truth consists in several properties $F\_1 , \ldots ,F\_n$. With (10), pluralists contend that the nature of truth is not a single property $F$ that is invariant across all regions of discourse; rather the true sentences in different regions of discourse may consist in different properties among the plurality $F\_1 , \ldots ,F\_n$ that constitute truth's nature. The idea that truth is grounded in various properties $F\_1 , \ldots ,F\_n$ might be further introduced by way of analogy. Consider water. We ordinarily think and talk about something's being water as if it were just one thing--able to exist in different states, but nevertheless consisting in just one property (H$\_2$O). But it would be a mistake to legislate in advance that we should be monists about water, since the nature of water is now known to vary more than our intuitions would initially have it. The isotopic distribution of water allows for different molecular structures, to include hydroxonium (H$\_3$O), deuterium oxide (D$\_2$O), and so-called 'semi-heavy water' (HDO). Or again, consider sugar, the nature of which includes glucose, fructose, lactose, cellulose, and similar other such carbohydrates. For the pluralist, so too might truth be grounded as a plurality of more basic properties. One reason to take pluralism about truth seriously, then, is that it provides a solution to the scope problem. In rejecting the 'one-size-fits-all' approach to truth, pluralists formulate a theory whose generality is guaranteed by accommodating the various properties $F\_1 , \ldots ,F\_n$ by which true sentences come to be true in different regions of discourse. A second and related reason is that the view promises to be explanatory. Variance in the nature of truth in turn explains why theories of truth perform unequally across various regions of discourse--i.e., why they are descriptively adequate and appropriate in certain regions of discourse, but not others. For pluralists, the existence of different kinds of truths is symptomatic of the non-uniform nature of truth itself. Subsequently, taxonomical differences among truths might be better understood by formulating descriptive models about how the nature of truth might vary between those taxa. ## 3. Prominent versions of pluralism ### 3.1 Platitude-based strategies Many pluralists have followed Wright (1992) in supposing that compliance with platitudes is what regiments and characterizes the behavior and content of truth-predicates. Given a corollary account of how differences in truth predicates relate to differences among truth properties, this supposition suggests a platitude-based strategy for positing many ways of being true. Generally, a strategy will be platitude-based if it is intended to show that a certain collection of platitudes $p\_1 , \ldots ,p\_n$ suffices for understanding the analysandum or explanandum. By 'platitude', philosophers generally mean certain uncontroversial expressions about a given topic or domain. Beyond that, conceptions about what more something must be or have to count as platitudinous vary widely. #### 3.1.1 Discourse pluralism/minimalism A well-known version of platitude-based pluralism is discourse pluralism. The simplest versions of this view make the following four claims. Firstly, discourse exhibits natural divisions, and so can be stably divided into different regions $D\_1 , \ldots ,D\_n$. Secondly, the platitudes subserving some $D\_i$ may be different than those subserving $D\_j$. Thirdly, for any pair $(D\_i, D\_j)$, compliance with different platitudes subserving each region of discourse can, in principle, result in numerically distinct truth predicates $(t\_i, t\_j)$. Finally, numerically distinct truth predicates designate different ways of being true. Discourse pluralism is frequently associated with Crispin Wright (1992, 1996, 2001), although others have held similar views (see, e.g., Putnam 1994: 515). Wright has argued that discourse pluralism is supported by what he calls 'minimalism'. According to minimalism, compliance with both the disquotational schema and the operator schema, * (11) '$p$' is true if, and only if, $p$. * (12) it is true that $p$ if, and only if, $p$. as well as other 'parent' platitudes, is both necessary and sufficient for some term $t\_i$ to qualify as expressing a concept worth regarding as TRUTH (1992: 34-5). Wright proposed that the parent platitudes, which basically serve as very superficial formal or syntactic constraints, fall into two subclasses: those connecting truth with assertion ('transparency'), * (13) To assert is to present as true. * (14) Any attitude to a proposition is an attitude towards its truth. and those connecting truth with logical operations ('embedding'), * (15) Any truth-apt content has a negation which is also truth-apt. * (16) Aptitude for truth is preserved under basic logical operations (disjunction, conjunction, etc.). Any such term complying with these parent platitudes, regardless of region of discourse, counts as what Wright called a 'lightweight' or 'minimal' truth predicate. Yet, the establishment of some $t$ as a minimal truth predicate is compatible, argued Wright, with the nature of truth consisting in different things in different domains (2001: 752). Wright (2001) has also suggested that lightweight truth predicates tend to comply with five additional subclasses of platitudes, including those connecting truth with reality ('correspondence') and eternity ('stability'), * (17) For a sentence to be true is for it to correspond with reality. * (18) True sentences accurately reflect how matters stand. * (19) To be true is to "tell it like it is". * (20) A sentence is always true if it ever is. * (21) Whatever may be asserted truly may be asserted truly at any time. and those disconnecting truth from epistemic state ('opacity'), justification ('contrast'), and scalar degree ('absoluteness'), * (22) A thinker may be so situated that a particular truth is beyond her ken. * (23) Some truths may never be known. * (24) Some truths may be unknowable in principle. * (25) A sentence may be true without being justified and vice-versa. * (26) Sentences cannot be more or less true. * (27) Sentences are completely true if at all. The idea is that $t$ may satisfy additional platitudes beyond these, and in doing so may increase its 'weight'. For example, some $t\_i$ may be a more heavyweight truth predicate than $t\_j$ in virtue of satisfying platitudes which entail that truth be evidence-transcendent or that there be mind-independent truth-makers. Finally, differences in what constitutes truth in $D\_1 , \ldots ,D\_n$ are tracked by differences in the weight of these predicates. In this way, Wright is able to accommodate the intuition that sentences about, e.g., macromolecules in biochemistry are amenable to realist truth in a way that sentences about distributive welfare in ethics may not be. Distinctions among truth predicates, according to the discourse pluralist, are due to more and less subtle differences among platitudes and principles with which they must comply. For example, assuming that accuracy of reflection is a matter of degree, predicates for truth and truthlikeness diverge because a candidate predicate may comply with either (18) or else either of (26) or (27); to accommodate both, two corollary platitudes must be included to make explicit that accurate reflection in the case of truth is necessarily maximal and that degrees of accuracy are not equivalent to degrees of truth. Indeed, it is not unusual for platitudes to presuppose certain attendant semantic or metaphysical views. For example, * (28) A sentence may be characterized as true just in case it expresses a true proposition. requires anti-nominalist commitments, an ontological commitment to propositions, and commitments to the expression relation (translation relations, an account of synonymy, etc.). Discourse pluralists requiring predicates to comply with (28) in order to count as truth-predicates must therefore be prepared to accommodate other claims that go along with (28) as a package-deal. #### 3.1.2 Functionalism about truth 'Functionalism about truth' names the thesis that truth is a functional kind. The most comprehensive and systematic development of a platitude-based version of functionalism comes from Michael Lynch, who has been at the forefront of ushering in pluralist themes and theses (see Lynch 1998, 2000, 2001, 2004c, 2005a, 2005b, 2006, 2009, 2012, 2013; Devlin 2003). Lynch has urged that we need to think about truth in terms of the 'job' or role, $F$, that true sentences stake out in our discursive practices (2005a: 29). Initially, Lynch's brand of functionalism attempted to implicitly define the denotation of 'truth' using the quasi-formal technique of Ramsification. The technique commences by treating 'true' as the theoretical term $\tau$ issued by the theory $T$ and targeted for implicit definition. Firstly, the platitudes and principles of the theory are amassed $(T: p\_1 , \ldots ,p\_n)$ so that the $F$-role can be specified holistically. Secondly, a certain subset $A$ of essential platitudes $(p\_i , \ldots ,p\_k)$ must be extracted from $T$, and are then conjoined. Thirdly, following David Lewis, $T$ is rewritten as * (29) $R(\tau\_1,\ldots , \tau\_n , \omicron\_1,\ldots , \omicron\_n )$ so as to isolate the $\tau$-terms from the non-theoretical ('old, original, other') $o$-terms. Fourthly, all instances of 'true' and other cognate or closely related $\tau$-terms are then replaced by subscripted variables $x\_1 , \ldots ,x\_n$. The resulting open sentence is prefixed with existential quantifiers to bind them. Next, the Ramsey sentence is embedded in a material biconditional; this allows functionalists to then specify the conditions by which a given truth-apt sentence $p$ has a property that plays the $F$-role: * (30) $p$ has some property $\varrho\_i$ realizing the $F$-role $\equiv \exists x\_1 , \ldots ,\exists x\_n [R(x\_1 , \ldots ,x\_n , \omicron\_1,\ldots , \omicron\_n ) \amp p$ has $x\_1$], where, say, the variable $x\_1$ is the one that replaced 'true'. Having specified the conditions under which $p$ has some property realizing $F$, functionalists can then derive another material biconditional stating that $p$ is true iff $p$ has some property realizing the $F$-role. However, as Lynch (2004: 394) cautioned, biconditionals that specify necessary and sufficient conditions for $p$ to be true still leave open questions about the 'deep' metaphysical nature of truth. Thus, given the choice, Lynch--following up on a suggestion from Pettit (1996: 886)--urged functionalists to identify truth, not with the properties realizing the $F$-role in a given region of discourse, but with the $F$-role itself. Doing so is one way to try to secure the 'unity' of truth (on the presumption that there is just one $F$-role). Hence, to say that truth is a functional kind $F$ is to say that the $\tau$-term 'truth' denotes the property of having a property that plays the $F$-role, where the $F$-role is tantamount to the single unique second-order property of being $F$. Accordingly, this theory proposes that something is true just in case it is $F$. Two consequences are apparent. Firstly, the functionalist's commitment to alethic properties realizing the $F$-role seems to be a commitment to a grounding thesis. This explains why Lynch's version of alethic functionalism fits the pattern typical of inflationary theories of truth, which are committed to (6) and (7) above. Secondly, however, like most traditional inflationary theories, Lynch's functionalism about truth appears to be monistic. Indeed, the functionalist commitment to identifying truth with and only with the unique property of being $F$ seems to entail a commitment to strong alethic monism in (5) rather than pluralism (Wright 2005). Nonetheless, it is clear that Lynch's version does emphasize that sentences can have the property of being $F$ in different ways. The theory thus does a great deal to accommodate the intuitions that initially motivate the pluralist thesis that there is more than one way of being true, and to finesse a fine line between monism and pluralism. For pluralists, this compromise may not be good enough, and critics of functionalism about truth have raised several concerns. One stumbling block for functionalist theories is a worry about epistemic circularity. As Wright (2010) observes, any technique for implicit definition, such as Ramsification, proceeds on the basis of explicit decisions that the platitudes and principles constitutive of the modified Ramsey sentence are themselves true, and making explicit decisions that they are true requires already knowing in advance what truth is. Lynch (2013a) notes that the problem is not peculiar to functionalism about truth, generalizing to virtually all approaches that attempt to fix the denotation of 'true' by appeal to implicit definition. Some might want to claim that it generalizes even further, namely to any theory of truth whatsoever. Another issue is that the $F$-role becomes disunified to the extent that $T$ can accommodate substantially different platitudes and principles. Recall that the individuation and identity conditions of the $F$-role--with which truth is identified--are determined holistically by the platitudes and principles constituting $T$. So where $T$ is constituted by expressions of the beliefs and commitments of ordinary folk, pluralists could try to show that these beliefs and commitments significantly differ across epistemic communities (see, e.g., Naess 1938a, b; Maffie 2002; Ulatowski 2017, Wyatt 2018). In that case, Ramsification over significantly different principles may yield implicit definitions of numerically distinct role properties $F\_1, F\_2 , \ldots ,F\_n$, each of which is a warranted claimant to being truth. ### 3.2 Correspondence pluralism The correspondence theory is often invoked as exemplary of traditional monistic theories of truth, and thus as a salient rival to pluralism about truth. Prima facie, however, the two are consistent. The most fundamental principle of any version of the correspondence theory, * (31) Truth consists in correspondence. specifies what truth consists in. Since it involves no covert commitment about how many ways of being true there are, it does not require denying that there is more than one (Wright & Pedersen 2010). In principle, there may be different ways of consisting in correspondence that yield different ways of being true. Subsequently, whether the two theories turn out to be genuine rivals depends on whether further commitments are made to explicitly rule out pluralism. Correspondence theorists have occasionally made proposals that combine their view with a version of pluralism. An early--although not fully developed--proposal of this kind was made by Henry Acton (1935: 191). Two recent proposals are noteworthy and have been developed in detail. Gila Sher (1998, 2004, 2005, 2013, 2015, 2016a) has picked up the project of expounding on the claim that sentence in domains like logic correspond to facts in a different way than do sentences in other domains, while Terence Horgan and colleagues (Horgan 2001; Horgan & Potrc 2000, 2006; Horgan & Timmons 2002; Horgan & Barnard 2006; Barnard & Horgan 2013) have elaborated a view that involves a defense of the claim that not all truths correspond to facts in the same way. For Sher, truth does not consist in different properties in different regions of discourse (e.g., superwarrant in macroeconomics, homomorphism in immunology, coherence in film studies, etc.). Rather, it always and everywhere consists in correspondence. Taking 'correspondence' to generally refer to an $n$-place relation $R$, Sher advances a version of correspondence pluralism by countenancing different 'forms', or ways of corresponding. For example, whereas the physical form of correspondence involves a systematic relation between the content of physical sentences and the physical structure of the world, the logical form of correspondence involves a systematic relation between the logical structure of sentences and the formal structure of the world, while the moral form of correspondence involves a relation between the moral content of sentences and (arguably) the psychological or sociological structure of the world. Sher's view can be regarded as a moderate form of pluralism. It combines the idea that truth is many with the idea that truth is one. Truth is many on Sher's view because there are different forms of correspondence. These are different ways of being true. At the same time, truth is one because these different ways of being true are all forms of correspondence. For Sher, a specific matrix of 'factors' determines the unique form of correspondence as well as the correspondence principles that govern our theorizing about them. Which factors are in play depends primarily on the satisfaction conditions of predicates. For example, the form of correspondence for logical truths of the form * (32) Every $x$ either is or is not self-identical. is determined solely by the logical factor, which is reflected by the universality of the union of the set of self-identical things and its complement. Or again, consider the categorical sentences * (33) Some humans are disadvantaged. and * (34) Some humans are vain. Both (33) and (34) involve a logical factor, which is reflected in their standard form as I-statements (i.e., some $S$ are $P)$, as well as the satisfaction conditions of the existential quantifier and copula; a biological factor, which is reflected in the satisfaction conditions for the predicate 'is human'; and a normative factor, which is reflected in the satisfaction conditions for the predicates 'is disadvantaged' and 'is vain'. But whereas (34) involves a psychological factor, which is reflected in the satisfaction conditions for 'is vain', (33) does not. Also, (33) may involve a socioeconomic factor, which is reflected in the satisfaction conditions for 'is disadvantaged', whereas (34) does not. By focusing on subsentential factors instead of supersentential regions of discourse, Sher offers a more fine-grained way to individuate ways in which true sentences correspond. (Sher supposes that we cannot name the correspondent of a given true sentence since there is no single discrete hypostatized entity beyond the $n$-tuples of objects, properties and relations, functions, structures (complexes, configurations), etc. that already populate reality.) The upshot is a putative solution to problems of mixed discourse (see SS4 below): the truth of sentences like * (35) Some humans are disadvantaged and vain. is determined by all of the above factors, and which is--despite the large overlap--a different kind of truth than either of the atomic sentences (33) and (34), according to Sher. For their part, Horgan and colleagues propose a twist on the correspondence theorist's claim that truth consists in a correspondence relation $R$ obtaining between a given truth-bearer and a fact. They propose that there are exactly two species of the relation $R$: 'direct' ($R\_{dir}$) and 'indirect correspondence' ($R\_{ind}$), and thus exactly two ways of being true. For Horgan and colleagues, which species of $R$--and thus which way of being true--obtains will depend on the austerity of ontological commitments involved in assessing sentences; in turn, which commitments are involved depends on discursive context and operative semantic standards. For example, an austere ontology commits to only a single extant object: namely, the world (affectionally termed the 'blobject'). Truths about the blobject, such as * (36) The world is all that is the case. if it is one, correspond to it directly. Truths about things other than the blobject correspond to them indirectly. For example, sentences such as * (37) Online universities are universities. may be true even if the extension of the predicate 'university' is--strictly speaking--empty or what is referred to by 'online universities' is not in the non-empty extension of 'university'. In short, $p$ is true$\_1$ iff $p$ is $R\_{dir}$-related to the blobject given contextually operative standards $c\_i, c\_j , \ldots ,c\_m$. Alternatively, $p$ is true$\_2$ iff $p$ is $R\_{ind}$-related to non-blobject entities given contextually operative standards $c\_j, c\_k , \ldots ,c\_n$. So, truth always consists in correspondence. But the two types of correspondence imply that there is more than one way of being true. ## 4. Objections to pluralism and responses ### 4.1 Ambiguity Some take pluralists to be committed to the thesis that 'true' is ambiguous: since the pluralist thinks that there is a range of alethically potent properties (correspondence, coherence, etc.), 'true' must be ambiguous between these different properties. This is thought to raise problems for pluralists. According to one objection, the pluralist appears caught in a grave dilemma. 'True' is either ambiguous or unambiguous. If it is, then there is a spate of further problems awaiting (see SS4.4-SS4.6 below). If it is not, then there is only one meaning of 'true' and thus only one property designated by it; so pluralism is false. Friends of pluralism have tended to self-consciously distance themselves from the claim that 'true' is ambiguous (e.g., Wright 1996: 924, 2001; Lynch 2001, 2004b, 2005c). Generally, however, the issue of ambiguity for pluralism has not been well-analyzed. Yet, one response has been investigated in some detail. According to this response, the ambiguity of 'true' is simply to be taken as a datum. 'True' is de facto ambiguous (see, e.g., Schiller 1906; Pratt 1908; Kaufmann 1948; Lucas 1969; Kolbel 2002, 2008; Sher 2005; Wright 2010). Alfred Tarski, for instance, wrote: > > > > The word 'true', like other words from our everyday > language, is certainly not unambiguous. [...] We should reconcile > ourselves with the fact that we are confronted, not with one concept, > but with several different concepts which are denoted by one word; we > should try to make these concepts as clear as possible (by means of > definition, or of an axiomatic procedure, or in some other way); to > avoid further confusion we should agree to use different terms for > different concepts [...]. (1944: 342, 355) > > > If 'true' is ambiguous de facto, as some authors have suggested, then the ambiguity objection may turn out to be--again--not so much an objection or disconfirmation of the theory, but rather just a datum about 'truth'-talk in natural language that should be explained or explained away by theories of truth. In that case, pluralists seem no worse off--and possibly better--than any number of other truth theorists. A second possible line of response from pluralists is that their view is not necessarily inconsistent with a monistic account of either the meaning of 'true' or the concept TRUTH. After all, 'true' is ambiguous only if it can be assigned more than one meaning or semantic structure; and it has more than one meaning only if there is more than one stable conceptualization or concept TRUTH supporting each numerically distinct meaning. Yet, nothing about the claim that there is more than one way of being true entails, by itself, that there is more than one concept TRUTH. In principle, the nature of properties like being true--whether homomorphism, superassertibility, coherence, etc.--may outstrip the concept thereof, just as the nature of properties like being water--such as H$\_2$O, H$\_3$O, XYZ, etc.--may outstrip the concept WATER (see, e.g., Wright 1996, 2001; Alston 2002; Lynch 2001, 2005c, 2006). Nor is monism about truth necessarily inconsistent with semantic or conceptual pluralism. The supposition that TRUTH is both many and one (i.e., 'moderate monism') neither rules out the construction of multiple concepts or meanings thereof, nor rules out the proliferation of uses to express those concepts or meanings. For example, suppose that the only way of being true turns out to be a structural relation $R$ between reality and certain representations thereof. Such a case is consistent with the existence of competing conceptions of what $R$ consists in: weak homomorphism, isomorphism, 'seriously dyadic' correspondence, a causal $n$-place correspondence relation, etc. A more sensitive conclusion, then, is just that the objection from ambiguity is an objection to conceptual or semantic pluralism, not to any alethic theory--pluralism or otherwise. ### 4.2 The scope problem as a pseudo-problem According to the so-called 'Quine-Sainsbury objection', pluralists' postulation of ambiguity in metalinguistic alethic terms is not actually necessary, and thus not well-motivated. This is because taxonomical differences among kinds of truths in different domains can be accounted for simply by doing basic ontology in object-level languages. > > > > [E]ven if it is one thing for 'this tree is an oak' to > be true, another thing for 'burning live cats is cruel' to > be true, and yet another for 'Buster Keaton is funnier than > Charlie Chaplin' to be true, this should not lead us to suppose > that 'true' is ambiguous; for we get a better explanation > of the differences by alluding to the differences between trees, > cruelty, and humor. (Sainsbury 1996: 900; see also Quine 1960: 131) > > > Generally, pluralists have not yet developed a response to the Quine-Sainsbury objection. And for some, this is because the real force of the Quine-Sainsbury objection lies in its exposure of the scope problem as a pseudo-problem (Dodd 2013; see also Asay 2018). Again, the idea is that traditional inflationary theories postulate some candidate for $F$ but the applicability and plausibility of $F$ differs across regions of discourse. No such theory handles the truths of moral, mathematical, comic, legal, etc. discourse equally well; and this suggests that these theories, by their monism, face limitations on their explanatory scope. Pluralism offers a non-deflationary solution. Yet, why think that these differences among domains mark an alethic difference in truth per se, rather than semantic or discursive differences among the sentences comprising those domains? There is more than one way to score a goal in soccer, for example (via corner kick, ricochet off the foot of an opposing player or the head of a teammate, obstruct the goalkeeper, etc.), but it is far from clear that this entails pluralism about the property of scoring a goal in soccer. (Analogy belongs to an anonymous referee.) Pluralists have yet to adequately address this criticism (although see Blackburn 2013; Lynch 2013b, 2018; Wright 1998 for further discussion). ### 4.3 The criteria problem Pluralists who invoke platitude-based strategies bear the burden of articulating inclusion and exclusion criteria for determining which expressions do, or do not, count as members of the essential subset of platitudes upon which this strategy is based (Wright, 2005). Candidates include: ordinariness, intuitiveness, uninformativeness, wide use or citation, uncontroversiality, a prioricity, analyticity, indefeasibility, incontrovertibility, and sundry others. But none has proven to be uniquely adequate, and there is nothing close to a consensus about which criteria to rely on. For instance, consider the following two conceptions. One conception takes platitudes about $x$ to be expressions that must be endorsed on pain of being linguistically incompetent with the application of the terms $t\_1 , \ldots ,t\_n$ used to talk about $x$ (Nolan 2009). However, this conception does not readily allow for disagreement: prima facie, it is not incoherent to think that two individuals, each of whom is competent with the application of $t\_1 (x), \ldots ,t\_n (x)$, may differ as to whether some $p$ must be endorsed or whether some expression is genuinely platitudinous. For instance, consider the platitude in (17), which connects being true with corresponding with reality. Being linguistically competent with terms for structural relations like correspondence does not force endorsement of claims that connect truth with correspondence; no one not already in the grip of the correspondence theory would suppose that they must endorse (17), and those who oppose it would certainly suppose otherwise. Further inadequacies beleaguer this conception. It makes no provision for degrees of either endorsement or linguistic incompetence. It makes no distinction between theoretical and non-theoretical terms, much less restrict $t\_1 (x), \ldots ,t\_n (x)$ to non-theoretical terms. Nor does it require that platitudes themselves be true. On one hand, this consequently leaves open the possibility that universally-endorsed but false or otherwise alethically defective expressions are included in the platitude-based analysis of 'true'. An old platitude about whales, for example--one which was universally endorsed on pain of being linguistically incompetent--prior to whales being classified as cetaceans--was that they are big fish. The worry, then, is that the criteria may allow us to screen in certain 'fish stories' about truth. This would be a major problem for advocates of Ramsification and other forms of implicit definition, since those techniques work only on the presupposition that all input being Ramsified over or implicitly defined is itself true (Wright 2010). On the other hand, making explicit that platitudes must also be true seems to entail that they are genuine 'truisms' (Lynch 2005c), though discovering which ones are truly indefeasible is a further difficulty--one made more difficult by the possibility of error theories (e.g., Devlin 2003) suggesting that instances of the $T$-schema are universally false. Indeed, we are inclined to say instances of disquotational, equivalence, and operator schemas are surely candidates for being platitudinous if anything is; but to say that they must be endorsed on pain of being linguistically incompetent is to rule out a priori error theories about instances of the $T$-schema. A second, closely related conception is that platitudes are expressions, which--in virtue of being banal, vacuous, elementary, or otherwise trivial--are acceptable by anyone who understands them (Horwich 1990). The interaction of banality or triviality with acceptance does rule out a wide variety of candidate expressions, however. For instance, claims that are acceptable by anyone who understands them may still be too substantive or informative to count as platitudinous, depending on what they countenance. Similarly, claims that are too 'thin' or neutral to vindicate any particular theory $T$ may still be too substantive or informative to count as genuinely platitudinous on this conception (Wright 1999). This is particularly so given that nothing about a conception of platitudes as 'pretheoretical claims' strictly entails that they reduce to mere banalities (Vision 2004). Nevertheless, criteria like banality or triviality plus acceptance might also screen in too few expressions (perhaps as few as one, such as a particular instance of the $T$-schema). Indeed, it is an open question whether any of the principles in (11)-(28) would count as platitudes on this conception. An alternative conception emphasizes that the criteria should instead be the interaction of informality, truth, a prioricity, or perhaps even analyticity (Wright 2001: 759). In particular, platitudes need not take the form of an identity claim, equational definition, or a material biconditional. At the extreme, expressions can be as colloquial as you please so long as they remain true a priori (or analytically). These latter criteria are commonly appealed to, but are also not without problems. Firstly, a common worry is whether there are any strictly analytic truths about truth, and, if there are, whether they can perform any serious theoretical work. Secondly, these latter criteria would exclude certain truths that are a posteriori but no less useful to a platitude-based strategist. ### 4.4 The instability challenge Another objection to pluralism is that it is an inherently instable view: i.e., as soon as the view is formulated, simple reasoning renders it untenable (Pedersen 2006, 2010; see also Tappolet 1997, 2000; Wright 2012). This so-called instability challenge can be presented as follows. According to the moderate pluralist, there is more than one truth property $F\_1 , \ldots ,F\_n$. Yet, given $F\_1 , \ldots ,F\_n$, it seems we should recognize another truth property: * (38) $\forall p[F\_U (p) \leftrightarrow F\_1 (p) \vee \cdots \vee F\_n (p)]$. Observe that $F\_U$ is not merely some property possessed by every $p$ which happens to have one of $F\_1 , \ldots ,F\_n$. (The property of being a sentence is one such a property, but it poses no trouble to the pluralist.) Rather, $F\_U$ must be an alethic property whose extension perfectly positively covaries with the combined extension of the pluralist truth properties $F\_1 , \ldots ,F\_n$. And since nothing is required for the existence of this new property other than the truth properties already granted by the pluralist, (38) gives a necessary and sufficient condition for $F\_U$ to be had by some $p$: a sentence $p$ is $F\_U$ just in case $p$ is $F\_1 \vee \cdots \vee F\_n$. Thus, any sentence that is any of $F\_1 , \ldots ,F\_n$ may be true in some more generic or universal way, $F\_U$. This suggests, at best, that strong pluralism is false, and moderate monism is true; and at worst, there seems to be something instable, or self-refuting, about pluralism. Pluralists can make concessive or non-concessive responses to the instability challenge. A concessive response grants that such a truth property exists, but maintains that it poses no serious threat to pluralism. A non-concessive response is one intended to rebut the challenge, e.g., by rejecting the existence of a common or universal truth property. One way of trying to motivate this rejection of $F\_U$ is by attending to the distinction between sparse and abundant properties, and then demonstrating that alethic properties like truth must be sparse and additionally argue that the would-be trouble-maker $F\_U$ is an abundant property. According to sparse property theorists, individuals must be unified by some qualitative similarity in order to share a property. For example, all even numbers are qualitatively similar in that they share the property of being divisible by two without remainder. Now, consider a subset of very diverse properties $G\_1 , \ldots ,G\_n$ possessed by an individual $a$. Is there some further, single property of being $G\_1$, or ..., or $G\_n$ that $a$ has? Such a further property, were it to exist, would be highly disjunctive; and it may seem unclear what, if anything, individuals that were $G\_1$, or ..., or $G\_n$ would have in common--other than being $G\_1$, or ..., or $G\_n$. According to sparse property theorists, the lack of qualitative similarity means that this putative disjunctive property is not a property properly so-called. Abundant property theorists, on the other hand, deny that qualitative similarity is needed in order for a range of individuals to share a property. Properties can be as disjunctive as you like. Indeed, for any set $A$ there is at least one property had by all members of $A$--namely, being a member of $A$. And since there is a set of all things that have some disjunctive property, there is a property--abundantly construed--had by exactly those things. It thus seems difficult to deny the existence of $F\_U$ if the abundant conception of properties is adopted. So pluralists who want to give a non-concessive response to the metaphysical instability challenge may want to endorse the sparse conception (Pedersen 2006). This is because the lack of uniformity in the nature of truth across domains is underwritten by a lack of qualitative similarity between the different truth properties that apply to specific domains of discourse. The truth property $F\_U$ does not exist, because truth properties are to be thought of in accordance with the sparse conception. Even if the sparse conception fails to ground pluralists' rejection of the existence of the universal truth property $F\_U$, a concessive response to the instability challenge is still available. Pluralists can make a strong case that the truth properties $F\_1 , \ldots ,F\_n$ are more fundamental than the universal truth property $F\_U$ (Pedersen 2010). This is because $F\_U$ is metaphysically dependent on $F\_1 , \ldots ,F\_n$, in the sense that $F\_U$ is introduced in virtue of its being one of $F\_1 , \ldots ,F\_n$, and not vice-versa. Hence, even if the pluralist commits to the existence of $F\_U$--and hence, to moderate metaphysical monism--there is still a clear sense in which her view is distinctively more pluralist than monist. ### 4.5 Problems regarding mixed discourse #### 4.5.1 Mixed atomic sentences The content of some atomic sentences seems to hark exclusively from a particular region of discourse. For instance, 'lactose is a sugar' concerns chemical reality, while '$7 + 5 = 12$' is solely about the realm of numbers (and operations on these). Not all discourse is pure or exclusive, however; we often engage in so-called 'mixed discourse', in which contents from different regions of discourse are combined. For example, consider: * (39) Causing pain is bad. Mixed atomic sentences such as (39) are thought to pose problems for pluralists. It seems to implicate concepts from the physical domain (causation), the mental domain (pain), and the moral domain (badness) (Sher 2005: 321-22). Yet, if pluralism is correct, then in which way is (39) true? Is it true in the way appropriate to talk of the physical, the mental, or the moral? Is it true in neither of these ways, or in all of these three ways, or in some altogether different way? The source of the problem may be the difficulty in classifying discursive content--a classificatory task that is an urgent one for pluralists. For it is unclear how they can maintain that regions of discourse $D\_1 , \ldots ,D\_n$ partially determine the ways in which sentences can be true without a procedure for determining which region of discourse $D\_i$ a given $p$ belongs to. One suggestion is that a mixed atomic sentence $p$ belongs to no particular domain. Another is that it belongs to several (Wyatt 2013). Lynch (2005b: 340-41) suggested paraphrasing mixed atomic sentences as sentences that are classifiable as belonging to particular domains. For example, (39) might be paraphrased as: * (40) We ought not cause pain. Unlike (39), the paraphrased (40) appears to be a pure atomic sentence belonging to the domain of morals. This proposal remains underdeveloped, however. It is not at all clear that (40) counts as a felicitous paraphrase of (39), and, more generally, unclear whether all mixed atomic sentences can be paraphrased such that they belong to just one domain without thereby altering their meaning, truth-conditions, or truth-values. Another possible solution addresses the problem head-on by questioning whether atomic sentences really are mixed, thereby denying the need for any such paraphrases. Consider the following sentences: * (41) The Mona Lisa is beautiful. * (42) Speeding is illegal. Prima facie, what determines the domain-membership of (41) and (42) is the aesthetic and legal predicates 'is beautiful' and 'is illegal', respectively. It is an aesthetic matter whether the Mona Lisa is beautiful; this is because (41) is true in some way just in case the Mona Lisa falls in the extension of the aesthetic predicate 'is beautiful' (and mutatis mutandis for (42)). In the same way, we might take (39) to exclusively belong to the moral domain given that the moral predicate 'is bad'. (This solution was presented in the first 2012 version of this entry; see Edwards 2018a for later, more detailed treatment.) It is crucial to the latter two proposals that any given mixed atomic sentence $p$ has its domain membership essentially, since such membership is what determines the relevant kind of truth. Sher (2005, 2011) deals with the problem of mixed atomic sentences differently. On her view, the truth of a mixed atomic sentence is not accounted for by membership to some specific domain; rather the 'factors' involved in the sentence determine a specific form of correspondence, and this specific form of correspondence is what accounts for the truth of $p$. The details about which specific form of correspondence obtains is determined at the sub-sentential levels of reference, satisfaction, and fulfillment. For example, the form of correspondence that accounts for the truth of (39) obtains as a combination of the physical fulfillment of 'the causing of $x$', the mental reference of 'pain', and the moral satisfaction of '$x$ is bad' (2005: 328). No paraphrase is needed. #### 4.5.2 Mixed compounds Another related problem pertains to two or more sentences joined by one or more logical connectives, as in * (43) Killing innocent people is wrong and $7 + 5 = 12$. Unlike atomic sentences, the mixing here takes place at the sentential rather than sub-sentential level: (43) is a conjunction, which mixes the pure sentence '$7 + 5 = 12$' with the pure sentence 'killing innocent people is wrong'. (There are, of course, also mixed compounds that involve mixed atomic sentences.) For many theorists, each conjunct seems to be true in a different way, if true at all: the first conjunct in whatever way is appropriate to moral theory, and the second conjunct in whatever way is appropriate to arithmetic. But then, how is the pluralist going to account for the truth of the conjunction (Tappolet 2000: 384)? Pluralists owe an answer to the question of which way, exactly, a conjunction is true when its conjuncts are true in different ways. Additional complications arise for pluralists who commit to facts being what make sentences true (e.g., Lynch 2001: 730), or other such truth-maker or -making theses. Prima facie, we would reasonably expect there to be different kinds of facts that make the conjuncts of (43) true, and which subsequently account for the differences in their different ways of being true. However, what fact or facts makes true the mixed compound? Regarding (43), is it the mathematical fact, the moral fact, or some further kind of fact? On one hand, the claims that mathematical or moral facts, respectively, make $p$ true seem to betray the thought that both facts contribute equally to the truth of the mixed compound. On the other hand, the claim that some third 'mixed' kind of fact makes $p$ true leaves the pluralist with the uneasy task of telling a rather alchemist story about fact-mixtures. Functionalists about truth (e.g., Lynch 2005b: 396-97) propose to deal with compounds by distinguishing between two kinds of realizers of the $F$-role. The first is an atomic realizer, such that an atomic proposition $p$ is true iff $p$ has a property that realizes the $F$-role. The second is a compound realizer, such that a compound $q \* r$ (where $q$ and $r$ may themselves be complex) is true iff * (44) ${\} = \wedge : q \wedge r$ has the property of being an instance of the truth-function for conjunction with conjuncts that both have a property that realizes the $F$-role. (45) ${\} = \vee : q \vee r$ has the property of being an instance of the truth-function for disjunction with at least one disjunct that has a property that realizes the $F$-role. (46) ${\} = \rightarrow : q \rightarrow r$ has the property of being an instance of the truth-function for material conditional with an antecedent that does not have the property that realizes the $F$-role for its domain or a consequent that has a property that realizes the $F$-role. The realizers for atomic sentences are properties like correspondence, coherence, and superwarrant. The realizer properties for compounds are special, in the sense that realizer properties for a given kind of compound are only had by compounds of that kind. Witness that each of these compound realizer properties requires any of its bearers to be an instance of a specific truth-function. Pure and mixed compounds are treated equally on this proposal: when true, they are true because they instantiate the truth-function for conjunction, having two or more conjuncts that have a property that realizes the $F$-role (and mutatis mutandis for disjunctions and material conditionals). However, this functionalist solution to the problem of mixed compounds relies heavily on that theory's monism--i.e., its insistence that the single role property $F$ is a universal truth property. This might leave one wondering whether a solution is readily available to someone who rejects the existence of such a property. One strategy is simply to identify the truth of conjunctions, disjunctions, and conditionals with the kind of properties specified by (44), (45), and (46), respectively (as opposed to taking them to be realizers of a single truth property). Thus, e.g., the truth of any conjunction simply $is$ to be an instance of the truth-function for conjunction with conjuncts that have the property that plays the $F$-role for them (Kim & Pedersen 2018, Pedersen & Lynch 2018 (Sect. 20.6.2.1). Another strategy is to try to use the resources of multi-valued logic. For example, one can posit an ordered set of designated values for each way of being true $F\_1 , \ldots ,F\_n$ (perhaps according to their status as 'heavyweight' or 'lightweight'), and then take conjunction to be a minimizing operation and disjunction a maximizing one, i.e., $v(p \wedge q) = \min\{v(p), v(q)\}$ and $v(p \vee q) = \max\{v(p), v(q)\}$. Resultingly, each conjunction and disjunction--whether pure or mixed--will be either true in some way or false in some way straightforwardly determined by the values of the constituents. For example, consider the sentences (47) Heat is mean molecular kinetic energy. * (48) Manslaughter is a felony. Suppose that (47) is true in virtue of corresponding to physical reality, while (48) true in virtue of cohering with a body of law; and suppose further that correspondence $(F\_1)$ is more 'heavyweight' than coherence $(F\_2)$. Since conjunction is a minimizing operation and $F\_2 \lt F\_1$, then 'heat is mean molecular kinetic energy and manslaughter is a felony' will be $F\_2$. Since disjunction is a maximizing operation, then 'heat is mean molecular kinetic energy or manslaughter is a felony' will be $F\_1$. The many-valued solution to the problem of mixed compounds just outlined is formally adequate because it determines a way that each compound is true. However, while interesting, the proposal needs to be substantially developed in several respects. For example, how is negation treated--are there several negations, one for each way of being true, or is there a single negation? Also, taking 'heat is mean molecular kinetic energy and manslaughter is a felony' to be true in the way appropriate to law betrays a thought that seems at least initially compelling, viz. that both conjuncts contribute to the truth of the conjunction. Alternatively, one could take mixed compounds to be true in some third way. However, this would leave the pluralist with the task of telling some story about how this third way of being true relates to the other two. Again substantial work needs to be done. Edwards (2008) proposed another solution to the problem of mixed conjunctions, the main idea of which is to appeal to the following biconditional schema: * (49) $p$ is true$\_i$ and $q$ is true$\_j$ iff $p \wedge q$ is true$\_k$. Edwards suggests that pluralists can answer the challenge that mixed conjunctions pose by reading the stated biconditional as having an order of determination: $p \wedge q$ is true$\_k$ in virtue of $p$'s being true$\_i$ and $q$'s being true$\_j$, but not vice-versa. This, he maintains, explains what kind of truth a conjunction $p \wedge q$ has when its conjuncts are true in different ways; for the conjunction is true$\_k$ in virtue of having conjuncts that are both true, where it is inessential whether the conjuncts are true in the same way. Truth$\_k$ is a further way of being true that depends on the conjuncts being true in some way without reducing to either of them. The property true$\_k$ is thus not a generic or universal truth property that applies to the conjuncts as well as the conjunction. As Cotnoir (2009) emphasizes, Edwards' proposal provides too little information about the nature of true$\_k$. What little is provided makes transparent the commitment to true$\_k$'s being a truth property had only by conjunctions, in which case it is unclear whether Edwards's solution can generalize. In this regard, Edwards' proposal is similar to Lynch's functionalist proposal, which is committed to there being a specific realizer property for each type of logical compound. #### 4.5.3 Mixed inferences Mixed inferences--inferences involving truth-apt sentences from different domains--appear to be yet another problem for the pluralist (Tappolet 1997, 2000; Pedersen 2006). One can illustrate the problem by supposing, with the pluralist, that there are two ways of being true, one of which is predicated of the antecedent of a conditional and the other as its consequent. It can be left open in what way the conditional itself is true. Consider the following inference: * (50) Satiated dogs are lazy. * (51) Our dog is satiated. * (52) Our dog is lazy. This inference would appear to be valid. However, it is not clear that pluralists can account for its validity by relying on the standard characterization of validity as necessary truth preservation from premises to conclusion. Given that the truth properties applicable to respectively (51) and (52) are different, what truth property is preserved in the inference? The pluralist owes an explanation of how the thesis that there are many ways of being true can account for the validity of mixed inferences. Beall (2000) argued that the account of validity used in multi-valued logics gives pluralists the resources to deal with the problem of mixed inferences. For many-valued logics, validity is accounted for in terms of preservation of designated value, where designated values can be thought of as ways of being true, while non-designated values can be thought of as ways of being false. Adopting a designated-value account of validity, pluralists can simply take $F\_1 , \ldots ,F\_n$ to be the relevant designated values and define an inference as valid just in case the conclusion is designated if each premise is designated (i.e., one of $F\_1 , \ldots ,F\_n)$. On this account, the validity of (mixed) arguments whose premises and conclusion concern different regions of discourse is evaluable in terms of more than one of $F\_1 , \ldots ,F\_n$; the validity of (pure) arguments whose premises and conclusion pertain to the same region of discourse is evaluable in terms of the same $F\_i$ (where $1 \le i \le n)$. An immediate rejoinder is that the term 'true' in 'ways of being true' refers to a universal way of being true--i.e., being designated simpliciter (Tappolet 2000: 384). If so, then the multi-valued solution comes at the cost of inadvertently acknowledging a universal truth property. Of course, as noted, the existence of a universal truth property poses a threat only to strong pluralism. ### 4.5 The problem of generalization Alethic terms are useful devices for generalizing. For instance, suppose we wish to state the law of excluded middle. A tedious way would be to produce a long--indeed, infinite--conjunction: * (53) Everything is either actual or non-actual, and thick or not thick, and red or not red, and ... However, given the equivalence schema for propositions, * (54) The proposition that $p$ is true iff $p$. there is a much shorter formula, which captures what (54) is meant to express by using 'true', but without loss of explanatory power (Horwich 1990: 4): * (55) Every proposition of the form $\langle$everything is $G$ or not$-G\rangle$ is true. Alethic terms are also useful devices for generalizing over what speakers say, as in * (56) What Chen said is true. The utility of a generalization like (56) is not so much that it eliminates the need to rely on an infinite conjunction, but that it is 'blind' (i.e., made under partial ignorance of what was said). Pluralists seem to have difficulty accounting for truth's use as a device for generalization. One response is to simply treat uses of 'is true' as elliptical for 'is true in one way or another'. In doing so, pluralists account for generalization without sacrificing their pluralism. A possible drawback, however, is that it may commit pluralists to the claim that 'true' designates the disjunctive property of being $F\_1 \vee \cdots \vee F\_n$. Granting the existence of such a property gives pluralists a story to tell about generalizations like (55) and (56), but the response is a concessive one available only to moderate pluralists. However, as noted in SS4.2.3, the existence of such a property is not a devastating blow to all pluralists, since the domain-specific truth properties $F\_1 , \ldots ,F\_n$ remain explanatorily basic in relation to the property of being $F\_1 \vee \cdots \vee F\_n$. ### 4.6 The problem of normativity As is often noted, truth appears to be normative--i.e., a positive standard governing immanent content (Sher 2004: 26). According to one prominent tradition (Engel 2002, 2013; Wedgwood 2002; Boghossian 2003; Shah 2003; Gibbard 2005; Ferrari 2018; Lynch 2009; Whiting 2013), truth is a doxastic norm because it is the norm of correctness for belief: * (57) $\forall p$(a belief that $p$ is correct iff $p$ is true). Indeed, many take it to be constitutive of belief that its norm of correctness is truth--i.e., part of what makes belief the kind of attitude that it is. If correctness is understood in prescriptive--rather than descriptive--terms, then (57) presumably gives way to the following schema: * (58) $\forall p$(one ought to believe that $p$ when $p$ is true). A third normative schema linking truth and belief classifies truth as a good of belief (Lynch, 2004a, 2005b: 390, 2009: 10; David 2005): * (59) $\forall p$(it is prima facie good to believe that $p$ when $p$ is true). What these schemas suggest is that the apparent doxastic and assertoric normativity of truth appears to be entirely general, in a manner analogous to which winning appears to be a general norm that applies to any competitive game (Dummett 1978: 8; Lynch 2005b: 390). Hence, if $p$ is true, then it is correct and good to believe that $p$, and one should believe that $p$--regardless of whether $p$ concerns fashion or physics, comedy or chemistry. And again, the generalized normativity of truth appears to make trouble for pluralists, insofar as the thesis that there are several ways of being true apparently implies a proliferation of doxastic truth norms. Yet, instead of truth being the single normative property mentioned in (57), (58), and (59), the pluralist commits to a wide variety of norms--one for each domain-specific truth property $F\_1 , \ldots ,F\_n$. For example, for any given trigonometric statement p, it is prima facie good to believe $p$ when $p$ is true in the way appropriate to trigonometry, while the prima facie goodness of believing a truth about antibodies is tied to whatever truth property is apropos to immunology. Hence, whereas the normative aspects of truth seem characterized by unity, pluralism renders disunity. As before, a concessive response can be given by granting the existence of a disjunctive, universal truth property: the normative property of being $F\_1 \vee \cdots \vee F\_n$. Although this amounts to an endorsement moderate pluralism, it poses no threat to the importance of the domain-specific norms $F\_1 , \ldots ,F\_n$, insofar as these properties are explanatorily more basic than the normative property of being $F\_1 \vee \cdots \vee F\_n$. However, at the same time, they do provide the unity needed to maintain that the predicate is true in (57), (58), and (59) denotes a single, universally applicable norm: * (60) $\forall p$(a belief that $p$ is correct iff $p$ is $F\_1 \vee \cdots \vee F\_n)$. * (61) $\forall p$(one ought to believe $p$ when $p$ is $F\_1 \vee \cdots \vee F\_n)$. * (62) $\forall p$(it is prima facie good to believe that $p$ when $p$ is $F\_1 \vee \cdots \vee F\_n)$. Likewise, functionalists once again respond to the challenge by invoking the monist aspect of their view. There is a single normative property--the property of having a property that realizes the $F$-role--that delivers a uniform understanding of (57), (58), and (59).
truth-pragmatic	## 1. History of the Pragmatic Theory of Truth The history of the pragmatic theory of truth is tied to the history of classical American pragmatism. According to one standard account, C.S. Peirce gets credit for first proposing a pragmatic theory of truth, William James is responsible for popularizing the pragmatic theory, and John Dewey subsequently reframed truth in terms of warranted assertibility (for this reading of Dewey see Burgess & Burgess 2011: 4). More specifically, Peirce is associated with the idea that true beliefs are those that will withstand future scrutiny; James with the idea that true beliefs are dependable and useful; Dewey with the idea that truth is a property of well-verified claims (or "judgments"). ### 1.1 Peirce's Pragmatic Theory of Truth The American philosopher, logician and scientist Charles Sanders Peirce (1839-1914) is generally recognized for first proposing a "pragmatic" theory of truth. Peirce's pragmatic theory of truth is a byproduct of his pragmatic theory of meaning. In a frequently-quoted passage in "How to Make Our Ideas Clear" (1878), Peirce writes that, in order to pin down the meaning of a concept, we must: > > > Consider what effects, which might conceivably have practical > bearings, we conceive the object of our conception to have. Then, our > conception of these effects is the whole of our conception of the > object. (1878 [1986: 266]) > > > The meaning of the concept of "truth" then boils down to the "practical bearings" of using this term: that is, of describing a belief as true. What, then, is the practical difference of describing a belief as "true" as opposed to any number of other positive attributes such as "creative", "clever", or "well-justified"? Peirce's answer to this question is that true beliefs eventually gain general acceptance by withstanding future inquiry. (Inquiry, for Peirce, is the process that takes us from a state of doubt to a state of stable belief.) This gives us the pragmatic meaning of truth and leads Peirce to conclude, in another frequently-quoted passage, that: > > > All the followers of science are fully persuaded that the processes of > investigation, if only pushed far enough, will give one certain > solution to every question to which they can be applied....The > opinion which is fated to be ultimately agreed to by all who > investigate, is what we mean by the truth. (1878 [1986: 273]) > > > Peirce realized that his reference to "fate" could be easily misinterpreted. In a less-frequently quoted footnote to this passage he writes that "fate" is not meant in a "superstitious" sense but rather as "that which is sure to come true, and can nohow be avoided" (1878 [1986: 273]). Over time Peirce moderated his position, referring less to fate and unanimous agreement and more to scientific investigation and general consensus (Misak 2004). The result is an account that views truth as what would be the result of scientific inquiry, if scientific inquiry were allowed to go on indefinitely. In 1901 Peirce writes that: > > > Truth is that concordance of an abstract statement with the ideal > limit towards which endless investigation would tend to bring > scientific belief. (1901a [1935: 5.565]) > > > Consequently, truth does not depend on actual unanimity or an actual end to inquiry: > > > If Truth consists in satisfaction, it cannot be any actual > satisfaction, but must be the satisfaction which would > ultimately be found if the inquiry were pushed to its ultimate and > indefeasible issue. (1908 [1935: 6.485], emphasis in original) > > > As these references to inquiry and investigation make clear, Peirce's concern is with how we come to have and hold the opinions we do. Some beliefs may in fact be very durable but would not stand up to inquiry and investigation (this is true of many cognitive biases, such as the Dunning-Kruger effect where people remain blissfully unaware of their own incompetence). For Peirce, a true belief is not simply one we will hold onto obstinately. Rather, a true belief is one that has and will continue to hold up to sustained inquiry. In the practical terms Peirce prefers, this means that to have a true belief is to have a belief that is dependable in the face of all future challenges. Moreover, to describe a belief as true is to point to this dependability, to signal the belief's scientific bona fides, and to endorse it as a basis for action. By focusing on the practical dimension of having true beliefs, Peirce plays down the significance of more theoretical questions about the nature of truth. In particular, Peirce is skeptical that the correspondence theory of truth--roughly, the idea that true beliefs correspond to reality--has much useful to say about the concept of truth. The problem with the correspondence theory of truth, he argues, is that it is only "nominally" correct and hence "useless" (1906 [1998: 379, 380]) as far as describing truth's practical value. In particular, the correspondence theory of truth sheds no light on what makes true beliefs valuable, the role of truth in the process of inquiry, or how best to go about discovering and defending true beliefs. For Peirce, the importance of truth rests not on a "transcendental" (1901a [1935: 5.572]) connection between beliefs on the one hand and reality on the other, but rather on the practical connection between doubt and belief, and the processes of inquiry that take us from the former to the latter: > > > If by truth and falsity you mean something not definable in terms of > doubt and belief in any way, then you are talking of entities of whose > existence you can know nothing, and which Ockham's razor would > clean shave off. Your problems would be greatly simplified, if, > instead of saying that you want to know the "Truth", you > were simply to say that you want to attain a state of belief > unassailable by doubt. (1905 [1998: 336]) > > > For Peirce, a true belief is one that is indefeasible and unassailable--and indefeasible and unassailable for all the right reasons: namely, because it will stand up to all further inquiry and investigation. In other words, > > > if we were to reach a stage where we could no longer improve upon a > belief, there is no point in withholding the title "true" > from it. (Misak 2000: 101) > > > ### 1.2 James' Pragmatic Theory of Truth Peirce's contemporary, the psychologist and philosopher William James (1842-1910), often gets credit for popularizing the pragmatic theory of truth. In a series of popular lectures and articles, James offers an account of truth that, like Peirce's, is grounded in the practical role played by the concept of truth. James, too, stresses that truth represents a kind of satisfaction: true beliefs are satisfying beliefs, in some sense. Unlike Peirce, however, James suggests that true beliefs can be satisfying short of being indefeasible and unassailable: short, that is, of how they would stand up to ongoing inquiry and investigation. In the lectures published as Pragmatism: A New Name for Some Old Ways of Thinking (1907) James writes that: > > > Ideas...become true just in so far as they help us get into > satisfactory relation with other parts of our experience, to summarize > them and get about among them by conceptual short-cuts instead of > following the interminable succession of particular phenomena. (1907 > [1975: 34]) > > > True ideas, James suggests, are like tools: they make us more efficient by helping us do the things that need to get done. James adds to the previous quote by making the connection between truth and utility explicit: > > > Any idea upon which we can ride, so to speak; any idea that will carry > us prosperously from any one part of our experience to any other part, > linking things satisfactorily, working securely, simplifying, saving > labor; is true for just so much, true in so far forth, true > instrumentally. This is the 'instrumental' view > of truth. (1907 [1975: 34]) > > > While James, here, credits this view to John Dewey and F.C.S. Schiller, it is clearly a view he endorses as well. To understand truth, he argues, we must consider the practical difference--or the pragmatic "cash-value" (1907 [1975: 97]) of having true beliefs. True beliefs, he suggests, are useful and dependable in ways that false beliefs are not: > > > you can say of it then either that "it is useful because it is > true" or that "it is true because it is useful". > Both these phrases mean exactly the same thing. (1907 [1975: 98]) > > > Passages such as this have cemented James' reputation for equating truth with mere utility (something along the lines of: "< p > is true just in case it is useful to believe that p" [see Schmitt 1995: 78]). (James does offer the qualification "in the long run and on the whole of course" (1907 [1975: 106]) to indicate that truth is different from instant gratification, though he does not say how long the long run should be.) Such an account might be viewed as a watered-down version of Peirce's account that substitutes "cash-value" or subjective satisfaction for indefeasibility and unassailability in the face of ongoing inquiry and investigation. Such an account might also be viewed as obviously wrong, given the undeniable existence of useless truths and useful falsehoods. In the early twentieth century Peirce's writings were not yet widely available. As a result, the pragmatic theory of truth was frequently identified with James' account--and, as we will see, many philosophers did view it as obviously wrong. James, in turn, accused his critics of willful misunderstanding: that because he wrote in an accessible, engaging style his critics "have boggled at every word they could boggle at, and refused to take the spirit rather than the letter of our discourse" (1909 [1975: 99]). However, it is also the case that James tends to overlook or intentionally blur--it is hard to say which--the distinction between (a) giving an account of true ideas and (b) giving an account of the concept of truth. This means that, while James' theory might give a psychologically realistic account of why we care about the truth (true ideas help us get things done) his theory fails to shed much light on what the concept of truth exactly is or on what makes an idea true. And, in fact, James often seems to encourage this reading. In the preface to The Meaning of Truth he doubles down by quoting many of his earlier claims and noting that "when the pragmatists speak of truth, they mean exclusively something about the ideas, namely their workableness" (1909 [1975: 6], emphasis added). James' point seems to be this: from a practical standpoint, we use the concept of truth to signal our confidence in a particular idea or belief; a true belief is one that can be acted upon, that is dependable and that leads to predictable outcomes; any further speculation is a pointless distraction. What then about the concept of truth? It often seems that James understands the concept of truth in terms of verification: thus, "true is the name for whatever idea starts the verification-process, useful is the name for its completed function in experience" (1907 [1975: 98]). And, more generally: > > > Truth for us is simply a collective name for verification-processes, > just as health, wealth, strength, etc., are names for other processes > connected with life, and also pursued because it pays to pursue them. > (1907 [1975: 104]) > > > James seems to claim that being verified is what makes an idea true, just as having a lot of money is what makes a person wealthy. To be true is to be verified: > > > Truth happens to an idea. It becomes true, is > made true by events. Its verity is in fact an event, > a process: the process namely of its verifying itself, its > veri-fication. Its validity is the process of its > valid-ation. (1907 [1975: 97], emphasis in original) > > > Like Peirce, James argues that a pragmatic account of truth is superior to a correspondence theory because it specifies, in concrete terms, what it means for an idea to correspond or "agree" with reality. For pragmatists, this agreement consists in being led "towards that reality and no other" in a way that yields "satisfaction as a result" (1909 [1975: 104]). By sometimes defining truth in terms of verification, and by unpacking the agreement of ideas and reality in pragmatic terms, James' account attempts to both criticize and co-opt the correspondence theory of truth. ### 1.3 Dewey's Pragmatic Theory of Truth John Dewey (1859-1952), the third figure from the golden era of classical American pragmatism, had surprisingly little to say about the concept of truth especially given his voluminous writings on other topics. On an anecdotal level, as many have observed, the index to his 527 page Logic: The Theory of Inquiry (1938 [2008]) has only one reference to "truth", and that to a footnote mentioning Peirce. Otherwise the reader is advised to "See also assertibility". At first glance, Dewey's account of truth looks like a combination of Peirce and James. Like Peirce, Dewey emphasizes the connection between truth and rigorous scientific inquiry; like James, Dewey views truth as the verified result of past inquiry rather than as the anticipated result of inquiry proceeding into an indefinite future. For example, in 1911 he writes that: > > > From the standpoint of scientific inquiry, truth indicates not just > accepted beliefs, but beliefs accepted in virtue of a certain > method....To science, truth denotes verified beliefs, > propositions that have emerged from a certain procedure of inquiry and > testing. By that I mean that if a scientific man were asked to point > to samples of what he meant by truth, he would pick...beliefs > which were the outcome of the best technique of inquiry available in > some particular field; and he would do this no matter what his > conception of the Nature of Truth. (1911 [2008: 28]) > > > Furthermore, like both Peirce and James, Dewey charges correspondence theories of truth with being unnecessarily obscure because these theories depend on an abstract (and unverifiable) relationship between a proposition and how things "really are" (1911 [2008: 34]). Finally, Dewey also offers a pragmatic reinterpretation of the correspondence theory that operationalizes the idea of correspondence: > > > Our definition of truth...uses correspondence as a mark of a > meaning or proposition in exactly the same sense in which it is used > everywhere else...as the parts of a machine correspond. (1911 > [2008: 45]) > > > Dewey has an expansive understanding of "science". For Dewey, science emerges from and is continuous with everyday processes of trial and error--cooking and small-engine repair count as "scientific" on his account--which means he should not be taken too strictly when he equates truth with scientific verification. (Peirce and James also had expansive understandings of science.) Rather, Dewey's point is that true propositions, when acted on, lead to the sort of predictable and dependable outcomes that are hallmarks of scientific verification, broadly construed. From a pragmatic standpoint, scientific verification boils down to the process of matching up expectations with outcomes, a process that gives us all the "correspondence" we could ask for. Dewey eventually came to believe that conventional philosophical terms such as "truth" and "knowledge" were burdened with so much baggage, and had become so fossilized, that it was difficult to grasp the practical role these terms had originally served. As a result, in his later writings Dewey largely avoids speaking of "truth" or "knowledge" while focusing instead on the functions played by these concepts. By his 1938 Logic: The Theory of Inquiry Dewey was speaking of "warranted assertibility" as the goal of inquiry, using this term in place of both "truth" and "knowledge" (1938 [2008: 15-16]). In 1941, in a response to Russell entitled "Propositions, Warranted Assertibility, and Truth", he wrote that "warranted assertibility" is a "definition of the nature of knowledge in the honorific sense according to which only true beliefs are knowledge" (1941: 169). Here Dewey suggests that "warranted assertibility" is a better way of capturing the function of both knowledge and truth insofar as both are goals of inquiry. His point is that it makes little difference, pragmatically, whether we describe the goal of inquiry as "acquiring more knowledge", "acquiring more truth", or better yet, "making more warrantably assertible judgments". Because it focuses on truth's function as a goal of inquiry, Dewey's pragmatic account of truth has some unconventional features. To begin with, Dewey reserves the term "true" only for claims that are the product of controlled inquiry. This means that claims are neither true nor false before they are tested but that, rather, it is the process of verification that makes them true: > > > truth and falsity are properties only of that subject-matter which is > the end, the close, of the inquiry by means of which it is > reached. (1941: 176) > > > Second, Dewey insists that only "judgments"--not "propositions"--are properly viewed as truth-bearers. For Dewey, "propositions" are the proposals and working hypotheses that are used, via a process of inquiry, to generate conclusions and verified judgments. As such, propositions may be more or less relevant to the inquiry at hand but they are not, strictly speaking true or false (1941: 176). Rather, truth and falsity are reserved for "judgments" or "the settled outcome of inquiry" (1941: 175; 1938 [2008: 124]; Burke 1994): reserved for claims, in other words, that are warrantedly assertible. Third, Dewey continues to argue that this pragmatic approach to truth is "the only one entitled to be called a correspondence theory of truth" (1941: 179) using terms nearly identical to those he used in 1911: > > > My own view takes correspondence in the operational sense...of > answering, as a key answers to conditions imposed by a lock, > or as two correspondents "answer" each other; or, in > general, as a reply is an adequate answer to a question or > criticism--; as, in short, a solution answers the > requirements of a problem. (1941: 178) > > > Thanks to Russell (e.g., 1941: Ch. XXIII) and others, by 1941 Dewey was aware of the problems facing pragmatic accounts of truth. In response, we see him turning to the language of "warranted assertibility", drawing a distinction between "propositions" and "judgments", and grounding the concept of truth (or warranted assertibility) in scientific inquiry (Thayer 1947; Burke 1994). These adjustments were designed to extend, clarify, and improve on Peirce's and James' accounts. Whether they did so is an open question. Certainly many, such as Quine, concluded that Dewey was only sidestepping important questions about truth: that Dewey's strategy was "simply to avoid the truth predicate and limp along with warranted belief" (Quine 2008: 165). Peirce, James, and Dewey were not the only ones to propose or defend a pragmatic theory of truth in the nineteenth and early twentieth centuries. Others, such as F.C.S. Schiller (1864-1937), also put forward pragmatic theories (though Schiller's view, which he called "humanism", also attracted more than its share of critics, arguably for very good reasons). Pragmatic theories of truth also received the attention of prominent critics, including Russell (1909, 1910 [1994]), Moore (1908), and Lovejoy (1908a,b) among others. Several of these criticisms will be considered later; suffice it to say that pragmatic theories of truth soon came under pressure that led to revisions and several successor approaches over the next hundred-plus years. Historically Peirce, James, and Dewey had the greatest influence in setting the parameters for what makes a theory of truth pragmatic--this despite the sometimes significant differences between their respective accounts, and that over time they modified and clarified their positions in response to both criticism and over-enthusiastic praise. While this can make it difficult to pin down a single definition of what, historically, counted as a pragmatic theory of truth, there are some common themes that cut across each of their accounts. First, each account begins from a pragmatic analysis of the meaning of the truth predicate. On the assumption that describing a belief, claim, or judgment as "true" must make some kind of practical difference, each of these accounts attempts to describe what this difference is. Second, each account then connects truth specifically to processes of inquiry: to describe a claim as true is to say that it either has or will stand up to scrutiny. Third, each account rejects correspondence theories of truth as overly abstract, "transcendental", or metaphysical. Or, more accurately, each attempts to redefine correspondence in pragmatic terms, as the agreement between a claim and a predicted outcome. While the exact accounts offered by Peirce, James, and Dewey found few defenders--by the mid-twentieth century pragmatic theories of truth were largely dormant--these themes did set a trajectory for future versions of the pragmatic theory of truth. ## 2. Neo-Pragmatic Theories of Truth Pragmatic theories of truth enjoyed a resurgence in the last decades of the twentieth century. This resurgence was especially visible in debates between Hilary Putnam (1926-2016) and Richard Rorty (1931-2007) though broadly pragmatic ideas were defended by other philosophers as well (Bacon 2012: Ch. 4). (One example is Crispin Wright's superassertibility theory (1992, 2001) which he claims is "as well equipped to express the aspiration for a developed pragmatist conception of truth as any other candidate" (2001: 781) though he does not accept the pragmatist label.) While these "neo-pragmatic" theories of truth sometimes resembled the classical pragmatic accounts of Peirce, James, or Dewey, they also differed significantly, often by framing the concept of truth in explicitly epistemic terms such as assertibility or by drawing on intervening philosophical developments. At the outset, neo-pragmatism was motivated by a renewed dissatisfaction with correspondence theories of truth and the metaphysical frameworks supporting them. Some neo-pragmatic theories of truth grew out of a rejection of metaphysical realism (e.g., Putnam 1981; for background see Khlentzos 2016). If metaphysical realism cannot be supported then this undermines a necessary condition for the correspondence theory of truth: namely, that there be a mind-independent reality to which propositions correspond. Other neo-pragmatic approaches emerged from a rejection of representationalism: if knowledge is not the mind representing objective reality--if we cannot make clear sense of how the mind could be a "mirror of nature" to use Rorty's (1979) term--then we are also well-advised to give up thinking of truth in realist, correspondence terms. Despite these similar starting points, neo-pragmatic theories took several different and evolving forms over the final decades of the twentieth century. At one extreme some neo-pragmatic theories of truth seemed to endorse relativism about truth (whether and in what sense they did remains a point of contention). This view was closely associated with influential work by Richard Rorty (1982, 1991a,b). The rejection of representationalism and the correspondence theory of truth could lead to the conclusion that inquiry is best viewed as aiming at agreement or "solidarity", not knowledge or truth as these terms are traditionally understood. This had the radical consequence of suggesting that truth is no more than "what our peers will, ceteris paribus, let us get away with saying" (Rorty 1979: 176; Rorty [2010a: 45] admits this phrase is provocative) or just "an expression of commendation" (Rorty 1991a: 23). Not surprisingly, many found this position deeply problematic since it appears to relativize truth to whatever one's audience will accept (Baghramian 2004: 147). A related concern is that this position also seems to conflate truth with justification, suggesting that if a claim meets contextual standards of acceptability then it also counts as true (Gutting 2003). Rorty for one often admitted as much, noting that he tended to "swing back and forth between trying to reduce truth to justification and propounding some form of minimalism about truth" (1998: 21). A possible response to the accusation of relativism is to claim that this neo-pragmatic approach does not aim to be a full-fledged theory of truth. Perhaps truth is actually a rather light-weight concept and does not need the heavy metaphysical lifting implied by putting forward a "theory". If the goal is not to describe what truth is but rather to describe how "truth" is used, then these uses are fairly straightforward: among other things, to make generalizations ("everything you said is true"), to commend ("so true!"), and to caution ("what you said is justified, but it might not be true") (Rorty 1998: 22; 2000: 4). None of these uses requires that we embark on a possibly fruitless hunt for the conditions that make a proposition true, or for a proper definition or theory of truth. If truth is "indefinable" (Rorty 2010b: 391) then Rorty's approach should not be described as a definition or theory of truth, relativist or otherwise. This approach differs in some noteworthy ways from earlier pragmatic accounts of truth. For one thing it is able to draw on, and draw parallels with, a range of well-developed non-correspondence theories of truth that begin (and sometimes end) by stressing the fundamental equivalence of "S is p" and "'S is p' is true". These theories, including disquotationalism, deflationism, and minimalism, simply were not available to earlier pragmatists (though Peirce does at times discuss the underlying notions). Furthermore, while Peirce and Dewey, for example, were proponents of scientific inquiry and scientific processes of verification, on this neo-pragmatic approach science is no more objective or rational than other disciplines: as Rorty put it, "the only sense in which science is exemplary is that it is a model of human solidarity" (1991b: 39). Finally, on this approach Peirce, James, and Dewey simply did not go far enough: they failed to recognize the radical implications of their accounts of truth, or else failed to convey these implications adequately. In turn much of the critical response to this kind of neo-pragmatism is that it goes too far by treating truth merely as a sign of commendation (plus a few other monor functions). In other words, this type of neo-pragmatism can be accused of going to unpragmatic extremes (e.g., Haack 1998; also the exchange in Rorty & Price 2010). A less extreme version of neo-pragmatism attempts to preserve truth's objectivity and independence while still rejecting metaphysical realism. This version was most closely associated with Hilary Putnam, though Putnam's views changed over time (see Hildebrand 2003 for an overview of Putnam's evolution). While this approach frames truth in epistemic terms--primarily in terms of justification and verification--it amplifies these terms to ensure that truth is more than mere consensus. For example, this approach might identify "being true with being warrantedly assertible under ideal conditions" (Putnam 2012b: 220). More specifically, it might demand "that truth is independent of justification here and now, but not independent of all justification" (Putnam 1981: 56). Rather than play up assertibility before one's peers or contemporaries, this neo-pragmatic approach frames truth in terms of ideal warranted assertibility: namely, warranted assertibility in the long run and before all audiences, or at least before all well-informed audiences. Not only does this sound much less relativist but it also bears a strong resemblance to Peirce's and Dewey's accounts (though Putnam, for one, resisted the comparison: "my admiration for the classical pragmatists does not extend to any of the different theories of truth that Peirce, James, and Dewey advanced" [2012c: 70]). To repeat, this neo-pragmatic approach is designed to avoid the problems facing correspondence theories of truth while still preserving truth's objectivity. In the 1980s this view was associated with Putnam's broader program of "internal realism": the idea that "what objects does the world consist of? is a question that it only makes sense to ask within a theory or description" (Putnam 1981: 49, emphasis in original). Internal realism was designed as an alternative to metaphysical realism that dispensed with achieving an external "God's Eye Point of View" while still preserving truth's objectivity, albeit internal to a given theory. (For additional criticisms of metaphysical realism see Khlentzos 2016.) In the mid-1990s Putnam's views shifted toward what he called "natural realism" (1999; for a critical discussion of Putnam's changing views see Wright 2000). This shift came about in part because of problems with defining truth in epistemic terms such as ideal warranted assertibility. One problem is that it is difficult to see how one can verify either what these ideal conditions are or whether they have been met: either one might attempt to do so by taking an external "god's eye view", which would be inconsistent with internal realism, or one might come to this determination from within one's current theory, which would be circular and relativistic. (As Putnam put it, "to talk of epistemically 'ideal' connections must either be understood outside the framework of internal realism or it too must be understood in a solipsistic manner " (2012d: 79-80).) Since neither option seems promising this does not bode well for internal realism or for any account of truth closely associated with it. If internal realism cannot be sustained then a possible fallback position is "natural realism"--the view "that the objects of (normal 'veridical') perception are 'external' things, and, more generally, aspects of 'external' reality" (Putnam 1999: 10)--which leads to a reconciliation of sorts with the correspondence theory of truth. A natural realism suggests "that true empirical statements correspond to states of affairs that actually obtain" (Putnam 2012a: 97), though this does not commit one to a correspondence theory of truth across the board. In other words, natural realism leaves open the possibility that not all true statements "correspond" to a state of affairs, and even those that do (such as empirical statements) do not always correspond in the same way (Putnam 2012c: 68-69; 2012a: 98). While not a ringing endorsement of the correspondence theory of truth, at least as traditionally understood, this neo-pragmatic approach is not a flat-out rejection either. Viewing truth in terms of ideal warranted assertibility has obvious pragmatic overtones of Peirce and Dewey. In contrast, viewing truth in terms of a commitment to natural realism is not so clearly pragmatic though some parallels still exist. Because natural realism allows for different types of truth-conditions--some but not all statements are true in virtue of correspondence--it is compatible with the truth-aptness of normative discourse: just because ethical statements, for example, do not correspond in an obvious way to ethical state of affairs is no reason to deny that they can be true (Putnam 2002). In addition, like earlier pragmatic theories of truth, this neo-pragmatic approach redefines correspondence: in this case, by taking a pluralist approach to the correspondence relation itself (Goodman 2013; see also Howat 2021 and Shields (forthcoming) for recent attempts to show the compatibility of pragmatic and correspondence theories of truth). These two approaches--one tending toward relativism, the other tending toward realism--represented the two main currents in late twentieth century neo-pragmatism. Both approaches, at least initially, framed truth in terms of justification, verification, or assertibility, reflecting a debt to the earlier accounts of Peirce, James, and Dewey. Subsequently they evolved in opposite directions. The first approach, associated with Rorty, flirts with relativism and implies that truth is not the important philosophical concept it has long been taken to be. Here, to take a neo-pragmatic stance toward truth is to recognize the relatively mundane functions this concept plays: to generalize, to commend, to caution and not much else. To ask for more, to ask for something "beyond the here and now", only commits us to "the banal thought that we might be wrong" (Rorty 2010a: 45). The second neo-pragmatic approach, associated with Putnam, attempts to preserve truth's objectivity and the important role it plays across scientific, mathematical, ethical, and political discourse. This could mean simply "that truth is independent of justification here and now" or "that to call a statement of any kind...true is to say that it has the sort of correctness appropriate to the kind of statement it is" (2012a: 97-98). On this account truth points to standards of correctness more rigorous than simply what our peers will let us get away with saying. ## 3. Truth as a Norm of Inquiry and Assertion More recently--since roughly the turn of the twenty-first century--pragmatic theories of truth have focused on truth's role as a norm of assertion or inquiry. These theories are sometimes referred to as "new pragmatic" theories to distinguish them from both classical and neo-pragmatic accounts (Misak 2007b; Legg and Hookway 2021). Like neo-pragmatic accounts, these theories often build on, or react to, positions besides the correspondence theory: for example, deflationary, minimal, and pluralistic theories of truth. Unlike some of the neo-pragmatic accounts discussed above, these theories give relativism a wide berth, avoid defining truth in terms of concepts such as warranted assertibility, and treat correspondence theories of truth with deep suspicion. On these accounts truth plays a unique and necessary role in assertoric discourse (Price 1998, 2003, 2011; Misak 2000, 2007a, 2015): without the concept of truth there would be no difference between making assertions and, to use Frank Ramsey's nice phrase, "comparing notes" (1925 [1990: 247]). Instead, truth provides the "convenient friction" that "makes our individual opinions engage with one another" (Price 2003: 169) and "is internally related to inquiry, reasons, and evidence" (Misak 2000: 73). Like all pragmatic theories of truth, these "new" pragmatic accounts focus on the use and function of truth. However, while classical pragmatists were responding primarily to the correspondence theory of truth, new pragmatic theories also respond to contemporary disquotational, deflationary, and minimal theories of truth (Misak 1998, 2007a). As a result, new pragmatic accounts aim to show that there is more to truth than its disquotational and generalizing function (for a dissenting view see Freedman 2006). Specifically, this "more" is that the concept of truth also functions as a norm that places clear expectations on speakers and their assertions. In asserting something to be true, speakers take on an obligation to specify the consequences of their assertion, to consider how their assertions can be verified, and to offer reasons in support of their claims: > > > once we see that truth and assertion are intimately > connected--once we see that to assert that p is true is > to assert p--we can and must look to our practices of > assertion and to the commitments incurred in them so as to say > something more substantial about truth. (Misak 2007a: 70) > > > This would mean that truth is not just a goal of inquiry, as Dewey claimed, but actually a norm of inquiry that sets expectations for how inquirers conduct themselves. More specifically, without the norm of truth assertoric discourse would be degraded nearly beyond recognition. Without the norm of truth, speakers could be held accountable only for either insincerely asserting things they don't themselves believe (thus violating the norm of "subjective assertibility") or for asserting things they don't have enough evidence for (thus violating the norm of "personal warranted assertibility") (Price 2003: 173-174). The norm of truth is a condition for genuine disagreement between people who speak sincerely and with, from their own perspective, good enough reasons. It provides the "friction" we need to treat disagreements as genuinely needing resolution: otherwise, "differences of opinion would simply slide past one another" (Price 2003: 180-181). In sum, the concept of truth plays an essential role in making assertoric discourse possible, ensuring that assertions come with obligations and that conflicting assertions get attention. Without truth, it is no longer clear to what degree assertions would still be assertions, as opposed to impromptu speculations or musings. (Correspondence theories should find little reason to object: they too can recognize that truth functions as a norm. Of course, correspondence theorists will want to add that truth also requires correspondence to reality, a step "new" pragmatists will resist taking.) It is important that this account of truth is not a definition or theory of truth, at least in the narrow sense of specifying necessary and sufficient conditions for a proposition being true. (That is, there is no proposal along the lines of "S is true iff..."; though see Brown (2015: 69) for a Deweyan definition of truth and Heney (2015) for a Peircean response.) As opposed to some versions of neo-pragmatism, which viewed truth as "indefinable" in part because of its supposed simplicity and transparency, this approach avoids definitions because the concept of truth is implicated in a complex range of assertoric practices. Instead, this approach offers something closer to a "pragmatic elucidation" of truth that gives "an account of the role the concept plays in practical endeavors" (Misak 2007a: 68; see also Wiggins 2002: 317). The proposal to treat truth as a norm of inquiry and assertion can be traced back to both classical and neo-pragmatist accounts. In one respect, this account can be viewed as adding on to neo-pragmatic theories that reduce truth to justification or "personal warranted assertibility". In this respect, these newer pragmatic accounts are a response to the problems facing neo-pragmatism. In another respect, new pragmatic accounts can be seen as a return to the insights of classical pragmatists updated for a contemporary audience. For example, while Peirce wrote of beliefs being "fated" to be agreed upon at the "ideal limit" of inquiry--conditions that to critics sounded metaphysical and unverifiable--a better approach is to treat true beliefs as those "that would withstand doubt, were we to inquire as far as we fruitfully could on the matter" (Misak 2000: 49). On this account, to say that a belief is true is shorthand for saying that it "gets thing right" and "stands up and would continue to stand up to reasons and evidence" (Misak 2015: 263, 265). This pragmatic elucidation of the concept of truth thus attempts to capture both what speakers say and what they do when they describe a claim as true. In a narrow sense the meaning of truth--what speakers are saying when they use this word--is that true beliefs are indefeasible. However, in a broader sense the meaning of truth is also what speakers are doing when they use this word, with the proposal here that truth functions as a norm that is constitutive of assertoric discourse. As we have seen, pragmatic accounts of truth focus on the function the concept plays: specifically, the practical difference made by having and using the concept of truth. Early pragmatic accounts tended to analyze this function in terms of the practical implications of labeling a belief as true: depending on the version, to say that a belief is true is to signal one's confidence, or that the belief is widely accepted, or that it has been scientifically verified, or that it would be assertible under ideal circumstances, among other possible implications. These earlier accounts focus on the function of truth in conversational contexts or in the context of ongoing inquiries. The newer pragmatic theories discussed in this section take a broader approach to truth's function, addressing its role not just in conversations and inquiries but in making certain kinds of conversations and inquiries possible in the first place. By viewing truth as a norm of assertion and inquiry, these more recent pragmatic theories make the function of truth independent of what individual speakers might imply in specific contexts. Truth is not just what is assertible or verifiable (under either ideal or non-ideal circumstances), but sets objective expectations for making assertions and engaging in inquiry. Unlike neo-pragmatists such as Rorty and Putnam, new pragmatists such as Misak and Price argue that truth plays a role entirely distinct from justification or warranted assertibility. This means that, without the concept of truth and the norm it represents, assertoric discourse (and inquiry in general) would dwindle into mere "comparing notes". ## 4. Common Features Pragmatic theories of truth have evolved to where a variety of different approaches are described as "pragmatic". These theories often disagree significantly with each other, making it difficult either to define pragmatic theories of truth in a simple and straightforward manner or to specify the necessary conditions that a pragmatic theory of truth must meet. As a result, one way to clarify what makes a theory of truth pragmatic is to say something about what pragmatic theories of truth are not. Given that pragmatic theories of truth have often been put forward in contrast to prevailing correspondence and other "substantive" theories of truth (Wyatt & Lynch, 2016), this suggests a common commitment shared by the pragmatic theories described above. One way to differentiate pragmatic accounts from other theories of truth is to distinguish the several questions that have historically guided discussions of truth. While some have used decision trees to categorize different theories of truth (Lynch 2001a; Kunne 2003), or have proposed family trees showing relations of influence and affinity (Haack 1978), another approach is to distinguish separate "projects" that examine different dimensions of the concept of truth (Kirkham 1992). (These projects also break into distinct subprojects; for a similar approach see Frapolli 1996.) On this last approach the first, "metaphysical", project aims to identify the necessary and sufficient conditions for "what it is for a statement...to be true" (Kirkham 1992: 20; Wyatt & Lynch call this the "essence project" [2016: 324]). This project often takes the form of identifying what makes a statement true: e.g., correspondence to reality, or coherence with other beliefs, or the existence of a particular state of affairs. A second, "justification", project attempts to specify "some characteristic, possessed by most true statements...by reference to which the probable truth or falsity of the statement can be judged" (Kirkham 1992: 20). This often takes the form of giving a criterion of truth that can be used to determine whether a given statement is true. Finally, the "speech-act" project addresses the question of "what are we doing when we make utterances" that "ascribe truth to some statement?" (Kirkham 1992: 28). Unfortunately, truth-theorists have not always been clear on which project they are pursuing, which can lead to confusion about what counts as a successful or complete theory of truth. It can also lead to truth-theorists talking past each other when they are pursuing distinct projects with different standards and criteria of success. In these terms, pragmatic theories of truth are best viewed as pursuing the speech-act and justification projects. As noted above, pragmatic accounts of truth have often focused on how the concept of truth is used and what speakers are doing when describing statements as true: depending on the version, speakers may be commending a statement, signaling its scientific reliability, or committing themselves to giving reasons in its support. Likewise, pragmatic theories often focus on the criteria by which truth can be judged: again, depending on the version, this may involve linking truth to verifiability, assertibility, usefulness, or long-term durability. With regard to the speech-act and justification projects pragmatic theories of truth seem to be on solid ground, offering plausible proposals for addressing these projects. They are on much less solid ground when viewed as addressing the metaphysical project (Capps 2020). As we will see, it is difficult to defend the idea, for example, that either utility, verifiability, or widespread acceptance are necessary and sufficient conditions for truth or are what make a statement true (though, to be clear, few pragmatists have defended their positions in these exact terms). This would suggest that the opposition between pragmatic and correspondence theories of truth is partly a result of their pursuing different projects. From a pragmatic perspective, the problem with the correspondence theory is its pursuit of the metaphysical project that, as its name suggests, invites metaphysical speculation about the conditions which make sentences true--speculation that can distract from more central questions of what makes true beliefs valuable, how the truth predicate is used, and how true beliefs are best recognized and acquired. (Pragmatic theories of truth are not alone in raising these concerns (David 2022).) From the standpoint of correspondence theories and other accounts that pursue the metaphysical project, pragmatic theories will likely seem incomplete, sidestepping the most important questions (Howat 2014). But from the standpoint of pragmatic theories, projects that pursue or prioritize the metaphysical project are deeply misguided and misleading. This supports the following truism: a common feature of pragmatic theories of truth is that they focus on the practical function that the concept of truth plays. Thus, whether truth is a norm of inquiry (Misak), a way of signaling widespread acceptance (Rorty), stands for future dependability (Peirce), or designates the product of a process of inquiry (Dewey), among other things, pragmatic theories shed light on the concept of truth by examining the practices through which solutions to problems are framed, tested, asserted, and defended--and, ultimately, come to be called true. Pragmatic theories of truth can thus be viewed as making contributions to the speech-act and justification projects by focusing especially on the practices people engage in when they solve problems, make assertions, and conduct scientific inquiry. (For another example, Chang has argued that claims are true "to the extent that there are operationally coherent activities that can be performed by relying on it" (2022: 167).) Of course, even though pragmatic theories of truth largely agree on which questions to address and in what order, this does not mean that they agree on the answers to these questions, or on how to best formulate the meaning and function of truth. Another common commitment of pragmatic theories of truth--besides prioritizing the speech-act and justification projects--is that they do not restrict truth to certain topics or types of inquiry. That is, regardless of whether the topic is descriptive or normative, scientific or ethical, pragmatists tend to view it as an opportunity for genuine inquiry that incorporates truth-apt assertions. The truth-aptness of ethical and normative statements is a notable feature across a range of pragmatic approaches, including Peirce's (at least in some of his moods, e.g., 1901b [1958: 8.158]), Dewey's theory of valuation (1939), Putnam's questioning of the fact-value dichotomy (2002), and Misak's claim that "moral beliefs must be in principle responsive to evidence and argument" (2000: 94; for a dissenting view see Frega 2013). This broadly cognitivist attitude--that normative statements are truth-apt--is related to how pragmatic theories of truth de-emphasize the metaphysical project. As a result, from a pragmatic standpoint one of the problems with the correspondence theory of truth is that it can undermine the truth-aptness of normative claims. If, as the correspondence theory proposes, a necessary condition for the truth of a normative claim is the existence of a normative fact to which it corresponds, and if the existence of normative facts is difficult to account for (normative facts seem ontologically distinct from garden-variety physical facts), then this does not bode well for the truth-aptness of normative claims or the point of posing, and inquiring into, normative questions (Lynch 2009). If the correspondence theory of truth leads to skepticism about normative inquiry, then this is all the more reason, according to pragmatists, to sidestep the metaphysical project in favor of the speech-act and justification projects. As we have seen, pragmatic theories of truth take a variety of different forms. Despite these differences, and despite often being averse to being called a "theory", pragmatic theories of truth do share some common features. To begin with, and unlike many theories of truth, these theories focus on the pragmatics of truth-talk: that is, they focus on how truth is used as an essential step toward an adequate understanding of the concept of truth (indeed, this comes close to being an oxymoron). More specifically, pragmatic theories look to how truth is used in epistemic contexts where people make assertions, conduct inquiries, solve problems, and act on their beliefs. By prioritizing the speech-act and justification projects, pragmatic theories of truth attempt to ground the concept of truth in epistemic practices as opposed to the abstract relations between truth-bearers (such as propositions or statements) and truth-makers (such as states of affairs) appealed to by correspondence theories (MacBride 2022). Pragmatic theories also recognize that truth can play a fundamental role in shaping inquiry and assertoric discourse--for example, by functioning as a norm of these practices--even when it is not explicitly mentioned. In this respect pragmatic theories are less austere than deflationary theories which limit the use of truth to its generalizing and disquotational roles. And, finally, pragmatic theories of truth draw no limits, at least at the outset, to the types of statements, topics, and inquiries where truth may play a practical role. If it turns out that a given topic is not truth-apt, this is something that should be discovered as a characteristic of that subject matter, not something determined by having chosen one theory of truth or another (Capps 2017). ## 5. Critical Assessments Pragmatic theories of truth have faced several objections since first being proposed. Some of these objections can be rather narrow, challenging a specific pragmatic account but not pragmatic theories in general (this is the case with objections raised by other pragmatic accounts). This section will look at more general objections: either objections that are especially common and persistent, or objections that pose a challenge to the basic assumptions underlying pragmatic theories more broadly. ### 5.1 Three Classic Objections and Responses Some objections are as old as the pragmatic theory of truth itself. The following objections were raised in response to James' account in particular. While James offered his own responses to many of these criticisms (see especially his 1909 [1975]), versions of these objections often apply to other and more recent pragmatic theories of truth (for further discussion see Haack 1976; Tiercelin 2014). One classic and influential line of criticism is that, if the pragmatic theory of truth equates truth with utility, this definition is (obviously!) refuted by the existence of useful but false beliefs, on the one hand, and by the existence of true but useless beliefs on the other (Russell 1910 [1994] and Lovejoy 1908a,b). In short, there seems to be a clear and obvious difference between describing a belief as true and describing it as useful: > > > when we say that a belief is true, the thought we wish to convey is > not the same thought as when we say that the belief furthers our > purposes; thus "true" does not mean "furthering our > purposes". (Russell 1910 [1994: 98]) > > > While this criticism is often aimed especially at James' account of truth, it plausibly carries over to any pragmatic theory. So whether truth is defined in terms of utility, long-term durability or assertibility (etc.), it is still an open question whether a useful or durable or assertible belief is, in fact, really true. In other words, whatever concept a pragmatic theory uses to define truth, there is likely to be a difference between that concept and the concept of truth (e.g., Bacon 2014 questions the connection between truth and indefeasibility). A second and related criticism builds on the first. Perhaps utility, long-term durability, and assertibility (etc.) should be viewed not as definitions but rather as criteria of truth, as yardsticks for distinguishing true beliefs from false ones. This seems initially plausible and might even serve as a reasonable response to the first objection above. Falling back on an earlier distinction, this would mean that appeals to utility, long-term durability, and assertibility (etc.) are best seen as answers to the justification and not the metaphysical project. However, without some account of what truth is, or what the necessary and sufficient conditions for truth are, any attempt to offer criteria of truth is arguably incomplete: we cannot have criteria of truth without first knowing what truth is. If so, then the justification project relies on and presupposes a successful resolution to the metaphysical project, the latter cannot be sidestepped or bracketed, and any theory which attempts to do so will give at best a partial account of truth (Creighton 1908; Stebbing 1914). And a third objection builds on the second. Putting aside the question of whether pragmatic theories of truth adequately address the metaphysical project (or address it at all), there is also a problem with the criteria of truth they propose for addressing the justification project. Pragmatic theories of truth seem committed, in part, to bringing the concept of truth down to earth, to explaining truth in concrete, easily confirmable, terms rather than the abstract, metaphysical correspondence of propositions to truth-makers, for example. The problem is that assessing the usefulness (etc.) of a belief is no more clear-cut than assessing its truth: beliefs may be more or less useful, useful in different ways and for different purposes, or useful in the short- or long-run. Determining whether a belief is really useful is no easier than determining whether it is really true: "it is so often harder to determine whether a belief is useful than whether it is true" (Russell 1910 [1994: 121]; also 1946: 817). Far from making the concept of truth more concrete, and the assessment of beliefs more straightforward, pragmatic theories of truth thus seem to leave the concept as opaque as ever. These three objections have been around long enough that pragmatists have, at various times, proposed a variety of responses. One response to the first objection, that there is a clear difference between utility (etc.) and truth, is to deny that pragmatic approaches are aiming to define the concept of truth in the first place. It has been argued that pragmatic theories are not about finding a word or concept that can substitute for truth but that they are, rather, focused on tracing the implications of using this concept in practical contexts. This is what Misak (2000, 2007a) calls a "pragmatic elucidation". Noting that it is "pointless" to offer a definition of truth, she concludes that "we ought to attempt to get leverage on the concept, or a fix on it, by exploring its connections with practice" (2007a: 69; see also Wiggins 2002). It is even possible that James--the main target of Russell and others--would agree with this response. As with Peirce, it often seems that James' complaint is not with the correspondence theory of truth, per se, as with the assumption that the correspondence theory, by itself, says much interesting or important about the concept of truth. (For charitable interpretations of what James was attempting to say see Ayer 1968, Chisholm 1992, Bybee 1984, Cormier 2001, 2011, Chang 2022, Pihlstrom 2021, and Perkins 1952; for a reading that emphasizes Peirce's commitment to correspondence idioms see Atkins 2010.) This still leaves the second objection: that the metaphysical project of defining truth cannot be avoided by focusing instead on finding the criteria for truth (the "justification project"). To be sure, pragmatic theories of truth have often been framed as providing criteria for distinguishing true from false beliefs. The distinction between offering a definition as opposed to offering criteria would suggest that criteria are separate from, and largely inferior to, a definition of truth. However, one might question the underlying distinction: as Haack (1976) argues, > > > the pragmatists' view of meaning is such that a dichotomy > between definitions and criteria would have been entirely unacceptable > to them. (1976: 236) > > > If meaning is related to use (as pragmatists generally claim) then explaining how a concept is used, and specifying criteria for recognizing that concept, may provide all one can reasonably expect from a theory of truth. Deflationists have often made a similar point though, as noted above, pragmatists tend to find deflationary accounts excessively austere. Even so, there is still the issue that pragmatic criteria of truth (whatever they are) do not provide useful insight into the concept of truth. If this concern is valid, then pragmatic criteria, ironically, fail the pragmatic test of making a difference to our understanding of truth. This objection has some merit: for example, if a pragmatic criterion of truth is that true beliefs will stand up to indefinite inquiry then, while it is possible to have true beliefs, "we are never in a position to judge whether a belief is true or not" (Misak 2000: 57). In that case it is not clear what good it serves to have a pragmatic criterion of truth. Pragmatic theories of truth might try to sidestep this objection by stressing their commitment to both the justification and the speech-act project. While pragmatic approaches to the justification project spell out what truth means in conversational contexts--to call a statement true is to cite its usefulness, durability, etc.--pragmatic approaches to the speech-act project point to what speakers do in using the concept of truth. This has the benefit of showing how the concept of truth--operating as a norm of assertion, say--makes a real difference to our understanding of the conditions on assertoric discourse. Pragmatic theories of truth are, as a result, wise to pursue both the justification and the speech-act projects. By itself, pragmatic approaches to the justification project are likely to disappoint. These classic objections to the pragmatic theory of truth raise several important points. For one thing, they make it clear that pragmatic theories of truth, or at least some historically prominent versions of it, do a poor job if viewed as providing a strict definition of truth. As Russell and others noted, defining truth in terms of utility or similar terms is open to obvious counter-examples. This does not bode well for pragmatic attempts to address the metaphysical project. As a result, pragmatic theories of truth have evolved often by focusing on the justification and speech-act projects instead. This is not to say that each of the above objections have been met. It is still an open question whether the metaphysical project can be avoided as many pragmatic theories attempt to do (e.g., Fox 2008 argues that epistemic accounts such as Putnam's fail to explain the value of truth as well as more traditional approaches do). It is also an open question whether, as they evolve in response to these objections, pragmatic theories of truth will invite new lines of criticism. ### 5.2 The Fundamental Objection One long-standing and still ongoing objection is that pragmatic theories of truth are anti-realist and, as such, violate basic intuitions about the nature and meaning of truth: call this "the fundamental objection". The source of this objection rests with the tendency of pragmatic theories of truth to treat truth epistemically, by focusing on verifiability, assertibility, and other related concepts. Some (see, e.g., Schmitt 1995; Nolt 2008) have argued that, by linking truth with verifiability or assertibility, pragmatic theories make truth too subjective and too dependent on our contingent ability to figure things out, as opposed to theories that, for example, appeal to objective facts as truth-makers. Others have argued that pragmatic theories cannot account for what Peirce called buried secrets: statements that would seem to be either true or false despite our inability to figure out which (see de Waal 1999, Howat 2013, and Talisse & Akin 2008 for discussions of this). For similar reasons, some have accused pragmatic theories of denying bivalence (Allen-Hermanson 2001). Whatever form the objection takes, it raises a common concern: that pragmatic theories of truth are insufficiently realist, failing both to account for truth's objectivity and to distinguish truth from the limitations of actual epistemic practice. What results, accordingly, is not a theory of truth, but rather a theory of justification, warranted assertibility, or some other epistemic concept. This objection has persisted despite inspiring a range of responses. At one extreme some, such as Rorty, have largely conceded the point while attempting to defuse its force. As noted earlier, Rorty grants that truth is not objective in the traditional sense while also attempting to undercut the very distinction between objectivity and relativism. Others, such as Putnam, have argued against metaphysical realist intuitions (such as "the God's Eye view" 1981: 55), while defending the idea of a more human-scale objectivity: "objectivity and rationality humanly speaking are what we have; they are better than nothing" (1981: 55). Another response is to claim that pragmatic accounts of truth are fully compatible with realism; any impression to the contrary is a result of confusing pragmatic "elucidations" of truth with more typical "definitions". For example Peirce's steadfast commitment to realism is perfectly compatible with his attempting to describe truth in terms of its practical role: hence, his notion of truth > > > is the ordinary notion, but he insists on this notion's being > philosophically characterized from the viewpoint of the practical > first order investigator. (Hookway 2002: 319; see also Hookway 2012 > and Legg 2014) > > > Even James claimed "my account of truth is realistic" (1909 [1975: 117]). Likewise, Chang argues that this objection "ignores the realist dimension of pragmatism, in terms of how pragmatism demands that our ideas answer to experience, and to realities" (2022: 203). Finally, others attempt to undercut the distinction between realism and antirealism though without making concessions to antirealism. Hildebrand argues for embracing a "practical starting point" (Hildebrand 2003: 185) as a way of going "beyond" the realism-antirealism debate (see also Fine 2007). Similarly, Price, while admitting that his theory might seem "fictionalist" about truth, argues that its bona fides are "impeccably pragmatist" (2003: 189) and, in fact, "deprive both sides of the realism-antirealism debate of conceptual resources on which the debate seems to depend" (2003: 188; but see Atkin 2015 for some caveats and Lynch 2015 for a pluralist amendment). Da Costa and French (2003) offer a formal account of pragmatic truth that, they argue, can benefit both sides of the realism-antirealism debate (though they themselves prefer structural realism). We find, in other words, an assortment of replies that run the gamut from embracing antirealism to defending realism to attempting to undermine the realist-antirealist distinction itself. Evidently, there is no consensus among pragmatic theories of truth as to the best line of response against this objection. In a way, this should be no surprise: the objection boils down to the charge that pragmatic theories of truth are too epistemic, when it is precisely their commitment to epistemic concepts that characterizes pragmatic theories of truth. Responding to this objection may involve concessions and qualifications that compromise the pragmatic nature of these approaches. Or responding may mean showing how pragmatic accounts have certain practical benefits--but these benefits as well as their relative importance are themselves contentious topics. As a result, we should not expect this objection to be easily resolvable, if it can be resolved at all. Despite being the target of significant criticism from nearly the moment of its birth, the pragmatic theory of truth has managed to survive and, at times, even flourish for over a century. Because the pragmatic theory of truth has come in several different versions, and because these versions often diverge significantly, it can be difficult to pin down and assess generally. Adding to the possible confusion, not all those identified as pragmatists have embraced a pragmatic theory of truth (e.g., Brandom 2011), while similar theories have been held by non-pragmatists (e.g., Dummett 1959; Wright 1992). Viewed more positively, pragmatic theories have evolved and matured to become more refined and, perhaps, more plausible over time. With the benefit of hind-sight we can see how pragmatic theories of truth have stayed focused on the practical function that the concept of truth plays: first, the role truth plays within inquiry and assertoric discourse by, for example, signaling those statements that are especially useful, well-verified, durable, or indefeasible and, second, the role truth plays in shaping inquiry and assertoric discourse by providing a necessary goal or norm. (While pragmatic theories agree on the importance of focusing on truth's practical function, they often disagree over what this practical function is.) The pragmatic theory of truth began with Peirce raising the question of truth's "practical bearings". It is also possible to ask this question of the pragmatic theory of truth itself: what difference does this theory make? Or to put it in James' terms, what is its "cash value"? One answer is that, by focusing on the practical function of the concept of truth, pragmatic theories highlight how this concept makes certain kinds of inquiry and discourse possible. In contrast, as Lynch (2009) notes, some accounts of truth make it difficult to see how certain claims are truth-apt: > > > consider propositions like two and two are four or > torture is wrong. Under the assumption that truth is always > and everywhere causal correspondence, it is a vexing question how > these true thoughts can be true. (Lynch 2009: 34, emphasis in > original) > > > If that is so, then pragmatic theories have the advantage of preserving the possibility and importance of various types of inquiry and discourse. While this does not guarantee that inquiry will always reach a satisfying or definite conclusion, this does suggest that pragmatic theories of truth do make a difference: in the spirit of Peirce's "first rule of reason", they "do not block the way of inquiry" (1898 [1992: 178]).
truth-revision	## 1. Semiformal introduction Let's take a closer look at the sentence (1), given above: (1) (1) is not true. It will be useful to make the paradoxical reasoning explicit. First, suppose that: (2) (1) is not true. It seems an intuitive principle concerning truth that, for any sentence $p$, we have the so-called T-biconditional (3) '$p$' is true iff $p$. (Here we are using 'iff' as an abbreviation for 'if and only if'.) In particular, we should have (4) '(1) is not true' is true iff (1) is not true. Thus, from (2) and (4), we get (5) '(1) is not true' is true. Then we can apply the identity, (6) (1) = '(1) is not true.' to conclude that (1) is true. This all shows that if (1) is not true, then (1) is true. Similarly, we can also argue that if (1) is true then (1) is not true. So (1) seems to be both true and not true: hence the paradox. As stated above, the three-valued approach to the paradox takes the liar sentence, (1), to be neither true nor false. Exactly how, or even whether, this move blocks the above reasoning is a matter for debate. The RTT is not designed to block reasoning of the above kind, but to model it -- or most of it.[2] As stated above, the central idea is the idea of a revision process: a process by which we revise hypotheses about the truth-value of one or more sentences. Consider the reasoning regarding the liar sentence, (1) above. Suppose that we hypothesize that (1) is not true. Then, with an application of the relevant T-biconditional, we might revise our hypothesis as follows: \| \| \| \| --- \| --- \| \| Hypothesis: \| (1) is not true. \| \| T-biconditional: \| '(1) is not true' is true iff (1) is not true. \| \| Therefore: \| '(1) is not true' is true. \| \| Known identity: \| (1) = '(1) is not true'. \| \| Conclusion: \| (1) is true. \| \| New revised hypothesis: \| (1) is true. \| We could continue the revision process, by revising our hypothesis once again, as follows: \| \| \| \| --- \| --- \| \| New hypothesis: \| (1) is true. \| \| T-biconditional: \| '(1) is not true' is true iff (1) is not true. \| \| Therefore: \| '(1) is not true' is not true. \| \| Known identity: \| (1) = '(1) is not true'. \| \| Conclusion: \| (1) is not true. \| \| New new revised hypothesis: \| (1) is not true. \| As the revision process continues, we flip back and forth between taking the liar sentence to be true and not true. Example 1.1 It is worth seeing how this kind of revision reasoning works in a case with several interconnected sentences. Let's apply the revision idea to the following three sentences: (7) (8) is true or (9) is true. (8) (7) is true. (9) (7) is not true. Informally, we might reason as follows. Either (7) is true or (7) is not true. Thus, either (8) is true or (9) is true. Thus, (7) is true. Thus (8) is true and (9) is not true, and (7) is still true. Iterating the process once again, we get (8) is true, (9) is not true, and (7) is true. More formally, consider any initial hypothesis, $h\_0$, about the truth values of (7), (8) and (9). Either $h\_0$ says that (7) is true or $h\_0$ says that (7) is not true. In either case, we can use the T-biconditional to generate our revised hypothesis $h\_1$: if $h\_0$ says that (7) is true, then $h\_1$ says that '(7) is true' is true, i.e. that (8) is true; and if $h\_0$ says that (7) is not true, then $h\_1$ says that '(7) is not true' is true, i.e. that (9) is true. So $h\_1$ says that either (8) is true or (9) is true. So $h\_2$ says that '(8) is true or (9) is true' is true. In other words, $h\_2$ says that (7) is true. So no matter what hypothesis $h\_0$ we start with, two iterations of the revision process lead to a hypothesis that (7) is true. Similarly, three or more iterations of the revision process, lead to the hypothesis that (7) is true, (8) is true and (9) is not true -- regardless of our initial hypothesis. In Section 3, we will reconsider this example in a more formal context. One thing to note is that, in Example 1.1, the revision process yields stable truth values for all three sentences. The notion of a sentence stably true in all revision sequences will be a central notion for the RTT. The revision-theoretic treatment contrasts, in this case, with the three-valued approach: on most ways of implementing the three-valued idea, all three sentences, (7), (8) and (9), turn out to be neither true nor false.[3] In this case, the RTT arguably better captures the correct informal reasoning than does the three-valued approach: the RTT assigns to the sentences (7), (8) and (9) the truth-values that were assigned to them by the informal reasoning given at the beginning of the example. ## 2. Framing the problem ### 2.1 Truth languages The goal of the RTT is not to give a paradox-free account of truth. Rather, the goal of the RTT is to give an account of our often unstable and often paradoxical reasoning about truth. RTT seeks, more specifically, to give a two-valued account that assigns stable classical truth values to sentences when intuitive reasoning would assign stable classical truth values. We will present a formal semantics for a formal language: we want that language to have both a truth predicate and the resources to refer to its own sentences. Let us consider a first-order language $L$, with connective &, $\vee$, and $\neg$, quantifiers $\forall$ and $\exists$, the equals sign =, variables, and some stock of names, function symbols and relation symbols. We will say that $L$ is a truth language, if it has a distinguished predicate $\boldsymbol{T}$ and quotation marks ' and ', which will be used to form quote names: if $A$ is a sentence of $L$, then '$A$' is a name. Let $\textit{Sent}\_L = \{A : A$ is a sentence of $L\}$. It will be useful to identify the $\boldsymbol{T}$-free fragment of a truth language $L$: the first-order language $L^-$ that has the same names, function symbols and relation symbols as $L$, except the unary predicate $\boldsymbol{T}$. Since $L^-$ has the same names as $L$, including the same quote names, $L^-$ will have a quote name '$A$' for every sentence $A$ of $L$. Thus $\forall x\boldsymbol{T}x$ is not a sentence of $L^-$, but '$\forall x\boldsymbol{T}x$' is a name of $L^-$ and $\forall x(x =$ '$\forall x\boldsymbol{T}x$') is a sentence of $L^-$. ### 2.2 Ground models Other than the truth predicate, we will assume that our language is interpreted classically. More precisely, let a ground model for $L$ be a classical model $M = \langle D,I\rangle$ for $L^-$, the $\boldsymbol{T}$-free fragment of $L$, satisfying the following: 1. $D$ is a nonempty domain of discourse; 2. $I$ is a function assigning 1. to each name of $L$ a member of $D$; 2. to each $n$-ary function symbol of $L$ a function from $D^n$ to $D$; and 3. to each $n$-ary relation symbol, other than $\boldsymbol{T}$, of $L$ a function from $D^n$ to one of the two truth-values in the set $\{\mathbf{t}, \mathbf{f}\}$;[4] 3. $\textit{Sent}\_L$ $\subseteq$ $D$; and 4. $I$('$A$') $= A$ for every $A \in$ $\textit{Sent}\_L$. Clauses (1) and (2) simply specify what it is for $M$ to be a classical model of the $\boldsymbol{T}$-free fragment of $L$. Clauses (3) and (4) ensure that $L$, when interpreted, can talk about its own sentences. Given a ground model, we will consider the prospects of providing a satisfying interpretation of $\boldsymbol{T}$. The most obvious desideratum is that the ground model, expanded to include an interpretation of $\boldsymbol{T}$, satisfy Tarski's T-biconditionals, i.e., the biconditionals of the form \[ \boldsymbol{T}\lsquo A\rsquo \text{ iff } A \] for each $A \in$ $\textit{Sent}\_L$. Some useful terminology: Given a ground model $M$ for $L$ and a name, function symbol or relation symbol $X$, we can think of $I(X)$ as the interpretation or, to borrow a term from Gupta and Belnap, the signification of $X$. Gupta and Belnap characterize an expression's or concept's signification in a world $w$ as "an abstract something that carries all the information about all the expression's [or concept's] extensional relations in $w$." If we want to interpret $\boldsymbol{T}x$ as '$x$ is true', then, given a ground model $M$, we would like to find an appropriate signification, or an appropriate range of significations, for $\boldsymbol{T}$. ### 2.3 Three ground models We might try to assign to $\boldsymbol{T}$ a classical signification, by expanding $M$ to a classical model $M' = \langle D',I'\rangle$ for all of $L$, including $\boldsymbol{T}$. Also recall that we want $M'$ to satisfy the T-biconditionals: for our immediate purposes, let us interpret these classically. Let us say that an expansion $M'$ of a ground model $M$ is Tarskian iff $M'$ is a classical model and all of the T-biconditionals, interpreted classically, are true in $M'$. We would like to expand ground models to Tarskian models. We consider three ground models in order to assess our prospects for doing this. Ground model $M\_1$ Our first ground model is a formalization of Example 1.1, above. Suppose that $L\_1$ contains three non-quote names, $\alpha , \beta$, and $\gamma$, and no predicates other than $\boldsymbol{T}$. Let $M\_1 = \langle D\_1 ,I\_1 \rangle$ be as follows: \[\begin{align} D\_1 &= \textit{Sent}\_{L\_1} \\ I\_1(\alpha) &= \boldsymbol{T}\beta \vee \boldsymbol{T}\gamma \\ I\_1 (\beta) &= \boldsymbol{T}\alpha \\ I\_1 (\gamma) &= \neg \boldsymbol{T}\alpha \end{align}\] Ground model $M\_2$ Suppose that $L\_2$ contains one non-quote name, $\tau$, and no predicates other than $\boldsymbol{T}$. Let $M\_2 = \langle D\_2 ,I\_2 \rangle$ be as follows: \[\begin{align} D\_2 &= \textit{Sent}\_{L\_2} \\ I\_2 (\tau) &= \boldsymbol{T}\tau \end{align}\] Ground model $M\_3$ Suppose that $L\_3$ contains one non-quote name, $\lambda$, and no predicates other than $\boldsymbol{T}$. Let $M\_3 = \langle D\_3 ,I\_3 \rangle$ be as follows: \[\begin{align} D\_3 &= \textit{Sent}\_{L\_3} \\ I\_3 (\lambda) &= \neg \boldsymbol{T}\lambda \end{align}\] Theorem 2.1 (1) $M\_1$ can be expanded to exactly one Tarskian model: in this model, the sentences $(\boldsymbol{T}\beta \vee \boldsymbol{T}\gamma)$ and $\boldsymbol{T}\alpha$ are true, while the sentence $\neg \boldsymbol{T}\alpha$ is false. (2) $M\_2$ can be expanded to exactly two Tarskian models, in one of which the sentence $\boldsymbol{T}\tau$ is true and in the other of which the sentence $\boldsymbol{T}\tau$ is false. (3) $M\_3$ cannot be expanded to a Tarskian model. The proofs of (1) and (2) are beyond the scope of this article, but some remarks are in order. Re (1): The fact that $M\_1$ can be expanded to a Tarskian model is not surprising, given the reasoning in Example 1.1, above: any initial hypothesis about the truth values of the three sentences in question leads, after three iterations of the revision process, to a stable hypothesis that $(\boldsymbol{T}\beta \vee \boldsymbol{T}\gamma)$ and $\boldsymbol{T}\alpha$ are true, while $\neg \boldsymbol{T}\alpha$ is false. The fact that $M\_1$ can be expanded to exactly one Tarskian model needs the so-called Transfer Theorem, Gupta and Belnap 1993, Theorem 2D.4. Remark: In the introductory remarks, above, we claim that there are consistent classical interpreted languages that refer to their own sentences and have their own truth predicates. Clauses (1) of Theorem 2.1 delivers an example. Let $M\_1 '$ be the unique Tarskian expansion of $M\_1$. Then the language $L\_1$, interpreted by $M\_1 '$ is an interpreted language that has its own truth predicate satisfying the T-biconditionals classically understood, obeys the rules of standard classical logic, and has the ability to refer to each of its own sentences. Thus Tarski was not quite right in his view that any language with its own truth predicate would be inconsistent, as long as it obeyed the rules of standard classical logic, and had the ability to refer to its own sentences. Re (2): The only potential problematic self-reference is in the sentence $\boldsymbol{T}\tau,$ the so-called truth teller, which says of itself that it is true. Informal reasoning suggests that the truth teller can consistently be assigned either classical truth value: if you assign it the value $\mathbf{t}$ then no paradox is produced, since the sentence now truly says of itself that it is true; and if you assign it the value $\mathbf{f}$ then no paradox is produced, since the sentence now falsely says of itself that it is true. Theorem 2.1 (2) formalizes this point, i.e., $M\_2$ can be expanded to one Tarskian model in which $\boldsymbol{T}\tau$ is true and one in which $\boldsymbol{T}\tau$ is false. The fact that $M\_2$ can be expanded to exactly two Tarskian models needs the Transfer Theorem, alluded to above. Note that the language $L\_2$, interpreted by either of these expansions, provides another example of an interpreted language that has its own truth predicate satisfying the T-biconditionals classically understood, obeys the rules of standard classical logic, and has the ability to refer to each of its own sentences. Proof of (3). Suppose that $M\_3 ' = \langle D\_3 ,I\_3 '\rangle$ is a classical expansion of $M\_3$ to all of $L\_3$. Since $M\_3 '$ is an expansion of $M\_3, I\_3$ and $I\_3 '$ agree on all the names of $L\_3$. So \[ I\_3 '(\lambda) = I\_3 (\lambda) = \neg \boldsymbol{T}\lambda = I\_3(\lsquo \neg \boldsymbol{T}\lambda \rsquo) = I\_3 '(\lsquo \neg \boldsymbol{T}\lambda\rsquo). \] So the sentences $\boldsymbol{T}\lambda$ and $\boldsymbol{T}$'$\neg \boldsymbol{T}\lambda$' have the same truth value in $M\_3 '$. So the T-biconditional \[ \boldsymbol{T}\lsquo \neg \boldsymbol{T}\lambda\rsquo \equiv \neg \boldsymbol{T}\lambda \] is false in $M\_3 '$. Remark: The language $L\_3$ interpreted by the ground model $M\_3$ formalizes the liar's paradox, with the sentence $\neg \boldsymbol{T}\lambda$ as the offending liar's sentence. Thus, despite Theorem 2.1, Clauses (1) and (2), Clause (3) strongly suggests that in a semantics for languages capable of expressing their own truth concepts, $\boldsymbol{T}$ cannot, in general, have a classical signification; and the 'iff' in the T-biconditionals will not be read as the classical biconditional. We take these suggestions up in Section 4, below. ## 3. Basic notions of the RTT ### 3.1 Revision rules In Section 1, we informally sketched the central thought of the RTT, namely, that we can use the T-biconditionals to generate a revision rule -- a rule for revising a hypothesis about the extension of the truth predicate. Here we will formalize this notion, and work through an example from Section 1. In general, let L be a truth language and $M$ be a ground model for $L$. An hypothesis is a function $h:D \rightarrow \{\mathbf{t}, \mathbf{f}\}$. A hypothesis will in effect be a hypothesized classical interpretation for $\boldsymbol{T}$. Let's work with an example that combines features from the ground models $M\_1$ and $M\_3$. We will state the example formally, but reason in a semiformal way, to transition from one hypothesized extension of $\boldsymbol{T}$ to another. Example 3.1 Suppose that $L$ contains four non-quote names, $\alpha , \beta , \gamma$ and $\lambda$ and no predicates other than $\boldsymbol{T}$. Also suppose that $M = \langle D,I\rangle$ is as follows: \[\begin{align} D &= \textit{Sent}\_L \\ I(\alpha) &= \boldsymbol{T}\beta \vee \boldsymbol{T}\gamma \\ I(\beta) &= \boldsymbol{T}\alpha \\ I(\gamma) &= \neg \boldsymbol{T}\alpha \\ I(\lambda) &= \neg \boldsymbol{T}\lambda \end{align}\] It will be convenient to let \[\begin{align} &A \text{ be the sentence } \boldsymbol{T}\beta \vee \boldsymbol{T}\gamma \\ &B \text{ be the sentence } \boldsymbol{T}\alpha \\ &C \text{ be the sentence } \neg \boldsymbol{T}\alpha \\ &X \text{ be the sentence } \neg \boldsymbol{T}\lambda \end{align}\] Thus: \[\begin{align} D &= \textit{Sent}\_L \\ I(\alpha) &= A \\ I(\beta) &= B \\ I(\gamma) &= C \\ I(\lambda) &= X \end{align}\] Suppose that the hypothesis $h\_0$ hypothesizes that $A$ is false, $B$ is true, $C$ is false and $X$ is false. Thus \[\begin{align} h\_0 (A) &= \mathbf{f} \\ h\_0 (B) &= \mathbf{t} \\ h\_0 (C) &= \mathbf{f} \\ h\_0 (X) &= \mathbf{f} \end{align}\] Now we will engage in some semiformal reasoning, on the basis of hypothesis $h\_0$. Among the four sentences, $A, B, C$ and $X, h\_0$ puts only $B$ in the extension of $\boldsymbol{T}$. Thus, reasoning from $h\_0$, we conclude that: \[\begin{align} \neg \boldsymbol{T}\alpha &\text{ since the referent of } \alpha \text{ is not in the extension of } \boldsymbol{T} \\ \boldsymbol{T}\beta &\text{ since the referent of } \beta \text{ is in the extension of } \boldsymbol{T} \\ \neg \boldsymbol{T}\gamma &\text{ since the referent of } \gamma \text{ is not in the extension of } \boldsymbol{T} \\ \neg \boldsymbol{T}\lambda &\text{ since the referent of } \lambda \text{ is not in the extension of } \boldsymbol{T}. \end{align}\] The T-biconditional for the four sentences $A, B, C$ and $X$ are as follows: \[\begin{align} \tag{T$\_A$} A \text{ is true iff }& \boldsymbol{T}\beta \vee \boldsymbol{T}\gamma \\ \tag{T$\_B$} B \text{ is true iff }& \boldsymbol{T}\alpha \\ \tag{T$\_C$} C \text{ is true iff }& \neg \boldsymbol{T}\alpha \\ \tag{T$\_X$} X \text{ is true iff }& \neg \boldsymbol{T}\lambda \end{align}\] Thus, reasoning from $h\_0$, we conclude that: \[\begin{align} &A \text{ is true} \\ &B \text{ is not true} \\ &C \text{ is true} \\ &X \text{ is true} \end{align}\] This produces our new hypothesis $h\_1$: \[\begin{align} h\_1 (A) &= \mathbf{t} \\ h\_1 (B) &= \mathbf{f} \\ h\_1 (C) &= \mathbf{t} \\ h\_1 (X) &= \mathbf{t} \end{align}\] Let's revise our hypothesis once again. So now we will engage in some semiformal reasoning, on the basis of hypothesis $h\_1$. Hypothesis $h\_1$ puts $A, C$ and $X$, but not $B$, in the extension of the $\boldsymbol{T}$. Thus, reasoning from $h\_1$, we conclude that: \[\begin{align} \boldsymbol{T}\alpha &\text{ since the referent of } \alpha \text{ is not in the extension of } \boldsymbol{T} \\ \neg \boldsymbol{T}\beta &\text{ since the referent of } \beta \text{ is in the extension of } \boldsymbol{T} \\ \boldsymbol{T}\gamma &\text{ since the referent of } \gamma \text{ is not in the extension of } \boldsymbol{T} \\ \boldsymbol{T}\lambda &\text{ since the referent of } \lambda \text{ is not in the extension of } \boldsymbol{T}. \end{align}\] Recall the T-biconditionals for the four sentences $A, B, C$ and $X$, given above. Reasoning from $h\_1$ and these T-biconditionals, we conclude that: \[\begin{align} &A \text{ is true} \\ &B \text{ is true} \\ &C \text{ is not true} \\ &X \text{ is not true} \end{align}\] This produces our new new hypothesis $h\_2$: \[\begin{align} h\_2 (A) &= \mathbf{t} \\ h\_2 (B) &= \mathbf{t} \\ h\_2 (C) &= \mathbf{f} \\ h\_2 (X) &= \mathbf{f} \end{align}\] $\Box$ Let's formalize the semiformal reasoning carried out in Example 3.1. First we hypothesized that certain sentences were, or were not, in the extension of $\boldsymbol{T}$. Consider ordinary classical model theory. Suppose that our language has a predicate $G$ and a name $a$, and that we have a model $M = \langle D,I\rangle$ which places the referent of $a$ inside the extension of $G$: \[ I(G)(I(\alpha)) = \mathbf{t} \] Then we conclude, classically, that the sentence $Ga$ is true in $M$. It will be useful to have some notation for the classical truth value of a sentence $S$ in a classical model $M$. We will write $\textit{Val}\_M (S)$. In this case, $\textit{Val}\_M (Ga) = \mathbf{t}$. In Example 3.1, we did not start with a classical model of the whole language $L$, but only a classical model of the $\boldsymbol{T}$-free fragment of $L$. But then we added a hypothesis, in order to get a classical model of all of $L$. Let's use the notation $M + h$ for the classical model of all of $L$ that you get when you extend $M$ by assigning $\boldsymbol{T}$ an extension via the hypothesis $h$. Once you have assigned an extension to the predicate $\boldsymbol{T}$, you can calculate the truth values of the various sentences of $L$. That is, for each sentence $S$ of $L$, we can calculate \[ \textit{Val}\_{M + h}(S) \] In Example 3.1, we started with hypothesis $h\_0$ as follows: \[\begin{align} h\_0 (A) &= \mathbf{f} \\ h\_0 (B) &= \mathbf{t} \\ h\_0 (C) &= \mathbf{f} \\ h\_0 (X) &= \mathbf{f} \end{align}\] Then we calculated as follows: \[\begin{align} \textit{Val}\_{M+h\_0}(\boldsymbol{T}\alpha) &= \mathbf{f} \\ \textit{Val}\_{M+h\_0}(\boldsymbol{T}\beta) &= \mathbf{t} \\ \textit{Val}\_{M+h\_0}(\boldsymbol{T}\gamma) &= \mathbf{f} \\ \textit{Val}\_{M+h\_0}(\boldsymbol{T}\lambda) &= \mathbf{f} \end{align}\] And then we concluded as follows: \[\begin{align} \textit{Val}\_{M+h\_0}(A) &= \textit{Val}\_{M+h\_0}(\boldsymbol{T}\beta \lor \boldsymbol{T}\gamma) = \mathbf{t} \\ \textit{Val}\_{M+h\_0}(B) &= \textit{Val}\_{M+h\_0}(\boldsymbol{T}\alpha) = \mathbf{f} \\ \textit{Val}\_{M+h\_0}(C) &= \textit{Val}\_{M+h\_0}(\neg\boldsymbol{T}\alpha) = \mathbf{t} \\ \textit{Val}\_{M+h\_0}(X) &= \textit{Val}\_{M+h\_0}(\neg\boldsymbol{T}\lambda) = \mathbf{t} \end{align}\] These conclusions generated our new hypothesis, $h\_1$: \[\begin{align} h\_1 (A) &= \mathbf{t} \\ h\_1 (B) &= \mathbf{f} \\ h\_1 (C) &= \mathbf{t} \\ h\_1 (X) &= \mathbf{t} \end{align}\] Note that, in general, \[ h\_1 (S) = \textit{Val}\_{M+h\_0}(S). \] We are now prepared to define the revision rule given by a ground model $M = \langle D,I\rangle$. In general, given an hypothesis $h$, let $M + h = \langle D,I'\rangle$ be the model of $L$ which agrees with $M$ on the $\boldsymbol{T}$-free fragment of $L$, and which is such that $I'(\boldsymbol{T}) = h$. So $M + h$ is just a classical model for all of $L$. For any model $M + h$ of all of $L$ and any sentence $S$ of $L$, let $\textit{Val}\_{M+h}(S)$ be the ordinary classical truth value of $S$ in $M + h$. Definition 3.2 Suppose that $L$ is a truth language and that $M = \langle D,I\rangle$ is a ground model for $L$. The revision rule, $\tau\_M$, is the function mapping hypotheses to hypotheses, as follows: \[\tau\_M (h)(d) = \begin{cases} \mathbf{t}, \text{ if } d\in D \text{ is a sentence of } L \text{ and } \textit{Val}\_{M+h}(d) = \mathbf{t} \\ \mathbf{f}, \text{ otherwise} \end{cases}\] The 'otherwise' clause tells us that if $d$ is not a sentence of $L$, then, after one application of revision, we stick with the hypothesis that $d$ is not true.[5] Note that, in Example 3.$1, h\_1 = \tau\_M (h\_0)$ and $h\_2 = \tau\_M (h\_1)$. We will often drop the subscripted '$M$' when the context make it clear which ground model is at issue. ### 3.2 Revision sequences Let's pick up Example 3.1 and see what happens when we iterate the application of the revision rule. Example 3.3 (Example 3.1 continued) Recall that $L$ contains four non-quote names, $\alpha , \beta , \gamma$ and $\lambda$ and no predicates other than $\boldsymbol{T}$. Also recall that $M = \langle D,I\rangle$ is as follows: \[\begin{align} D &= \textit{Sent}\_L \\ I(\alpha) &= A = \boldsymbol{T}\beta \vee \boldsymbol{T}\gamma \\ I(\beta) &= B = \boldsymbol{T}\alpha \\ I(\gamma) &= C = \neg \boldsymbol{T}\alpha \\ I(\lambda) &= X = \neg \boldsymbol{T}\lambda \end{align}\] The following table indicates what happens with repeated applications of the revision rule $\tau\_M$ to the hypothesis $h\_0$ from Example 3.1. In this table, we will write $\tau$ instead of $\tau\_M$: \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| $S$ \| $h\_0 (S)$ \| $\tau(h\_0)(S)^{}$ \| $\tau^2 (h\_0)(S)$ \| $\tau^3 (h\_0)(S)$ \| $\tau^4 (h\_0)(S)$ \| $\cdots$ \| \| $A$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| \| $B$ \| $\mathbf{t}$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| \| $C$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\cdots$ \| \| $X$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{f}$ \| $\cdots$ \| So $h\_0$ generates a revision sequence (see Definition 3.7, below). And $A$ and $B$ are stably true in that revision sequence (see Definition 3.6, below), while $C$ is stably false. The liar sentence $X$ is, unsurprisingly, neither stably true nor stably false: the liar sentence is unstable. A similar calculation would show that $A$ is stably true, regardless of the initial hypothesis: thus $A$ is categorically true (see Definition 3.8). Before giving a precise definition of a revision sequence, we give an example where we would want to carry the revision process beyond the finite stages, $h, \tau^1 (h), \tau^2 (h), \tau^3 (h)$, and so on. Example 3.4 Suppose that $L$ contains nonquote names $\alpha\_0, \alpha\_1, \alpha\_2, \alpha\_3,\ldots$, and unary predicates $G$ and $\boldsymbol{T}$. Now we will specify a ground model $M = \langle D,I\rangle$ where the name $\alpha\_0$ refers to some tautology, and where \[\begin{align} &\text{the name } \alpha\_1 \text{ refers to the sentence } \boldsymbol{T}\alpha\_0 \\ &\text{the name } \alpha\_2 \text{ refers to the sentence } \boldsymbol{T}\alpha\_1 \\ &\text{the name } \alpha\_3 \text{ refers to the sentence } \boldsymbol{T}\alpha\_2 \\ &\ \vdots \end{align}\] More formally, let $A\_0$ be the sentence $\boldsymbol{T}\alpha\_0 \vee \neg \boldsymbol{T}\alpha\_0$, and for each $n \ge 0$, let $A\_{n+1}$ be the sentence $\boldsymbol{T}\alpha\_n$. Thus $A\_1$ is the sentence $\boldsymbol{T}\alpha\_0$, and $A\_2$ is the sentence $\boldsymbol{T}\alpha\_1$, and $A\_3$ is the sentence $\boldsymbol{T}\alpha\_2$, and so on. Our ground model $M = \langle D,I\rangle$ is as follows: \[\begin{align} D &= \textit{Sent}\_L \\ I(\alpha\_n) &= A\_n \\ I(G)(A) &= \mathbf{t} \text{ iff } A = A\_n \text{ for some } n \end{align}\] Thus, the extension of $G$ is the following set of sentences: \[\{A\_0, A\_1, A\_2, A\_3 , \ldots \} = \{(\boldsymbol{T}\alpha\_0 \vee \neg \boldsymbol{T}\alpha\_0), \boldsymbol{T}\alpha\_0, \boldsymbol{T}\alpha\_1, \boldsymbol{T}\alpha\_2, \boldsymbol{T}\alpha\_3 , \ldots \}. \] Finally let $B$ be the sentence $\forall x(Gx \supset \boldsymbol{T}x)$. Let $h$ be any hypothesis for which we have, for each natural number $n$, \[ h(A\_n) = h(B) = \mathbf{f}. \] The following table indicates what happens with repeated applications of the revision rule $\tau\_M$ to the hypothesis $h$. In this table, we will write $\tau$ instead of $\tau\_M$: \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| $S$ \| $h(S)$ \| $t(h)(S)$ \| $\tau^2 (h)(S)$ \| $\tau^3 (h)(S)$ \| $\tau^4 (h)(S)$ \| $\cdots$ \| \| $A\_0$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| \| $A\_1$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| \| $A\_2$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| \| $A\_3$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\cdots$ \| \| $A\_4$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\cdots$ \| \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| \| $B$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\cdots$ \| At the $0^{\text{th}}$ stage, each $A\_n$ is outside the hypothesized extension of $\boldsymbol{T}$. But from the $n{+}1^{\text{th}}$ stage onwards, $A\_n$ is $in$ the hypothesized extension of $\boldsymbol{T}$. So, for each $n$, the sentence $A\_n$ is eventually stably hypothesized to be true. Despite this, there is no finite stage at which all the $A\_n$'s are hypothesized to be true: as a result the sentence $B = \forall x(Gx \supset \boldsymbol{T}x)$ remains false at each finite stage. This suggests extending the process as follows: \| \| \| \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| $S$ \| $h(S)$ \| $\tau(h)(S)$ \| $\tau^2 (h)(S)$ \| $\tau^3 (h)(S)$ \| $\cdots$ \| $\omega$ \| $\omega +1$ \| $\omega +2$ \| $\cdots$ \| \| $A\_0$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| \| $A\_1$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| \| $A\_2$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\cdots$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| \| $A\_3$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\cdots$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| \| $A\_4$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\cdots$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| \| $B$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\cdots$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| Thus, if we allow the revision process to proceed beyond the finite stages, then the sentence $B = \forall x(Gx \supset \boldsymbol{T}x)$ is stably true from the $\omega{+}1^{\text{th}}$ stage onwards. $\Box$ In Example 3.4, the intuitive verdict is that not only should each $A\_n$ receive a stable truth value of $\mathbf{t}$, but so should the sentence $B = \forall x(Gx \supset \boldsymbol{T}x)$. The only way to ensure this is to carry the revision process beyond the finite stages. So we will consider revision sequences that are very long: not only will a revision sequence have a $n^{\text{th}}$ stage for each finite number $n$, but a $\eta^{\text{th}}$ stage for every ordinal number $\eta$. (The next paragraph is to help the reader unfamiliar with ordinal numbers.) One way to think of the ordinal numbers is as follows. Start with the finite natural numbers: \[ 0, 1, 2, 3, \ldots \] Add a number, $\omega$, greater than all of these but not the immediate successor of any of them: \[ 0, 1, 2, 3, \ldots ,\omega \] And then take the successor of $\omega$, its successor, and so on: \[ 0, 1, 2, 3, \ldots ,\omega , \omega +1, \omega +2, \omega +3\ldots \] Then add a number $\omega +\omega$, or $\omega \times 2$, greater than all of these (and again, not the immediate successor of any), and start over, reiterating this process over and over: \[\begin{align} &0, 1, 2, 3, \ldots \\ &\omega , \omega +1, \omega +2, \omega +3,\ldots, \\ &\omega \times 2, (\omega \times 2)+1, (\omega \times 2)+2, (\omega \times 2)+3,\ldots, \\ &\omega \times 3, (\omega \times 3)+1, (\omega \times 3)+2, (\omega \times 3)+3,\ldots \\ &\ \vdots \end{align}\] At the end of this, we add an ordinal number $\omega \times \omega$ or $\omega^2$: \[\begin{align} &0, 1, 2, \ldots ,\omega , \omega +1, \omega +2, \ldots ,\omega \times 2, (\omega \times 2)+1,\ldots, \\ &\omega \times 3, \ldots ,\omega \times 4, \ldots ,\omega \times 5, \ldots ,\omega^2, \omega^2 +1,\ldots \end{align}\] The ordinal numbers have the following structure: every ordinal number has an immediate successor known as a successor ordinal; and for any infinitely ascending sequence of ordinal numbers, there is a limit ordinal which is greater than all the members of the sequence and which is not the immediate successor of any member of the sequence. Thus the following are successor ordinals: $5, 178, \omega +12, (\omega \times 5)+56, \omega^2 +8$; and the following are limit ordinals: $\omega , \omega \times 2, \omega^2 , (\omega^2 +\omega)$, etc. Given a limit ordinal $\eta$, a sequence $S$ of objects is an $\eta$-long sequence if there is an object $S\_{\delta}$ for every ordinal $\delta \lt \eta$. We will denote the class of ordinals as $\textsf{On}$. Any sequence $S$ of objects is an $\textsf{On}$-long sequence if there is an object $S\_{\delta}$ for every ordinal $\delta$. When assessing whether a sentence receives a stable truth value, the RTT considers sequences of hypotheses of length $\textsf{On}$. So suppose that $S$ is an $\textsf{On}$-long sequence of hypotheses, and let $\zeta$ and $\eta$ range over ordinals. Clearly, in order for $S$ to represent the revision process, we need the $\zeta{+}1^{\text{th}}$ hypothesis to be generated from the $\zeta^{\text{th}}$ hypothesis by the revision rule. So we insist that $S\_{\zeta +1} = \tau\_M(S\_{\zeta})$. But what should we do at a limit stage? That is, how should we set $S\_{\eta}$(d) when $\eta$ is a limit ordinal? Clearly any object that is stably true [false] up to that stage should be true [false] $at$ that stage. Thus consider Example 3.4. The sentence $A\_2$, for example, is true up to the $\omega^{\text{th}}$ stage; so we set $A\_2$ to be true $at$ the $\omega^{\text{th}}$ stage. For objects that do not stabilize up to that stage, Gupta and Belnap 1993 adopt a liberal policy: when constructing a revision sequence $S$, if the value of the object $d \in D$ has not stabilized by the time you get to the limit stage $\eta$, then you can set $S\_{\eta}$(d) to be whichever of $\mathbf{t}$ or $\mathbf{f}$ you like. Before we give the precise definition of a revision sequence, we continue with Example 3.3 to see an application of this idea. Example 3.5 (Example 3.3 continued) Recall that $L$ contains four non-quote names, $\alpha , \beta , \gamma$ and $\lambda$ and no predicates other than $\boldsymbol{T}$. Also recall that $M = \langle D,I\rangle$ is as follows: \[\begin{align} D &= \textit{Sent}\_L \\ I(\alpha) &= A = \boldsymbol{T}\beta \vee \boldsymbol{T}\gamma \\ I(\beta) &= B = \boldsymbol{T}\alpha \\ I(\gamma) &= C = \neg \boldsymbol{T}\alpha \\ I(\lambda) &= X = \neg \boldsymbol{T}\lambda \end{align}\] The following table indicates what happens with repeated applications of the revision rule $\tau\_M$ to the hypothesis $h\_0$ from Example 3.3. For each ordinal $\eta$, we will indicate the $\eta^{\text{th}}$ hypothesis by $S\_{\eta}$ (suppressing the index $M$ on $\tau)$. Thus $S\_0 = h\_0,$ $S\_1 = \tau(h\_0),$ $S\_2 = \tau^2 (h\_0),$ $S\_3 = \tau^3 (h\_0),$ and $S\_{\omega},$ the $\omega^{\text{th}}$ hypothesis, is determined in some way from the hypotheses leading up to it. So, starting with $h\_0$ from Example 3.3, our revision sequence begins as follows: \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| $S$ \| $S\_0 (S)$ \| $S\_1 (S)$ \| $S\_2 (S)$ \| $S\_3 (S)$ \| $S\_4 (S)$ \| $\cdots$ \| \| $A$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| \| $B$ \| $\mathbf{t}$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| \| $C$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\cdots$ \| \| $X$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{f}$ \| $\cdots$ \| What happens at the $\omega^{\text{th}}$ stage? $A$ and $B$ are stably true up to the $\omega^{\text{th}}$ stage, and $C$ is stably false up to the $\omega^{\text{th}}$ stage. So $at$ the $\omega^{\text{th}}$ stage, we must have the following: \| \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| $S$ \| $S\_0 (S)$ \| $S\_1 (S)$ \| $S\_2 (S)$ \| $S\_3 (S)$ \| $S\_4 (S)$ \| $\cdots$ \| $S\_{\omega}(S)$ \| \| $A$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| $\mathbf{t}$ \| \| $B$ \| $\mathbf{t}$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| $\mathbf{t}$ \| \| $C$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\cdots$ \| $\mathbf{f}$ \| \| $X$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{f}$ \| $\cdots$ \| ? \| But the entry for $S\_{\omega}(X)$ can be either $\mathbf{t}$ or $\mathbf{f}$. In other words, the initial hypothesis $h\_0$ generates at least two revision sequences. Every revision sequence $S$ that has $h\_0$ as its initial hypothesis must have $S\_{\omega}(A) = \mathbf{t}, S\_{\omega}(B) = \mathbf{t}$, and $S\_{\omega}(C) = \mathbf{f}$. But there is some revision sequence $S$, with $h\_0$ as its initial hypothesis, and with $S\_{\omega}(X) = \mathbf{t}$; and there is some revision sequence $S'$, with $h\_0$ as its initial hypothesis, and with $S\_{\omega}'(X) = \mathbf{f}. \Box$ We are now ready to define the notion of a revision sequence: Definition 3.6 Suppose that $L$ is a truth language, and that $M = \langle D,I\rangle$ is a ground model. Suppose that $S$ is an $\textsf{On}$-long sequence of hypotheses. Then we say that $d \in D$ is stably $\mathbf{t} [\mathbf{f}]$ in $S$ iff for some ordinal $\theta$ we have \[ S\_{\zeta}(d) = \mathbf{t}\ [\mathbf{f}], \text{ for every ordinal } \zeta \ge \theta. \] Suppose that $S$ is a $\eta$-long sequence of hypothesis for some limit ordinal $\eta$. Then we say that $d \in D$ is stably $\mathbf{t}$ $[\mathbf{f}]$ in $S$ iff for some ordinal $\theta \lt \eta$ we have \[ S\_{\zeta}(d) = \mathbf{t}\ [\mathbf{f}], \text{ for every ordinal } \zeta \text{ such that } \zeta \ge \theta \text{ and } \zeta \lt \eta. \] If $S$ is an $\textsf{On}$-long sequence of hypotheses and $\eta$ is a limit ordinal, then $S\|\_{\eta}$ is the initial segment of $S$ up to but not including $\eta$. Note that $S\|\_{\eta}$ is a $\eta$-long sequence of hypotheses. Definition 3.7 Suppose that $L$ is a truth language, and that $M = \langle D,I\rangle$ is a ground model. Suppose that $S$ is an $\textsf{On}$-long sequence of hypotheses. $S$ is a revision sequence for $M$ iff * $S\_{\zeta +1} = \tau\_M(S\_{\zeta})$, for each $\zeta \in \textsf{On}$, and * for each limit ordinal $\eta$ and each $d \in D$, if $d$ is stably $\mathbf{t}$ $[\mathbf{f}$] in $S\|\_{\eta}$, then $S\_{\eta}(d) = \mathbf{t} [\mathbf{f}$]. Definition 3.8 Suppose that $L$ is a truth language, and that $M = \langle D,I\rangle$ is a ground model. We say that the sentence $A$ is categorically true [false] in $M$ iff $A$ is stably $\mathbf{t}$ $[\mathbf{f}]$ in every revision sequence for $M$. We say that $A$ is categorical in $M$ iff $A$ is either categorically true or categorically false in $M$. We now illustrate these concepts with an example. The example will also illustrate a new concept to be defined afterwards. Example 3.9 Suppose that $L$ is a truth language containing nonquote names $\beta , \alpha\_0, \alpha\_1, \alpha\_2, \alpha\_3,\ldots$, and unary predicates $G$ and $\boldsymbol{T}$. Let $B$ be the sentence \[ \boldsymbol{T}\beta \vee \forall x\forall y(Gx \amp \neg \boldsymbol{T}x \amp Gy \amp \neg \boldsymbol{T}y \supset x=y). \] Let $A\_0$ be the sentence $\exists x(Gx \amp \neg \boldsymbol{T}x)$. And for each $n \ge 0$, let $A\_{n+1}$ be the sentence $\boldsymbol{T}\alpha\_n$. Consider the following ground model $M = \langle D,I\rangle$ \[\begin{align} D &= \textit{Sent}\_L \\ I(\beta) &= B \\ I(\alpha\_n) &= A\_n \\ I(G)(A) &= \mathbf{t} \text{ iff } A = A\_n \text{ for some } n \end{align}\] Thus, the extension of $G$ is the following set of sentences: $\{A\_0, A\_1, A\_2, A\_3 , \ldots \} = \{\exists x(Gx \amp \neg \boldsymbol{T}x), \boldsymbol{T}\alpha\_0, \boldsymbol{T}\alpha\_1, \boldsymbol{T} \alpha\_2, \boldsymbol{T}\alpha\_3 , \ldots \}$. Let $h$ be any hypothesis for which we have, $h(B) = \mathbf{f}$ and for each natural number $n$, \[ h(A\_n) = \mathbf{f}. \] And let $S$ be a revision sequence whose initial hypothesis is $h$, i.e., $S\_0 = h$. The following table indicates some of the values of $S\_{\gamma}(C)$, for sentences $C \in \{B, A\_0, A\_1, A\_2, A\_3 , \ldots \}$. In the top row, we indicate only the ordinal number representing the stage in the revision process. \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| --- \| \| \| 0 \| 1 \| 2 \| 3 \| $\cdots$ \| $\omega$ \| $\omega{+}1$ \| $\omega{+}2$ \| $\omega{+}3$ \| $\cdots$ \| $\omega{\times}2$ \| $(\omega{\times}2){+}1$ \| $(\omega{\times}2){+}2$ \| $\cdots$ \| \| $B$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\cdots$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| \| $A\_0$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| $\mathbf{t}$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| $\mathbf{t}$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\cdots$ \| \| $A\_1$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\cdots$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{f}$ \| $\cdots$ \| \| $A\_2$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{t}$ \| $\cdots$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{f}$ \| $\cdots$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| \| $A\_3$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\cdots$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| \| $A\_4$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\mathbf{f}$ \| $\cdots$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\mathbf{t}$ \| $\cdots$ \| \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\vdots$ \| $\ddots$ \| It is worth contrasting the behaviour of the sentence B and the sentence $A\_0$. From the $\omega{+}1^{\text{th}}$ stage on, $B$ stabilizes as true. In fact, $B$ is stably true in every revision sequence for $M$. Thus, $B$ is categorically true in $M$. The sentence $A\_0$, however, never quite stabilizes: it is usually true, but within a few finite stages of a limit ordinal, the sentence $A\_0$ can be false. In these circumstances, we say that $A\_0$ is nearly stably true (See Definition 3.10, below.) In fact, $A\_0$ is nearly stably true in every revision sequence for $M. \Box$ Example 3.9 illustrates not only the notion of stability in a revision sequence, but also of near stability, which we define now: Definition 3.10. Suppose that $L$ is a truth language, and that $M = \langle D,I\rangle$ is a ground model. Suppose that $S$ is an $\textsf{On}$-long sequence of hypotheses. Then we say that $d \in D$ is nearly stably $\mathbf{t}$ $[\mathbf{f}]$ in $S$ iff for some ordinal $\theta$ we have for every $\zeta \ge \theta$, there is a natural number $n$ such that, for every $m \ge n, S\_{\zeta +m}(d) = \mathbf{t}$ $[\mathbf{f}]$. Gupta and Belnap 1993 characterize the difference between stability and near stability as follows: "Stability simpliciter requires an element [in our case a sentence] to settle down to a value $\mathbf{x}$ [in our case a truth value] after some initial fluctuations say up to [an ordinal $\eta$]... In contrast, near stability allows fluctuations after $\eta$ also, but these fluctuations must be confined to finite regions just after limit ordinals" (p. 169). Gupta and Belnap 1993 introduce two theories of truth, $\boldsymbol{T}^\$ and $\boldsymbol{T}^{\#}$, based on stability and near stability. Theorems 3.12 and 3.13, below, illustrate an advantage of the system $\boldsymbol{T}^{\#}$, i.e., the system based on near stability. Definition 3.11* Suppose that $L$ is a truth language, and that $M = \langle D,I\rangle$ is a ground model. We say that a sentence $A$ is valid in $M$ by $\boldsymbol{T}^\$ iff $A$ is stably true in every revision sequence[6]. And we say that a sentence $A$ is valid in* $M$ by $\boldsymbol{T}^{\#}$ iff $A$ is nearly stably true in every revision sequence. Theorem 3.12 Suppose that $L$ is a truth language, and that $M = \langle D,I\rangle$ is a ground model. Then, for every sentence $A$ of $L$, the following is valid in $M$ by $\boldsymbol{T}^{\#}$: \[ \boldsymbol{T}\lsquo \neg A\rsquo \equiv \neg \boldsymbol{T}\lsquo A\rsquo. \] Theorem 3.13 There is a truth language $L$ and a ground model $M = \langle D,I\rangle$ and a sentence $A$ of $L$ such that the following is not valid in $M$ by $\boldsymbol{T}^\$: \[ \boldsymbol{T}\lsquo \neg A\rsquo \equiv \neg \boldsymbol{T}\lsquo A\rsquo. \] Gupta and Belnap 1993, Section 6C, note similar advantages of $\boldsymbol{T}^{\#}$ over $\boldsymbol{T}^\$. For example, $\boldsymbol{T}^{\#}$ does, but $\boldsymbol{T}^\$ does not, validate the following semantic principles: \[\begin{align} \boldsymbol{T}\lsquo A \amp B\rsquo &\equiv \boldsymbol{T}\lsquo A\rsquo \amp \boldsymbol{T}\lsquo B\rsquo \\ \boldsymbol{T}\lsquo A \vee B\rsquo &\equiv \boldsymbol{T}\lsquo A\rsquo \vee \boldsymbol{T}\lsquo B\rsquo \end{align}\] Gupta and Belnap remain noncommittal about which of $\boldsymbol{T}^{\#}$ and $\boldsymbol{T}^\$ (and a further alternative that they define, $\boldsymbol{T}^c)$ is preferable. ## 4. Interpreting the formalism The main formal notions of the RTT are the notion of a revision rule (Definition 3.2), i.e., a rule for revising hypotheses; and a revision sequence (Definition 3.7), a sequence of hypotheses generated in accordance with the appropriate revision rule. Using these notions, we can, given a ground model, specify when a sentence is stably (or nearly stably) true or stably (or nearly stably) false (Definition 3.6 and 3.10, respectively) in a particular revision sequence. Thus we could define two theories of truth, $\boldsymbol{T}^\$ and $\boldsymbol{T}^{\#}$ (Definition 3.11), based on stability and near stability (respectively). The final idea is that each of these theories delivers a verdict on which sentences of the language are valid* (Definition 3.11), given a ground model. Recall the suggestions made at the end of Section 2: > > In a semantics for languages capable of expressing their own truth > concepts, $\boldsymbol{T}$ will not, in general, have a classical > signification; and the 'iff' in the T-biconditionals will > not be read as the classical biconditional. > Gupta and Belnap fill out these suggestions in the following way. ### 4.1 The signification of $\boldsymbol{T}$ First, they suggest that the signification of $\boldsymbol{T}$, given a ground model $M$, is the revision rule $\tau\_M$ itself. As noted in the preceding paragraph, we can give a fine-grained analysis of sentences' statuses and interrelations on the basis of notions generated directly and naturally from the revision rule $\tau\_M$. Thus, $\tau\_M$ is a good candidate for the signification of $\boldsymbol{T}$, since it does seem to be "an abstract something that carries all the information about all [of $\boldsymbol{T}$'s] extensional relations" in $M$. (See Gupta and Belnap's characterization of an expression's signification, given in Section 2, above.) ### 4.2 The 'iff' in the T-biconditionals Gupta and Belnap's related suggestion concerning the 'iff' in the T-biconditionals is that, rather than being the classical biconditional, this 'iff' is the distinctive biconditional used to define a previously undefined concept. In 1993, Gupta and Belnap present the revision theory of truth as a special case of a revision theory of circularly defined concepts. Suppose that $L$ is a language with a unary predicate $F$ and a binary predicate $R$. Consider a new concept expressed by a predicate $G$, introduced through a definition like this: \[ Gx =\_{df} \forall y(Ryx \supset Fx) \vee \exists y(Ryx \amp Gx). \] Suppose that we start with a domain of discourse, $D$, and an interpretation of the predicate $F$ and the relation symbol $R$. Gupta and Belnap's revision-theoretic treatment of concepts thus circularly introduced allows one to give categorical verdicts, for certain $d \in D$ about whether or not $d$ satisfies $G$. Other objects will be unstable relative to $G$: we will be able categorically to assert neither that $d$ satisfies $G$ nor that d does not satisfy $G$. In the case of truth, Gupta and Belnap take the set of T-biconditionals of the form \[\tag{10} \boldsymbol{T}\lsquo A\rsquo =\_{df} A \] together to give the definition of the concept of truth. It is their treatment of '$=\_{df}$' (the 'iff' of definitional concept introduction), together with the T-biconditionals of the form (10), that determine the revision rule $\tau\_M$. ### 4.3 The paradoxical reasoning Recall the liar sentence, (1), from the beginning of this article: (1) (1) is not true In Section 1, we claimed that the RTT is designed to model, rather than block, the kind of paradoxical reasoning regarding (1). But we noted in footnote 2 that the RTT does avoid contradictions in these situations. There are two ways to see this. First, while the RTT does endorse the biconditional (1) is true iff (1) is not true, the relevant 'iff' is not the material biconditional, as explained above. Thus, it does not follow that both (1) is true and (1) is not true. Second, note that on no hypothesis can we conclude that both (1) is true and (1) is not true. If we keep it firmly in mind that revision-theoretical reasoning is hypothetical rather than categorical, then we will not infer any contradictions from the existence of a sentence such as (1), above. ### 4.4 The signification thesis Gupta and Belnap's suggestions, concerning the signification of $\boldsymbol{T}$ and the interpretation of the 'iff' in the T-biconditionals, dovetail nicely with two closely related intuitions articulated in Gupta & Belnap 1993. The first intuition, loosely expressed, is "that the T-biconditionals are analytic and fix the meaning of 'true'" (p. 6). More tightly expressed, it becomes the "Signification Thesis" (p. 31): "The T-biconditionals fix the signification of truth in every world [where a world is represented by a ground model]."[7] Given the revision-theoretic treatment of the definition 'iff', and given a ground model $M$, the T-biconditionals (10) do, as noted, fix the suggested signification of $\boldsymbol{T}$, i.e., the revision rule $\tau\_M$. ### 4.5 The supervenience of semantics The second intuition is the supervenience of the signification of truth. This is a descendant of M. Kremer's 1988 proposed supervenience of semantics. The idea is simple: which sentences fall under the concept truth should be fixed by (1) the interpretation of the nonsemantic vocabulary, and (2) the empirical facts. In non-circular cases, this intuition is particularly strong: the standard interpretation of "snow" and "white" and the empirical fact that snow is white, are enough to determine that the sentence "snow is white" falls under the concept truth. The supervenience of the signification of truth is the thesis that the signification of truth, whatever it is, is fixed by the ground model $M$. Clearly, the RTT satisfies this principle. It is worth seeing how a theory of truth might violate this principle. Consider a truth-teller sentence, i.e., the sentence that says of itself that it is true: (11) (11) is true As noted above, Kripke's three-valued semantics allows three truth values, true $(\mathbf{t})$, false $(\mathbf{f})$, and neither $(\mathbf{n})$. Given a ground model $M = \langle D,I\rangle$ for a truth language $L$, the candidate interpretations of $\boldsymbol{T}$ are three-valued interpretations, i.e., functions $h:D \rightarrow \{\mathbf{t}, \mathbf{f}, \mathbf{n}\}$. Given a three-valued interpretation of $\boldsymbol{T}$, and a scheme for evaluating the truth value of composite sentences in terms of their parts, we can specify a truth value $\textit{Val}\_{M+h}(A) = \mathbf{t}, \mathbf{f}$ or $\mathbf{n}$, for every sentence $A$ of $L$. The central theorem of the three-valued semantics is that, given any ground model $M$, there is a three-valued interpretation h of $\boldsymbol{T}$ so that, for every sentence $A$, we have $\textit{Val}\_{M+h}(\boldsymbol{T}\lsquo A\rsquo) = \textit{Val}\_{M+h}(A)$.[8] We will call such an interpretation of $\boldsymbol{T}$ an acceptable interpretation. Our point here is this: if there's a truth-teller, as in (11), then there is not only one acceptable interpretation of $\boldsymbol{T}$; there are three: one according to which (11) is true, one according to which (11) is false, and one according to which (11) is neither. Thus, there is no single "correct" interpretation of $\boldsymbol{T}$ given a ground model M. Thus the three-valued semantics seems to violate the supervenience of semantics.[9] The RTT does not assign a truth value to the truth-teller, (11). Rather, it gives an analysis of the kind of reasoning that one might engage in with respect to the truth-teller: If we start with a hypothesis $h$ according to which (11) is true, then upon revision (11) remains true. And if we start with a hypothesis $h$ according to which (11) is not true, then upon revision (11) remains not true. And that is all that the concept of truth leaves us with. Given this behaviour of (11), the RTT tells us that (11) is neither categorically true nor categorically false, but this is quite different from a verdict that (11) is neither true nor false. ### 4.6 A nonsupervenient interpretation of the formalism We note an alternative interpretation of the revision-theoretic formalism. Yaqub 1993 agrees with Gupta and Belnap that the T-biconditionals are definitional rather than material biconditionals, and that the concept of truth is therefore circular. But Yaqub interprets this circularity in a distinctive way. He argues that, > > since the truth conditions of some sentences involve reference to > truth in an essential, irreducible manner, these conditions can only > obtain or fail in a world that already includes an extension of the > truth predicate. Hence, in order for the revision process to determine > an extension of the truth predicate, an initial extension of > the predicate must be posited. This much follows from circularity and > bivalence. (1993, 40) > Like Gupta and Belnap, Yaqub posits no privileged extension for $\boldsymbol{T}$. And like Gupta and Belnap, he sees the revision sequences of extensions of $\boldsymbol{T}$, each sequence generated by an initial hypothesized extension, as "capable of accommodating (and diagnosing) the various kinds of problematic and unproblematic sentences of the languages under consideration" (1993, 41). But, unlike Gupta and Belnap, he concludes from these considerations that "truth in a bivalent language is not supervenient" (1993, 39). He explains in a footnote: for truth to be supervenient, the truth status of each sentence must be "fully determined by nonsemantical facts". Yaqub does not explicitly use the notion of a concept's signification. But Yaqub seems committed to the claim that the signification of $\boldsymbol{T}$ -- i.e., that which determines the truth status of each sentence -- is given by a particular revision sequence itself. And no revision sequence is determined by the nonsemantical facts, i.e., by the ground model, alone: a revision sequence is determined, at best, by a ground model and an initial hypothesis.[10] ### 4.7 Classifying sentences To obtain a signification of $\boldsymbol{T}$ and a notion of validity based on the concept of stability (or near stability) is by no means the only use we can make of revision sequences. For one thing, we could use revision-theoretic notions to make rather fine-grained distinctions among sentences: Some sentences are unstable in every revision sequence; others are stable in every revision sequence, though stably true in some and stably false in others; and so on. Thus, we can use revision-theoretic ideas to give a fine-grained analysis of the status of various sentences, and of the relationships of various sentences to one another. Hsiung (2017) explores further this possibility by generalizing the notion of a revision sequence to a revision mapping on a digraph, in order to extend this analysis to sets of sentences of a certain kind, called Boolean paradoxes. As shown in Hsiung 2022 this procedure can also be reversed, at least to some extent: Not only we can use revision sequences (or mappings) to classify paradoxical sentences by means of their revision-theoretic patterns, but we can also construct "new" paradoxes from given revision-theoretic patterns. Rossi (2019) combines the revision-theoretic technique with graph-theoretic tools and fixed-point constructions in order to represent a threefold classification of paradoxical sentences (liar-like, truth-teller-like, and revenge sentences) within a single model. ## 5. Further issues ### 5.1 Three-valued semantics We have given only the barest exposition of the three-valued semantics, in our discussion of the supervenience of the signification of truth, above. Given a truth language $L$ and a ground model $M$, we defined an acceptable three-valued interpretation of $\boldsymbol{T}$ as an interpretation $h:D \rightarrow \{\mathbf{t}, \mathbf{f}, \mathbf{n}\}$ such that $\textit{Val}\_{M+h}(\boldsymbol{T}\lsquo A\rsquo) = \textit{Val}\_{M+h}(A)$ for each sentence $A$ of $L$. In general, given a ground model $M$, there are many acceptable interpretations of $\boldsymbol{T}$. Suppose that each of these is indeed a truly acceptable interpretation. Then the three-valued semantics violates the supervenience of the signification of $\boldsymbol{T}$. Suppose, on the other hand, that, for each ground model $M$, we can isolate a privileged acceptable interpretation as the correct interpretation of $\boldsymbol{T}$. Gupta and Belnap present a number of considerations against the three-valued semantics, so conceived. (See Gupta & Belnap 1993, Chapter 3.) One principal argument is that the central theorem, i.e., that for each ground model there is an acceptable interpretation, only holds when the underlying language is expressively impoverished in certain ways: for example, the three-valued approach fails if the language has a connective ${\sim}$ with the following truth table: \| \| \| \| --- \| --- \| \| $A$ \| ${\sim}A$ \| \| $\mathbf{t}$ \| $\mathbf{f}$ \| \| $\mathbf{f}$ \| $\mathbf{t}$ \| \| $\mathbf{n}$ \| $\mathbf{t}$ \| The only negation operator that the three-valued approach can handle has the following truth table: \| \| \| \| --- \| --- \| \| $A$ \| $\neg A$ \| \| $\mathbf{t}$ \| $\mathbf{f}$ \| \| $\mathbf{f}$ \| $\mathbf{t}$ \| \| $\mathbf{n}$ \| $\mathbf{n}$ \| But consider the liar that says of itself that it is 'not' true, in this latter sense of 'not'. Gupta and Belnap urge the claim that this sentence "ceases to be intuitively paradoxical" (1993, 100). The claimed advantage of the RTT is its ability to describe the behaviour of genuinely paradoxical sentences: the genuine liar is unstable under semantic evaluation: "No matter what we hypothesize its value to be, semantic evaluation refutes our hypothesis." The three-valued semantics can only handle the "weak liar", i.e., a sentence that only weakly negates itself, but that is not guaranteed to be paradoxical: "There are appearances of the liar here, but they deceive." We've thus far reviewed two of Gupta and Belnap's complaints against three-valued approaches, and now we raise a third: in the three-valued theories, truth typically behaves like a nonclassical concept even when there's no vicious reference in the language. Without defining terms here, we note that one popular precisification of the three-valued approach, is to take the correct interpretation of T to be that given by the 'least fixed point' of the 'strong Kleene scheme': putting aside details, this interpretation always assigns the truth value $\mathbf{n}$ to the sentence $\forall$x$(\boldsymbol{T}$x $\vee \neg \boldsymbol{T}$x), even when the ground model allows no circular, let alone vicious, reference. Gupta and Belnap claim an advantage for the RTT: according to revision-theoretic approach, they claim, truth always behaves like a classical concept when there is no vicious reference. Kremer 2010 challenges this claim by precisifying it as a formal claim against which particular revision theories (e.g. $\boldsymbol{T}^\$ or $\boldsymbol{T}^{\#}$, see Definition 3.11, above) and particular three-valued theories can be tested. As it turns out, on many three-valued theories, truth does in fact behave like a classical concept when there's no vicious reference: for example, the least fixed point of a natural variant of the supervaluation scheme always assigns $\boldsymbol{T}$ a classical interpretation in the absence of vicious reference. When there's no vicious reference, it is granted that truth behaves like a classical concept if we adopt Gupta and Belnap's theory $\boldsymbol{T}^\$, however, so Kremer argues, this is not the case if we instead adopt Gupta and Belnap's theory $\boldsymbol{T}^{\#}$. This discussion is further taken up by Wintein 2014. A general assessment of the relative merits of the three-valued approaches and the revision-theoretic approaches, from a metasemantic point of view, is at the core of Pinder 2018. ### 5.2 Two values? A contrast presupposed by this entry is between allegedly two-valued theories, like the RTT, and allegedly three-valued or other many-valued rivals. One might think of the RTT itself as providing infinitely many semantic values, for example one value for every possible revision sequence. Or one could extract three semantic values for sentences: categorical truth, categorical falsehood, and uncategoricalness. In reply, it must be granted that the RTT generates many statuses available to sentences. Similarly, three-valued approaches also typically generate many statuses available to sentences. The claim of two-valuedness is not a claim about statuses available to sentences, but rather a claim about the truth values presupposed in the whole enterprise. ### 5.3 Amendments to the RTT We note three ways to amend the RTT. First, we might put constraints on which hypotheses are acceptable. For example, Gupta and Belnap 1993 introduce a theory, $\mathbf{T}^c$, of truth based on consistent hypotheses: an hypothesis $h$ is consistent iff the set $\{A:h(A) = \mathbf{t}\}$ is a complete consistent set of sentences. The relative merits of $\mathbf{T}^\, \mathbf{T}^{\#}$ and $\mathbf{T}^c$ are discussed in Gupta & Belnap 1993, Chapter 6. Second, we might adopt a more restrictive limit policy* than Gupta and Belnap adopt. Recall the question asked in Section 3: How should we set $S\_{\eta}(d)$ when $\eta$ is a limit ordinal? We gave a partial answer: any object that is stably true [false] up to that stage should be true [false] at that stage. We also noted that for an object $d \in D$ that does not stabilize up to the stage $\eta$, Gupta and Belnap 1993 allow us to set $S\_{\eta}(d)$ as either $\mathbf{t}$ or $\mathbf{f}$. In a similar context, Herzberger 1982a and 1982b assigns the value $\mathbf{f}$ to the unstable objects. And Gupta originally suggested, in Gupta 1982, that unstable elements receive whatever value they received at the initial hypothesis $S\_0$. These first two ways of amending the RTT both, in effect, restrict the notion of a revision sequence, by putting constraints on which of our revision sequences really count as acceptable revision sequences. The constraints are, in some sense local: the first constraint is achieved by putting restrictions on which hypotheses can be used, and the second constraint is achieved by putting restrictions on what happens at limit ordinals. A third option would be to put more global constraints on which putative revision sequences count as acceptable. Yaqub 1993 suggests, in effect, a limit rule whereby acceptable verdicts on unstable sentences at some limit stage $\eta$ depend on verdicts rendered at other limit stages. Yaqub argues that these constraints allow us to avoid certain "artifacts". For example, suppose that a ground model $M = \langle D,I\rangle$ has two independent liars, by having two names $\alpha$ and $\beta$, where $I(\alpha) = \neg \boldsymbol{T}\alpha$ and $I(\beta) = \neg \boldsymbol{T}\beta$. Yaqub argues that it is a mere "artifact" of the revision semantics, naively presented, that there are revision sequences in which the sentence $\neg \boldsymbol{T}\alpha \equiv \neg \boldsymbol{T}\beta$ is stably true, since the two liars are independent. His global constraints are developed to rule out such sequences. (See Chapuis 1996 for further discussion.) The first and the second way of amending the RTT are in some sense put together by Campbell-Moore (2019). Here, the notion of stability for objects is extended to sets of hypotheses: A set $H$ of hypotheses is stable in a sequence $S$ of hypotheses if for some ordinal $\theta$ all hypotheses $S\_{\zeta}$, with $\zeta \ge \theta$, belong to $H$. With this, we can introduce the notion of $P$-revision sequence: If $P$ is a class of sets of hypotheses, a sequence $S$ is a $P$-revision sequence just in case that, at every limit ordinal $\eta$, if a set of hypotheses $H$ belongs to $P$ and is stable in $S$, then $S\_{\eta}$ belongs to $H$. It can be shown that, for a suitable choice of $P$, all limit stages of $P$-revision sequences are maximal consistent hypotheses. ### 5.4 Revision theory for circularly defined concepts As indicated in our discussion, in Section 4, of the 'iff' in the T-biconditionals, Gupta and Belnap present the RTT as a special case of a revision theory of circularly defined concepts. To reconsider the example from Section 4. Suppose that $L$ is a language with a unary predicate F and a binary predicate R. Consider a new concept expressed by a predicate $G$, introduced through a definition, $D$, like this: \[ Gx =\_{df} A(x,G) \] where $A(x,G)$ is the formula \[ \forall y(Ryx \supset Fx) \vee \exists y(Ryx \amp Gx). \] In this context, a ground model is a classical model $M = \langle D,I\rangle$ of the language $L$: we start with a domain of discourse, $D$, and an interpretation of the predicate $F$ and the relation symbol $R$. We would like to extend $M$ to an interpretation of the language $L + G$. So, in this context, an hypothesis will be thought of as an hypothesized extension for the newly introduced concept $G$. Formally, a hypothesis is simply a function $h:D \rightarrow \{\mathbf{t}, \mathbf{f}\}$. Given a hypothesis $h$, we take $M+h$ to be the classical model $M+h = \langle D,I'\rangle$, where $I'$ interprets $F$ and $R$ in the same way as $I$, and where $I'(G) = h$. Given a hypothesized interpretation $h$ of $G$, we generate a new interpretation of $G$ as follows: and object $d \in D$ is in the new extension of $G$ just in case the defining formula $A(x,G)$ is true of $d$ in the model $M+h$. Formally, we use the ground model $M$ and the definition $D$ to define a revision rule, $\delta\_{D,M}$, mapping hypotheses to hypotheses, i.e., hypothetical interpretations of $G$ to hypothetical interpretations of $G$. In particular, for any formula $B$ with one free variable $x$, and $d \in D$, we can define the truth value $\textit{Val}\_{M+h,d}(B)$ in the standard way. Then, \[ \delta\_{D,M}(h)(d) = \textit{Val}\_{M+h,d}(A) \] Given a revision rule $\delta\_{D,M}$, we can generalize the notion of a revision sequence, which is now a sequence of hypothetical extensions of $G$ rather than $\boldsymbol{T}$. We can generalize the notion of a sentence $B$ being stably true, nearly stably true, etc., relative to a revision sequence. Gupta and Belnap introduce the systems $\mathbf{S}^\$ and $\mathbf{S}^{\#}$, analogous to $\mathbf{T}^\$ and $\mathbf{T}^{\#}$, as follows:[11] > > Definition 5.1. > * A sentence $B$ is valid on the definition $D$ in > the ground model $M$ in the system $\mathbf{S}^\$ > (notation $M \vDash\_{\,D} B)$ iff $B$ is stably true relative to > each revision sequence for the revision rule $\delta\_{D,M}$. > * A sentence $B$ is valid on the definition $D$ in > the ground model $M$ in the system $\mathbf{S}^{\#}$ > (notation $M \vDash\_{\#,D} B)$ iff $B$ is nearly stably > true relative to each revision sequence for the revision rule > $\delta\_{D,M}$. > * A sentence $B$ is valid on the definition $D$ in > the system $\mathbf{S}^\$ (notation $\vDash\_{\,D} B)$ iff for > all classical ground models $M$, we have $M \vDash\_{\,D} B$. > A sentence $B$ is valid on the definition $D$ in > the system $\mathbf{S}^{\#}$ (notation $\vDash\_{\#,D} B)$ iff > for all classical ground models $M$, we have $M \vDash\_{\#,D} > B$. > > > One of Gupta and Belnap's principle open questions is whether there is a complete calculus for these systems: that is, whether, for each definition $D$, either of the following two sets of sentences is recursively axiomatizable: $\{B:\vDash\_{\,D} B\}$ and $\{B:\vDash\_{\#,D} B\}$. Kremer 1993 proves that the answer is no: he shows that there is a definition $D$ such that each of these sets of sentences is of complexity at least $\Pi^{1}\_2$, thereby putting a lower limit on the complexity of $\mathbf{S}^\$ and $\mathbf{S}^{\#}$. (Antonelli 1994a and 2002 shows that this is also an upper limit.) Kremer's proof exploits an intimate relationship between circular definitions understood revision-theoretically and circular definitions understood as inductive definitions: the theory of inductive definitions has been quite well understood for some time. In particular, Kremer proves that every inductively defined concept can be revision-theoretically defined. The expressive power and other aspects of the revision-theoretic treatment of circular definitions is the topic of much interesting work: see Welch 2001, Lowe 2001, Lowe and Welch 2001, and Kuhnberger et al. 2005. Alongside Kremer's limitative result there is the positive observation that, for some semantic system of definitions based on restricted kinds of revision sequences, sound and complete calculi do exist. For instance, Gupta and Belnap give some examples of calculi and revision-theoretic systems which use only finite revision sequences. Further investigation on proof-theoretic calculi capturing some revision-theoretic semantic systems is done in Bruni 2013, Standefer 2016, Bruni 2019, and Fjellstad 2020. ### 5.5 Axiomatic Theories of Truth and the Revision Theory The RTT is a clear example of a semantically motivated theory of truth. Quite a different tradition seeks to give a satisfying axiomatic theory of truth. Granted we cannot retain all of classical logic and all of our intuitive principles regarding truth, especially if we allow vicious self-reference. But maybe we can arrive at satisfying axiom systems for truth, that, for example, maintain consistency and classical logic, but give up only a little bit when it comes to our intuitive principles concerning truth, such as the T-biconditionals (interpreted classically); or maintain consistency and all of the T-biconditionals, but give up only a little bit of classical logic. Halbach 2011 comprehensively studies such axiomatic theories (mainly those that retain classical logic), and Horsten 2011 is in the same tradition. Both Chapter 14 of Halbach 2011 and Chapter 8 of Horsten 2011 study the relationship between the Friedman-Sheard theory FS and the revision semantics, with some interesting results. For more work on axiomatic systems and the RTT, see Horsten et al 2012. Field 2008 makes an interesting contribution to axiomatic theorizing about truth, even though most of the positive work in the book consists of model building and is therefore semantics. In particular, Field is interested in producing a theory as close to classical logic as possible, which retains all T-biconditionals (the conditional itself will be nonclassical) and which at the same time can express, in some sense, the claim that such and such a sentence is defective. Field uses tools from multivalued logic, fixed-point semantics, and revision theory to build models showing, in effect, that a very attractive axiomatic system is consistent. Field's construction is an intricate interplay between using fixed-point constructions for successively interpreting T, and revision sequences for successively interpreting the nonclassical conditional -- the final interpretation being determined by a sort of super-revision-theoretic process. The connection between revision and Field's theory is explored further in Standefer 2015b and in Gupta and Standefer 2017. ### 5.6 Applications Given Gupta and Belnap's general revision-theoretic treatment of circular definitions -- of which their treatment of truth is a special case -- one would expect revision-theoretic ideas to be applied to other concepts. Antonelli 1994b applies these ideas to non-well-founded sets: a non-well-founded set $X$ can be thought of as circular, since, for some $X\_0 , \ldots ,X\_n$ we have $X \in X\_0 \in \ldots \in X\_n \in X$. Chapuis 2003 applies revision-theoretic ideas to rational decision making. This connection is further developed by Bruni 2015 and by Bruni and Sillari 2018. For a discussion of revision theory and abstract objects see Wang 2011. For a discussion of revision theory and vagueness, see Asmus 2013. Standefer (2015a) studies the connection between the circular definitions of revision theory and a particular modal logic RT (for "Revision Theory"). Campbell-Moore et al. 2019 and Campbell-Moore 2021 use revision sequences to model probabilities and credences, respectively. Cook 2019 employs a revision-theoretic analysis to find a new possible solution of Benardete's version of the Zeno paradox. In recent times, there has been increasing interest in bridging the gap between classic debates on the nature of truth -- deflationism, the correspondence theory, minimalism, pragmatism, and so on -- and formal work on truth, motivated by the liar's paradox. The RTT is tied to pro-sententialism by Belnap 2006; deflationism, by Yaqub 2008; and minimalism, by Restall 2005. We must also mention Gupta 2006. In this work, Gupta argues that an experience provides the experiencer, not with a straightforward entitlement to a proposition, but rather with a hypothetical entitlement: as explicated in Berker 2011, if subject S has experience $e$ and is entitled to hold view $v$ (where S's view is the totality of S's concepts, conceptions, and beliefs), then S is entitled to believe a certain class of perceptual judgements, $\Gamma(v)$. (Berker uses "propositions" instead of "perceptual judgements" in his formulation.) But this generates a problem: how is S entitled to hold a view? There seems to be a circular interdependence between entitlements to views and entitlements to perceptual judgements. Here, Gupta appeals to a general form of revision theory -- generalizing beyond both the revision theory of truth and the revision theory of circularly defined concepts (Section 5.4, above) -- to given an account of how "hypothetical perceptual entitlements could yield categorical entitlements" (Berker 2011). ### 5.7 An open question We close with an open question about $\mathbf{T}^\$ and $\mathbf{T}^{\#}$. Recall Definition 3.11, above, which defines when a sentence $A$ of a truth language $L$ is valid in the ground model* $M$ by $\mathbf{T}^\$ or by* $\mathbf{T}^{\#}$. We will say that $A$ is valid by $\mathbf{T}^\$ [alternatively, by* $\mathbf{T}^{\#}$] iff $A$ is valid in the ground model $M$ by $\mathbf{T}^\$ [alternatively, by $\mathbf{T}^{\#}$] for every ground model $M$. Our open question is this: What is the complexity of the set of sentences valid by $\mathbf{T}^\ [\mathbf{T}^{\#}$]?
tarski-truth	## 1. The 1933 programme and the semantic conception In the late 1920s Alfred Tarski embarked on a project to give rigorous definitions for notions useful in scientific methodology. In 1933 he published (in Polish) his analysis of the notion of a true sentence. This long paper undertook two tasks: first to say what should count as a satisfactory definition of 'true sentence' for a given formal language, and second to show that there do exist satisfactory definitions of 'true sentence' for a range of formal languages. We begin with the first task; Section 2 will consider the second. We say that a language is fully interpreted if all its sentences have meanings that make them either true or false. All the languages that Tarski considered in the 1933 paper were fully interpreted, with one exception described in Section 2.2 below. This was the main difference between the 1933 definition and the later model-theoretic definition of 1956, which we shall examine in Section 3. Tarski described several conditions that a satisfactory definition of truth should meet. ### 1.1 Object language and metalanguage If the language under discussion (the object language) is $L$, then the definition should be given in another language known as the metalanguage, call it $M$. The metalanguage should contain a copy of the object language (so that anything one can say in $L$ can be said in $M$ too), and $M$ should also be able to talk about the sentences of $L$ and their syntax. Finally Tarski allowed $M$ to contain notions from set theory, and a 1-ary predicate symbol True with the intended reading 'is a true sentence of $L$'. The main purpose of the metalanguage was to formalise what was being said about the object language, and so Tarski also required that the metalanguage should carry with it a set of axioms expressing everything that one needs to assume for purposes of defining and justifying the truth definition. The truth definition itself was to be a definition of True in terms of the other expressions of the metalanguage. So the definition was to be in terms of syntax, set theory and the notions expressible in $L$, but not semantic notions like 'denote' or 'mean' (unless the object language happened to contain these notions). Tarski assumed, in the manner of his time, that the object language $L$ and the metalanguage $M$ would be languages of some kind of higher order logic. Today it is more usual to take some kind of informal set theory as one's metalanguage; this would affect a few details of Tarski's paper but not its main thrust. Also today it is usual to define syntax in set-theoretic terms, so that for example a string of letters becomes a sequence. In fact one must use a set-theoretic syntax if one wants to work with an object language that has uncountably many symbols, as model theorists have done freely for over half a century now. ### 1.2 Formal correctness The definition of True should be 'formally correct'. This means that it should be a sentence of the form For all $x$, True$(x)$ if and only if $\phi(x)$, where True never occurs in $\phi$; or failing this, that the definition should be provably equivalent to a sentence of this form. The equivalence must be provable using axioms of the metalanguage that don't contain True. Definitions of the kind displayed above are usually called explicit, though Tarski in 1933 called them normal. ### 1.3 Material adequacy The definition should be 'materially adequate' (trafny - a better translation would be 'accurate'). This means that the objects satisfying $\phi$ should be exactly the objects that we would intuitively count as being true sentences of $L$, and that this fact should be provable from the axioms of the metalanguage. At first sight this is a paradoxical requirement: if we can prove what Tarski asks for, just from the axioms of the metalanguage, then we must already have a materially adequate formalisation of 'true sentence of $L$' within the metalanguage, suggesting an infinite regress. In fact Tarski escapes the paradox by using (in general) infinitely many sentences of $M$ to express truth, namely all the sentences of the form \[ \phi(s) \text{ if and only if } \psi \] whenever $s$ is the name of a sentence $S$ of $L$ and $\psi$ is the copy of $S$ in the metalanguage. So the technical problem is to find a single formula $\phi$ that allows us to deduce all these sentences from the axioms of $M$; this formula $\phi$ will serve to give the explicit definition of True. Tarski's own name for this criterion of material adequacy was Convention T. More generally his name for his approach to defining truth, using this criterion, was the semantic conception of truth. As Tarski himself emphasised, Convention $T$ rapidly leads to the liar paradox if the language $L$ has enough resources to talk about its own semantics. (See the entry on the revision theory of truth.) Tarski's own conclusion was that a truth definition for a language $L$ has to be given in a metalanguage which is essentially stronger than $L$. There is a consequence for the foundations of mathematics. First-order Zermelo-Fraenkel set theory is widely regarded as the standard of mathematical correctness, in the sense that a proof is correct if and only if it can be formalised as a formal proof in set theory. We would like to be able to give a truth definition for set theory; but by Tarski's result this truth definition can't be given in set theory itself. The usual solution is to give the truth definition informally in English. But there are a number of ways of giving limited formal truth definitions for set theory. For example Azriel Levy showed that for every natural number $n$ there is a $\Sigma\_n$ formula that is satisfied by all and only the set-theoretic names of true $\Sigma\_n$ sentences of set theory. The definition of $\Sigma\_n$ is too technical to give here, but three points are worth making. First, every sentence of set theory is provably equivalent to a $\Sigma\_n$ sentence for any large enough $n$. Second, the class of $\Sigma\_n$ formulas is closed under adding existential quantifiers at the beginning, but not under adding universal quantifiers. Third, the class is not closed under negation; this is how Levy escapes Tarski's paradox. (See the entry on set theory.) Essentially the same devices allow Jaakko Hintikka to give an internal truth definition for his independence friendly logic; this logic shares the second and third properties of Levy's classes of formulas. ## 2. Some kinds of truth definition on the 1933 pattern In his 1933 paper Tarski went on to show that many fully interpreted formal languages do have a truth definition that satisfies his conditions. He gave four examples in that paper. One was a trivial definition for a finite language; it simply listed the finitely many true sentences. One was a definition by quantifier elimination; see Section 2.2 below. The remaining two, for different classes of language, were examples of what people today think of as the standard Tarski truth definition; they are forerunners of the 1956 model-theoretic definition. ### 2.1 The standard truth definitions The two standard truth definitions are at first glance not definitions of truth at all, but definitions of a more complicated relation involving assignments $a$ of objects to variables: \[ a \text{ satisfies the formula } F \] (where the symbol '$F$' is a placeholder for a name of a particular formula of the object language). In fact satisfaction reduces to truth in this sense: $a$ satisfies the formula $F$ if and only if taking each free variable in $F$ as a name of the object assigned to it by $a$ makes the formula $F$ into a true sentence. So it follows that our intuitions about when a sentence is true can guide our intuitions about when an assignment satisfies a formula. But none of this can enter into the formal definition of truth, because 'taking a variable as a name of an object' is a semantic notion, and Tarski's truth definition has to be built only on notions from syntax and set theory (together with those in the object language); recall Section 1.1. In fact Tarski's reduction goes in the other direction: if the formula $F$ has no free variables, then to say that $F$ is true is to say that every assignment satisfies it. The reason why Tarski defines satisfaction directly, and then deduces a definition of truth, is that satisfaction obeys recursive conditions in the following sense: if $F$ is a compound formula, then to know which assignments satisfy $F$, it's enough to know which assignments satisfy the immediate constituents of $F$. Here are two typical examples: * The assignment $a$ satisfies the formula '$F$ and $G$' if and only if $a$ satisfies $F$ and $a$ satisfies $G$. * The assignment $a$ satisfies the formula 'For all $x$, $G$' if and only if for every individual $i$, if $b$ is the assignment that assigns $i$ to the variable $x$ and is otherwise exactly like $a$, then $b$ satisfies $G$. We have to use a different approach for atomic formulas. But for these, at least assuming for simplicity that $L$ has no function symbols, we can use the metalanguage copies $\#(R)$ of the predicate symbols $R$ of the object language. Thus: * The assignment $a$ satisfies the formula $R(x,y)$ if and only if $\#(R)(a(x),a(y))$. (Warning: the expression $\#$ is in the metametalanguage, not in the metalanguage $M$. We may or may not be able to find a formula of $M$ that expresses $\#$ for predicate symbols; it depends on exactly what the language $L$ is.) Subject to the mild reservation in the next paragraph, Tarski's definition of satisfaction is compositional, meaning that the class of assignments which satisfy a compound formula $F$ is determined solely by (1) the syntactic rule used to construct $F$ from its immediate constituents and (2) the classes of assignments that satisfy these immediate constituents. (This is sometimes phrased loosely as: satisfaction is defined recursively. But this formulation misses the central point, that (1) and (2) don't contain any syntactic information about the immediate constituents.) Compositionality explains why Tarski switched from truth to satisfaction. You can't define whether 'For all $x, G$' is true in terms of whether $G$ is true, because in general $G$ has a free variable $x$ and so it isn't either true or false. The reservation is that Tarski's definition of satisfaction in the 1933 paper doesn't in fact mention the class of assignments that satisfy a formula $F$. Instead, as we saw, he defines the relation '$a$ satisfies $F$', which determines what that class is. This is probably the main reason why some people (including Tarski himself in conversation, as reported by Barbara Partee) have preferred not to describe the 1933 definition as compositional. But the class format, which is compositional on any reckoning, does appear in an early variant of the truth definition in Tarski's paper of 1931 on definable sets of real numbers. Tarski had a good reason for preferring the format '$a$ satisfies $F$' in his 1933 paper, namely that it allowed him to reduce the set-theoretic requirements of the truth definition. In sections 4 and 5 of the 1933 paper he spelled out these requirements carefully. The name 'compositional(ity)' first appears in papers of Putnam in 1960 (published 1975) and Katz and Fodor in 1963 on natural language semantics. In talking about compositionality, we have moved to thinking of Tarski's definition as a semantics, i.e. a way of assigning 'meanings' to formulas. (Here we take the meaning of a sentence to be its truth value.) Compositionality means essentially that the meanings assigned to formulas give at least enough information to determine the truth values of sentences containing them. One can ask conversely whether Tarski's semantics provides only as much information as we need about each formula, in order to reach the truth values of sentences. If the answer is yes, we say that the semantics is fully abstract (for truth). One can show fairly easily, for any of the standard languages of logic, that Tarski's definition of satisfaction is in fact fully abstract. As it stands, Tarski's definition of satisfaction is not an explicit definition, because satisfaction for one formula is defined in terms of satisfaction for other formulas. So to show that it is formally correct, we need a way of converting it to an explicit definition. One way to do this is as follows, using either higher order logic or set theory. Suppose we write $S$ for a binary relation between assignments and formulas. We say that $S$ is a satisfaction relation if for every formula $G, S$ meets the conditions put for satisfaction of $G$ by Tarski's definition. For example, if $G$ is '$G\_1$ and $G\_2$', $S$ should satisfy the following condition for every assignment $a$: \[ S(a,G) \text{ if and only if } S(a,G\_1) \text{ and } S(a,G\_2). \] We can define 'satisfaction relation' formally, using the recursive clauses and the conditions for atomic formulas in Tarski's recursive definition. Now we prove, by induction on the complexity of formulas, that there is exactly one satisfaction relation $S$. (There are some technical subtleties, but it can be done.) Finally we define $a$ satisfies $F$ if and only if: there is a satisfaction relation $S$ such that $S(a,F)$. It is then a technical exercise to show that this definition of satisfaction is materially adequate. Actually one must first write out the counterpart of Convention $T$ for satisfaction of formulas, but I leave this to the reader. ### 2.2 The truth definition by quantifier elimination The remaining truth definition in Tarski's 1933 paper - the third as they appear in the paper - is really a bundle of related truth definitions, all for the same object language $L$ but in different interpretations. The quantifiers of $L$ are assumed to range over a particular class, call it $A$; in fact they are second order quantifiers, so that really they range over the collection of subclasses of $A$. The class $A$ is not named explicitly in the object language, and thus one can give separate truth definitions for different values of $A$, as Tarski proceeds to do. So for this section of the paper, Tarski allows one and the same sentence to be given different interpretations; this is the exception to the general claim that his object language sentences are fully interpreted. But Tarski stays on the straight and narrow: he talks about 'truth' only in the special case where $A$ is the class of all individuals. For other values of $A$, he speaks not of 'truth' but of 'correctness in the domain $A$'. These truth or correctness definitions don't fall out of a definition of satisfaction. In fact they go by a much less direct route, which Tarski describes as a 'purely accidental' possibility that relies on the 'specific peculiarities' of the particular object language. It may be helpful to give a few more of the technical details than Tarski does, in a more familiar notation than Tarski's, in order to show what is involved. Tarski refers his readers to a paper of Thoralf Skolem in 1919 for the technicalities. One can think of the language $L$ as the first-order language with predicate symbols $\subseteq$ and =. The language is interpreted as talking about the subclasses of the class $A$. In this language we can define: * '$x$ is the empty set' (viz. $x \subseteq$ every class). * '$x$ is an atom' (viz. $x$ is not empty, but every subclass of $x$ not equal to $x$ is empty). * '$x$ has exactly $k$ members' (where $k$ is a finite number; viz. there are exactly $k$ distinct atoms $\subseteq x)$. * 'There are exactly $k$ elements in $A$' (viz. there is a class with exactly $k$ members, but there is no class with exactly $k+1$ members). Now we aim to prove: Lemma. Every formula $F$ of $L$ is equivalent to (i.e. is satisfied by exactly the same assignments as) some boolean combination of sentences of the form 'There are exactly $k$ elements in $A$' and formulas of the form 'There are exactly $k$ elements that are in $v\_1$, not in $v\_2$, not in $v\_3$ and in $v\_4$' (or any other combination of this type, using only variables free in $F)$. The proof is by induction on the complexity of formulas. For atomic formulas it is easy. For boolean combinations of formulas it is easy, since a boolean combination of boolean combinations is again a boolean combination. For formulas beginning with $\forall$, we take the negation. This leaves just one case that involves any work, namely the case of a formula beginning with an existential quantifier. By induction hypothesis we can replace the part after the quantifier by a boolean combination of formulas of the kinds stated. So a typical case might be: $\exists z$ (there are exactly two elements that are in $z$ and $x$ and not in $y)$. This holds if and only if there are at least two elements that are in $x$ and not in $y$. We can write this in turn as: The number of elements in $x$ and not in $y$ is not 0 and is not 1; which is a boolean combination of allowed formulas. The general proof is very similar but more complicated. When the lemma has been proved, we look at what it says about a sentence. Since the sentence has no free variables, the lemma tells us that it is equivalent to a boolean combination of statements saying that $A$ has a given finite number of elements. So if we know how many elements $A$ has, we can immediately calculate whether the sentence is 'correct in the domain $A$'. One more step and we are home. As we prove the lemma, we should gather up any facts that can be stated in $L$, are true in every domain, and are needed for proving the lemma. For example we shall almost certainly need the sentence saying that $\subseteq$ is transitive. Write $T$ for the set of all these sentences. (In Tarski's presentation $T$ vanishes, since he is using higher order logic and the required statements about classes become theorems of logic.) Thus we reach, for example: Theorem. If the domain $A$ is infinite, then a sentence $S$ of the language $L$ is correct in $A$ if and only if $S$ is deducible from $T$ and the sentences saying that the number of elements of $A$ is not any finite number. The class of all individuals is infinite (Tarski asserts), so the theorem applies when $A$ is this class. And in this case Tarski has no inhibitions about saying not just 'correct in $A$' but 'true'; so we have our truth definition. The method we have described revolves almost entirely around removing existential quantifiers from the beginnings of formulas; so it is known as the method of quantifier elimination. It is not as far as you might think from the two standard definitions. In all cases Tarski assigns to each formula, by induction on the complexity of formulas, a description of the class of assignments that satisfy the formula. In the two previous truth definitions this class is described directly; in the quantifier elimination case it is described in terms of a boolean combination of formulas of a simple kind. At around the same time as he was writing the 1933 paper, Tarski gave a truth definition by quantifier elimination for the first-order language of the field of real numbers. In his 1931 paper it appears only as an interesting way of characterising the set of relations definable by formulas. Later he gave a fuller account, emphasising that his method provided not just a truth definition but an algorithm for determining which sentences about the real numbers are true and which are false. ## 3. The 1956 definition and its offspring In 1933 Tarski assumed that the formal languages that he was dealing with had two kinds of symbol (apart from punctuation), namely constants and variables. The constants included logical constants, but also any other terms of fixed meaning. The variables had no independent meaning and were simply part of the apparatus of quantification. Model theory by contrast works with three levels of symbol. There are the logical constants $(=, \neg$, & for example), the variables (as before), and between these a middle group of symbols which have no fixed meaning but get a meaning through being applied to a particular structure. The symbols of this middle group include the nonlogical constants of the language, such as relation symbols, function symbols and constant individual symbols. They also include the quantifier symbols $\forall$ and $\exists$, since we need to refer to the structure to see what set they range over. This type of three-level language corresponds to mathematical usage; for example we write the addition operation of an abelian group as +, and this symbol stands for different functions in different groups. So one has to work a little to apply the 1933 definition to model-theoretic languages. There are basically two approaches: (1) Take one structure $A$ at a time, and regard the nonlogical constants as constants, interpreted in $A$. (2) Regard the nonlogical constants as variables, and use the 1933 definition to describe when a sentence is satisfied by an assignment of the ingredients of a structure $A$ to these variables. There are problems with both these approaches, as Tarski himself describes in several places. The chief problem with (1) is that in model theory we very frequently want to use the same language in connection with two or more different structures - for example when we are defining elementary embeddings between structures (see the entry on first-order model theory). The problem with (2) is more abstract: it is disruptive and bad practice to talk of formulas with free variables being 'true'. (We saw in Section 2.2 how Tarski avoided talking about truth in connection with sentences that have varying interpretations.) What Tarski did in practice, from the appearance of his textbook in 1936 to the late 1940s, was to use a version of (2) and simply avoid talking about model-theoretic sentences being true in structures; instead he gave an indirect definition of what it is for a structure to be a 'model of' a sentence, and apologised that strictly this was an abuse of language. (Chapter VI of Tarski 1994 still contains relics of this old approach.) By the late 1940s it had become clear that a direct model-theoretic truth definition was needed. Tarski and colleagues experimented with several ways of casting it. The version we use today is based on that published by Tarski and Robert Vaught in 1956. See the entry on classical logic for an exposition. The right way to think of the model-theoretic definition is that we have sentences whose truth value varies according to the situation where they are used. So the nonlogical constants are not variables; they are definite descriptions whose reference depends on the context. Likewise the quantifiers have this indexical feature, that the domain over which they range depends on the context of use. In this spirit one can add other kinds of indexing. For example a Kripke structure is an indexed family of structures, with a relation on the index set; these structures and their close relatives are fundamental for the semantics of modal, temporal and intuitionist logic. Already in the 1950s model theorists were interested in formal languages that include kinds of expression different from anything in Tarski's 1933 paper. Extending the truth definition to infinitary logics was no problem at all. Nor was there any serious problem about most of the generalised quantifiers proposed at the time. For example there is a quantifier $Qxy$ with the intended meaning: $QxyF(x,y)$ if and only if there is an infinite set $X$ of elements such that for all $a$ and $b$ in $X, F(a,b)$. This definition itself shows at once how the required clause in the truth definition should go. In 1961 Leon Henkin pointed out two sorts of model-theoretic language that didn't immediately have a truth definition of Tarski's kind. The first had infinite strings of quantifiers: \[ \forall v\_1 \exists v\_2 \forall v\_3 \exists v\_4\ldots R(v\_1,v\_2,v\_3, v\_4,\ldots). \] The second had quantifiers that are not linearly ordered. For ease of writing I use Hintikka's later notation for these: \[ \forall v\_1 \exists v\_2 \forall v\_3 (\exists v\_4 /\forall v\_1) R(v\_1,v\_2,v\_3, v\_4). \] Here the slash after $\exists v\_4$ means that this quantifier is outside the scope of the earlier quantifier $\forall v\_1$ (and also outside that of the earlier existential quantifier). Henkin pointed out that in both cases one could give a natural semantics in terms of Skolem functions. For example the second sentence can be paraphrased as \[ \exists f\exists g \forall v\_1 \forall v\_3 R(v\_1,f(v\_1),v\_3,g(v\_3)), \] which has a straightforward Tarski truth condition in second order logic. Hintikka then observed that one can read the Skolem functions as winning strategies in a game, as in the entry on logic and games. In this way one can build up a compositional semantics, by assigning to each formula a game. A sentence is true if and only if the player Myself (in Hintikka's nomenclature) has a winning strategy for the game assigned to the sentence. This game semantics agrees with Tarski's on conventional first-order sentences. But it is far from fully abstract; probably one should think of it as an operational semantics, describing how a sentence is verified rather than whether it is true. The problem of giving a Tarski-style semantics for Henkin's two languages turned out to be different in the two cases. With the first, the problem is that the syntax of the language is not well-founded: there is an infinite descending sequence of subformulas as one strips off the quantifiers one by one. Hence there is no hope of giving a definition of satisfaction by recursion on the complexity of formulas. The remedy is to note that the explicit form of Tarski's truth definition in Section 2.1 above didn't require a recursive definition; it needed only that the conditions on the satisfaction relation $S$ pin it down uniquely. For Henkin's first style of language this is still true, though the reason is no longer the well-foundedness of the syntax. For Henkin's second style of language, at least in Hintikka's notation (see the entry on independence friendly logic), the syntax is well-founded, but the displacement of the quantifier scopes means that the usual quantifier clauses in the definition of satisfaction no longer work. To get a compositional and fully abstract semantics, one has to ask not what assignments of variables satisfy a formula, but what sets of assignments satisfy the formula 'uniformly', where 'uniformly' means 'independent of assignments to certain variables, as shown by the slashes on quantifiers inside the formula'. (Further details of revisions of Tarski's truth definition along these lines are in the entry on dependence logic.) Henkin's second example is of more than theoretical interest, because clashes between the semantic and the syntactic scope of quantifiers occur very often in natural languages.
truthlikeness	## 1. The Logical Problem Truth, perhaps even more than beauty and goodness, has been the target of an extraordinary amount of philosophical dissection and speculation. By comparison with truth, the more complex and much more interesting concept of truthlikeness has only recently become the subject of serious investigation. The logical problem of truthlikeness is to give a consistent and materially adequate account of the concept. But first, we have to make it plausible that there is a coherent concept in the offing to be investigated. ### 1.1 What's the problem? Suppose we are interested in what is the number of planets in the solar system. With the demotion of Pluto to planetoid status, the truth of this matter is that there are precisely 8 planets. Now, the proposition the number of planets in our solar system is 9 may be false, but quite a bit closer to the truth than the proposition that the number of planets in our solar system is 9 billion. (One falsehood may be closer to the truth than another falsehood.) The true proposition the number of the planets is between 7 and 9 inclusive is closer to the truth than the true proposition that the number of the planets is greater than or equal to 0. (So a truth may be closer to the truth than another truth.) Finally, the proposition that the number of the planets is either less than or greater than 9 may be true but it is arguably not as close to the whole truth as its highly accurate but strictly false negation: that there are 9 planets. This particular numerical example is admittedly simple, but a wide variety of judgments of relative likeness to truth crop up both in everyday parlance as well as in scientific discourse. While some involve the relative accuracy of claims concerning the value of numerical magnitudes, others involve the sharing of properties, structural similarity, or closeness among putative laws. Consider a non-numerical example, also highly simplified but quite topical in the light of the recent rise in status of the concept of fundamentality. Suppose you are interested in the truth about which particles are fundamental. At the outset of your inquiry all you know are various logical truths, like the tautology either electrons are fundamental or they are not. Tautologies are pretty much useless in helping you locate the truth about fundamental particles. Suppose that the standard model is actually on the right track. Then, learning that electrons are fundamental edges you a little bit closer to your goal. It is by no means the complete truth about fundamental particles, but it is a piece of it. If you go on to learn that electrons, along with muons and tau particles, are a kind of lepton and that all leptons are fundamental, you have presumably edged closer. If this is right, then some truths are closer to the truth about fundamental particles than others. The discovery that atoms are not fundamental, that they are in fact composite objects, displaced the earlier hypothesis that atoms are fundamental. For a while the proposition that protons, neutrons and electrons are the fundamental components of atoms was embraced, but unfortunately it too turned out to be false. Still, this latter falsehood seems closer to the truth than its predecessor (assuming, again, that the standard model is true). And even if the standard model contains errors, as surely it does, it is presumably closer to the truth about fundamental particles than these other falsehoods. So again, some falsehoods may be closer to the truth about fundamental particles than other falsehoods. As we have seen, a tautology is not a terrific truth locator, but if you moved from the tautology that electrons either are or are not fundamental to embrace the false proposition that electrons are not fundamental you would have moved further from your goal. So, some truths are closer to the truth than some falsehoods. But it is by no means obvious that all truths about fundamental particles are closer to the whole truth than any falsehood. The false proposition that electrons, protons and neutrons are the fundamental components of atoms, for instance, may well be an improvement over the tautology. If this is right, certain falsehoods are closer to the truth than some truths. Investigations into the concept of truthlikeness only began in earnest in the early nineteen sixties. Why was truthlikeness such a latecomer to the philosophical scene? It wasn't until the latter half of the twentieth century that mainstream philosophers gave up on the Cartesian goal of infallible knowledge. The idea that we are quite possibly, even probably, mistaken in our most cherished beliefs, that they might well be just false, was mostly considered tantamount to capitulation to the skeptic. By the middle of the twentieth century, however, it was clear that many of our commonsense beliefs, as well as previous scientific theories, are strictly speaking, false. Further, the increasingly rapid turnover of scientific theories suggested that, far from being certain, they are ever vulnerable to refutation, and typically are eventually refuted and replaced by some new theory. The history of inquiry is one of a parade of refuted theories, replaced by other theories awaiting their turn at the guillotine. (This is the "dismal induction", see also the entry on realism and theory change in science.) Realism holds that the constitutive aim of inquiry is the truth of some matter. Optimism holds that the history of inquiry is one of progress with respect to its constitutive aim. But fallibilism holds that our theories are false or very likely to be false, and to be replaced by other false theories. To combine these three ideas, we must affirm that some false propositions better realize the goal of truth - are closer to the truth - than others. We are thus stuck with the logical problem of truthlikeness. While a multitude of apparently different solutions to the problem have been proposed, they can be classified into three main approaches, each with its own heuristic - the content approach, the consequence approach and the likeness approach. Before exploring these possible solutions to the logical problem, it could be useful to dispel a couple of common confusions, since truthlikeness should not be conflated with either epistemic probability or with vagueness. We discuss this latter notion in the supplement Why truthlikeness is not probability or vagueness (see also the entry on vagueness); as for the former, we shall discuss the difference between (expected) truthlikeness and probability when discussing the epistemological problem (SS2). ### 1.2 The content approach Karl Popper was the first philosopher to take the logical problem of truthlikeness seriously enough to make an assay on it. This is not surprising, since Popper was also the first prominent realist to embrace a very radical fallibilism about science while also trumpeting the epistemic superiority of the enterprise. In his early work, he implied that the only kind of progress an inquiry can make consists in falsification of theories. This is a little depressing, to say the least. It is almost as depressing as the pessimistic induction. What it lacks is a positive account of how a succession of falsehoods might constitute positive cognitive progress. Perhaps this is why Popper's early work received a pretty short shrift from other philosophers. If a miss is as good as a mile, and all we can ever establish with confidence is that our inquiry has missed its target once again, then epistemic pessimism seems inevitable. Popper eventually realized that falsificationism is compatible with optimism provided we have an acceptable notion of verisimilitude (or truthlikeness). If some false hypotheses are closer to the truth than others, then the history of inquiry may turn out to be one of progress towards the goal of truth. Moreover, it may even be reasonable, on the basis of our evidence, to conjecture that our theories are in fact making such progress, even though we know they are all false or highly likely to be false. Popper saw clearly that the concept of truthlikeness should not be confused with the concept of epistemic probability, and that it has often been so confused. (See Popper 1963 for a history of the confusion and the supplement Why truthlikeness is not probability or vagueness for an explanation of the difference between the two concepts.) Popper's insight here was facilitated by his deep but largely unjustified antipathy to epistemic probability. He thought that his starkly falsificationist account favored bold, contentful theories. Degree of informative content varies inversely with probability - the greater the content the less likely a theory is to be true. So if you are after theories which seem, on the evidence, to be true, then you will eschew those which make bold - that is, highly improbable - predictions. On this picture, the quest for theories with high probability is simply misguided. To see this distinction between truthlikeness and probability clearly, and to articulate it, was one of Popper's most significant contributions, not only to the debate about truthlikeness, but to philosophy of science and logic in general. However, his deep antagonism to probability, combined with his love of boldness, was both a blessing and a curse. The blessing: it led him to produce not only the first interesting and important account of truthlikeness, but to initiate an approach to the problem in terms of content. The curse: content alone, as Popper envisaged it, is insufficient to characterize truthlikeness. Popper made the first attempt to solve the problem in his famous collection Conjectures and Refutations. As a great admirer of Tarski's assay on the concept of truth, he modelled his theory of truthlikeness on Tarski's theory. First, let a matter for investigation be circumscribed by a formalized language $L$ adequate for discussing it. Tarski showed us how each possible world, or model of the language, induces a partition of sentences of $L$ into those that are true and those that are false. The set of all sentences true in the actual world is thus a complete true account of the world, as far as that language goes. It is aptly called the Truth, $T$. $T$ is the target of the investigation couched in $L$. It is the theory (relative to the resources in $L)$ that we are seeking. If truthlikeness is to make sense, theories other than $T$, even false theories, come more or less close to capturing $T$. $T$, the Truth, is a theory only in the technical Tarskian sense, not in the ordinary everyday sense of that term. It is a set of sentences closed under the consequence relation: $T$ may not be finitely axiomatizable, or even axiomatizable at all. However, it is a perfectly good set of sentences all the same. In general, we will follow the Tarski-Popper usage here and call any set of sentences closed under consequence a theory, and we will assume that each proposition we deal with is identified with a theory in this sense. (Note that theory $A$ logically entails theory $B$ just in case $B$ is a subset of $A$.) The complement of $T$, the set of false sentences $F$, is not a theory even in this technical sense. Since falsehoods always entail truths, $F$ is not closed under the consequence relation. (This may be the reason why we have no expression like the Falsth: the set of false sentences does not describe a possible alternative to the actual world.) But $F$ too is a perfectly good set of sentences. The consequences of any theory $A$ that can be formulated in $L$ will thus divide between $T$ and $F$. Popper called the intersection of $A$ and $T$, the truth content of $A$ ($A\_T$), and the intersection of $A$ and $F$, the falsity content of $A$ ($A\_T$). Any theory $A$ is thus the union of its non-overlapping truth content and falsity content. Note that since every theory entails all logical truths, these will constitute a special set, at the center of $T$, which will be included in every theory, whether true or false. ![missing text, please inform](diagram1.png) Diagram 1. Truth and falsity contents of false theory $A$ A false theory will cover some of $F$, but because every false theory has true consequences, including all logical truths, it will also overlap with $T$ (Diagram 1). A true theory, however, will only overlap $T$ (Diagram 2): ![missing text, please inform](diagram2.png) Diagram 2. True theory $A$ is identical to its own truth content Amongst true theories, then, it seems that the more true sentences that are entailed, the closer we get to $T$, hence the more truthlike. Set theoretically that simply means that, where $A$ and $B$ are both true, $A$ will be more truthlike than $B$ just in case $B$ is a proper subset of $A$ (which for true theories means that $B\_T$ is a proper subset of $A\_T$). Call this principle: the value of content for truths. ![missing text, please inform](diagram3.png) Diagram 3. True theory $A$ has more truth content than true theory $B$ This essentially syntactic account of truthlikeness has some nice features. It induces a partial ordering of truths, with the whole Truth $T$ at the top of the ordering: $T$ is closer to the Truth than any other true theory. The set of logical truths is at the bottom: further from the Truth than any other true theory. In between these two extremes, true theories are ordered simply by logical strength: the more logical content, the closer to the Truth. Since probability varies inversely with logical strength, amongst truths the theory with the greatest truthlikeness $(T)$ must have the smallest probability, and the theory with the largest probability (the logical truth) is the furthest from the Truth. Popper made a simple and perhaps plausible generalization of this. Just as truth content (coverage of $T)$ counts in favor of truthlikeness, falsity content (coverage of $F)$ counts against. In general then, a theory $A$ is closer to the truth if it has more truth content without engendering more falsity content, or has less falsity content without sacrificing truth content (diagram 4): ![missing text, please inform](diagram4.png) Diagram 4. False theory $A$ closer to the Truth than false theory $B$ The generalization of the truth content comparison, by incorporating falsity content comparisons, also has some nice features. It preserves the comparisons of true theories mentioned above. The truth content $A\_T$ of a false theory $A$ (itself a theory in the Tarskian sense) will clearly be closer to the truth than $A$ (Diagram 1). And the whole truth $T$ will be closer to the truth than any falsehood $B$ because the truth content of $B$ must be contained within $T$, and the falsity content of $T$ (the empty class) must be properly contained within the non-empty falsity content of $B$. Despite its attractive features, the account has a couple of disastrous consequences. Firstly, since a falsehood has some false consequences, and no truth has any, it follows that no falsehood can be as close to the truth as a logical truth - the weakest of all truths. A logical truth leaves the location of the truth wide open, so it is rather worthless as an approximation to the whole truth. On Popper's account, no falsehood can ever be more worthwhile than a worthless logical truth. (We could call this result the absolute worthlessness of falsehoods). Furthermore, it is impossible to add a true consequence to a false theory without thereby adding additional false consequences (or subtract a false consequence without subtracting true consequences). So the account entails that no false theory is closer to the truth than any other. We could call this result the relative worthlessness of all falsehoods. These worthlessness results were proved independently by Pavel Tichy and David Miller (Miller 1974, and Tichy 1974) - for a proof, see the supplement on Why Popper's definition of truthlikeness fails: the Tichy-Miller theorem. It is tempting (and Popper was so tempted) to retreat in the face of these results to something like the comparison of truth contents alone. That is to say, $A$ is as close to the truth as $B$ if $B\_T$ is contained in $A\_T$, and $A$ is closer to the truth than $B$ just in case $B\_T$ is properly contained in $B\_T$. Call this the Simple Truth Content account. This Simple Truth Content account preserves what many consider to be the chief virtue of Popper's account: the value of content for truths. And while it delivers the absolute worthlessness of falsehoods (no falsehood is closer to the truth than a tautology) it avoids the relative worthlessness of falsehoods. If $A$ and $B$ are both false, then $A\_T$ may well properly contain $B\_T$. But that holds if and only if $A$ is logically stronger than $B$. That is to say, a false proposition is the closer to the truth the stronger it is. According to this principle - call it the value of content for falsehoods - the false proposition that there are nine planets, and all of them are made of green cheese is more truthlike than the false proposition there are nine planets. And so once one knows that a certain theory is false one can be confident that tacking on any old arbitrary proposition, no matter how inaccurate it is, will lead us inexorably closer to the truth. This is sometimes called the child's play objection. Among false theories, brute logical strength becomes the sole criterion of a theory's likeness to truth. Even though Popper's particular proposals were flawed, his idea of comparing truth-content and falsity-content is nevertheless worth exploring. Several philosophers have developed variations on the idea. Some stay within Popper's essentially syntactic paradigm, elucidating content in terms of consequence classes (e.g., Newton Smith 1981; Schurz and Weingartner 1987, 2010; Cevolani and Festa 2020). Others have switched to a semantic conception of content, construing semantic content in terms of classes of possibilities, and searching for a plausible theory of distance between those. A variant of this approach takes the class of models of a language as a surrogate for possible states of affairs (Miller 1978a). The other utilizes a semantics of incomplete possible states like those favored by structuralist accounts of scientific theories (Kuipers 1987b, Kuipers 2019). The idea which these accounts have in common is that the distance between two propositions $A$ and $B$ is measured by the symmetric difference $A\mathbin{\Delta} B$ of the two sets of possibilities: $(A - B)\cup(B - A)$. Roughly speaking, the larger the symmetric difference, the greater the distance between the two propositions. Symmetric differences might be compared qualitatively - by means of set-theoretic inclusion - or quantitatively, using some kind of probability measure. Both can be shown to have the general features of a measure of distance. The fundamental problem with the content approach lies not in the way it has been articulated, but rather in the basic underlying assumption: that truthlikeness is a function of just two variables - content and truth value. This assumption has several rather problematic consequences. Firstly, any given proposition $A$ can have only two degrees of verisimilitude: one in case it is false and the other in case it is true. This is obviously wrong. A theory can be false in very many different ways. The proposition that there are eight planets is false whether there are nine planets or a thousand planets, but its degree of truthlikeness is much higher in the first case than in the latter. Secondly, if we combine the value of content for truths and the value of content for falsehoods, then if we fix truth value, verisimilitude will vary only according to amount of content. So, for example, two equally strong false theories will have to have the same degree of verisimilitude. That's pretty far-fetched. That there are ten planets and that there are ten billion planets are (roughly) equally strong, and both are false in fact, but the latter seems much further from the truth than the former. Finally, how might strength determine verisimilitude amongst false theories? There seem to be just two plausible candidates: that verisimilitude increases with increasing strength (the principle of the value of content for falsehoods) or that it decreases with increasing strength (the principle of the disvalue of content for falsehoods). Both proposals are at odds with attractive judgements and principles, which suggest that the original content approach is in need of serious revision (see, e.g., Kuipers 2019 for a recent proposal). ### 1.3 The Consequence Approach Popper crafted his initial proposal in terms of the true and false consequences of a theory. Any sentence at all that follows from a theory is counted as a consequence that, if true, contributes to its overall truthlikeness, and if false, detracts from that. But it has struck many that this both involves an enormous amount of double counting, and that it is the indiscriminate counting of arbitrary consequences that lies behind the Tichy-Miller trivialization result. Consider a very simple framework with three primitive sentences: $h$ (for the state hot), $r$ (for rainy) and $w$ (for windy). This framework generates a very small space of eight possibilities. The eight maximal conjunctions (like $h \amp r \amp w, {\sim}h \amp r \amp w,$ etc.) of the three primitive sentences and of their negations express those possibilities. Suppose that in fact it is hot, rainy and windy (expressed by the maximal conjunction $h \amp r \amp w)$. Then the claim that it is cold, dry and still (expressed by the sentence ${\sim}h \amp{\sim}r \amp{\sim}w)$ is further from the truth than the claim that it is cold, rainy and windy (expressed by the sentence ${\sim}h \amp r \amp w)$. And the claim that it is cold, dry and windy (expressed by the sentence ${\sim}h \amp{\sim}r \amp w)$ is somewhere between the two. These kinds of judgements, which seem both innocent and intuitively correct, Popper's theory cannot accommodate. And if they are to be accommodated we cannot treat all true and false consequences alike. For the three false claims mentioned here have exactly the same number of true and false consequences (this is the problem we called the relative worthlessness of all falsehoods). Clearly, if we are going to measure closeness to truth by counting true and false consequences, some true consequences should count more than others. For example, $h$ and $r$ are both true, and ${\sim}h$ and ${\sim}r$ are false. The former should surely count in favor of a claim, and the latter against. But ${\sim}h\rightarrow{\sim}r$ is true and $h\rightarrow{\sim}r$ is false. After we have counted the truth $h$ in favor of a claim's truthlikeness and the falsehood ${\sim}r$ against it, should we also count the true consequence ${\sim}h\rightarrow{\sim}r$ in favor, and the falsehood $h\rightarrow{\sim}r$ against? Surely this is both unnecessary and misleading. And it is precisely counting sentences like these that renders Popper's account susceptible to the Tichy-Miller argument. According to the consequence approach, Popper was right in thinking that truthlikeness depends on the relative sizes of classes of true and false consequences, but erred in thinking that all consequences of a theory count the same. Some consequences are relevant, some aren't. Let $R$ be some criterion of relevance of consequences; let $A\_R$ be the set of relevant consequences of $A$. Whatever the criterion $R$ is it has to satisfy the constraint that $A$ be recoverable from (and hence equivalent to) $A\_R$. Popper's account is the limiting one - all consequences are relevant. (Popper's relevance criterion is the empty one, $P$, according to which $A\_P$ is just $A$ itself.) The relevant truth content of A (abbreviated $A\_R^T$) can be defined as $A\_R\cap T$ (or $A\cap T\_R$), and similarly the relevant falsity content of $A$ can be defined as $A\_R\cap F$. Since $A\_R = (A\_R\cap T)\cup(A\_R\cap F)$ it follows that the union of true and false relevant consequences of $A$ is equivalent to $A$. And where $A$ is true $A\_R\cap F$ is empty, so that A is equivalent to $A\_R\cap T$ alone. With this restriction to relevant consequences we can basically apply Popper's definitions: one theory is more truthlike than another if its relevant truth content is larger and its relevant falsity content no larger; or its relevant falsity content is smaller, and its relevant truth content is no smaller. This idea was first explored by Mortensen in his 1983, but he abandoned the basic idea as unworkable. Subsequent proposals within the broad program have been offered by Burger and Heidema 1994, Schurz and Weingartner 1987 and 2010, and Gemes 2007. (Gerla 2007 also uses the notion of the relevance of a "test" or factor, but his account is best located more squarely within the likeness approach.) One possible relevance criterion that the $h$-$r$-$w$ framework might suggest is atomicity. This amounts to identifying relevant consequences as basic ones, i.e., atomic sentences or their negations (Cevolani, Crupi and Festa 2011; Cevolani, Festa and Kuipers 2013). But even if we could avoid the problem of saying what it is for a sentence to be atomic, since many distinct propositions imply the same atomic sentences, this criterion would not satisfy the requirement that $A$ be equivalent to $A\_R$. For example, $(h\vee r)$ and $({\sim}h\vee{\sim}r)$, like tautologies, imply no atomic sentences at all. This latter problem can be solved by resorting to the notion of partial consequence; interestingly, the resulting account becomes virtually indentical to one version of the likeness approach (Cevolani and Festa 2020). Burger and Heidema 1994 compare theories by positive and negative sentences. A positive sentence is one that can be constructed out of $\amp,$ $\vee$ and any true basic sentence. A negative sentence is one that can be constructed out of $\amp,$ $\vee$ and any false basic sentence. Call a sentence pure if it is either positive or negative. If we take the relevance criterion to be purity, and combine that with the relevant consequence schema above, we have Burger and Heidema's proposal, which yields a reasonable set of intuitive judgments. Unfortunately purity (like atomicity) does not quite satisfy the constraint that $A$ be equivalent to the class of its relevant consequences. For example, if $h$ and $r$ are both true then $({\sim}h\vee r)$ and $(h\vee{\sim}r)$ both have the same pure consequences (namely, none). Schurz and Weingartner 2010 use the following notion of relevance $S$: being equivalent to a disjunction of atomic propositions or their negations. With this criterion they can accommodate a range of intuitive judgments in the simple weather framework that Popper's account cannot. For example, where $\gt\_S$ is the relation of greater S-truthlikeness we capture the following relations among false claims, which, on Popper's account, are mostly incommensurable: \[ (h \amp{\sim}r) \gt\_S ({\sim}r) \gt\_S ({\sim}h \amp{\sim}r). \] and \[ (h\vee r) \gt\_S ({\sim}r) \gt\_S ({\sim}h\vee{\sim}r) \gt\_S ({\sim}h \amp{\sim}r). \] The relevant consequence approach faces three major hurdles. The first is an extension problem: the approach does produce some intuitively acceptable results in a finite propositional framework, but it needs to be extended to more realistic frameworks - for example, first-order and higher-order frameworks (see Gemes 2007 for an attempt along these lines). The second is that, like Popper's original proposal, it judges no false proposition to be closer to the truth than any truth, including logical truths. Schurz and Weingartner (2010) have answered this objection by quantitatively extending their qualitative account by assigning weights to relevant consequences and summing; one problem with this is that it assumes finite consequence classes. The third involves the language-dependence of any adequate relevance criterion. This problem will be outlined and discussed below in connection with the likeness approach (SS1.4.3). ### 1.4 The Likeness Approach In the wake of the difficulties facing Popper's approach, two philosophers, working quite independently, suggested a radically different approach: one which takes the likeness in truthlikeness seriously (Tichy 1974, Hilpinen 1976). This shift from content to likeness was also marked by an immediate shift from Popper's essentially syntactic approach (something it shares with the consequence program) to a semantic approach. Traditionally the semantic contents of sentences have been taken to be non-linguistic, or rather non-syntactic, items - propositions. What propositions are is highly contested, but most agree that a proposition carves the class of possibilities into two sub-classes - those in which the proposition is true and those in which it is false. Call the class of worlds in which the proposition is true its range. Some have proposed that propositions be identified with their ranges (for example, David Lewis, in his 1986). This is implausible since, for example, the informative content of $7+5=12$ seems distinct from the informative content of $12=12$, which in turn seems distinct from the informative content of Godel's first incompleteness theorem - and yet all three have the same range: they are all true in all possible worlds. Clearly, if semantic content is supposed to be sensitive to informative content, classes of possible worlds will are not discriminating enough. We need something more fine-grained for a full theory of semantic content. Despite this, the range of a proposition is certainly an important aspect of informative content, and it is not immediately obvious why truthlikeness should be sensitive to differences in the way a proposition picks out its range. (Perhaps there are cases of logical falsehoods some of which seem further from the truth than others. For example $7+5=113$ might be considered further from the truth than $7+5=13$ though both have the same range - namely, the empty set of worlds; see Sorensen 2007.) But as a first approximation, we will assume that it is not hyperintensional and that logically equivalent propositions have the same degree of truthlikess. The proposition that the number of planets is eight for example, should have the same degree of truthlikeness as the proposition that the square of the number of the planets is sixty four. Leaving apart the controversy over the nature of possible worlds, we shall call the complete collection of possibilities, given some array of features, the logical space, and call the array of properties and relations which underlie that logical space, the framework of the space. Familiar logical relations and operations correspond to well-understood set-theoretic relations and operations on ranges. The range of the conjunction of two propositions is the intersection of the ranges of the two conjuncts. Entailment corresponds to the subset relation on ranges. The actual world is a single point in logical space - a complete specification of every matter of fact (with respect to the framework of features) - and a proposition is true if its range contains the actual world, false otherwise. The whole Truth is a true proposition that is also complete: it entails all true propositions. The range of the Truth is none other than the singleton of the actual world. That singleton is the target, the bullseye, the thing at which the most comprehensive inquiry is aiming. Without additional structure on the logical space we have just three factors for a theorist of truthlikeness to work with - the size of a proposition (content factor), whether it contains the actual world (truth factor), and which propositions it implies (consequence factor). The likeness approach requires some additional structure to the logical space. For example, worlds might be more or less like other worlds. There might be a betweenness relation amongst worlds, or even a fully-fledged distance metric. If that's the case, we can start to see how one proposition might be closer to the Truth - the proposition whose range contains just the actual world - than another. The core of the likeness approach is that truthlikeness supervenes on the likeness between worlds. The likeness theorist has two tasks: firstly, making it plausible that there is an appropriate likeness or distance function on worlds; and secondly, extending likeness between individual worlds to likeness of propositions (i.e., sets of worlds) to the actual world. Suppose, for example, that worlds are arranged in similarity spheres nested around the actual world, familiar from the Stalnaker-Lewis approach to counterfactuals. Consider Diagram 5. ![missing text, please inform](diagram5.png) Diagram 5. Verisimilitude by similarity circles The bullseye is the actual world and the small sphere which includes it is $T$, the Truth. The nested spheres represent likeness to the actual world. A world is less like the actual world the larger the first sphere of which it is a member. Propositions $A$ and $B$ are false, $C$ and $D$ are true. A carves out a class of worlds which are rather close to the actual world - all within spheres two to four - whereas $B$ carves out a class rather far from the actual world - all within spheres five to seven. Intuitively $A$ is closer to the bullseye than is $B$. The largest sphere which does not overlap at all with a proposition is plausibly a measure of how close the proposition is to being true. Call that the truth factor. A proposition $X$ is closer to being true than $Y$ if the truth factor of $X$ is included in the truth factor of $Y$. The truth factor of $A$, for example, is the smallest non-empty sphere, $T$ itself, whereas the truth factor of $B$ is the fourth sphere, of which $T$ is a proper subset: so $A$ is closer to being true than $B$. If a proposition includes the bullseye, then of course it is true simpliciter and it has the maximal truth factor (the empty set). So all true propositions are equally close to being true. But truthlikeness is not just a matter of being close to being true. The tautology, $D,$ $C$ and the Truth itself are equally true, but in that order they increase in their closeness to the whole truth. Taking a leaf out of Popper's book, Hilpinen argued that closeness to the whole truth is in part a matter of degree of informativeness of a proposition. In the case of the true propositions, this correlates roughly with the smallest sphere which totally includes the proposition. The further out the outermost sphere, the less informative the proposition is, because the larger the area of the logical space which it covers. So, in a way which echoes Popper's account, we could take truthlikeness to be a combination of a truth factor (given by the likeness of that world in the range of a proposition that is closest to the actual world) and a content factor (given by the likeness of that world in the range of a proposition that is furthest from the actual world): > > $A$ is closer to the truth than $B$ if and only if $A$ does as > well as $B$ on both truth factor and content factor, and better on > at least one of those. > Applying Hilpinen's definition we capture two more particular judgements, in addition to those already mentioned, that seem intuitively acceptable: that $C$ is closer to the truth than $A$, and that $D$ is closer than $B$. (Note, however, that we have here a partial ordering: $A$ and $D$, for example, are not ranked.) We can derive from this various apparently desirable features of the relation closer to the truth: for example, that the relation is transitive, asymmetric and irreflexive; that the Truth is closer to the Truth than any other theory; that the tautology is at least as far from the Truth as any other truth; that one cannot make a true theory worse by strengthening it by a truth (a weak version of the value of content for truths); that a falsehood is not necessarily improved by adding another falsehood, or even by adding another truth (a repudiation of the value of content for falsehoods). But there are also some worrying features here. While it avoids the relative worthlessness of falsehoods, Hilpinen's account, just like Popper's, entails the absolute worthlessness of all falsehoods: no falsehood is closer to the truth than any truth. So, for example, Newton's theory is deemed to be no more truthlike, no closer to the whole truth, than the tautology. Characterizing Hilpinen's account as a combination of a truth factor and an information factor seems to mask its quite radical departure from Popper's account. The incorporation of similarity spheres signals a fundamental break with the pure content approach, and opens up a range of possible new accounts: what such accounts have in common is that the truthlikeness of a proposition is a non-trivial function of the likeness to the actual world of worlds in the range of the proposition. There are three main problems for any concrete proposal within the likeness approach. The first concerns an account of likeness between states of affairs - in what does this consist and how can it be analyzed or defined? The second concerns the dependence of the truthlikeness of a proposition on the likeness of worlds in its range to the actual world: what is the correct function? (This can be called "the extension problem".) And finally, there is the famous problem of "translation variance" or "framework dependence" of judgements of likeness and of truthlikeness. This last problem will be taken up in SS1.4.3. #### 1.4.1 Likeness of worlds in a simple propositional framework One objection to Hilpinen's proposal (like Lewis's proposal for counterfactuals) is that it assumes the similarity relation on worlds as a primitive, there for the taking. At the end of his 1974 paper, Tichy not only suggested the use of similarity rankings on worlds, but also provided a ranking in propositional frameworks and indicated how to generalize this to more complex frameworks. Examples and counterexamples in Tichy 1974 are exceedingly simple, utilizing the little propositional framework introduced above, with three primitives - $h$ (for the state hot), $r$ (for rainy) and $w$ (for windy). Corresponding to the eight-members of the logical space generated by distributions of truth values through the three basic conditions, there are eight maximal conjunctions (or constituents): \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| \| $w\_1$ \| $h \amp r \amp w$ \| \| $w\_5$ \| ${\sim}h \amp r \amp w$ \| \| $w\_2$ \| $h \amp r \amp{\sim}w$ \| \| $w\_6$ \| ${\sim}h \amp r \amp{\sim}w$ \| \| $w\_3$ \| $h \amp{\sim}r \amp w$ \| \| $w\_7$ \| ${\sim}h \amp{\sim}r \amp w$ \| \| $w\_4$ \| $h \amp{\sim}r \amp{\sim}w$ \| \| $w\_8$ \| ${\sim}h \amp{\sim}r \amp{\sim}w$ \| Worlds differ in the distributions of these traits, and a natural, albeit simple, suggestion is to measure the likeness between two worlds by the number of agreements on traits. This is tantamount to taking distance to be measured by the size of the symmetric difference of generating states - the so-called city-block measure. As is well known, this will generate a genuine metric, in particular satisfying the triangular inequality. If $w\_1$ is the actual world this immediately induces a system of nested spheres, but one in which the spheres come with numbers attached: ![missing text, please inform](diagram6.png) Diagram 6. Similarity circles for the weather space Those worlds orbiting on the sphere $n$ are of distance $n$ from the actual world. In fact, the structure of the space is better represented not by similarity circles, but rather by a three-dimensional cube: ![missing text](diagram7.png) Diagram 7. The three-dimensional weather space This way of representing the space makes a clearer connection between distances between worlds and the role of the atomic propositions in generating those distances through the city-block metric. It also eliminates inaccuracies in the relations between the worlds that are not at the center that the similarity circle diagram suggests. #### 1.4.2 The likeness of a proposition to the truth Now that we have numerical distances between worlds, numerical measures of propositional likeness to, and distance from, the truth can be defined as some function of the distances, from the actual world, of worlds in the range of a proposition. But which function is the right one? This is the extension problem. Suppose that $h \amp r \amp w$ is the whole truth about the weather. Following Hilpinen, we might consider overall distance of a propositions from the truth to be some function of the distances from actuality of two extreme worlds. Let truth$(A)$ be the truth value of $A$ in the actual world. Let min$(A)$ be the distance from actuality of that world in $A$ closest to the actual world, and max$(A)$ be the distance from actuality of that world in $A$ furthest from the actual world. Table 1 display the values of the min and max functions for some representative propositions. \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| $\boldsymbol{A}$ \| **truth$(\boldsymbol{A})$ \| min$(\boldsymbol{A})$ \| max$(\boldsymbol{A})$** \| \| $h \amp r \amp w$ \| true \| 0 \| 0 \| \| $h \amp r$ \| true \| 0 \| 1 \| \| $h \amp r \amp{\sim}$w \| false \| 1 \| 1 \| \| $h$ \| true \| 0 \| 2 \| \| $h \amp{\sim}r$ \| false \| 1 \| 2 \| \| ${\sim}h$ \| false \| 1 \| 3 \| \| ${\sim}h \amp{\sim}r \amp w$ \| false \| 2 \| 2 \| \| ${\sim}h \amp{\sim}r$ \| false \| 2 \| 3 \| \| ${\sim}h \amp{\sim}r \amp{\sim}w$ \| false \| 3 \| 3 \| Table 1. The min and max functions. The simplest proposal (made first in Niiniluoto 1977) would be to take the average of the min and the max (call this measure min-max-average). This would remedy a rather glaring shortcoming which Hilpinen's qualitative proposal shares with Popper's proposal, namely that no falsehood is closer to the truth than any truth (even the worthless tautology). This numerical equivalent of Hilpinen's proposal renders all propositions comparable for truthlikeness, and some falsehoods it deems more truthlike than some truths. But now that we have distances between all worlds, why take only the extreme worlds in a proposition into account? Why shouldn't every world in a proposition potentially count towards its overall distance from the actual world? A simple measure which does count all worlds is average distance from the actual world. Average delivers all of the particular judgements we used above to motivate Hilpinen's proposal in the first place, and in conjunction with the simple metric on worlds it delivers the following ordering of propositions in our simple framework: \| \| \| \| \| --- \| --- \| --- \| \| $\boldsymbol{A}$ \| **truth$(\boldsymbol{A})$ \| average$(\boldsymbol{A})$** \| \| $h \amp r \amp w$ \| true \| 0 \| \| $h \amp r$ \| true \| 0.5 \| \| $h \amp r \amp{\sim}$w \| false \| 1.0 \| \| $h$ \| true \| 1.3 \| \| $h \amp{\sim}r$ \| false \| 1.5 \| \| ${\sim}h$ \| false \| 1.7 \| \| ${\sim}h \amp{\sim}r \amp w$ \| false \| 2.0 \| \| ${\sim}h \amp{\sim}r$ \| false \| 2.5 \| \| ${\sim}h \amp{\sim}r \amp{\sim}w$ \| false \| 3.0 \| Table 2. The average function. This ordering looks promising. Propositions are closer to the truth the more they get the basic weather traits right, further away the more mistakes they make. A false proposition may be made either worse or better by strengthening $(h \amp r \amp{\sim}w$ is better than ${\sim}w$ while ${\sim}h \amp{\sim}r \amp{\sim}w$ is worse). A false proposition (like $h \amp r \amp{\sim}w)$ can be closer to the truth than some true propositions (like $h)$. And so on. These judgments may be sufficient to show that average is superior to min-max-average, at least on this group of propositions, but they are clearly not sufficient to show that averaging is the right procedure. What we need are some straightforward and compelling general desiderata which jointly yield a single correct function. In the absence of such a proof, we can only resort to case by case comparisons. Furthermore average has not found universal favor. Notably, there are pairs of true propositions such that average deems the stronger of the two to be the further from the truth. According to average, the tautology is not the true proposition furthest from the truth. Averaging thus violates the Popperian principle of the value of content for truths, see Table 3 for an example. \| \| \| \| \| --- \| --- \| --- \| \| $\boldsymbol{A}$ \| **truth$(\boldsymbol{A})$ \| average$(\boldsymbol{A})$** \| \| $h \vee{\sim}r \vee w$ \| true \| 1.4 \| \| $h \vee{\sim}r$ \| true \| 1.5 \| \| $h \vee{\sim}h$ \| true \| 1.5 \| Table 3. average violates the value of content for truths. Let's then consider other measures, like the sum function - the sum of the distances of worlds in the range of a proposition from the actual world (Table 4). \| \| \| \| \| --- \| --- \| --- \| \| $\boldsymbol{A}$ \| **truth$(\boldsymbol{A})$ \| sum$(\boldsymbol{A})$** \| \| $h \amp r \amp w$ \| true \| 0 \| \| $h \amp r$ \| true \| 1 \| \| $h \amp r \amp{\sim}$w \| false \| 1 \| \| $h$ \| true \| 4 \| \| $h \amp{\sim}r$ \| false \| 3 \| \| ${\sim}h$ \| false \| 8 \| \| ${\sim}h \amp{\sim}r \amp w$ \| false \| 2 \| \| ${\sim}h \amp{\sim}r$ \| false \| 5 \| \| $h \amp{\sim}r \amp{\sim}w$ \| false \| 3 \| Table 4. The sum function. The sum function is an interesting measure in its own right. While, like average, it is sensitive to the distances of all worlds in a proposition from the actual world, it is not plausible as a measure of distance from the truth, and indeed no one has proposed it as such a measure. What sum does measure is a certain kind of distance-weighted logical weakness. In general the weaker a proposition is, the larger its sum value. But adding words far from the actual world makes the sum value larger than adding worlds closer to it. This guarantees, for example, that of two truths the sum of the logically weaker is always greater than the sum of the stronger. Thus sum might play a role in capturing the value of content for truths. But it also delivers the implausible value of content for falsehoods. If you think that there is anything to the likeness program it is hardly plausible that the falsehood ${\sim}h \amp{\sim}r \amp{\sim}w$ is closer to the truth than its consequence ${\sim}h$. Niiniluoto argues that sum is a good likeness-based candidate for measuring Hilpinen's "information factor". It is obviously much more sensitive than is max to the proposition's informativeness about the location of the truth. Niiniluoto thus proposes, as a measure of distance from the truth, the average of this information factor and Hilpinen's truth factor: min-sum-average. Averaging the more sensitive information factor (sum) and the closeness-to-being-true factor (min) yields some interesting results (see Table 5). \| \| \| \| \| --- \| --- \| --- \| \| $\boldsymbol{A}$ \| **truth$(\boldsymbol{A})$ \| min-sum-average$(\boldsymbol{A})$** \| \| $h \amp r \amp w$ \| true \| 0 \| \| $h \amp r$ \| true \| 0.5 \| \| $h \amp r \amp{\sim}$w \| false \| 1 \| \| $h$ \| true \| 2 \| \| $h \amp{\sim}r$ \| false \| 2 \| \| ${\sim}h$ \| false \| 4.5 \| \| ${\sim}h \amp{\sim}r \amp w$ \| false \| 2 \| \| ${\sim}h \amp{\sim}r$ \| false \| 3.5 \| \| ${\sim}h \amp{\sim}r \amp{\sim}w$ \| false \| 3 \| Table 5. The min-sum-average function. For example, this measure deems $h \amp r \amp w$ more truthlike than $h \amp r$, and the latter more truthlike than $h$. And in general min-sum-average delivers the value of content for truths. For any two truths the min factor is the same (0), and the sum factor increases as content decreases. Furthermore, unlike the symmetric difference measures, min-sum-average doesn't deliver the objectionable value of contents for falsehoods. For example, ${\sim}h \amp{\sim}r \amp{\sim}w$ is deemed further from the truth than ${\sim}h$. But min-sum-average is not quite home free, at least from an intuitive point of view. For example, ${\sim}h \amp{\sim}r \amp{\sim}w$ is deemed closer to the truth than ${\sim}h \amp{\sim}r$. This is because what ${\sim}h \amp{\sim}r \amp{\sim}w$ loses in closeness to the actual world (min) it makes up for by an increase in strength (sum). In deciding how to proceed here we confront a methodological problem. The methodology favored by Tichy is very much bottom-up. For the purposes of deciding between rival accounts it takes the intuitive data very seriously. Popper (along with Popperians like Miller) favor a more top-down approach. They are suspicious of folk intuitions, and sometimes appear to be in the business of constructing a new concept rather than explicating an existing one. They place enormous weight on certain plausible general principles, largely those that fit in with other principles of their overall theory of science: for example, the principle that strength is a virtue and that the stronger of two true theories (and maybe even of two false theories) is the closer to the truth. A third approach, one which lies between these two extremes, is that of reflective equilibrium. This recognizes the claims of both intuitive judgements on low-level cases, and plausible high-level principles, and enjoins us to bring principle and judgement into equilibrium, possibly by tinkering with both. Neither intuitive low-level judgements nor plausible high-level principles are given advance priority. The protagonist in the truthlikeness debate who has argued most consistently for this approach is Niiniluoto. How might reflective equilibrium be employed to help resolve the current dispute? Consider a different space of possibilities, generated by a single magnitude like the number of the planets $(N).$ Suppose that $N$ is in fact 8 and that the further $n$ is from 8, the further the proposition that $N=n$ from the Truth. Consider the three sets of propositions in Table 6. In the left-hand column we have a sequence of false propositions which, intuitively, decrease in truthlikeness while increasing in strength. In the middle column we have a sequence of corresponding true propositions, in each case the strongest true consequence of its false counterpart on the left (Popper's "truth content"). Again members of this sequence steadily increase in strength. Finally on the right we have another column of falsehoods. These are also steadily increasing in strength, and like the left-hand falsehoods, seem (intuitively) to be decreasing in truthlikeness as well. \| \| \| \| \| --- \| --- \| --- \| \| Falsehood (1) \| Strongest True Consequence \| Falsehood (2) \| \| $10 \le N \le 20$ \| $N=8$ or $10 \le N \le 20$ \| $N=9$ or $10 \le N \le 20$ \| \| $11 \le N \le 20$ \| $N=8$ or $11 \le N \le 20$ \| $N=9$ or $11 \le N \le 20$ \| \| ...... \| ...... \| ...... \| \| $19 \le N \le 20$ \| $N=8$ or $19 \le N \le 20$ \| $N=9$ or $19 \le N \le 20$ \| \| $N= 20$ \| $N=8$ or $N= 20$ \| $N=9$ or $N= 20$ \| Table 6. Judgements about the closeness of the true propositions in the center column to the truth may be less intuitively clear than are judgments about their left-hand counterparts. However, it would seem highly incongruous to judge the truths in Table 6 to be steadily increasing in truthlikeness, while the falsehoods both to the left and the right, both marginally different in their overall likeness relations to truth, steadily decrease in truthlikeness. This suggests that that all three are sequences of steadily increasing strength combined with steadily decreasing truthlikeness. And if that's right, it might be enough to overturn Popper's principle that amongst true theories strength and truthlikeness must covary (even while granting that this is not so for falsehoods). If this argument is sound, it removes an objection to averaging distances, but it does not settle the issue in its favor, for there may still be other more plausible counterexamples to averaging that we have not considered. Schurz and Weingartner argue that this extension problem is the main defect of the likeness approach: > > the problem of extending truthlikeness from possible worlds to > propositions is intuitively underdetermined. Even if we are granted an > ordering or a measure of distance on worlds, there are many very > different ways of extending that to propositional distance, and > apparently no objective way to decide between them. (Schurz and > Weingartner 2010, 423) > One way of answering this objection head on is to identify principles that, given a distance function on worlds, constrain the distances between worlds and sets of worlds, principles perhaps powerful enough to identify a unique extension. Apart from the extension problem, two other issues affect the likeness approach. The first is how to apply it beyond simple propositional examples as the ones considered above (Popper's content approach, whatever else its shortcomings, can be applied in principle to theories expressible in any language, no matter how sophisticated). We discuss this in the supplement on Extending the likeness approach to first-order and higher-order frameworks. The second has to do with the fact that assessments of relative likeness are sensitive to how the framework underlying the logical space is defined. This "framework dependence" issue is discussed in the next section. #### 1.4.3 The framework dependence of likeness The single most powerful and influential argument against the whole likeness approach is the charge that it is "language dependent" or "framework dependent" (Miller 1974a, 197 a, 1976, and most recently defended, vigorously as usual, in his 2006). Early formulations of the likeness approach (Tichy 1974, 1976, Niiniluoto 1976) proceeded in terms of syntactic surrogates for their semantic correlates - sentences for propositions, predicates for properties, constituents for partitions of the logical space, and the like. The question naturally arises, then, whether we obtain the same measures if all the syntactic items are translated into an essentially equivalent language - one capable of expressing the same propositions and properties with a different set of primitive predicates. Newton's theory can be formulated with a variety of different primitive concepts, but these formulations are typically taken to be equivalent. If the degree of truthlikeness of Newton's theory were to vary from one such formulation to another, then while such a concept might still might have useful applications, it would hardly help to vindicate realism. Take our simple weather-framework above. This trafficks in three primitives - hot, rainy, and windy. Suppose, however, that we define the following two new weather conditions: > > > minnesotan $ =\_{df}$ hot if and only if > rainy > > > > arizonan $ =\_{df}$ hot if and only if > windy > > > Now it appears as though we can describe the same sets of weather states in the new $h$-$m$-$a$-ese language based on the above conditions. Table 7 shows the translations of four representative theories between the two languages. \| \| \| \| \| --- \| --- \| --- \| \| \| $h$-$r$-$w$-ese \| $h$-$m$-$a$-ese \| \| $T$ \| $h \amp r \amp w$ \| $h \amp m \amp a$ \| \| $A$ \| ${\sim}h \amp r \amp w$ \| ${\sim}h \amp{\sim}m \amp{\sim}a$ \| \| $B$ \| ${\sim}h \amp{\sim}r \amp w$ \| ${\sim}h \amp m \amp{\sim}a$ \| \| $C$ \| ${\sim}h \amp{\sim}r \amp{\sim}w$ \| ${\sim}h \amp m \amp a$ \| Table 7. If $T$ is the truth about the weather then theory $A$, in $h$-$r$-$w$-ese, seems to make just one error concerning the original weather states, while $B$ makes two and $C$ makes three. However, if we express these two theories in $h$-$m$-$a$-ese however, then this is reversed: $A$ appears to make three errors and $B$ still makes two and $C$ makes only one error. But that means the account makes truthlikeness, unlike truth, radically language-relative. There are two live responses to this criticism. But before detailing them, note a dead one: the likeness theorist cannot object that $h$-$m$-$a$ is somehow logically inferior to $h$-$r$-$w$, on the grounds that the primitives of the latter are essentially "biconditional" whereas the primitives of the former are not. This is because there is a perfect symmetry between the two sets of primitives. Starting within $h$-$m$-$a$-ese we can arrive at the original primitives by exactly analogous definitions: > > rainy $ =\_{df}$ hot if and only if > minnesotan > > > windy $ =\_{df}$ hot if and only if > arizonan > Thus if we are going to object to $h$-$m$-$a$-ese it will have to be on other than purely logical grounds. Firstly, then, the likeness theorist could maintain that certain predicates (presumably "hot", "rainy" and "windy") are primitive in some absolute, realist, sense. Such predicates "carve reality at the joints" whereas others (like "minnesotan" and "arizonan") are gerrymandered affairs. With the demise of predicate nominalism as a viable account of properties and relations this approach is not as unattractive as it might have seemed in the middle of the last century. Realism about universals is certainly on the rise. While this version of realism presupposes a sparse theory of properties - that is to say, it is not the case that to every definable predicate there corresponds a genuine universal - such theories have been championed both by those doing traditional a priori metaphysics of properties (e.g. Bealer 1982) as well as those who favor a more empiricist, scientifically informed approach (e.g. Armstrong 1978, Tooley 1977). According to Armstrong, for example, which predicates pick out genuine universals is a matter for developed science. The primitive predicates of our best fundamental physical theory will give us our best guess at what the genuine universals in nature are. They might be predicates like electron or mass, or more likely something even more abstruse and remote from the phenomena - like the primitives of String Theory. One apparently cogent objection to this realist solution is that it would render the task of empirically estimating degree of truthlikeness completely hopeless. If we know a priori which primitives should be used in the computation of distances between theories it will be difficult to estimate truthlikeness, but not impossible. For example, we might compute the distance of a theory from the various possibilities for the truth, and then make a weighted average, weighting each possible true theory by its probability on the evidence. That would be the credence-mean estimate of truthlikeness (see SS2). However, if we don't even know which features should count towards the computation of similarities and distances then it appears that we cannot get off first base. To see this consider our simple weather frameworks. Suppose that all I learn is that it is rainy. Do I thereby have some grounds for thinking $A$ is closer to the truth than $B$? I would if I also knew that $h$-$r$-$w$-ese is the language for calculating distances. For then, whatever the truth is, $A$ makes one fewer mistake than $B$ makes. $A$ gets it right on the rain factor, while $B$ doesn't, and they must score the same on the other two factors whatever the truth of the matter. But if we switch to $h$-$m$-$a$-ese then $A$'s epistemic superiority is no longer guaranteed. If, for example, $T$ is the truth then $B$ will be closer to the truth than $A$. That's because in the $h$-$m$-$a$ framework raininess as such doesn't count in favor or against the truthlikeness of a proposition. This objection would fail if there were empirical indicators not just of which atomic states obtain, but also of which are the genuine ones, the ones that really carve reality at the joints. Obviously the framework would have to contain more than just $h, m$ and $a$. It would have to contain resources for describing the states that indicate whether these were genuine universals. Maybe whether they enter into genuine causal relations will be crucial, for example. Once we can distribute probabilities over the candidates for the real universals, then we can use those probabilities to weight the various possible distances which a hypothesis might be from any given theory. The second live response is both more modest and more radical. It is more modest in that it is not hostage to the objective priority of a particular conceptual scheme, whether that priority is accessed a priori or a posteriori. It is more radical in that it denies a premise of the invariance argument that at first blush is apparently obvious. It denies the equivalence of the two conceptual schemes. It denies that $h \amp r \amp w$, for example, expresses the very same proposition as $h \amp m \amp a$ expresses. If we deny translatability then we can grant the invariance principle, and grant the judgements of distance in both cases, but remain untroubled. There is no contradiction (Tichy 1978). At first blush this response seems somewhat desperate. Haven't the respective conditions been defined in such a way that they are simple equivalents by fiat? That would, of course, be the case if $m$ and $a$ had been introduced as defined terms into $h$-$r$-$w$. But if that were the intention then the likeness theorist could retort that the calculation of distances should proceed in terms of the primitives, not the introduced terms. However, that is not the only way the argument can be read. We are asked to contemplate two partially overlapping sequences of conditions, and two spaces of possibilities generated by those two sequences. We can thus think of each possibility as a point in a simple three dimensional space. These points are ordered triples of 0s and 1s, the $n$th entry being 0 if the $n$th condition is satisfied and 1 if it isn't. Thinking of possibilities in this way, we already have rudimentary geometrical features generated simply by the selection of generating conditions. Points are adjacent if they differ on only one dimension. A path is a sequence of adjacent points. A point $q$ is between two points $p$ and $r$ if $q$ lies on a shortest path from $p$ to $r$. A region of possibility space is convex if it is closed under the betweenness relation - anything between two points in the region is also in the region (Oddie 1987, Goldstick and O'Neill 1988). Evidently we have two spaces of possibilities, S1 and S2, and the question now arises whether a sentence interpreted over one of these spaces expresses the very same thing as any sentence interpreted over the other. Does $h \amp r \amp w$ express the same thing as $h \amp m \amp a$? $h \amp r \amp w$ expresses (the singleton of) $u\_1$ (which is the entity $\langle 1,1,1\rangle$ in S1 or $\langle 1,1,1\rangle\_{S1})$ and $h \amp m \amp a$ expresses $v\_1$ (the entity $\langle 1,1,1\rangle\_{S2}). {\sim}h \amp r \amp w$ expresses $u\_2 (\langle 0,1,1\rangle\_{S1})$, a point adjacent to that expressed by $h \amp r \amp w$. However ${\sim}h \amp{\sim}m \amp{\sim}a$ expresses $v\_8 (\langle 0,0,0\rangle\_{S2})$, which is not adjacent to $v\_1 (\langle 1,1,1\rangle\_{S2})$. So now we can construct a simple proof that the two sentences do not express the same thing. * $u\_1$ is adjacent to $u\_2$. * $v\_1$ is not adjacent to $v\_8$. * Therefore, either $u\_1$ is not identical to $v\_1$, or $u\_2$ is not identical $v\_8$. * Therefore, either $h \amp r \amp w$ and $h \amp m \amp a$ do not express the same proposition, or ${\sim}h \amp r \amp w$ and ${\sim}h \amp{\sim}m \amp{\sim}a$ do not express the same proposition. Thus at least one of the two required intertranslatability claims fails, and $h$-$r$-$w$-ese is not intertranslatable with $h$-$m$-$a$-ese. The important point here is that a space of possibilities already comes with a structure and the points in such a space cannot be individuated without reference to rest of the space and its structure. The identity of a possibility is bound up with its geometrical relations to other possibilities. Different relations, different possibilities. This kind of response has also been endorsed in the very different truth-maker proposal put forward in Fine (2021, 2022). This kind of rebuttal to the Miller argument would have radical implications for the comparability of actual theories that appear to be constructed from quite different sets of primitives. Classical mechanics can be formulated using mass and position as basic, or it can be formulated using mass and momentum. The classical concepts of velocity and of mass are different from their relativistic counterparts, even if they were "intertranslatable" in the way that the concepts of $h$-$r$-$w$-ese are intertranslatable with $h$-$m$-$a$-ese. This idea meshes well with recent work on conceptual spaces in Gardenfors (2000). Gardenfors is concerned both with the semantics and the nature of genuine properties, and his bold and simple hypothesis is that properties carve out convex regions of an $n$-dimensional quality space. He supports this hypothesis with an impressive array of logical, linguistic and empirical data. (Looking back at our little spaces above it is not hard to see that the convex regions are those that correspond to the generating (or atomic) conditions and conjunctions of those. See Burger and Heidema 1994.) While Gardenfors is dealing with properties, it is not hard to see that similar considerations apply to propositions, since propositions can be regarded as 0-ary properties. Ultimately, however, this response may seem less than entirely satisfactory by itself. If the choice of a conceptual space is merely a matter of taste then we may be forced to embrace a radical kind of incommensurability. Those who talk $h$-$r$-$w$-ese and conjecture ${\sim}h \amp r \amp w$ on the basis of the available evidence will be close to the truth. Those who talk $h$-$m$-$a$-ese while exposed to the "same" circumstances would presumably conjecture ${\sim}h \amp{\sim}m \amp{\sim}a$ on the basis of the "same" evidence (or the corresponding evidence that they gather). If in fact $h \amp r \amp w$ is the truth (in $h$-$r$-$w$-ese) then the $h$-$r$-$w$ weather researchers will be close to the truth. But the $h$-$m$-$a$ researchers will be very far from the truth. This may not be an explicit contradiction, but it should be worrying all the same. Realists started out with the ambition of defending a concept of truthlikeness which would enable them to embrace both fallibilism and optimism. But what the likeness theorists seem to have ended up with here is something that suggests a rather unpalatable incommensurability of competing conceptual frameworks. To avoid this, the realist will need to affirm that some conceptual frameworks really are better than others. Some really do "carve reality at the joints" and others don't. But is that something the realist should be reluctant to affirm? ## 2. The Epistemological Problem The quest to nail down a viable concept of truthlikeness is motivated, at least in part, by fallibilism (SS1.1). It is certainly true that a viable notion of distance from the truth renders progress in an inquiry through a succession of false theories at least possible. It is also true that if there is no such viable notion, then truth can be retained as the goal of inquiry only at the cost of making partial progress towards it virtually impossible. But does the mere possibility of making progress towards the truth improve our epistemic lot? Some have argued that it doesn't (see for example Laudan 1977, Cohen 1980, Newton-Smith 1981). One common argument can be recast in the form of a simple dilemma. Either we can ascertain the truth, or we can't. If we can ascertain the truth then we do not need a concept of truthlikeness - it is an entirely useless addition to our intellectual repertoire. But if we cannot ascertain the truth, then we cannot ascertain the degree of truthlikeness of our theories either. So again, the concept is useless for all practical purposes. (See the entry on scientific progress, especially SS2.4.) Consider the second horn of this dilemma. Is it true that if we can't know what the (whole) truth of some matter is, we also cannot ascertain whether or not we are making progress towards it? Suppose you are interested in the truth about the weather tomorrow. Suppose you learn (from a highly reliable source) that it will be hot. Even though you don't know the whole truth about the weather tomorrow, you do know that you have added a truth to your existing corpus of weather beliefs. One does not need to be able to ascertain the whole truth to ascertain some less-encompassing truths. And it seems to follow that you can also know you have made at least some progress towards the whole weather truth. This rebuttal is too swift. It presupposes that the addition of a new truth $A$ to an existing corpus $K$ guarantees that your revised belief $K$\$A$ constitutes progress towards the truth. But whether or not $K$\$A$ is closer to the truth than $K$ depends not only on a theory of truthlikeness but also on a theory of belief revision. (See also the entry on the logic of belief revision.) Let's consider a simple case. Suppose $A$ is some newly discovered truth, and that $A$ is compatible with $K$. Assume that belief revision in such cases is simply a matter of so-called expansion - i.e., conjoining $A$ to $K$. Consider the case in which $K$ also happens to be true. Then any account of truthlikeness that endorses the value of content for truths (e.g. Niiniluoto's min-sum-average) guarantees that $K$\$A$ is closer to the truth than $K$. That's a welcome result, but it has rather limited application. Typically one doesn't know that $K$ is true: so even if one knows that $A$ is true, one cannot use this fact to celebrate progress. The situation is more dire when it comes to falsehoods. If $K$ is in fact false then, without the disastrous principle of the value of content for falsehoods, there is certainly no guarantee that $K$\$A$ will constitute a step toward the truth. (And even if one endorsed the disastrous principle one would hardly be better off. For then the addition of any proposition, whether true or false, would constitute an improvement on a false theory.) Consider again the number of the planets, $N$. Suppose that the truth is N$=8$, and that your existing corpus $K$ is ($N=7 \vee N=100)$. Suppose you somehow acquire the truth $A$: $N\gt 7$. Then $K$\$A$ is $N=100$, which (on average, min-max-average* and min-sum-average) is further from the truth than $K$. So revising a false theory by adding truths by no means guarantees progress towards the truth. For theories that reject the value of content for truths (e.g., the average proposal) the situation is worse still. Even if $K$ happens to be true, there is no guarantee that expanding $K$ with truths will constitute progress. Of course, there will be certain general conditions under which the value of content for truths holds. For example, on the average proposal, the expansion of a true $K$ by an atomic truth (or, more generally, by a convex truth) will guarantee progress toward the truth. So under very special conditions, one can know that the acquisition of a truth will enhance the overall truthlikeness of one's theories, but these conditions are exceptionally narrow and provide at best a very weak defense against the dilemma. (See Niiniluoto 2011. For rather more optimistic views of the relation between truthlikeness and belief revision see Kuipers 2000, Lavalette, Renardel & Zwart 2011, Cevolani, Crupi and Festa 2011, and Cevolani, Festa and Kuipers 2013.) A different tack is to deny that a concept is useless if there is no effective empirical decision procedure for ascertaining whether it applies. For even if we cannot know for sure what the value of a certain unobservable magnitude is, we might well have better or worse estimates of the value of the magnitude on the evidence. And that may be all we need for the concept to be of practical value. Consider, for example, the propensity of a certain coin-tossing set-up to produce heads - a magnitude which, for the sake of the example, we assume to be not directly observable. Any non-extreme value of this magnitude is compatible with any number of heads in a sequence of $n$ tosses. So we can never know with certainty what the actual propensity is, no matter how many tosses we observe. But we can certainly make rational estimates of the propensity on the basis of the accumulating evidence. Suppose one's initial state of ignorance of the propensity is represented by an even distribution of credences over the space of possibilities for the propensity (i.e., the unit interval). Using Bayes theorem and the Principal Principle, after a fairly small number of tosses we can become quite confident that the propensity lies in a small interval around the observed relative frequency. Our best estimate of the value of the magnitude is its expected value on the evidence. Similarly, suppose we don't and perhaps cannot know which constituent is in fact true. But suppose that we do have a good measure of distance between constituents (or the elements of some salient partition of the space) $C\_i$ and we have selected the right extension function. So we have a measure $TL(A\mid C\_i)$of the truthlikeness of a proposition $A$ given that constituent $C\_i$ is true. Provided we also have a measure of epistemic probability $P$ (where $P(C\_i\mid e)$ is the degree of rational credence in $C\_i$ given evidence $e)$ we also have a measure of the expected degree of truthlikeness of $A$ on the evidence (call this $\mathbf{E}TL(A\mid e))$ which we can identify with the best epistemic estimate of truthlikeness. (Niiniluoto, who first explored this concept in his 1977, calls the epistemic estimate of degree of truthlikeness on the evidence, or expected degree of truthlikeness, verisimilitude. Since verisimilitude is typically taken to be a synonym for truthlikeness, we will not follow him in this, and will stick instead with expected truthlikeness for the epistemic notion. See also Maher (1993).) \[ \mathbf{E}TL(A\mid e) = \sum\_i TL(A\mid C\_i)\times P(C\_i \mid e). \] Clearly, the expected degree of truthlikeness of a proposition $is$ epistemically accessible, and it can serve as our best empirical estimate of the objective degree of truthlikeness. Progress occurs in an inquiry when actual truthlikeness increases. And apparent progress occurs when the expected degree of truthlikeness increases. (See the entry on scientific progress, SS2.5.) This notion of expected truthlikeness is comparable to, but sharply different from, that of epistemic probability: the supplement on Expected Truthlikeness discusses some instructive differences between the two concepts. With this proposal, Niiniluoto also made a connection with the application of decision theory to epistemology. Decision theory is an account of what it is rational to do in light of one's beliefs and desires. One's goal, it is assumed, is to maximize utility. But given that one does not have perfect information about the state of the world, one cannot know for sure how to accomplish that. Given uncertainty, the rule to maximize utility or value cannot be applied in normal circumstances. So under conditions of uncertainty, what it is rational to do, according to the theory, is to maximize subjective expected utility. Starting with Hempel's classic 1960 essay, epistemologists conjectured that decision-theoretic tools might be applied to the problem of theory acceptance - which hypothesis it is rational to accept on the basis of the total available evidence to hand. But, as Hempel argued, the values or utilities involved in a decision to accept a hypothesis cannot be simply regular practical values. These are typically thought to be generated by one's desires for various states of affairs to obtain in the world. But the fact that one would very much like a certain favored hypothesis to be true does not increase the cognitive value of accepting that hypothesis. > > This much is clear: the utilities should reflect the value or disvalue > which the different outcomes have from the point of view of pure > scientific research rather than the practical advantages or > disadvantages that might result from the application of an accepted > hypothesis, according as the latter is true or false. Let me refer to > the kind of utilities thus vaguely characterized as purely scientific, > or epistemic, utilities. (Hempel 1960, 465) > If we had a decent theory of epistemic utility (also known as cognitive utility or cognitive value), perhaps what hypotheses one ought to accept, or what experiments one ought to perform, or how one ought to revise one's corpus of belief in the light of new information, could be determined by the rule: maximize expected epistemic utility (or maximize expected cognitive value). Thus decision-theoretic epistemology was born. Hempel went on to ask what epistemic utilities are implied in the standard conception of scientific inquiry - "...construing the proverbial 'pursuit of truth' in science as aimed at the establishment of a maximal system of true statements..." Hempel (1960), p 465) Already we have here the germ of the idea central to the truthlikeness program: that the goal of an inquiry is to end up accepting (or "establishing") the theory that yields the whole truth of some matter. It is interesting that, around the same time that Popper was trying to articulate a content-based account of truthlikeness, Hempel was attempting to characterize partial fulfillment of the goal (that is, of maximally contentful truth) in terms of some combination of truth and content. These decision-theoretic considerations lead naturally to a brief consideration of the axiological problem of truthlikeness. ## 3. The Axiological Problem Our interest in the concept of truthlikeness seems grounded in the value of highly truthlike theories. And that, in turn, seems grounded in the value of truth. Let's start with the putative value of truth. Truth is not, of course, a good-making property of the objects of belief states. The proposition $h \amp r \amp w$ is not a better proposition when it happens to be hot, rainy and windy than when it happens to be cold, dry and still. Rather, the cognitive state of believing $h \amp r \amp w$ is often deemed to be a good state to be in if the proposition is true, not good if it is false. So the state of believing truly is better than the state of believing falsely. At least some, perhaps most, of the value of believing truly can be accounted for instrumentally. Desires mesh with beliefs to produce actions that will best achieve what is desired. True beliefs will generally do a better job of this than false beliefs. If you are thirsty and you believe the glass in front of you contains safe drinking water rather than lethal poison then you will be motivated to drink. And if you do drink, you will be better off if the belief you acted on was true (you quench your thirst) rather than false (you end up dead). We can do more with decision theory utilizing purely practical or non-cognitive values. For example, there is a well-known, decision-theoretic, value-of-learning theorem, the Ramsey-Good theorem, which partially vindicates the value of gathering new information, and it does so in terms of "practical" values, without assuming any purely cognitive values. (Good 1967.) Suppose you have a choice to make, and you can either choose now, or choose after doing some experiment. Suppose further that you are rational (you always choose by expected value) and that the experiment is cost-free. It follows that performing the experiment and then choosing always has at least as much expected value as choosing without further ado. Further, doing the experiment has higher expected value if one possible outcome of the experiment would alter the relative values of some your options. So, you should do the experiment just in case the outcome of the experiment could make a difference to what you choose to do. Of course, the expected gain of doing the experiment has to be worth the expected cost. Not all information is worth pursuing when you have limited time and resources. David Miller notes the following serious limitation of the Ramsey-Good value-of-learning theorem: > > The argument as it stands simply doesn't impinge on the question > of why evidence is collected in those innumerable cases in theoretical > science in which no decisions will be, or anyway are envisaged to be, > affected. (Miller 1994, 141). > There are some things that we think are worth knowing about, the knowledge of which would not change the expected value of any action that one might be contemplating performing. We spent billions of dollars conducting an elaborate experiment to determine whether the Higgs boson exists, and the results were promising enough that Higgs was awarded the Nobel Prize for his prediction. The discovery may also yield practical benefits, but it is the cognitive change it induces that makes it worthwhile. It is valuable simply for what it did to our credal state - not just our credence in the existence of the Higgs, but our overall credal state. We may of course be wrong about the Higgs - the results might be misleading - but from our new epistemic viewpoint it certainly appears to us that we have made cognitive gains. We have a little bit more evidence for the truth, or at least the truthlikeness, of the standard model. What we need, it seems, is an account of pure cognitive value, one that embodies the idea that getting at the truth is itself valuable whatever its practical benefits. As noted this possibility was first explicitly raised by Hempel, and developed in various difference ways in the 1960s and 1970s by Levi (1967) and Hilpinen (1968), amongst others. (For recent developments see the entry epistemic utility arguments for probabilism, SS6.) We could take the cognitive value of believing a single proposition $A$ to be positive when $A$ is true and negative when $A$ is false. But acknowledging that belief comes in degrees, the idea might be that the stronger one's belief in a truth (or of one's disbelief in a falsehood), the greater the cognitive value. Let $V$ be the characteristic function of answers, both complete and partial, to the question $\{C\_1, C\_2 , \ldots ,C\_n , \ldots \}$. That is to say, where $A$ is equivalent to some disjunction of complete answers: * $V\_i (A) = 1$ if $A$ is entailed by the complete answer $C\_i$ (i.e $A$ is true according to $C\_i)$; * $V\_i (A) = 0$ if the negation of $A$ is entailed by the complete answer $C\_i$ (i.e., $A$ is false according to $C\_i)$. Then the simple view is that the cognitive value of believing $A$ to degree $P(A)$ (where $C\_i$ is the complete true answer) is greater the closer $P(A)$ is to the actual value of $V\_i (A)$- that is, the smaller $\|V\_i (A)-P(A)\|$ is. Variants of this idea have been endorsed, for example by Horwich (1982, 127-9), and Goldman (1999), who calls this "veristic value". $\|V\_i (A)-P(A)\|$ is a measure of how far $P(A)$ is from $V\_i (A)$, and $-\|V\_i (A)-P(A)\|$ of how close it is. So here is the simplest linear realization of this desideratum: \[ Cv^1 \_i (A, P) = - \|V\_i (A)-P(A)\|. \] But there are of course many other measures satisfying the basic idea. For example we have the following quadratic measure: \[ Cv^2 \_i (A, P) = -((V\_i (A)- P(A))^2. \] Both $Cv^1$ and $Cv^2$ reach a maximum when $P(A)$ is maximal and $A$ is true, or $P(A)$ is minimal and $A$ is false; and a minimum when $P(A)$ is maximal and $A$ is false, or $P(A)$ is minimal and $A$ is true. These are measures of local value - the value of investing a certain degree of belief in a single answer $A$ to the question $Q$. But we can agglomerate these local values into the value of a total credal state $P$. This involves a substantive assumption: that the value of a credal state is some additive function of the values of the individual beliefs states it underwrites. A credal state, relative to inquiry $Q$, is characterized by its distribution of credences over the elements of the partition $Q$. An opinionated credal state is one that assigns credence 1 to just one of the complete answers. The best credal state to be in, relative to the inquiry $Q$, is the opinionated state that assigns credence 1 to the complete correct answer. This will also assign credence 1 to every true answer, and 0 to every false answer, whether partial or complete. In other words, if the correct answer to $Q$ is $C\_i$ then the best credal state to be in is the one identical to $V\_i$: for each partial or complete answer $A$ to $Q, P(A) = V\_i (A)$. This credal state is the state of believing the truth, the whole truth and nothing but the truth about $Q$. Other total credal states (whether opinionated or not) should turn out to be less good than this perfectly accurate credal state. But there will, of course, have to be more constraints on the accuracy and value of total cognitive states. Assuming additivity, the value of the credal state $P$ can be taken to be the (weighted) sum of the cognitive values of local belief states. > > $CV\_i(P) = \sum\_A \lambda\_A Cv\_i(A,P)$ (where $A$ ranges over all > the answers, both complete and incomplete, to question $Q)$, and the > $\lambda$-terms assign a fixed (non-contingent) weight to the > contribution of each answer $A$ to overall accuracy. > Plugging in either of the above local cognitive value measures, total cognitive value is maximal for an assignment of maximal probability to the true complete answer, and falls off as confidence in the correct answer falls away. Since there are many different measures that reflect the basic idea of the value of true belief how are we to decide between them? We have here a familiar problem of underdetermination. Joyce (1998) lays down a number of desiderata, most of them very plausible, for a measure of what he calls the accuracy of a cognitive state. These desiderata are satisfied by the $Cv^2$ but not by $Cv^1$. In fact all the members of a family of closely related measures, of which $Cv^2$ is a member, satisfy Joyce's desiderata: \[ CV\_i(P) = \sum\_A -\lambda\_A [(V\_i (A) - P(A))^2]. \] Giving equal weight to all propositions $(\lambda\_A =$ some non-zero constant $c$ for all $A)$, this is equivalent to the Brier measure (see the entry on epistemic utility arguments for probabilism, SS6). Given the $\lambda$-weightings can vary, there is considerable flexibility in the family of admissible measures. Our main interest here is whether any of these so-called scoring rules (rules which score the overall accuracy of a credal state) might constitute an acceptable measure of the cognitive value of a credal state. Absent specific refinements to the $\lambda$-weightings, the quadratic measures seem unsatisfactory as a solution either to the problem of accuracy or to the problem of cognitive value. Suppose you are inquiring into the number of the planets and you end up fully believing that the number of the planets is 9. Given that the correct answer is 8, your credal state is not perfect. But it is pretty good, and it is surely a much better credal state to be in than the opinionated state that sets probability 1 on the number of planets being 9 billion. It seems rather natural to hold that the cognitive value of an opinionated credal state is sensitive to the degree of truthlikeness of the complete answer that it fixes on, not just to its truth value. But this is not endorsed by quadratic measures. Joyce's desiderata build in the thesis that accuracy is insensitive to the distances of various different complete answers from the complete true answer. The crucial principle is Extensionality. > > Our next constraint stipulates that the "facts" which a > person's partial beliefs must "fit" are exhausted by > the truth-values of the propositions believed, and that the only > aspect of her opinions that matter is their strengths. (Joyce 1998, > 591) > We have already seen that this seems wrong for opinionated states that confine all probability to a false hypothesis. Being convinced that the number of planets in the solar system is 9 is better than being convinced that it is 9 billion. But the same idea holds for truths. Suppose you believe truly that the number of the planets is either 7, 8, 9 or 9 billion. This can be true in four different ways. If the number of the planets is 8 then, intuitively, your belief is a little bit closer to the truth than if it is 9 billion. (This judgment is endorsed by both the average and the min-sum-average proposals.) So even in the case of a true answer to some query, the value of believing that truth can depend not just on its truth and the strength of the belief, but on where the truth lies. If this is right Extensionality is misguided. Joyce considers a variant of this objection: namely, that given Extensionality Kepler's beliefs about planetary motion would be judged to be no more accurate than Copernicus's. His response is that there will always be propositions which distinguish between the accuracy of these falsehoods: > > I am happy to admit that Kepler held more accurate beliefs than > Copernicus did, but I think the sense in which they were more accurate > is best captured by an extensional notion. While Extensionality rates > Kepler and Copernicus as equally inaccurate when their false beliefs > about the earth's orbit are considered apart from their effects > on other beliefs, the advantage of Kepler's belief has to do > with the other opinions it supports. An agent who strongly believes > that the earth's orbit is elliptical will also strongly believe > many more truths than a person who believes that it is circular (e.g., > that the average distance from the earth to the sun is different in > different seasons). This means that the overall effect of > Kepler's inaccurate belief was to improve the extensional > accuracy of his system of beliefs as a whole. Indeed, this is why his > theory won the day. I suspect that most intuitions about falsehoods > being "close to the truth" can be explained in this way, > and that they therefore pose no real threat to Extensionality. (Joyce > 1998, 592) > Unfortunately, this contention - that considerations of accuracy over the whole set of answers which the two theories give will sort them into the right ranking - isn't correct (Oddie 2019). Wallace and Greaves (2005) assert that the weighted quadratic measures "can take account of the value of verisimilitude." They suggest this can be done "by a judicious choice of the coefficients" (i.e., the $\lambda$-values). They go on to say: "... we simply assign high $\lambda\_A$ when $A$ is a set of 'close' states." (Wallace and Greaves 2005, 628). 'Close' here presumably means 'close to the actual world'. But whether an answer $A$ contains complete answers that are close to the actual world - that is, whether $A$ is truthlike - is clearly a world-dependent matter. The coefficients $\lambda\_A$ were not intended by the authors to be world-dependent. (But whether or not the co-efficients of world-dependent or world-independent the quadratic measures cannot capture truthlikeness. See Oddie 2019.) One defect of the quadratic class of measures, combined with world-independent weights, corresponds to a problem familiar from the investigation of attempts to capture truthlikeness simply in terms of classes of true and false consequences, or in terms of truth value and content alone. Any two complete false answers yield precisely the same number of true consequences and the same number of false consequences. The local quadratic measure accords the same value to each true answer and the same disvalue to each false answer. So if, for example, all answers are given the same $\lambda$-weighting in the global quadratic measure, any two opinionated false answers will be accorded the same degree of accuracy by the corresponding global measure. It may be that there are ways of tinkering with the $\lambda$-terms to avoid the objection, but on the face of it the notions of local cognitive value embodied in both the linear and quadratic rules seem to be deficient because they omit considerations of truthlikeness even while valuing truth. It seems that any adequate measure of the cognitive value of investing a certain degree of belief in $A$ should take into account not just whether $A$ is true or false, but here the truth lies in relation to the worlds in $A$. Whatever our account of cognitive value, each credal state $P$ assigns an expected cognitive value to every credal state $Q$ (including $P$ itself of course). \[ \mathbf{E}CV\_P(Q) = \sum\_i P(C\_i) CV\_i(Q). \] Suppose we accept the injunction that one ought to maximize expected value as calculated from the perspective of one's current credal state. Let us say that a credal state $P$ is self-defeating if to maximize expected value from the perspective of $P$ itself one would have to adopt some distinct credal state $Q$, without the benefit of any new information: > > $P$ is self-defeating = for some $Q$ distinct from $P$, > $\mathbf{E}$CV$\_P$$(Q) \gt$ > $\mathbf{E}$CV$\_P$$(P)$. > The requirement of propriety requires that no credal state be self-defeating. No credal state demands that you shift to another credal state without new information. One feature of the quadratic family of proper scoring rules is that they guarantee propriety, and propriety is an extremely attractive feature of cognitive value. From propriety alone, one can construct arguments for the various elements of probabilism. For example, propriety effectively guarantees that the credence function must obey the standard axioms governing additivity (Leitgeb and Pettigrew 2010); propriety guarantees a version of the value-of-learning theorem in purely cognitive terms (Oddie, 1997); propriety provides an argument that conditionalization is the only method of updating compatible with the maximization of expected cognitive value (Oddie 1997, Greaves and Wallace 2005, and Leitgeb and Pettigrew 2010). Not only does propriety deliver these goods for probabilism, but any account of cognitive value that doesn't obey propriety entails that self-defeating probability distributions should be eschewed a priori. (And that might appear to violate a fundamental commitment of empiricism.) There is however a powerful argument that any measure of cognitive value that satisfies propriety cannot deliver a fundamental desideratum on an adequate theory of truthlikeness (or accuracy): the principle of proximity. Every serious account of truthlikeness satisfies proximity. But proximity and propriety turn out to be incompatible, given only very weak assumptions (see Oddie 2019 and, for an extended critique, also Schoenfield 2022). If this is right then the best account of genuine accuracy cannot provide a vindication of pure probabilism. ## 4. Conclusion We are all fallibilists now, but we are not all skeptics, or antirealists or nihilists. Most of us think inquiries can and do progress even when they fall short of their goal of locating the truth of the matter. We think that an inquiry can progress by moving from one falsehood to another falsehood, or from one imperfect credal state to another. To reconcile epistemic optimism with realism in the teeth of the dismal induction we need a viable concept of truthlikeness, a viable account of the empirical indicators of truthlikeness, and a viable account of the role of truthlikeness in cognitive value. And all three accounts must fit together appropriately. There are a number of approaches to the logical problem of truthlikeness but, unfortunately, there is as yet little consensus on what constitutes the best or most promising approach, and prospects for combining the best features of each approach do not at this stage seem bright. In fact, recent work on whether the three main approaches to the logical problem of truthlikeness presented above are compatible seems to point to a negative answer (see the supplement on The compatibility of the approaches). There is, however, much work to be done on both the epistemological and the axiological aspects of truthlikeness, and it may well be that new constraints will emerge from those investigations that will help facilitate a fully adequate solution to the logical problem as well.
truthmakers	## 1. What is a Truth-maker? Truth-makers are often introduced in the following terms (Bigelow 1988: 125; Armstrong 1989c: 88): (Virtue-T) a truth-maker is that in virtue of which something is true The sense in which a truth-maker "makes" something true is said to be different from the causal sense in which, e.g., a potter makes a pot. It is added that the primary notion of a truth-maker is that of a minimal one: a truth-maker for a truth-bearer p none of whose proper parts or constituents are truth-makers for p. (Whether every proposition has a minimal truth-maker is contentious.) We are also cautioned that even though people often speak as if there is a unique truth-maker for each truth, it is usually the case that one truth is made true by many things (collectively or severally). Whether (Virtue-T) provides a satisfactory elucidation of truth-making depends on whether we have a clear understanding of "in virtue of". Some philosophers argue this notion is an unavoidable primitive (Rodriguez-Pereyra 2006a: 960-1). But others are wary. Thus, for example, Bigelow finds the locution "in virtue of" both obscure and, as will see, avoidable "we should not rest content with an explanation which turns on the notion of virtue!" (Bigelow 1988). These kinds of concern provides a significant motivation for establishing whether it is possible to elucidate the concept of truth-making in terms of notions that enjoy a life independently of the circle of notions to which in virtue of and truth-making belong. ### 1.1 Truth-making as Entailment One influential proposal for making an elucidatory advance upon (Virtue-T) appeals to the notion of entailment (Fox 1987: 189; Bigelow 1988: 125-7): (Entailment-T) a truth-maker is a thing the very existence of which entails that something is true. So x is a truth-maker for a truth p iff x exists and another representation that says x exists entails the representation that p. It is an attraction of this principle that the key notion it deploys, namely entailment, is ubiquitous, unavoidable and enjoys a rich life outside philosophy--both in ordinary life and in scientific and mathematical practice. Unfortunately this account threatens to over-generate truth-makers for necessary truths--at least if the notion of entailment it employs is classical. It's a feature of this notion that anything whatsoever entails a necessary truth p. It follows as a special case of this that any claim that a given object exists must entail p too. So it also follows--if (Entailment-T) is granted--that any object makes any necessary truth true. But this runs counter to the belief that, e.g., the leftovers in your refrigerator aren't truth-makers for the representation that 2+2=4. Even worse, Restall has shown how to plausibly reason from (Entailment-T) to "truth-maker monism": the doctrine that every truth-maker makes every truth true (whether necessary or contingent). Every claim of the form p [?] ~p is a necessary truth. So every existing thing s is a truth-maker for each instance of this form (see preceding paragraph). Now let p be some arbitrary truth (grass is green) and s any truth-maker for p [?] ~p (a particular ice floe in the Antarctic ocean). Now it is intuitively plausible that something makes a disjunction true either by making one disjunct true or by making the other disjunct true (Mulligan, Simons, & Smith 1984: 316). This is what Restall calls "the disjunction thesis". It follows from this thesis that either s makes p true or s makes ~p true. Given that p is true, neither s nor anything else makes it true that ~p. So s (the ice floe) must make it true that p (grass is green)! But since s and p were chosen arbitrarily it follows that all truth-makers are on a par, making true every truth (Restall 1996: 333-4). The unacceptability of these results indicates that insofar as we have an intuitive grip upon the concept of a truth-maker it is constrained by the requirement that a truth-maker for a truth must be relevant to or about what it represents as being the case. For example, the truths of pure arithmetic are not about what's at the back of your refrigerator; what's there isn't relevant to their being true. So we're constrained to judge that what's there cannot be truth-makers for them. This suggests that operating at the back of our minds when we issue these snap judgements there must be something like the principle: (Relevance) what makes something true must--in some sense--be what it is "about". Of course the notions of "about" and "relevance" are notoriously difficult to pin down (Goodman 1961). And a speaker can know something is true without knowing everything about what makes it true. Often it will require empirical research to settle what makes a statement true. Moreover, what is determined a posteriori to be a truth-maker may exhibit a complexity quite different from that of the statement it makes true (Mulligan, Simons, & Smith 1984: 299). Nevertheless, it is clear that unless (Entailment-T) is constrained in some way it will generate truth-makers that are unwanted because their presence conflicts with (Relevance). One way to restore accord between them would be to abandon the Disjunction Thesis that together with (Entailment-T) led us down the path to Truth-maker Monism. Indeed isn't the Disjunction Thesis dubious anyway? Consider examples involving the open future. Can't we imagine a situation arising that makes it true that one or other of the horses competing in a race will win, but which neither makes it true that one of the horses in particular will win nor makes it true that another will (Read 2000; Restall 2009)? For further discussion of issues surrounding truthmaking entailment, the Disjunctive Thesis and the Conjunctive Thesis, i.e. the principle that a truthmaker for a conjunction is a truthmaker for its conjuncts, see Rodriguez-Pereyra 2006a, 2009, Jago 2009, Lopez de Sa (2009) and Talasiewicz (et al) (2012). But even if the Disjunction Thesis is given up still this leaves in place the embarrassing consequence of defining what it is to be a truth-maker in terms of classical entailment, viz. that any existing thing turns out to be a truth-maker for any necessary truth. Some more radical overhaul of (Entailment-T) is needed to avoid over-generation. One possibility is to redefine what is to be a truth-maker in terms of a more restrictive notion of "relevant entailment" (in the tradition of Anderson & Belnap) that requires what is entailed to be relevant to what it is entailed by (Restall 1996, 2000; Armstrong 2004: 10-12). Whether exploring this avenue will take us very far remains a matter of dispute. Simons, for example, reflects, > > > In truth however I suspect our intuitions about truth-makers may be at > least as robust as our intuitions about what is good for a logical > system of implication, and I would not at present attempt to explicate > truth-making via the arrow of the relevance logic system > R. (2008: 13) > > > Nevertheless since truth-making concerns the bestowal of truth, entailment its preservation, there must at some level be an important connection to be made out between truth-making and entailment; the effort expended to make out such a connection will be effort spent to the advantage of metaphysicians and logicians alike. But even granted this is so there are reasons to be doubtful that any overhaul of (Entailment-T), however radical, will capture what it is to be a truth-maker. One vital motivation for believing in truth-makers is this. Positing truth-makers enables us to make sense of the fact that the truth of something depends on how things stand with an independently given reality. This is how Bigelow reports what happens to him when he stops believing in truth-makers: > > > I find I have no adequate anchor to hold me from drifting onto the > shoals of some sort of pragmatism or idealism. And that is altogether > uncongenial to me; I am a congenital realist. (1988: 123) > > > Truth-makers are posited to provide the point of semantic contact whereby true representations touch upon an independent reality, upon something non-representational. Since entailment is a relation between representations it follows that the notion of a truth-maker cannot be fully explicated in terms of the relation of entailment--regardless of whether representations are best understood as sentences or propositions or some other candidate truth-bearer (Heil 2000: 233-4; 2003: 62-5; Merricks 2007: 12-13). Ultimately (Entailment-T), or a relevance logic version if it, will leave us wanting an account of what makes a representation of the existence of a truth-maker--whatever it entails--itself beholden to an independent reality. ### 1.2 Truth-making as Necessitation This difficulty arises because entailment is a relation that lights upon representations at both ends. Appreciating this Armstrong recognised that truth-making > > > cannot be any form of entailment. Both terms of an entailment relation > must be propositions, but the truth-making term of the truth-maker > relation is a portion of reality, and, in general at least, portions > of reality are not propositions. (2004: 5-6; see also Lowe 2006: > 185) > > > Accordingly Armstrong made a bold maneuver. He posited a metaphysically primitive relation of necessitation (i.e., one not itself defined in terms of possible worlds because Armstrong is "opposed to the extensional view proposed by those who put metaphysical faith in possible worlds" preferring an 'intensional account' of modality instead (1997: 151, 2004: 96)). The relation in question lights upon a portion of reality at one end and upon a truth at the other. In the simplest case that means that the truth making relation is one that "holds between any truthmaker, T, which is something in the world, and the proposition" that T exists (2004: 6). Armstrong then defined what is to be a truth-maker in terms of this metaphysical bridging relation: (Necessitation-T) a truth-maker is a thing that necessitates something's being true. This conception of truth making avoids the "category" mistake that results from attempting to define truth-making in terms of entailment. It also makes some advance upon (Virtue-T) and its primitive use of "in virtue of" because at least (Necessitation-T) relates the notion of truth making to other modal notions, like that of necessity, upon which we have some independent handle. But what is there positively to be said in favour of conceiving of truth-makers in terms of "necessitation"? Suppose that T is a candidate truth-maker for a truth p even though T fails to necessitate p. Then it is possible for T to exist even when p is false. Armstrong now reflects "we will surely think that the alleged truth-maker was insufficient by itself and requires to be supplemented in some way" (1997: 116). Suppose this supplementary condition is the existence of another entity, U. Then "T+U would appear to be the true and necessitating truth-maker for p" (2004: 7). Armstrong concludes that a truth-maker for a truth must necessitate the truth in question. Unfortunately this argument takes us nowhere except around in a circle. (Necessitation-T) embodies the doctrine that it is both necessary and sufficient for being a truth-maker that a thing necessitates the truth it makes true. Armstrong's argument for this doctrine relies upon the dual assumptions: (1) anything that fails to necessitate p (witness T) cannot be a truth-maker for p,whereas (2) anything that succeeds in necessitating p (witness T+U) must be. But (1) just is the claim that it is necessary, and (2) just the claim that it is sufficient for being a truth-maker that a thing necessitates the truth it makes true. Since it relies upon (1) and (2), and (1) and (2) are just equivalent to (Necessitation-T), it follows that Armstrong's argument is incapable of providing independent support for the conception he favours of what it is to be a truth-maker. Even though this argument may be circular, does (Necessitation-T) at least have the favourable feature that adopting it enables us to avoid the other difficulty that beset (Entailment-T), viz. over-generation? Not if there are things that necessitate a truth whilst still failing to be sufficiently relevant to be plausible truth-makers for it. If the necessitation relation is so distributed that it holds between any contingently existing portion of reality, e.g., an ice-floe, and any necessary truth, e.g., 2+2=4, then we shall be no further forward than we were before. So Armstrong needs to tell us more about the cross-categorial relation in question to assure us that such cases cannot arise. Smith suggests another problem case for (Necessitarian-T). > > > Suppose that God wills that John kiss Mary now. God's willing > act thereby necessitates the truth of "John is kissing > Mary". (For Malebranche, all necessitation is of this sort.) But > God's act is not a truth-maker for this judgement. (Smith 1999: > 6) > > > If such cases are possible then (Necessitation-T) fails to provide a sufficient condition for being a truth-maker. In fact the commitment already emerges from (Necessitarian-T) if the necessitation relation it embodies is conceived as internal in the following sense: "An internal relation is one where the existence of the terms entails the existence of the relation" strung out between the terms; otherwise a relation is external (Armstrong 1997: 87). Armstrong argues that the relation of truth-making has to be internal (in just this sense) because if it weren't then we would have to allow that, absurdly, "anything may be a truth-maker for any truth" (1997: 198). But then nothing but propositions--conceived in the self-interpreting sense--can be truth-bearers that are internally related to their truth-makers. Any other candidate for this representational role, as we've already reflected--a token belief state or utterance--could have been endowed with a different representational significance than the one it possesses. So the other eligible candidates, by contrast to propositions, aren't internally related to what makes them true. Armstrong's commitment to the truth-making relation's being internal clashes with his naturalism (David 2005: 156-9). According to Armstrong, > > > Truth attaches in the first place to propositions, those propositions > which have a truth-maker. But no Naturalist can be happy with a realm > of propositions. (Armstrong 1997: 131) > > > So Armstrong counsels that we don't take propositions with "ontological seriousness": > > > What exists are classes of intentionally equivalent tokens. The > fundamental correspondence, therefore, is not between > entities called truths and their truth-makers, but between the token > beliefs and thoughts, on the one hand, and truth-makers on the other. > (Armstrong 1997: 131) > > > But naturalistically kosher token beliefs and thoughts aren't internally related to what makes them true. So Armstrong's naturalism commits him to denying that the truth-making relation is internal after all. Heil has also inveighed in a naturalistic spirit against incurring a commitment to propositions that are designed to have their own "built-in intentionality" whilst continuing to maintain that truth-making is an internal relation (2006: 240-3). Since we have no idea of what a naturalistic representation would look like that was internally related to what made it true, this appears to be an impossible combination of views. Our inability to conjure up a credible class of truth-bearers that are internally related to their truth-makers provides us with a very strong incentive for supposing that the truth-making relation is external. The long and short of it: if we are wary, as many naturalists are, of the doctrine that truth-bearers are propositions, then we should also be wary of thinking that the truth-making relation is internal. This spells trouble for (Necessitarian-T) and any other account of truth making which invoke internal relations to propositions. ### 1.3 Truth-making as Projection Smith tries to get around this problem by adding to (Necessitation-T) the further constraint that a truth-maker for a given truth should be part of what that truth is about: "roughly: it should fall within the mereological fusion of all the objects to which reference is made in the judgment" (1999: 6). His suggestion is that a typical contingent judgement p not only makes singular reference to the objects whose antics it describes but also "generic reference" to the events, processes and states that are associated with the main verb of the sentence that is used to express the judgement in question. So, for example, the judgement that John is kissing Mary incorporates not only singular reference to John and Mary but also generic reference to all kisses. Smith identifies the projection of a judgement with the fusion of all these things to which it refers (singularly or generically). He then defines what is to be a truth-maker in terms of this notion: (Projection T) a truth-maker for a judgement p is something that necessitates p and falls within its projection Since God's act of willing isn't one of the things to which the judgement that John is kissing Mary makes singular or generic reference--it doesn't fall within the projection of this judgement--His act isn't a truth-maker for this judgement. But that fateful kiss does fall within the generic reference of the judgement that John is kissing Mary and necessitates the truth of that judgement; so the kiss is a truth-maker for it. Whether taking up (Projection-T) will avoid classifying malignant cases of necessitating things as truth-makers will depend upon whether the notion of projection can be made sufficiently precise to vindicate the intuition that the net cast by a judgement won't catch these unwanted fishes with the rest of its haul; whether if the judgement that phx incorporates singular reference to x and generic reference to ph-ings, it doesn't incorporate (even more) generic reference to anything else that may (waywardly) necessitate phx. (Here "ph" indicates the place for a verb phrase and "x" a place for a name.) Smith attempts to make the notion of projection precise using just mereology and classical entailment. Something belongs to the projection of a judgement that phx only if it is one of the things to which the judgement is existentially committed (Smith 1999: 7). In order to capture the "generic" as well as the singular consequences of a judgement, Smith includes amongst its existential consequences the fusions that result from applying the mereological comprehension principle (T[?]) ([?]xphx iff there exists a unique fusion of all and only those x such that phx) to the immediate commitments of the judgement. For example, the judgement that Nikita meows is immediately committed to the existence of a meow and thereby to the unique fusion of all and only meows. Because God's act of willing John to kiss Mary doesn't fall within the class of logical-cum-mereological consequences of the judgement that John is kissing Mary, His act isn't a truth-maker for it. But this still threatens to over-generate truth-makers (Gregory 2001). The judgement that John is kissing Mary has amongst it existential consequences that someone exists. Since Smith assumes that "x exists", like "x meows" and "x kisses Mary", is a predicate, it also follows via (T[?]) that the projection of this judgement includes the fusion of all and only existing things (Smith 1999: 10). But since (ex hypothesi) God's act exists--and so belongs to the aforementioned fusion and thereby the projection of the judgement in question--and necessitates that John is kissing Mary, it fails to rule out God's act as a truth-maker for it! The projection of a judgement needs to be made far more relevant to what a judgement is intuitively about; the logical-mereological notion of a projection supplied doesn't even provide a basis for affirming that singular judgements don't generically refer to every existing thing. An initial attractive move to avoid this particular problem would be to deny that "exists" is a predicate. But more generally, it remains to be seen whether an appropriate notion of a judgement's referential net, its projection, can be made out that's isn't too permissive--thereby including illegitimate truth-makers--without having to deploy resources that are not obviously clearer or more problematic than that of truth-making itself (see Smith 2002 and Schnieder 2006b for contrasting prognoses). ### 1.4 Truthmaking in terms of Essentialism Others attempt to avoid over-generating truth-makers by appealing to the notion of essence, a notion that purports to be far more discriminating than necessitation (Fine 1994; Lowe 1994). The difference in grain between these notions becomes evident when we reflect that although it is (supposed to be) necessarily the case that if Socrates exists then his singleton does too, it isn't part of the essence of Socrates that he belongs to this set. If these notions really are different then we will need to distinguish between those entities that merely necessitate a true claim on the one hand and those that are also part of its essence on the other. This suggests a strategy for ruling out spurious truth-makers: they're the ones that only necessitate true claims, whereas the real ones are also implicated (somehow) in the essences of the claims they make true. This brings us back to the question of truth-bearers. If truth-makers are implicated in the essences of truth-bearers then truth-bearers can neither be sentences nor judgements. Truth-bearers of these kinds only bear their representational features accidentally; they could have been used to say or think something different, or occurred in contexts where they lacked significance altogether. Since they could have meant something different, or nothing at all, the truth-makers of these truth-bearers can hardly be implicated in their essences. Accordingly truth-bearers that essentially implicate their truth-makers must be creatures that could not have shifted or lacked their representational features. They must be propositions in the deep sense of being items that are incapable of meaning anything other than they do. Conceiving of propositions only in this sense and appealing to their essences, what is to be a truth-maker admits of the following definition (Mulligan 2003: 547, 2006: 39, 2007; Lowe 2006: 203-10, 2009: 209-15): (Essential-T) a truth-maker of a proposition is something such that it is part of the essence of that proposition that it is true if that thing exists. If this definition is adopted then it is plausible to maintain that many of the spurious cases of truth-makers, which have afflicted accounts of truth-making in terms of entailment, necessitation or projection, will thereby be weeded out (Lowe 2006: 202-3). It isn't part of the essence of the proposition that John is kissing Mary that it is true if there exists an act of God's willing it. Nor is it part of the essence of the proposition that 2+2=4 that it is true if there is a particular ice floe in the Antarctic. It isn't even part of the essence of the proposition that 2+2=4 that it is true if p exists. So none of these things are (spuriously) classified, if (Essential-T) is our touchstone, as truth-makers for these propositions. In this respect, the essentialist conception of truth-making has conspicuous advantages over its aforementioned rivals. Nevertheless, on the downside, it may be questioned whether our grip upon the notion of the essence of a proposition is any firmer than the notion of truth-maker for it. Moreover, the benefits of adopting (Essential-T) come at a cost. The idea of a truth-maker is introduced as "intuitively attractive" (Lowe 2006: 207). But (Essential-T) requires propositions that are not only abstract--already anathema to naturalists--but also mysterious, because they are, so to speak, self-interpreting, i.e. propositions that mean what they do irrespective of what speakers or thinkers ever do with the signs or judgements that express them. So the idea of a truth-maker turns out to be far less intuitive and attractive than it initially seemed. Nonetheless, this is a commitment some may accept anyway (Lowe 2009: 178-80), or may be willing to accept if it gives them a good account of truth-making. Essentialist conceptions of truth-making have proved less influential in the recent literature than grounding conceptions (1.6 below). This is surprising in light of the affinities between the notions of ground and essence, especially given the possibility, favoured by some, of explaining grounding in terms of essence (Correia 2013). This also means that at least some of the attractions and some of the objections to a grounding approach to truth making, discussed below, are liable to carry over to an essentialist approach. ### 1.5 Axiomatic Truth-making The difficulties that have beset the definitions of truth-making proffered so far have suggested to some philosophers that a pessimistic induction is in order: that the project of defining truth-making in more basic terms is misconceived, much as the project of defining knowledge in more basic terms has come to seem misconceived because e.g., of the Gettier cases. But there's no gain to be had from, doing nothing more than declaring the notion of a truth-maker to be primitive. If it's primitive then we also need to know how the notion may be fruitfully applied in association with other concepts we already deploy--entailment, existence, truth etc.--to describe the interplay of truth-bearers and the world. As Simons remarks, > > > The signs are that truth-making is not analysable in terms of anything > more primitive, but we need to be able to say more than just that. So > we ought to consider it as specified by principles of truth-making. > (2000: 20) > > > In other words, the notion needs to be introduced non-reductively but still informatively and this is to be achieved by appealing to its systematic liaisons with other concepts. This is the approach that was originally shared by Mulligan, Simons and Smith (1984: 312-8). The principal schemata they employed to convey to us an articulate grasp of what truth-making means in non-reductive terms included: (i) (Factive) If A makes it true that p, then p (ii) (Existence) If A exists, then A makes it true that A exists (iii) (Entailment) If A makes it true that p, and that p entails that q, then A makes it true that q Each of these schemata specifies a definite linkage between the application of the notion of truth-making and some other condition. Truth-making is introduced as the notion that sustains all of these linkages. Putting the schemata together what it is to be a truth-maker is then definable intra-theoretically as follows, (Axiomatic-T) A truth-maker is something x such that (i) if x makes it true that p then p, (ii) if x exists then x makes it true that x exists... and so on for each of the axiom schemata of our favoured theory of truth-makers. It is important to appreciate that adopting this approach to truth-making doesn't have the benefits of theft over honest toil. For one thing it doesn't obviate the threat of superfluous truth-makers for necessary truths. (Entailment) is the principle that truth-making tracks entailment: if A makes p true then it makes all the consequences of p true too. It's a principle that recommends itself irrespective of whether truth-making can be defined. This is because it dovetails smoothly with the idea that one truth-maker can make many truths true. For example, suppose that a particular a has some absolutely determinate mass. It is entailed by this description that various determinable descriptions are also truly predicable of a. Some of these truths say more than others, nonetheless they all have the same truth-maker. Why so? Because they are entailed by a's having the mass it does (Armstrong 1997: 130; 2004: 10-11). To answer so is to appeal to (Entailment). But if the entailment that truth-making tracks is classical then we are back to flouting (Relevance). If q is necessary then any contingent p classically entails q. So if something at the back of your refrigerator makes it true that p then by (Entailment) it makes q true too. To avoid flouting (Relevance) in this way (Entailment) had better be a principle that links truth-making to a more restrictive, non-classical notion of entailment (Mulligan, Simons, & Smith 1984: 316). So we won't be saved the logical labour of figuring out which non-classical connective it is that contributes to capturing what it is to be a truth-maker. Nor will appealing to (Axiomatic-T) save us the hard work of figuring out what truth-makers and truth-bearers must be like in order to collectively realise the structure described by the axiom schemata for truth-makers we favour. But there is nothing about these schemata that demands truth-bearer and truth-maker be internally related. It is striking that the Axiomatic approach to truth-making has proved less popular in the recent literature than other more metaphysically-loaded conceptions, in terms of necessitation, essence or grounding. A principal attraction of the approach is its metaphysical neutrality. By contrast, in more recent debate, philosophers have preferred approaches which rely upon a distinctive metaphysical vision. ### 1.6 Truth-Making as Grounding Many recent approaches define truth-making in terms of grounding or what is sometimes described as non-causal metaphysical dependence. The notion of truth-making is typically introduced, as we have seen, in terms of the ideology of "in virtue of". As we have also seen, many philosophers then ask: but what independent content can be given to this notion? In response to this challenge, a theory of grounding may be conceived as a general theory of "in virtue of" in terms of which truth-making may then be explained. A further consideration which motivates those already committed to grounding is the general methodological principle that favours theoretical unification. Their idea is that by conceiving of truth-making as a kind of grounding, we are able to understand truth-making better, i.e. not as an isolated phenomenon but as an instance of a more general pattern, thereby illuminating not only the nature of truth-making but grounding too. Early exponents of grounding within analytic philosophy date from the 1960s (Bergmann 1961: 229, Hochberg 1967: 416-7). Recently, grounding has been explained in a variety of different ways (Correia and Schnieder 2012, Raven 2015). But, broadly speaking, we can distinguish between two conceptions which assign different logical forms to grounding statements: (1) the "predicate" or "relational" view whereby grounding statements are assigned the logical form "x grounds y" where "x" and "y" mark singular term positions and the verb expresses a relation; (2) the "operator" view whereby the canonical form of a grounding statement is characterised by a sentence operator. So statements about something grounding something else have the logical form "p because q" where "p" and "q" mark positions for sentences. The predicate view of grounding is held by, amongst others, Schaffer 2009, Audi 2012, Raven 2012; the operator view is advanced by Fine 2001, 2012, Schnieder 2006a, Correia 2011 [2014]. (A third approach, owed to Mulligan 2007, recognises a plurality of different forms of grounding statements.) The choice between (1) and (2) is often made either on the basis of an assessment of the logical form of typical grounding claims, or on the basis of ontological commitment, for example, the claim that the operator approach to grounding is "ontologically neutral" because (allegedly) operators don't stand for anything as (allegedly) predicates do (Correia 2010: 254, Fine 2012: 47). We can now distinguish two broad conceptions of truth-making, defined, respectively, in terms of the operator and predicate approach to grounding. (Grounding-Predicate-T) A truth-maker is an entity x which makes a proposition y true iff the fact that x exists grounds the fact that y is true. (Grounding-Operator-T) A truth-maker is an entity x which makes a proposition y true iff y is true because x exists. For the former, see Rodriguez-Pereyra 2005, Schaffer 2009, Jago 2018: 184-202, for the latter, Correia 2005 (sec. 3.2), 2011 [2014], Schnieder 2006a, Mulligan 2007, Caputo 2007. One attraction of conceiving truth-making as based on grounding, aside from the more general methodological virtues of theoretical unification mentioned above, arises from taking grounding to be "hyperintensional" nature. Grounding is intended to be a notion that's fine-grained enough to distinguish those entities, e.g., the cardinal numbers, that ground a necessary truth, such as 2+2=4, from those that don't, e.g. a contingent existent such as the aforementioned ice floe. Because grounding is so fine-grained neither Grounding-Predicate-T or Grounding-Operator-T will give rise to the over-generation that beset Entailment-T and Necessitation-T but without having to appeal, like Projection-T, to an underdeveloped notion of "about " or "relevance". Of course, in light of sec, 1.2 and 1.4 above, we can foresee a downside to this. Both Grounding-Predicate-T and Grounding-Operator-T quantify over propositions that stand hyperintensionally to their truth-makers. It follows that both Grounding-Predicate-T and Grounding-Operator-T share a commitment with Necessitarian-T and Essentialist-T, viz. an ontology of abstract propositions which have their meanings essentially. But, again, this may be a commitment many grounding theorists have already made or be willing to accept if it gives them a good account of truth-making. Another issue is that there is no consensus upon the formal properties of grounding. Whilst, for example, Schaffer (2009: 376) has maintained that grounding is irreflexive, asymmetric and transitive, Rodriguez-Pereyra (2015) has argued that the relation of truth-making isn't any of these things. But Rodriguez-Pereyra also argues that grounding isn't any of those things either. So far we have considered efforts to define truth-making in terms of grounding. But there are other approaches in the literature which seek to endorse one notion at the expense of the other. One way accepts grounding as a welcome theoretical innovation but argues that we are better off without truth-making, at least in the forms which have become familiar to us. A second way continues to endorse truth-making whilst rejecting grounding. The first approach, taken by Fine, involves embracing grounding but rejecting truth-making. It's truth-making, he argues, we can do without; grounding does a better job (Fine 2012; 43-6). Here truth-making is conceived as a relation between a worldly entity and a representation: the existence of the entity guarantees the truth of the representation. But, Fine argues, it is more theoretically fruitful to take grounding as the central notion for metaphysics. This is partly because, Fine maintains, grounding is a less restrictive notion than truth-making, because grounding does not require that the ultimate source of what is true should lie in what exists, rather we can remain open about what form grounding takes. Moreover, whilst there may be genuine questions about what the ground is for the truth of the representation that p there may be further questions about what makes it the case that p and these questions will typically have nothing to do with representation as such. Because truth-making is typically understood in modal terms, i.e. necessitation is conceived to be sufficient for truth-making, Fine is also sceptical that there is any way of avoiding the over-generation of truth-makers we have already discussed (see 1.1 and 1.2 above). Finally, Fine thinks that we should be able to conceive of the world hierarchically, whereby (e.g.) the normative is grounded in the natural, the natural in the physical and the physical in the micro-physical. But this is difficult to do if we think in terms of truth-making, because if (e.g.) we conceive of a normative representation being made true by the existence of something natural, that natural thing is just the wrong sort of thing to be made true by something at the physical level because it is not a representation. Liggins (2012) maintains, like Fine, that the theory of truth-making is inevitably less general than a theory of grounding. But Liggins also argues that appealing to grounding obviates the need to appeal to truth-makers in the sense that (e.g.) Armstrong maintained. This is because one significant reason for positing truth-makers is that doing so enables us to explain the asymmetric sense in which truth depends upon being (see sec. 3.2 below). But, Liggins maintains, this can already be done in terms of grounding, by saying that the fact that p grounds the fact that a certain proposition is true, namely the proposition that p. According to Liggins, this isn't a case of truth-making because the relation of grounding invoked holds solely between two facts, albeit one of them a fact about a proposition, that it is true, rather than, as Liggins reads the requirements of truth-maker theory, a relation holding between something and a proposition. Fine's critique of truth-making presupposes that we can only avail ourselves of truth-making if we conceive of truth-making in monolithic terms, as providing a unique method for metaphysics. But whilst, for example, Armstrong may have had a tendency to think in such terms, this doesn't appear to be mandated by the nature of truth-making itself. One might, as Asay (2017) argues, instead conceive of grounding and truth-making as complementary projects, whereby a theory of truth-making doesn't purport to do everything a theory of grounding does and so reciprocal illumination remains a live possibility. Indeed one might already think of the aforementioned definitions or elucidations of truth-making, which are designed in terms of a hyperintensional notion of grounding in order to avoid over-generation of truthmakers as one way in which grounding is already being used to illuminate truth-making. The second approach, outlined above, refusing grounding but favouring truth-making, is advanced by Heil (2016). Heil is generally suspicious of grounding because, he points out, grounding is often characterized in different and incompatible ways, so he is sceptical that there is a univocal concept of grounding to be had. But, more specifically, Heil has long maintained that hierarchical conceptions of reality spell trouble because of the difficulties explaining the causal and nomological relationship between the layers (Heil 2003: 31-9). He argues that this was the problem identified for supervenience and realization in the '90s and it's a problem inherited by the hierarchical conception of reality to which grounding gives rise. So Heil does a modus tollens where Fine does a modus ponens. Whilst Fine conceives of the hierarchical styles of explanation grounding provides as one of the principal virtues of grounding, Heil considers this is one of the deepest drawbacks of grounding. By contrast with Fine, Heil instead recommends truth-making as the proper methodology for metaphysics because it enables us to give a non-hierarchical description of reality whereby fundamental physics gives us the truth-makers for truths that have truth-makers. Further and independent objections to grounding include whether the notion of grounding is intelligible (Daly 2012), theoretically unified (MacBride 2013b), theoretically illuminating (Wilson 2014) or the result of a cognitive illusion (Miller and Norton 2017). Other available positions include: that truth making is a special kind of grounding which involves a unique form of dependence (Griffith 2013), and that truth making isn't grounding but that grounding is key to truthmaking (Saenz 2018). ## 2. Which range of truths are eligible to be made true (if any are)? Even when truth-maker panegyrists agree about what it is to be a truth-maker, they often still disagree about the range of truths that are eligible to be made true. This results in further disagreement about what kinds of entities truth-makers are. There is potential for disagreement here because of the appearance that different ranges of truths require different kinds of truth-makers. Sometimes, discovering themselves unable to countenance the existence of one or other kind of truth-maker, panegyrists may find themselves obliged to reconsider what truths really require truth-makers or to reconsider what it is to be a truth-maker. We can get a sense of the complex interplay of forces at work here by starting out from the most simple and general principle about truth-making (maximalism) and then seeing what pressures are generated to make us step back from it. ### 2.1 Maximalism Truth-maker maximalists demand that every truth has a truth-maker--no exceptions granted. So they advance the completely general principle, (Maximalism) For every truth, then there must be something in the world that makes it true. The principle lies at one end of the spectrum of positions we can potentially occupy. At the other end, we find truth-maker nihilism, the idea that no truth needs to be made true because (roughly) the very idea of a truth-maker is a corrupt one: there is no such role as making something true for anything to perform. Truth-maker optimalism is the intermediate position that only some truths stand in need of truth-makers: not so few that truth fails to be anchored in reality but not so many that we strain credulity about the kinds of things there are. #### 2.1.1 The Liar? Milne (2005) has offered the following knock-down, if not knock-out, argument against maximalism. Take the sentence, (M) This sentence has no truth-maker. Suppose that (M) has a truth-maker. Since it's made true, M must be true. And since it's true, what (M) says must also be the case: that M has no truth-maker. So if (M) has a truth-maker then it doesn't have a truth-maker. By reductio ad absurdum (M) therefore has no truth-maker. But this is just what (M) says. So, contra (Maximalism), there is at least one sentence, (M), that is true without benefit of a truth-maker. Rodriguez-Pereyra (2006c) has responded to this argument by maintaining that (M) is like the Liar sentence (L) This sentence is false. It's a familiar position in the philosophy of logic to respond to the inconsistency that arises from supposing that (L) is either true or false--if it is, it isn't and if it isn't, it is--that the sentence (L) isn't meaningful, despite superficial appearances (see, for a useful introduction to these issues, Sainsbury 1995: 111-33). According to Rodriguez-Pereyra, because (M) is akin to the Liar sentence there's no reason to suppose that (M) is meaningful either. But if (M) isn't meaningful then (M) certainly can't be true; in which case (M) can't be a counterexample to maximalism either. This response is flawed because, as Milne points out, (M) is importantly unlike (L): (L) gives rise to an outright inconsistency when only elementary logic rules are applied to it; whereas (M) isn't inconsistent per se but only when combined with a substantive metaphysical principle: maximalism. If, by contrast, you don't believe in truth-makers then you have every right to treat (M) as just another true sentence--just as you have every right to think (P) is true if you don't believe in propositions, (P) This sentence does not express a proposition It's only if one was (absurdly) to think that maximalism was a logical truth that (M) could be intelligibly thought to be "Liar-like". Of course, one person's modus ponens is another's modus tollens. So if one already had very strong independent reasons for being committed to maximalism, Milne's argument would provide a reason for thinking that (M) is meaningless (even though it isn't Liar-like). But even if there are such reasons--or appear to be--we can't claim to have control of our subject matter until we have established what it is about the constitutive connections that obtain between truth-making, truth and truth-bearers that determines (M) to be meaningless (if it is); until we've established the lie of the land, we can't be sure that we're not kidding ourselves thinking (M) to be meaningless rather than maximalism to be false. The lesson repeats itself: a convincing theory of truth-makers requires a coeval theory of truth bearers. For further discussion of the Liar Paradox in relation to truth-making, see Milne 2013 and Barrio and Rodriguez-Pereyra 2015. #### 2.1.2 Could there be nothing rather than something? Here's another shot across the bows, this time from Lewis. Take the most encompassing negative existential of all: absolutely nothing exists. Surely this statement is possibly true. But if it were true then something would have to exist to make it true if the principle that every truth has a truth-maker is to be upheld. But then there would have to be something rather than nothing. So combining maximalism with the conviction that there could have been nothing rather than something leads to contradiction (Lewis 1998: 220, 2001: 611). So unless we already have reason to think there must be something rather than nothing--as both Armstrong (1989b: 24-5) and Lewis (1986: 73-4) think they do--maximalism is already in trouble. #### 2.1.3 Maximalism is distinct from 1-1 Correspondence What is there to be said in defence of maximalism? Even though he favours it, Armstrong finds himself obliged to admit that "I do not have any direct argument" for recommending the position (2004: 7). Instead he expresses the hope that > > > philosophers of realist inclinations will be immediately attracted to > the idea that a truth, any truth, should depend for its truth for > something "outside" it, in virtue of which it is true. > > > > Let us follow Armstrong's lead and treat maximalism as a "hypothesis to be tested". (See Schaffer 2008b: 308 for a more direct argument that relies upon (Grounding-T).) Maximalism needs to be distinguished from the even stronger claim, (Correspondence) For each truth there is exactly one thing that makes it true and for each truth-maker there is exactly one truth made true by it. But that there's clear blue water between these claims is evident if we combine maximalism with (Entailment)--that whatever makes a truth p true makes what p entails true too. So p and all of its consequences are not only all made true (as maximalism demands) but they also share a truth-maker (as Correspondence denies). This takes us halfway to appreciating that so far from being an inevitably profligate doctrine--as its name suggests--maximalism is compatible with denying that some logically complex claims have their own bespoke truth-makers. (The inspiration for thinking this way comes from the logical atomism of Russell (1918-19) that admitted some logically complex facts but not others--it is to be contrasted with Wittgenstein's version of the doctrine (1921) which admitted only atomic facts.) Suppose, for the sake of expounding the view, that some truth-bearers are atomic. Also suppose that P and Q are atomic and t makes P true. Then, by Entailment, t makes P [?] Q true too. Similarly, if s makes P true and s\* makes Q true then, by Entailment, s and s\* together make P & Q true. Since the task of making P [?] Q and P & Q true has already been discharged by the truth-makers for the atomic truth bearers, there is no need to posit additional truth-makers for making these disjunctive and conjunctive truths true. Similar reasoning suggests that there is no need to posit bespoke truth-makers for existential generalisations or truths of identity either (Mulligan, Simons, & Smith 1984: 313; Simons 1992: 161-3; Armstrong 2004: 54-5). But maximalism is not thereby compromised: even though disjunctive and conjunctive truths lack specific truth-makers of their own, they're still made true by the truth-makers of the basic claims from which they're compounded by the logical operations of disjunction and conjunction. #### 2.1.4 The need for bespoke truth-makers But positing truth-makers for atomic truths doesn't obviate the need--supposing maximalism--to posit additional truth-makers for negative and universal truths. This becomes apparent in the case of negative truths when we compare the truth-tables for conjunction and disjunction with the truth-table for negation. The former tell us that the truth of a disjunctive formula is determined by the truth of one or other of its disjuncts, whilst the truth of a conjunctive formula is determined by the truth of both its conjuncts. But the truth-table for negation doesn't tell us how the truth of ~P is determined by the truth of some other atomic formula Q from which ~P follows; it only tells us that ~P is true iff P is false. The strategy for avoiding bespoke truth-makers for logically complex truths can't get a grip in this case: there's no Q such that supplying a truth-maker for it obviates the necessity of positing an additional truth-maker for ~P (Russell 1918-19: 209-11; Hochberg 1969: 325-7). The problem is even starker for universal truths: there's no truth-table for them because there is no set of atomic formulae whose truth determines that a universally quantified formula is true too. Why so? Because whatever true atomic formulae we light upon (Fa, Fb... Fn), it doesn't follow from them that [?]xFx is true. To extract the general conclusion one would need to add to the premises that a, b... n are all the things there are; that there's no extra thing waiting in the wings to appear on stage that isn't F. But this extra premise is itself universally quantified, not atomic. So it can't be argued that the truth-makers for Fa, Fb... Fn put together already discharge the task of making [?]xFx true because they entail it (Russell 1918-9: 236-7; Hochberg 1969: 335-7). ##### 2.1.4.1 Truth-makers for Negative Truths The most straightforward response--since we are treating maximalism as a working hypothesis--is to find more fitting truth-makers for those truths that aren't already made true by the truth-makers for the atomic truths that entail them. Whilst looking around for truth-makers for negative truths Russell reflected, > > > There is implanted in the human breast an almost unquenchable desire > to find some way of avoiding the admission that negative facts are as > ultimate as those that are positive. (1919: 287) > > > He was right that our desire for positive facts and things makes us awkward about acknowledging that negative facts or things are the truth-makers of negative truths. Nonetheless, discussions about whether there are truth-makers for a given range of truth bearers on one or the other side of the positive-negative divide are apt to appear nebulous. This is because, as Russell had himself previously noted, there is "no formal test" or "general definition" for being a negative fact; we "must go into the meanings of words" (1918-19: 215-6). Statements of the form "a is F" aren't invariably positive ("so-and-so is dead"), nor are statements of the form "a isn't F" ("so-and-so isn't blind") always negative. But it doesn't follow from the fact that a syntactic test cannot be given that there is nothing to the contrast between positive and negative. Molnar suggests that the contrast can be put on a sound scientific footing. For Molnar, natural kinds are paradigm instances of the positive, to be identified on a posteriori grounds (2000: 73). To say that a thing belongs to a natural kind identified in this way is to state a positive fact. To state a negative fact is to negate a statement of a positive fact. It's a very natural suggestion that if the negative claim that a isn't F is true it's made true by the existence of something positive that's incompatible with a's being F (Demos 1917). For example, the truth-maker for the claim that kingfishers aren't yellow is the fact that they're blue because their being blue is incompatible with their being yellow. But what makes it true that these colours are incompatible? The notion of incompatibility appears itself to be negative--a relation that obtains between two states when it's not possible for them to obtain together. So this proposal threatens to become regressive: we'll need to find another positive truth-maker for the further negative claim that yellow and blue are incompatible, something whose obtaining is incompatible with the state of yellow and blue's being compatible, and so on (Russell 1918-9: 213-5, 1919: 287-9; Taylor 1952: 438-40; Hochberg 1969: 330-1; Grossman 1992: 130-1; Molnar 2000: 74-5; Simons 2008). There's another worry: it's not obvious that there are enough positive states out there to underwrite all the negative truths there are. Even though it may be true that this liquid is odourless this needn't be because there's something further about it that excludes its being odorous (Taylor 1952: 447; Mulligan, Simons, & Smith 1984: 314; Molnar 2000: 75; Armstrong 2004: 62-3; Dodd 2007: 387). One could circumvent the threatened regress by denying that the incompatibilities in question require truth-makers of their own because they're necessary truths and such truths are a legitimate exception to maximalism--because "they are true come (or exist) what may" (Simons 2005: 254; Mellor 2003: 213). There is some plausibility to the idea that formal truths (tautologies) don't stand in need of truth-makers; their truth is settled by the truth-tables of the logical constants. But material necessary truths--such as that expressed by "yellow is incompatible with blue"--appear to make just as substantive demands upon the world as contingent truths do (Molnar 2000: 74). In a sense they appear to make even more of a demand since the world must be so endowed that it could not in any circumstances have failed to live up to the expectations of material necessary truths. It's a peculiar feature of our philosophical culture that even though it's almost universally acknowledged that Wittgenstein's plan (1921: 6.37) to show all necessity is logical necessity ended in failure--indeed foundered upon the very problem of explaining colour incompatibilities--that so many philosophers continue to think and talk as though the only necessities were formal ones so that necessary truths don't need truth-makers (MacBride 2011: 176-7). Russell reluctantly chose to acknowledge negative facts as truth-makers for negative truths. He just couldn't see any way of living without them. But negative facts are an unruly bunch. Try to think of all the ways you are. Contrast that with the even harder task of thinking of all the ways you aren't! If negative truths are acknowledged as truth-makers they will have to be indefinitely numerous, unbounded in their variety; choosing to live with them is a heavy commitment to make (Armstrong 2004: 55). What's worse, if negative facts are akin to positive facts--as their name suggests--then they must be made up out of things, properties and relations arranged together. But, prima facie, many of these things, properties and relations aren't existing elements of reality. So unless, like Meinong, we believe in the non-existent, we'll have to admit that negative facts aren't configurations of their constituents and so an entirely different kind of entity from positive facts altogether (Molnar 2000: 77; Dodd 2007: 388). It is for such reasons that Armstrong counsels us to adopt a more parsimonious account of what makes negative truths true (2004: 56-9). Armstrong's own account lies at the opposite extreme to Russell's. Whereas Russell posited indefinitely many negative facts to make negative truths true, Armstrong posits just one thing that's responsible for making them all true, viz. a totality fact. We should register C.B. Martin's doubt that this is throwing out the baby with the bath water (1996: 59). According to Martin, we already recognise in ordinary discourse that different negative truths have different truth-makers--not just one as Armstrong proposes. For example, we recognise that what makes it true that there is no oil in this engine is different from what makes it true that there are no dodos left. What makes claims like these true are absences, lacks, limits, holes and voids, where these are conceived not as things but as "localised states of the world", robustly first-order and "causally relevant" to what goes on (Martin 1996: 58, 65-6; Taylor 1952: 443-5). But, as many philosophers have argued, when we talk about an absence having causal effects what we're really saying can be understood without reifying negative states and appealing instead to the actual effects, or the counterfactual effects, of a positive state (Molnar 2000: 77-80; Armstrong 2004: 64-7; Lewis 2004; Beebee 2004). ##### 2.1.4.2 Truth-makers for general truths Armstrong's grand design is to sweep away the difficulties that attend the admission of negative facts by positing a special kind of general fact that also serves as the truth-maker for general truths. Russell admitted general facts too but he acknowledged that, "I do not profess to know what the right analysis of general facts is" (Russell 1918-9: 236-7). But Armstrong has gone further and assigned to general facts the following structure: a general, or totality fact consists in a binary relation T of totality that holds between an aggregate on the one hand and a property on the other when the aggregate comprises all the items that fall under the property in question (Armstrong 1989b: 93-4, 1997: 199-200, 2004: 72-5). For example, China, France, the Russian Federation, the U.K. and the U.S.A. comprise the permanent membership of the UN Security Council. So the aggregate (A) of them bears the T relation to the property (P) of being a permanent member of the Council (T(A, P)). Since the aggregate bears that relation to that property, there can be other permanent members of the Council who aren't already included in it. So the totality fact T(A, P) suffices for the general truth that China, France, the Russian Federation, the U.K. and the U.S.A are all its members. It also suffices for the truth of the negative existential that there are no other members of the Council. So once we've recognised that T(A, P) exists, there's no need to recognise additional bespoke truth-makers for these negative truths. In a similar way Armstrong endeavours to sweep away the need for negative facts by affirming "the biggest totality state of all, the one embracing all lower-order states of affairs", i.e., the existence of a totality state that consists in an aggregate of all the (1st order) states of affairs there are related by T to the property of being a (1st order) states of affairs (2004: 75). It's one of Armstrong's abiding contentions that the world is "a world of states of affairs, with particulars and universals only having existence within states of affairs" (1989: 94). Consequently this totality state comprises a vast swathe of what exists--whether particulars, universals or states of affairs that consist in particulars having universals. So it follows from the existence of this totality fact that there are no more (1st order) states of affairs that are not already included in the aggregate of states that T relates to the property of being a (1st order) states of affairs. Nor are there any particulars or (1st order) universals had by those particulars that are not constituents of the state of affairs included in that aggregate. It also follows that there are no more particulars or (1st order) universals. So this totality fact serves as truth-maker for all these negative truths. It is sometimes objected that such totality facts are just negative facts in disguise: "Totality statements state the non-existence of certain entities, they state 'no more facts'"; so we should reject totality facts if we are dissatisfied with negative ones (Molnar 2000: 81-2; Dodd 2007: 389). Armstrong responds to this charge with equanimity: "It is not denied, of course, that the totalling or alling relation involves negation. It sets a limit to the things of that sort" (2004: 73). But if negation has indeed been smuggled into the description of the role that T performs in comprising a totality state then it is difficult to avoid the suspicion that Armstrong has simply exchanged many negative facts for one big one. But we may think of Armstrong's contribution in a different way. There are two ostensible concerns that negative states of affairs present. First, there is a concern about their number. Second, there is a concern about, so to speak, their negativity. Armstrong has addressed the first concern by showing how we may reduce the number of negative states of affairs. But the second concern neither cannot be met--because, as Armstrong reflects, we cannot eliminate negation from our description of the world. It has also been objected that Armstrong's position gives rise to a "paradox of totality" (Armstrong 1989: 94, 1997: 198-9, 2004: 78-9; Cox 1997: 53-60; Molnar 2000: 81). Take the totality state of affairs that comprises all the (1st order) state of affairs. Since this (2nd order) state is itself a state of affairs it follows that the initial aggregate of (1st order) states of affairs failed to comprise all the states of affairs there are. So there must be a further totality state that comprised the aggregate of all of them. But this (3rd order) state is also a state of affairs so it will need to be added to the mix too, and so ad infinitum. Armstrong responds to this objection with equanimity too: > > > we can afford to be casual about this infinite series. For after the > first fact of totality these "extra" states of affairs are > all supervenient. As such, we do not have to take them with > ontological seriousness. (1989b: 94) > > > To understand this we need to appreciate that Armstrong's notion of supervenience is non-standard: "an entity Q supervenes upon entity P if and only if it is impossible that P should exist and Q not exist" or, in other words that the existence of P entails the existence of Q. (Armstrong 1997: 11). He also holds a non-standard conception of ontological commitment, viz. that "What supervenes is no addition of being" (Armstrong 1997: 12, 2004: 23-4). Armstrong's idea is (roughly) that to be a genuine addition to being is to be a net (indispensable) contributor to the schedule of truth-makers for all the truths. But supervenient entities are superfluous as truth-makers. If the existence of an R entails the existence of a certain S which in turn entails the truth of P, then R already makes P true so there is no need to include S in the inventory of truth-makers. It follows that supervenient entities, like S, are no addition of being (Lewis 1992: 202-3). Now bring this non-standard conception of ontological commitment to bear upon the envisaged infinite series of totality facts. It is impossible that the 2nd order totality fact comprising all the 1st order states of affairs exist and the 3rd order totality fact comprising all the 1st and 2nd order states not exist. So the 3rd totality fact--or any other state higher-order than it--is entailed by the existence of the 2nd order totality state. So all of these n > 2 higher-order totality states supervene on it. That's why Armstrong doesn't think we need to take any of them with ontological seriousness. What should we make of this non-standard conception of ontological commitment in terms of truth-making: that to be is to be a truth-maker? Cameron (2008c) has proposed that this conception should replace the standard one that to be is to be the value of a variable. But this threatens to cut off the branch the advocates of truth-maker theory are sitting on: if they disavow that existential quantification is ontologically committing then they will be left without a means of determining the ontological commitments of truth-maker theory itself, i.e., the theory that gives the inventory, using existential quantification, of what makes all the truths true (Schaffer 2010: 16-7). Of course, if someone grants that existential quantification is ontologically committing in the context of a theory of truth-makers then they won't stultify themselves in this way (Heil 2003: 9, 2009: 316-7, Simons 2010: 200). But this just seems like special pleading. An argument is owed that we can't legitimately commit ourselves to the existence of things that perform theoretical roles outside the theory of truth-makers (MacBride 2011: 169). Why should there be only one theoretical dance allowed in town? Why shouldn't we allow that there are other theoretical roles for existing things to perform? Indeed it's a very real possibility that when we come to understand the capacity of the truth-makers to make truth-bearers true we will find ourselves embroiled in commitment to the existence of other things in their explanatory wake that aren't truth-makers themselves. In fact Armstrong himself should have been one of the first to recognise this. For it has been an abiding feature of Armstrong's world view that we are obliged to acknowledge not only states affairs, which are truth makers, but also properties and relations, constituents of states of affairs, which aren't. More generally, can we make any sense of the idea of "an ontological free lunch"? Why is something supervenient no addition to being? Even if we only ever came to recognise the existence of one supervening entity would we not thereby have added at least one extra item to our inventory of things that exist? Of course in the special case where the supervening entity is constituted by the entities it supervenes upon--i.e., the things that are already there--it makes some sense to say that it's not adding anything new; but it's not at all obvious that what Armstrong et al. declare to be ontological free lunches are constituted from the entities upon which they supervene (Melia 2005: 74-5). Accordingly Schulte (2011a, 2011b) argues that no necessitarian or grounding account of truth-making can make sense of the notion of an ontological free lunch, maintaining instead that truthmaker theory needs to be augmented with the idea that we are offering a reductive explanation when we make an explanation in terms of truthmakers Where does this leave us? So long as Armstrong's non-standard conception of ontological commitment remains controversial, it also remains controversial whether the infinite series of totality facts to which Armstrong is "committed" may be dismissed as mere ontological frippery. ### 2.2 Optimalism One way to respond to these difficulties is to abandon maximalism in favour of optimalism, to deny that universal and negative statements need truth-makers. But Merricks argues the optimalist "way out" is blocked. Negative truths need truth-makers if any truths do, but they can't have them. So we must give up thinking that truths need truth-makers in the first place (2007: 39-67). Here's why Merricks thinks so. In order to avoid the over-generation of truth-makers for necessary truths (etc.) Merricks imposes a relevance constraint: "a truth-maker must be that which its truth is about" (2007: 28). But can this constraint be satisfied in the case of negative existentials, such as the statement there are no hobbits? This statement isn't about hobbits, because there aren't any, and other apparent candidates, such as the universe's exhibiting the global property of being such that there are no hobbits in it, appear hoaxed up and artificial (2007: 43-55). Even if one follows Merricks this far, one may still think that negative existentials are the principled exception that proves the rule that a truth-maker for a positive truth must be what its truth is about. But, so far as Merricks is concerned, this is throwing the baby out with the bathwater: > > > I deny that if we set aside the intuition that "a truth, any > truth" depends on being we are left with the equally compelling > intuition that all truths except negative existentials depend > on being. (2007: 41) > > > He also suggests that this position is theoretically disingenuous because no one would consider retreating to it from full-blown maximalism unless he or she had already been "spooked" by his or her failure to find truth-makers for negative truths; or if they held onto the view that truth is correspondence (Merricks 2007: 40-1; Dodd 2007: 394; Cameron 2008a: 411). Merricks surmises that if we have any reason to commit to truth-makers, we have only reason to commit outright to a truth-maker for every truth (maximalism). But since maximalism cannot be sustained because of the lack of things for negative existentials to be about, Merricks recommends the rejection of truth-makers altogether. But optimalists aren't just "spooked" or "timid" maximalists. They stand on their two feet with a principled position of their own that need neither be based upon "gerrymandered intuition" nor adopted as a consequence of a forced retreat from maximalism. If maximalism is intellectual heir to Russell's logical atomism, then optimalism (at least in the form under consideration) is heir to Wittgenstein's version of the doctrine according to which it is only atomic propositions that represent the existence of states of affairs. The optimalists' idea is that once truth-makers have been supplied for the atomic truths there is simply no need to posit further truth-makers for the molecular ones. All we need to recognise is that an atomic statement P is true whenever a truth-maker for P exists, that P is false if and only if no truth-maker for P exists. Once the existence and non-existence of the truth-makers has settled the truth-values of all the atomic statements, the logical operations described by the truth-tables then settle the truth and falsity of all the molecular statements (another story must be told about what the truth-makers are for the non-extensional constructions--another elephant in the room). In particular, the truth-table for negation--that tells us what "~" means--assures us that if P is false then ~P is true. So all it takes to make ~P true is that no truth-maker for P exists. Thus Mulligan, Simons, and Smith: > > > it seems more adequate to regard sentences of the given kind as true > not in virtue of any truth-maker of their own, but simply in virtue of > the fact that the corresponding positive sentences have no > truth-maker. (1984: 315; see also Mellor 2003: 213-4, 2009; > Simons 2000:7-8, 2005: 255-6) > > > Simons offers the illuminating reflection: > > > This is the truth-maker end of Wittgenstein's insight that > propositions are bi-polar: if a proposition has one truth-value, > however it gets it, its contradictory opposite has the opposite > truth-value without further ado; > > > Simons dubs the truth functional mechanism whereby the negation of an atomic statement gets the value true, "truth by default" (2008: 14). Optimalists also think that general truths are true by default so there is no need for bespoke truth-makers for them (like totality facts). Universal quantifications [?]xFx are logically equivalent to negative statements of the form [?][?]x[?]Fx. Since the latter are negative, they're true, if they are, only by default. Then because the former statements are logically equivalent to them, the optimalists surmise that universal quantifications are true by default too (Mellor 2003: 214; Simons 2008: 14-5). Further challenges for the optimalist or non-maximalist are discussed in Jago 2012, Simpson 2014, Jago 2018 (81-102). ### 2.3 Truth Supervenes Upon Being Optimalism retains the original demand for truth-makers but restricts it to atomic statements. Optimalism accordingly disavows a commitment to truth-makers for negative statements and statements of generality. But Bigelow[?]also wary of the commitments that maximalism engenders to negative and totality facts[?] weakens what truth-making means to a point where negative and general statements don't require bespoke truth-makers of their own (1988: 131-3). He offers the following principle to capture the kernel of truth in truth-making worth saving. (Truth Supervenes on Being) If something is true then it would not be possible for it to be false unless either certain things were to exist which don't, or else certain things had not existed which do (1988: 133). This principle allows atomic truths to have truth-makers[?]because it would only be possible for them to be false if certain things had not existed: their truth-makers. But it also allows negative truths to be true without them. The statement that there are no dragons gets to be true because it would only be possible for that statement to be false if something which hadn't existed (dragons) did exist. Such statements are true not because they have truth-makers but because they have no counterexamples, "they lack false-makers" (Lewis 1992: 216, 2001: 610). General truths are also true for lack of false-makers. The statement that everything is physical (if true) is true because it would only be possible for the statement to be false if something that hadn't existed did, viz. something that wasn't physical. So when truth-making is understood in this weakened sense there is no need to acknowledge, e.g., an additional totality fact to make it true that there are only five coins in my pocket; it's enough that if I hadn't stopped adding coins that statement would have been false because then there would have been at least one other coin in my pocket (Heil 2000: 236-240, 2003: 68-72, 2006: 238-40; Bricker 2006). Despite their differences, optimalism and (Truth Supervenes Upon Being) share a key idea in common--that a negative existential truth isn't true because something exists, but because something doesn't, viz. a truth-maker for its contradictory, a false-maker for it. C.B. Martin has objected that this doesn't obviate the need to posit truth-makers for negative existentials. This is because a statement that there are no false-makers for a negative existential truth is itself a negative existential truth. So this statement "can't be used to explain or show how the latter needs no truth-making state of the world for it to be true" (1996: 61). Lewis takes Martin's objection to be that this account of the truth of negative truths isn't informative. Take the statement that there are no unicorns. Why is it true? Well because there are no unicorns. That's not much of an explanation! But, Lewis retorts, the positive existential statement that there is a cat is true because there is a cat. That's "No explanation at all, and none the worse for that" (Lewis 2001: 611-12). Fair enough; but perhaps Martin meant that optimalists and their fellow-travellers were presupposing what they set out to show--because they assumed without argument that a special class of negative claims stood in no need of truth-makers, those according to which other negative statements lacked false makers. ### 2.4 Humean Scepticism Lewis has anyway argued that the doctrine that truth supervenes upon being--and, by implication, optimalism--are an uncomfortable halfway house. He began by trying to persuade us that the retreat from maximalism was already mandated. He pointed out how deeply counterintuitive it is to suppose that negative existentials are true because their truth-makers exist. It seems, offhand, that they are true not because some things do exist but because some don't. "Why defy this first impression?" (1992: 204). But, more importantly, Lewis pointed out that maximalism was incompatible with what he took to be the master principle governing our thought about modality: Hume's denial of necessary connection between distinct existences. It is a consequence of this principle that anything can co-exist with anything else that's distinct. But it is the raison d'etre of any truth-maker for the negative existential truth that there are no unicorns--if there is one and whatever else it is like--that even though it is distinct from all unicorns it cannot co-exist with any of them, else it would make it true that there are no unicorns even in circumstances where unicorns existed. Similarly it is the raison d'etre of any truth-maker for a general truth that such-and-such are all the facts there are that it refuses to co-exist with any other facts even though it is distinct from them. So if we're to hang onto Hume's denial of necessary connexions, we'd better give up the demand for truth-makers for negative and general truths, i.e., maximalism (Lewis 1992: 205, 2001: 610-11). Lewis also argues that we can't stop here--just giving up maximalism for optimalism--because even the truth-makers for atomic statements conflict with Humeanism. Suppose that the statement that Dog Harry is golden is atomic. Also suppose that Harry is only accidentally golden, so the statement is contingent. What is its truth-maker? It can't be Harry because it's possible for him to exist even in circumstances where the statement is false, i.e., when he has a different coloured coat. But it can't be the property of being golden either since there are possible circumstances in which plenty of things are golden but not Harry. It can't even be the mereological fusion Harry + being golden because it's a feature of fusions that they exist even in circumstances where their parts are otherwise unrelated--e.g., when Harry is black whilst Harriet is golden. For this reason, many maximalists make common cause with optimalists to posit another fundamental kind of thing to perform the truth-making role in the case of contingent (atomic) predications: facts that consist in objects, properties and relations bound together; in this case the fact Harry's being golden (Armstrong 1989a: 41-2, 1997: 115-6; Mellor 1995: 24). Now facts in general and Harry's being golden in particular cannot be built up mereologically from Harry and being golden; otherwise Harry's being golden would be no more fit a candidate for making the statement that Harry is golden true than the fusion Harry + being golden. Since Harry's being golden isn't built up mereologically from Harry and being golden, they cannot be parts of it. But if they aren't parts of it they must be entirely distinct from this state of affairs. This is where Lewis pounced. Even though Harry's being golden is entirely distinct from its "constituents" it cannot obtain without Harry and being golden existing. More generally, the obtaining of a fact necessitates the existence of its constituents even though the fact and its constituents are entirely distinct. So we cannot posit facts as truth-makers for contingent (atomic) statements without going against Humeanism (Lewis 1992: 200, 1998: 215-6, 2001: 611). Other maximalists and optimalists, often those wary of facts, posit (non-transferrable) tropes as truth-makers for contingent predications (Martin 1980; Mulligan, Simons, & Smith 1984: 295-304, Lowe 2006: 186-7, 204-5) Tropes, in the non-transferrable sense, are particular instances of properties that are existentially dependent upon their bearers. For example, the particular golden colour g of Harry's coat is a non-transferrable trope because g could not have existed except as the colour of his coat. Since g is non-transferrable, this trope only exists in circumstances where Harry's coat is golden; hence g's eligibility to be a truth-maker for the statement that Harry is golden. (If g was transferrable, then g could have existed in circumstances where whilst borne by Harriet, Harry bore another black trope d; so if g is transferrable it isn't eligible to be a truth-maker for the statement that Harry is golden.) But Lewis' reasoning can be easily extended to show that non-transferrable tropes cannot serve as the truth-makers for contingent statements without contradicting Humeanism. If trope g is a property that Harry bears rather than a part of him, then g is wholly distinct from Harry. Nevertheless, the existence of g necessitates the existence of Harry even though they are distinct (MacBride 2005: 121). Alternatively if g is a part of the bundle of tropes that constitute Harry, if g is non-transferrable, then the existence of g necessitates the existence of some other distinct tropes that are also parts of this bundle. So Humeanism is violated either way. #### 2.4.1 Subject Matter In order to avoid contradicting Humeanism, Lewis recommended a further weakening of (Truth Supervenes upon Being). According to Lewis, the kernel of truth in truth-making is the idea that propositions have a subject matter. They are about things so whether they are true or false depends on how those things stand. This led Lewis to endorse (2003: 25): (Subject matter) Any proposition has a subject matter, on which its truth value supervenes. Equivalently, there cannot be a difference in the truth-value of a proposition without a difference in its subject matter. This might consist in (1) an increase or decrease in the population of things that fall within the subject matter; or (2) a shift in the pattern of fundamental properties and relations those things exhibit (Lewis 2001: 612). Now the truth value of "Harry is golden" supervenes upon its subject matter without there needing to be any existing thing distinct from Harry or being golden that necessitates their existence; it is enough that the statement would have been false if Harry had lost his hair or been dyed. So we can avoid contradicting Humeanism by abandoning optimalism in favour of (Subject matter). In an intriguing twist to the plot, Lewis subsequently appeared to withdraw his doubts about truth-makers (2003: 30, Lewis & Rosen 2003: 39). When building the case for (Subject Matter) Lewis had deliberately remained neutral about the metaphysics of modality (2001: 605). But Lewis recognised that if he gave up his neutrality and availed himself of counterpart theory then he could supply truth-makers aplenty whilst still adhering to his Humeanism. According to Lewis's counterpart theory, "a is essentially F just in case all of a's counterparts (including a itself) are F" (2003: 27). Counterparts of x are objects that are similar to x in certain salient respects. Different conversational contexts make different respects salient. This means that the truth (or falsity) of essentialist judgements is always relative to the counterpart relation conversational cues select; so it will often be the case that whilst something is essentially F with respect to one counterpart relation it won't be with respect to another (Lewis 1968). In a conversation that makes it salient to think about Harry under the counterpart relation of being a golden Labrador then every relevant counterpart of him will be golden. So relative to this counterpart relation he is essentially golden. But in a different conversation that selects the counterpart relation of being a dog because then whilst all of his counterparts will be dogs Harry won't be essentially golden because some of his counterparts will be other colours. What's important is that it's one and the same thing, our beloved Harry, that's essentially golden with respect to the former counterpart relation but only accidentally so with respect to the latter. To make these essentialist judgements count as true in the right contexts we don't need one dog that's essentially golden, and another that's only accidentally so; we just need to think about Harry in two different (but compatible) ways. According to (Necessitarian-T) and other related views, the property a truth-maker x has of making a given proposition P true is an essential feature of it. Why so? Because x is supposed to be something the mere existence of which suffices for the truth of P; it couldn't have existed without P being true (Armstrong 1997: 115). But Lewis' proposal is to treat essentialist attributions about truth-making in the same relativistic spirit that counterpart theory treats every other modal attribution (2003: 27-32). Think of Harry qua golden Labrador, i.e., under the counterpart relation of being a golden Labrador. All his counterparts selected by this relation are golden. So every world in which Harry or one of the counterparts exists is a world in which "Harry is golden" is true. In other words, Harry qua golden is truth-maker for the proposition that Harry is golden. But this doesn't commit us to necessary connexions between distinct existences--to a dog that couldn't have failed to be golden. In another context it will be just as legitimate to talk about Harry in a different way, as Harry qua dog. In that context his counterparts will include dogs that aren't golden. So Harry qua dog will be the truth-maker for the statement that Harry is a dog, but not the statement that he is golden. In the same way that ordinary objects like Harry can serve as truth-makers for contingent predications, Lewis & Rosen (2003) suggest that the entire world--"the totality of everything there actually is"--can serve as the truth-maker for negative existentials. Take the world qua unaccompanied by unicorns. In the conversational context just set up, the world has no counterparts that are inhabited by unicorns. So the world qua unaccompanied by unicorns is essentially lacking in unicorns and therefore qualifies as a truth-maker for the statement that there are no unicorns. Has Lewis shown how it is possible to garner truth-makers for contingent predications and negative existentials without positing totality facts or tropes that ensnare us in a web of necessary connections? Indeed has he shown how we can get by using only ordinary objects and collections of them to serve as truth-makers? We need to be clear about what Lewis is trying to do in this paper. He isn't reporting upon a damascene conversion, belatedly recognising what he had previously denied, that the truth-making role is genuine, but then ingeniously coming up with the idea that qua-versions of things perform this role just as effectively as states of affairs, only without necessary connections between distinct existences. Rather, Lewis' aim in this paper is to damn the very idea of truth-makers with faint praise. By showing how qua versions of things performed the truth-making role just as effectively as facts or tropes Lewis aimed to show how explanatorily bankrupt the truth-making role truly was (MacBride 2005: 134-6, Bricker 2015: 180-3). There are various criticisms of detail that might be made. MacBride (2005: 130-2) points out that Lewis failed to provide any account of the truthmakers for relational predications and argues that it is very difficult to see how the lacuna can be filled when the class of eligible truthmakers is restricted by Lewis to the class of ordinary things and only intrinsic counterpart relations can be evoked. Lewis considers a relaxation of the latter requirement when it came to negative existentials (qua unaccompanied by unicorns does not express an intrinsic counterpart relation). But MacBride argues removing the latter requirement risks trivialising Lewis's account because for any object a such that p is true, for any true p, a satisfies the description F: "inhabiting a world where p is true", hence a qua F makes it true that p. Bricker (2015: 179-80) argues that Lewis can provide for relational predications by allowing sequences to be truthmakers, i.e. not just ordinary things as Lewis required. But Bricker finds this solution ultimately unattractive because it conflicts with another truthmaking principle he finds plausible: that distinct atomic truths have distinct truthmakers. One might also object to Lewis' controversial modal metaphysical assumptions of counterpart theory more generally (Rodriguez-Pereyra 2006b: 193; Dodd 2007: 385). More broadly, one might question whether Lewis was right to place so much weight upon Hume's denial of necessary connections between distinct existences or question whether Lewis had interpreted it correctly (Daly 2000: 97-8; MacBride 2005: 123-7; Hoffman 2006; Cameron 2008b). It's one thing to say that we shouldn't multiply necessities without necessity but it's another thing to demand that we purge our world-view of them altogether. What about dispositions, chances, laws etc? Hume relied upon an empiricist theory of content to underpin his rejection of necessary connections. But Lewis certainly didn't advocate such a view. One is left wondering therefore whether Hume's denial of necessary connexions should be abandoned along with the theory of content that inspired it, an antiquated relic. If so, then we have yet to be given a reason to retreat from either optimalism or (Truth Supervenes Upon Being) to (Subject Matter)(MacBride 2005: 126-7). ## 3. What Motivates the Doctrine of Truth-makers? ### 3.1 To catch a cheater What is the motivation for adopting a theory of truth-makers, whether maximalist or optimalist? Following Sider, it has become customary to describe the "whole point" of adopting a theory of truth-makers thus: to "Catch the cheaters" who don't believe in truth-makers (Sider 2001: 40-1; see also Merricks 2007: 2-3). But this is a bit like saying that the point of a benefit system is to catch out benefit frauds. It's only if there is some antecedent point in favour of truth-makers that there can be anything wrong with leaving them out. Nonetheless, it is possible that we can come to appreciate the need for truth-makers by appreciating what is lacking in theories that neglect them. This is how C.B. Martin and Armstrong came to recognise the necessity for admitting truth-makers (Armstrong 1989b: 8-11, 2004: 1-3). Consider phenomenalism: the view that the physical world is a construction out of sense-impressions. One obstacle for this view is the need to make sense in sense-impression terms of true claims we make about the unobserved world. Martin noticed that the phenomenalist could only appeal to brute, ungrounded counterfactuals about possible experience to do so. But counterfactuals do not float free in a void. They must be responsive to an underlying reality, the reality that makes them true. For Martin this was the phenomenalists' fatal error: their inability to supply credible truth-makers for truths about unobserved objects. The same error afflicted Ryle's behaviourism with its brute counterfactuals about non-existent behaviour and Prior's presentism according to which there is nothing outside the present and so there is nothing for past-tensed and future-tensed truths to be held responsible to. We come to appreciate the need for truth-makers as the common need these different theories fail to fulfil. As Lewis points out, this demand for truth-makers represented an overreaction to what was undoubtedly a flaw in phenomenalism (1992: 217). We can already understand what's wrong with phenomenalism when we appreciate that it fails to provide an account of the things upon which the truth of counterfactuals about sense impressions supervene, i.e., when we see that it fails to satisfy (Subject Matter). This leaves under-motivated the claim that phenomenalism should also be faulted for failing to supply truth-makers to ground counterfactuals about sense-impressions. The same reasoning applies to behaviourism and presentism; they also fail to satisfy (Subject Matter). So there's no need to adopt truth makers to catch cheats. All we need to do is to recognise the strictures (Subject Matter) places upon us. It's a further flaw of Martin and Armstrong's reasoning that even if we accede to the demand for truth-makers it doesn't follow that even idealism is ruled out. According to Armstrong if we forsake the demand for truth-makers we thereby challenge "the realistic insight that there is a world that exists independently of our thoughts and statements, making the latter true or false" (1997: 128). But even an idealist could accept that there are truth-makers whilst thinking of them as mind-dependent entities. So acceding to the demand for truth-makers doesn't tell us what's wrong with idealism (Daly 2005: 95-7). Bergmann, the grandfather of the contemporary truth-maker movement, was explicit about this: "the truth of S must be grounded ontologically. On this first move idealists and realists agree" (Bergmann 1961: 229). The realist-idealist distinction cuts across the distinction between philosophers who admit truth-makers and those that don't (Dodd 2002: 83-4). The demand for truth-makers doesn't help "catch cheaters" at all. Just acknowledge, e.g., some brute counterfactual facts about sense data, or brute dispositions to behave in one way or another, or brute facts about what happened in the past or will happen in the future. What's unsatisfactory about these posits isn't that if they existed they'd fail to make true statements about unobserved objects or statements about mental states that failed to manifest themselves in actual behaviour or statements about the past or future. The problem is that we have difficulty in understanding how such facts or dispositions could be explanatorily sound (Horwich 2009: 195-7). Whatever we find lacking in theories that posit such items, it isn't that they fail to provide truth-makers, because they do. Of course if we are already committed to the need for truth makers then we will likely conceive of the demand for them as, e.g., drawing the phenomenalist out into the open to reveal the unfitness of their explanatory posits. But unless we already have independent reasons for recognising the demand for truth makers, catching cheaters cannot provide a motivation for positing them. ### 3.2 Truth Depends on Being Does this mean that we should join Bigelow in his retreat to (Truth Supervenes Upon Being) or step back with Lewis to (Subject Matter)? The problem is that these supervenience principles don't seem a satisfactory resting place either, not if our concern is to understand how true representations can touch upon an independent reality, i.e., something non-representational. First, it is familiar point that just appealing to a systematic pattern of modal co-variation between truths and their subject matters--so that there is no difference in the former without a difference in the latter--doesn't provide us within any insight into the underlying mechanism or mechanisms that sustains this dependency (Molnar 2000: 82-3; Heil 2003: 67; Daly 2005: 97-8; Melia 2005: 82-3). Second, if possible worlds are conceived as maximal representations then these supervenience principles (understood in the idiom of possible worlds) fail to articulate what it is for a representation to touch upon being, something non-representational: "For the idea that every truth depends on being is not the idea that every truth is entailed by propositions of a certain sort" (Merricks 2007: 86). A third reason for being unsatisfied with Lewis' (Subject Matter) is based upon the observation that the supervenience involved is symmetric. Just as there is no difference in the distribution of truth-values amongst the body of propositions without a reciprocating difference in their subject matter, there is no difference in their subject matter without a difference in the distribution of their truth-values (Armstrong 2004: 7-8). But we also have the firm intuition that the truth or falsity of a proposition depends upon the state of the world in a way in which the state of the world doesn't depend upon the proposition's truth or falsity. This means that a supervenience principle like (Subject Matter) cannot be used to articulate the asymmetric way in which truth so depends upon being; for this, it is argued, we need to rely upon a robust asymmetric notion of truth-making (Rodriguez-Pereyra 2005: 18-19). Bricker (2015: 180-183) replies on Lewis's behalf that the oft-heard complaint that supervenience by itself is not a dependence or ontological priority relation is besides the point. Rather, Bricker argues, the metaphysical punch of Lewis's doctrine that truth supervenes upon being "has little to do with the 'supervenience part' and everything to do with the 'being' part and the conception of fundamentality that informs it", where by 'fundamental' Bricker means Lewis's theory of natural properties and relations. But is there really anything so problematic or mysterious about the asymmetric dependence of truth upon being that we need anything as heavy duty as truth-making or supervenience+naturalness to explain it? A number of proposals have been made for explaining the felt asymmetry in other terms. Horwich (1998: 105; 2008: 265-66) argues that asymmetric dependence of truth upon being arises from the fact that we can deduce p from the proposition that p is true. But we can equally deduce p from p is true so this fails to establish an asymmetric dependence (Kunne 2003: 151-52; Rodriguez-Pereyra 2005: 27). Hornsby argues instead that there is explanatory asymmetry between its being the case that p and the proposition that p is true, because the latter requires more to be the case than the former, (e.g.) that there are propositions (Hornsby 2005: 44). Dodd suggests a cognate alternative: that the felt asymmetry of truth upon being arises from the fact that the identity of the proposition that p is partially determined by the worldly items it's about, so we can't understand what is required for the proposition that p to be true without already having an understanding of what it is for p to be the case (Dodd 2007: 398-400). MacBride (2014) offers a purely semantic account of the felt asymmetry in terms of the interlocking mechanism of reference and satisfaction that explains the truth of a proposition. Whilst we can deduce from the truth of the proposition that p that p is the case, MacBride argues that we can't (ultimately) explain its being the case that p in terms of the truth of the proposition that p because its being the case that p is already assumed by the truth of that proposition, that is, has already performed a role in the semantic mechanism whereby the truth-value is determined. By contrast, MacBride continues, its being the case that p isn't determined by a semantic mechanism but by whatever worldly factors, if any, that gave rise to p's being the case. Saenz (2018) replies that whilst reference and satisfaction may be used to explain that propositions are true, they still "do not, in any sense, account for the world 'making it' [sic] that propositions are some way" (a similar point is made by Mulligan et al 1984: 288, see sec. 3.5 below for further discussion). But Saenz risks begging the question by making the additional demand on the character of the felt asymmetry of truth and being that it consists in more than an asymmetric relationship between truth and being. Saenz holds that the felt asymmetry consists in the requirement that the world makes propositions true but whether we need to appeal to the idea of the world making propositions true in the first place is the point at issue. Finally, Liggins (2012) argues that the felt asymmetry between truth and being is best explained in terms of the asymmetry of grounding, although, as we have seen, the asymmetry of grounding is contested (see sec. 1.6 above for further discussion). ### 3.3 Correspondence "Catching cheats" doesn't supply a motivation for adopting truth-makers. Are there other motivations for so doing? One such motivation is that truth-making falls out naturally from the correspondence theory of truth. > > > Anybody who is attracted to the correspondence theory of truth should > be drawn to the truth-maker [sic]. Correspondence demands a > correspondent, and a correspondent for a truth is a truth-maker. > (Armstrong 1997: 14; see also 1997: 128-9, 2004: 16-7) > > > > According to Armstrong, the "truth-maker principle" (aka maximalism) just is what we are left with once we drop the assumption from the correspondence theory that the relation between truth bearers and truth-makers is one-one. Of course, the truth-making relation, however it is explicated, cannot be the relation of correspondence in terms of which truth is defined--for whereas the latter is symmetric, the former isn't (Beebee & Dodd 2005b: 13-4). But this doesn't prevent truth being defined using the truth-making relation, albeit in a more roundabout way according to the following pattern, (C) S is true = ([?]x)(S is-made-true-by x). Of course such a definition will only succeed if the relation expressed by "is-made-true-by" can itself be explicated without relying upon the notion of truth itself or other notions that implicitly presuppose it, e.g., entailment (Merricks 2007: 15). This is no doubt one of the reasons that principles of truth-making are typically not put forward as definitions of truth (David 2009: 144). Another objection made by Horwich is that it is implausible to suppose that an ordinary person understands the notion of truth via the heterogeneous principles that govern the provision of truth-makers for different types of propositions. It's more plausible to suppose that we first grasp what truth is, and only subsequently figure out what truth-makers are required for the various kinds of propositions there are (Horwich 2009: 188). ### 3.4 Deflationism Several philosophers have argued that so far from presupposing the notion of truth, the various truth-making principles we have discussed operate at a far more subterranean level. Bigelow writes, "The force behind Truth-maker lies deeper than worries about the nature of truth and of truth-bearers" (Bigelow 1988: 127, 2009: 396-7; Robinson 2000: 152; Lewis 2001: 605-6; Horwich 2009: 188-9, Bricker 2015: 169). Suppose we understand what it is to be truth-maker in terms of entailment. Then whatever the range of truths we think are capable of being rendered true, the "guts" of our truth-maker principle can be stated using the schema, (Schema) If P, then there must be something in the world whose existence entails that P. This schema is equivalent to the infinitely many conditionals that fall under it: (i1) If the donkey is brown, then there must be something in the world whose existence entails that the donkey is brown; (i2) if the Parthenon is on the Acropolis then there must be something in the world whose existence entails that the Parthenon is on the Acropolis; and so on. Note that neither "true" nor "truth" shows up anywhere in the schema or its instances. Arguably the only role that the notion of truth performs is the one that minimalist or deflationary theories of truth emphasize, that of enabling us to capture all of that unending string of conditionals in a single slogan, (Slogan) For any truth, there must be something in the world whose existence entails that truth. But since truth is serving here only as a device of generalization, the real subject matter of (Slogan) is already expressed by the conditionals (i1), (i2) etc. Since they make no explicit mention of either propositions or truth, a theory of truth-makers is neither a theory about propositions nor a theory about truth. If so, then the theory of truth-makers can neither gain inspiration from, nor be tarred by the same brush as the correspondence theory of truth. Nor need the theory of truth-makers be bedeviled by concerns about the nature of truth bearers. One may nevertheless wonder whether Bigelow and Lewis have thrown the baby out with the bathwater. MacBride (2013a) argues that if we do not allow ourselves the general thought that truth-bearers are true or false depending upon how things stand in the world then it is unclear what, if any, credibility (i1), (i2) etc. have. For why should there be something in the world whose existence necessitates that (e.g.) cats purr? Why can't cats purr, even though there is nothing whose existence entails that truth? Since Slogan is intended to be just shorthand for its instances, (i 1), (i 2) etc. it follows Slogan can't be any more credible or motivated than its instances. If, however, we allow ourselves a conception of truth which isn't deflationary, i.e. a substantial conception whereby truth is conceived as a relation borne by a truth-bearer to something worldly that exists independently of us, then, MacBride argues, we will have an independent reason for positing something worldly the existence of which necessitates the truth that cats purr etc. (see also Armstrong 1997: 128). For further discussion of deflationism in connection with truth-making see McGrath 2003, Vision 2005 and Thomas 2011. ### 3.5 Truth-making and Conceptual Explanation Other philosophers appeal to the idea that when we say that a statement is true because a truth-maker for it exists, the "because" we employ is a connective (Kunne 2003: 150-6; Hornsby 2005: 35-7; Melia 2005: 78-9; Schnieder 2006a: 29-37; Mulligan 2007). They think of this connective on the model of logical operators ("&", "[?]" etc.) i.e., expressions that link sentences but without expressing a relation that holds between the states of affairs, facts or tropes that these sentences denote; but unlike "&", "[?]" etc., "because" is not truth-functional. According to these philosophers, the truth-maker panegyrists have misconstrued the logical form of "makes true". They have taken it to be a verb like "x hits y" when really it is akin to the connective "-" or "because" (Melia 2005: 78). There's no need to introduce truth-makers as the special things that stand at one end of the truth-making relation with true statements at the other, no need to because it's only superficial features of the grammar of our language that suggest there is a truth-making relation to stand at the ends of. Typically philosophers who maintain that "makes true" is a connective do not argue for this conclusion directly but encourage us to dwell upon the fact that it natural to hear the "because" that occurs in equivalent constructions as a connective. When we hear, e.g., "It is true that the rose is red because the rose is red" we don't naturally hear it as expressing a binary relation between two things. So we shouldn't hear, or at least don't have to hear, "It's true that the rose is red in virtue of the rose's being red" or "the rose's being red makes it true that the rose is red" as a expressing a binary relation between a truth and truth-maker either. Kunne has gone further and suggested that the truth-making connective is really the "because" of conceptual explanation (2003: 154-5; see also Schnieder 2006a: 36-7). He encourages us to hear the equivalence between, (R) He is a child of a sibling of one of your parents, which makes him your first cousin and, (R\) He is your first cousin because he is a child of a sibling of one of your parents. He tells us that the "because" in (R\*) is the "because" of "conceptual explanation". Since (R) and (R\*) are equivalent, he concludes that the "makes" construction in (R) is just a cumbersome way of expressing the "because" of conceptual explanation in (R\). Kunne then invites us take the use of "makes" in (R) as a model for understanding the truth-making construction, (S) The fact that snow is white makes the statement that snow is white true. This enables us to see that (S) does "not affirm a relation of any kind between a truth vehicle and something in the world" (Kunne 2003: 155). Why? Because it is equivalent to, (S\) The statement that snow is white is true because snow is white. a claim that makes no (explicit) mention of facts. Moreover, the second clause of (S\), like the second clause of (R\), tells us when it is correct to affirm its first clause. In both cases we have an explanation that draws respectively upon our understanding of the concepts cousin* and truth. Drawing upon these materials, we are then able to construct an explanation of the felt asymmetry whereby the truth of what we say about the world depends upon what the world is like, but not the other way around. What is written on the left-hand-side of (S\) is conceptually more sophisticated than what is written upon the right-hand-side because the former requires us to be able to grasp the concepts of statement* and truth to understand it whereas the latter does not. Accordingly the latter requires less of both the world and us than the former to be both understood and true. That's why it makes sense to deploy the right-hand-side (a claim about reality) to explain the left-hand-side (a claim about a truth bearer) but not the other way around (Hornsby 2005: 44; Dodd 2007: 399-400). It may still be that this is just wishful thinking, until it has been established that the "because" of truth making is a connective rather than a relational expression--i.e., not just because it would be technically convenient for us to believe it to be so or because it sounds like a connective to someone with the ears of a naive grammarian. As we saw in section 1.6, some advocates of truth-making as grounding also consider "because" to be a connective, although there is much disagreement about its implications. Some early exponents of the connective view of truth-making have embraced grounding (e.g. Schnieder), while others have not (e.g. Hornsby and Dodd). ### 3.6 Truth Making and Quantification A final supporting suggestion: that the grand truth-maker projects of the late twentieth century arose (partly) out of a failure to appreciate the logical variety of natural language quantifiers that we unreflectively employ when expressing the intuitions that speak in favour of truth- makers. Consider how Armstrong expressed himself when he started out: > > > It seems obvious that for every true contingent proposition there must > be something in the world (in the largest sense of > "something") which makes the proposition true. For > consider any true contingent proposition and imagine that it is false. > We must automatically imagine some difference in the world. (Armstrong > 1973: 11) > > > Bigelow asks us to compare two worlds, one in which A is true, the other in which it is false. According to Bigelow, > > > There must surely be some difference between these two possible > worlds! They are clearly different in that A is true in one > but not the other. So there must be something in one of these worlds > which is lacking in the other and which accounts for the difference in > truth. (1988: 126) > > > Armstrong and Bigelow make the same assumption about the "some" they employ in their expression of what makes the difference between the world in which A is true and the world in which A is false; they assume it is an objectual quantifier in name position. With this assumption in place it is an easy move to make to think that there must be some thing that we quantify over that constitutes the difference between these circumstances, viz. the truth-maker for A. But as Williamson remarks "We should not assume that all quantification is either objectual or substitutional" (1999: 262-3; cf. Prior 1971: 31-4). Williamson argues that that the truth maker principle in fact > > > involves irreducible non-substitutional quantification into sentence > position... We should not even assume that all non-substitutional > quantification is interpreted in terms of assignments of values to the > variables. For "value" is a noun, not a sentence. (1999: > 263) > > > But suppose the "some" Armstrong and Bigelow employ is understood not as an objectual quantifier that comes equipped with a domain of entities over which it ranges but is understood in some other way. Then it would be open to us to acknowledge the force of Armstrong and Bigelow's intuition that the circumstances in which A is true are somehow different from those in which it isn't, but without our having to think that there exists something that we've quantified over than makes it so. Evidently it will not be until we have arrived at a settled view of the admissible interpretations our quantifiers may bear that this issue can be settled in a principled rather than dogmatic manner.
truth-values	## 1. Truth values as objects and referents of sentences ### 1.1 Functional analysis of language and truth values The approach to language analysis developed by Frege rests essentially on the idea of a strict discrimination between two main kinds of expressions: proper names (singular terms) and functional expressions. Proper names designate (signify, denote, or refer to) singular objects, and functional expressions designate (signify, denote, or refer to) functions. [Note: In the literature, the expressions 'designation', 'signification', 'denotation', and 'reference' are usually taken to be synonymous. This practice is used throughout the present entry.] The name 'Ukraine', for example, refers to a certain country, and the expression 'the capital of' denotes a one-place function from countries to cities, in particular, a function that maps Ukraine to Kyiv (Kiev). Whereas names are "saturated" (complete) expressions, functional expressions are "unsaturated" (incomplete) and may be saturated by applying them to names, producing in this way new names. Similarly, the objects to which singular terms refer are saturated and the functions denoted by functional expression are unsaturated. Names to which a functional expression can be applied are called the arguments of this functional expression, and entities to which a function can be applied are called the arguments of this function. The object which serves as the reference for the name generated by an application of a functional expression to its arguments is called the value of the function for these arguments. Particularly, the above mentioned functional expression 'the capital of' remains incomplete until applied to some name. An application of the function denoted by 'the capital of' to Ukraine (as an argument) returns Kyiv as the object denoted by the compound expression 'the capital of Ukraine' which, according to Frege, is a proper name of Kyiv. Note that Frege distinguishes between an $n$-place function $f$ as an unsaturated entity that can be completed by and applied to arguments $a\_1$,..., $a\_n$ and its course of values, which can be seen as the set-theoretic representation of this function: the set \[\{\langle a\_1, \ldots, a\_n, a\rangle \mid a = f(a\_1,\ldots , a\_n)\}.\] Pursuing this kind of analysis, one is very quickly confronted with two intricate problems. First, how should one treat declarative sentences? Should one perhaps separate them into a specific linguistic category distinct from the ones of names and function symbols? And second, how--from a functional point of view--should one deal with predicate expressions such as 'is a city', 'is tall', 'runs', 'is bigger than', 'loves', etc., which are used to denote classes of objects, properties of objects, or relations between them and which can be combined with (applied to) singular terms to obtain sentences? If one considers predicates to be a kind of functional expressions, what sort of names are generated by applying predicates to their arguments, and what can serve as referents of these names, respectively values of these functions? A uniform solution of both problems is obtained by introducing the notion of a truth value. Namely, by applying the criterion of "saturatedness" Frege provides a negative answer to the first of the above problems. Since sentences are a kind of complete entities, they should be treated as a sort of proper names, but names destined to denote some specific objects, namely the truth values: the True and the False. In this way one also obtains a solution of the second problem. Predicates are to be interpreted as some kind of functional expressions, which being applied to these or those names generate sentences referring to one of the two truth values. For example, if the predicate 'is a city' is applied to the name 'Kyiv', one gets the sentence 'Kyiv is a city', which designates the True (i.e., 'Kyiv is a city' is true). On the other hand, by using the name 'Mount Everest', one obtains the sentence 'Mount Everest is a city' which clearly designates the False, since 'Mount Everest is a city' is false. Functions whose values are truth values are called propositional functions. Frege also referred to them as concepts (Begriffe). A typical kind of such functions (besides the ones denoted by predicates) are the functions denoted by propositional connectives. Negation, for example, can be interpreted as a unary function converting the True into the False and vice versa, and conjunction is a binary function that returns the True as a value when both its argument positions are filled in by the True, etc. Propositional functions mapping $n$-tuples of truth values into truth values are also called truth-value functions. Frege thus in a first step extended the familiar notion of a numerical function to functions on singular objects in general and, moreover, introduced a new kind of singular objects that can serve as arguments and values of functions on singular objects, the truth values. In a further step, he considered propositional functions taking functions as their arguments. The quantifier phrase 'every city', for example, can be applied to the predicate 'is a capital' to produce a sentence. The argument of the second-order function denoted by 'every city' is the first-order propositional function on singular objects denoted by 'is a capital'. The functional value denoted by the sentence 'Every city is a capital' is a truth value, the False. Truth values thus prove to be an extremely effective instrument for a logical and semantical analysis of language.[1] Moreover, Frege provides truth values (as proper referents of sentences) not merely with a pragmatical motivation but also with a strong theoretical justification. The idea of such justification, that can be found in Frege 1892, employs the principle of substitutivity of co-referential terms, according to which the reference of a complex singular term must remain unchanged when any of its sub-terms is replaced by an expression having the same reference. This is actually just an instance of the compositionality principle mentioned above. If sentences are treated as a kind of singular terms which must have designations, then assuming the principle of substitutivity one "almost inevitably" (as Kurt Godel (1944: 129) explains) is forced to recognize truth values as the most suitable entities for such designations. Accordingly, Frege asks: > > > What else but the truth value could be found, that belongs quite > generally to every sentence if the reference of its components is > relevant, and remains unchanged by substitutions of the kind in > question? (Geach and Black 1952: 64) > > > The idea underlying this question has been neatly reconstructed by Alonzo Church in his Introduction to Mathematical Logic (1956: 24-25) by considering the following sequence of four sentences: C1. Sir Walter Scott is the author of Waverley. C2. C2. Sir Walter Scott is the man who wrote 29 Waverley Novels altogether. C3. The number, such that Sir Walter Scott is the man who wrote that many Waverley Novels altogether is 29. C4. The number of counties in Utah is 29. C1-C4 present a number of conversion steps each producing co-referential sentences. It is claimed that C1 and C2 must have the same designation by substitutivity, for the terms 'the author of Waverley' and 'the man who wrote 29 Waverley Novels altogether' designate one and the same object, namely Walter Scott. And so must C3 and C4, because the number, such that Sir Walter Scott is the man who wrote that many Waverley Novels altogether is the same as the number of counties in Utah, namely 29. Next, Church argues, it is plausible to suppose that C2, even if not completely synonymous with C3, is at least so close to C3 "so as to ensure its having the same denotation". If this is indeed the case, then C1 and C4 must have the same denotation (designation) as well. But it seems that the only (semantically relevant) thing these sentences have in common is that both are true. Thus, taken that there must be something what the sentences designate, one concludes that it is just their truth value. As Church remarks, a parallel example involving false sentences can be constructed in the same way (by considering, e.g., 'Sir Walter Scott is not the author of Waverley'). This line of reasoning is now widely known as the "slingshot argument", a term coined by Jon Barwise and John Perry (in Barwise and Perry 1981: 395), who stressed thus an extraordinary simplicity of the argument and the minimality of presuppositions involved. Stated generally, the pattern of the argument goes as follows (cf. Perry 1996). One starts with a certain sentence, and then moves, step by step, to a completely different sentence. Every two sentences in any step designate presumably one and the same thing. Hence, the starting and the concluding sentences of the argument must have the same designation as well. But the only semantically significant thing they have in common seems to be their truth value. Thus, what any sentence designates is just its truth value. A formal version of this argument, employing the term-forming, variable-binding class abstraction (or property abstraction) operator l$x$ ("the class of all $x$ such that" or "the property of being such an $x$ that"), was first formulated by Church (1943) in his review of Carnap's Introduction to Semantics. Quine (1953), too, presents a variant of the slingshot using class abstraction, see also (Shramko and Wansing 2009a). Other remarkable variations of the argument are those by Kurt Godel (1944) and Donald Davidson (1967, 1969), which make use of the formal apparatus of a theory of definite descriptions dealing with the description-forming, variable-binding iota-operator (i$x$, "the $x$ such that"). It is worth noticing that the formal versions of the slingshot show how to move--using steps that ultimately preserve reference--from any true (false) sentence to any other such sentence. In view of this result, it is hard to avoid the conclusion that what the sentences refer to are just truth values. The slingshot argument has been analyzed in detail by many authors (see especially the comprehensive study by Stephen Neale (Neale 2001) and references therein) and has caused much controversy notably on the part of fact-theorists, i.e., adherents of facts, situations, propositions, states of affairs, and other fact-like entities conceived as alternative candidates for denotations of declarative sentences. Also see the supplement on the slingshot argument. ### 1.2 Truth as a property versus truth as an object Truth values evidently have something to do with a general concept of truth. Therefore it may seem rather tempting to try to incorporate considerations on truth values into the broader context of traditional truth-theories, such as correspondence, coherence, anti-realistic, or pragmatist conceptions of truth. Yet, it is unlikely that such attempts can give rise to any considerable success. Indeed, the immense fruitfulness of Frege's introduction of truth values into logic to a large extent is just due to its philosophical neutrality with respect to theories of truth. It does not commit one to any specific metaphysical doctrine of truth. In one significant respect, however, the idea of truth values contravenes traditional approaches to truth by bringing to the forefront the problem of its categorial classification. In most of the established conceptions, truth is usually treated as a property. It is customary to talk about a "truth predicate" and its attribution to sentences, propositions, beliefs or the like. Such an understanding corresponds also to a routine linguistic practice, when one operates with the adjective 'true' and asserts, e.g., 'That 5 is a prime number is true'. By contrast with this apparently quite natural attitude, the suggestion to interpret truth as an object may seem very confusing, to say the least. Nevertheless this suggestion is also equipped with a profound and strong motivation demonstrating that it is far from being just an oddity and has to be taken seriously (cf. Burge 1986). First, it should be noted that the view of truth as a property is not as natural as it appears on the face of it. Frege brought into play an argument to the effect that characterizing a sentence as true adds nothing new to its content, for 'It is true that 5 is a prime number' says exactly the same as just '5 is a prime number'. That is, the adjective 'true' is in a sense redundant and thus is not a real predicate expressing a real property such as the predicates 'white' or 'prime' which, on the contrary, cannot simply be eliminated from a sentence without an essential loss for its content. In this case a superficial grammatical analogy is misleading. This idea gave an impetus to the deflationary conception of truth (advocated by Ramsey, Ayer, Quine, Horwich, and others, see the entry on the deflationary theory of truth). However, even admitting the redundancy of truth as a property, Frege emphasizes its importance and indispensable role in some other respect. Namely, truth, accompanying every act of judgment as its ultimate goal, secures an objective value of cognition by arranging for every assertive sentence a transition from the level of sense (the thought expressed by a sentence) to the level of denotation (its truth value). This circumstance specifies the significance of taking truth as a particular object. As Tyler Burge explains: > > > Normally, the point of using sentences, what "matters to > us", is to claim truth for a thought. The object, in the sense > of the point or objective, of sentence use was truth. It is > illuminating therefore to see truth as an object. (Burge 1986: 120) > > > > As it has been observed repeatedly in the literature (cf., e.g., Burge 1986, Ruffino 2003), the stress Frege laid on the notion of a truth value was, to a great extent, pragmatically motivated. Besides an intended gain for his system of "Basic Laws" (Frege 1893/1903) reflected in enhanced technical clarity, simplicity, and unity, Frege also sought to substantiate in this way his view on logic as a theoretical discipline with truth as its main goal and primary subject-matter. Incidentally, Gottfried Gabriel (1986) demonstrated that in the latter respect Frege's ideas can be naturally linked up with a value-theoretical tradition in German philosophy of the second half of the 19th century; see also (Gabriel 2013) on the relation between Frege's value-theoretically inspired conception of truth values and his theory of judgement. More specifically, Wilhelm Windelband, the founder and the principal representative of the Southwest school of Neo-Kantianism, was actually the first who employed the term "truth value" ("Wahrheitswert") in his essay "What is Philosophy?" published in 1882 (see Windelband 1915: 32), i.e., nine years before Frege 1891, even if he was very far from treating a truth value as a value of a function. Windelband defined philosophy as a "critical science about universal values". He considered philosophical statements to be not mere judgements but rather assessments, dealing with some fundamental values, the value of truth being one of the most important among them. This latter value is to be studied by logic as a special philosophical discipline. Thus, from a value-theoretical standpoint, the main task of philosophy, taken generally, is to establish the principles of logical, ethical and aesthetical assessments, and Windelband accordingly highlighted the triad of basic values: "true", "good" and "beautiful". Later this triad was taken up by Frege in 1918 when he defined the subject-matter of logic (see below). Gabriel points out (1984: 374) that this connection between logic and a value theory can be traced back to Hermann Lotze, whose seminars in Gottingen were attended by both Windelband and Frege. The decisive move made by Frege was to bring together a philosophical and a mathematical understanding of values on the basis of a generalization of the notion of a function on numbers. While Frege may have been inspired by Windelband's use of the word 'value' (and even more concretely - 'truth value'), it is clear that he uses the word in its mathematical sense. If predicates are construed as a kind of functional expressions which, being applied to singular terms as arguments, produce sentences, then the values of the corresponding functions must be references of sentences. Taking into account that the range of any function typically consists of objects, it is natural to conclude that references of sentences must be objects as well. And if one now just takes it that sentences refer to truth values (the True and the False), then it turns out that truth values are indeed objects, and it seems quite reasonable to generally explicate truth and falsity as objects and not as properties. As Frege explains: > > > A statement contains no empty place, and therefore we must take its > Bedeutung as an object. But this Bedeutung is a > truth-value. Thus the two truth-values are objects. (Frege 1891, > trans. Beaney 1997: 140) > > > Frege's theory of sentences as names of truth values has been criticized, for example, by Michael Dummett who stated rather dramatically: > > > This was the most disastrous of the effects of the misbegotten > doctrine that sentences are a species of complex singular terms, which > dominated Frege's later period: to rob him of the insight that > sentences play a unique role, and that the role of almost every other > linguistic expression ... consists in its part in forming > sentences. (Dummett 1981: 196) > > > But even Dummett (1991: 242) concedes that "to deny that truth-values are objects ... seems a weak response". ### 1.3 The ontology of truth values If truth values are accepted and taken seriously as a special kind of objects, the obvious question as to the nature of these entities arises. The above characterization of truth values as objects is far too general and requires further specification. One way of such specification is to qualify truth values as abstract objects. Note that Frege himself never used the word 'abstract' when describing truth values. Instead, he has a conception of so called "logical objects", truth values being primary and the most fundamental of them (Frege 1976: 121). Among the other logical objects Frege pays particular attention to are sets and numbers, emphasizing thus their logical nature (in accordance with his logicist view). Church (1956: 25), when considering truth values, explicitly attributes to them the property of being abstract. Since then it is customary to label truth values as abstract objects, thus allocating them into the same category of entities as mathematical objects (numbers, classes, geometrical figures) and propositions. One may pose here an interesting question about the correlation between Fregean logical objects and abstract objects in the modern sense (see the entry on abstract objects). Obviously, the universe of abstract objects is much broader than the universe of logical objects as Frege conceives them. The latter are construed as constituting an ontological foundation for logic, and hence for mathematics (pursuant to Frege's logicist program). Generally, the class of abstracta includes a wide diversity of platonic universals (such as redness, youngness, justice or triangularity) and not only those of them which are logically necessary. Nevertheless, it may safely be said that logical objects can be considered as paradigmatic cases of abstract entities, or abstract objects in their purest form. It should be noted that finding an adequate definition of abstract objects is a matter of considerable controversy. According to a common view, abstract entities lack spatio-temporal properties and relations, as opposed to concrete objects which exist in space and time (Lowe 1995: 515). In this respect truth values obviously are abstract as they clearly have nothing to do with physical spacetime. In a similar fashion truth values fulfill another requirement often imposed upon abstract objects, namely the one of a causal inefficacy (see, e.g., Grossmann 1992: 7). Here again, truth values are very much like numbers and geometrical figures: they have no causal power and make nothing happen. Finally, it is of interest to consider how truth values can be introduced by applying so-called abstraction principles, which are used for supplying abstract objects with criteria of identity. The idea of this method of characterizing abstract objects is also largely due to Frege, who wrote: > > > If the symbol a is to designate an object for us, then we must > have a criterion that decides in all cases whether b is the > same as a, even if it is not always in our power to apply this > criterion. (Frege 1884, trans. Beaney 1997: 109) > > > More precisely, one obtains a new object by abstracting it from some given kind of entities, in virtue of certain criteria of identity for this new (abstract) object. This abstraction is performed in terms of an equivalence relation defined on the given entities (see Wrigley 2006: 161). The celebrated slogan by Quine (1969: 23) "No entity without identity" is intended to express essentially the same understanding of an (abstract) object as an "item falling under a sortal concept which supplies a well-defined criterion of identity for its instances" (Lowe 1997: 619). For truth values such a criterion has been suggested in Anderson and Zalta (2004: 2), stating that for any two sentences $p$ and $q$, the truth value of $p$ is identical with the truth value of $q$ if and only if $p$ is (non-logically) equivalent with $q$ (cf. also Dummett 1959: 141). This idea can be formally explicated following the style of presentation in Lowe (1997: 620): \[ \forall p\forall q[(\textit{Sentence}(p) \mathbin{\&} \textit{Sentence}(q)) \Rightarrow(tv(p)=tv(q) \Leftrightarrow(p\leftrightarrow q))], \] where &, $\Rightarrow, \Leftrightarrow, \forall$ stand correspondingly for 'and', 'if... then', 'if and only if' and 'for all' in the metalanguage, and $\leftrightarrow$ stands for some object language equivalence connective (biconditional). Incidentally, Carnap (1947: 26), when introducing truth-values as extensions of sentences, is guided by essentially the same idea. Namely, he points out a strong analogy between extensions of predicators and truth values of sentences. Carnap considers a wide class of designating expressions ("designators") among which there are predicate expressions ("predicators"), functional expressions ("functors"), and some others. Applying the well-known technique of interpreting sentences as predicators of degree 0, he generalizes the fact that two predicators of degree $n$ (say, $P$ and $Q)$ have the same extension if and only if $\forall x\_1\forall x\_2 \ldots \forall x\_n(Px\_1 x\_2\ldots x\_n \leftrightarrow Qx\_1 x\_2\ldots x\_n)$ holds. Then, analogously, two sentences (say, $p$ and $q)$, being interpreted as predicators of degree 0, must have the same extension if and only if $p\leftrightarrow q$ holds, that is if and only if they are equivalent. And then, Carnap remarks, it seems quite natural to take truth values as extensions for sentences. Note that this criterion employs a functional dependency between an introduced abstract object (in this case a truth value) and some other objects (sentences). More specifically, what is considered is the truth value of a sentence (or proposition, or the like). The criterion of identity for truth values is formulated then through the logical relation of equivalence holding between these other objects--sentences, propositions, or the like (with an explicit quantification over them). It should also be remarked that the properties of the object language biconditional depend on the logical system in which the biconditional is employed. Biconditionals of different logics may have different logical properties, and it surely matters what kind of the equivalence connective is used for defining truth values. This means that the concept of a truth value introduced by means of the identity criterion that involves a biconditional between sentences is also logic-relative. Thus, if '$\leftrightarrow$' stands for material equivalence, one obtains classical truth values, but if the intuitionistic biconditional is employed, one gets truth values of intuitionistic logic, etc. Taking into account the role truth values play in logic, such an outcome seems to be not at all unnatural. Anderson and Zalta (2004: 13), making use of an object theory from Zalta (1983), propose the following definition of 'the truth value of proposition $p$' ('$tv(p)$' [notation adjusted]): \[ tv(p) =\_{df} i x(A!x \wedge \forall F(xF \leftrightarrow \exists q(q\leftrightarrow p \wedge F= [l y\ q]))), \] where $A$! stands for a primitive theoretical predicate 'being abstract', $xF$ is to be read as "$x$ encodes $F$" and [ly q] is a propositional property ("being such a $y$ that $q$"). That is, according to this definition, "the extension of $p$ is the abstract object that encodes all and only the properties of the form [ly q] which are constructed out of propositions $q$ materially equivalent to $p$" (Anderson and Zalta 2004: 14). The notion of a truth value in general is then defined as an object which is the truth value of some proposition: \[TV(x) =\_{df} \exists p(x = tv(p)).\] Using this apparatus, it is possible to explicitly define the Fregean truth values the True $(\top)$ and the False $(\bot)$: \[ \begin{align} \top &=\_{df} i x(A!x \wedge \forall F(xF \leftrightarrow \exists p(p \wedge F= [l y\ p])));\\ \bot &=\_{df} ix (A!x \wedge \forall F(xF \leftrightarrow \exists p(\neg p \wedge F= [l y\ p]))).\\ \end{align} \] Anderson and Zalta prove then that $\top$ and $\bot$ are indeed truth values and, moreover, that there are exactly two such objects. The latter result is expected, if one bears in mind that what the definitions above actually introduce are the classical truth values (as the underlying logic is classical). Indeed, $p\leftrightarrow q$ is classically equivalent to $(p\wedge q)\vee(\neg p\wedge \neg q)$, and $\neg(p\leftrightarrow q)$ is classically equivalent to $(p\wedge \neg q)\vee(\neg p\wedge q)$. That is, the connective of material equivalence divides sentences into two distinct collections. Due to the law of excluded middle these collections are exhaustive, and by virtue of the law of non-contradiction they are exclusive. Thus, we get exactly two equivalence classes of sentences, each being a hypostatized representative of one of two classical truth values. ## 2. Truth values as logical values ### 2.1 Logic as the science of logical values In a late paper Frege (1918) claims that the word 'true' determines the subject-matter of logic in exactly the same way as the word 'beautiful' does for aesthetics and the word 'good' for ethics. Thus, according to such a view, the proper task of logic consists, ultimately, in investigating "the laws of being true" (Sluga 2002: 86). By doing so, logic is interested in truth as such, understood objectively, and not in what is merely taken to be true. Now, if one admits that truth is a specific abstract object (the corresponding truth value), then logic in the first place has to explore the features of this object and its interrelations to other entities of various other kinds. A prominent adherent of this conception was Jan Lukasiewicz. As he paradigmatically put it: > > > All true propositions denote one and the same object, namely truth, > and all false propositions denote one and the same object, namely > falsehood. I consider truth and falsehood to be singular > objects in the same sense as the number 2 or 4 is. ... > Ontologically, truth has its analogue in being, and falsehood, in > non-being. The objects denoted by propositions are called logical > values. Truth is the positive, and falsehood is the negative > logical value. ... Logic is the science of objects of a special > kind, namely a science of logical values. (Lukasiewicz > 1970: 90) > > > This definition may seem rather unconventional, for logic is usually treated as the science of correct reasoning and valid inference. The latter understanding, however, calls for further justification. This becomes evident, as soon as one asks, on what grounds one should qualify this or that pattern of reasoning as correct or incorrect. In answering this question, one has to take into account that any valid inference should be based on logical rules which, according to a commonly accepted view, should at least guarantee that in a valid inference the conclusion(s) is (are) true if all the premises are true. Translating this demand into the Fregean terminology, it would mean that in the course of a correct inference the possession of the truth value The True should be preserved from the premises to the conclusion(s). Thus, granting the realistic treatment of truth values adopted by Frege, the understanding of logic as the science of truth values in fact provides logical rules with an ontological justification placing the roots of logic in a certain kind of ideal entities (see Shramko 2014). These entities constitute a certain uniform domain, which can be viewed as a subdomain of Frege's so-called "third realm" (the realm of the objective content of thoughts, and generally abstract objects of various kinds, see Frege 1918, cf. Popper 1972 and also Burge 1992: 634). Among the subdomains of this third realm one finds, e.g., the collection of mathematical objects (numbers, classes, etc.). The set of truth values may be regarded as forming another such subdomain, namely the one of logical values, and logic as a branch of science rests essentially on this logical domain and on exploring its features and regularities. ### 2.2 Many-valued logics, truth degrees and valuation systems According to Frege, there are exactly two truth values, the True and the False. This opinion appears to be rather restrictive, and one may ask whether it is really indispensable for the concept of a truth value. One should observe that in elaborating this conception, Frege assumed specific requirements of his system of the Begriffsschrift, especially the principle of bivalence taken as a metatheoretical principle, viz. that there exist only two distinct logical values. On the object-language level this principle finds its expression in the famous classical laws of excluded middle and non-contradiction. The further development of modern logic, however, has clearly demonstrated that classical logic is only one particular theory (although maybe a very distinctive one) among the vast variety of logical systems. In fact, the Fregean ontological interpretation of truth values depicts logical principles as a kind of ontological postulations, and as such they may well be modified or even abandoned. For example, by giving up the principle of bivalence, one is naturally led to the idea of postulating many truth values. It was Lukasiewicz, who as early as 1918 proposed to take seriously other logical values different from truth and falsehood (see Lukasiewicz 1918, 1920). Independently of Lukasiewicz, Emil Post in his dissertation from 1920, published as Post 1921, introduced $m$-valued truth tables, where $m$ is any positive integer. Whereas Post's interest in many-valued logic (where "many" means "more than two") was almost exclusively mathematical, Lukasiewicz's motivation was philosophical (see the entry on many-valued logic). He contemplated the semantical value of sentences about the contingent future, as discussed in Aristotle's De interpretatione. Lukasiewicz introduced a third truth value and interpreted it as "possible". By generalizing this idea and also adopting the above understanding of the subject-matter of logic, one naturally arrives at the representation of particular logical systems as a certain kind of valuation systems (see, e.g., Dummett 1981, 2000; Ryan and Sadler 1992). Consider a propositional language $\mathcal{L}$ built upon a set of atomic sentences $\mathcal{P}$ and a set of propositional connectives $\mathcal{C}$ (the set of sentences of $\mathcal{L}$ being the smallest set containing $\mathcal{P}$ and being closed under the connectives from $\mathcal{C})$. Then a valuation system $\mathbf{V}$ for the language $\mathcal{L}$ is a triple $\langle \mathcal{V}, \mathcal{D}, \mathcal{F}\rangle$, where $\mathcal{V}$ is a non-empty set with at least two elements, $\mathcal{D}$ is a subset of $\mathcal{V}$, and $\mathcal{F} = \{f\_{c \_1},\ldots, f\_{c \_m}\}$ is a set of functions such that $f\_{c \_i}$ is an $n$-place function on $\mathcal{V}$ if $c\_i$ is an $n$-place connective. Intuitively, $\mathcal{V}$ is the set of truth values, $\mathcal{D}$ is the set of designated truth values, and $\mathcal{F}$ is the set of truth-value functions interpreting the elements of $\mathcal{C}$. If the set of truth values of a valuation system $\mathbf{V}$ has $n$ elements, $\mathbf{V}$ is said to be $n$-valued. Any valuation system can be equipped with an assignment function which maps the set of atomic sentences into $\mathcal{V}$. Each assignment $a$ relative to a valuation system $\mathbf{V}$ can be extended to all sentences of $\mathcal{L}$ by means of a valuation function $v\_a$ defined in accordance with the following conditions: \[ \begin{align} \forall p &\in \mathcal{P} , &v\_a (p) &= a(p) ; \tag{1}\\ \forall c\_i &\in \mathcal{C} , & v\_a ( c\_i ( A\_1 ,\ldots , A\_n )) &= f\_{c\_i} ( v\_a ( A\_1 ),\ldots , v\_a ( A\_n )) \tag{2} \\ \end{align} \] It is interesting to observe that the elements of $\mathcal{V}$ are sometimes referred to as quasi truth values. Siegfried Gottwald (1989: 2) explains that one reason for using the term 'quasi truth value' is that there is no convincing and uniform interpretation of the truth values that in many-valued logic are assumed in addition to the classical truth values the True and the False, an understanding that, according to Gottwald, associates the additional values with the naive understanding of being true, respectively the naive understanding of degrees of being true (cf. also the remark by Font (2009: 383) that "[o]ne of the main problems in many-valued logic, at least in its initial stages, was the interpretation of the 'intermediate' or 'non-classical' values", et seq.). In later publications, Gottwald has changed his terminology and states that > > > [t]o avoid any confusion with the case of classical logic one prefers > in many-valued logic to speak of truth degrees and to use the > word "truth value" only for classical logic. (Gottwald > 2001: 4) > > > Nevertheless in what follows the term 'truth values' will be used even in the context of many-valued logics, without any commitment to a philosophical conception of truth as a graded notion or a specific understanding of semantical values in addition to the classical truth values. Since the cardinality of $\mathcal{V}$ may be greater than 2, the notion of a valuation system provides a natural foundational framework for the very idea of a many-valued logic. The set $\mathcal{D}$ of designated values is of central importance for the notion of a valuation system. This set can be seen as a generalization of the classical truth value the True in the sense that it determines many central logical notions and thereby generalizes some of the important roles played by Frege's the True (cf. the introductory remarks about uses of truth values). For example, the set of tautologies (logical laws) is directly specified by the given set of designated truth values: a sentence $A$ is a tautology in a valuation system $\mathbf{V}$ iff for every assignment $a$ relative to $\mathbf{V}$, $v\_a(A) \in \mathcal{D}$. Another fundamental logical notion--that of an entailment relation--can also be defined by referring to the set $\mathcal{D}$. For a given valuation system $\mathbf{V}$ a corresponding entailment relation $(\vDash\_V)$ is usually defined by postulating the preservation of designated values from the premises to the conclusion: \[ \tag{3} D\vDash\_V A \textrm{ iff }\forall a[(\forall B \in D: v\_a (B) \in \mathcal{D}) \Rightarrow v \_a (A) \in \mathcal{D}]. \] A pair $\mathcal{M} = \langle \mathbf{V}, v\_a\rangle$, where $\mathbf{V}$ is an $(n$-valued) valuation system and $v\_a$ a valuation in $\mathbf{V}$, may be called an $(n$-valued) model based on $\mathbf{V}$. Every model $\mathcal{M} = \langle \mathbf{V}, v\_a\rangle$ comes with a corresponding entailment relation $\vDash\_{\mathcal{M}}$ by defining $D\vDash\_{\mathcal{M} }A\textrm{ iff }(\forall B \in D: v\_a (B) \in \mathcal{D}) \Rightarrow v\_a(A) \in \mathcal{D}$. Suppose $\mathfrak{L}$ is a syntactically defined logical system $\mathfrak{L}$ with a consequence relation $\vdash\_{ \mathfrak{L} }$, specified as a relation between the power-set of $\mathcal{L}$ and $\mathcal{L}$. Then a valuational system $\mathbf{V}$ is said to be strictly characteristic for $\mathfrak{L}$ just in case $D\vDash\_V A \textrm{ iff } D\vdash\_{ \mathfrak{L} }A$ (see Dummett 1981: 431). Conversely, one says that $\mathfrak{L}$ is characterized by $\mathbf{V}$. Thus, if a valuation system is said to determine a logic, the valuation system by itself is, properly speaking, not a logic, but only serves as a semantic basis for some logical system. Valuation systems are often referred to as (logical) matrices. Note that in Urquhart 1986, the set $\mathcal{D}$ of designated elements of a matrix is required to be non-empty, and in Dunn & Hardegree 2001, $\mathcal{D}$ is required to be a non-empty proper subset of $\mathbf{V}$. With a view on semantically defining a many-valued logic, these restrictions are very natural and have been taken up in Shramko & Wansing 2011 and elsewhere. For the characterization of consequence relations (see the supplementary document Suszko's Thesis), however, the restrictions do not apply. In this way Fregean, i.e., classical, logic can be presented as determined by a particular valuation system based on exactly two elements: $\mathbf{V}\_{cl} = \langle \{T, F\}, \{T\}, \{ f\_{\wedge}, f\_{\vee}, f\_{\rightarrow}, f\_{\sim}\}\rangle$, where $f\_{\wedge}, f\_{\vee}, f\_{\rightarrow},f\_{\sim}$ are given by the classical truth tables for conjunction, disjunction, material implication, and negation. As an example for a valuation system based on more that two elements, consider two well-known valuation systems which determine Kleene's (strong) "logic of indeterminacy" $K\_3$ and Priest's "logic of paradox" $P\_3$. In a propositional language without implication, $K\_3$ is specified by the Kleene matrix $\mathbf{K}\_3 = \langle \{T, I, F\}, \{T\}, \{ f\_c: c \in \{\sim , \wedge , \vee \}\} \rangle$, where the functions $f\_c$ are defined as follows: \[ \begin{array}{c\|c} f\_\sim & \\\hline T & F \\ I & I \\ F & T \\ \end{array}\quad \begin{array}{c\|c\|c\|c} f\_\wedge & T & I & F \\\hline T & T & I & F \\ I & I & I & F \\ F & F & F & F \\ \end{array}\quad \begin{array}{c\|c\|c\|c} f\_\vee & T & I & F \\\hline T & T & T & T \\ I & T & I & I \\ F & T & I & F \\ \end{array} \] The Priest matrix $\mathbf{P}\_3$ differs from $\mathbf{K}\_3$ only in that $\mathcal{D} = \{T, I\}$. Entailment in $\mathbf{K}\_3$ as well as in $\mathbf{P}\_3$ is defined by means of (3). There are natural intuitive interpretations of $I$ in $\mathbf{K}\_3$ and in $\mathbf{P}\_3$ as the underdetermined and the overdetermined value respectively--a truth-value gap and a truth-value glut. Formally these interpretations can be modeled by presenting the values as certain subsets of the set of classical truth values $\{T, F\}$. Then $T$ turns into $\mathbf{T} = \{T\}$ (understood as "true only"), $F$ into $\mathbf{F} = \{F\}$ ("false only"), $I$ is interpreted in $K\_3$ as $\mathbf{N} = \{\} = \varnothing$ ("neither true nor false"), and in $P\_3$ as $\mathbf{B} = \{T, F\}$ ("both true and false"). (Note that also Asenjo (1966) considers the same truth-tables with an interpretation of the third value as "antinomic".) The designatedness of a truth value can be understood in both cases as containment of the classical $T$ as a member. If one combines all these new values into a joint framework, one obtains the four-valued logic $B\_4$ introduced by Dunn and Belnap (Dunn 1976; Belnap 1977a,b). A Gentzen-style formulation can be found in Font (1997: 7)). This logic is determined by the Belnap matrix $\mathbf{B}\_4 = \langle \{\mathbf{N}, \mathbf{T}, \mathbf{F}, \mathbf{B}\}, \{\mathbf{T}, \mathbf{B}\}, \{ f\_c: c \in \{\sim , \wedge , \vee \}\}\rangle$, where the functions $f\_c$ are defined as follows: \[ \begin{array}{c\|c} f\_\sim & \\\hline T & F \\ B & B \\ N & N \\ F & T \\ \end{array}\quad \begin{array}{c\|c\|c\|c\|c} f\_\wedge & T & B & N & F \\\hline T & T & B & N & F \\ B & B & B & F & F \\ N & N & F & N & F \\ F & F & F & F & F\\ \end{array}\quad \begin{array}{c\|c\|c\|c\|c} f\_\vee & T & B & N & F \\\hline T & T & T & T & T\\ B & T & B & T & B \\ N & T & T & N & N \\ F & T & B & N & F \\ \end{array} \] Definition (3) applied to the Belnap matrix determines the entailment relation of $\mathbf{B}\_4$. This entailment relation is formalized as the well-known logic of "first-degree entailment" ($E\_{fde}$) introduced in Anderson & Belnap 1975 (see also Omori and Wansing 2017). The syntactic notion of a single-conclusion consequence relation has been extensively studied by representatives of the Polish school of logic, most notably by Alfred Tarski, who in fact initiated this line of research (see Tarski 1930a,b; cf. also Wojcicki 1988). In view of certain key features of a standard consequence relation it is quite remarkable--as well as important--that any entailment relation $\vDash\_V$ defined as above has the following structural properties (see Ryan and Sadler 1992: 34): \[\begin{align} \tag{4} D\cup \{A\}&\vDash\_V A &\textrm{(Reflexivity)} \\ \tag{5} \textrm{If } D\vDash\_V A \textrm{ then } D\cup G &\vDash\_V A &\textrm{(Monotonicity)}\\ \tag{6} \textrm{If } D\vDash\_V A \textrm{ for every } A \in G &\\ \textrm{ and } G\cup D \vDash\_V B, \textrm{ then } D &\vDash\_VB & \textrm{(Cut)} \end{align}\] Moreover, for every $A \in \mathcal{L}$, every $D \subseteq \mathcal{L}$, and every uniform substitution function $s$ on $\mathcal{L}$ the following Substitution property holds ($s(D)$ stands for $\{ s(B) \mid B \in D\})$: \[ \tag{7} D\vDash\_V A \textrm{ implies } s(D)\vDash\_Vs(A). \] (The function of uniform substitution $s$ is defined as follows. Let $B$ be a formula in $\mathcal{L}$, let $p\_1,\ldots, p\_n$ be all the propositional variables occurring in $B$, and let $s(p\_1) = A\_1,\ldots , s(p\_n) = A\_n$ for some formulas $A\_1 ,\ldots ,A\_n$. Then $s(B)$ is the formula that results from B by substituting simultaneously $A\_1$,..., $A\_n$ for all occurrences of $p\_1,\ldots, p\_n$, respectively.) If $\vDash\_V$ in the conditions (4)-(6) is replaced by $\vdash\_{ \mathfrak{L} }$, then one obtains what is often called a Tarskian consequence relation. If additionally a consequence relation has the substitution property (7), then it is called structural. Thus, any entailment relation defined for a given valuation system $\mathbf{V}$ presents an important example of a consequence relation, in that $\mathbf{V}$ is strictly characteristic for some logical system $\mathfrak{L}$ with a structural Tarskian consequence relation. Generally speaking, the framework of valuation systems not only perfectly suits the conception of logic as the science of truth values, but also turns out to be an effective technical tool for resolving various sophisticated and important problems in modern logic, such as soundness, completeness, independence of axioms, etc. ### 2.3 Truth values, truth degrees, and vague concepts The term 'truth degrees', used by Gottwald and many other authors, suggests that truth comes by degrees, and these degrees may be seen as truth values in an extended sense. The idea of truth as a graded notion has been applied to model vague predicates and to obtain a solution to the Sorites Paradox, the Paradox of the Heap (see the entry on the Sorites Paradox). However, the success of applying many-valued logic to the problem of vagueness is highly controversial. Timothy Williamson (1994: 97), for example, holds that the phenomenon of higher-order vagueness "makes most work on many-valued logic irrelevant to the problem of vagueness". In any case, the vagueness of concepts has been much debated in philosophy (see the entry on vagueness) and it was one of the major motivations for the development of fuzzy logic (see the entry on fuzzy logic). In the 1960s, Lotfi Zadeh (1965) introduced the notion of a fuzzy set. A characteristic function of a set $X$ is a mapping which is defined on a superset $Y$ of $X$ and which indicates membership of an element in $X$. The range of the characteristic function of a classical set $X$ is the two-element set $\{0,1\}$ (which may be seen as the set of classical truth values). The function assigns the value 1 to elements of $X$ and the value 0 to all elements of $Y$ not in $X$. A fuzzy set has a membership function ranging over the real interval [0,1]. A vague predicate such as 'is much earlier than March 20th, 1963', 'is beautiful', or 'is a heap' may then be regarded as denoting a fuzzy set. The membership function $g$ of the fuzzy set denoted by 'is much earlier than March 20th, 1963' thus assigns values (seen as truth degrees) from the interval [0, 1] to moments in time, for example $g$(1p.m., August 1st, 2006) $= 0$, $g$(3a.m., March 19th, 1963) $= 0$, $g$(9:16a.m., April 9th, 1960) $= 0.005$, $g$(2p.m., August 13th, 1943) $= 0.05$, $g$(7:02a.m., December 2nd, 1278) $= 1$. The application of continuum-valued logics to the Sorites Paradox has been suggested by Joseph Goguen (1969). The Sorites Paradox in its so-called conditional form is obtained by repeatedly applying modus ponens in arguments such as: * A collection of 100,000 grains of sand is a heap. * If a collection of 100,000 grains of sand is a heap, then a collection 99,999 grains of sand is a heap. * If a collection of 99,999 grains of sand is a heap, then a collection 99,998 grains of sand is a heap. * ... * If a collection of 2 grains of sand is a heap, then a collection of 1 grain of sand is a heap. * Therefore: A collection of 1 grain of sand is a heap. Whereas it seems that all premises are acceptable, because the first premise is true and one grain does not make a difference to a collection of grains being a heap or not, the conclusion is, of course, unacceptable. If the predicate 'is a heap' denotes a fuzzy set and the conditional is interpreted as implication in Lukasiewicz's continuum-valued logic, then the Sorites Paradox can be avoided. The truth-function $f\_{\rightarrow}$ of Lukasiewicz's implication $\rightarrow$ is defined by stipulating that if $x \le y$, then $f\_{\rightarrow}(x, y) = 1$, and otherwise $f\_{\rightarrow}(x, y) = 1 - (x - y)$. If, say, the truth value of the sentence 'A collection of 500 grains of sand is a heap' is 0.8 and the truth value of 'A collection of 499 grains of sand is a heap' is 0.7, then the truth value of the implication 'If a collection of 500 grains of sand is a heap, then a collection 499 grains of sand is a heap.' is 0.9. Moreover, if the acceptability of a statement is defined as having a value greater than $j$ for $0 \lt j \lt 1$ and all the conditional premises of the Sorites Paradox do not fall below the value $j$, then modus ponens does not preserve acceptability, because the conclusion of the Sorites Argument, being evaluated as 0, is unacceptable. Alasdair Urquhart (1986: 108) stresses > > > the extremely artificial nature of the attaching of precise numerical > values to sentences like ... "Picasso's > Guernica is beautiful". > > > To overcome the problem of assigning precise values to predications of vague concepts, Zadeh (1975) introduced fuzzy truth values as distinct from the numerical truth values in [0, 1], the former being fuzzy subsets of the set [0, 1], understood as true, very true, not very true, etc. The interpretation of continuum-valued logics in terms of fuzzy set theory has for some time be seen as defining the field of mathematical fuzzy logic. Susan Haack (1996) refers to such systems of mathematical fuzzy logic as "base logics" of fuzzy logic and reserves the term 'fuzzy logics' for systems in which the truth values themselves are fuzzy sets. Fuzzy logic in Zadeh's latter sense has been thoroughly criticized from a philosophical point of view by Haack (1996) for its "methodological extravagances" and its linguistic incorrectness. Haack emphasizes that her criticisms of fuzzy logic do not apply to the base logics. Moreover, it should be pointed out that mathematical fuzzy logics are nowadays studied not in the first place as continuum-valued logics, but as many-valued logics related to residuated lattices (see Hajek 1998; Cignoli et al. 2000; Gottwald 2001; Galatos et al. 2007), whereas fuzzy logic in the broad sense is to a large extent concerned with certain engineering methods. A fundamental concern about the semantical treatment of vague predicates is whether an adequate semantics should be truth-functional, that is, whether the truth value of a complex formula should depend functionally on the truth values of its subformulas. Whereas mathematical fuzzy logic is truth-functional, Williamson (1994: 97) holds that "the nature of vagueness is not captured by any approach that generalizes truth-functionality". According to Williamson, the degree of truth of a conjunction, a disjunction, or a conditional just fails to be a function of the degrees of truth of vague component sentences. The sentences 'John is awake' and 'John is asleep', for example, may have the same degree of truth. By truth-functionality the sentences 'If John is awake, then John is awake' and 'If John is awake, then John is asleep' are alike in truth degree, indicating for Williamson the failure of degree-functionality. One way of in a certain sense non-truthfunctionally reasoning about vagueness is supervaluationism. The method of supervaluations has been developed by Henryk Mehlberg (1958) and Bas van Fraassen (1966) and has later been applied to vagueness by Kit Fine (1975), Rosanna Keefe (2000) and others. Van Fraassen's aim was to develop a semantics for sentences containing non-denoting singular terms. Even if one grants atomic sentences containing non-denoting singular terms and that some attributions of vague predicates are neither true nor false, it nevertheless seems natural not to preclude that compound sentences of a certain shape containing non-denoting terms or vague predications are either true or false, e.g., sentences of the form 'If $A$, then $A$'. Supervaluational semantics provides a solution to this problem. A three-valued assignment $a$ into $\{T, I, F\}$ may assign a truth-value gap (or rather the value $I)$ to the vague sentence 'Picasso's Guernica is beautiful'. Any classical assignment $a'$ that agrees with $a$ whenever $a$ assigns $T$ or $F$ may be seen as a precisification (or superassignment) of $a$. A sentence may than be said to be supertrue under assignment $a$ if it is true under every precisification $a'$ of $a$. Thus, if $a$ is a three-valued assignment into $\{T, I, F\}$ and $a'$ is a two-valued assignment into $\{T, F\}$ such that $a(p) = a'(p)$ if $a(p) \in \{T, F\}$, then $a'$ is said to be a superassignment of $a$. It turns out that if $a$ is an assignment extended to a valuation function $v\_a$ for the Kleene matrix $\mathbf{K}\_3$, then for every formula $A$ in the language of $\mathbf{K}\_3$, $v\_a (A) = v\_{a'}(A)$ if $v\_a (A) \in \{T, F\}$. Therefore, the function $v\_{a'}$ may be called a supervaluation of $v\_a$. A formula is then said to be supertrue under a valuation function $v\_a$ for $\mathbf{K}\_3$ if it is true under every supervaluation $v\_{a'}$ of $v\_a$, i.e., if $v\_{a'}(A) = T$ for every supervaluation $v\_{a'}$ of $v\_a$. The property of being superfalse is defined analogously. Since every supervaluation is a classical valuation, every classical tautology is supertrue under every valuation function in $\mathbf{K}\_3$. Supervaluationism is, however, not truth-functional with respect to supervalues. The supervalue of a disjunction, for example, does not depend on the supervalue of the disjuncts. Suppose $a(p) = I$. Then $a(\neg p) = I$ and $v\_{a'} (p\vee \neg p) = T$ for every supervaluation $v\_{a'}$ of $v\_a$. Whereas $(p\vee \neg p)$ is thus supertrue under $v\_a,p\vee p$ is not, because there are superassignments $a'$ of $a$ with $a'(p) = F$. An argument against the charge that supervaluationism requires a non-truth-functional semantics of the connectives can be found in MacFarlane (2008) (cf. also other references given there). Although the possession of supertruth is preserved from the premises to the conclusion(s) of valid inferences in supervaluationism, and although it might be tempting to consider supertruth an abstract object on its own, it seems that it has never been suggested to hypostatize supertruth in this way, comparable to Frege's the True. A sentence supertrue under a three-valued valuation $v$ just takes the Fregean value the True under every supervaluation of $v$. The advice not to confuse supertruth with "real truth" can be found in Belnap (2009). ### 2.4 Suszko's thesis and anti-designated values One might, perhaps, think that the mere existence of many-valued logics shows that there exist infinitely, in fact, uncountably many truth values. However, this is not at all clear (recall the more cautious use of terminology advocated by Gottwald). In the 1970's Roman Suszko (1977: 377) declared many-valued logic to be "a magnificent conceptual deceit". Suszko actually claimed that "there are but two logical values, true and false" (Caleiro et al. 2005: 169), a statement now called Suszko's Thesis. For Suszko, the set of truth values assumed in a logical matrix for a many-valued logic is a set of "admissible referents" (called "algebraic values") of formulas but not a set of logical values. Whereas the algebraic values are elements of an algebraic structure and referents of formulas, the logical value true is used to define valid consequence: If every premise is true, then so is (at least one of) the conclusion(s). The other logical value, false, is preserved in the opposite direction: If the (every) conclusion is false, then so is at least one of the premises. The logical values are thus represented by a bi-partition of the set of algebraic values into a set of designated values (truth) and its complement (falsity). Essentially the same idea has been taken up earlier by Dummett (1959) in an influential paper, where he asks > > > what point there may be in distinguishing between different ways in > which a statement may be true or between different ways in which it > may be false, or, as we might say, between degrees of truth and > falsity. (Dummett 1959: 153) > > > Dummett observes that, first, > > > the sense of a sentence is determined wholly by knowing the case in > which it has a designated value and the cases in which it has an > undesignated one, > > > and moreover, > > > finer distinctions between different designated values or different > undesignated ones, however naturally they come to us, are justified > only if they are needed in order to give a truth-functional account of > the formation of complex statements by means of operators. (Dummett > 1959: 155) > > > Suszko's claim evidently echoes this observation by Dummett. Suszko's Thesis is substantiated by a rigorous proof (the Suszko Reduction) showing that every structural Tarskian consequence relation and therefore also every structural Tarskian many-valued propositional logic is characterized by a bivalent semantics. (Note also that Richard Routley (1975) has shown that every logic based on a l-categorical language has a sound and complete bivalent possible worlds semantics.) The dichotomy between designated values and values which are not designated and its use in the definition of entailment plays a crucial role in the Suszko Reduction. Nevertheless, while it seems quite natural to construe the set of designated values as a generalization of the classical truth value $T$ in some of its significant roles, it would not always be adequate to interpret the set of non-designated values as a generalization of the classical truth value $F$. The point is that in a many-valued logic, unlike in classical logic, "not true" does not always mean "false" (cf., e.g., the above interpretation of Kleene's logic, where sentences can be neither true nor false). In the literature on many-valued logic it is sometimes proposed to consider a set of antidesignated values which not obligatorily constitute the complement of the set of designated values (see, e.g., Rescher 1969, Gottwald 2001). The set of antidesignated values can be regarded as representing a generalized concept of falsity. This distinction leaves room for values that are neither designated nor antidesignated and even for values that are both designated and antidesignated. Grzegorz Malinowski (1990, 1994) takes advantage of this proposal to give a counterexample to Suszko's Thesis. He defines the notion of a single-conclusion quasi-consequence $(q$-consequence) relation. The semantic counterpart of $q$-consequence is called $q$-entailment. Single-conclusion $q$-entailment is defined by requiring that if no premise is antidesignated, the conclusion is designated. Malinowski (1990) proved that for every structural $q$-consequence relation, there exists a characterizing class of $q$-matrices, matrices which in addition to a subset $\mathcal{D}^{+}$ of designated values comprise a disjoint subset $\mathcal{D}^-$ of antidesignated values. Not every $q$-consequence relation has a bivalent semantics. In the supplementary document Suszko's Thesis, Suszko's reduction is introduced, Malinowski's counterexample to Suszko's Thesis is outlined, and a short analysis of these results is presented. Can one provide evidence for a multiplicity of logical values? More concretely, $is$ there more than one logical value, each of which may be taken to determine its own (independent) entailment relation? A positive answer to this question emerges from considerations on truth values as structured entities which, by virtue of their internal structure, give rise to natural partial orderings on the set of values. ## 3. Ordering relations between truth-values ### 3.1 The notion of a logical order As soon as one admits that truth values come with valuation systems, it is quite natural to assume that the elements of such a system are somehow interrelated. And indeed, already the valuation system for classical logic constitutes a well-known algebraic structure, namely the two-element Boolean algebra with $\cap$ and $\cup$ as meet and join operators (see the entry on the mathematics of Boolean algebra). In its turn, this Boolean algebra forms a lattice with a partial order defined by $a\le\_t b \textrm{ iff } a\cap b = a$. This lattice may be referred to as TWO. It is easy to see that the elements of TWO are ordered as follows: $F\le\_t T$. This ordering is sometimes called the truth order (as indicated by the corresponding subscript), for intuitively it expresses an increase in truth: $F$ is "less true" than $T$. It can be schematically presented by means of a so-called Hasse-diagram as in Figure 1. ![[a horizontal line segment with the left endpoint labeled 'F' and the right endpoint labeled 'T', below an arrow goes from left to right with the arrowhead labeled 't'.]](Figure1.png) Figure 1: Lattice TWO It is also well-known that the truth values of both Kleene's and Priest's logic can be ordered to form a lattice (THREE), which is diagrammed in Figure 2. ![[The same as figure 1 except the line segment has a point near the middle labeled 'I'.]](Figure2.png) Figure 2: Lattice THREE Here $\le\_t$ orders $T, I$ and $F$ so that the intermediate value $I$ is "more true" than $F$, but "less true" than $T$. The relation $\le\_t$ is also called a logical order, because it can be used to determine key logical notions: logical connectives and an entailment relation. Namely, if the elements of the given valuation system $\mathbf{V}$ form a lattice, then the operations of meet and join with respect to $\le\_t$ are usually seen as the functions for conjunction and disjunction, whereas negation can be represented by the inversion of this order. Moreover, one can consider an entailment relation for $\mathbf{V}$ as expressing agreement with the truth order, that is, the conclusion should be at least as true as the premises taken together: \[ \tag{8} D\vDash B\textrm{ iff }\forall v\_a[\Pi\_t\{ v\_a (A) \mid A \in D\} \le\_t v\_a (B)], \] where $\Pi\_t$ is the lattice meet in the corresponding lattice. The Belnap matrix $\mathbf{B}\_4$ considered above also can be represented as a partially ordered valuation system. The set of truth values $\{\mathbf{N}, \mathbf{T}, \mathbf{F}, \mathbf{B}\}$ from $\mathbf{B}\_4$ constitutes a specific algebraic structure - the bilattice FOUR$\_2$ presented in Figure 3 (see, e.g., Ginsberg 1988, Arieli and Avron 1996, Fitting 2006). ![[a graph with the y axis labeled 'i' and the x axis labeled 't'. A square with the corners labeled 'B' (top), 'T' (right), 'N' (bottom), and 'F' (left).]](Figure3.png) Figure 3: The bilattice FOUR$\_2$ This bilattice is equipped with two partial orderings; in addition to a truth order, there is an information order $(\le\_i )$ which is said to order the values under consideration according to the information they give concerning a formula to which they are assigned. Lattice meet and join with respect to $\le\_t$ coincide with the functions $f\_{\wedge}$ and $f\_{\vee}$ in the Belnap matrix $\mathbf{B}\_4$, $f\_{{\sim}}$ turns out to be the truth order inversion, and an entailment relation, which happens to coincide with the matrix entailment, is defined by (8). FOUR$\_2$ arises as a combination of two structures: the approximation lattice $A\_4$ and the logical lattice $L\_4$ which are discussed in Belnap 1977a and 1977b (see also, Anderson, Belnap and Dunn 1992: 510-518)). ### 3.2 Truth values as structured entities. Generalized truth values Frege (1892: 30) points out the possibility of "distinctions of parts within truth values". Although he immediately specifies that the word 'part' is used here "in a special sense", the basic idea seems nevertheless to be that truth values are not something amorphous, but possess some inner structure. It is not quite clear how serious Frege is about this view, but it seems to suggest that truth values may well be interpreted as complex, structured entities that can be divided into parts. There exist several approaches to semantic constructions where truth values are represented as being made up from some primitive components. For example, in some explications of Kripke models for intuitionistic logic propositions (identified with sets of "worlds" in a model structure) can be understood as truth values of a certain kind. Then the empty proposition is interpreted as the value false, and the maximal proposition (the set of all worlds in a structure) as the value true. Moreover, one can consider non-empty subsets of the maximal proposition as intermediate truth values. Clearly, the intuitionistic truth values so conceived are composed from some simpler elements and as such they turn out to be complex entities. Another prominent example of structured truth values are the "truth-value objects" in topos models from category theory (see the entry on category theory). For any topos $C$ and for a $C$-object O one can define a truth value of $C$ as an arrow $1 \rightarrow O$ ("a subobject classifier for $C$"), where 1 is a terminal object in $C$ (cf. Goldblatt 2006: 81, 94). The set of truth values so defined plays a special role in the logical structure of $C$, since arrows of the form $1 \rightarrow O$ determine central semantical notions for the given topos. And again, these truth values evidently have some inner structure. One can also mention in this respect the so-called "factor semantics" for many-valued logic, where truth values are defined as ordered $n$-tuples of classical truth values $(T$-$F$ sequences, see Karpenko 1983). Then the value $3/5$, for example, can be interpreted as a $T$-$F$ sequence of length 5 with exactly 3 occurrences of $T$. Here the classical values $T$ and $F$ are used as "building blocks" for non-classical truth values. Moreover, the idea of truth values as compound entities nicely conforms with the modeling of truth values considered above in three-valued (Kleene, Priest) and four-valued (Belnap) logics as certain subsets of the set of classical truth values. The latter approach stems essentially from Dunn (1976), where a generalization of the notion of a classical truth-value function has been proposed to obtain so-called "underdetermined" and "overdetermined" valuations. Namely, Dunn considers a valuation to be a function not from sentences to elements of the set $\{T, F\}$ but from sentences to subsets of this set (see also Dunn 2000: 7). By developing this idea, one arrives at the concept of a generalized truth value function, which is a function from sentences into the subsets of some basic set of truth values (see Shramko and Wansing 2005). The values of generalized truth value functions can be called generalized truth values. By employing the idea of generalized truth value functions, one can obtain a hierarchy of valuation systems starting with a certain set-theoretic representation of the valuation system for classical logic. The representation in question is built on a single initial value which serves then as the designated value of the resulting valuation system. More specifically, consider the singleton $\{\varnothing \}$ taken as the basic set subject to a further generalization procedure. Set-theoretically the basic set can serve as the universal set (the complement of the empty set) for the valuation system $\mathbf{V}^{\varnothing}\_{cl}$ introduced below. At the first stage $\varnothing$ comes out with no specific intuitive interpretation, it is only important to take it as some distinct unit. Consider then the power-set of $\{\varnothing \}$ consisting of exactly two elements: $\{\{\varnothing \}, \varnothing \}$. Now, these elements can be interpreted as Frege's the True and the False, and thus it is possible to construct a valuation system for classical logic, $\mathbf{V}^{\varnothing}\_{cl} = \langle \{\{\varnothing \}, \varnothing \}, \{\{\varnothing \}\}, \{f\_{\wedge}, f\_{\vee}, f\_{\rightarrow}, f\_{\sim}\}\rangle$, where the functions $f\_{\wedge}, f\_{\vee}, f\_{\rightarrow}, f\_{\sim}$ are defined as follows (for \[ \begin{align} X, Y \in \{\{\varnothing \}, \varnothing \}:\quad & f\_{\wedge}(X, Y) = X\cap Y; \\ & f\_{\vee}(X, Y) = X\cup Y; \\ & f\_{\rightarrow}(X, Y) = X^{c}\cup Y; \\ & f\_{\sim}(X) = X^{c}. \end{align} \] It is not difficult to see that for any assignment $a$ relative to $\mathbf{V}^{\varnothing}\_{cl}$, and for any formulas $A$ and $B$, the following holds: \[\begin{align} v\_a (A\wedge B) = \{\varnothing \}&\Leftrightarrow v\_a (A) = \{\varnothing \} \text{ and } v\_a (B) = \{\varnothing \}; \\ v\_a (A\vee B) = \{\varnothing \}&\Leftrightarrow v\_a (A) = \{\varnothing \} \text{ or } v\_a (B) = \{\varnothing \}; \\ v\_a (A\rightarrow B) = \{\varnothing \}&\Leftrightarrow v\_a (A) = \varnothing \text{ or } v\_a (B) = \{\varnothing \}; \\ v\_a (\sim A) = \{\varnothing \}&\Leftrightarrow v\_a (A) = \varnothing. \end{align}\] This shows that $f\_{\wedge}, f\_{\vee}, f\_{\rightarrow}$ and $f\_{\sim}$ determine exactly the propositional connectives of classical logic. One can conveniently mark the elements $\{\varnothing \}$ and $\varnothing$ in the valuation system $\mathbf{V}^{\varnothing}\_{cl}$ by the classical labels $T$ and $F$. Note that within $\mathbf{V}^{\varnothing}\_{cl}$ it is fully justifiable to associate $\varnothing$ with falsity, taking into account the virtual monism of truth characteristic for classical logic, which treats falsity not as an independent entity but merely as the absence of truth. Then, by taking the set $\mathbf{2} = \{F, T\}$ of these classical values as the basic set for the next valuation system, one obtains the four truth values of Belnap's logic as the power-set of the set of classical values $\mathcal{P}(\mathbf{2}) = \mathbf{4}: \mathbf{N} = \varnothing$, $\mathbf{F} = \{F\} (= \{\varnothing \})$, $\mathbf{T} = \{T\} (= \{\{\varnothing \}\})$ and $\mathbf{B} = \{F, T\} (= \{\varnothing, \{\varnothing \}\})$. In this way, Belnap's four-valued logic emerges as a certain generalization of classical logic with its two Fregean truth values. In Belnap's logic truth and falsity are considered to be full-fledged, self-sufficient entities, and therefore $\varnothing$ is now to be interpreted not as falsity, but as a real truth-value gap (neither true nor false). The dissimilarity of Belnap's truth and falsity from their classical analogues is naturally expressed by passing from the corresponding classical values to their singleton-sets, indicating thus their new interpretations as false only and true only. Belnap's interpretation of the four truth values has been critically discussed in Lewis 1982 and Dubois 2008 (see also the reply to Dubois in Wansing and Belnap 2010). Generalized truth values have a strong intuitive background, especially as a tool for the rational explication of incomplete and inconsistent information states. In particular, Belnap's heuristic interpretation of truth values as information that "has been told to a computer" (see Belnap 1977a,b; also reproduced in Anderson, Belnap and Dunn 1992, SS81) has been widely acknowledged. As Belnap points out, a computer may receive data from various (maybe independent) sources. Belnap's computers have to take into account various kinds of information concerning a given sentence. Besides the standard (classical) cases, when a computer obtains information either that the sentence is (1) true or that it is (2) false, two other (non-standard) situations are possible: (3) nothing is told about the sentence or (4) the sources supply inconsistent information, information that the sentence is true and information that it is false. And the four truth values from $\mathbf{B}\_4$ naturally correspond to these four situations: there is no information that the sentence is false and no information that it is true $(\mathbf{N})$, there is merely information that the sentence is false $(\mathbf{F})$, there is merely information that the sentence is true $(\mathbf{T})$, and there is information that the sentence is false, but there is also information that it is true $(\mathbf{B})$. Joseph Camp in 2002: 125-160 provides Belnap's four values with quite a different intuitive motivation by developing what he calls a "semantics of confused thought". Consider a rational agent, who happens to mix up two very similar objects (say, $a$ and $b)$ and ambiguously uses one name (say, '$C$') for both of them. Now let such an agent assert some statement, saying, for instance, that $C$ has some property. How should one evaluate this statement if $a$ has the property in question whereas $b$ lacks it? Camp argues against ascribing truth values to such statements and puts forward an "epistemic semantics" in terms of "profitability" and "costliness" as suitable characterizations of sentences. A sentence $S$ is said to be "profitable" if one would profit from acting on the belief that $S$, and it is said to be "costly" if acting on the belief that $S$ would generate costs, for example as measured by failure to achieve an intended goal. If our "confused agent" asks some external observers whether $C$ has the discussed property, the following four answers are possible: 'yes' (mark the corresponding sentence with $\mathbf{Y})$, 'no' (mark it with $\mathbf{N})$, 'cannot say' (mark it with ?), 'yes' and 'no' (mark it with Y&N). Note that the external observers, who provide answers, are "non-confused" and have different objects in mind as to the referent of '$C$', in view of all the facts that may be relevant here. Camp conceives these four possible answers concerning epistemic properties of sentences as a kind of "semantic values", interpreting them as follows: the value $\mathbf{Y}$ is an indicator of profitability, the value $\mathbf{N}$ is an indicator of costliness, the value ? is no indicator either way, and the value Y&N is both an indicator of profitability and an indicator of costliness. A strict analogy between this "semantics of confused reasoning" and Belnap's four valued logic is straightforward. And indeed, as Camp (2002: 157) observes, the set of implications valid according to his semantics is exactly the set of implications of the entailment system $E\_{fde}$. In Zaitsev and Shramko 2013 it is demonstrated how ontological and epistemic aspects of truth values can be combined within a joint semantical framework. Kapsner (2019) extends Belnap's framework by two additional values "Contestedly-True" and "Contestedly-False" which allows for new outcomes for disjunctions and conjunctions between statements with values $\mathbf{B}$ and $\mathbf{N}$. The conception of generalized truth values has its purely logical import as well. If one continues the construction and applies the idea of generalized truth value functions to Belnap's four truth values, then one obtains further valuation systems which can be represented by various multilattices. One arrives, in particular, at SIXTEEN$\_3$ - the trilattice of 16 truth-values, which can be viewed as a basis for a logic of computer networks (see Shramko and Wansing 2005, 2006; Kamide and Wansing 2009; Odintsov 2009; Wansing 2010; Odintsov and Wansing 2015; cf. also Shramko, Dunn, Takenaka 2001). The notion of a multilattice and SIXTEEN$\_3$ are discussed further in the supplementary document Generalized truth values and multilattices. A comprehensive study of the conception of generalized logical values can be found in Shramko and Wansing 2011. ## 4. Concluding remarks Gottlob Frege's notion of a truth value has become part of the standard philosophical and logical terminology. The notion of a truth value is an indispensable instrument of realistic, model-theoretic approaches to semantics. Indeed, truth values play an essential role in applications of model-theoretic semantics in areas such as, for example, knowledge representation and theorem proving based on semantic tableaux, which could not be treated in the present entry. Moreover, considerations on truth values give rise to deep ontological questions concerning their own nature, the feasibility of fact ontologies, and the role of truth values in such ontological theories. There also exist well-motivated theories of generalized truth values that lead far beyond Frege's classical values the True and the False. (For various directions of further logical and philosophical investigations in the area of truth values see Shramko & Wansing 2009b, 2009c.)
tsongkhapa	## 1. Tsongkhapa's Life ### 1.1 Overview Biographical information about Tsongkhapa comes from clues in his own writing, and primarily from the hagiography written by his student Kedrup Pelzangpo (mKhas grub dpal bzang po) (1385-1438) called Stream of Faith (Dad pa'i 'jug ngog) (partial translation in Thurman 1982).[1] Tsongkhapa's life falls roughly into an earlier and later period. The later period is defined by a series of publications, beginning in about 1400, which systematically present his mature philosophy. Tsongkhapa, influenced by the divisions used by the editors of the Kanjur, treats non-tantric and tantric sources separately. His philosophical views on tantra, and, to a certain extent his work on ethics fall naturally into separate categories. The later period of his life includes a period of institution building, possibly with an eye to the founding of a new school or sect. ### 1.2 Detailed account The name Tsongkhapa derives from Tsong kha, an ancient name for a part of the A mdo region of Greater Tibet (Bod chen) now included in Qinghai Province of the People's Republic of China, and the Tibetan suffix pa that works as an agentive nominalizing particle. His given name is Losang (sometimes written Lozang) Drakpa (bLo bzang grags pa). He is also known by the honorific title Je Rinpoche (rJe rin po che) ("Precious Lord"). He was born, probably to semi-nomadic farmers, in a settlement now incorporated into the outskirts of the Chinese city of Xi ning. His birthplace is marked by the popular Kumbum (sKu 'bum) monastery, In his teens (1372-73), Tsongkhapa travelled from Tsong kha to Central (dBus) Tibet (Bod) where he remained until his death in 1419. He arrived as a young man in Central Tibet at the end of a long flowering of intellectual activity called "the later diffusion [of Buddhism]" (Tib. spyi dar) that began with translator Rin chen bzang po (958-1055) and the scholar-saint Dipamkara-srijnana Atisa (980-1054), and ended with the most important editor of the Kanjur, Buton (Bu ston Rin chen grub), (1290-1364). According to Kedrup, on his arrival in Central Tibet Tsongkhapa first studied Tibetan medicine, then a traditional Buddhist curriculum of abhidharma, tenet systems (Tib. grub mtha') focusing on the Middle Way and Mind Only (Sk. cittamatra also called yogacara) philosophy, Buddhist ethics (Tib. sdom gsum), and epistemology (Sk. pramana). His early studies were pursued mainly in institutions affiliated with the two dominant scholarly traditions (Tib. lugs) of the time: the Sangphu (gSang phu ne'u tog) tradition founded by Ngok (rNgog bLo ldan shes rab) (1059-1109), and the Sakya (Sa skya) epistemological tradition based primarily on the works of the Sa skya pandi ta (1182-1251). He also studied and practiced tantric Buddhism. In his early years he wrote a number of essays on topics in abhidharma (Apple 2008), a detailed investigation of the alaya-vijnana (Sk.) ("foundation, storehouse, basis-of all consciousness") (Sparham 1993), and an important treatise, Golden Garland (Legs bshad gser phreng), on the Perfection of Wisdom (Sk. prajna-paramita) literature based on the codification of its topics in the Ornament for the Clear Realizations (Abhisamayalamkara) (Sparham 2007-2013). After completing his Golden Garland in 1388-89, Tsongkhapa spent a period of some ten years removed from the hub of intellectual activity. He engaged in meditation and ritual religious exercises in Southern Tibet (Lho kha) where, as evidenced in his short autobiographical writing from this transitional period (Thurman 1982), and corroborated by traditional sources (Kaschewsky 1971, Vostrikov 1970), he resorted to a communion or dialogue with Manjusri--the iconographic representation of perfect knowledge in the Buddhist pantheon. Through the voice of Manjusri he articulated a hierarchical system of philosophies culminating in what he characterized as a pristine, Middle Way, \Prasangika-madhyamaka (Tib. dBu ma thal 'gyur pa) that avoided over-reification (of the absolute) and over-negation (of the conventional). Tsongkhapa framed his insights not as original contributions, but as a rediscovery of meanings already revealed by the Buddha. In all his works he characterizes his philosophy as identical to the Buddha's. Further, he says his philosophy is based on Nagarjuna's and Nagarjuna's follower Aryadeva's (third-fourth century) explanation of what the Buddha said. His philosophy gets its name \Prasangika ("those who reveal the unwelcome consequences [inherent in other's assertions]") from the importance it gives to Buddhapalita's (late fifth century) explication of Nagarjuna's work, and to Candrakirti's defense of Buddhapalita's explanation, in the face of criticism by Bhaviveka. Tsongkhapa presented his mature philosophy in a series of volumes during the later period of his life, beginning with the publication of his most famous work, the Great Exposition of the Stages of the Path (Lam rim chen mo) in 1402, at the age of 46. This was followed by Essence of Eloquence (Legs bshad snying po) and Ocean of Reasoning (Rigs pa'i rgya mtsho) (1407-1408), the Medium-Length Exposition of the Stages of the Path (Lam rim 'bring) in 1415, and, in the last years of his life, his Elucidation of the Intention (dGongs pa rab gsal). Together, in this series of works written over a seventeen year period, he succeeded in setting the agenda for the following centuries, to a great extent, with later writers taking positions pro and con his arguments. As is so often the case, his stature pushed much of earlier Tibetan intellectual history into the shadows, until its rediscovery by the Eclectic Movement (Tib. Ris med) in Derge (sDe dge) in the late 18th century. Tsongkhapa wrote his major works on tantra and ethics at the same time as his major philosophical works. Volumes 3-12 of his collected works in 18 or 19 volumes deal exclusively with topics based on tantric sources. In 1405 he completed his Great Exposition of Tantra (sNgags rim chen mo) (Hopkins 1980, Yarnall 2013) a companion volume to his Great Exposition, where, in harmony with his philosophy, he argued that tantra is defined neither by special insight (Sk. vipasyana) or Mahayana altruism (Sk. bodhicitta), but solely by deity yoga (Tib. lha'i rnal 'byor) (Hopkins 1980). In 1402, Tsongkhapa, with his teacher Rendawa (Red mda' ba gzhon nu blo gros), (1349-1412) (Roloff 2009) and others, at the temple of Namstedeng (rNam rtsed ldeng/lding) convened a gathering of monks with the intention of reinvigorating the Buddhist order. From this came Tsongkhapa's short, but influential works on basic Buddhist morality (Sk. pratimoksa) that in turn laid the foundation for the importance some Gelukpa monasteries would place on a stricter adherence to the monastic code. In 1403 his influential works on ethical codes for bodhisattvas (Byang chub gzhung lam) (Tatz 1987) and for tantric practitioners (rTsa ltung rnam bshad) (Sparham 2005) appeared. His separate explanations of the three ethical codes (Tib. sdom gsum) are distinguished by the centrality they give to basic morality, the importance the bodhisattva's code retains in tantra, and in the antinomian tantras the presence of a separate pratimoksa-like ordination ritual built around a codification of of ethical conduct specific to tantric practitioners. By 1409, at the age of 52, his place in Tibetan society was sufficiently established that he could garner the support and sponsorship necessary for a successful rejuvenation of the central temple in Lhasa, for establishing a new-year prayer festival (smon lam chen mo), and for completion of a large new monastery, Ganden (dGa' ldan), where he lived for much of his subsequent life until his death there in 1419. He inspired two of his students Tashi Palden (bKra shis dpal ldan) (1379-1449) and Shakya Yeshey (Ye shes) (1354-1435) to found Drepung ('Bras spungs) monastery in 1416, and Sera Monastery in 1419, respectively. These, with Ganden, would later become the three largest and most powerful Gelukpa monasteries, and indeed, the biggest monasteries in the world. In 1407 and 1413 the Ming dynasty Yongle Emperor recognized Tsongkhapa's growing fame and importance by inviting him to the Chinese court, as was the custom. ## 2. Early Period ### 2.1 Explanation of the Difficult Points (Kun gzhi dka' 'grel) Tsongkhapa's most important works from his early period are his Explanation of the Difficult Points (Kun gzhi dka' 'grel) and Golden Garland (Legs bshad gser phreng). In the former he gives a detailed explanation of the alaya-vijnana ("foundation consciousness")--according to Tsongkhapa the distinctive eighth consciousness in the eight-consciousness system of Asanga's (fourth century) Indian Yogacara Buddhism. Tsongkhapa opens his work with a brief discussion of the relationship between the views of Nagarjuna and Asanga. Tsongkhapa then discovers in the different usages of the term alaya-vijnana and its near synonyms a negative connotation conveying the cause and effect nature of repeated suffering existences (Sk. samsara). He surveys the relevant literature, limiting his discussion of gotra ("lineage," in the sense of the innate ability a person has to grow up into a fully enlightened being) to the presentation given in Asanga's Bodhisattva Levels (Bodhisattva-bhumi). There the "residual impression left by listening [to the correct exposition of the truth]" (Sk. sruta-vasana) is the "seed" (Sk. bija) that matures into enlightenment. It is little more than the innate clarity of mind when it is freed from limitation by habituation to a correct vision of reality. Tsongkhapa limits the sources for his explanation to non-tantric works, almost all by persons he will later describe as Indian Yogacara writers, demonstrating thereby the influence the categories used in the recently redacted Kanjur and Tenjur (Tib. bsTan 'gyur, the name for the commentaries in Tibetan translation included in the canon) had on his thinking. When Tsongkhapa arrived in Central Tibet, the pressing philosophical issue of the day was Dolpopa's (Dol po pa Shes rab rgyal mtshan) (1292-1361) Great Middle Way (Tib. dBu ma chen po, Sk. \*Maha-madhayamaka) philosophy. It propounded a hermeneutics based on the principle that a Buddha will always tell you clearly what he means. Based on a wide array of sources, and without making a clear distinction between tantric and non-tantric works, the central tenet of the Great Middle Way is that an absolute, pure, transcendental mind endowed with all good qualities exists radically other than the ordinary world of appearance. It is the archetypical Tibetan Buddhist gzhan stong (pronounced shentong, "emptiness of other") philosophy. Dolpopa, stressing the hermeneutic stated above, says Nagarjuna and Asanga, writing during the golden age, have the same philosophy, the Great Middle Way clearly articulated by the Buddha. Dolpopa distinguishes a pure, transcendental knowledge (Tib. kun gzhi ye shes) that is quite other than the defiled foundation consciousness (Tib. kun gzhi rnam shes), and equates the former with his single absolute, endowed with all the qualities of a Buddha. Later, in his mature period, as explained below, Tsongkhapa will be explicit. He will state categorically that Asanga's Yogacara Buddhism is quite separate from, and in profundity inferior to, Nagarjuna's Middle Way school (Wangchuk 2013), removing all references to a Great Middle Way school. He will say the alaya-vijnana, devoid of subject-object bifurcation, provides the essential mind-stuff for defilement (=samsara) and carries seeds for purification (=nirvana) in a Yogacara system that is fundamentally wrong, and at odds with the way things actually are. He will give the alaya-vijnana only a heuristic value, and say a correct view (the \Prasangika-madhyamaka view) of the workings of psycho-physical reality (the Buddhist skandhas) totally invalidates it. As Jeffrey Hopkins (2002, 2006b) has shown conclusively, such statements constitute an explicit rejection of Dolpopa's view, a position Tsongkhapa still is formulating in early works like the Explanation of the Difficult Points. In his Explanation of the Difficult Points, Tsongkhapa is certain he does not accept the view of Dolpopa, but he is still unsure exactly what he will put in its place. He ignores Dolpopa's pure, transcendental knowledge called alaya, mentioning it obliquely, only in passing, during a final brief survey of views based on Chinese sources; he leaves unanswered the question of the final ontological status of the eighth consciousness, but appears to accept it as having a provisional reality; and he allows the views of Asanga set forth in the Bodhisattva-bhumi* to retain an authority he will later explicitly deny. ### 2.2 Golden Garland (Legs bshad gser phreng) Tsongkhapa's Golden Garland is his most important early work. It takes the form of a long explanation of the Perfection of Wisdom sutras, given pride of place in the Kanjur as the foremost words of the Buddha, after the Vinaya section (the codifications of ethical conduct). It is a word-by-word commentary on the topics in the Ornament for the Clear Realizations (Abhisamayalamkara). Tsongkhapa bases his explanation on two sub-commentaries by Arya Vimuktisena (sixth century?) and Hari Bhadra (end of the eighth century). It propounds a philosophy that later Gelukpas, following the taxonomy developed in the mature works of Tsongkhapa, call Yogacara-svatantrika-madhyamaka, in essence a Middle Way that incorporates many of the categories of Yogacara Buddhism, yet does not have the authority of Candrakirti's Prasangika interpretation. A great deal has been written on the difference between Prasangika and Svatantrika (Dreyfus and McClintock 2003). Historically, the Gelukpa's Yogacara-svatantrika-madhyamaka is another name for the mainstream philosophical school that developed in Tibet, based primarily on the teaching of Santaraksita and his student Kamalasila (both fl. end of the eight century), the most influential of all the Indian panditas involved in the dissemination of Buddhist ideas to Tibet. Later Gelukpa scholastics, basing themselves on the mature works of Tsongkhapa, are unequivocal that Yogacara-svatantrika-madhyamaka, like the Yogacara philosophy of Asanga, is wrong, and has only heuristic value. At the same time, they tacitly acknowledge its centrality by taking this "wrong" philosophy as the foundation of their studies. In the Golden Garland, Tsongkhapa, following Santaraksita's school as it developed in Tibet, asserts that all bases (Sk. vastu) (the psycho-physical factors locating the person as they begin their philosophical career), all mental states along the course of the path (Sk. marga), and the final result, are illusory, because they lack any essential nature. But it is not yet clear how the basis relates to the final outcome (the result). As explained below, Tsongkhapa will solve the problem in his mature works by a clear, radical, nihilistic leap denying any essential nature whatsoever, anywhere, to anything, thereby implicitly and explicitly devaluing (in absolutist terms) the ontological status of the final, pure, unified state, relegating it to just one more conventional, dependently-originated thing. In the Golden Garland, Tsongkhapa has not yet fully developed his mature, systematic philosophy. He is still content with discrediting Dolpopa and surveying different points of view. He uses Yogacara terminology to present his own opinion, and uses a language to describe enlightened (and enlightening) knowledge that retains for it a separateness (through its freedom from all mental construction) from all other mental states. The primary object of philosophical inquiry in the Golden Garland is the unity of the perceiving subject in enlightenment, a topic that reflects the language and concerns of the Yogacara-svatantrika-madhyamaka school. Tsongkhapa in this early work is still comfortable with characterizing the ultimate as an absence of elaboration (Tib. spros bral, Sk. nisprapanca) beyond the four extremes, later set forth as the orthodox view of the Sakya sect. Tsongkhapa's rejection of this characterization in his later work occasions a strong rejection of his view by writers like Gorampa (Go rab 'byams pa bsod nams sge ge) (1429-1489) (Cabezon and Dargyay 2006). The Golden Garland is driven in no small measure by an agenda dedicated to discrediting Dolpopa's Perfection of Wisdom commentaries. This is evident from a comparison of Dolpopa and Tsongkhapa's sources and views. While Dolpopa excoriates Arya Vimuktisena and Hari Bhadra and says they belong to the degenerate age, propounding doctrines harmful to golden age Buddhism, the Golden Garland privileges the commentaries by Arya Vimuktisena and Hari Bhadra. Dolpopa says the two Detailed Explanations of the Perfection of Wisdom Sutras (Sk. Brhattika, Tib. Gnod 'jom) by "the foremost, Great Middle Way master Vasubandhu" stand as the primary scriptural authority for the Great Middle Way doctrine, superior even to the partial truths revealed by Nagarjuna. The Golden Garland disputes that Vasubandhu is the author of the two Detailed Explanations, and says, regardless of their author, they are simply restatements of Nagarjuna's Middle Way. Finally, Dolpopa propounds a doctrine that holds the basis, path, and result are eternally the same, and all else is thoroughly imaginary, but the Golden Garland is certain that such a view is wrong, and directly cites a passage from Dolpopa's work, though without naming its author, saying, "since no other great path-breaker, [i.e., founder of an original philosophy] besides him has ever asserted [what Dolpopa says], learned persons are right to cast out [what he says] like a gob of spit" (Sparham 2007-2013 vol. 1, 425). ## 3. Mature Period Tsongkhapa formulated the philosophy for which he is best known some ten years after finishing the Golden Garland. He characterized the vision that led him to his philosophy as a pristine Middle Way. Reflecting back on his insight he would write in his In Praise of Dependent Origination (brTen 'brel bstod pa) (translation by Tupten Jinpa, undated) > > > "Nonetheless, before the stream of this life > > > Flowing towards death has come to cease > > > That I have found slight faith in you-- > > > Even this I think is fortunate. > > > Among teachers, the teacher of dependent origination, > > > Amongst wisdoms, the knowledge of dependent origination-- > > > You, who're most excellent like the kings in the worlds, > > > Know this perfectly well, not others." > > > Tsongkhapa first sets forth this mature philosophy linking dependent origination and emptiness in a special section at the end of his Great Exposition. There, in the context of an investigation into the end-product of an authentic, intellectual investigation into the truly real (Sk. tattva, Tib. de kho na), and into the way things finally are at their deepest level (Sk. tathata, Tib. de bzhin nyid),[2] he says you have to identify the object of negation (Tib. dgag bya), i.e., the last false projection to appear as reality, by avoiding two errors: going too far (Tib. khyab che ba) and not going far enough (Tib. khyab chung ba) (Drefus et al, 2011, chapter 5). That his project does not presuppose a privileged, soul-like, state of consciousness that surveys and categorizes phenomena is evident from his assertion that these two errors originate in a latent psychological tendency in philosophers: first, to hold on to vestiges of truth (reality) where there is in fact an absence of it, and second, to fall back on some version of truth (reality) in ordinary appearance after failing to avoid nihilism in a futile quest for meaning. This section of Tsongkhapa's work is based on the opening verse of Nagarjuna's Root Verses on the Middle Way (Mula-madhyamaka-karika). Tsongkhapa's Ocean of Reasoning (Garfield and Samten 2006) is a detailed exegesis of this work. In it, Nagarjuna says nothing is produced because it is not produced from itself, other, both or neither. In Tsongkhapa's Great Exposition, the subject of Nagarjuna's syllogism is primarily the embodied person, reflecting the trend that runs throughout Buddhist philosophical literature, in general, to stress the application of philosophical analysis to praxis. Furthermore, for Tsongkhapa, Nagarjuna's statement is about a first moment in the continuum of the investigating person. By implication, reflecting the continuing influence of Santaraksita, it is the person engaged in the philosophical investigation in the immediate moment. It is axiomatic for Buddhist philosophers that there is no independent subject (soul) other than the five heaps (skandhas), so the subject of the syllogism becomes the complex or continuum of sense-faculties in the present instant. Such a subject makes excellence sense of Tsongkhapa's otherwise confusing juxtaposition of Nagarjuna's analysis with Dharmakirti's epistemology, in which it is axiomatic that direct sense perception is an authoritative means of knowledge (Sk. pramana). Nagarjuna's statement that such a person was never produced seems paradoxical, to say the least, because it appears to undercut the reality of the very intellectual act in which the thinker is self-evidently engaged. Tsongkhapa makes it abundantly clear that in his view not only is the intellectual act itself utterly devoid of any essential reality, even the sense-faculties (the complex eye, ear and so on) also lack any essential reality, never mind the sense data they convey to the knowing individual. Tsongkhapa's choice of the opening line's of Nagarjuna's work as a point of departure for his philosophy stems from his belief that he has gained a true insight into dependent origination (pratitya-samutpada). This insight, he believes, resolves the paradox between his apparent nihilism and his insistence on the weight of authoritative statements and cognitions. Thus he says in his Great Exposition, > > > "Therefore, the intelligent should develop an unshakeable > certainty that the very meaning of emptiness is > dependent-arising... Specifically, this is the subtle point > that the noble Nagarjuna and his spiritual son Aryadeva > had in mind and upon which the master Buddhapalita and the > glorious Candrakirti gave fully comprehensive commentary. This is > how dependent-arising bestows certain knowledge of the absence of > intrinsic existence; this is how it dawns on you that it is things > which are devoid of intrinsic existence that are causes and > effects" (Lamrim Chenmo Translation Committee Vol. 3, 139). > > > Tsongkhapa set out the ramifications of his view in "eight points that are difficult to understand" (Tib. dka' gnas brgyad) first enumerated in notes (rjes byang) to one of his lectures by his contemporary and disciple, Darma rin chen (also called rGyal tshab rje "the regent"). Tillemans (1998) has nicely rendered the opening of the text as follows, > > > "Concerning the [ontological] bases, there are the following > [three points]: (1) the conventional nonacceptance of particulars and > of (2) the storehouse consciousness, and (3) the acceptance of > external objects. Concerning the path, there are the following [four > points]: (4) the nonacceptance of autonomous reasonings as being means > for understanding reality and (5) the nonacceptance of self-awareness; > (6) the way in which the two obscurations exist; (7) how it is > accepted that the Buddha's disciples and those who become awakened > without a Buddha's help realize that things are without any > own-nature. Concerning the result, there is: (8) the way in which the > buddhas know [conventional] things in their full extent. > Thus, there are four accepted theses and four unaccepted > theses." > > > They have been discussed by a number of writers (Ruegg 2002, Cabezon 1992, 397, Cabezon and Dargyay 2006, 76-93, 114-201). I have surveyed Tsongkhapa's early work on the alaya-vijnana (Tilleman's "storehouse consciousness") above. For Tsongkhapa, until the correct view is understood, it is necessary to assert the alaya-vijnana in order to save cause and effect, and avoid falling into nihilism. When the correct view is obtained, the alaya-vijnana stands invalidated (Tillemans's second point). For Tsongkhapa, it is only possible to understand cause and effect (in particular, as experienced in the immediate moment by an embodied person), by correctly understanding that dependent origination precludes any essential existence whatsoever. Tsongkhapa asserts that a specific mark (Sk. sva-laksana), for instance a mark that makes blue blue, instead of red or any other color, has not even a nominal existence. This is the first of Tillemans's eight difficult points. In his monograph Essence of Eloquence, Tsongkhapa employs a hermeneutics that treats language and knowledge as equally semiotic in nature. This is consistent with Tsongkhapa's view that any intellectual act is itself utterly devoid of any essential reality, yet functions on a conventional level through the natural workings of dependent origination (Sk. dharmata). This view allows him to conclude that Dignaga and Dharmakirti's Logico-epistemological school is equivalent to Asanga's Yogacara school insofar as the former school asserts a specific mark, seen by direct sense perception, that is necessary in order to retain the reality of the conventional world. Tsongkhapa takes the strong position that no datum that appears to (or is there but unknown to) thought or sense perception, has any essential reality. All are, equally, simply labeled by thought construction (Tib. rtog pas btags tsam). Only convention makes the actual sense-faculties, for example, real, and success or failure experienced in a dream, for example, false. This dependent origination (between a label and what is labeled) precludes the essential existence implicit in the Epistemologist's sva-laksana. Tsongkhapa's strong rejection of Yogacara idealism leads him to assert the existence of external objects (the third point). His (discredited) Yogacara school, based on either the works of Asanga or the Epistemologists, explains the absence of subject-object bifurcation as non-dual with thought or mind (Sk. citta). Realizing this in a non-conceptual, meditational state constitutes a liberating vision. Ultimately, therefore, external objects are projections of a deluded mind. Tsongkhapa rejects this, though he suggests only those who have understood the emptiness of inherent existence through reflecting on the natural workings of dependent origination can set it aside. As he says in his Ocean of Reasoning, > > > "The meaning of the statement that the conventional designation > of subject and objects stops is that the designation of these two > stops from the perspective of meditative equipoise, but it does not > mean that the insight in meditative equipoise and the ultimate truth > are rejected as subject and object. This is because their being > subject and object is not posited from the perspective of analytic > insight, but from the perspective of conventional understanding" > (Garfield and Samten 2006, p. 26). > > > From the perspective of ordinary convention there are external objects, so it is sufficient, on that level, to assert that they are there. Tsongkhapa does not accept svatantra ("autonomous") reasoning (the fourth point). He asserts that it is enough, when proving that any given subject is empty of intrinsic existence, to lead the interlocutor, through reasoning, to the unwelcome consequences (prasanga) in their own untenable position; it is not necessary to demonstrate the thesis based on reasoning that presupposes any sort of intrinsic (=autonomous) existence. This gives Tsongkhapa's philosophy its name \Prasangika-madhyamaka, i.e., a philosophy of a middle way (between nihilism and eternalism) arrived at through demonstrating the unwelcome consequences (in any given position that presupposes intrinsic existence). In the context of this assertion, Tsongkhapa offers a distinctive explanation of the well-known Nayayika objection to Nagarjuna's philosophy, namely, if the statements he uses to prove his thesis are themselves without any final intrinsic reality, they will be ineffective as proofs. Tsongkhapa says Prasangika-madhyamakas do not simply find faults in all positions and reject all positions as their own. They only deny any thesis that presupposes an intrinsic existence. They do hold a specific thesis, to wit, that all phenomena lack an intrinsic existence. They use reasoning and logic that lacks any essential reality to establish this thesis. Such reasoning derives its efficacity on a conventional level through the natural workings of dependent origination. This is one of the most contentious assertions of Tsongkhapa. Tsongkhapa's rejection of any form of self-referential consciousness (Sk. sva-samvitti, sva-samvedana) (the fifth point) is in essence a rejection of Santaraksita's position that such self-referential or reflexive awareness is necessary to explain the self-evidential nature of consciousness, and to explain the privileged access the conscious person has to their own consciousness as immediate and veridical (Garfield 2006).[3] At issue is the status of the knowing subject engaged in the intellectual pursuit of the truly real. Tsongkhapa holds that such a knowing subject has no essential reality at all. Such a position requires of Tsongkhapa an explanation of memory. His solution is to deny that when you remember seeing object X* you also remember the conscious act of seeing it. (Were you to do so, there would have to be an aspect of the earlier consciousness of the object that was equally conscious of itself, i.e., self-referential.) Instead, Tsongkhapa argues, memory is simply the earlier consciousness of object X, now designated "past." When designated past, inexorably (or inferentially, as it were) the presence of a consciousness of the past object X is required to make sense of the present reality. Tsongkhapa characterizes basic ignorance (Sk. avidya), the root cause of suffering in Buddhist philosophy, not as a latent tendency, but as an active defiling agency (Sk. klesavarana) that projects a reality onto objects that is in fact absent from them. This ignorance affects even sense perception and explains the veridical aspect that is, in fact, just error. The residual impressions left by distortion (literally "perfumings" Sk. vasana) explain the mere appearance of things as real. Beyond that, he asserts that habituation to this distortion prevents conventional and ultimate reality from appearing united in an appearing object. This explanation of the psychology of error differs markedly from earlier Tibetan explanations and constitutes the sixth difficult point. In Tsongkhapa's mature philosophy the Mahayana altruistic principle (bodhicitta) is the sole criterion for distinguishing authentic Mahayana views and practices from non-Mahayana ones. By privileging the principle in this way he is able to assert that any authentic realization of truth is a realization of the way things are, namely, a realization of no sva-bhava (Sk.) ("own-being, own-nature, intrinsic identity"). For Tsongkhapa, therefore, Hinayanists (by which he intends followers of the basic Buddhist doctrine of the Four Noble Truths set forth in the earliest scriptures), necessarily have the same authentic knowledge of reality. Were they not to have such knowledge, he argues, they could not have reached the goals they reached (the seventh point). Finally, Tsongkhapa has a robust explanation of the difference between true and false on the covering or conventional level. He denies any difference between a false object (a dream lottery ticket, for example) and a real one; as appearances, he asserts, both are equally false, only convention decides which is true. All phenomena equally lack truth. In Tsongkhapa's mature philosophy, therefore, all appearance is false--to appear is to appear as being truly what the appearance is of, and the principle of dependent origination precludes such truth from according with the way things actually are. Tsongkhapa holds that sentences and their content and minds and what appears to them function in the same way. Saying (1) of a set of "true" and "false" statements that they are all equally untrue, and (2) saying of unmediated sense-based perception or mistaken ideas that they are equally untrue is to say the same thing. The truth in both is decided by convention, not by something inhering in the true statement (or its content), or the valid perception (or its object). For Tsongkhapa, therefore, since all appearance is false, the Buddha knows, but without any appearance of truth (the eighth difficult point). Towards the very end of his life, in his Medium-Length Exposition of the Path and Elucidation of the Intention Tsongkhapa defends his views against Rongton's (rong stong shes bya kun rig) (1367-1449) criticism that his earlier and later works are inconsistent. Tsongkhapa extends his comparison of Candrakirti's and Bhaviveka's interpretations of Nagarjuna to more clearly identify a less subtle object of negation in the works of Yogacara-svatantrika-madhyamaka writers, particularly Jnanagarbha (8th century) and Santaraksita (Hopkins 2008, Part One, Yi 2015). Tsongkhapa finds in their works an analysis leading beyond the Yogacara's mere absence of a substantial difference between known object and knowing subject, but limited insofar as it negates the ultimate truth of all objects appearing to conventional awareness while still allowing room for an object of negation that does not appear to sense perception. Dreyfus et al (2011) and Falls (2010) suggest parallels between the issues raised by the Svatantrika-Prasangika difference and some recent work in Western philosophy. They are helpful for those familar with Western philosophy but unfamiliar with Tsongkhapa, surveying questions that arise from the interaction of philosophical traditions from different cultures. Falls argues Tsongkhapa's philosophy is a therapeutea not a theoria and draws parallels between issues raised by Tsongkhapa and John McDowell's interpretation of Wilfrid Sellars in particular. ## 4. Hermeneutics A distinctive feature of Tsongkhapa's mature philosophy is the centrality he accords hermeneutics, and the particular stratification of philosophical systems it occasions. His focus on hermeneutics stems in no small measure from his acceptance of the Kanjur as a true record of authentic statements of the Buddha, and also from his wish to discredit Dolpopa's views. According to Tsongkhapa, and worked out in detail in his Essence of Eloquence, the given record of the Buddha's diverse statements seems to contain contradictions, so a reader must decide on criteria for interpreting them. No statement of the Buddha can serve as a primary hermeneutic principle, so that principle necessarily becomes human reason (Sk. yukti, Tib. rigs pa). When human reason is brought to bear on the diverse statements of the Buddha, it concludes that all statements that an essential or intrinsic identity exists, or any statements that presupposes that, cannot be taken literally, at face value. This is because human reason, when it analyzes the ultimate truth of any object or statement, finds it empty of anything intrinsic to it that would make it true. For Tsongkhapa, the line of reasoning that leads most clearly to this conclusion is dependent origination. Statements that require interpretation Tsongkhapa groups into those of different Buddhist philosophical schools: in ascending order, Listener (Sk. Sravaka) schools based on older Buddhist sources (the Vaibhasika and Sautrantika), Yogacara schools following Asanga and Dignaga/Dharmakirti, and non-Prasangika Madhyamaka schools following Bhaviveka and Santaraksita. The Prasangika-madhyamaka school of Candrakirti alone captures the final intention of the Buddha. Tsongkhapa was certainly not the first doxographer, but his clear and definitive categories had immense influence on later writers, so much so they have anachronistically been read back into works that were written before his time. Tsongkhapa's particular hermeneutics, the primary means he employs to lead readers to his philosophical insight, allows him to characterize particular, authentic, Buddhist philosophies as wrong (because they are wrong from a Prasangika perspective), yet right from the perspective of those particular systems. They are right because the philosophies have particular roles to play in a larger, grander scheme. This scheme is part of the larger philosophy of a perfect person whose views are in perfect accord (Sk. tathagata) with the way things are, i.e., dependently originated, and whose statements are motivated solely by the benefit they have to those who hear them. In this way, Tsongkhapa's hermeneutics lead to, or incorporate, a second principle, namely, bodhicitta (Sk.). The word bodhicitta has at least six different, but interrelated meanings in different contexts (Wangchuk 2007). In his philosophical works Tsongkhapa uses it to mean a universal, altruistic principle (not unlike the logos) that explains, primarily, the genesis of the Buddha's diverse statements, i.e., explains why a person with perfect intellect and powers of expression would make statements that seem to contain contradictions. This principle plays a central role in Tsongkhapa's assertion that all authentic attainments, without distinction, are based on an authentic insight into emptiness (the seventh of the eight difficult points listed above), and it leads him to assert that the "origin" of the Mahayana is located in bodhicitta, and bodhicitta alone ## 5. Ethics Tsongkhapa is part of a long, shared, Indo-Tibetan Buddhist tradition that conceives of ethical conduct not in absolute terms, but in the context of different individuals in different situations. There is an unspoken presupposition that ethical statements are grounded in reality, in the sense that suffering (Sk. duhkha) comes from actions (Sk. karma) that are not in accord with the way things actually are (Tib. dngos po'i gnas lugs). The person is ultimately not present as an ethical subject, but is so, on a conventional or covering level, through the natural working of dependent origination. Tsongkhapa groups such individuals into three separate categories (those who privilege basic Buddhist codes, bodhisattvas, and tantrikas). Each is governed by an ethical code (Tib. sdom gsum), each superior code subsuming the points of the lesser. Beyond these three, his Great Exposition suggests ordinary ethical conduct is codified as well, in the main, in the ten ethical points (Sk. dasa-kusala-patha) basic to any human life that rises above the mere animal. The first of the three specifically Buddhist codes, the basic code, primarily governs the behavior of monks and nuns. Since for Tsongkhapa each higher code incorporates all the rules in the lower code, he conceives of the seven (or even twelve) sub-codes making up the basic code as, in descending order, containing fewer and fewer of the rules that constitute the full code. Each of the sub-codes is designed for particular people in particular human situations. Tsongkhapa does not expect a butcher to be governed by the rule of not killing, for example, and he does not expect a lawyer to be governed by the rule of not lying. He avoids gross devaluation of the basic code by privileging spiritual elites. In this, he takes a pragmatic approach to ethics in line with his non-absolutist stance. For Tsongkhapa, it is axiomatic that human life is not ipso facto privileged above any other form of life, but his detailed explanation gives greater "weight" to the karmic retribution that comes from killing humans, and amongst humans, noble persons, for example, than less fortunate forms of life. Tsongkhapa's explanation of the bodhisattva moral code presented in the Bodhisattva-bhumi (Tatz 1987) follows naturally from the importance he gives to the altruistic principle (bodhicitta) in his other writing. He reconciles rules in the bodhisattva code that enjoin behavior contradicting the basic code by positing an elite that, through a noblesse oblige of the spiritually advanced, are required to do things which would be unethical in an ordinary person. Their not doing so constitutes an ethical lapse. For example, the basic code prohibits actions that influence what food a donor puts in the begging bowl (with a few exceptions for human flesh and so on), and prohibits eating after noon. The bodhisattva code contradicts the basic code insofar as it prohibits eating meat, even though, by so doing, the donor does not get the opportunity to make charity. Tsongkhapa argues that if a monk at the bodhisattva stage of development eats meat it clashes with the dictates of the altruistic principle (bodhicitta). Tsongkhapa accepts that the diverse body of literature, including the historically latest Buddhist tantras (some of which are distinctly antinomian in character), are the work of a fully awakened being (Sk. buddha). The last of the three codes systematizes the conduct espoused in these tantras. Tsongkhapa's presentation is distinctive for finding a code complete with a full ordination ceremony. The practical result of his presentation is to revalue the basic code for monks, and devalue the ritual of tantric consecration as it pertains to ethical conduct. It stresses ethical conduct even in the context of works that appear, in line with the nihilistic drift of Buddhist philosophy, to count ethical codes as a block to the highest spiritual and philosophical attainment. For Tsongkhapa the tantric code is only for the very highest spiritual elite, mainly monks and nuns. Tsongkhapa divides tantras into two sections, and says the lower section is governed exclusively by the bodhisattva code, supplemented by specific ritual injunctions (Sk. vrata). He gives two interpretations of the rules for the higher tantras, reserving the truly antinomian behavior for a theoretical elite whose altruism is so strong, and whose understanding and status is so ennobled, that they engage in what ordinarily would be condemned as gross immorality. According to this code, it is an ethical lapse not to eat meat, perhaps even human flesh, the logic being that in certain specific and unusual circumstances, in a person at this stage of development, such behavior would constitute a skillful means to benefit others. ## 6. Tantra Tsongkhapa does not have a different tantric philosophy. His Prasangika Middle Way is the philosophical position he articulates in his works on tantra. He does however, unlike in his non-tantric works, accept that those propounding Idealistic philosophies (Yogacara, Svatantrika Middle Way) can have success in their practice. In this he reveals again the importance of the central Tibetan philosophical tradition going back to Santaraksita. Tsongkhapa conceives of tantra as a subset of the Mahayana, and to that extent all authentic Buddhist tantric activities are, necessarily, authentically altruistic. What differentiates tantric activity from other ordinary Mahayana activities is deity yoga (Tib. lha'i rnal 'byor), i.e., whether or not, from a first person perspective, the person is acting as a perfect, "divine" subject when engaging in (primarily ritual) behavior. Tsongkhapa wrote works on the Vajrabhairava, Cakrasamvara (Gray 2017), and Kalacakra tantras, but is best known for his exposition of the Guhyasamaja tantra based on Indian commentaries associated with the names of Nagarjuna and Aryadeva, particularly his short but influential Commentary on the Jnana-vajra-samuccaya (Ye shes rdo rje kun las 'dus pa'i ti ka) (1410), and magisterial Clear Exposition of the Five Stages of Guhyasamaja (gSang 'dus rim lnga gsal sgron) (1411) (Thurman 2011, Kilty 2013). He accords great importance to esoteric yoga. In this respect, his work is firmly located in the mainstream Tibetan tradition, influenced in particular by the translator Mar pa (Mar pa Chos kyi blo gros) (1012-1097), who spread the Six Teachings of Naropa (Na ro chos drug), a later Indian synthesis of diverse tantric practices, in Tibet. Based on this Tsongkhapa gives detailed explanations of the nadi (channels for the energy or feelings that run through the tantric practitioner's body), cakra (circles of channels in the heart, throat, and other central points up the center of the body from the bottom of the spine, or tip of the sex organ, to the top of the head) and candali (an intense pleasure experienced as heat that spreads through the channels and fills the body). Tsongkhapa is praised, in particular, for his explanation of theory and praxis associated with the illusory body (Sk. maya-kaya, Tib. sgyu lus) and clear light (Sk. prabhasvara, Tib. 'od gsal), a skillful adaptation of his understanding of the two truths to yogic praxis (Natanya 2017). This is summed up nicely in the statement, "The Pancakramasamgrahaprakasa (Illumination of the Summary of the Five Steps), a short treatise attributed to Naropa (956-1040), combining the Six Teachings with the 'Five Steps' (pancakrama) of the Arya tradition provided Tsong kha pa with the basic ideas of his Tantric system" (Tillemans 1998).
turing	## 1. Outline of Life Alan Turing's short and extraordinary life has attracted wide interest. It has inspired his mother's memoir (E. S. Turing 1959), a detailed biography (Hodges 1983), a play and television film (Whitemore 1986), and various other works of fiction and art. There are many reasons for this interest, but one is that in every sphere of his life and work he made unexpected connections between apparently unrelated areas. His central contribution to science and philosophy came through his treating the subject of symbolic logic as a new branch of applied mathematics, giving it a physical and engineering content. Unwilling or unable to remain within any standard role or department of thought, Alan Turing continued a life full of incongruity. Though a shy, boyish, man, he had a pivotal role in world history through his role in Second World War cryptology. Though the founder of the dominant technology of the twentieth century, he variously impressed, charmed or disturbed people with his unworldly innocence and his dislike of moral or intellectual compromise. Alan Mathison Turing was born in London, 23 June 1912, to upper-middle-class British parents. His schooling was of a traditional kind, dominated by the British imperial system, but from earliest life his fascination with the scientific impulse--expressed by him as finding the 'commonest in nature'--found him at odds with authority. His scepticism, and disrespect for worldly values, were never tamed and became ever more confidently eccentric. His moody humour swung between gloom and vivacity. His life was also notable as that of a gay man with strong emotions and a growing insistence on his identity. His first true home was at King's College, Cambridge University, noted for its progressive intellectual life centred on J. M. Keynes. Turing studied mathematics with increasing distinction and was elected a Fellow of the college in 1935. This appointment was followed by a remarkable and sudden debut in an area where he was an unknown figure: that of mathematical logic. The paper "On Computable Numbers..." (Turing 1936-7) was his first and perhaps greatest triumph. It gave a definition of computation and an absolute limitation on what computation could achieve, which makes it the founding work of modern computer science. It led him to Princeton for more advanced work in logic and other branches of mathematics. He had the opportunity to remain in the United States, but chose to return to Britain in 1938, and was immediately recruited for the British communications war. From 1939 to 1945 Turing was almost totally engaged in the mastery of the German enciphering machine, Enigma, and other cryptological investigations at now-famous Bletchley Park, the British government's wartime communications headquarters. Turing made a unique logical contribution to the decryption of the Enigma and became the chief scientific figure, with a particular responsibility for reading the U-boat communications. As such he became a top-level figure in Anglo-American liaison, and also gained exposure to the most advanced electronic technology of the day. Combining his ideas from mathematical logic, his experience in cryptology, and some practical electronic knowledge, his ambition, at the end of the war in Europe, was to create an electronic computer in the full modern sense. His plans, commissioned by the National Physical Laboratory, London, were overshadowed by the more powerfully supported American projects. Turing also laboured under the disadvantage that his wartime achievements remained totally secret. His ideas led the field in 1946, but this was little recognised. Frustrated in his work, he emerged as a powerful marathon runner, and almost qualified for the British team in the 1948 Olympic games. Turing's motivations were scientific rather than industrial or commercial, and he soon returned to the theoretical limitations of computation, this time focussing on the comparison of the power of computation and the power of the human brain. His contention was that the computer, when properly programmed, could rival the brain. It founded the 'Artificial Intelligence' program of coming decades. In 1948 he moved to Manchester University, where he partly fulfilled the expectations placed upon him to plan software for the pioneer computer development there, but still remained a free-ranging thinker. It was here that his famous 1950 paper, "Computing Machinery and Intelligence," (Turing 1950b) was written. In 1951 he was elected a Fellow of the Royal Society for his 1936 achievement, yet at the same time he was striking into entirely new territory with a mathematical theory of biological morphogenesis (Turing 1952). This work was interrupted by Alan Turing's arrest in February 1952 for his sexual affair with a young Manchester man, and he was obliged, to escape imprisonment, to undergo the injection of oestrogen intended to negate his sexual drive. He was disqualified from continuing secret cryptological work. His general libertarian attitude was enhanced rather than suppressed by the criminal trial, and his intellectual individuality also remained as lively as ever. While remaining formally a Reader in the Theory of Computing, he not only embarked on more ambitious applications of his biological theory, but advanced new ideas for fundamental physics. For this reason his death, on 7 June 1954, at his home in Wilmslow, Cheshire, came as a general surprise. In hindsight it is obvious that Turing's unique status in Anglo-American secret communication work meant that there were pressures on him of which his contemporaries were unaware; there was certainly another 'security' conflict with government in 1953 (Hodges 1983, p. 483). Some commentators, e.g. Dawson (1985), have argued that assassination should not be ruled out. But he had spoken of suicide, and his death, which was by cyanide poisoning, was most likely by his own hand, contrived so as to allow those who wished to do so to believe it a result of his penchant for chemistry experiments. The symbolism of its dramatic element--a partly eaten apple--has continued to haunt the intellectual Eden from which Alan Turing was expelled. ## 2. The Turing Machine and Computability Alan Turing drew much between 1928 and 1933 from the work of the mathematical physicist and populariser A. S. Eddington, from J. von Neumann's account of the foundations of quantum mechanics, and then from Bertrand Russell's mathematical logic. Meanwhile, his lasting fascination with the problems of mind and matter was heightened by emotional elements in his own life (Hodges 1983, p. 63). In 1934 he graduated with an outstanding degree in mathematics from Cambridge University, followed by a successful dissertation in probability theory which won him a Fellowship of King's College, Cambridge, in 1935. This was the background to his learning, also in 1935, of the problem which was to make his name. It was from the lectures of the topologist M. H. A. (Max) Newman in that year that he learnt of Godel's 1931 proof of the formal incompleteness of logical systems rich enough to include arithmetic, and of the outstanding problem in the foundations of mathematics as posed by Hilbert: the "Entscheidungsproblem" (decision problem). Was there a method by which it could be decided, for any given mathematical proposition, whether or not it was provable? The principal difficulty of this question lay in giving an unassailably correct and general definition of what was meant by such expressions as 'definite method' or 'effective procedure.' Turing worked on this alone for a year until April 1936; independence and isolation was to be both his strength, in formulating original ideas, and his weakness, when it came to promoting and implementing them. The word 'mechanical' had often been used of the formalist approach lying behind Hilbert's problem, and Turing seized on the concept of the machine.Turing's solution lay in defining what was soon to be named the Turing machine. With this he defined the concept of 'the mechanical' in terms of simple atomic operations. The Turing machine formalism was modelled on the teleprinter, slightly enlarged in scope to allow a paper tape that could move in both directions and a 'head' that could read, erase and print new symbols, rather than only read and punch permanent holes. The Turing machine is 'theoretical,' in the sense that it is not intended actually to be engineered (there being no point in doing so), although it is essential that its atomic components (the paper tape, movement to left and right, testing for the presence of a symbol) are such as could actually be implemented. The whole point of the formalism is to reduce the concept of 'method' to simple operations that can unquestionably be 'effected.' Nevertheless Turing's purpose was to embody the most general mechanical process as carried out by a human being. His analysis began not with any existing computing machines, but with the picture of a child's exercise book marked off in squares. From the beginning, the Turing machine concept aimed to capture what the human mind can do when carrying out a procedure. In speaking of 'the' Turing machine it should be made clear that there are infinitely many Turing machines, each corresponding to a different method or procedure, by virtue of having a different 'table of behaviour.' Nowadays it is almost impossible to avoid imagery which did not exist in 1936: that of the computer. In modern terms, the 'table of behaviour' of a Turing machine is equivalent to a computer program. If a Turing machine corresponds to a computer program, what is the analogy of the computer? It is what Turing described as a universal machine (Turing 1936, p. 241). Again, there are infinitely many universal Turing machines, forming a subset of Turing machines; they are those machines with 'tables of behaviour' complex enough to read the tables of other Turing machines, and then do what those machines would have done. If this seems strange, note the modern parallel that any computer can be simulated by software on another computer. The way that tables can read and simulate the effect of other tables is crucial to Turing's theory, going far beyond Babbage's ideas of a hundred years earlier. It also shows why Turing's ideas go to the heart of the modern computer, in which it is essential that programs are themselves a form of data which can be manipulated by other programs. But the reader must always remember that in 1936 there were no such computers; indeed the modern computer arose out of the formulation of 'behaving mechanically' that Turing found in this work. Turing's machine formulation allowed the precise definition of the computable: namely, as what can be done by a Turing machine acting alone. More exactly, computable operations are those which can be effected by what Turing called automatic machines. The crucial point here is that the action of an automatic Turing machine is totally determined by its 'table of behaviour'. (Turing also allowed for 'choice machines' which call for human inputs, rather than being totally determined.) Turing then proposed that this definition of 'computable' captured precisely what was intended by such words as 'definite method, procedure, mechanical process' in stating the Entscheidungsproblem. In applying his machine concept to the Entscheidungsproblem, Turing took the step of defining computable numbers. These are those real numbers, considered as infinite decimals, say, which it is possible for a Turing machine, starting with an empty tape, to print out. For example, the Turing machine which simply prints the digit 1 and moves to the right, then repeats that action for ever, can thereby compute the number .1111111... A more complicated Turing machine can compute the infinite decimal expansion of p. Turing machines, like computer programs, are countable; indeed they can be ordered in a complete list by a kind of alphabetical ordering of their 'tables of behaviour'. Turing did this by encoding the tables into 'description numbers' which can then be ordered in magnitude. Amongst this list, a subset of them (those with 'satisfactory' description numbers) are the machines which have the effect of printing out infinite decimals. It is readily shown, using a 'diagonal' argument first used by Cantor and familiar from the discoveries of Russell and Godel, that there can be no Turing machine with the property of deciding whether a description number is satisfactory or not. The argument can be presented as follows. Suppose that such a Turing machine exists. Then it is possible to construct a new Turing machine which works out in turn the Nth digit from the Nth machine possessing a satisfactory description number. This new machine then prints an Nth digit differing from that digit. As the machine proceeds, it prints out an infinite decimal, and therefore has a 'satisfactory' description number. Yet this number must by construction differ from the outputs of every Turing machine with a satisfactory description number. This is a contradiction, so the hypothesis must be false (Turing 1936, p. 246). From this, Turing was able to answer Hilbert's Entscheidungsproblem in the negative: there can be no such general method. Turing's proof can be recast in many ways, but the core idea depends on the self-reference involved in a machine operating on symbols, which is itself described by symbols and so can operate on its own description. Indeed, the self-referential aspect of the theory can be highlighted by a different form of the proof, which Turing preferred (Turing 1936, p. 247). Suppose that such a machine for deciding satisfactoriness does exist; then apply it to its own description number. A contradiction can readily be obtained. However, the 'diagonal' method has the advantage of bringing out the following: that a real number may be defined unambiguously, yet be uncomputable. It is a non-trivial discovery that whereas some infinite decimals (e.g. p) may be encapsulated in a finite table, other infinite decimals (in fact, almost all) cannot. Likewise there are decision problems such as 'is this number prime?' in which infinitely many answers are wrapped up in a finite recipe, while there are others (again, almost all) which are not, and must be regarded as requiring infinitely many different methods. 'Is this a provable proposition?' belongs to the latter category. This is what Turing established, and into the bargain the remarkable fact that anything that is computable can in fact be computed by one machine, a universal Turing machine. It was vital to Turing's work that he justified the definition by showing that it encompassed the most general idea of 'method'. For if it did not, the Entscheidungproblem remained open: there might be some more powerful type of method than was encompassed by Turing computability. One justification lay in showing that the definition included many processes a mathematician would consider to be natural in computation (Turing 1936, p. 254). Another argument involved a human calculator following written instruction notes. (Turing 1936, p. 253). But in a bolder argument, the one he placed first, he considered an 'intuitive' argument appealing to the states of mind of a human computer. (Turing 1936, p. 249). The entry of 'mind' into his argument was highly significant, but at this stage it was only a mind following a rule. To summarise: Turing found, and justified on very general and far-reaching grounds, a precise mathematical formulation of the conception of a general process or method. His work, as presented to Newman in April 1936, argued that his formulation of 'computability' encompassed 'the possible processes which can be carried out in computing a number.' (Turing 1936, p. 232). This opened up new fields of discovery both in practical computation, and in the discussion of human mental processes. However, although Turing had worked as what Newman called 'a confirmed solitary' (Hodges 1983, p 113), he soon learned that he was not alone in what Gandy (1988) has called 'the confluence of ideas in 1936.' The Princeton logician Alonzo Church had slightly outpaced Turing in finding a satisfactory definition of what he called 'effective calculability.' Church's definition required the logical formalism of the lambda-calculus. This meant that from the outset Turing's achievement merged with and superseded the formulation of Church's Thesis, namely the assertion that the lambda-calculus formalism correctly embodied the concept of effective process or method. Very rapidly it was shown that the mathematical scope of Turing computability coincided with Church's definition (and also with the scope of the general recursive functions defined by Godel). Turing wrote his own statement (Turing 1939, p. 166) of the conclusions that had been reached in 1938; it is in the Ph.D. thesis that he wrote under Church's supervision, and so this statement is the nearest we have to a joint statement of the 'Church-Turing thesis': > A function is said to be 'effectively > calculable' if its values can be found by some purely mechanical > process. Although it is fairly easy to get an intuitive grasp of this > idea, it is nevertheless desirable to have some more definite, > mathematically expressible definition. Such a definition was first > given by Godel at Princeton in 1934... These functions were > described as 'general recursive' by Godel... > Another definition of effective calculability has been given by > Church... who identifies it with lambda-definability. The author > [i.e. Turing] has recently suggested a definition corresponding more > closely to the intuitive idea... It was stated above that 'a > function is effectively calculable if its values can be found by a > purely mechanical process.' We may take this statement literally, > understanding by a purely mechanical process one which could be carried > out by a machine. It is possible to give a mathematical description, in > a certain normal form, of the structures of these machines. The > development of these ideas leads to the author's definition of a > computable function, and to an identification of computability with > effective calculability. It is not difficult, though somewhat > laborious, to prove that these three definitions are > equivalent. Church accepted that Turing's definition gave a compelling, intuitive reason for why Church's thesis was true. The recent exposition by Davis (2000) emphasises that Godel also was convinced by Turing's argument that an absolute concept had been identified (Godel 1946). The situation has not changed since 1937. (For further comment, see the article on the Church-Turing Thesis. The recent selection of Turing's papers edited by Copeland (2004), and the review of Hodges (2006), continue this discussion.) Turing himself did little to evangelise his formulation in the world of mathematical logic and early computer science. The textbooks of Davis (1958) and Minsky (1967) did more. Nowadays Turing computability is often reformulated (e.g. in terms of 'register machines'). However, computer simulations (e.g., Turing's World, from Stanford) have brought Turing's original imagery to life. Turing's work also opened new areas for decidability questions within pure mathematics. From the 1970s, Turing machines also took on new life in the development of complexity theory, and as such underpin one of the most important research areas in computer science. This development exemplifies the lasting value of Turing's special quality of giving concrete illustration to abstract concepts. ## 3. The Logical and the Physical As put by Gandy (1988), Turing's paper was 'a paradigm of philosophical analysis,' refining a vague notion into a precise definition. But it was more than being an analysis within the world of mathematical logic: in Turing's thought the question that constantly recurs both theoretically and practically is the relationship of the logical Turing machine to the physical world. 'Effective' means doing, not merely imagining or postulating. At this stage neither Turing nor any other logician made a serious investigation into the physics of such 'doing.' But Turing's image of a teleprinter-like machine does inescapably refer to something that could actually be physically 'done.' His concept is a distillation of the idea that one can only 'do' one simple action, or finite number of simple actions, at a time. How 'physical' a concept is it? The tape never holds more than a finite number of marked squares at any point in a computation. Thus it can be thought of as being finite, but always capable of further extension as required. Obviously this unbounded extendibility is unphysical, but the definition is still of practical use: it means that anything done on a finite tape, however large, is computable. (Turing himself took such a finitistic approach when explaining the practical relevance of computability in his 1950 paper.) One aspect of Turing's formulation, however, involves absolute finiteness: the table of behaviour of a Turing machine must be finite, since Turing allows only a finite number of 'configurations' of a Turing machine, and only a finite repertoire of symbols which can be marked on the tape. This is essentially equivalent to allowing only computer programs with finite lengths of code. 'Calculable by finite means' was Turing's characterisation of computability, which he justified with the argument that 'the human memory is necessarily limited.' (Turing 1936, p. 231). The whole point of his definition lies in encoding infinite potential effects, (e.g. the printing of an infinite decimal) into finite 'tables of behaviour'. There would be no point in allowing machines with infinite 'tables of behaviour'. It is obvious, for instance, that any real number could be printed by such a 'machine', by letting the Nth configuration be 'programmed' to print the Nth digit, for example. Such a 'machine' could likewise store any countable number of statements about all possible mathematical expressions, and so make the Entscheidungsproblem trivial. Church (1937), when reviewing Turing's paper while Turing was in Princeton under his supervision, actually gave a bolder characterisation of the Turing machine as an arbitrary finite machine. > The author [i.e. Turing] proposes as a criterion that an > infinite sequence of digits 0 and 1 be "computable" that it shall be > possible to devise a computing machine, occupying a finite space and > with working parts of finite size, which will write down the sequence > to any desired number of terms if allowed to run for a sufficiently > long time. As a matter of convenience, certain further restrictions are > imposed on the character of the machine, but these are of such a nature > as obviously to cause no loss of generality--in particular, a > human calculator, provided with pencil and paper and explicit > instructions, can be regarded as a kind of Turing machine. Church (1940) repeated this characterisation. Turing neither endorsed it nor said anything to contradict it, leaving the general concept of 'machine' itself undefined. The work of Gandy (1980) did more to justify this characterisation, by refining the statement of what is meant by 'a machine.' His results support Church's statement; they also argue strongly for the view that natural attempts to extend the notion of computability lead to trivialisation: if Gandy's conditions on a 'machine' are significantly weakened then every real number becomes calculable (Gandy 1980, p. 130ff.). (For a different interpretation of Church's statement, see the article on the Church-Turing Thesis.) Turing did not explicitly discuss the question of the speed of his elementary actions. It is left implicit in his discussion, by his use of the word 'never,' that it is not possible for infinitely many steps to be performed in a finite time. Others have explored the effect of abandoning this restriction. Davies (2001), for instance, describes a 'machine' with an infinite number of parts, requiring components of arbitrarily small size, running at arbitrarily high speeds. Such a 'machine' could perform uncomputable tasks. Davies emphasises that such a machine cannot be built in our own physical world, but argues that it could be constructed in a universe with different physics. To the extent that it rules out such 'machines', the Church-Turing thesis must have at least some physical content. True physics is quantum-mechanical, and this implies a different idea of matter and action from Turing's purely classical picture. It is perhaps odd that Turing did not point this out in this period, since he was well versed in quantum physics. Instead, the analysis and practical development of quantum computing was left to the 1980s. Quantum computation, using the evolution of wave-functions rather than classical machine states, is the most important way in which Turing machine model has been challenged. The standard formulation of quantum computing (Deutsch 1985, following Feynman 1982) does not predict anything beyond computable effects, although within the realm of the computable, quantum computations may be very much more efficient than classical computations. It is possible that a deeper understanding of quantum mechanical physics may further change the picture of what can be physically 'done.' ## 4. The Uncomputable Turing turned to the exploration of the uncomputable for his Princeton Ph.D. thesis (1938), which then appeared as Systems of Logic based on Ordinals (Turing 1939). It is generally the view, as expressed by Feferman (1988), that this work was a diversion from the main thrust of his work. But from another angle, as expressed in (Hodges 1997), one can see Turing's development as turning naturally from considering the mind when following a rule, to the action of the mind when not following a rule. In particular this 1938 work considered the mind when seeing the truth of one of Godel's true but formally unprovable propositions, and hence going beyond rules based on the axioms of the system. As Turing expressed it (Turing 1939, p. 198), there are 'formulae, seen intuitively to be correct, but which the Godel theorem shows are unprovable in the original system.' Turing's theory of 'ordinal logics' was an attempt to 'avoid as far as possible the effects of Godel's theorem' by studying the effect of adding Godel sentences as new axioms to create stronger and stronger logics. It did not reach a definitive conclusion. In his investigation, Turing introduced the idea of an 'oracle' capable of performing, as if by magic, an uncomputable operation. Turing's oracle cannot be considered as some 'black box' component of a new class of machines, to be put on a par with the primitive operations of reading single symbols, as has been suggested by (Copeland 1998). An oracle is infinitely more powerful than anything a modern computer can do, and nothing like an elementary component of a computer. Turing defined 'oracle-machines' as Turing machines with an additional configuration in which they 'call the oracle' so as to take an uncomputable step. But these oracle-machines are not purely mechanical. They are only partially mechanical, like Turing's choice-machines. Indeed the whole point of the oracle-machine is to explore the realm of what cannot be done by purely mechanical processes. Turing emphasised (Turing 1939, p. 173): > We shall not go any further into the nature of this oracle > apart from saying that it cannot be a machine. Turing's oracle can be seen simply as a mathematical tool, useful for exploring the mathematics of the uncomputable. The idea of an oracle allows the formulation of questions of relative rather than absolute computability. Thus Turing opened new fields of investigation in mathematical logic. However, there is also a possible interpretation in terms of human cognitive capacity. On this interpretation, the oracle is related to the 'intuition' involved in seeing the truth of a Godel statement. M. H. A. Newman, who introduced Turing to mathematical logic and continued to collaborate with him, wrote in (Newman 1955) that the oracle resembles a mathematician 'having an idea', as opposed to using a mechanical method. However, Turing's oracle cannot actually be identified with a human mental faculty. It is too powerful: it immediately supplies the answer as to whether any given Turing machine is 'satisfactory,' something no human being could do. On the other hand, anyone hoping to see mental 'intuition' captured completely by an oracle, must face the difficulty that Turing showed how his argument for the incompleteness of Turing machines could be applied with equal force to oracle-machines (Turing 1939, p. 173). This point has been emphasised by Penrose (1994, p. 380). Newman's comment might better be taken to refer to the different oracle suggested later on (Turing 1939, p. 200), which has the property of recognising 'ordinal formulae.' One can only safely say that Turing's interest at this time in uncomputable operations appears in the general setting of studying the mental 'intuition' of truths which are not established by following mechanical processes (Turing 1939, p. 214ff.). In Turing's presentation, intuition is in practice present in every part of a mathematician's thought, but when mathematical proof is formalised, intuition has an explicit manifestation in those steps where the mathematician sees the truth of a formally unprovable statement. Turing did not offer any suggestion as to what he considered the brain was physically doing in a moment of such 'intuition'; indeed the word 'brain' did not appear in his writing in this era. This question is of interest because of the views of Penrose (1989, 1990, 1994, 1996) on just this issue: Penrose holds that the ability of the mind to see formally unprovable truths shows that there must be uncomputable physical operations in the brain. It should be noted that there is widespread disagreement about whether the human mind is really seeing the truth of a Godel sentence; see for instance the discussion in (Penrose 1990) and the reviews following it. However Turing's writing at this period accepted without criticism the concept of intuitive recognition of the truth. It was also at this period that Turing met Wittgenstein, and there is a full record of their 1939 discussions on the foundations of mathematics in (Diamond 1976). To the disappointment of many, there is no record of any discussions between them, verbal or written, on the problem of Mind. In 1939 Turing's various energetic investigations were broken off for war work. This did, however, have the positive feature of leading Turing to turn his universal machine into the practical form of the modern digital computer. ## 5. Building a Universal Machine When apprised in 1936 of Turing's idea for a universal machine, Turing's contemporary and friend, the economist David Champernowne, reacted by saying that such a thing was impractical; it would need 'the Albert Hall.' If built from relays as then employed in telephone exchanges, that might indeed have been so, and Turing made no attempt at it. However, in 1937 Turing did work with relays on a smaller machine with a special cryptological function (Hodges 1983, p. 138). World history then led Turing to his unique role in the Enigma problem, to his becoming the chief figure in the mechanisation of logical procedures, and to his being introduced to ever faster and more ambitious technology as the war continued. After 1942, Turing learnt that electronic components offered the speed, storage capacity and logical functions required to be effective as 'tapes' and instruction tables. So from 1945, Turing tried to use electronics to turn his universal machine into practical reality. Turing rapidly composed a detailed plan for a modern stored-program computer: that is, a computer in which data and instructions are stored and manipulated alike. Turing's ideas led the field, although his report of 1946 postdated von Neumann's more famous EDVAC report (von Neumann 1945). It can however be argued, as does Davis (2000), that von Neumann gained his fundamental insight into the computer through his pre-war familiarity with Turing's logical work. At the time, however, these basic principles were not much discussed. The difficulty of engineering the electronic hardware dominated everything. It therefore escaped observers that Turing was ahead of von Neumann and everyone else on the future of software, or as he called it, the 'construction of instruction tables.' Turing (1946) foresaw at once: > Instruction tables will have to be made up by > mathematicians with computing experiences and perhaps a certain > puzzle-solving ability. There will probably be a great deal of work to > be done, for every known process has got to be translated into > instruction table form at some stage. > > > > The process of constructing instruction tables should be very > fascinating. There need be no real danger of it ever becoming a drudge, > for any processes that are quite mechanical may be turned over to the > machine itself. > > > These remarks, reflecting the universality of the computer, and its ability to manipulate its own instructions, correctly described the future trajectory of the computer industry. However, Turing had in mind something greater: 'building a brain.' ## 6. Building a Brain The provocative words 'building a brain' from the outset announced the relationship of Turing's technical computer engineering to a philosophy of Mind. Even in 1936, Turing had given an interpretation of computability in terms of 'states of mind'. His war work had shown the astounding power of the computable in mechanising expert human procedures and judgments. From 1941 onwards, Turing had also discussed the mechanisation of chess-playing and other 'intelligent' activities with his colleagues at Bletchley Park (Hodges 1983, p. 213). But more profoundly, it appears that Turing emerged in 1945 with a conviction that computable operations were sufficient to embrace all mental functions performed by the brain. As will become clear from the ensuing discussion, the uncomputable 'intuition' of 1938 disappeared from Turing's thought, and was replaced by new ideas all lying within the realm of the computable. This change shows even in the technical prospectus of (Turing 1946), where Turing referred to the possibility of making a machine calculate chess moves, and then continued: > This ... raises the question 'Can a machine play > chess?' It could fairly easily be made to play a rather bad game. > It would be bad because chess requires intelligence. We stated ... > that the machine should be treated as entirely without intelligence. > There are indications however that it is possible to make the machine > display intelligence at the risk of its making occasional serious > mistakes. By following up this aspect the machine could probably be > made to play very good chess. The puzzling reference to 'mistakes' is made clear by a talk Turing gave a year later (Turing 1947), in which the issue of mistakes is linked to the issue of the significance of seeing the truth of formally unprovable statements. > ...I would say that fair play must be given to the > machine. Instead of it giving no answer we could arrange that it gives > occasional wrong answers. But the human mathematician would likewise > make blunders when trying out new techniques... In other words > then, if a machine is expected to be infallible, it cannot also be > intelligent. There are several mathematical theorems which say almost > exactly that. But these theorems say nothing about how much > intelligence may be displayed if a machine makes no pretence at > infallibility. Turing's post-war view was that mathematicians make mistakes, and so do not in fact see the truth infallibly. Once the possibility of mistakes is admitted, Godel's theorem become irrelevant. Mathematicians and computers alike apply computable processes to the problem of judging the correctness of assertions; both will therefore sometimes err, since seeing the truth is known not to be a computable operation, but there is no reason why the computer need do worse than the mathematician. This argument is still very much alive. For instance, Davis (2000) endorses Turing's view and attacks Penrose (1989, 1990, 1994, 1996) who argues against the significance of human error on the grounds of a Platonist account of mathematics. Turing also pursued more constructively the question of how computers could be made to perform operations which did not appear to be 'mechanical' (to use common parlance). His guiding principle was that it should be possible to simulate the operation of human brains. In an unpublished report (Turing 1948), Turing explained that the question was that of how to simulate 'initiative' in addition to 'discipline'--comparable to the need for 'intuition' as well as mechanical ingenuity expressed in his pre-war work. He announced ideas for how to achieve this: he thought 'initiative' could arise from systems where the algorithm applied is not consciously designed, but is arrived at by some other means. Thus, he now seemed to think that the mind when not actually following any conscious rule or plan, was nevertheless carrying out some computable process. He suggested a range of ideas for systems which could be said to modify their own programs. These ideas included nets of logical components ('unorganised machines') whose properties could be 'trained' into a desired function. Thus, as expressed by (Ince 1989), he predicted neural networks. However, Turing's nets did not have the 'layered' structure of the neural networks that were to be developed from the 1950s onwards. By the expression 'genetical or evolutionary search', he also anticipated the 'genetic algorithms' which since the late 1980s have been developed as a less closely structured approach to self-modifying programs. Turing's proposals were not well developed in 1948, and at a time when electronic computers were only barely in operation, could not have been. Copeland and Proudfoot (1996) have drawn fresh attention to Turing's connectionist ideas, which have since been tried out (Teuscher 2001). It is important to note that Turing identified his prototype neural networks and genetic algorithms as computable. This has to be emphasised since the word 'nonalgorithmic' is often now confusingly employed for computer operations that are not explicitly planned. Indeed, his ambition was explicit: he himself wanted to implement them as programs on a computer. Using the term Universal Practical Computing Machine for what is now called a digital computer, he wrote in (Turing 1948): > It should be easy to make a model of any particular machine > that one wishes to work on within such a UPCM instead of having to work > with a paper machine as at present. If one also decided on quite > definite 'teaching policies' these could also be programmed > into the machine. One would then allow the whole system to run for an > appreciable period, and then break in as a kind of 'inspector of > schools' and see what progress had been made. One might also be > able to make some progress with unorganised > machines... The upshot of this line of thought is that all mental operations are computable and hence realisable on a universal machine: the computer. Turing advanced this view with increasing confidence in the late 1940s, perfectly aware that it represented what he enjoyed calling 'heresy' to the believers in minds or souls beyond material description. Turing was not a mechanical thinker, or a stickler for convention; far from it. Of all people, he knew the nature of originality and individual independence. Even in tackling the U-boat Enigma problem, for instance, he declared that he did so because no-one else was looking at it and he could have it to himself. Far from being trained or organised into this problem, he took it on despite the prevailing wisdom in 1939 that it was too difficult to attempt. His arrival at a thesis of 'machine intelligence' was not the outcome of some dull or restricted mentality, or a lack of appreciation of individual human creativity. ## 7. Machine Intelligence Turing relished the paradox of 'Machine Intelligence': an apparent contradiction in terms. It is likely that he was already savouring this theme in 1941, when he read a theological book by the author Dorothy Sayers (Sayers 1941). In (Turing 1948) he quoted from this work to illustrate his full awareness that in common parlance 'mechanical' was used to to mean 'devoid of intelligence.' Giving a date which no doubt had his highly sophisticated Enigma-breaking machines secretly in mind, he wrote that 'up to 1940' only very limited machinery had been used, and this 'encouraged the belief that machinery was necessarily limited to extremely straightforward, possibly even to repetitious, jobs.' His object was to dispel these connotations. In 1950, Turing wrote on the first page of his Manual for users of the Manchester University computer (Turing 1950a): > Electronic computers are intended to carry out any definite > rule of thumb process which could have been done by a human operator > working in a disciplined but unintelligent manner. This is, of course, just the 1936 universal Turing machine, now in electronic form. On the other hand, he also wrote in the more famous paper of that year (Turing 1950b, p. 460) > We may hope that machines will eventually compete with men > in all purely intellectual fields. How could the intelligent arise from operations which were themselves totally routine and mindless--'entirely without intelligence'? This is the core of the problem Turing faced, and the same problem faces Artificial Intelligence research today. Turing's underlying argument was that the human brain must somehow be organised for intelligence, and that the organisation of the brain must be realisable as a finite discrete-state machine. The implications of this view were exposed to a wider circle in his famous paper, "Computing Machinery and Intelligence," which appeared in Mind in October 1950. The appearance of this paper, Turing's first foray into a journal of philosophy, was stimulated by his discussions at Manchester University with Michael Polanyi. It also reflects the general sympathy of Gilbert Ryle, editor of Mind, with Turing's point of view. Turing's 1950 paper was intended for a wide readership, and its fresh and direct approach has made it one of the most frequently cited and republished papers in modern philosophical literature. Not surprisingly, the paper has attracted many critiques. Not all commentators note the careful explication of computability which opens the paper, with an emphasis on the concept of the universal machine. This explains why if mental function can be achieved by any finite discrete state machine, then the same effect can be achieved by programming a computer (Turing 1950b, p. 442). (Note, however, that Turing makes no claim that the nervous system should resemble a digital computer in its structure.) Turing's treatment has a severely finitistic flavour: his argument is that the relevant action of the brain is not only computable, but realisable as a totally finite machine, i.e. as a Turing machine that does not use any 'tape' at all. In his account, the full range of computable functions, defined in terms of Turing machines that use an infinite tape, only appears as being of 'special theoretical interest.' (Of uncomputable functions there is, a fortiori, no mention.) Turing uses the finiteness of the nervous system to give an estimate of about 109 bits of storage required for a limited simulation of intelligence (Turing 1950b, p. 455). The wit and drama of Turing's 'imitation game' has attracted more fame than his careful groundwork. Turing's argument was designed to bypass discussions of the nature of thought, mind, and consciousness, and to give a criterion in terms of external observation alone. His justification for this was that one only judges that other human beings are thinking by external observation, and he applied a principle of 'fair play for machines' to argue that the same should hold for machine intelligence. He dramatised this viewpoint by a thought-experiment (which nowadays can readily be tried out). A human being and a programmed computer compete to convince an impartial judge, using textual messages alone, as to which is the human being. If the computer wins, it must be credited with intelligence. Turing introduced his 'game' confusingly with a poor analogy: a party game in which a man pretends to be a woman. His loose wording (Turing 1950b, p. 434) has led some writers wrongly to suppose that Turing proposed an 'imitation game' in which a machine has to imitate a man imitating a woman. Others, like Lassegue (1998), place much weight on this game of gender pretence and its real or imaginary connotations. In fact, the whole point of the 'test' setting, with its remote text-message link, was to separate intelligence from other human faculties and properties. But it may fairly be said that this confusion reflects Turing's richly ambitious concept of what is involved in human 'intelligence'. It might also be said to illustrate his own human intelligence, in particular a delight in the Wildean reversal of roles, perhaps reflecting, as in Wilde, his homosexual identity. His friends knew an Alan Turing in whom intelligence, humour and sex were often intermingled. Turing was in fact sensitive to the difficulty of separating 'intelligence' from other aspects of human senses and actions; he described ideas for robots with sensory attachments and raised questions as to whether they might enjoy strawberries and cream or feel racial kinship. In contrast, he paid scant attention to the questions of authenticity and deception implicit in his test, essentially because he wished to by-pass questions about the reality of consciousness. A subtle aspect of one of his imagined 'intelligent' conversations (Turing 1950b, p. 434) is where the computer imitates human intelligence by giving the wrong answer to a simple arithmetic problem. But in Turing's setting we are not supposed to ask whether the computer 'consciously' deceives by giving the impression of innumerate humanity, nor why it should wish to do so. There is a certain lack of seriousness in this approach. Turing took on a second-rank target in countering the published views of the brain surgeon G. Jefferson, as regards the objectivity of consciousness. Wittgenstein's views on Mind would have made a more serious point of departure. Turing's imitation principle perhaps also assumes (like 'intelligence tests' of that epoch) too much of a shared language and culture for his imagined interrogations. Neither does it address the possibility that there may be kinds of thought, by animals or extra-terrestrial intelligences, which are not amenable to communication. A more positive feature of the paper lies in its constructive program for research, culminating in Turing's ideas for 'learning machines' and educating 'child' machines (Turing 1950b, p. 454). It is generally thought (e.g. in Dreyfus and Dreyfus 1990) that there was always an antagonism between programming and the 'connectionist' approach of neural networks. But Turing never expressed such a dichotomy, writing that both approaches should be tried. Donald Michie, the British AI research pioneer profoundly influenced by early discussions with Turing, has called this suggestion 'Alan Turing's Buried Treasure', in an allusion to a bizarre wartime episode in which Michie was himself involved (Hodges 1983, p. 345). The question is still highly pertinent. It is also a commonly expressed view that Artificial Intelligence ideas only occurred to pioneers in the 1950s after the success of computers in large arithmetical calculations. It is hard to see why Turing's work, which was rooted from the outset in the question of mechanising Mind, has been so much overlooked. But through his failure to publish and promote work such as that in (Turing 1948) he largely lost recognition and influence. It is also curious that Turing's best-known paper should appear in a journal of philosophy, for it may well be said that Turing, always committed to materialist explanation, was not really a philosopher at all. Turing was a mathematician, and what he had to offer philosophy lay in illuminating its field with what had been discovered in mathematics and physics. In the 1950 paper this was surprisingly cursory, apart from his groundwork on the concept of computability. His emphasis on the sufficiency of the computable to explain the action of the mind was stated more as a hypothesis, even a manifesto, than argued in detail. Of his hypothesis he wrote (Turing 1950b, p. 442): > ...I believe that at the end of the century the use of > words and general educated opinion will have altered so much that one > will be able to speak of machines thinking without expecting to be > contradicted. I believe further that no useful purpose is served by > concealing these beliefs. The popular view that scientists proceed > inexorably from established fact to established fact, never being > influenced by any unproved conjecture, is quite mistaken. Provided it > is made clear which are proved facts and which are conjecture, no harm > can result. Conjectures are of great importance since they suggest > useful lines of research. Penrose (1994, p.21), probing into Turing's conjecture, has presented it as 'Turing's thesis' thus: > It seems likely that he viewed physical action in general--which would include the action of a human brain--to be > always reducible to some kind of Turing-machine action. The statement that all physical action is in effect computable goes beyond Turing's explicit words, but is a fair characterisation of the implicit assumptions behind the 1950 paper. Turing's consideration of 'The Argument from Continuity in the Nervous System,' in particular, simply asserts that the physical system of the brain can be approximated as closely as is desired by a computer program (Turing 1950b, p. 451). Certainly there is nothing in Turing's work in the 1945-50 period to contradict Penrose's interpretation. The more technical precursor papers (Turing 1947, 1948) include wide-ranging comments on physical processes, but make no reference to the possibility of physical effects being uncomputable. In particular, a section of (Turing 1948) is devoted to a general classification of 'machines.' The period between 1937 and 1948 had given Turing much more experience of actual machinery than he had in 1936, and his post-war remarks reflected this in a down-to-earth manner. Turing distinguished 'controlling' from 'active' machinery, the latter being illustrated by 'a bulldozer'. Naturally it is the former--in modern terms 'information-based machinery'--with which Turing's analysis is concerned. It is noteworthy that in 1948 as in 1936, despite his knowledge of physics, Turing made no mention of how quantum mechanics might affect the concept of 'controlling'. His concept of 'controlling' remained entirely within the classical framework of the Turing machine (which he called a Logical Computing Machine in this paper.) The same section of (Turing 1948) also drew the distinction between discrete and continuous machinery, illustrating the latter with 'the telephone' as a continuous, controlling machine. He made light of the difficulty of reducing continuous physics to the discrete model of the Turing machine, and though citing 'the brain' as a continuous machine, stated that it could probably be treated as if discrete. He gave no indication that physical continuity threatened the paramount role of computability. In fact, his thrust in (Turing 1947) was to promote the digital computer as more powerful than analog machines such as the differential analyser. When he discussed this comparison, he gave the following informal version of the Church-Turing thesis: > One of my conclusions was that the idea of a 'rule of > thumb' process and a 'machine process' were > synonymous. The expression 'machine process' of course > means one which could be carried out by the type of machine I was > considering [i.e. Turing machines] Turing gave no hint that the discreteness of the Turing machine constituted a real limitation, or that the non-discrete processes of analog machines might be of any deep significance. Turing also introduced the idea of 'random elements' but his examples (using the digits of p) showed that he considered pseudo-random sequences (i.e. computable sequences with suitable 'random' properties) quite adequate for his discussion. He made no suggestion that randomness implied something uncomputable, and indeed gave no definition of the term 'random'. This is perhaps surprising in view of the fact that his work in pure mathematics, logic and cryptography all gave him considerable motivation to approach this question at a serious level. ## 8. Unfinished Work From 1950 Turing worked on a new mathematical theory of morphogenesis, based on showing the consequences of non-linear equations for chemical reaction and diffusion (Turing 1952). He was a pioneer in using a computer for such work. Some writers have referred to this theory as founding Artificial Life (A-life), but this is a misleading description, apt only to the extent that the theory was intended, as Turing saw it, to counter the Argument from Design. A-life since the 1980s has concerned itself with using computers to explore the logical consequences of evolutionary theory without worrying about specific physiological forms. Morphogenesis is complementary, being concerned to show which physiological pathways are feasible for evolution to exploit. Turing's work was developed by others in the 1970s and is now regarded as central to this field. It may well be that Turing's interest in morphogenesis went back to a primordial childhood wonder at the appearance of plants and flowers. But in another late development, Turing went back to other stimuli of his youth. For in 1951 Turing did consider the problem, hitherto avoided, of setting computability in the context of quantum-mechanical physics. In a BBC radio talk of that year (Turing 1951) he discussed the basic groundwork of his 1950 paper, but this time dealing rather less certainly with the argument from Godel's theorem, and this time also referring to the quantum-mechanical physics underlying the brain. Turing described the universal machine property, applying it to the brain, but said that its applicability required that the machine whose behaviour is to be imitated > ...should be of the sort whose behaviour is in > principle predictable by calculation. We certainly do not know how any > such calculation should be done, and it was even argued by Sir Arthur > Eddington that on account of the indeterminacy principle in quantum > mechanics no such prediction is even theoretically > possible. Copeland (1999) has rightly drawn attention to this sentence in his preface to his edition of the 1951 talk. However, Copeland's critical context suggests some connection with Turing's 'oracle.' There is is in fact no mention of oracles here (nor anywhere in Turing's post-war discussion of mind and machine.) Turing here is discussing the possibility that, when seen as as a quantum-mechanical machine rather than a classical machine, the Turing machine model is inadequate. The correct connection to draw is not with Turing's 1938 work on ordinal logics, but with his knowledge of quantum mechanics from Eddington and von Neumann in his youth. Indeed, in an early speculation, influenced by Eddington, Turing had suggested that quantum mechanical physics could yield the basis of free-will (Hodges 1983, p. 63). Von Neumann's axioms of quantum mechanics involve two processes: unitary evolution of the wave function, which is predictable, and the measurement or reduction operation, which introduces unpredictability. Turing's reference to unpredictability must therefore refer to the reduction process. The essential difficulty is that still to this day there is no agreed or compelling theory of when or how reduction actually occurs. (It should be noted that 'quantum computing,' in the standard modern sense, is based on the predictability of the unitary evolution, and does not, as yet, go into the question of how reduction occurs.) It seems that this single sentence indicates the beginning of a new field of investigation for Turing, this time into the foundations of quantum mechanics. In 1953 Turing wrote to his friend and student Robin Gandy that he was 'trying to invent a new Quantum Mechanics but it won't really work.' At Turing's death in June 1954, Gandy reported in a letter to Newman on what he knew of Turing's current work (Gandy 1954). He wrote of Turing having discussed a problem in understanding the reduction process, in the form of > ...'the Turing Paradox'; it is easy to > show using standard theory that if a system start in an eigenstate of > some observable, and measurements are made of that observable N times a > second, then, even if the state is not a stationary one, the > probability that the system will be in the same state after, say, 1 > second, tends to one as N tends to infinity; i.e. that continual > observation will prevent motion. Alan and I tackled one or two > theoretical physicists with this, and they rather pooh-poohed it by > saying that continual observation is not possible. But there is nothing > in the standard books (e.g., Dirac's) to this effect, so that at least the > paradox shows up an inadequacy of Quantum Theory as usually > presented. Turing's investigations take on added significance in view of the assertion of Penrose (1989, 1990, 1994, 1996) that the reduction process must involve something uncomputable. Probably Turing was aiming at the opposite idea, of finding a theory of the reduction process that would be predictive and computable, and so plug the gap in his hypothesis that the action of the brain is computable. However Turing and Penrose are alike in seeing this as an important question affecting the assumption that all mental action is computable; in this they both differ from the mainstream view in which the question is accorded little significance. Alan Turing's last postcards to Robin Gandy, in March 1954, headed 'Messages from the Unseen World' in allusion to Eddington, hinted at new ideas in the fundamental physics of relativity and particle physics (Hodges 1983, p. 512). They illustrate the wealth of ideas with which he was concerned at that last point in his life, but which apart from these hints are entirely lost. A review of such lost ideas is given in (Hodges 2004), as part of a larger volume on Turing's legacy (Teuscher 2004). ## 9. Alan Turing: the Unknown Mind It is a pity that Turing did not write more about his ethical philosophy and world outlook. As a student he was an admirer of Bernard Shaw's plays of ideas, and to friends would openly voice both the hilarities and frustrations of his many difficult situations. Yet the nearest he came to serious personal writing, apart from occasional comments in private letters, was in penning a short story about his 1952 crisis (Hodges 1983, p. 448). His last two years were particularly full of Shavian drama and Wildean irony. In one letter (to his friend Norman Routledge; the letter is now in the Turing Archive at King's College, Cambridge) he wrote: > Turing believes machines think > > > Turing lies with men > > > Therefore machines do not think The syllogistic allusion to Socrates is unmistakeable, and his demise, with cyanide rather than hemlock, may have signalled something similar. A parallel figure in World War II, Robert Oppenheimer, suffered the loss of his reputation during the same week that Turing died. Both combined the purest scientific work and the most effective application of science in war. Alan Turing was even more directly on the receiving end of science, when his sexual mind was treated as a machine, against his protesting consciousness and will. But amidst all this human drama, he left little to say about what he really thought of himself and his relationship to the world of human events. Alan Turing did not fit easily with any of the intellectual movements of his time, aesthetic, technocratic or marxist. In the 1950s, commentators struggled to find discreet words to categorise him: as 'a scientific Shelley,' as possessing great 'moral integrity'. Until the 1970s, the reality of his life was unmentionable. He is still hard to place within twentieth-century thought. He exalted the science that according to existentialists had robbed life of meaning. The most original figure, the most insistent on personal freedom, he held originality and will to be susceptible to mechanisation. The mind of Alan Turing continues to be an enigma. But it is an enigma to which the twenty-first century seems increasingly drawn. The year of his centenary, 2012, witnessed numerous conferences, publications, and cultural events in his honor. Some reasons for this explosion of interest are obvious. One is that the question of the power and limitations of computation now arises in virtually every sphere of human activity. Another is that issues of sexual orientation have taken on a new importance in modern democracies. More subtly, the interdisciplinary breadth of Turing's work is now better appreciated. A landmark of the centenary period was the publication of Alan Turing, his work and impact (eds. Cooper and van Leeuwen, 2013), which made available almost all aspects of Turing's scientific oeuvre, with a wealth of modern commentary. In this new climate, fresh attention has been paid to Turing's lesser-known work, and new light shed upon his achievements. He has emerged from obscurity to become one of the most intensely studied figures in modern science.
turing-machine	## 1. Definitions of the Turing Machine ### 1.1 Turing's Definition Turing introduced Turing machines in the context of research into the foundations of mathematics. More particularly, he used these abstract devices to prove that there is no effective general method or procedure to solve, calculate or compute every instance of the following problem: > > > *Entscheidungsproblem* The problem to decide > for every statement in first-order logic (the so-called restricted > functional calculus, see the entry on > classical logic > for an introduction) whether or not it is derivable in that > logic. > > > Note that in its original form (Hilbert & Ackermann 1928), the problem was stated in terms of validity rather than derivability. Given Godel's completeness theorem (Godel 1929) proving that there is an effective procedure (or not) for derivability is also a solution to the problem in its validity form. In order to tackle this problem, one needs a formalized notion of "effective procedure" and Turing's machines were intended to do exactly that. A Turing machine then, or a computing machine as Turing called it, in Turing's original definition is a machine capable of a finite set of configurations $q\_{1},\ldots,q\_{n}$ (the states of the machine, called m-configurations by Turing). It is supplied with a one-way infinite and one-dimensional tape divided into squares each capable of carrying exactly one symbol. At any moment, the machine is scanning the content of one square r which is either blank (symbolized by $S\_0$) or contains a symbol $S\_{1},\ldots ,S\_{m}$ with $S\_1 = 0$ and $S\_2 = 1$. The machine is an automatic machine (a-machine) which means that at any given moment, the behavior of the machine is completely determined by the current state and symbol (called the configuration) being scanned. This is the so-called determinacy condition (Section 3). These a-machines are contrasted with the so-called choice machines for which the next state depends on the decision of an external device or operator (Turing 1936-7: 232). A Turing machine is capable of three types of action: 1. Print $S\_i$, move one square to the left (L) and go to state $q\_{j}$ 2. Print $S\_i$, move one square to the right (R) and go to state $q\_{j}$ 3. Print $S\_i$, do not move (N) and go to state $q\_{j}$ The 'program' of a Turing machine can then be written as a finite set of quintuples of the form: \[q\_{i}S\_{j}S\_{i,j}M\_{i,j}q\_{i,j}\] Where $q\_i$ is the current state, $S\_j$ the content of the square being scanned, $S\_{i,j}$ the new content of the square; $M\_{i,j}$ specifies whether the machine is to move one square to the left, to the right or to remain at the same square, and $q\_{i,j}$ is the next state of the machine. These quintuples are also called the transition rules of a given machine. The Turing machine $T\_{\textrm{Simple}}$ which, when started from a blank tape, computes the sequence $S\_0S\_1S\_0S\_1\ldots$ is then given by Table 1. Table 1: Quintuple representation of $T\_{\textrm{Simple}}$ \[ \begin{align}\hline ;q\_{1}S\_{0}S\_{0}Rq\_{2}\\ ;q\_{1}S\_{1}S\_{0}Rq\_{2}\\ ;q\_{2}S\_{0}S\_{1}Rq\_{1}\\ ;q\_{2}S\_{1}S\_{1}Rq\_{1}\\\hline \end{align} \] Note that $T\_{\textrm{Simple}}$ will never enter a configuration where it is scanning $S\_1$ so that two of the four quintuples are redundant. Another typical format to represent Turing machines and which was also used by Turing is the transition table. Table 2 gives the transition table of $T\_{\textrm{Simple}}$. Table 2: Transition table for $T\_{\textrm{Simple}}$ \| \| \| \| \| --- \| --- \| --- \| \| \| $S\_0$ \| $S\_1$ \| \| $q\_1$ \| $S\_{0}\opR q\_{2}$ \| $S\_{0}\opR q\_{2}$ \| \| $q\_2$ \| $S\_{1}\opR q\_{1}$ \| $S\_{1}\opR q\_{1}$ \| Where current definitions of Turing machines usually have only one type of symbols (usually just 0 and 1; it was proven by Shannon that any Turing machine can be reduced to a binary Turing machine (Shannon 1956)) Turing, in his original definition of so-called computing machines, used two kinds of symbols: the figures which consist entirely of 0s and 1s and the so-called symbols of the second kind. These are differentiated on the Turing machine tape by using a system of alternating squares of figures and symbols of the second kind. One sequence of alternating squares contains the figures and is called the sequence of F-squares. It contains the sequence computed by the machine; the other is called the sequence of E-squares. The latter are used to mark F-squares and are there to "assist the memory" (Turing 1936-7: 232). The content of the E-squares is liable to change. F-squares however cannot be changed which means that one cannot implement algorithms whereby earlier computed digits need to be changed. Moreover, the machine will never print a symbol on an F-square if the F-square preceding it has not been computed yet. This usage of F and E-squares can be quite useful (see Sec. 2.3) but, as was shown by Emil L. Post, it results in a number of complications (see Sec. 1.2). There are two important things to notice about the Turing machine setup. The first concerns the definition of the machine itself, namely that the machine's tape is potentially infinite. This corresponds to an assumption that the memory of the machine is (potentially) infinite. The second concerns the definition of Turing computable, namely that a function will be Turing computable if there exists a set of instructions that will result in a Turing machine computing the function regardless of the amount of time it takes. One can think of this as assuming the availability of potentially infinite time to complete the computation. These two assumptions are intended to ensure that the definition of computation that results is not too narrow. This is, it ensures that no computable function will fail to be Turing-computable solely because there is insufficient time or memory to complete the computation. It follows that there may be some Turing computable functions which may not be carried out by any existing computer, perhaps because no existing machine has sufficient memory to carry out the task. Some Turing computable functions may not ever be computable in practice, since they may require more memory than can be built using all of the (finite number of) atoms in the universe. If we moreover assume that a physical computer is a finite realization of the Turing machine, and so that the Turing machine functions as a good formal model for the computer, a result which shows that a function is not Turing computable is very strong, since it implies that no computer that we could ever build could carry out the computation. In Section 2.4, it is shown that there are functions which are not Turing-computable. ### 1.2 Post's Definition Turing's definition was standardized through (some of) Post's modifications of it in Post 1947. In that paper Post proves that a certain problem from mathematics known as Thue's problem or the word problem for semi-groups is not Turing computable (or, in Post's words, recursively unsolvable). Post's main strategy was to show that if it were decidable then the following decision problem from Turing 1936-7 would also be decidable: > > > PRINT? The problem to decide for every Turing machine > M whether or not it will ever print some symbol (for instance, > 0). > > > It was however proven by Turing that PRINT? is not Turing computable and so the same is true of Thue's problem. While the uncomputability of PRINT? plays a central role in Post's proof, Post believed that Turing's proof of that was affected by the "spurious Turing convention" (Post 1947: 9), viz. the system of F and E-squares. Thus, Post introduced a modified version of the Turing machine. The most important differences between Post's and Turing's definition are: 1. Post's Turing machine, when in a given state, either prints or moves and so its transition rules are more 'atomic' (it does not have the composite operation of moving and printing). This results in the quadruple notation of Turing machines, where each quadruple is in one of the three forms of Table 3: Table 3: Post's Quadruple notation \[ \begin{aligned}\hline & q\_iS\_jS\_{i,j}q\_{i,j}\\ & q\_iS\_jLq\_{i,j}\\ & q\_iS\_jRq\_{i,j}\\\hline \end{aligned} \] 2. Post's Turing machine has only one kind of symbol and so does not rely on the Turing system of F and E-squares. 3. Post's Turing machine has a two-way infinite tape. 4. Post's Turing machine halts when it reaches a state for which no actions are defined. Note that Post's reformulation of the Turing machine is very much rooted in his Post 1936. (Some of) Post's modifications of Turing's definition became part of the definition of the Turing machine in standard works such as Kleene 1952 and Davis 1958. Since that time, several (logically equivalent) definitions have been introduced. Today, standard definitions of Turing machines are, in some respects, closer to Post's Turing machines than to Turing's machines. In what follows we will use a variant on the standard definition from Minsky 1967 which uses the quintuple notation but has no E and F-squares and includes a special halting state H. It also has only two move operations, viz., L and R and so the action whereby the machine merely prints is not used. When the machine is started, the tape is blank except for some finite portion of the tape. Note that the blank square can also be represented as a square containing the symbol $S\_0$ or simply 0. The finite content of the tape will also be called the dataword on the tape. ### 1.3 The Definition Formalized Talk of "tape" and a "read-write head" is intended to aid the intuition (and reveals something of the time in which Turing was writing) but plays no important role in the definition of Turing machines. In situations where a formal analysis of Turing machines is required, it is appropriate to spell out the definition of the machinery and program in more mathematical terms. Purely formally a Turing machine can be specified as a quadruple $T = (Q,\Sigma, s, \delta)$ where: * Q is a finite set of states q * $\Sigma$ is a finite set of symbols * s is the initial state $s \in Q$ * $\delta$ is a transition function determining the next move: \[\delta : (Q \times \Sigma) \rightarrow (\Sigma \times \{L,R\} \times Q)\] The transition function for the machine T is a function from computation states to computation states. If $\delta(q\_i,S\_j) = (S\_{i,j},D,q\_{i,j})$, then when the machine's state is $q\_j$, reading the symbol $S\_j$, $T$ replaces $S\_j$ by $S\_{i,j}$, moves in direction $D \in \{L,R\}$ and goes to state $q\_{i,j}$. ### 1.4 Describing the Behavior of a Turing Machine We introduce a representation which allows us to describe the behavior or dynamics of a Turing machine $T\_n$, relying on the notation of the complete configuration (Turing 1936-7: 232) also known today as instantaneous description (ID) (Davis 1982: 6). At any stage of the computation of $T\_{i}$ its ID is given by: * (1)the content of the tape, that is, its data word * (2)the location of the reading head * (3)the machine's internal state So, given some Turing machine T which is in state $q\_{i}$ scanning the symbol $S\_{j}$, its ID is given by $Pq\_{i}S\_{j}Q$ where P and Q are the finite words to the left and right hand side of the square containing the symbol $S\_{j}$. Figure 1 gives a visual representation of an ID of some Turing machine T in state $q\_i$ scanning the tape. ![a horizontal strip of concatenated boxed 0s and 1s with the left and right ends of the strip being ragged. The numbers from left to right are 01001100000010000000. The sixth 0 from the left is red and label points to it stating $q_{i} 0 : 0 R q_{i}, 0$ ](TM_basic0.svg) Figure 1: A complete configuration of some Turing machine T The notation thus allows us to capture the developing behavior of the machine and its tape through its consecutive IDs. Figure 2 gives the first few consecutive IDs of $T\_{\textrm{Simple}}$ using a graphical representation. ![a horizontal strip of concatenated boxed 0s with the left and right ends of the strip being ragged. The third 0 from the left is red and there is a label pointing to it stating $q_1 0 : 0 R q_2$ ](Simple1.svg) Figure 2: The dynamics of $T\_{\textrm{Simple}}$ graphical representation The animation can be started by clicking on the picture. One can also explicitly print the consecutive IDs, using their symbolic representations. This results in a state-space diagram of the behavior of a Turing machine. So, for $T\_{\textrm{Simple}}$ we get (Note that $\overline{0}$ means the infinite repetition of 0s): \[\begin{matrix} \overline{0}q\_1{\bf 0}\overline{0}\\ \overline{0}{\color{blue} 0}q\_2{\bf 0}\overline{0}\\ \overline{0}{\color{blue}01}q\_1{\bf 0}\overline{0}\\ \overline{0}{\color{blue}010}q\_2{\bf 0}\overline{0}\\ \overline{0}{\color{blue}0101}q\_1{\bf 0}\overline{0}\\ \overline{0}{\color{blue}01010}q\_2{\bf 0}\overline{0}\\ \vdots \end{matrix}\] ## 2. Computing with Turing Machines As explained in Sec. 1.1, Turing machines were originally intended to formalize the notion of computability in order to tackle a fundamental problem of mathematics. Independently of Turing, Alonzo Church gave a different but logically equivalent formulation (see Sec. 4). Today, most computer scientists agree that Turing's, or any other logically equivalent, formal notion captures all computable problems, viz. for any computable problem, there is a Turing machine which computes it. This is known as the Church-Turing thesis, Turing's thesis (when the reference is only to Turing's work) or Church's thesis (when the reference is only to Church's work). It implies that, if accepted, any problem not computable by a Turing machine is not computable by any finite means whatsoever. Indeed, since it was Turing's ambition to capture "[all] the possible processes which can be carried out in computing a number" (Turing 1936-7: 249), it follows that, if we accept Turing's analysis: * Any problem not computable by a Turing machine is not "computable" in the absolute sense (at least, absolute relative to humans, see Section 3). * For any problem that we believe is computable, we should be able to construct a Turing machine which computes it. To put it in Turing's wording: > > It is my contention that [the] operations [of a computing machine] > include all those which are used in the computation of a number. > (Turing 1936-7: 231) > In this section, examples will be given which illustrate the computational power and boundaries of the Turing machine model. Section 3 then discusses some philosophical issues related to Turing's thesis. ### 2.1 Some (Simple) Examples In order to speak about a Turing machine that does something useful from the human perspective, we will have to provide an interpretation of the symbols recorded on the tape. For example, if we want to design a machine which will compute some mathematical function, addition say, then we will need to describe how to interpret the ones and zeros appearing on the tape as numbers. In the examples that follow we will represent the number n as a block of $n+1$ copies of the symbol '1' on the tape. Thus we will represent the number 0 as a single '1' and the number 3 as a block of four '1's. This is called unary notation. We will also have to make some assumptions about the configuration of the tape when the machine is started, and when it finishes, in order to interpret the computation. We will assume that if the function to be computed requires n arguments, then the Turing machine will start with its head scanning the leftmost '1' of a sequence of n blocks of '1's. The blocks of '1's representing the arguments must be separated by a single occurrence of the symbol '0'. For example, to compute the sum $3+4$, a Turing machine will start in the configuration shown in Figure 3. ![a horizontal strip of concatenated boxed 0s and 1s with the left and right ends of the strip being ragged. The numbers from left to right are 00111101111100000000. The first 1 from the left is red with a label pointing to it stating $q_{1} : j R q_{1}$ ](Initial.svg) Figure 3: Initial configuration for a computation over two numbers n and m Here the supposed addition machine takes two arguments representing the numbers to be added, starting at the leftmost 1 of the first argument. The arguments are separated by a single 0 as required, and the first block contains four '1's, representing the number 3, and the second contains five '1's, representing the number 4. A machine must finish in standard configuration too. There must be a single block of symbols (a sequence of 1s representing some number or a symbol representing another kind of output) and the machine must be scanning the leftmost symbol of that sequence. If the machine correctly computes the function then this block must represent the correct answer. Adopting this convention for the terminating configuration of a Turing machine means that we can compose machines by identifying the final state of one machine with the initial state of the next. ##### Addition of two numbers n and m Table 4 gives the transition table of a Turing machine $T\_{\textrm{Add}\_2}$ which adds two natural numbers n and m. We assume the machine starts in state $q\_1$ scanning the leftmost 1 of $n+1$. Table 4: Transition table for $T\_{\textrm{Add}\_2}$ \| \| \| \| \| --- \| --- \| --- \| \| \| 0 \| 1 \| \| $q\_1$ \| / \| $0\opR q\_2$ \| \| $q\_2$ \| $1\opL q\_3$ \| $1\opR q\_2$ \| \| $q\_3$ \| $0\opR q\_{4}$ \| $1\opL q\_3$ \| \| $q\_4$ \| $/$ \| $0\opR q\_{\textrm{halt}}$ \| The idea of doing an addition with Turing machines when using unary representation is to shift the leftmost number n one square to the right. This is achieved by erasing the leftmost 1 of $n +1$ (this is done in state $q\_1$) and then setting the 0 between $n+1$ and $m+1$ to 1 (state $q\_2$). We then have $n + m + 2$ and so we still need to erase one additional 1. This is done by erasing the leftmost 1 (states $q\_3$ and $q\_4$). Figure 4 shows this computation for $3 + 4$. ![a horizontal strip of concatenated boxed 0s and 1s with the left and right ends of the strip being ragged. The numbers from left to right are 00111101111100000000. The first 1 from the left is red and labeled $q_{1} 1 : 0 R q_{2}$ ](tm1.svg) Figure 4: The computation of $3+4$ by $T\_{\textrm{Add}\_2}$ ##### Addition of n numbers We can generalize $T\_{\textrm{Add}\_2}$ to a Turing machine $T\_{\textrm{Add}\_i}$ for the addition of an arbitrary number i of integers $n\_1, n\_2,\ldots, n\_j$. We assume again that the machine starts in state $q\_1$ scanning the leftmost 1 of $n\_1+1$. The transition table for such a machine $T\_{\textrm{Add}\_i}$ is given in Table 5. Table 5: Transition table for $T\_{\textrm{Add}\_i}$ \| \| \| \| \| --- \| --- \| --- \| \| \| 0 \| 1 \| \| $q\_1$ \| / \| $0\opR q\_2$ \| \| $q\_2$ \| $1\opR q\_3$ \| $1\opR q\_2$ \| \| $q\_3$ \| $0\opL q\_{6}$ \| $1\opL q\_4$ \| \| $q\_4$ \| $0\opR q\_5$ \| $1\opL q\_4$ \| \| $q\_5$ \| / \| $0\opR q\_1$ \| \| $q\_6$ \| $0\opR q\_{\textrm{halt}}$ \| $1\opL q\_6$ \| The machine $T\_{\textrm{Add}\_i}$ uses the principle of shifting the addends to the right which was also used for $T\_{\textrm{Add}\_2}$. More particularly, $T\_{add\_i}$ computes the sum of $n\_1 + 1$, $n\_2 + 1$,... $n\_i+1$ from left to right, viz. it computes this sum as follows: \[\begin{align} N\_1 & = n\_1 + n\_2 + 1\\ N\_2 & = N\_1 + n\_3 \\ N\_3 &= N\_2 + n\_4\\ &\vdots\\ N\_i &= N\_{i-1} + n\_i + 1 \end{align} \] The most important difference between $T\_{\textrm{Add}\_2}$ and $T\_{\textrm{Add}\_i}$ is that $T\_{\textrm{Add}\_i}$ needs to verify if the leftmost addend $N\_j, 1 < j \leq i$ is equal to $N\_i$. This is achieved by checking whether the first 0 to the right of $N\_j$ is followed by another 0 or not (states $q\_2$ and $q\_3$). If it is not the case, then there is at least one more addend $n\_{j+1}$ to be added. Note that, as was the case for $T\_{\textrm{Add}\_2}$, the machine needs to erase an additional one from the addend $n\_{j+1}$ which is done via state $q\_5$. It then moves back to state $q\_1$. If, on the other hand, $N\_j = N\_i$, the machine moves to the leftmost 1 of $N\_i = n\_1 + n\_2 + \ldots + n\_i + 1 $ and halts. ### 2.2 Computable Numbers and Problems Turing's original paper is concerned with computable (real) numbers. A (real) number is Turing computable if there exists a Turing machine which computes an arbitrarily precise approximation to that number. All of the algebraic numbers (roots of polynomials with algebraic coefficients) and many transcendental mathematical constants, such as e and $\pi$ are Turing-computable. Turing gave several examples of classes of numbers computable by Turing machines (see section 10 Examples of large classes of numbers which are computable of Turing 1936-7) as a heuristic argument showing that a wide diversity of classes of numbers can be computed by Turing machines. One might wonder however in what sense computation with numbers, viz. calculation, captures non-numerical but computable problems and so how Turing machines capture all general and effective procedures which determine whether something is the case or not. Examples of such problems are: * "decide for any given x whether or not x denotes a prime" * "decide for any given x whether or not x is the description of a Turing machine". In general, these problems are of the form: * "decide for any given x whether or not x has property X" An important challenge of both theoretical and concrete advances in computing (often at the interface with other disciplines) has become the problem of providing an interpretation of X such that it can be tackled computationally. To give just one concrete example, in daily computational practices it might be important to have a method to decide for any digital "source" whether or not it can be trusted and so one needs a computational interpretation of trust. The characteristic function of a predicate is a function which has the value TRUE or FALSE when given appropriate arguments. In order for such functions to be computable, Turing relied on Godel's insight that these kind of problems can be encoded as a problem about numbers (See Godel's incompleteness theorem and the next Sec. 2.3) In Turing's wording: > > > The expression "there is a general process for determining > ..." has been used [here] [...] as equivalent to > "there is a machine which will determine ...". This > usage can be justified if and only if we can justify our definition of > "computable". For each of these "general > process" problems can be expressed as a problem concerning a > general process for determining whether a given integer n has a > property $G(n)$ [e.g. $G(n)$ might mean "n is > satisfactory" or "n is the Godel > representation of a provable formula"], and this is equivalent > to computing a number whose n-th figure is 1 if $G(n)$ is > true and 0 if it is false. (1936-7: 248) > > > It is the possibility of coding the "general process" problems as numerical problems that is essential to Turing's construction of the universal Turing machine and its use within a proof that shows there are problems that cannot be computed by a Turing machine. ### 2.3 Turing's Universal Machine The universal Turing machine which was constructed to prove the uncomputability of certain problems, is, roughly speaking, a Turing machine that is able to compute what any other Turing machine computes. Assuming that the Turing machine notion fully captures computability (and so that Turing's thesis is valid), it is implied that anything which can be "computed", can also be computed by that one universal machine. Conversely, any problem that is not computable by the universal machine is considered to be uncomputable. This is the rhetorical and theoretical power of the universal machine concept, viz. that one relatively simple formal device captures all "the possible processes which can be carried out in computing a number" (Turing 1936-7). It is also one of the main reasons why Turing has been retrospectively identified as one of the founding fathers of computer science (see Section 5). So how to construct a universal machine U out of the set of basic operations we have at our disposal? Turing's approach is the construction of a machine U which is able to (1) 'understand' the program of any other machine $T\_{n}$ and, based on that "understanding", (2) 'mimic' the behavior of $T\_{n}$. To this end, a method is needed which allows to treat the program and the behavior of $T\_n$ interchangeably since both aspects are manipulated on the same tape and by the same machine. This is achieved by Turing in two basic steps: the development of (1) a notational method (2) a set of elementary functions which treats that notation--independent of whether it is formalizing the program or the behavior of $T\_n$--as text to be compared, copied down, erased, etc. In other words, Turing develops a technique that allows to treat program and behavior on the same level. #### 2.3.1 Interchangeability of program and behavior: a notation Given some machine $T\_n$, Turing's basic idea is to construct a machine $T\_n'$ which, rather than directly printing the output of $T\_n$, prints out the successive complete configurations or instantaneous descriptions of $T\_n$. In order to achieve this, $T\_n'$: > > > [...] could be made to depend on having the rules of operation > [...] of [$T\_n$] written somewhere within itself [...] > each step could be carried out by referring to these rules. (Turing > 1936-7: 242) > > > In other words, $T\_n'$ prints out the successive complete configurations of $T\_n$ by having the program of $T\_n$ written on its tape. Thus, Turing needs a notational method which makes it possible to 'capture' two different aspects of a Turing machine on one and the same tape in such a way they can be treated by the same machine, viz.: * (1) its description in terms of what it should do--the quintuple notation * (2) its description in terms of what it is doing--the complete configuration notation Thus, a first and perhaps most essential step, in the construction of U are the quintuple and complete configuration notation and the idea of putting them on the same tape. More particularly, the tape is divided into two regions which we will call the A and B region here. The A region contains a notation of the 'program' of $T\_n$ and the B region a notation for the successive complete configurations of $T\_n$. In Turing's paper they are separated by an additional symbol "::". To simplify the construction of U and in order to encode any Turing machine as a unique number, Turing develops a third notation which permits to express the quintuples and complete configurations with letters only. This is determined by [Note that we use Turing's original encoding. Of course, there is a broad variety of possible encodings, including binary encodings]: * Replacing each state $q\_i$ in a quintuple of $T\_n$ by \[D\underbrace{A\ldots A}\_i,\] so, for instance $q\_3$ becomes $DAAA$. * Replacing each symbol $S\_{j}$ in a quintuple of $T\_n$ by \[D\underbrace{C\ldots C}\_j,\] so, for instance, $S\_1$ becomes $DC$. Using this method, each quintuple of some Turing machine $T\_n$ can be expressed in terms of a sequence of capital letters and so the 'program' of any machine $T\_{n}$ can be expressed by the set of symbols A, C, D, R, L, N and ;. This is the so-called Standard Description (S.D.) of a Turing machine. Thus, for instance, the S.D. of $T\_{\textrm{Simple}}$ is: ;DADDRDAA;DADCDRDAA;DAADDCRDA;DAADCDCRDA This is, essentially, Turing's version of Godel numbering. Indeed, as Turing shows, one can easily get a numerical description representation or Description Number (D.N.) of a Turing machine $T\_{n}$ by replacing: * "A" by "1" * "C" by "2" * "D" by "3" * "L" by "4" * "R" by "5" * "N" by "6" * ";" by "7" Thus, the D.N. of $T\_{\textrm{Simple}}$ is: 7313353117313135311731133153173113131531 Note that every machine $T\_n$ has a unique D.N.; a D.N. represents one and one machine only. Clearly, the method used to determine the $S.D.$ of some machine $T\_n$ can also be used to write out the successive complete configurations of $T\_n$. Using ":" as a separator between successive complete configurations, the first few complete configurations of $T\_{\textrm{Simple}}$ are: :DAD:DDAAD:DDCDAD:DDCDDAAD:DDCDDCDAD #### 2.3.2 Interchangeability of program and behavior: a basic set of functions Having a notational method to write the program and successive complete configurations of some machine $T\_n$ on one and the same tape of some other machine $T\_n'$ is the first step in Turing's construction of U. However, U should also be able to "emulate" the program of $T\_n$ as written in region A so that it can actually write out its successive complete configurations in region B. Moreover it should be possible to "take out and exchange[...] [the rules of operations of some Turing machine] for others" (Turing 1936-7: 242). Viz., it should be able not just to calculate but also to compute, an issue that was also dealt with by others such as Church, Godel and Post using their own formal devices. It should, for instance, be able to "recognize" whether it is in region A or B and it should be able to determine whether or not a certain sequence of symbols is the next state $q\_i$ which needs to be executed. This is achieved by Turing through the construction of a sequence of Turing computable problems such as: * Finding the leftmost or rightmost occurrence of a sequence of symbols * Marking a sequence of symbols by some symbol a (remember that Turing uses two kinds of alternating squares) * Comparing two symbol sequences * Copying a symbol sequence Turing develops a notational technique, called skeleton tables, for these functions which serves as a kind of shorthand notation for a complete Turing machine table but can be easily used to construct more complicated machines from previous ones. The technique is quite reminiscent of the recursive technique of composition (see: recursive functions). To illustrate how such functions are Turing computable, we discuss one such function in more detail, viz. the compare function. It is constructed on the basis of a number of other Turing computable functions which are built on top of each other. In order to understand how these functions work, remember that Turing used a system of alternating F and E-squares where the F-squares contain the actual quintuples and complete configurations and the E-squares are used as a way to mark off certain parts of the machine tape. For the comparing of two sequences $S\_1$ and $S\_2$, each symbol of $S\_1$ will be marked by some symbol a and each symbol of $S\_2$ will be marked by some symbol b. Turing defined nine different functions to show how the compare function can be computed with Turing machines: * FIND$(q\_{i}, q\_{j},a)$: this machine function searches for the leftmost occurrence of a. If a is found, the machine moves to state $q\_{i}$ else it moves to state $q\_{j}$. This is achieved by having the machine first move to the beginning of the tape (indicated by a special mark) and then to have it move right until it finds a or reaches the rightmost symbol on the tape. * FINDL$(q\_{i}, q\_{j},a)$: the same as FIND but after a has been found, the machine moves one square to the left. This is used in functions which need to compute on the symbols in F-squares which are marked by symbols a in the E-squares. * ERASE$(q\_{i},q\_{j},a)$: the machine computes FIND. If a is found, it erases a and goes to state $q\_{i}$ else it goes to state $q\_{j}$ * ERASE\_ALL$(q\_j,a) = \textrm{ERASE}(\textrm{ERASE}\\_\textrm{ALL}, q\_j,a)$: the machines computes ERASE on a repeatedly until all a's have been erased. Then it moves to $q\_{j}$. * EQUAL$(q\_i,q\_j,a)$: the machine checks whether or not the current symbol is a. If yes, it moves to state $q\_i$ else it moves to state $q\_j$ * CMP\_XY$(q\_i,q\_j,b) = \textrm{FINDL(EQUAL}(q\_i,q\_j,x), q\_j, b)$: whatever the current symbol x, the machine computes FINDL on b (and so looks for the symbol marked by b). If there is a symbol y marked with b, the machine computes $\textrm{EQUAL}$ on x and y, else, the machine goes to state $q\_j$. In other words, CMP\_XY$(q\_i,q\_j,b)$ compares whether the current symbol is the same as the leftmost symbol marked b. * COMPARE\_MARKED$(q\_i,q\_j,q\_n,a,b)$: the machine checks whether the leftmost symbols marked a and b respectively are the same. If there is no symbol marked a nor b, the machine goes to state $q\_{n}$; if there is a symbol marked a and one marked b and they are the same, the machine goes to state $q\_i$, else the machine goes to state $q\_j$. The function is computed as $\textrm{FINDL(CMP}\\_XY(q\_i,q\_j,b), \textrm{FIND}(q\_j,q\_n,b),a)$ * $\textrm{COMPARE}\\_\textrm{ERASE}(q\_iq\_j,q\_n,a,b)$: the same as COMPARE\_MARKED but when the symbols marked a and b are the same, the marks a and b are erased. This is achieved by computing $\textrm{ERASE}$ first on a and then on b. * $\textrm{COMPARE}\\_\textrm{ALL}(q\_j,q\_n,a,b)$ The machine compares the sequences A and B marked with a and b respectively. This is done by repeatedly computing COMPARE\_ERASE on a and b. If A and B are equal, all a's and b's will have been erased and the machine moves to state $q\_j$, else, it will move to state $q\_n$. It is computed by \[\textrm{COMPARE}\\_\textrm{ERASE}(\textrm{COMPARE}\\_\textrm{ALL}(q\_j,q\_n,a,b),q\_j,q\_n,a,b)\] and so by recursively calling $\textrm{COMPARE}\\_\textrm{ALL}$. In a similar manner, Turing defines the following functions: * $\textrm{COPY}(q\_i,a)$: copy the sequence of symbols marked with a's to the right of the last complete configuration and erase the marks. * $\textrm{COPY}\_{n}(q\_i, a\_1,a\_2,\ldots ,a\_n)$: copy down the sequences marked $a\_1$ to $a\_n$ to the right of the last complete configuration and erase all marks $a\_i$. * $\textrm{REPLACE}(q\_i, a,b)$: replace all letters a by b * $\textrm{MARK\_NEXT\_CONFIG}(q\_i,a)$: mark the first configuration $q\_iS\_j$ to the right of the machine's head with the letter a. * $\textrm{FIND}\\_\textrm{RIGHT}(q\_i,a)$: find the rightmost symbol a. Using the basic functions COPY, REPLACE and COMPARE, Turing constructs a universal Turing machine. Below is an outline of the universal Turing machine indicating how these basic functions indeed make possible universal computation. It is assumed that upon initialization, U has on its tape the S.D. of some Turing machine $T\_n$. Remember that Turing uses the system of alternating F and E-squares and so, for instance, the S.D. of $T\_{\textrm{Simple}}$ will be written on the tape of U as: ;\_D\_A\_D\_D\_R\_D\_A\_A\_;\_D\_A\_D\_C\_D\_R\_D\_A\_A\_;\_D\_A\_A\_D\_D\_C\_R\_D\_A\_;\_D\_A\_A\_D\_C\_D\_C\_R\_D\_A\_ where "\_" indicates an unmarked E-square. * INIT: To the right of the rightmost quintuple of T\_n, U prints ::\_:\_D\_A\_, where \_ indicates an unmarked E-square. * FIND\_NEXT\_STATE: The machine first marks (1) with y the configuration $q\_{CC,i}S\_{CC,j}$ of the rightmost (and so last) complete configuration computed by U in the B part of the tape and (2) with x the configuration $q\_{q,m}S\_{q,n}$ of the leftmost quintuple which is not preceded by a marked (with the letter z) semicolon in the A part of the tape. The two configurations are compared. If they are identical, the machine moves to MARK\_OPERATIONS, if not, it marks the semicolon preceding $q\_{q,m}S\_{q,n}$ with z and goes to FIND\_NEXT\_STATE. This is easily achieved using the function COMPARE\_ALL which means that, whatever the outcome of the comparison, the marks x and y will be erased. For instance, suppose that $T\_n = T\_{\textrm{Simple}}$ and that the last complete configuration of $T\_{\textrm{Simple}}$ as computed by U is: \[\tag{1} \label{CC\_univ} :\\_\underbrace{D\\_}\_{S\_0}\underbrace{D\\_C\\_}\_{S\_1}\underbrace{D\\_}\_{S\_0}\textcolor{orange}{\underbrace{D\\_A\\_A\\_}\_{q\_{2}}\underbrace{D\\_}\_{S\_0}} \] Then U will move to region A and determine that the corresponding quintuple is: \[\tag{2}\label{quint\_univ} \textcolor{orange}{\underbrace{D\\_A\\_A\\_}\_{q\_{2}}\underbrace{D\\_}\_{S\_{0}}}\underbrace{D\\_C\\_}\_{S\_1}\underbrace{R\\_}\underbrace{D\\_A\\_}\_{q\_1}\] * MARK\_OPERATIONS: The machine U marks the operations that it needs to execute in order to compute the next complete configuration of $T\_n$. The printing and move (L,R, N) operations are marked with u and the next state with y. All marks z are erased. Continuing with our example, U will mark $\eqref{quint\_univ}$ as follows: \[D\\_A\\_A\\_D\\_\textcolor{magenta}{DuCuRu}\textcolor{green}{DyAy}\] * MARK\_COMPCONFIG: The last complete configuration of $T\_n$ as computed by U is marked into four regions: the configuration $q\_{CC,i}S\_{CC,j}$ itself is left unmarked; the symbol just preceding it is marked with an x and the remaining symbols to the left or marked with v. Finally, all symbols to the right, if any, are marked with w and a ":" is printed to the right of the rightmost symbol in order to indicate the beginning of the next complete configuration of $T\_n$ to be computed by U. Continuing with our example, $\eqref{CC\_univ}$ will be marked as follows by U: \[\textcolor{red}{\underbrace{Dv}\_{S\_0}\underbrace{DvCv}\_{S\_1}}\textcolor{blue}{\underbrace{Dx}\_{S\_0}}\underbrace{D\\_A\\_A\\_}\_{q\_2}\underbrace{D\\_}\_{S\_0}:\\_\] U then goes to PRINT * PRINT. It is determined if, in the instructions that have been marked in MARK\_OPERATIONS, there is an operation Print 0 or Print 1. If that is the case, $0:$ respectively $1:$ is printed to the right of the last complete configuration. This is not a necessary function but Turing insisted on having U print out not just the (coded) complete configurations computed by $T\_n$ but also the actual (binary) real number computed by $T\_n$. * PRINT\_COMPLETE\_CONFIGURATION. U prints the next complete configuration and erases all marks u, v, w, x, y. It then returns to FIND\_NEXT\_STATE. U first searches for the rightmost letter u, to check which move is needed (R, L, N) and erases the mark u for R, L, N. Depending on the value L, R or N will then write down the next complete configuration by applying COPY$\_5$ to u, v, w, x, y. The move operation (L, R, N) is accounted for by the particular combination of u, v, w, x, y: \[\begin{array}{ll} \textrm{When ~} L: & \textrm{COPY}\_5(\textrm{FIND}\\_\textrm{NEXT}\\_\textrm{STATE}, \textcolor{red}{v},\textcolor{green}{y},\textcolor{blue}{x},\textcolor{magenta}{u},\textcolor{RawSienna}{w})\\ \textrm{When ~} R: & \textrm{COPY}\_5(\textrm{FIND}\\_\textrm{NEXT}\\_\textrm{STATE}, \textcolor{red}{v},\textcolor{blue}{x},\textcolor{magenta}{u},\textcolor{green}{y},\textcolor{RawSienna}{w})\\ \textrm{When ~} N: & \textrm{COPY}\_5(\textrm{FIND}\\_\textrm{NEXT}\\_\textrm{STATE}, \textcolor{red}{v},\textcolor{blue}{x},\textcolor{green}{y},\textcolor{magenta}{u},\textcolor{RawSienna}{w}) \end{array}\] Following our example, since $T\_{\textrm{Simple}}$ needs to move right, the new rightmost complete configursiation of $T\_{\textrm{Simple}}$ written on the tape of U is: \[\textcolor{red}{\underbrace{D\\_}\_{S\_0}\underbrace{D\\_C\\_}\_{S\_1}}\textcolor{blue}{\underbrace{D\\_}\_{S\_0}}\textcolor{magenta}{\underbrace{D\\_C\\_}\_{S\_1}}\textcolor{green}{\underbrace{D\\_A\\_}\_{q\_1}} \] Since we have that for this complete configuration the square being scanned by $T\_{\textrm{Simple}}$ is one that was not included in the previous complete configuration (viz. $T\_{\textrm{Simple}}$ has reached beyond the rightmost previous point) the complete configuration as written out by U is in fact incomplete. This small defect was corrected by Post (Post 1947) by including an additional instruction in the function used to mark the complete configuration in the next round. As is clear, Turing's universal machine indeed requires that program and 'data' produced by that program are manipulated interchangeably, viz. the program and its productions are put next to each other and treated in the same manner, as sequences of letters to be copied, marked, erased and compared. Turing's particular construction is quite intricate with its reliance on the F and E-squares, the use of a rather large set of symbols and a rather arcane notation used to describe the different functions discussed above. Since 1936 several modifications and simplifications have been implemented. The removal of the difference between F and E-squares was already discussed in Section 1.2 and it was proven by Shannon that any Turing machine, including the universal machine, can be reduced to a binary Turing machine (Shannon 1956). Since the 1950s, there has been quite some research on what could be the smallest possible universal devices (with respect to the number of states and symbols) and quite some "small" universal Turing machines have been found. These results are usually achieved by relying on other equivalent models of computability such as, for instance, tag systems. For a survey on research into small universal devices (see Margenstern 2000; Woods & Neary 2009). ### 2.4 The Halting Problem and the Entscheidungsproblem As explained, the purpose of Turing's paper was to show that the Entscheidungsproblem for first-order logic is not computable. The same result was achieved independently by Church (1936a, 1936b) using a different kind of formal device which is logically equivalent to a Turing machine (see Sec. 4). The result went very much against what Hilbert had hoped to achieve with his finitary and formalist program. Indeed, next to Godel's incompleteness results, they broke much of Hilbert's dream of making mathematics void of Ignorabimus and which was explicitly expressed in the following words of Hilbert: > > > The true reason why Comte could not find an unsolvable problem, lies > in my opinion in the assertion that there exists no unsolvable > problem. Instead of the stupid Ignorabimus, our solution should be: We > must know. We shall know. (1930: 963) [translation by the author] > > > Note that the solvability Hilbert is referring to here concerns solvability of mathematical problems in general and not just mechanically solvable. It is shown however in Mancosu et al. 2009 (p. 94), that this general aim of solving every mathematical problem, underpins two particular convictions of Hilbert namely that (1) the axioms of number theory are complete and (2) that there are no undecidable problems in mathematics. #### 2.4.1 Direct and indirect proofs of uncomputable decision problems So, how can one show, for a particular decision problem $\textrm{D}\_i$, that it is not computable? There are two main methods: * Indirect proof: take some problem $\textrm{D}\_{\textrm{uncomp}}$ which is already known to be uncomputable and show that the problem "reduces" to $\textrm{D}\_{i}$. * Direct proof: prove the uncomputability of $\textrm{D}\_{i}$ directly by assuming some version of the Church-Turing thesis. Today, one usually relies on the first method while it is evident that in the absence of a problem $\textrm{D}\_{\textrm{uncomp}}$, Turing but also Church and Post (see Sec. 4) had to rely on the direct approach. The notion of reducibility has its origins in the work of Turing and Post who considered several variants of computability (Post 1947; Turing 1939). The concept was later appropriated in the context of computational complexity theory and is today one of the basic concepts of both computability and computational complexity theory (Odifreddi 1989; Sipser 1996). Roughly speaking, a reduction of a problem $D\_i$ to a problem $D\_j$ comes down to providing an effective procedure for translating every instance $d\_{i,m}$ of the problem $D\_i$ to an instance $d\_{j,n}$ of $D\_j$ in such a way that an effective procedure for solving $d\_{j,n}$ also yields an effective procedure for solving $d\_{i,m}$. In other words, if $D\_i$ reduces to $D\_j$ then, if $D\_i$ is uncomputable so is $D\_j$. Note that the reduction of one problem to another can also be used in decidability proofs: if $D\_i$ reduces to $D\_j$ and $D\_j$ is known to be computable then so is $D\_i$. In the absence of D$\_{\textrm{uncomp}}$ a very different approach was required and Church, Post and Turing each used more or less the same approach to this end (Gandy 1988). First of all, one needs a formalism which captures the notion of computability. Turing proposed the Turing machine formalism to this end. A second step is to show that there are problems that are not computable within the formalism. To achieve this, a uniform process U needs to be set-up relative to the formalism which is able to compute every computable number. One can then use (some form of) diagonalization in combination with U to derive a contradiction. Diagonalization was introduced by Cantor to show that the set of real numbers is "uncountable" or not denumerable. A variant of the method was used also by Godel in the proof of his first incompleteness theorem. #### 2.4.2 Turing's basic problem CIRC?, PRINT? and the Entscheidungsproblem Recall that in Turing's original version of the Turing machine, the machines are computing real numbers. This implied that a "well-behaving" Turing machine should in fact never halt and print out an infinite sequence of figures. Such machines were identified by Turing as circle-free. All other machines are called circular machines. A number n which is the D.N. of a circle-free machine is called satisfactory. This basic difference is used in Turing's proof of the uncomputability of: > > > CIRC? The problem to decide for every number n > whether or not it is satisfactory > > > The proof of the uncomputability of CIRC? uses the construction of a hypothetical and circle-free machine $T\_{decide}$ which computes the diagonal sequence of the set of all computable numbers computed by the circle-free machines. Hence, it relies for its construction on the universal Turing machine and a hypothetical machine that is able to decide CIRC? for each number n given to it. It is shown that the machine $T\_{decide}$ becomes a circular machine when it is provided with its own description number, hence the assumption of a machine which is capable of solving CIRC? must be false. Based on the uncomputability of CIRC?, Turing then shows that also PRINT? is not computable. More particularly he shows that if PRINT? were to be computable, also CIRC? would be decidable, viz. he rephrases PRINT? in such a way that it becomes the problem to decide for any machine whether or not it will print an infinity of symbols which would amount to deciding CIRC?. Finally, based on the uncomputability of PRINT? Turing shows that the Entscheidungsproblem is not decidable. This is achieved by showing: 1. how for each Turing machine T, it is possible to construct a corresponding formula T in first-order logic and 2. if there is a general method for determining whether T is provable, then there is a general method for proving that T will ever print 0. This is the problem PRINT? and so cannot be decidable (provided we accept Turing's thesis). It thus follows from the uncomputability of PRINT?, that the Entscheidungsproblem is not computable. #### 2.4.3 The halting problem Given Turing's focus on computable real numbers, his base decision problem is about determining whether or not some Turing machine will not halt and so is not quite the same as the more well-known halting problem: * HALT? The problem to decide for every Turing machine T whether or not T will halt. Turing's problem PRINT? is in fact very close to HALT? (see Davis 1958: Chapter 5, Theorem 2.3). A popular proof of HALT? goes as follows. Assume that HALT? is computable. Then it should be possible to construct a Turing machine which decides, for each machine $T\_i$ and some input w for $T\_i$ whether or not $T\_i$ will halt on w. Let us call this machine $T\_{H}$. More particularly, we have: \[ T\_H(T\_i,w) = \left\{ \begin{array}{ll} \textrm{HALT} & \textrm{if $T\_i$ halts on } w\\ \textrm{LOOP} & \textrm{if $T\_i$ loops on } w \end{array} \right. \] We now define a second machine $T\_D$ which relies on the assumption that the machine $T\_H$ can be constructed. More particularly, we have: \[ T\_D(T\_i,D.N.~of~ T\_i) = \left\{ \begin{array}{ll} \textrm{HALT} & \textrm{if $T\_i$ does not halt on its own} \\ & \qquad \textrm{description number}\\ \textrm{LOOP} & \textrm{if $T\_i$ halts on its own} \\ & \qquad \textrm{description number}\\ \end{array} \right. \] If we now set $T\_i$ to $T\_D$ we end up with a contradiction: if $T\_D$ halts it means that $T\_D$ does not halt and vice versa. A popular but quite informal variant of this proof was given by Christopher Strachey in the context of programming (Strachey 1965). ### 2.5 Variations on the Turing machine As is clear from Sections 1.1 and 1.2, there is a variety of definitions of the Turing machine. One can use a quintuple or quadruple notation; one can have different types of symbols or just one; one can have a two-way infinite or a one-way infinite tape; etc. Several other less obvious modifications have been considered and used in the past. These modifications can be of two kinds: generalizations or restrictions. These do not result in "stronger" or "weaker" models. Viz. these modified machines compute no more and no less than the Turing computable functions. This adds to the robustness of the Turing machine definition. ##### Binary machines In his short 1936 note Post considers machines that either mark or unmark a square which means we have only two symbols $S\_0$ and $S\_1$ but he did not prove that this formulation captures exactly the Turing computable functions. It was Shannon who proved that for any Turing machine T with n symbols there is a Turing machine with two symbols that simulates T (Shannon 1956). He also showed that for any Turing machine with m states, there is a Turing machine with only two states that simulates it. ##### Non-erasing machines Non-erasing machines are machines that can only overprint $S\_0$. In Moore 1952, it was mentioned that Shannon proved that non-erasing machines can compute what any Turing machine computes. This result was given in a context of actual digital computers of the 50s which relied on punched tape (and so, for which, one cannot erase). Shannon's result however remained unpublished. It was Wang who published the result (Wang 1957). ##### Non-writing machines It was shown by Minsky that for every Turing machine there is a non-writing Turing machine with two tapes that simulates it. ##### Multiple tapes Instead of one tape one can consider a Turing machine with multiple tapes. This turned out the be very useful in several different contexts. For instance, Minsky, used two-tape non-writing Turing machines to prove that a certain decision problem defined by Post (the decision problem for tag systems) is non-Turing computable (Minsky 1961). Hartmanis and Stearns then, in their founding paper for computational complexity theory, proved that any n-tape Turing machine reduces to a single tape Turing machine and so anything that can be computed by an n-tape or multitape Turing machine can also be computed by a single tape Turing machine, and conversely (Hartmanis & Stearns 1965). They used multitape machines because they were considered to be closer to actual digital computers. ##### n-dimensional Turing machines Another variant is to consider Turing machines where the tape is not one-dimensional but n-dimensional. This variant too reduces to the one-dimensional variant. ##### Non-deterministic machines An apparently more radical reformulation of the notion of Turing machine is that of non-deterministic Turing machines. As explained in 1.1, one fundamental condition of Turing's machines is the so-called determinacy condition, viz. the idea that at any given moment, the machine's behavior is completely determined by the configuration or state it is in and the symbol it is scanning. Next to these, Turing also mentions the idea of choice machines for which the next state is not completely determined by the state and symbol pair. Instead, some external device makes a random choice of what to do next. Non-deterministic Turing machines are a kind of choice machines: for each state and symbol pair, the non-deterministic machine makes an arbitrary choice between a finite (possibly zero) number of states. Thus, unlike the computation of a deterministic Turing machine, the computation of a non-deterministic machine is a tree of possible configuration paths. One way to visualize the computation of a non-deterministic Turing machine is that the machine spawns an exact copy of itself and the tape for each alternative available transition, and each machine continues the computation. If any of the machines terminates successfully, then the entire computation terminates and inherits that machine's resulting tape. Notice the word successfully in the preceding sentence. In this formulation, some states are designated as accepting states and when the machine terminates in one of these states, then the computation is successful, otherwise the computation is unsuccessful and any other machines continue in their search for a successful outcome. The addition of non-determinism to Turing machines does not alter the extent of Turing-computability. Non-determinism was introduced for finite automata in the paper, Rabin & Scott 1959, where it is also shown that adding non-determinism does not result in more powerful automata. Non-deterministic Turing machines are an important model in the context of computational complexity theory. ##### Weak and semi-weak machines Weak Turing machines are machines where some word over the alphabet is repeated infinitely often to the left and right of the input. Semi-weak machines are machines where some word is repeated infinitely often either to the left or right of the input. These machines are generalizations of the standard model in which the initial tape contains some finite word (possibly nil). They were introduced to determine smaller universal machines. Watanabe was the first to define a universal semi-weak machine with six states and five symbols (Watanabe 1961). Recently, a number of researchers have determined several small weak and semi-weak universal Turing machines (e.g., Woods & Neary 2007; Cook 2004) Besides these variants on the Turing machine model, there are also variants that result in models which capture, in some well-defined sense, more than the (Turing)-computable functions. Examples of such models are oracle machines (Turing 1939), infinite-time Turing machines (Hamkins & Lewis 2008) and accelerating Turing machines (Copeland 2002). There are various reasons for introducing such stronger models. Some are well-known models of computability or recursion theory and are used in the theory of higher-order recursion and relative computability (oracle machines); others, like the accelerating machines, were introduced in the context of supertasks and the idea of providing physical models that "compute" functions which are not Turing-computable. ## 3. Philosophical Issues Related to Turing Machines ### 3.1 Human and Machine Computations In its original context, Turing's identification between the computable numbers and Turing machines was aimed at proving that the Entscheidungsproblem is not a computable problem and so not a so-called "general process" problem (Turing 1936-7: 248). The basic assumption to be made for this result is that our "intuitive" notion of computability can be formally defined as Turing computability and so that there are no "computable" problems that are not Turing computable. But what was Turing's "intuitive" notion of computability and how can we be sure that it really covers all computable problems, and, more generally, all kinds of computations? This is a very basic question in the philosophy of computer science. At the time Turing was writing his paper, the modern computer was not developed yet and so rephrasings of Turing's thesis which identify Turing computability with computability by a modern computer are interpretations rather than historically correct statements of Turing's thesis. The existing computing machines at the time Turing wrote his paper, such as the differential analyzer or desk calculators, were quite restricted in what they could compute and were used in a context of human computational practices (Grier 2007). It is thus not surprising that Turing did not attempt to formalize machine computation but rather human computation and so computable problems in Turing's paper become computable by human means. This is very explicit in Section 9 of Turing 1936-7 where he shows that Turing machines are a 'natural' model of (human) computation by analyzing the process of human computation. The analysis results in a kind of abstract human 'computor' who fulfills a set of different conditions that are rooted in Turing's recognition of a set of human limitations which restrict what we can compute (of our sensory apparatus but also of our mental apparatus). This 'computor' computes (real) numbers on an infinite one-dimensional tape divided into squares [Note: Turing assumed that the reduction of the 2-dimensional character of the paper a human mathematician usually works on "is not essential of computation" (Turing 1936-7: 249)]. It has the following restrictions (Gandy 1988; Sieg 1994): * Determinacy condition D "The behaviour of the computer at any moment is determined by the symbols which he is observing and his 'state of mind' at that moment." (Turing 1936-7: 250) * Boundedness condition B1 "there is a bound B to the number of symbols or squares which the computer can observe at one moment. If he wishes to observe more, he must use successive observations." (Turing 1936-7: 250) * Boundedness condition B2 "the number of states of mind which need be taken into account is finite" (Turing 1936-7: 250) * Locality condition L1 "We may [...] assume that the squares whose symbols are changed are always 'observed' squares." (Turing 1936-7: 250) * Locality condition L2 "each of the new observed squares is within L squares of an immediately previously observed square." (Turing 1936-7: 250) It is this so-called "direct appeal to intuition" (1936-7: 249) of Turing's analysis and resulting model that explain why the Turing machine is today considered by many as the best standard model of computability (for a strong statement of this point of view, see Soare 1996). Indeed, from the above set of conditions one can quite easily derive Turing's machines. This is achieved basically by analyzing the restrictive conditions into "'simple operations' which are so elementary that it is not easy to imagine them further divided" (Turing 1936-7: 250). Note that while Turing's analysis focuses on human computation, the application of his identification between (human) computation and Turing machine computation to the Entscheidungsproblem suggests that he did not consider the possibility of a model of computation that somehow goes "beyond" human computation and is capable of providing an effective and general procedure which solves the Entscheidungsproblem. If that would have been the case, he would not have considered the Entscheidungsproblem to be uncomputable. The focus on human computation in Turing's analysis of computation, has led researchers to extend Turing's analysis to computation by physical devices. This results in (versions of) the physical Church-Turing thesis. Robin Gandy focused on extending Turing's analysis to discrete mechanical devices (note that he did not consider analog machines). More particularly, like Turing, Gandy starts from a basic set of restrictions of computation by discrete mechanical devices and, on that basis, develops a new model which he proved to be reducible to the Turing machine model. This work is continued by Wilfried Sieg who proposed the framework of Computable Dynamical Systems (Sieg 2008). Others have considered the possibility of "reasonable" models from physics which "compute" something that is not Turing computable. See for instance Aaronson, Bavarian, & Gueltrini 2016 (Other Internet Resources) in which it is shown that if closed timelike curves would exist, the halting problem would become solvable with finite resources. Others have proposed alternative models for computation which are inspired by the Turing machine model but capture specific aspects of current computing practices for which the Turing machine model is considered less suited. One example here are the persistent Turing machines intended to capture interactive processes. Note however that these results do not show that there are "computable" problems that are not Turing computable. These and other related proposals have been considered by some authors as reasonable models of computation that somehow compute more than Turing machines. It is the latter kind of statements that became affiliated with research on so-called hypercomputation resulting in the early 2000s in a rather fierce debate in the computer science community, see, e.g., Teuscher 2004 for various positions. ### 3.2 Thesis, Definition, Axioms or Theorem As is clear, strictly speaking, Turing's thesis is not provable, since, in its original form, it is a claim about the relationship between a formal and a vague or intuitive concept. By consequence, many consider it as a thesis or a definition. The thesis would be refuted if one would be able to provide an intuitively acceptable effective procedure for a task that is not Turing-computable. This far, no such counterexample has been found. Other independently defined notions of computability based on alternative foundations, such as recursive functions and abacus machines have also been shown to be equivalent to Turing computability. These equivalences between quite different formulations indicate that there is a natural and robust notion of computability underlying our understanding. Given this apparent robustness of our notion of computability, some have proposed to avoid the notion of a thesis altogether and instead propose a set of axioms used to sharpen the informal notion. There are several approaches, most notably, an approach of structural axiomatization where computability itself is axiomatized (Sieg 2008) and one whereby an axiomatization is given from which the Church-Turing thesis can be derived (Dershowitz & Gurevich 2008). ## 4. Alternative Historical Models of Computability Besides the Turing machine, several other models were introduced independently of Turing in the context of research into the foundation of mathematics which resulted in theses that are logically equivalent to Turing's thesis. For each of these models it was proven that they capture the Turing computable functions. Note that the development of the modern computer stimulated the development of other models such as register machines or Markov algorithms. More recently, computational approaches in disciplines such as biology or physics, resulted in bio-inspired and physics-inspired models such as Petri nets or quantum Turing machines. A discussion of such models, however, lies beyond the scope of this entry. ### 4.1 General Recursive Functions The original formulation of general recursive functions can be found in Godel 1934, which built on a suggestion by Herbrand. In Kleene 1936 a simpler definition was given and in Kleene 1943 the standard form which uses the so-called minimization or $\mu$-operator was introduced. For more information, see the entry on recursive functions. Church used the definition of general recursive functions to state his thesis: > > > Church's thesis Every effectively calculable > function is general recursive > > > In the context of recursive function one uses the notion of recursive solvability and unsolvability rather than Turing computability and uncomputability. This terminology is due to Post (1944). ### 4.2 l-Definability Church's l-calculus has its origin in the papers (Church 1932, 1933) and which were intended as a logical foundation for mathematics. It was Church's conviction at that time that this different formal approach might avoid Godel incompleteness (Sieg 1997: 177). However, the logical system proposed by Church was proven inconsistent by his two PhD students Stephen C. Kleene and Barkley Rosser and so they started to focus on a subpart of that logic which was basically the l-calculus. Church, Kleene and Rosser started to l-define any calculable function they could think of and quite soon Church proposed to define effective calculability in terms of l-definability. However, it was only after Church, Kleene and Rosser had established that general recursiveness and l-definability are equivalent that Church announced his thesis publicly and in terms of general recursive functions rather than l-definability (Davis 1982; Sieg 1997). In l-calculus there are only two types of symbols. The three primitive symbols l, (, ) also called the improper symbols, and an infinite list of variables. There are three rules to define the well-formed formulas of l-calculus, called l-formulas. 1. The l-formulas are first of all the variables themselves. 2. If P is a l-formula containing x as a free variable then $\lambda x[\textbf{P}]$ is also a l-formula. The l-operator is used to bind variables and it thus converts an expression containing free variables into one that denotes a function 3. If M and N are l-formulas then so is {M}(N), where {M}(N) is to be understood as the application of the function M to N. The l-formulas, or well-formed formulas of l-calculus are all and only those formulas that result from (repeated) application of these three rules. There are three operations or rules of conversion. Let us define $\textrm{S}\_{\mathbf{N}}^{x}\mathbf{M}\|$ as standing for the formula that results by substitution of N for x in M. 1. Reduction. To replace any part $\{\lambda x \mathbf{[M]}\} (\mathbf{N})$ of a formula by $\textrm{S}\_{\mathbf{N}}^{x}\mathbf{M}\|$ provided that the bound variables of M are distinct both from x and from the free variables of N. For example $\{\lambda x [x^{2}]\}(2)$ reduces to $2^{2}$ 2. Expansion To replace any part $\textrm{S}\_{\mathbf{N}}^{x}\mathbf{M}\|$ of a formula by $\{\lambda x \mathbf{[M]}\} (\mathbf{N})$ provided that $((\lambda x \mathbf{M}) \mathbf{N})$ is well-formed and the bound variables of M are distinct both from x and from the free variables in N. For example, $2^{2}$ can be expanded to $\{\lambda x [x^{2}]\}(2)$ 3. Change of bound variable To replace any part M of a formula by $\textrm{S}\_{\textrm{y}}^{x}\mathbf{M}\|$ provided that x is not a free variable of M and y does not occur in M. For example changing $\{\lambda x [x^{2}]\}$ to $\{\lambda y [y^{2}]\}$ Church introduces the following abbreviations to define the natural numbers in l-calculus: \[\begin{array}{l} 1 \rightarrow \lambda yx.yx,\\ 2 \rightarrow \lambda yx.y(yx),\\ 3 \rightarrow \lambda yx.y(y(yx)),\\ \ldots \end{array}\] Using this definition, it is possible to l-define functions over the positive integers. A function F of one positive integer is l-definable if we can find a l-formula F, such that if $F(m) = n$ and m and n are l-formulas standing for the integers m and n, then the l-formula $\{\mathbf{F}\} (\mathbf{m})$ can be converted to n by applying the conversion rules of l-calculus. Thus, for example, the successor function S, first introduced by Church, can be l-defined as follows: \[S \rightarrow \lambda abc. b(abc)\] To give an example, applying S to the l-formula standing for 2, we get: \[\begin{align} \big(\lambda abc. b(abc)\big ) \big(\lambda yx. y(yx)\big) \\ \rightarrow \lambda bc. b\big( \big(\lambda yx. y(yx)\big) bc\big)\\ \rightarrow \lambda bc. b\big( \big(\lambda x. b(bx)\big) c\big)\\ \rightarrow \lambda bc. b (b(bc)) \end{align}\] Today, l-calculus is considered to be a basic model in the theory of programming. ### 4.3 Post Production Systems Around 1920-21 Emil Post developed different but related types of production systems in order to develop a syntactical form which would allow him to tackle the decision problem for first-order logic. One of these forms are Post canonical systems C which became later known as Post production systems. A canonical system consists of a finite alphabet $\Sigma$, a finite set of initial words $W\_{0,0}$, $W\_{0,1}$,..., $W\_{0,n}$ and a finite set of production rules of the following form: \[ \begin{array}{c} g\_{11}P\_{i\_{1}^{1}}g\_{12}P\_{i\_{2}^{1}} \ldots g\_{1m\_{1}}P\_{i^{1}\_{m\_{1}}}g\_{1 {(m\_{1} + 1)}}\\ g\_{21}P\_{i\_{1}^{2}}g\_{22}P\_{i\_{2}^{2}} \ldots g\_{2m\_{2}}P\_{i^{2}\_{m\_{2}}}g\_{2 {(m\_{2} + 1)}}\\ .................................\\ g\_{k1}P\_{i\_{1}^{k}}g\_{k2}P\_{i\_{2}^{k}} \ldots g\_{km\_{k}}P\_{i^{k}\_{m\_{k}}}g\_{k {(m\_{k} + 1)}}\\ \textit{produce}\\ g\_{1}P\_{i\_{1}}g\_{2}P\_{i\_{2}} \ldots g\_{m}P\_{i\_{m}}g\_{(m + 1)}\\ \end{array} \] The symbols g are a kind of metasymbols: they correspond to actual sequences of letters in actual productions. The symbols P are the operational variables and so can represent any sequence of letters in a production. So, for instance, consider a production system over the alphabet $\Sigma = \{a,b\}$ with initial word: \[W\_0 = ababaaabbaabbaabbaba\] and the following production rule: \[ \begin{array}{c} P\_{1,1}bbP\_{1,2}\\ \textit{produces}\\ P\_{1,3}aaP\_{1,4}\\ \end{array} \] Then, starting with $W\_0$, there are three possible ways to apply the production rule and in each application the variables $P\_{1,i}$ will have different values but the values of the g's are fixed. Any set of finite sequences of words that can be produced by a canonical system is called a canonical set. A special class of canonical forms defined by Post are normal systems. A normal system N consists of a finite alphabet $\Sigma$, one initial word $W\_0 \in \Sigma^{\ast}$ and a finite set of production rules, each of the following form: \[ \begin{array}{c} g\_iP\\ \textit{produces}\\ Pg\_i'\\ \end{array} \] Any set of finite sequences of words that can be produced by a normal system is called a normal set. Post was able to show that for any canonical set C over some alphabet $\Sigma$ there is a normal set N over an alphabet $\Delta$ with $\Sigma \subseteq \Delta$ such that $C = N \cap \Sigma^{\ast}$. It was his conviction that (1) any set of finite sequences that can be generated by finite means can be generated by canonical systems and (2) the proof that for every canonical set there is a normal set which contains it, which resulted in Post's thesis I: > > > Post's thesis I (Davis 1982) Every set of > finite sequences of letters that can be generated by finite processes > can also be generated by normal systems. More particularly, any set of > words on an alphabet $\Sigma$ which can be generated by a finite > process is of the form $N \cap \Sigma^{\ast}$, with N a > normal set. > > > Post realized that "[for the thesis to obtain its full generality] a complete analysis would have to be made of all the possible ways in which the human mind could set up finite processes for generating sequences" (Post 1965: 408) and it is quite probable that the formulation 1 given in Post 1936 and which is almost identical to Turing's machines is the result of such an analysis. Post production systems became important formal devices in computer science and, more particularly, formal language theory (Davis 1989; Pullum 2011). ### 4.4 Formulation 1 In 1936 Post published a short note from which one can derive Post's second thesis (De Mol 2013): > > > Post's thesis II Solvability of a > problem in the intuitive sense coincides with solvability by > formulation 1 > > > Formulation 1 is very similar to Turing machines but the 'program' is given as a list of directions which a human worker needs to follow. Instead of a one-way infinite tape, Post's 'machine' consists of a two-way infinite symbol space divided into boxes. The idea is that a worker is working in this symbol space, being capable of a set of five primitive acts ($O\_{1}$ mark a box, $O\_{2}$ unmark a box, $O\_{3}$ move one box to the left, $O\_{4}$ move one box to the right, $O\_{5}$ determining whether the box he is in is marked or unmarked), following a finite set of directions $d\_{1}$,..., $d\_{n}$ where each direction $d\_{i}$ always has one of the following forms: 1. Perform one of the operations ($O\_{1}$-$O\_4$) and go to instruction $d\_{j}$ 2. Perform operation $O\_{5}$ and according as the box the worker is in is marked or unmarked follow direction $d\_{j'}$ or $d\_{j''}$. 3. Stop. Post also defined a specific terminology for his formulation 1 in order to define the solvability of a problem in terms of formulation 1. These notions are applicability, finite-1-process, 1-solution and 1-given. Roughly speaking these notions assure that a decision problem is solvable with formulation 1 on the condition that the solution given in the formalism always terminates with a correct solution. ## 5. Impact of Turing Machines on Computer Science Turing is today one of the most celebrated figures of computer science. Many consider him as the father of computer science and the fact that the main award in the computer science community is called the Turing award is a clear indication of that (Daylight 2015). This was strengthened by the Turing centenary celebrations from 2012, which were largely coordinated by S. Barry Cooper. This resulted not only in an enormous number of scientific events around Turing but also a number of initiatives that brought the idea of Turing as the father of computer science also to the broader public (Bullynck, Daylight, & De Mol 2015). Amongst Turing's contributions which are today considered as pioneering, the 1936 paper on Turing machines stands out as the one which has the largest impact on computer science. However, recent historical research shows also that one should treat the impact of Turing machines with great care and that one should be careful in retrofitting the past into the present. ### 5.1 Impact on Theoretical Computer Science Today, the Turing machine and its theory are part of the theoretical foundations of computer science. It is a standard reference in research on foundational questions such as: * What is an algorithm? * What is a computation? * What is a physical computation? * What is an efficient computation? * etc. It is also one of the main models for research into a broad range of subdisciplines in theoretical computer science such as: variant and minimal models of computability, higher-order computability, computational complexity theory, algorithmic information theory, etc. This significance of the Turing machine model for theoretical computer science has at least two historical roots. First of all, there is the continuation of the work in mathematical logic from the 1920s and 1930s by people like Martin Davis--who is a student of Post and Church--and Kleene. Within that tradition, Turing's work was of course well-known and the Turing machine was considered as the best model of computability given. Both Davis and Kleene published a book in the 1950s on these topics (Kleene 1952; Davis 1958) which soon became standard references not just for early computability theory but also for more theoretical reflections in the late 1950s and 1960s on computing. Secondly, one sees that in the 1950s there is a need for theoretical models to reflect on the new computing machines, their abilities and limitations and this in a more systematic manner. It is in that context that the theoretical work already done was picked up. One important development is automata theory in which one can situate, amongst others, the development of other machine models like the register machine model or the Wang B machine model which are, ultimately, rooted in Turing's and Post's machines; there are the minimal machine designs discussed in Section 5.2; and there is the use of Turing machines in the context of what would become the origins of formal language theory, viz the study of different classes of machines with respect to the different "languages" they can recognize and so also their limitations and strengths. It are these more theoretical developments that contributed to the establishment of computational complexity theory in the 1960s. Of course, besides Turing machines, other models also played and play an important role in these developments. Still, within theoretical computer science it is mostly the Turing machine which remains the model, even today. Indeed, when in 1965 one of the founding papers of computational complexity theory (Hartmanis & Stearns 1965) is published, it is the multitape Turing machine which is introduced as the standard model for the computer. ### 5.2 Turing Machines and the Modern Computer In several accounts, Turing has been identified not just as the father of computer science but as the father of the modern computer. The classical story for this more or less goes as follows: the blueprint of the modern computer can be found in von Neumann's EDVAC design and today classical computers are usually described as having a so-called von Neumann architecture. One fundamental idea of the EDVAC design is the so-called stored-program idea. Roughly speaking this means the storage of instructions and data in the same memory allowing the manipulation of programs as data. There are good reasons for assuming that von Neumann knew the main results of Turing's paper (Davis 1988). Thus, one could argue that the stored-program concept originates in Turing's notion of the universal Turing machine and, singling this out as the defining feature of the modern computer, some might claim that Turing is the father of the modern computer. Another related argument is that Turing was the first who "captured" the idea of a general-purpose machine through his notion of the universal machine and that in this sense he also "invented" the modern computer (Copeland & Proudfoot 2011). This argument is then strengthened by the fact that Turing was also involved with the construction of an important class of computing devices (the Bombe) used for decrypting the German Enigma code and later proposed the design of the ACE (Automatic Computing Engine) which was explicitly identified as a kind of physical realization of the universal machine by Turing himself: > > > Some years ago I was researching on what might now be described as an > investigation of the theoretical possibilities and limitations of > digital computing machines. [...] Machines such as the ACE may be > regarded as practical versions of this same type of machine. (Turing > 1947) > > > Note however that Turing already knew the ENIAC and EDVAC designs and proposed the ACE as a kind of improvement on that design (amongst others, it had a simpler hardware architecture). These claims about Turing as the inventor and/or father of the computer have been scrutinized by some historians of computing (Daylight 2014; Haigh 2013; Haigh 2014; Mounier-Kuhn 2012), mostly in the wake of the Turing centenary and this from several perspectives. Based on that research it is clear that claims about Turing being the inventor of the modern computer give a distorted and biased picture of the development of the modern computer. At best, he is one of the many who made a contribution to one of the several historical developments (scientific, political, technological, social and industrial) which resulted, ultimately, in (our concept of) the modern computer. Indeed, the "first" computers are the result of a wide number of innovations and so are rooted in the work of not just one but several people with diverse backgrounds and viewpoints. In the 1950s then the (universal) Turing machine starts to become an accepted model in relation to actual computers and is used as a tool to reflect on the limits and potentials of general-purpose computers by both engineers, mathematicians and logicians. More particularly, with respect to machine designs, it was the insight that only a few number of operations were required to built a general-purpose machine which inspired in the 1950s reflections on minimal machine architectures. Frankel, who (partially) constructed the MINAC stated this as follows: > > > One remarkable result of Turing's investigation is that he was > able to describe a single computer which is able to compute > any computable number. He called this machine a universal > computer. It is thus the "best possible" computer > mentioned. > > > > [...] This surprising result shows that in examining the question > of what problems are, in principle, solvable by computing machines, we > do not need to consider an infinite series of computers of greater and > greater complexity but may think only of a single machine. > > > > Even more surprising than the theoretical possibility of such a > "best possible" computer is the fact that it need not be > very complex. The description given by Turing of a universal computer > is not unique. Many computers, some of quite modest complexity, > satisfy the requirements for a universal computer. (Frankel 1956: > 635) > > > The result was a series of experimental machines such as the MINAC, TX-0 (Lincoln Lab) or the ZERO machine (van der Poel) which in their turn became predecessors of a number of commercial machines. It is worth pointing out that also Turing's ACE machine design fits into this philosophy. It was also commercialized as the BENDIX G15 machine (De Mol, Bullynck, & Daylight 2018). Of course, by minimizing the machine instructions, coding or programming became a much more complicated task. To put it in Turing's words who clearly realized this trade-off between code and (hard-wired) instructions when designing the ACE: "[W]e have often simplified the circuit at the expense of the code" (Turing 1947). And indeed, one sees that with these early minimal designs, much effort goes into developing more efficient coding strategies. It is here that one can also situate one historical root of making the connection between the universal Turing machine and the important principle of the interchangeability between hardware and programs. Today, the universal Turing machine is by many still considered as the main theoretical model of the modern computer especially in relation to the so-called von Neumann architecture. Of course, other models have been introduced for other architectures such as the Bulk synchronous parallel model for parallel machines or the persistent Turing machine for modeling interactive problems. ### 5.3 Theories of Programming The idea that any general-purpose machine can, in principle, be modeled as a universal Turing machine also became an important principle in the context of automatic programming in the 1950s (Daylight 2015). In the machine design context it was the minimizing of the machine instructions that was the most important consequence of that viewpoint. In the programming context then it was about the idea that one can built a machine that is able to 'mimic'' the behavior of any other machine and so, ultimately, the interchangeability between machine hardware and language implementations. This is introduced in several forms in the 1950s by people like John W. Carr III and Saul Gorn--who were also actively involved in the shaping of the Association for Computing Machinery (ACM)--as the unifying theoretical idea for automatic programming which indeed is about the (automatic) "translation" of higher-order to lower-level, and, ultimately, machine code. Thus, also in the context of programming, the universal Turing machine starts to take on its foundational role in the 1950s (Daylight 2015). Whereas the Turing machine is and was a fundamental theoretical model delimiting what is possible and not on the general level, it did not have a real impact on the syntax and semantics of programming languages. In that context it were rather l-calculus and Post production systems that had an effect (though also here one should be careful in overstating the influence of a formal model on a programming practice). In fact, Turing machines were often regarded as machine models rather than as a model for programming: > > > Turing machines are not conceptually different from the automatic > computers in general use, but they are very poor in their control > structure. [...] Of course, most of the theory of computability > deals with questions which are not concerned with the particular ways > computations are represented. It is sufficient that computable > functions be represented somehow by symbolic expressions, e.g., > numbers, and that functions computable in terms of given functions be > somehow represented by expressions computable in terms of the > expressions representing the original functions. However, a practical > theory of computation must be applicable to particular algorithms. > (McCarthy 1963: 37) > > > Thus one sees that the role of the Turing machine for computer science should be situated rather on the theoretical level: the universal machine is today by many still considered as the model for the modern computer while its ability to mimic machines through its manipulation of programs-as-data is one of the basic principles of modern computing. Moreover, its robustness and naturalness as a model of computability have made it the main model to challenge if one is attacking versions of the so-called (physical) Church-Turing thesis.
turing-test	## 1. Turing (1950) and the Imitation Game Turing (1950) describes the following kind of game. Suppose that we have a person, a machine, and an interrogator. The interrogator is in a room separated from the other person and the machine. The object of the game is for the interrogator to determine which of the other two is the person, and which is the machine. The interrogator knows the other person and the machine by the labels 'X' and 'Y'--but, at least at the beginning of the game, does not know which of the other person and the machine is 'X'--and at the end of the game says either 'X is the person and Y is the machine' or 'X is the machine and Y is the person'. The interrogator is allowed to put questions to the person and the machine of the following kind: "Will X please tell me whether X plays chess?" Whichever of the machine and the other person is X must answer questions that are addressed to X. The object of the machine is to try to cause the interrogator to mistakenly conclude that the machine is the other person; the object of the other person is to try to help the interrogator to correctly identify the machine. About this game, Turing (1950) says: > > I believe that in about fifty years' time it will be possible to > programme computers, with a storage capacity of about 109, > to make them play the imitation game so well that an average > interrogator will not have more than 70 percent chance of making the > right identification after five minutes of questioning. ... I > believe that at the end of the century the use of words and general > educated opinion will have altered so much that one will be able to > speak of machines thinking without expecting to be contradicted. > There are at least two kinds of questions that can be raised about Turing's predictions concerning his Imitation Game. First, there are empirical questions, e.g., Is it true that we now--or will soon--have made computers that can play the imitation game so well that an average interrogator has no more than a 70 percent chance of making the right identification after five minutes of questioning? Second, there are conceptual questions, e.g., Is it true that, if an average interrogator had no more than a 70 percent chance of making the right identification after five minutes of questioning, we should conclude that the machine exhibits some level of thought, or intelligence, or mentality? There is little doubt that Turing would have been disappointed by the state of play at the end of the twentieth century. Participants in the Loebner Prize Competition--an annual event in which computer programmes are submitted to the Turing Test-- had come nowhere near the standard that Turing envisaged. A quick look at the transcripts of the participants for the preceding decade reveals that the entered programs were all easily detected by a range of not-very-subtle lines of questioning. Moreover, major players in the field regularly claimed that the Loebner Prize Competition was an embarrassment precisely because we were still so far from having a computer programme that could carry out a decent conversation for a period of five minutes--see, for example, Shieber (1994). It was widely conceded on all sides that the programs entered in the Loebner Prize Competition were designed solely with the aim of winning the minor prize of best competitor for the year, with no thought that the embodied strategies would actually yield something capable of passing the Turing Test. At the end of the second decade of the twenty-first century, it is unclear how much has changed. On the one hand, there have been interesting developments in language generators. In particular, the release of Open AI's GPT-3 (Brown, et al. 2020, Other Internet Resources) has prompted a flurry of excitement. GPT-3 is quite good at generating fiction, poetry, press releases, code, music, jokes, technical manuals, and news articles. Perhaps, as Chalmers speculates (2020, Other Internet Resources), GPT-3 "suggests a potential mindless path to artificial general intelligence". But, of course, GPT-3 is not close to passing the Turing Test: GPT-3 neither perceives nor acts, and it is, at best, highly contentious whether it is a site of understanding. What remains to be seen is whether, within the next couple of generations of language generators - GPT-4 or GPT-5 - we have something that can be linked to perceptual inputs and behavioural outputs in a way that does produce something capable of passing the Turing Test. (For further discussion, see Floridi and Chiriatti (2020).) On the other hand, as, for example, Floridi (2008) complains, there are other ways in which progress has been frustratingly slow. In 2014, claims emerged that, because the computer program Eugene Goostman had fooled 33% of judges in the Turing Test 2014 competition, it had "passed the Turing Test". But there have been other one-off competitions in which similar results have been achieved. Back in 1991, PC Therapist had 50% of judges fooled. And, in a 2011 demonstration, Cleverbot had an even higher success rate. In all three of these cases, the size of the trial was very small, and the result was not reliably projectible: in no case were there strong grounds for holding that an average interrogator had no more than a 70% chance of making the right determination about the relevant program after five minutes of questioning. Moreover--and much more importantly--we must distinguish between the test the Turing proposed, and the particular prediction that he made about how things would be by the end of the twentieth century. The percentage chance of making the correct identification, the time interval over which the test takes place, and the number of conversational exchanges required are all adjustable parameters in the Test, despite the fact that they are fixed in the particular prediction that Turing made. Even if Turing was very far out in the prediction that he made about how things would be by the end of the twentieth century, it remains possible that the test that he proposes is a good one. However, before one can endorse the suggestion that the Turing Test is good, there are various objections that ought to be addressed. Some people have suggested that the Turing Test is chauvinistic: it only recognizes intelligence in things that are able to sustain a conversation with us. Why couldn't it be the case that there are intelligent things that are unable to carry on a conversation, or, at any rate, unable to carry on a conversation with creatures like us? (See, for example, French (1990).) Perhaps the intuition behind this question can be granted; perhaps it is unduly chauvinistic to insist that anything that is intelligent has to be capable of sustaining a conversation with us. (On the other hand, one might think that, given the availability of suitably qualified translators, it ought to be possible for any two intelligent agents that speak different languages to carry on some kind of conversation.) But, in any case, the charge of chauvinism is completely beside the point. What Turing claims is only that, if something can carry out a conversation with us, then we have good grounds to suppose that that thing has intelligence of the kind that we possess; he does not claim that only something that can carry out a conversation with us can possess the kind of intelligence that we have. Other people have thought that the Turing Test is not sufficiently demanding: we already have anecdotal evidence that quite unintelligent programs (e.g., ELIZA--for details of which, see Weizenbaum (1966)) can seem to ordinary observers to be loci of intelligence for quite extended periods of time. Moreover, over a short period of time--such as the five minutes that Turing mentions in his prediction about how things will be in the year 2000--it might well be the case that almost all human observers could be taken in by cunningly designed but quite unintelligent programs. However, it is important to recall that, in order to pass Turing's Test, it is not enough for the computer program to fool "ordinary observers" in circumstances other than those in which the test is supposed to take place. What the computer program has to be able to do is to survive interrogation by someone who knows that one of the other two participants in the conversation is a machine. Moreover, the computer program has to be able to survive such interrogation with a high degree of success over a repeated number of trials. (Turing says nothing about how many trials he would require. However, we can safely assume that, in order to get decent evidence that there is no more than a 70% chance that a machine will be correctly identified as a machine after five minutes of conversation, there will have to be a reasonably large number of trials.) If a computer program could do this quite demanding thing, then it does seem plausible to claim that we would have at least prima facie reason for thinking that we are in the presence of intelligence. (Perhaps it is worth emphasizing again that there might be all kinds of intelligent things--including intelligent machines--that would not pass this test. It is conceivable, for example, that there might be machines that, as a result of moral considerations, refused to lie or to engage in pretence. Since the human participant is supposed to do everything that he or she can to help the interrogator, the question "Are you a machine?" would quickly allow the interrogator to sort such (pathological?) truth-telling machines from humans.) Another contentious aspect of Turing's paper (1950) concerns his restriction of the discussion to the case of "digital computers." On the one hand, it seems clear that this restriction is really only significant for the prediction that Turing makes about how things will be in the year 2000, and not for the details of the test itself. (Indeed, it seems that if the test that Turing proposes is a good one, then it will be a good test for any kinds of entities, including, for example, animals, aliens, and analog computers. That is: if animals, aliens, analog computers, or any other kinds of things, pass the test that Turing proposes, then there will be as much reason to think that these things exhibit intelligence as there is reason to think that digital computers that pass the test exhibit intelligence.) On the other hand, it is actually a highly controversial question whether "thinking machines" would have to be digital computers; and it is also a controversial question whether Turing himself assumed that this would be the case. In particular, it is worth noting that the seventh of the objections that Turing (1950) considers addresses the possibility of continuous state machines, which Turing explicitly acknowledges to be different from discrete state machines. Turing appears to claim that, even if we are continuous state machines, a discrete state machine would be able to imitate us sufficiently well for the purposes of the Imitation Game. However, it seems doubtful that the considerations that he gives are sufficient to establish that, if there are continuous state machines that pass the Turing Test, then it is possible to make discrete state machines that pass the test as well. (Turing himself was keen to point out that some limits had to be set on the notion of "machine" in order to make the question about "thinking machines" interesting: > > It is natural that we should wish to permit every kind of engineering > technique to be used in our machine. We also wish to allow the > possibility that an engineer or team of engineers may construct a > machine which works, but whose manner of operation cannot be > satisfactorily described by its constructors because they have applied > a method which is largely experimental. Finally, we wish to exclude > from the machines men born in the usual manner. It is difficult to > frame the definitions so as to satisfy these three conditions. One > might for instance insist that the team of engineers should all be of > one sex, but this would not really be satisfactory, for it is probably > possible to rear a complete individual from a single cell of the skin > (say) of a man. To do so would be a feat of biological technique > deserving of the very highest praise, but we would not be inclined to > regard it as a case of 'constructing a thinking machine'. > (435/6) > But, of course, as Turing himself recognized, there is a large class of possible "machines" that are neither digital nor biotechnological.) More generally, the crucial point seems to be that, while Turing recognized that the class of machines is potentially much larger than the class of discrete state machines, he was himself very confident that properly engineered discrete state machines could succeed in the Imitation Game (and, moreover, at the time that he was writing, there were certain discrete state machines--"electronic computers"--that loomed very large in the public imagination). ## 2. Turing (1950) and Responses to Objections Although Turing (1950) is pretty informal, and, in some ways rather idiosyncratic, there is much to be gained by considering the discussion that Turing gives of potential objections to his claim that machines--and, in particular, digital computers--can "think". Turing gives the following labels to the objections that he considers: (1) The Theological Objection; (2) The "Heads in the Sand" Objection; (3) The Mathematical Objection; (4) The Argument from Consciousness; (5) Arguments from Various Disabilities; (6) Lady Lovelace's Objection; (7) Argument from Continuity of the Nervous System; (8) The Argument from Informality of Behavior; and (9) The Argument from Extra-Sensory Perception. We shall consider these objections in the corresponding subsections below. (In some--but not all--cases, the counter-arguments to these objections that we discuss are also provided by Turing.) ### 2.1 The Theological Objection Substance dualists believe that thinking is a function of a non-material, separately existing, substance that somehow "combines" with the body to make a person. So--the argument might go--making a body can never be sufficient to guarantee the presence of thought: in themselves, digital computers are no different from any other merely material bodies in being utterly unable to think. Moreover--to introduce the "theological" element--it might be further added that, where a "soul" is suitably combined with a body, this is always the work of the divine creator of the universe: it is entirely up to God whether or not a particular kind of body is imbued with a thinking soul. (There is well known scriptural support for the proposition that human beings are "made in God's image". Perhaps there is also theological support for the claim that only God can make things in God's image.) There are several different kinds of remarks to make here. First, there are many serious objections to substance dualism. Second, there are many serious objections to theism. Third, even if theism and substance dualism are both allowed to pass, it remains quite unclear why thinking machines are supposed to be ruled out by this combination of views. Given that God can unite souls with human bodies, it is hard to see what reason there is for thinking that God could not unite souls with digital computers (or rocks, for that matter!). Perhaps, on this combination of views, there is no especially good reason why, amongst the things that we can make, certain kinds of digital computers turn out to be the only ones to which God gives souls--but it seems pretty clear that there is also no particularly good reason for ruling out the possibility that God would choose to give souls to certain kinds of digital computers. Evidence that God is dead set against the idea of giving souls to certain kinds of digital computers is not particularly thick on the ground. ### 2.2 The 'Heads in the Sand' Objection If there were thinking machines, then various consequences would follow. First, we would lose the best reasons that we have for thinking that we are superior to everything else in the universe (since our cherished "reason" would no longer be something that we alone possess). Second, the possibility that we might be "supplanted" by machines would become a genuine worry: if there were thinking machines, then very likely there would be machines that could think much better than we can. Third, the possibility that we might be "dominated" by machines would also become a genuine worry: if there were thinking machines, who's to say that they would not take over the universe, and either enslave or exterminate us? As it stands, what we have here is not an argument against the claim that machines can think; rather, we have the expression of various fears about what might follow if there were thinking machines. Someone who took these worries seriously--and who was persuaded that it is indeed possible for us to construct thinking machines--might well think that we have here reasons for giving up on the project of attempting to construct thinking machines. However, it would be a major task--which we do not intend to pursue here--to determine whether there really are any good reasons for taking these worries seriously. ### 2.3 The Mathematical Objection Some people have supposed that certain fundamental results in mathematical logic that were discovered during the 1930s--by Godel (first incompleteness theorem) and Turing (the halting problem)--have important consequences for questions about digital computation and intelligent thought. (See, for example, Lucas (1961) and Penrose (1989); see, too, Hodges (1983:414) who mentions Polanyi's discussions with Turing on this matter.) Essentially, these results show that within a formal system that is strong enough, there are a class of true statements that can be expressed but not proven within the system (see the entry on Godel's incompleteness theorems). Let us say that such a system is "subject to the Lucas-Penrose constraint" because it is constrained from being able to prove a class of true statements expressible within the system. Turing (1950:444) himself observes that these results from mathematical logic might have implications for the Turing test: > > There are certain things that [any digital computer] cannot do. If it > is rigged up to give answers to questions as in the imitation game, > there will be some questions to which it will either give a wrong > answer, or fail to give an answer at all however much time is allowed > for a reply. (444) > So, in the context of the Turing test, "being subject to the Lucas-Penrose constraint" implies the existence of a class of "unanswerable" questions. However Turing noted that in the context of the Turing test, these "unanswerable" questions are only a concern if humans can answer them. His "short" reply was that it is not clear that humans are free from such a constraint themselves. Turing then goes on to add that he does not think that the argument can be dismissed "quite so lightly." To make the argument more precise, we can write it as follows: 1. Let C be a digital computer. 2. Since C is subject to the Lucas-Penrose constraint, there is an "unanswerable" question q for C. 3. If an entity, E, is not subject to the Lucas-Penrose constraint, then there are no "unanswerable" questions for E. 4. The human intellect is not subject to the Lucas-Penrose constraint. 5. Thus, there are no "unanswerable" questions for the human intellect. 6. The question q is therefore "answerable" to the human intellect. 7. By asking question q, a human could determine if the responder is a computer or a human. 8. Thus C may fail the Turing test. Once the argument is laid out as above, it becomes clear that premise (3) should be challenged. Putting that aside, we note that one interpretation of Turing's "short" reply is that claim (4) is merely asserted--without any kind of proof. The "short" reply then leads us to examine whether humans are free from the Lucas-Penrose constraint. If humans are subject to the Lucas-Penrose constraint then the constraint does not provide any basis for distinguishing humans from digital computers. If humans are free from the Lucas-Penrose constraint, then (granting premise 3) it follows that digital computers may fail the Turing test and thus, it seems, cannot think. However, there remains a question as to whether being free from the constraint is necessary for the capacity to think. It may be that the Turing test is too strict. Since, by hypothesis, we are free from the Lucas-Penrose constraint, we are, in some sense, too good at asking and answering questions. Suppose there is a thinking entity that is subject to the Lucas-Penrose constraint. By an argument analogous to the one above, it can fail the Turing test. Thus, an entity which can think would fail the Turing test. We can respond to this concern by noting that the construction of questions suggested by the results from mathematical logic--Godel, Turing, etc.--are extremely complicated, and require extremely detailed information about the language and internal programming of the digital computer (which, of course, is not available to the interrogators in the Imitation Game). At the very least, much more argument is required to overthrow the view that the Turing Test could remain a very high quality statistical test for the presence of mind and intelligence even if digital computers differ from human beings in being subject to the Lucas-Penrose constraint. (See Bowie 1982, Dietrich 1994, Feferman 1996, Abramson 2008, and Section 6.3 of the entry on Godel's incompleteness theorems, for further discussion.) ### 2.4 The Argument from Consciousness Turing cites Professor Jefferson's Lister Oration for 1949 as a source for the kind of objection that he takes to fall under this label: > > Not until a machine can write a sonnet or compose a concerto because > of thoughts and emotions felt, and not by the chance fall of symbols, > could we agree that machine equals brain--that is, not only write > it but know that it had written it. No mechanism could feel (and not > merely artificially signal, an easy contrivance) pleasure at its > successes, grief when its valves fuse, be warmed by flattery, be made > miserable by its mistakes, be charmed by sex, be angry or depressed > when it cannot get what it wants. (445/6) > There are several different ideas that are being run together here, and that it is profitable to disentangle. One idea--the one upon which Turing first focuses--is the idea that the only way in which one could be certain that a machine thinks is to be the machine, and to feel oneself thinking. A second idea, perhaps, is that the presence of mind requires the presence of a certain kind of self-consciousness ("not only write it but know that it had written it"). A third idea is that it is a mistake to take a narrow view of the mind, i.e. to suppose that there could be a believing intellect divorced from the kinds of desires and emotions that play such a central role in the generation of human behavior ("no mechanism could feel ..."). Against the solipsistic line of thought, Turing makes the effective reply that he would be satisfied if he could secure agreement on the claim that we might each have just as much reason to suppose that machines think as we have reason to suppose that other people think. (The point isn't that Turing thinks that solipsism is a serious option; rather, the point is that following this line of argument isn't going to lead to the conclusion that there are respects in which digital computers could not be our intellectual equals or superiors.) Against the other lines of thought, Turing provides a little "viva voce" that is intended to illustrate the kind of evidence that he supposes one might have that a machine is intelligent. Given the right kinds of responses from the machine, we would naturally interpret its utterances as evidence of pleasure, grief, warmth, misery, anger, depression, etc. Perhaps--though Turing doesn't say this--the only way to make a machine of this kind would be to equip it with sensors, affective states, etc., i.e., in effect, to make an artificial person. However, the important point is that if the claims about self-consciousness, desires, emotions, etc. are right, then Turing can accept these claims with equanimity: his claim is then that a machine with a digital computing "brain" can have the full range of mental states that can be enjoyed by adult human beings. ### 2.5 Arguments from Various Disabilities Turing considers a list of things that some people have claimed machines will never be able to do: (1) be kind; (2) be resourceful; (3) be beautiful; (4) be friendly; (5) have initiative; (6) have a sense of humor; (7) tell right from wrong; (8) make mistakes; (9) fall in love; (10) enjoy strawberries and cream; (11) make someone fall in love with one; (12) learn from experience; (13) use words properly; (14) be the subject of one's own thoughts; (15) have as much diversity of behavior as a man; (16) do something really new. An interesting question to ask, before we address these claims directly, is whether we should suppose that intelligent creatures from some other part of the universe would necessarily be able to do these things. Why, for example, should we suppose that there must be something deficient about a creature that does not enjoy--or that is not able to enjoy--strawberries and cream? True enough, we might suppose that an intelligent creature ought to have the capacity to enjoy some kinds of things--but it seems unduly chauvinistic to insist that intelligent creatures must be able to enjoy just the kinds of things that we do. (No doubt, similar considerations apply to the claim that an intelligent creature must be the kind of thing that can make a human being fall in love with it. Yes, perhaps, an intelligent creature should be the kind of thing that can love and be loved; but what is so special about us?) Setting aside those tasks that we deem to be unduly chauvinistic, we should then ask what grounds there are for supposing that no digital computing machine could do the other things on the list. Turing suggests that the most likely ground lies in our prior acquaintance with machines of all kinds: none of the machines that any of us has hitherto encountered has been able to do these things. In particular, the digital computers with which we are now familiar cannot do these things. (Except perhaps for make mistakes: after all, even digital computers are subject to "errors of functioning." But this might be set aside as an irrelevant case.) However, given the limitations of storage capacity and processing speed of even the most recent digital computers, there are obvious reasons for being cautious in assessing the merits of this inductive argument. (A different question worth asking concerns the progress that has been made until now in constructing machines that can do the kinds of things that appear on Turing's list. There is at least room for debate about the extent to which current computers can: make mistakes, use words properly, learn from experience, be beautiful, etc. Moreover, there is also room for debate about the extent to which recent advances in other areas may be expected to lead to further advancements in overcoming these alleged disabilities. Perhaps, for example, recent advances in work on artificial sensors may one day contribute to the production of machines that can enjoy strawberries and cream. Of course, if the intended objection is to the notion that machines can experience any kind of feeling of enjoyment, then it is not clear that work on particular kinds of artificial sensors is to the point.) ### 2.6 Lady Lovelace's Objection One of the most popular objections to the claim that there can be thinking machines is suggested by a remark made by Lady Lovelace in her memoir on Babbage's Analytical Engine: > > The Analytical Engine has no pretensions to originate anything. It can > do whatever we know how to order it to perform (cited by Hartree, > p. 70) > The key idea is that machines can only do what we know how to order them to do (or that machines can never do anything really new, or anything that would take us by surprise). As Turing says, one way to respond to these challenges is to ask whether we can ever do anything "really new." Suppose, for instance, that the world is deterministic, so that everything that we do is fully determined by the laws of nature and the boundary conditions of the universe. There is a sense in which nothing "really new" happens in a deterministic universe--though, of course, the universe's being deterministic would be entirely compatible with our being surprised by events that occur within it. Moreover--as Turing goes on to point out--there are many ways in which even digital computers do things that take us by surprise; more needs to be said to make clear exactly what the nature of this suggestion is. (Yes, we might suppose, digital computers are "constrained" by their programs: they can't do anything that is not permitted by the programs that they have. But human beings are "constrained" by their biology and their genetic inheritance in what might be argued to be just the same kind of way: they can't do anything that is not permitted by the biology and genetic inheritance that they have. If a program were sufficiently complex--and if the processor(s) on which it ran were sufficiently fast--then it is not easy to say whether the kinds of "constraints" that would remain would necessarily differ in kind from the kinds of constraints that are imposed by biology and genetic inheritance.) Bringsjord et al. (2001) claim that Turing's response to the Lovelace Objection is "mysterious" at best, and "incompetent" at worst (p.4). In their view, Turing's claim that "computers do take us by surprise" is only true when "surprise" is given a very superficial interpretation. For, while it is true that computers do things that we don't intend them to do--because we're not smart enough, or because we're not careful enough, or because there are rare hardware errors, or whatever--it isn't true that there are any cases in which we should want to say that a computer has originated something. Whatever merit might be found in this objection, it seems worth pointing out that, in the relevant sense of origination, human beings "originate something" on more or less every occasion in which they engage in conversation: they produce new sentences of natural language that it is appropriate for them to produce in the circumstances in which they find themselves. Thus, on the one hand--for all that Bringsjord et al. have argued--The Turing Test is a perfectly good test for the presence of "origination" (or "creativity," or whatever). Moreover, on the other hand, for all that Bringsjord et al. have argued, it remains an open question whether a digital computing device is capable of "origination" in this sense (i.e. capable of producing new sentences that are appropriate to the circumstances in which the computer finds itself). So we are not overly inclined to think that Turing's response to the Lovelace Objection is poor; and we are even less inclined to think that Turing lacked the resources to provide a satisfactory response on this point. ### 2.7 Argument from Continuity of the Nervous System The human brain and nervous system is not much like a digital computer. In particular, there are reasons for being skeptical of the claim that the brain is a discrete-state machine. Turing observes that a small error in the information about the size of a nervous impulse impinging on a neuron may make a large difference to the size of the outgoing impulse. From this, Turing infers that the brain is likely to be a continuous-state machine; and he then notes that, since discrete-state machines are not continuous-state machines, there might be reason here for thinking that no discrete-state machine can be intelligent. Turing's response to this kind of argument seems to be that a continuous-state machine can be imitated by discrete-state machines with very small levels of error. Just as differential analyzers can be imitated by digital computers to within quite small margins of error, so too, the conversation of human beings can be imitated by digital computers to margins of error that would not be detected by ordinary interrogators playing the imitation game. It is not clear that this is the right kind of response for Turing to make. If someone thinks that real thought (or intelligence, or mind, or whatever) can only be located in a continuous-state machine, then the fact--if, indeed, it is a fact--that it is possible for discrete-state machines to pass the Turing Test shows only that the Turing Test is no good. A better reply is to ask why one should be so confident that real thought, etc. can only be located in continuous-state machines (if, indeed, it is right to suppose that we are not discrete-state machines). And, before we ask this question, we would do well to consider whether we really do have such good reason to suppose that, from the standpoint of our ability to think, we are not essentially discrete-state machines. (As Block (1981) points out, it seems that there is nothing in our concept of intelligence that rules out intelligent beings with quantised sensory devices; and nor is there anything in our concept of intelligence that rules out intelligent beings with digital working parts.) ### 2.8 Argument from Informality of Behavior This argument relies on the assumption that there is no set of rules that describes what a person ought to do in every possible set of circumstances, and on the further assumption that there is a set of rules that describes what a machine will do in every possible set of circumstances. From these two assumptions, it is supposed to follow--somehow!--that people are not machines. As Turing notes, there is some slippage between "ought" and "will" in this formulation of the argument. However, once we make the appropriate adjustments, it is not clear that an obvious difference between people and digital computers emerges. Suppose, first, that we focus on the question of whether there are sets of rules that describe what a person and a machine "will" do in every possible set of circumstances. If the world is deterministic, then there are such rules for both persons and machines (though perhaps it is not possible to write down the rules). If the world is not deterministic, then there are no such rules for either persons or machines (since both persons and machines can be subject to non-deterministic processes in the production of their behavior). Either way, it is hard to see any reason for supposing that there is a relevant difference between people and machines that bears on the description of what they will do in all possible sets of circumstances. (Perhaps it might be said that what the objection invites us to suppose is that, even though the world is not deterministic, humans differ from digital machines precisely because the operations of the latter are indeed deterministic. But, if the world is non-deterministic, then there is no reason why digital machines cannot be programmed to behave non-deterministically, by allowing them to access input from non-deterministic features of the world.) Suppose, instead, that we focus on the question of whether there are sets of rules that describe what a person and a machine "ought" to do in every possible set of circumstances. Whether or not we suppose that norms can be codified--and quite apart from the question of which kinds of norms are in question--it is hard to see what grounds there could be for this judgment, other than the question-begging claim that machines are not the kinds of things whose behavior could be subject to norms. (And, in that case, the initial argument is badly mis-stated: the claim ought to be that, whereas there are sets of rules that describe what a person ought to do in every possible set of circumstances, there are no sets of rules that describe what machines ought to do in all possible sets of circumstances!) ### 2.9 Argument from Extra-Sensory Perception The strangest part of Turing's paper is the few paragraphs on ESP. Perhaps it is intended to be tongue-in-cheek, though, if it is, this fact is poorly signposted by Turing. Perhaps, instead, Turing was influenced by the apparently scientifically respectable results of J. B. Rhine. At any rate, taking the text at face value, Turing seems to have thought that there was overwhelming empirical evidence for telepathy (and he was also prepared to take clairvoyance, precognition and psychokinesis seriously). Moreover, he also seems to have thought that if the human participant in the game was telepathic, then the interrogator could exploit this fact in order to determine the identity of the machine--and, in order to circumvent this difficulty, Turing proposes that the competitors should be housed in a "telepathy-proof room." Leaving aside the point that, as a matter of fact, there is no current statistical support for telepathy--or clairvoyance, or precognition, or telekinesis--it is worth asking what kind of theory of the nature of telepathy would have appealed to Turing. After all, if humans can be telepathic, why shouldn't digital computers be so as well? If the capacity for telepathy were a standard feature of any sufficiently advanced system that is able to carry out human conversation, then there is no in-principle reason why digital computers could not be the equals of human beings in this respect as well. (Perhaps this response assumes that a successful machine participant in the imitation game will need to be equipped with sensors, etc. However, as we noted above, this assumption is not terribly controversial. A plausible conversationalist has to keep up to date with goings-on in the world.) After discussing the nine objections mentioned above, Turing goes on to say that he has "no very convincing arguments of a positive nature to support my views. If I had I should not have taken such pains to point out the fallacies in contrary views." (454) Perhaps Turing sells himself a little short in this self-assessment. First of all--as his brief discussion of solipsism makes clear--it is worth asking what grounds we have for attributing intelligence (thought, mind) to other people. If it is plausible to suppose that we base our attributions on behavioral tests or behavioral criteria, then his claim about the appropriate test to apply in the case of machines seems apt, and his conjecture that digital computing machines might pass the test seems like a reasonable--though controversial--empirical conjecture. Second, subsequent developments in the philosophy of mind--and, in particular, the fashioning of functionalist theories of the mind--have provided a more secure theoretical environment in which to place speculations about the possibility of thinking machines. If mental states are functional states--and if mental states are capable of realisation in vastly different kinds of materials--then there is some reason to think that it is an empirical question whether minds can be realised in digital computing machines. Of course, this kind of suggestion is open to challenge; we shall consider some important philosophical objections in the later parts of this review. ## 3. Some Minor Issues Arising There are a number of much-debated issues that arise in connection with the interpretation of various parts of Turing (1950), and that we have hitherto neglected to discuss. What has been said in the first two sections of this document amounts to our interpretation of what Turing has to say (perhaps bolstered with what we take to be further relevant considerations in those cases where Turing's remarks can be fairly readily improved upon). But since some of this interpretation has been contested, it is probably worth noting where the major points of controversy have been. ### 3.1 Interpreting the Imitation Game Turing (1950) introduces the imitation game by describing a game in which the participants are a man, a woman, and a human interrogator. The interrogator is in a room apart from the other two, and is set the task of determining which of the other two is a man and which is a woman. Both the man and the woman are set the task of trying to convince the interrogator that they are the woman. Turing recommends that the best strategy for the woman is to answer all questions truthfully; of course, the best strategy for the man will require some lying. The participants in this game also use teletypewriter to communicate with one another--to avoid clues that might be offered by tone of voice, etc. Turing then says: "We now ask the question, 'What will happen when a machine takes the part of A in this game?' Will the interrogator decide wrongly as often when the game is played like this as he does when the game is played between a man and a woman?" (434). Now, of course, it is possible to interpret Turing as here intending to say what he seems literally to say, namely, that the new game is one in which the computer must pretend to be a woman, and the other participant in the game is a woman. (For discussion, see, for example, Genova (1994) and Traiger (2000).) And it is also possible to interpret Turing as intending to say that the new game is one in which the computer must pretend to be a woman, and the other participant in the game is a man who must also pretend to be a woman. However, as Copeland (2000), Piccinini (2000), and Moor (2001) convincingly argue, the rest of Turing's article, and material in other articles that Turing wrote at around the same time, very strongly support the claim that Turing actually intended the standard interpretation that we gave above, viz. that the computer is to pretend to be a human being, and the other participant in the game is a human being of unspecified gender. Moreover, as Moor (2001) argues, there is no reason to think that one would get a better test if the computer must pretend to be a woman and the other participant in the game is a man pretending to be a woman; and, indeed, there is some reason to think that one would get a worse test. Perhaps it would make no difference to the effectiveness of the test if the computer must pretend to be a woman, and the other participant is a woman (any more than it would make a difference if the computer must pretend to be an accountant and the other participant is an accountant); however, this consideration is simply insufficient to outweigh the strong textual evidence that supports the standard interpretation of the imitation game that we gave at the beginning of our discussion of Turing (1950). (For a dissenting view about many of the matters discussed in this paragraph, see Sterrett (2000; 2020).) ### 3.2 Turing's Predictions As we noted earlier, Turing (1950) makes the claim that: > > I believe that in about fifty years' time it will be possible to > programme computers, with a storage capacity of about 109, > to make them play the imitation game so well that an average > interrogator will not have more than 70 percent chance of making the > right identification after five minutes of questioning. ... I > believe that at the end of the century the use of words and general > educated opinion will have altered so much that one will be able to > speak of machines thinking without expecting to be contradicted. > Most commentators contend that this claim has been shown to be mistaken: in the year 2000, no-one was able to program computers to make them play the imitation game so well that an average interrogator had no more than a 70% chance of making the correct identification after five minutes of questioning. Copeland (2000) argues that this contention is seriously mistaken: "about fifty years" is by no means "exactly fifty years," and it remains open that we may soon be able to do the required programming. Against this, it should be noted that Turing (1950) goes on immediately to refer to how things will be "at the end of the century," which suggests that not too much can be read into the qualifying "about." However, as Copeland (2000) points out, there are other more cautious predictions that Turing makes elsewhere (e.g., that it would be "at least 100 years" before a machine was able to pass an unrestricted version of his test); and there are other predictions that are made in Turing (1950) that seem to have been vindicated. In particular, it is plausible to claim that, in the year 2000, educated opinion had altered to the extent that, in many quarters, one could speak of the possibility of machines' thinking--and of machines' learning--without expecting to be contradicted. As Moor (2001) points out, "machine intelligence" is not the oxymoron that it might have been taken to be when Turing first started thinking about these matters. ### 3.3 A Useful Distinction There are two different theoretical claims that are run together in many discussions of The Turing Test that can profitably be separated. One claim holds that the general scheme that is described in Turing's Imitation Game provides a good test for the presence of intelligence. (If something can pass itself off as a person under sufficiently demanding test conditions, then we have very good reason to suppose that that thing is intelligent.) Another claim holds that an appropriately programmed computer could pass the kind of test that is described in the first claim. We might call the first claim "The Turing Test Claim" and the second claim "The Thinking Machine Claim". Some objections to the claims made in Turing (1950) are objections to the Thinking Machine Claim, but not objections to the Turing Test Claim. (Consider, for example, the argument of Searle (1982), which we discuss further in Section 6.) However, other objections are objections to the Turing Test Claim. Until we get to Section 6, we shall be confining our attention to discussions of the Turing Test Claim. ### 3.4 A Further Note In this article, we follow the standard philosophical convention according to which "a mind" means "at least one mind". If "passing the Turing Test" implies intelligence, then "passing the Turing Test" implies the presence of at least one mind. We cannot here explore recent discussions of "swarm intelligence", "collective intelligence", and the like. However, it is surely clear that two people taking turns could "pass the Turing Test" in circumstances in which we should be very reluctant to say that there is a "collective mind" that has the minds of the two as components. ## 4. Assessment of the Current Standing of The Turing Test Given the initial distinction that we made between different ways in which the expression The Turing Test gets interpreted in the literature, it is probably best to approach the question of the assessment of the current standing of The Turing Test by dividing cases. True enough, we think that there is a correct interpretation of exactly what test it is that is proposed by Turing (1950); but a complete discussion of the current standing of The Turing Test should pay at least some attention to the current standing of other tests that have been mistakenly supposed to be proposed by Turing (1950). There are a number of main ideas to be investigated. First, there is the suggestion that The Turing Test provides logically necessary and sufficient conditions for the attribution of intelligence. Second, there is the suggestion that The Turing Test provides logically sufficient--but not logically necessary--conditions for the attribution of intelligence. Third, there is the suggestion that The Turing Test provides "criteria"--defeasible sufficient conditions--for the attribution of intelligence. Fourth--and perhaps not importantly distinct from the previous claim--there is the suggestion that The Turing Test provides (more or less strong) probabilistic support for the attribution of intelligence. We shall consider each of these suggestions in turn. ### 4.1 (Logically) Necessary and Sufficient Conditions It is doubtful whether there are very many examples of people who have explicitly claimed that The Turing Test is meant to provide conditions that are both logically necessary and logically sufficient for the attribution of intelligence. (Perhaps Block (1981) is one such case.) However, some of the objections that have been proposed against The Turing Test only make sense under the assumption that The Turing Test does indeed provide logically necessary and logically sufficient conditions for the attribution of intelligence; and many more of the objections that have been proposed against The Turing Test only make sense under the assumption that The Turing Test provides necessary and sufficient conditions for the attribution of intelligence, where the modality in question is weaker than the strictly logical, e.g., nomic or causal. Consider, for example, those people who have claimed that The Turing Test is chauvinistic; and, in particular, those people who have claimed that it is surely logically possible for there to be something that possesses considerable intelligence, and yet that is not able to pass The Turing Test. (Examples: Intelligent creatures might fail to pass The Turing Test because they do not share our way of life; intelligent creatures might fail to pass The Turing Test because they refuse to engage in games of pretence; intelligent creatures might fail to pass The Turing Test because the pragmatic conventions that govern the languages that they speak are so very different from the pragmatic conventions that govern human languages. Etc.) None of this can constitute objections to The Turing Test unless The Turing Test delivers necessary conditions for the attribution of intelligence. French (1990) offers ingenious arguments that are intended to show that "the Turing Test provides a guarantee not of intelligence, but of culturally-oriented intelligence." But, of course, anything that has culturally-oriented intelligence has intelligence; so French's objections cannot be taken to be directed towards the idea that The Turing Test provides sufficient conditions for the attribution of intelligence. Rather--as we shall see later--French supposes that The Turing Test establishes sufficient conditions that no machine will ever satisfy. That is, in French's view, what is wrong with The Turing Test is that it establishes utterly uninteresting sufficient conditions for the attribution of intelligence. Floridi and Chiriatti (2020: 683) say that The Turing Test provides necessary but insufficient conditions for intelligence: not passing The Turing Test disqualifies an AI from being intelligent, but passing The Turing Test is not sufficient to qualify an AI as intelligent. However, they also say that "any reader ... will be well acquainted with the nature of the test, so we shall not describe it." The account that they would give of The Turing Test must be quite different from the account of The Turing Test that we have been presenting. ### 4.2 Logically Sufficient Conditions There are many philosophers who have supposed that The Turing Test is intended to provide logically sufficient conditions for the attribution of intelligence. That is, there are many philosophers who have supposed that The Turing Test claims that it is logically impossible for something that lacks intelligence to pass The Turing Test. (Often, this supposition goes with an interpretation according to which passing The Turing Test requires rather a lot, e.g., producing behavior that is indistinguishable from human behavior over an entire lifetime.) There are well-known arguments against the claim that passing The Turing Test--or any other purely behavioral test--provides logically sufficient conditions for the attribution of intelligence. The standard objection to this kind of analysis of intelligence (mind, thought) is that a being whose behavior was produced by "brute force" methods ought not to count as intelligent (as possessing a mind, as having thoughts). Consider, for example, Ned Block's Blockhead. Blockhead is a creature that looks just like a human being, but that is controlled by a "game-of-life look-up tree," i.e. by a tree that contains a programmed response for every discriminable input at each stage in the creature's life. If we agree that Blockhead is logically possible, and if we agree that Blockhead is not intelligent (does not have a mind, does not think), then Blockhead is a counterexample to the claim that the Turing Test provides a logically sufficient condition for the ascription of intelligence. After all, Blockhead could be programmed with a look-up tree that produces responses identical with the ones that you would give over the entire course of your life (given the same inputs). There are perhaps only two ways in which someone who claims that The Turing Test offers logically sufficient conditions for the attribution of intelligence can respond to Block's argument. First, it could be denied that Blockhead is a logical possibility; second, it could be claimed that Blockhead would be intelligent (have a mind, think). In order to deny that Blockhead is a logical possibility, it seems that what needs to be denied is the commonly accepted link between conceivability and logical possibility: it certainly seems that Blockhead is conceivable, and so, if (properly circumscribed) conceivability is sufficient for logical possibility, then it seems that we have good reason to accept that Blockhead is a logical possibility. Since it would take us too far away from our present concerns to explore this issue properly, we merely note that it remains a controversial question whether (properly circumscribed) conceivability is sufficient for logical possibility. (For further discussion of this issue, see Crooke (2002).) The question of whether Blockhead is intelligent (has a mind, thinks) may seem straightforward, but--despite Block's confident assertion that Blockhead "has all of the intelligence of a toaster"--it is not obvious that we should deny that Blockhead is intelligent. Blockhead may not be a particularly efficient processor of information; but it is at least a processor of information, and that--in combination with the behavior that is produced as a result of the processing of information--might well be taken to be sufficient grounds for the attribution of some level of intelligence to Blockhead. For further critical discussion of the argument of Block (1981), see McDermott (2014), and Pautz and Stoljar (2019). ### 4.3 Criteria In his Philosophical Investigations, Wittgenstein famously writes: "An 'inner process' stands in need of outward criteria" (580). Exactly what Wittgenstein meant by this remark is unclear, but one way in which it might be interpreted is as follows: in order to be justified in ascribing a "mental state" to some entity, there must be some true claims about the observable behavior of that entity that, (perhaps) together with other true claims about that entity (not themselves couched in "mentalistic" vocabulary), entail that the entity has the mental state in question. If no true claims about the observable behavior of the entity can play any role in the justification of the ascription of the mental state in question to the entity, then there are no grounds for attributing that kind of mental state to the entity. The claim that, in order to be justified in ascribing a mental state to an entity, there must be some true claims about the observable behavior of that entity that alone--i.e. without the addition of any other true claims about that entity--entail that the entity has the mental state in question, is a piece of philosophical behaviorism. It may be--for all that we are able to argue--that Wittgenstein was a philosophical behaviorist; it may be--for all that we are able to argue--that Turing was one, too. However, if we go by the letter of the account given in the previous paragraph, then all that need follow from the claim that the Turing Test is criterial for the ascription of intelligence (thought, mind) is that, when other true claims (not themselves couched in terms of mentalistic vocabulary) are conjoined with the claim that an entity has passed the Turing Test, it then follows that the entity in question has intelligence (thought, mind). (Note that the parenthetical qualification that the additional true claims not be couched in terms of mentalistic vocabulary is only one way in which one might try to avoid the threat of trivialization. The difficulty is that the addition of the true claim that an entity has a mind will always produce a set of claims that entails that that entity has a mind, no matter what other claims belong to the set!) To see how the claim that the Turing Test is merely criterial for the ascription of intelligence differs from the logical behaviorist claim that the Turing Test provides logically sufficient conditions for the ascription of intelligence, it suffices to consider the question of whether it is nomically possible for there to be a "hand simulation" of a Turing Test program. Many people have supposed that there is good reason to deny that Blockhead is a nomic (or physical) possibility. For example, in The Physics of Immortality, Frank Tipler provides the following argument in defence of the claim that it is physically impossible to "hand simulate" a Turing-Test-passing program: > > If my earlier estimate that the human brain can code as much as > 1015 bits is correct, then since an average book codes > about 106 bits ... it would require more than 100 > million books to code the human brain. It would take at least thirty > five-story main university libraries to hold this many books. We know > from experience that we can access any memory in our brain in about > 100 seconds, so a hand simulation of a Turing Test-passing program > would require a human being to be able to take off the shelf, glance > through, and return to the shelf all of these 100 million books in 100 > seconds. If each book weighs about a pound (0.5 kilograms), and on the > average the book moves one yard (one meter) in the process of taking > it off the shelf and returning it, then in 100 seconds the energy > consumed in just moving the books is 3 x 1019 joules; the > rate of energy consumption is 3 x 1011 megawatts. Since a > human uses energy at a normal rate of 100 watts, the power required is > the bodily power of 3 x 1015 human beings, about a million > times the current population of the entire earth. A typical large > nuclear power plant has a power output of 1,000 megawatts, so a hand > simulation of the human program requires a power output equal to that > of 300 million large nuclear power plants. As I said, a man can no > more hand-simulate a Turing Test-passing program than he can jump to > the Moon. In fact, it is far more difficult. (40) > While there might be ways in which the details of Tipler's argument could be improved, the general point seems clearly right: the kind of combinatorial explosion that is required for a look-up tree for a human being is ruled out by the laws and boundary conditions that govern the operations of the physical world. But, if this is right, then, while it may be true that Blockhead is a logical possibility, it follows that Blockhead is not a nomic or physical possibility. And then it seems natural to hold that The Turing Test does indeed provide nomically sufficient conditions for the attribution of intelligence: given everything else that we already know--or, at any rate, take ourselves to know--about the universe in which we live, we would be fully justified in concluding that anything that succeeds in passing The Turing Test is, indeed, intelligent (possessed of a mind, and so forth). There are ways in which the argument in the previous paragraph might be resisted. At the very least, it is worth noting that there is a serious gap in the argument that we have just rehearsed. Even if we can rule out "hand simulation" of intelligence, it does not follow that we have ruled out all other kinds of mere simulation of intelligence. Perhaps--for all that has been argued so far--there are nomically possible ways of producing mere simulations of intelligence. But, if that's right, then passing The Turing Test need not be so much as criterial for the possession of intelligence: it need not be that given everything else that we already know--or, at any rate, take ourselves to know--about the universe in which we live, we would be fully justified in concluding that anything that succeeds in passing The Turing Test is, indeed, intelligent (possessed of a mind, and so forth). (McDermott (2014) calculates that a look-up table for a participant who makes 50 conversational exchanges would have about 1022278 nodes. It is tempting to take this calculation to establish that it is neither nomically nor physically possible for there to be a "hand simulation" of a Turing Test program, on the grounds that the required number of nodes could not be fitted into a space much much larger than the entire observable universe.) ### 4.4 Probabilistic Support When we look at the initial formulation that Turing provides of his test, it is clear that he thought that the passing of the test would provide probabilistic support for the hypothesis of intelligence. There are at least two different points to make here. First, the prediction that Turing makes is itself probabilistic: Turing predicts that, in about fifty years from the time of his writing, it will be possible to programme digital computers to make them play the imitation game so well that an average interrogator will have no more than a seventy per cent chance of making the right identification after five minutes of questioning. Second, the probabilistic nature of Turing's prediction provides good reason to think that the test that Turing proposes is itself of a probabilistic nature: a given level of success in the imitation game produces--or, at any rate, should produce--a specifiable level of increase in confidence that the participant in question is intelligent (has thoughts, is possessed of a mind). Since Turing doesn't tell us how he supposes that levels of success in the imitation game correlate with increases in confidence that the participant in question is intelligent, there is a sense in which The Turing Test is greatly underspecified. Relevant variables clearly include: the length of the period of time over which the questioning in the game takes place (or, at any rate, the "amount" of questioning that takes place); the skills and expertise of the interrogator (this bears, for example, on the "depth" and "difficulty" of the questioning that takes place); the skills and expertise of the third player in the game; and the number of independent sessions of the game that are run (particularly when the other participants in the game differ from one run to the next). Clearly, a machine that is very successful in many different runs of the game that last for quite extended periods of time and that involve highly skilled participants in the other roles has a much stronger claim to intelligence than a machine that has been successful in a single, short run of the game with highly inexpert participants. That a machine has succeeded in one short run of the game against inexpert opponents might provide some reason for increase in confidence that the machine in question is intelligent: but it is clear that results on subsequent runs of the game could quickly overturn this initial increase in confidence. That a machine has done much better than chance over many long runs of the imitation game against a variety of skilled participants surely provides much stronger evidence that the machine is intelligent. (Given enough evidence of this kind, it seems that one could be quite confident indeed that the machine is intelligent, while still--of course--recognizing that one's judgment could be overturned by further evidence, such as a series of short runs in which it does much worse than chance against participants who use the same strategy over and over to expose the machine as a machine.) The probabilistic nature of The Turing Test is often overlooked. True enough, Moor (1976, 2001)--along with various other commentators--has noted that The Turing Test is "inductive," i.e. that "The Turing Test" provides no more than defeasible evidence of intelligence. However, it is one thing to say that success in "a rigorous Turing test" provides no more than defeasible evidence of intelligence; it is quite another to note the probabilistic features to which we have drawn attention in the preceding paragraph. Consider, for example, Moor's observation (Moor 2001:83) that "... inductive evidence gathered in a Turing test can be outweighed by new evidence. ... If new evidence shows that a machine passed the Turing Test by remote control run by a human behind the scenes, then reassessment is called for." This--and other similar passages--seems to us to suggest that Moor supposes that a "rigorous Turing test" is a one-off event in which the machine either succeeds or fails. But this interpretation of The Turing Test is vulnerable to the kind of objection lodged by Bringsjord (1994): even on a moderately long single run with relatively expert participants, it may not be all that unlikely that an unintelligent machine serendipitously succeeds in the imitation game. In our view, given enough sufficiently long runs with different sufficiently expert participants, the likelihood of serendipitous success can be made as small as one wishes. Thus, while Bringsjord's "argument from serendipity" has force against some versions of The Turing Test, it has no force against the most plausible interpretation of the test that Turing actually proposed. It is worth noting that it is quite easy to construct more sophisticated versions of "The Imitation Game" that yield more fine-grained statistical data. For example, rather than getting the judges to issue Yes/No verdicts about both of the participants in the game, one could get the judges to provide probabilistic answers. ("I give a 75% probability to the claim that A is the machine, and only 25% probability to the claim that B is the machine.") This point is important when one comes to consider criticisms of the "methodology" implicit in "The Turing Test". (For further discussion of the probabilistic nature of "The Turing Test", see Shieber (2007).) ## 5. Alternative Tests Some of the literature about The Turing Test is concerned with questions about the framing of a test that can provide a suitable guide to future research in the area of Artificial Intelligence. The idea here is very simple. Suppose that we have the ambition to produce an artificially intelligent entity. What tests should we take as setting the goals that putatively intelligent artificial systems should achieve? Should we suppose that The Turing Test provides an appropriate goal for research in this field? In assessing these proposals, there are two different questions that need to be borne in mind. First, there is the question whether it is a useful goal for AI research to aim to make a machine that can pass the given test (administered over the specified length of time, at the specified degree of success). Second, there is the question of the appropriate conclusion to draw about the mental capacities of a machine that does manage to pass the test (administered over the specified length of time, at the specified degree of success). Opinion on these questions is deeply divided. Some people suppose that The Turing Test does not provide a useful goal for research in AI because it is far too difficult to produce a system that can pass the test. Other people suppose that The Turing Test does not provide a useful goal for research in AI because it sets a very narrow target (and thus sets unnecessary restrictions on the kind of research that gets done). Some people think that The Turing Test provides an entirely appropriate goal for research in AI; while other people think that there is a sense in which The Turing Test is not really demanding enough, and who suppose that The Turing Test needs to be extended in various ways in order to provide an appropriate goal for AI. We shall consider some representatives of each of these positions in turn. There are some people who continue to endorse The Turing Test. For example, Neufeld and Finnestad (2020a) (2020b) argue that The Turing Test is no barrier to progress in AI, requires no significant redefinition, and does not shut down other avenues of investigation. Maybe we do better just to take The Turing Test to define a watershed rather than a threshold towards which we might hope to make incremental progression. ### 5.1 The Turing Test is Too Hard Some people have claimed that The Turing Test doesn't set an appropriate goal for current research in AI because we are plainly so far away from attaining this goal. Amongst these people there are some who have gone on to offer reasons for thinking that it is doubtful that we shall ever be able to create a machine that can pass The Turing Test--or, at any rate, that it is doubtful that we shall be able to do this at any time in the foreseeable future. Perhaps the most interesting arguments of this kind are due to French (1990); at any rate, these are the arguments that we shall go on to consider. (Cullen (2009) sets out similar considerations.) According to French, The Turing Test is "virtually useless" as a real test of intelligence, because nothing without a "human subcognitive substrate" could pass the test, and yet the development of an artificial "human cognitive substrate" is almost impossibly difficult. At the very least, there are straightforward sets of questions that reveal "low-level cognitive structure" and that--in French's view--are almost certain to be successful in separating human beings from machines. First, if interrogators are allowed to draw on the results of research into, say, associative priming, then there is data that will very plausibly separate human beings from machines. For example, there is research that shows that, if humans are presented with series of strings of letters, they require less time to recognize that a string is a word (in a language that they speak) if it is preceded by a related word (in the language that they speak), rather than by an unrelated word (in the language that they speak) or a string of letters that is not a word (in the language that they speak). Provided that the interrogator has accurate data about average recognition times for subjects who speak the language in question, the interrogator can distinguish between the machine and the human simply by looking at recognition times for appropriate series of strings of letters. Or so says French. It isn't clear to us that this is right. After all, the design of The Turing Test makes it hard to see how the interrogator will get reliable information about response times to series of strings of symbols. The point of putting the computer in a separate room and requiring communication by teletype was precisely to rule out certain irrelevant ways of identifying the computer. If these requirements don't already rule out identification of the computer by the application of tests of associative priming, then the requirements can surely be altered to bring it about that this is the case. (Perhaps it is also worth noting that administration of the kind of test that French imagines is not ordinary conversation; nor is it something that one would expect that any but a few expert interrogators would happen upon. So, even if the circumstances of The Turing Test do not rule out the kind of procedure that French here envisages, it is not clear that The Turing Test will be impossibly hard for machines to pass.) Second, at a slightly higher cognitive level, there are certain kinds of "ratings games" that French supposes will be very reliable discriminators between humans and machines. For instance, the "Neologism Ratings Game"--which asks participants to rank made-up words on their appropriateness as names for given kinds of entities--and the "Category Rating Game"--which asks participants to rate things of one category as things of another category--are both, according to French, likely to prove highly reliable in discriminating between humans and machines. For, in the first case, the ratings that humans make depend upon large numbers of culturally acquired associations (which it would be well-nigh impossible to identify and describe, and hence which it would (arguably) be well-nigh impossible to program into a computer). And, in the second case, the ratings that people actually make are highly dependent upon particular social and cultural settings (and upon the particular ways in which human life is experienced). To take French's examples, there would be widespread agreement amongst competent English speakers in the technologically developed Western world that "Flugblogs" is not an appropriate name for a breakfast cereal, while "Flugly" is an appropriate name for a child's teddy bear. And there would also be widespread agreement amongst competent speakers of English in the developed world that pens rate higher as weapons than grand pianos rate as wheelbarrows. Again, there are questions that can be raised about French's argument here. It is not clear to us that the data upon which the ratings games rely is as reliable as French would have us suppose. (At least one of us thinks that "Flugly" would be an entirely inappropriate name for a child's teddy bear, a response that is due to the similarity between the made-up word "Flugly" and the word "Fugly," that had some currency in the primarily undergraduate University college that we both attended. At least one of us also thinks that young children would very likely be delighted to eat a cereal called "Flugblogs," and that a good answer to the question about ratings pens and grand pianos is that it all depends upon the pens and grand pianos in question. What if the grand piano has wheels? What if the opponent has a sword or a sub-machine gun? It isn't obvious that a refusal to play this kind of ratings game would necessarily be a give-away that one is a machine.) Moreover, even if the data is reliable, it is not obvious that any but a select group of interrogators will hit upon this kind of strategy for trying to unmask the machine; nor is it obvious that it is impossibly hard to build a machine that is able to perform in the way in which typical humans do on these kinds of tests. In particular, if--as Turing assumes--it is possible to make learning machines that can be "trained up" to learn how to do various kinds of tasks, then it is quite unclear why these machines couldn't acquire just the same kinds of "subcognitive competencies" that human children acquire when they are "trained up" in the use of language. There are other reasons that have been given for thinking that The Turing Test is too hard (and, for this reason, inappropriate in setting goals for current research into artificial intelligence). In general, the idea is that there may well be features of human cognition that are particularly hard to simulate, but that are not in any sense essential for intelligence (or thought, or possession of a mind). The problem here is not merely that The Turing Test really does test for human intelligence; rather, the problem here is the fact--if indeed it is a fact--that there are quite inessential features of human intelligence that are extraordinarily difficult to replicate in a machine. If this complaint is justified--if, indeed, there are features of human intelligence that are extraordinarily difficult to replicate in machines, and that could and would be reliably used to unmask machines in runs of The Turing Test--then there is reason to worry about the idea that The Turing Test sets an appropriate direction for research in artificial intelligence. However, as our discussion of French shows, there may be reason for caution in supposing that the kinds of considerations discussed in the present section show that we are already in a position to say that The Turing Test does indeed set inappropriate goals for research in artificial intelligence. ### 5.2 The Turing Test is Too Narrow There are authors who have suggested that The Turing Test does not set a sufficiently broad goal for research in the area of artificial intelligence. Amongst these authors, there are many who suppose that The Turing Test is too easy. (We go on to consider some of these authors in the next sub-section.) But there are also some authors who have supposed that, even if the goal that is set by The Turing Test is very demanding indeed, it is nonetheless too restrictive. Objection to the notion that the Turing Test provides a logically sufficient condition for intelligence can be adapted to the goal of showing that the Turing Test is too restrictive. Consider, for example, Gunderson (1964). Gunderson has two major complaints to make against The Turing Test. First, he thinks that success in Turing's Imitation Game might come for reasons other than the possession of intelligence. But, second, he thinks that success in the Imitation Game would be but one example of the kinds of things that intelligent beings can do and--hence--in itself could not be taken as a reliable indicator of intelligence. By way of analogy, Gunderson offers the case of a vacuum cleaner salesman who claims that his product is "all-purpose" when, in fact, all it does is to suck up dust. According to Gunderson, Turing is in the same position as the vacuum cleaner salesman if he is prepared to say that a machine is intelligent merely on the basis of its success in the Imitation Game. Just as "all purpose" entails the ability to do a range of things, so, too, "thinking" entails the possession of a range of abilities (beyond the mere ability to succeed in the Imitation Game). There is an obvious reply to the argument that we have here attributed to Gunderson, viz. that a machine that is capable of success in the Imitation Game is capable of doing a large range of different kinds of things. In order to carry out a conversation, one needs to have many different kinds of cognitive skills, each of which is capable of application in other areas. Apart from the obvious general cognitive competencies--memory, perception, etc.--there are many particular competencies--rudimentary arithmetic abilities, understanding of the rules of games, rudimentary understanding of national politics, etc.--which are tested in the course of repeated runs of the Imitation Game. It is inconceivable that that there be a machine that is startlingly good at playing the Imitation Game, and yet unable to do well at any other tasks that might be assigned to it; and it is equally inconceivable that there is a machine that is startlingly good at the Imitation Game and yet that does not have a wide range of competencies that can be displayed in a range of quite disparate areas. To the extent that Gunderson considers this line of reply, all that he says is that there is no reason to think that a machine that can succeed in the Imitation Game must have more than a narrow range of abilities; we think that there is no reason to believe that this reply should be taken seriously. More recently, Erion (2001) has defended a position that has some affinity to that of Gunderson. According to Erion, machines might be "capable of outperforming human beings in limited tasks in specific environments, [and yet] still be unable to act skillfully in the diverse range of situations that a person with common sense can" (36). On one way of understanding the claim that Erion makes, he too believes that The Turing Test only identifies one amongst a range of independent competencies that are possessed by intelligent human beings, and it is for this reason that he proposes a more comprehensive "Cartesian Test" that "involves a more careful examination of a creature's language, [and] also tests the creature's ability to solve problems in a wide variety of everyday circumstances" (37). In our view, at least when The Turing Test is properly understood, it is clear that anything that passes The Turing Test must have the ability to solve problems in a wide variety of everyday circumstances (because the interrogators will use their questions to probe these--and other--kinds of abilities in those who play the Imitation Game). ### 5.3 The Turing Test is Too Easy There are authors who have suggested that The Turing Test should be replaced with a more demanding test of one kind or another. It is not at all clear that any of these tests actually proposes a better goal for research in AI than is set by The Turing Test. However, in this section, we shall not attempt to defend that claim; rather, we shall simply describe some of the further tests that have been proposed, and make occasional comments upon them. (One preliminary point upon which we wish to insist is that Turing's Imitation Game was devised against the background of the limitations imposed by then current technology. It is, of course, not essential to the game that tele-text devices be used to prevent direct access to information about the sex or genus of participants in the game. We shall not advert to these relatively mundane kinds of considerations in what follows.) #### 5.3.1 The Total Turing Test Harnad (1989, 1991) claims that a better test than The Turing Test will be one that requires responses to all of our inputs, and not merely to text-formatted linguistic inputs. That is, according to Harnad, the appropriate goal for research in AI has to be to construct a robot with something like human sensorimotor capabilities. Harnad also considers the suggestion that it might be an appropriate goal for AI to aim for "neuromolecular indistinguishability," but rejects this suggestion on the grounds that once we know how to make a robot that can pass his Total Turing Test, there will be no problems about mind-modeling that remain unsolved. It is an interesting question whether the test that Harnad proposes sets a more appropriate goal for AI research. In particular, it seems worth noting that it is not clear that there could be a system that was able to pass The Turing Test and yet that was not able to pass The Total Turing Test. Since Harnad himself seems to think that it is quite likely that "full robotic capacities [are] ... necessary to generate ... successful linguistic performance," it is unclear why there is reason to replace The Turing Test with his extended test. (This point against Harnad can be found in Hauser (1993:227), and elsewhere.) #### 5.3.2 The Lovelace Test Bringsjord et al. (2001) propose that a more satisfactory aim for AI is provided by a certain kind of meta-test that they call the Lovelace Test. They say that an artificial agent A, designed by human H, passes the Lovelace Test just in case three conditions are jointly satisfied: (1) the artificial agent A produces output O; (2) A's outputting O is not the result of a fluke hardware error, but rather the result of processes that A can repeat; and (3) H--or someone who knows what H knows and who has H's resources--cannot explain how A produced O by appeal to A's architecture, knowledge-base and core functions. Against this proposal, it seems worth noting that there are questions to be raised about the interpretation of the third condition. If a computer program is long and complex, then no human agent can explain in complete detail how the output was produced. (Why did the computer output 3.16 rather than 3.17?) But if we are allowed to give a highly schematic explanation--the computer took the input, did some internal processing and then produced an answer--then it seems that it will turn out to be very hard to support the claim that human agents ever do anything genuinely creative. (After all, we too take external input, perform internal processing, and produce outputs.) What is missing from the account that we are considering is any suggestion about the appropriate level of explanation that is to be provided. It is quite unclear why we should suppose that there is a relevant difference between people and machines at any level of explanation; but, if that's right, then the test in question is trivial. (One might also worry that the proposed test rules out by fiat the possibility that creativity can be best achieved by using genuine randomising devices.) #### 5.3.3 The Truly Total Turing Test Schweizer (1998) claims that a better test than The Turing Test will advert to the evolutionary history of the subjects of the test. When we attribute intelligence to human beings, we rely on an extensive historical record of the intellectual achievements of human beings. On the basis of this historical record, we are able to claim that human beings are intelligent; and we can rely upon this claim when we attribute intelligence to individual human beings on the basis of their behavior. According to Schweizer, if we are to attribute intelligence to machines, we need to be able to advert to a comparable historical record of cognitive achievements. So, it will only be when machines have developed languages, written scientific treatises, composed symphonies, invented games, and the like, that we shall be in a position to attribute intelligence to individual machines on the basis of their behavior. Of course, we can still use The Turing Test to determine whether an individual machine is intelligent: but our answer to the question won't depend merely upon whether or not the machine is successful in The Turing Test; there is the further "evolutionary" condition that also must be satisfied. Against Schweizer, it seems worth noting that it is not at all clear that our reason for granting intelligence to other humans on the basis of their behavior is that we have prior knowledge of the collective cognitive achievements of human beings. 5.3.4 Further Proposals Damassino (2020) suggests that it would be better to require test subjects to produce an enquiry in which performance is assessed along three dimensions: (a) comparison with human performance; (b) success in completing the enquiry; and (c) efficiency in completing the enquiry (minimisation of the number of questions asked in completing the enquiry). The motivation given for this proposal is that, because The Turing Test attracts projects whose primary ambition is to fool judges, it is concerned with whether or how well test subjects perform on their allocated tasks. It seems to us that there is nothing here that impugns The Turing Test. It does not count against The Turing Test that public competitions based on it with prizes attached lead to gaming, given that everyone knows that those prizes are being awarded to entries that clearly do not pass The Turing Test. If anything is impugned here, it is the public competitions, rather than The Turing Test. Kulikov (2020) suggests that there is value in considering Preferential Engagement Tests or Meaningful Engagement Tests. Even though computers can now beat the best humans at chess, many people prefer to play chess with humans rather than with expert chess-playing computers. Perhaps, even if computers could pass The Turing Test, people would prefer to carry on conversations with humans rather than with expert conversational computers. We think that this kind of speculation relies upon assumptions about what could make for expert conversational partners. If our conversational partners need to be able to update information about their surroundings in real time--for example, while watching a game of football--then we will not think that there is a direct path from GPT-3 to expert conversational partners. If only androids can be expert conversational partners, then it is less clear that Preferential Engagement Tests or Meaningful Engagement Tests will track anything other than anthropocentric bias. ### 5.4 Should the Turing Test be Considered Harmful? Perhaps the best known attack on the suggestion that The Turing Test provides an appropriate research goal for AI is due to Hayes and Ford (1995). Among the controversial claims that Hayes and Ford make, there are at least the following: 1. Turing suggested the imitation game as a definite goal for program of research. 2. Turing intended The Turing Test to be a gender test rather than a species test. 3. The task of trying to make a machine that is successful in The Turing Test is so extremely difficult that no one could seriously adopt the creation of such a machine as a research goal. 4. The Turing Test suffers from the basic design flaw that it sets out to confirm a "null hypothesis", viz. that there is no difference in behavior between certain machines and humans. 5. No null effect experiment can provide an adequate criterion for intelligence, since the question can always arise that the judges did not look hard enough (and did not raise the right kinds of questions). But, if this question is left open, then there is no stable endpoint of enquiry. 6. Null effect experiments cannot measure anything: The Turing Test can only test for complete success. ("A man who failed to seem feminine in 10% of what he said would almost always fail the Imitation game.") 7. The Turing Test is really a test of the ability of the human species to discriminate its members from human imposters. ("The gender test ... is a test of making a mechanical transvestite.") 8. The Turing Test is circular: what it fails to detect cannot be "intelligence" or"humanity", since many humans would fail The Turing Test. Indeed, "since one of the players must be judged to be a machine, half the human population would fail the species test". 9. The perspective of The Turing Test is arrogant and parochial: it mistakenly assumes that we can understand human cognition without first obtaining a firm grasp of the basic principles of cognition. 10. The Turing Test does not admit of weaker, different, or even stronger forms of intelligence than those deemed human. Some of these claims seem straightforwardly incorrect. Consider (h), for example. In what sense can it be claimed that 50% of the human population would fail "the species test"? If "the species test" requires the interrogator to decide which of two people is a machine, why should it be thought that the verdict of the interrogator has any consequences for the assessment of the intelligence of the person who is judged to be a machine? (Remember, too, that one of the conditions for "the species test"--as it is originally described by Hayes and Ford--is that one of the contestants is a machine. While the machine can "demonstrate" its intelligence by winning the imitation game, a person cannot "demonstrate" their lack of intelligence by failing to win.) It seems wrong to say that The Turing Test is defective because it is a "null effect experiment". True enough, there is a sense in which The Turing Test does look for a "null result": if ordinary judges in the specified circumstances fail to identify the machine (at a given level of success), then there is a given likelihood that the machine is intelligent. But the point of insisting on "ordinary judges" in the specified circumstances is precisely to rule out irrelevant ways of identifying the machine (i.e. ways of identifying the machine that are not relevant to the question whether it is intelligent). There might be all kinds of irrelevant differences between a given kind of machine and a human being--not all of them rendered undetectable by the experimental set-up that Turing describes--but The Turing Test will remain a good test provided that it is able to ignore these irrelevant differences. It also seems doubtful that it is a serious failing of The Turing Test that it can only test for "complete success". On the one hand, if a man has a one in ten chance of producing a claim that is plainly not feminine, then we can compute the chance that he will be discovered in a game in which he answers N questions--and, if N is sufficiently small, then it won't turn out that "he would almost always fail to win". On the other hand, as we noted at the end of Section 4.4 above, if one were worried about the "YES/NO" nature of "The Turing Test", then one could always get the judges to produce probabilistic verdicts instead. This change preserves the character of The Turing Test, but gives it scope for greater statistical sophistication. While there are (many) other criticisms that can be made of the claims defended by Hayes and Ford (1995), it should be acknowledged that they are right to worry about the suggestion that The Turing Test provides the defining goal for research in AI. There are various reasons why one should be loathe to accept the proposition that the one central ambition of AI research is to produce artificial people. However it is worth pointing out that there is no reason to think that Turing supposed that The Turing Test defined the field of AI research (and there is not much evidence that any other serious thinkers have thought so either). Turing himself was well aware that there might be non-human forms of intelligence--cf. (j) above. However, all of this remains consistent with the suggestion that it is quite appropriate to suppose that The Turing Test sets one long term goal for AI research: one thing that we might well aim to do eventually is to produce artificial people. If--as Hayes and Ford claim--that task is almost impossibly difficult, then there is no harm in supposing that the goal is merely an ambit goal to which few resources should be committed; but we might still have good reason to allow that it is a goal. Others who have argued that we need to "move beyond" The Turing Test include Hernandez-Orallo (2000) (2020) and Marcus (2020). ## 6. The Chinese Room There are many different objections to The Turing Test which have surfaced in the literature during the past fifty years, but which we have not yet discussed. We cannot hope to canvass all of these objections here. However, there is one argument--Searle's "Chinese Room" argument--that is mentioned so often in connection with the Turing Test that we feel obliged to end with some discussion of it. In Minds, Brains and Programs and elsewhere, John Searle argues against the claim that "appropriately programmed computers literally have cognitive states" (64). Clearly enough, Searle is here disagreeing with Turing's claim that an appropriately programmed computer could think. There is much that is controversial about Searle's argument; we shall just consider one way of understanding what it is that he is arguing for. The basic structure of Searle's argument is very well known. We can imagine a "hand simulation" of an intelligent agent--in the case described, a speaker of a Chinese language--in circumstances in which we might well be very reluctant to allow that there is any appropriate intelligence lying behind the simulated behavior. (Thus, what we are invited to suppose is a logical possibility is not so very different from what Block invites us to suppose is a logical possibility. However, the argument that Searle goes on to develop is rather different from the argument that Block defends.) Moreover--and this is really the key point for Searle's argument--the "hand simulation" in question is, in all relevant respects, simply a special kind of digital computation. So, there is a possible world--doubtless one quite remote from the actual world--in which a digital computer simulates intelligence but in which the digital computer does not itself possess intelligence. But, if we consider any digital computer in the actual world, it will not differ from the computer in that remote possible world in any way which could make it the case that the computer in the actual world is more intelligent than the computer in that remote possible world. Given that we agree that the "hand simulating" computer in the Chinese Room is not intelligent, we have no option but to conclude that digital computers are simply not the kinds of things that can be intelligent. So far, the argument that we have described arrives at the conclusion that no appropriately programmed computer can think. While this conclusion is not one that Turing accepted, it is important to note that it is compatible with the claim that The Turing Test is a good test for intelligence. This is because, for all that has been argued, it may be that it is not nomically possible to provide any "hand simulation" of intelligence (and, in particular, that it is not possible to simulate intelligence using any kind of computer). In order to turn Searle's argument--at least in the way in which we have developed it--into an objection to The Turing Test, we need to have some reason for thinking that it is at least nomically possible to simulate intelligence using computers. (If it is nomically impossible to simulate intelligence using computers, then the alleged fact that digital computers cannot genuinely possess intelligence casts no doubt at all on the usefulness of the Turing Test, since digital computers are nomically disqualified from the range of cases in which there is mere simulation of intelligence.) In the absence of reason to believe this, the most that Searle's argument yields is an objection to Turing's confidently held belief that digital computing machines will one day pass The Turing Test. (Here, as elsewhere, we are supposing that, for any kind of creature C, there is a version of The Turing Test in which C takes the role of the machine in the specific test that Turing describes. This general format for testing for the presence of intelligence would not necessarily be undermined by the success of Searle's Chinese Room argument.) There are various responses that might be made to the argument that we have attributed to Searle. One kind of response is to dispute the claim that there is no intelligence present in the case of the Chinese Room. (Suppose that the "hand simulation" is embedded in a robot that is equipped with appropriate sensors, etc. Suppose, further, that the "hand simulation" involves updating the process of "hand simulation," etc. If enough details of this kind are added, then it becomes quite unclear whether we do want to say that we still haven't described an intelligent system.) Another kind of response is to dispute the claim that digital computers in the actual world could not be relevantly different from the system that operates in the Chinese Room in that remote possible world. (If we suppose that the core of the Chinese Room is a kind of giant look-up table, then it may well be important to note that digital computers in the actual world do not work with look-up tables in that kind of way.) Doubtless there are other possible lines of response as well. However, it would take us out of our way to try to take this discussion further. (One good place to look for further discussion of these matters is Braddon-Mitchell and Jackson (1996).) ## 7. Brief Notes on Intelligence There are radically different views about the measurement of intelligence that have not been canvassed in this article. Our concern has been to discuss Turing (1950) and its legacy. But, of course, a more wide-ranging discussion would also consider, for example, research on the measurement of intelligence using the mathematical and computational resources of Algorithmic Information Theory, Kolmogorov Complexity Theory, Minimum Message Length (MML) Theory, and so forth. (For an introduction to this literature, see Hernandez-Orallo and Dowe (2010), and the list of references contained therein. For a more general introduction to research into AI, see Marquis et al. (2020).) More broadly, there are radically different views about our concept--or concepts--of intelligence that have not been canvassed in this article. There is a dispute, for example, about whether Turing is best interpreted as working with a response-dependent concept of intelligence. (Pro: Proudfoot (2013) (2020); contra: Wheeler (2020).) Relatedly, there is a dispute about whether intelligence bears some kind of necessary relationship to symmetrical relations of recognition between agents, as suggested in Mallory (2020) There is also a broader dispute about whether we should think that useful notions of intelligence are always domain specific, or whether we should rather suppose that there is something important in the idea of general, domain independent intelligence. And there are radically different views about the most likely paths to building general intelligence (assuming that there is such a thing as general intelligence). For example, Crosby (2020) suggests that the best way forwards may be to try to make machines that can pass animal cognition tests, i.e. that can create predictive models of their environment from sensory input. (There are clear precusors to this line of thought in, for example, Brooks (1990).)
scottish-18th	## 1. Major figures The major figures in Scottish eighteenth century philosophy were Francis Hutcheson, David Hume, Adam Smith, Thomas Reid and Adam Ferguson. Others who produced notable works included Gershom Carmichael, Archibald Campbell, George Turnbull, George Campbell, James Beattie, Hugh Blair, Alexander Gerard, Henry Home (Lord Kames), John Millar and Dugald Stewart. ## 2. Carmichael on Natural Law Gershom Carmichael (1672-1729) studied at Edinburgh University (1687-1691), taught at St Andrews University (1693-1694), and spent the rest of his life at Glasgow, first as a regent in arts and then as professor of moral philosophy. He was a main conduit into Scotland of the European natural law tradition, a tradition of scientific investigation of human nature with a view to constructing an account of the principles that are morally binding on us. Among the great figures of that tradition were Hugo Grotius (1583-1645) and Samuel Pufendorf (1632-1694), thinkers whose writings played a major role in moral philosophical activity in Scotland during the Age of Enlightenment. In 1718, during the first stirrings of the Scottish Enlightenment, Carmichael published Supplements and Observations upon the two books of the distinguished Samuel Pufendorf's On the Duty of Man and Citizen. In 1724 he published a second edition containing extensive additional material. Carmichael affirms: "when God prescribes something to us, He is simply signifying that he requires us to do such and such an action, and regards it, when offered with that intention, as a sign of love and veneration towards him, while failure to perform such actions, and, still worse, commission of the contrary acts, he interprets as an indication of contempt or hatred" (Carmichael, [NR], p. 46). Hence we owe God love and veneration, and on this basis Carmichael distinguishes between immediate and mediate duties. Our immediate duty is formulated in the first precept of natural law, that God is to be worshipped. He seeks a sign of our love and veneration for him, and worship is the clearest manifestation of these feelings. The second precept, which identifies our mediate duties, is: "Each man should promote, so far as it is in his power, the common good of the whole human race, and, so far as this allows, the private good of individuals" ([NR], p. 48). This relates to our 'mediate' duties since we indirectly signify our love and veneration of God by treating his creatures well. On this basis, Carmichael deploys the distinction between self and others in two subordinate precepts: "Each man should take care to promote his own interest without harming others" and "Sociability is to be cultivated and preserved by every man so far as in him lies." These precepts, concerning duties to God, to self and to others, are the fundamental precepts of natural law, and though the precept that God is to be worshipped is prior to and more evident than the precept that one should live sociably with men, the requirement that we cultivate sociability is a foundation of the well-lived life. Carmichael therefore rejects an important aspect of Pufendorf's doctrine on the cultivation of sociability, for the latter argues that the demand "that every man must cultivate and preserve sociability so far as he can" is that to which all our duties are subordinate. Yet for Carmichael the precept that we worship God is not traceable back to the duty to cultivate sociability, and therefore the requirement that we cultivate and preserve sociability cannot precede the laws binding us to behave appropriately towards God. For instance, God is central to the narrative concerning the duty to cultivate our mind, for performance of this duty requires that we cultivate in ourselves the conviction that God is creator and governor of the universe and of us. Carmichael criticises Pufendorf for paying too little attention to the subject of cultivation of the mind, and indicates some features that might profitably have been considered by Pufendorf, for example the following. Due cultivation of the mind involves filling it with sound opinion regarding our duty, learning to judge well the objects which commonly stimulate our desires, and acquiring rational control of our passions. It also involves our learning to act on the knowledge that, as regards our humanity, we are neither superior nor inferior to other people. Finally, a person with a well cultivated mind is aware of how little he knows of what the future holds, and consequently is neither arrogant at his present happy circumstances nor excessively anxious about ills that might yet assail him. The Stoic character of this text is evident, as is Carmichael's injunction that we not be disturbed on account of evils which have befallen us, or which might befall us, due to no fault of ours. The deliberate infringement of the moral law is said however to be another matter; it prompts a discomfort peculiarly hard to bear. In full concord with the Stoic tendency here observed, we find him supporting, under the heading 'duty to oneself', a Stoic view of anger. Though not expressing unconditional disapproval of anger, he does point out that it is difficult to keep an outburst of anger within just limits, and that such an outburst is problematic in relation to natural law, for: "it must be regarded as one of the things which most of all makes human life unsocial, and has pernicious effects for the human race. Thus we can scarcely be too diligent in restraining our anger" ([NR], p. 65). Anger conflicts with sociability and it is only by due cultivation of the mind that our sociability can be fortified and enhanced. ## 3. Hutcheson and Archibald Campbell The first of the major philosophers was Francis Hutcheson (1694-1746). His reputation rests chiefly on his earlier writings, especially An Inquiry into the Original of our Ideas of Beauty and Virtue (London 1725), Reflections upon Laughter and Remarks on the Fable of the Bees (both in the Dublin Journal 1725-1726), and Essay on the Nature and Conduct of the Passions with Illustrations on the Moral Sense (London 1728). His magnum opus, A System of Moral Philosophy, was published posthumously in Glasgow in 1755. During his period as a student at Glasgow University (c. 1711-1717) Gershom Carmichael taught moral philosophy and jurisprudence there and there are clear signs in Hutcheson's writings of Carmichael's influence though it is not known whether he studied under Carmichael during his student days. In 1730 he took up the moral philosophy chair left vacant on Carmichael's death. Hutcheson is known principally for his ideas on moral philosophy and aesthetics. First moral philosophy. Hutcheson reacted against both the psychological egoism of Thomas Hobbes and the rationalism of Samuel Clarke and William Wollaston. As regards Hobbes, Hutcheson thought his doctrine was both wrong and dangerous; wrong because by the frame of our nature we have compassionate, generous and benevolent affections which owe nothing at all to calculations of self-interest, and dangerous because people may be discouraged from the morally worthy exercise of cultivating generous affections in themselves on the grounds that the exercise of such affections is really an exercise in dissimulation or pretence. As against Hobbes Hutcheson held that a morally good act is one motivated by benevolence, a desire for the happiness of others. Indeed the wider the scope of the act the better, morally speaking, the act is; Hutcheson was the first to speak of "the greatest happiness for the greatest numbers". He believed that moral knowledge is gained via our moral sense. A sense, as the term is deployed by Hutcheson, is every determination of our minds to receive ideas independently of our will, and to have perceptions of pleasure and pain. In accordance with this definition, the five external senses determine us to receive ideas which please or pain us, and the will does not intervene -- we open our eyes and by natural necessity see whatever it is that we see. But Hutcheson thought that there were far more senses than the five external ones. Three in particular play a role in our moral life. The public sense is that by which we are pleased with the happiness of others, and are uneasy at their misery. The moral sense is that by which we perceive virtue or vice in ourselves or others, and derive pleasure, or pain, from the perception. And the sense of honour is that which makes the approbation, or gratitude of others, for any good actions we have done, the necessary occasion of pleasure. In each of these cases the will is not involved. We see a person acting with the intention of bringing happiness to someone else, and by the frame of our nature pleasure wells up in us. Hutcheson emphasises both the complexity of the relations between our natural affections and also the need, in the name of morality, to exercise careful management of the relations between the affections. We must especially be careful not to let any of our affections get too 'passionate', for a passionate affection might become an effective obstacle to other affections that should be given priority. Above all the selfish affections must not be allowed to over-rule 'calm universal benevolence'. Hutcheson's opposition to Hobbesian egoism is matched by his opposition to ethical rationalism, an opposition which emerges in the Illustrations upon the Moral Sense, where he demonstrates that his account of the affections and the moral sense makes sense of the moral facts whereas the doctrines of Clarke and Wollaston totally fail to do so. Hutcheson's main thesis against ethical rationalism is that all exciting reasons presuppose instincts and affections, while justifying reasons presuppose a moral sense. An exciting reason is a motive which actually prompts a person to act; a justifying reason is one which grounds moral approval of the act. Hutcheson demonstrates that reason, unlike affection, cannot furnish an exciting motive, and that there can be no exciting reason previous to affection. Reason does of course play a role in our moral life, but only as helping to guide us to an end antecedently determined by affection, in particular the affection of universal benevolence. On this basis, an act can be called 'reasonable', but this is not a point on the side of the rationalists, since they hold that reason by itself can motivate, and in this case it is affection, not reason that motivates, that is, that gets us doing something rather than nothing. If we add to this the fact, as Hutcheson sees it, that it has never been demonstrated that reason is a fit faculty to determine what the ends are that we are obliged to seek, we shall see that Hutcheson's criticism of rationalism is that it can account for neither moral motivation nor moral judgment. On the other hand our natural affections, in particular benevolence, account fully for our moral motivation and our faculty of moral sense accounts fully for our ability to make an assessment of actions whether our own or others'. Certain features of Hutcheson's moral philosophy appear in his aesthetic theory also. Indeed the two fields are inextricably related, as witness Hutcheson's reference to the 'moral sense of beauty'. Two features especially work hard. He contends that we sense the beauty, sublimity or grandeur of a sight or of a sound. The sense of the thing's beauty, so to say, wells up unbidden. And associated with that sense, and perhaps even part of it -- Hutcheson does not give us a clear account of the matter -- is a pleasure that we take in the thing. We enjoy beautiful things and that enjoyment is not merely incidental to our sensing their beauty. A question arises here regarding the features of a thing that cause us to see it as beautiful and to take pleasure in it. Hutcheson suggests that a beautiful thing displays unity (or uniformity) amidst variety. If a work has too much uniformity it is simply boring. If it has too much variety it is a jumble. An object, whether visual or audible, requires therefore to occupy the intermediate position if it is to give rise to a sense of beauty in the object. But if Hutcheson is right about the basis of aesthetic judgment how does disagreement arise? Hutcheson's reply is that our aesthetic response is affected in part by the associations that the thing arouses in our mind. If an object that we had found beautiful comes to be associated in our mind with something disagreeable this will affect our aesthetic response; we might even find the thing ugly. Hutcheson gives an example of wines to which men acquire an aversion after they have taken them in an emetic preparation. On this matter his position may seem extreme, for he holds that if two people have the same experience and if the thing experienced carries the same identical associations for the two people, then they will have the same aesthetic response to the object. The position is however difficult to disprove, since if two people do in fact disagree about the aesthetic merit of an object, Hutcheson can say that the object produces different associations in the two spectators. Nevertheless, Hutcheson does believe aesthetic misjudgments are possible, and in the course of explaining their occurrence he deploys Locke's doctrine of association of ideas, a doctrine according to which ideas linked solely by chance or custom come to be associated in our minds and become almost inseparable from each other though they are 'not at all of kin'. Hutcheson holds that an art connoisseur's judgment can be distorted through his tendency to associate ideas, and notes in particular that a connoisseur's aesthetic response to a work of art is likely to be affected by the fact that he owns it, for the pleasure of ownership will tend to intermix with and distort the affective response he would otherwise have to the object. Hutcheson, it should be added, is equally sensitive to the danger to our moral judgments that is posed by our associative tendency. And in both types of case the best defence against the threat is reflection, understood as a mental probing, an examination and then cross-examination, whether of a work of art or of an action, and of the elements in and aspects of our situation that motivate our judgments, all this with a view to factoring out irrelevant considerations. Without such mental exercises we cannot, in his view, obtain what he terms 'true liberty and self-command'. This position, which he presents several times, points to a doctrine of free will not otherwise readily discernible in his writings. Our free will, on this account, is a habit of reflection through which we form a judgment which we are in a position to defend. We stand back from the object of reflection, do not allow ourselves to be overwhelmed by it, but instead adjudicate it in the light of whatever considerations we judge it appropriate to bring to bear (Broadie 2016). A philosopher of the early period of the Scottish Enlightenment with whom Hutcheson may helpfully be compared and contrasted is his close contemporary Archibald Campbell (1691-1756), professor of divinity and church history at St Andrews University. Both men studied at Glasgow University under John Simson, a professorial divine much harassed by conservative Reformists in the Kirk on account of his rejection of the Kirk's doctrine that, since the Fall, human nature through the generations has suffered from total depravity. In due course Hutcheson and Campbell were both harassed by the Kirk on account of their claim that human beings are by nature inclined to virtue, for the Kirk took that claim to imply that the two men did not fully embrace the doctrine of total depravity. However, beyond this agreement Campbell opposes Hutcheson on certain essential points. Most especially, in his Enquiry into the Original of Moral Virtue (1733) Campbell rejects Hutcheson's claim that for an action to be virtuous it must be motivated by a disinterested benevolence, and argues to the contrary that all human acts are and can only be motivated by self-love. From which Campbell concludes that all virtuous human acts also are motivated by self-love. This claim, though at first sight Hobbesian, is however not at all of a Hobbesian stripe, for Campbell holds that the self-love that motivates us to perform a virtuous act takes the form of a desire for esteem, where the desired esteem derives from our gratification of another person's self-love. As well as writing against Hobbes and Hutcheson, Campbell also directs his fire at Bernard Mandeville, who held that a virtuous act must involve an exercise of self-denial, in the sense that to act virtuously we have to cut across, or frustrate our natural principles, whereas for Campbell, as also for Hutcheson, virtuous acts are performed in realisation of, and not at all in conflict with, our nature. The Kirk set up a committee of purity of doctrine to investigate the teachings of Campbell, a Kirk minister who painted such a distressingly agreeable picture of human nature. The conservatively orthodox Reformists on the Committee wished to move against him, but the Kirk's General Assembly, which was already beginning to display an Enlightenment spirit, prevented such a move, and Campbell retained his professorship of divinity and his position as minister of the Kirk. Campbell's ideas, till now neglected, are at last beginning to receive serious consideration (Maurer 2012, 2016). ## 4. Hutcheson, Hume and Turnbull Hutcheson influenced most of the Scottish philosophers who succeeded him, perhaps all of them, whether because he helped to set their agenda or because they appropriated, in a form suitable to their needs, certain of his doctrines. In the field of aesthetics for example, where Hutcheson led, many, including Hume, Reid, and Archibald Alison, followed. But influences can be hard to pin down and there is much dispute in particular concerning his influence on David Hume (1711-1776). It is widely held that Hume's moral philosophy is essentially Hutchesonian, and that Hume took a stage further Hutcheson's projects of internalisation and of grounding our experience of the world on sentiment or feeling. For Hume agreed with Hutcheson that moral and aesthetic qualities are really sentiments existing in our minds, but he also argued that the necessary connection between any pair of events E1 and E2 which are related as cause to effect is also in our minds, for it is nothing more than a determination of the mind, due to custom or habit, to have a belief (a kind of feeling) that an event of kind E2 will occur next when we experience an event of kind E1. Furthermore Hume argues that what we think of as the 'external' world is almost entirely a product of our own imaginative activity. As against these reasons for thinking Hume indebted to Hutcheson there are the awkward facts that Hutcheson greatly disapproved of the draft of Treatise Book III that he saw in 1739 and that Hutcheson did his best to prevent Hume being appointed to the moral philosophy chair at Edinburgh University in 1744-1745. In addition many of their contemporaries, such as Adam Smith and Thomas Reid, held that Hume's moral philosophy was significantly different from Hutcheson's (Moore 1990). One close contemporary of Hutcheson, who also stands in interesting relations to Hume, is George Turnbull (1698-1748), regent at Marischal College, Aberdeen (1721-1727), and teacher of Thomas Reid at Marischal. He describes Hutcheson as "one whom I think not inferior to any modern writer on morals in accuracy and perspicuity, but rather superior to almost all" (Principles of Moral Philosophy, p. 14), and no doubt Hutcheson was an influence on Turnbull in several ways. But it has to be borne in mind that the earliest of Turnbull's writings, Theses philosophicae de scientiae naturalis cum philosophia morali conjunctione (Philosophical theses on the unity of natural science and moral philosophy), a graduation oration delivered in 1723 (Education for Life, pp. 45-57), shows Turnbull already working on a grand project that might be thought of as roughly Hutchesonian, but doing so several years before Hutcheson's earliest published work (Turnbull [EL]). As regards Turnbull's relationship with Hume, we should recall that the subtitle of Hume's Treatise of Human Nature is "An attempt to introduce the experimental method of reasoning into moral subjects". As with Hume's Treatise, so also Turnbull's Principles of Moral Philosophy, published in 1740 (the year of publication of Bk. III of the Treatise) but based on lectures given in Aberdeen in the mid-1720s, contains a defence of the claim that natural and moral philosophy are very similar types of enquiry. When Turnbull tells us that all enquiries into fact, reality, or any part of nature must be set about, and carried on in the same way, he is bearing in mind the fact, as he sees it to be one, that there are moral facts and a moral reality, and that our moral nature is part of nature and therefore to be investigated by the methods appropriate to the investigation of the natural world. As the natural philosopher relies on experience of the external world, so the moral philosopher relies on his experience of the internal world. Likewise, writing in Humean terms, but uninfluenced by Hume, Turnbull affirms: "every Enquiry about the Constitution of the human Mind, is as much a question of Fact or natural History, as Enquiries about Objects of Sense are: It must therefore be managed and carried on in the same way of Experiment, and in the one case as well as in the other, nothing ought to be admitted as Fact, till it is clearly found to be such from unexceptionable Experience and Observation" (Education for Life, pp. 342-3). It is, in Turnbull's judgment, the failure to respect this experimental method that led to the moral scepticism (as Turnbull thought it to be) of Hobbes and Mandeville, whose reduction of morality to self-love flies in the face of experience and shocks common sense. The experience in question is of the reality of the public affection in our nature, the immediate object of which is the good of others, and the reality of the moral sense by which we are determined to approve such affections. This moral sense, of whose workings we are all aware, is the faculty by which, without the mediation of rational activity, we approve of virtuous acts and disapprove of vicious ones; and the approval and disapproval rise up in us without any regard for self-love or self-interest. In a very Hutchesonian way Turnbull invites us to consider the difference we feel when faced with two acts which are the same except for the fact that one of them is performed from love of another person and the other act is performed from self-interest. These facts about our nature have to be accommodated within moral philosophy just as the fact that heavy bodies tend to fall has to be accommodated within natural philosophy. Turnbull is committed to a form of reliabilism according to which the faculties that we have by the frame or constitution of our nature are trustworthy. It is not simply that we are so constructed that we cannot but accept their deliverances; it is that we are also entitled to accept them. Turnbull, a deeply committed Christian, believed that the author of our nature would not have so constituted us as to accept the deliverances of our nature if our nature could not be relied upon to deliver up truth. We are in the hands of providence, and live directed towards the truth for that reason. This doctrine has been termed 'providential naturalism', and bears a marked resemblance to the language and also to the substance of the position held by Turnbull's pupil Thomas Reid. ## 5. Kames on aesthetics and religion Henry Home, Lord Kames, likewise held a version of providential naturalism. In his Essays on the Principles of Morality and Natural Religion (1779) he has a good deal to say about the senses external and internal, treating them as enabling us, by the original frame of our nature, to gain access, without the use of reasoning processes, to the realities in the corresponding domains, including the moral domain. Kames's moral sense has as much to do with aesthetics as with morality; or rather, for Kames, no less than for Hutcheson, virtue is a kind of beauty, moral beauty, as vice is moral deformity. Beauty itself is ascribed to anything that gives pleasure. And as there are degrees of pleasure and pain, so also there are degrees of beauty and ugliness. In the lowest rank are things considered without regard to an end or a designing agent. The possibility of greater pleasure, and of the ascription of greater beauty, arises when an object is considered with respect to the object's end. A house, considered in itself, might be beautiful, but how much more beautiful is it judged to be if it is seen to be well designed for human occupancy. Approbation, as applied to works of art, is our pleasure at them when we consider them to be well fitted or suited to an end. The approbation is greater if the end for which the object is well suited also gives pleasure. A ship may give pleasure because it is so shapely, and also give pleasure because it is well suited to trade, and also give pleasure because trade also is a fine thing. If these further things are taken into account the beauty of the ship is enhanced. Kames argues that these kinds of pleasure can also be taken in human actions, and that human acts can cause pleasure additionally by the special fact about them that they proceed from intention, deliberation and choice. In the case of, for example, an act of generosity towards a worthy person, the act is intentionally well suited, or fitted, to an end whose beauty is recognised by the agent. The fact that observation of acts displaying generosity, and other virtues, gives us pleasure is due to the original constitution of our nature. The pleasure arises unbidden, and no exercise of will or reason is required, any more than we require to use our reason to see the beauty of a landscape or a work of art. Kames wrote extensively on revealed and natural theology. As regards the latter, he often has Hume in his sights, particularly Hume's Dialogues Concerning Natural Religion (1779), with whose contents Kames was familiar decades before the work's publication. Hume held that in an inference from effect to cause no more should be assigned to the cause than is sufficient to explain the effect. In particular, if we argue from the existence of the natural world to the existence of God we should ascribe to God only such attributes as are requisite for the explanation of the world. And since the world is imperfect, why not say that we are not constrained by the facts in the natural order to ascribe perfection to God? Kames, on the other hand, holds that there are principles implanted in our nature that permit us to draw conclusions that reason alone does not sanction. If something is a tendency of our nature then we have to rely on it as a source of truth. Invoking just such a tendency Kames affirms that though we see both good and evil around us we do not conclude that the cause of the world must also be a mixture of good and evil: "it is a tendency of our nature to reject a mixed character of benevolence and malevolence, unless where it is necessarily pressed home upon us by an equality of opposite effects; and in every subject that cannot be reached by the reasoning faculty, we justly rely on the tendency of our nature" (Essays, p. 353). In any case Kames sees a world which is predominantly good even though it has 'a few cross instances'. But the few cross instances might not look so cross, or even at all cross, if we had a fuller perspective, and Kames anticipates the time when that perspective will be granted us. This latter position did not raise the hackles of the zealots among the Presbyterian clergy in Scotland, but Kames's position on free will caused a furore and he had to defend himself from attempts to expel him from the Kirk. Kames, accepting the concept of history, natural and human, as the gradual realisation of a divine plan, believed in universal necessity. The laws ordained by God "produce a regular train of causes and effects in the moral as well as material world, bringing about those events which are comprehended in the original plan, and admitting the possibility of none other" (Essays, p. 192). On the other hand, if we are to fulfill our role in the grand scheme we must see ourselves as able to initiate things, that is, to be the free cause of their occurrence. God has therefore, according to Kames, concealed from us the necessity of our acts and he is therefore a deceitful God. Kames sought to explain how this divine deceit enables us to live as morally accountable beings, but this latter part of his philosophy did nothing to placate those in the Kirk for whom the affirmation of a deceitful God was a sacrilege. Kames, however, could not see any difference between the deception by which we believe ourselves to be free when in fact we are necessitated and the deception by which we believe secondary qualities, such as colours and sounds, to be in the external world and able to get along without us, when in fact they depend for their existence upon the exercise of our own sensory powers. ## 6. George Campbell on miracles Kames did not dedicate an entire book to an attack on Hume on religion, but George Campbell (1719-1796) did. This interesting man, a student at Marischal College, Aberdeen, of which in 1759 he became Principal, was a founder-member of the Aberdeen Philosophical Society, the 'Wise Cub', which also included Thomas Reid, John Gregory, David Skene, Alexander Gerard, James Beattie and James Dunbar. It is probable that many of Campbell's writings began life as papers to the Club. In 1763 he published A Dissertation on Miracles which was intended as a demolition of Hume's essay 'On miracles', Chapter Ten in An Enquiry Concerning Human Understanding. Miracles were commonly discussed in eighteenth century Scotland. On the one hand the Kirk required people to accept miracle claims on the basis either of eyewitness reports or of reports of such reports, and on the other hand the spirit of Enlightenment required that claims based on the authority of others be put before the tribunal of reason. Hume focuses especially on the credibility of testimony, and argues that the credence we place in testimony is based entirely on experience, the experience of the occasions when testimony has turned out to be true as against those experiences where it has not. Likewise it is on the basis of experience that we judge whether a reported event occurred. If the reported event is improbable we ask how probable it is that the eyewitness is speaking truly. We have to balance the probability that the eyewitness is speaking truly against the improbability of the occurrence of the event. Hume held that the improbability of a miracle is always so great that no testimony could tell effectively in its favour. The wise man, proportioning his belief to the evidence, would believe that the testimony in favour of the miracle is false. Campbell's opening move against this argument is to reject Hume's premiss that we believe testimony solely on the basis of experience. For, according to Campbell, there is in all of us a natural tendency to believe other people. This is not a learned response based on repeated experience but an innate disposition. In practice this principle of credulity is gradually finessed in the light of experience. Once testimony is placed before us it becomes the default position, something that is true unless or until proved false, not false unless or until proved true. The credence we give to testimony is much like the credence we give to memory. It is the default position as regards beliefs about the past, even though in the light of experience we might withhold belief from some of its deliverances. Because our tendency to accept testimony is innate, it is harder to overturn than Hume believes it to be. Campbell considers the case of a ferry that has safely made a crossing two thousand times. I, who have seen these safe crossings, meet a stranger who tells me solemnly that he has just seen the boat sink taking with it all on board. The likelihood of my believing this testimony is greater than would be implied by Hume's formula for determining the balance of probabilities. Reid, a close friend of Campbell's, likewise gave massive emphasis to the role of testimony, stressing both the innate nature of the credence we give to testimony and also the very great proportion of our knowledge of the world that we gain, not through perception or reason, but through the testimony of others. Reid's comparison of the credence we naturally give to the testimony of others and the credence we naturally give to the deliverances of our senses, is one of the central features of his Inquiry into the Human Mind (1764). ## 7. George Campbell and the rhetorical tradition A number of eighteenth century Scots, including James Burnett (Lord Monboddo), Adam Smith, Thomas Reid, Hugh Blair and James Dunbar, made significant contributions in the field of language and rhetoric. George Campbell's The Philosophy of Rhetoric (London 1776) is a large-scale essay in which he takes a roughly Aristotelian position on the relation between logic and rhetoric, since he holds that convincing an audience, which is the province of rhetoric or eloquence, is a particular application of the logician's art. The central insight from which Campbell is working is that the orator seeks to persuade people, and in general the best way to persuade is to produce perspicuous arguments. Good orators have to be good logicians. Their grammar also must be sound. This double requirement of orators leads Campbell to make a sharp distinction between logic and grammar, on the grounds that though both have rules, the rules of logic are universal and those of grammar particular. Though there are many natural languages there is but one set of rules of logic, and on the other hand different languages have different rules of grammar. It is against a background of discussion by prominent writers on language such as Locke and James ('Hermes') Harris that Campbell takes his stand with the claim that there cannot be such a thing as a universal grammar. His argument is that there cannot be a universal grammar unless there is a universal language, and there is no such thing as a universal language, just many particular languages. There are, he grants, collections of rules that some have presented under the heading 'universal grammar'. But, protests, Campbell, "such collections convey the knowledge of no tongue whatever". His position stands in interesting relation to Reid's frequent appeals to universals of language in support of the claim that given beliefs are held by all humankind. ## 8. Common sense Campbell was a leading member of the school of common sense philosophy. For him common sense is an original source of knowledge common to humankind, by which we are assured of a number of truths that cannot be evinced by reason and "it is equally impossible, without a full conviction of them, to advance a single step in the acquisition of knowledge" (Philosophy of Rhetoric, vol. 1, p. 114). His account is much in line with that of his colleague James Beattie: "that power of the mind which perceives truth, or commands belief, not by progressive argumentation, but by an instantaneous, instinctive, and irresistible impulse; derived neither from education nor from habit, but from nature; acting independently on our will, whenever its object is presented, according to an established law, and therefore properly called Sense; and acting in a similar manner upon all, or at least upon a great majority of mankind, and therefore properly called Common Sense" (An Essay on the Nature and Immutability of Truth, p. 40). We are plainly in the same territory as Reid's account: "there are principles common to [philosophers and the vulgar], which need no proof, and which do not admit of direct proof", and these common principles "are the foundation of all reasoning and science" (Reid [EIP]). These philosophers do however disagree about substantive matters. In particular, Reid lists as the first principle of common sense: "The operations of our minds are attended with consciousness; and this consciousness is the evidence, the only evidence which we have or can have of their existence" (Reid [EIP], p. 41). Campbell on the other hand lists three sorts of intuitive evidence. The first concerns our unmediated insight into the truth of mathematical axioms and the third concerns common sense principles. The second concerns the deliverances of consciousness, consciousness being the faculty through which we learn directly of the occurrence of mental acts -- thinking, remembering, being in pain, and so on. What is listed as a principle of common sense by Reid is, therefore, according to Campbell, to be contrasted with such principles. Aside from this, however, it is clear that Campbell is philosophically very close to Reid, even if Reid is unquestionably the greater philosopher. ## 9. Smith on moral sentiments Reid and Hume both owed an immense debt to Hutcheson. So also did Adam Smith (1723-1790) who, unlike the others, studied under Hutcheson at Glasgow University. In 1751 Smith was appointed to the chair of logic and rhetoric at Glasgow and the following year transferred to the chair of moral philosophy that Hutcheson had occupied. Smith's An Inquiry into the Nature and Causes of the Wealth of Nations appeared in 1776. Essays on Philosophical Subjects appeared posthumously in 1795. He also published an essay on the first formation of languages, and student notes of his lectures on rhetoric and belles lettres, and on jurisprudence have survived. But much his most important work in philosophy is the Theory of Moral Sentiments which appeared in 1759 and of which six authorised editions appeared during Smith's lifetime. The second and sixth editions contain significant revisions and additions. Smith sets out to provide a theory that will address the weaknesses of existing systems of moral philosophy (critcised in Part VII of the Moral Sentiments). He does so by rejecting attempts to reduce morality to a single principle and instead seeks to provide an account of the operation of ordinary moral judgment that recognises the central role of socialisation. The concepts of sympathy and spectatorship, central to the doctrine of TMS, had already been put to work by Hutcheson and Hume, but Smith's account is distinct. As spectator of an agent's suffering we form in our imagination a copy of such 'impression of our own senses' as we have experienced when we have been in a situation of the kind the agent is in: "we enter as it were into his body, and become in some measure the same person with the agent" (Smith 1790, p. 9). Smith gives two spectacular examples of cases where the spectator has a sympathetic feeling that does not correspond to the agent's. The first concerns the agent who has lost his reason. He is happy, unaware of his tragic situation. The spectator imagines how he himself would feel if reduced to this same situation. In this imaginative experiment, in which the spectator is operating on the edge of a contradiction, the spectator's idea of the agent's situation plays a large role while his idea of the agent's actual feelings has a role only in that the agent's happiness is itself evidence of his tragedy. The second of Smith's examples is the spectator's sympathy for the dead, deprived of sunshine, conversation and company. Again Smith emphasises the agent's situation, and asks how the spectator would feel if in the agent's situation, deprived of everything that matters to people. Smith relates sympathy to approval. For a spectator to approve of an agent's feelings is for him to observe that he sympathises with the agent. This account is used as the basis of the analysis of propriety. For a spectator to judge that an agent's act is proper or appropriate is for him to approve of the agent's act. The agent's act lacks propriety, in the judgment of the spectator, if the spectator does not sympathise with the agent's performance. Propriety and impropriety are based on a bilateral relation, between spectator and agent. Smith attends also to a trilateral relation, between a spectator, an agent who acts on someone, and the person who is acted on, the 'recipient' of the act. There are several kinds of response that the recipient may make to the agent's act, and Smith focuses on two, gratitude and resentment. If the spectator judges the recipient's gratitude proper or appropriate then he approves of the agent's act and judges it meritorious, or worthy of reward. If he judges the recipient's resentment proper or appropriate then he disapproves of the agent's act and judges it demeritorious, or worthy of punishment. Judgments of merit or demerit concerning a person's act are therefore made on the basis of an antecedent judgment concerning the propriety or impropriety of another person's reaction to that act. Sympathy underlies all these judgments, for in the cases just mentioned the spectator sympathises with the recipient's gratitude and with his resentment. He has direct sympathy with the affections and motives of the agent and indirect sympathy with the recipient's gratitude; or in judging the agent's behaviour improper the spectator has indirect sympathy with the agent's resentment (Smith 1790, p. 74). We have supposed, in each of these cases, that the recipient really does have the feeling in question, whether of gratitude or resentment. However, in Smith's account the spectator's belief about what the recipient actually feels about the agent is not important for the spectator's judgment concerning the merit and demerit of the agent. The recipient may, for whatever reason, resent an act that was kindly intentioned and in all other ways admirable, and the spectator, knowing the situation better than the recipient does, puts himself imaginatively in the shoes of the recipient while taking with him into this spectatorial role information about the agent's behaviour that the recipient lacks. The spectator judges that were he himself in the recipient's situation he would be grateful for the agent's act; and on that basis, and independently of the recipient's actual reaction, he approves of the agent's act and judges it meritorious. Here the spectator considers himself as a better (because better informed) spectator of the agent's act than the recipient is. As regards judgments of merit and demerit, Smith sets up a model of three people, but the three differ in respect of the weight that has to be given to their work, for the recipient does almost nothing. He is acted on by the agent, but apart from that he is no more than a place holder for the spectator who will imaginatively occupy his shoes and make a judgment concerning merit or demerit on the basis solely of his conception of how he would respond to the agent if he were in the place of the recipient. He does not judge on the basis of the actual reaction of the recipient, who might approve of the agent's act or disapprove or have no feelings about it one way or the other. Up to this point we have attended to the spectator's moral judgment of the acts of others. What of his judgment of his own acts? In judging the other the spectator has the advantage of disinterest, but he may lack requisite information and much of the work of creative imagination goes into his rectifying the lack. In judging himself he has, or may be presumed to have, the requisite information but he has the problem of overcoming the tendency to a distorted judgment caused by self-love or self-interest. He must therefore factor out of his judgment those features that are due to self-love. He does this by setting up, by an act of creative imagination, a spectator, an other who, qua spectator, is at a distance from him.The point about the distance is that it creates the possibility of disinterest or impartiality, but it is still necessary to ask how disinterest or impartiality is achieved if it is the agent himself who imagines the spectator into existence. Let us move to an answer by wondering who or what it is that is imagined into existence? Is it the voice of society, representing established social attitudes? At times in the first edition of The Theory of Moral Sentiments Smith comes close to saying that it is. In the second edition Smith is clear that this is not the role of the impartial spectator for the latter can, and occasionally does, speak against established social attitudes. Nor can the judgment of the impartial spectator be reduced to the judgment of society, even where those two judgments coincide. Nevertheless the impartial spectator exists because of real live spectators. Were it not for our discovery that while we are judging other people, those same people are judging us, we would not form the idea of a spectator judging us impartially. Smith's account of justice is built upon his account of the spectator's sympathetic response to the recipient of an agent's act. If a spectator sympathises with a recipient's resentment at the agent's act then he judges the act demeritorious and the agent worthy of punishment. In the latter case the moral quality attributed to the act is injustice. An act of injustice "does a real and positive hurt to some particular persons, from motives which are naturally disapproved of" (Smith 1790, p. 79). Since a failure to act justly has a tendency to result in injury, while a failure to act charitably or generously does not, a distinction is drawn by Smith, in line with Humean thinking, between justice and the other social virtues, on the basis that it is so much more important to prevent injury than promote positive good that the proper response to injustice is punishment, whereas we do not feel it appropriate to punish someone who does not act charitably or gratefully. In a word, we have a stricter obligation to justice than to the other virtues. Though there are important points of contact between Smith's account of justice and Hume's, the differences are considerable, chief of them being the fact that Hume grounds our approval of justice on our recognition of its utility, and Smith does not. We do sometimes take it into account in coming to a judgment, but more often than not it is something of a quite different nature that wells up in us: "All men, even the most stupid and unthinking, abhor fraud, perfidy, and injustice, and delight to see them punished. But few men have reflected upon the necessity of justice to the existence of society, how obvious soever that necessity may appear to be" (Smith 1790, p. 89). There are a few cases where utility is plainly involved in our judgment, but they are few, and they are in a distinct psychological class. Smith instances the sentinel who fell asleep while on watch and was executed because such carelessness might endanger the whole army. Smith's comment is: "When the preservation of an individual is inconsistent with the safety of a multitude, nothing can be more just than that the many should be preferred to the one. Yet this punishment, how necessary soever, always appears to be excessively severe. The natural atrocity of the crime seems to be so little, and the punishment so great, that it is with great difficulty that our heart can reconcile itself to it" (Smith 1790, p. 90). And our reaction in this kind of case is to be contrasted with our reaction to the punishment of 'an ungrateful murderer or parricide', where we applaud the punishment with ardour and would be enraged and disappointed if the murderer escaped punishment. These very different reactions demonstrate that our approval of punishment in the one case and in the other are founded on very different principles. Smith extends the discussion of merit and demerit into a naturalised account of the development of religious belief. In his Essays on Philosophical Subjects Smith provides an account of early religious belief as a precursor to science and philosophy. In the case of both, the desire to explain the world is driven by anxiety created through novel experience. Here the appeal to heaven is a psychological coping mechanism that we develop to address injustices that go unrecognised in this world. Smith then argues that our views on the rules of justice more generally come to be associated with religion and this lends a further social sanction to morality. ## 10. Blair's Christian stoicism Smith devotes considerable space to the Stoic virtue of self-command. Another eighteenth century Scottish thinker who devotes considerable space to it is Hugh Blair (1718-1800), minister of the High Kirk of St Giles in Edinburgh and first professor of rhetoric and belles lettres at Edinburgh University. Blair's sermons bear ample witness to his interest in Stoic virtue. For example, in the sermon 'On our imperfect knowledge of a future state' he wonders why we have been left in the dark about our future state. Blair replies that to see clearly into our future would have disastrous consequences. We would be so spellbound by the sight that we would neglect the arts and labours which support social order and promote the happiness of society. We are, believes Blair, in 'the childhood of existence', being educated for immortality. The education is of such a nature as to enable us to develop virtues such as self-control and self-denial. These are Stoic virtues, and Blair's sermons are full of the need to be Stoical. In his sermon 'Of the proper estimate of human life' he says: "if we cannot control fortune, [let us] study at least to control ourselves." Only through exercise of self- control is a virtuous life possible, and only through virtue can we attain happiness. He adds that the search for worldly pleasure is bound to end in disappointment and that that is just as well. For it is through the failure of the search that we come to a realisation both of the essential vanity of the life we have been living and also of the need to turn to God and to virtue. For many, the fact of suffering is the strongest argument there is against the existence of God. Blair on the contrary holds that our suffering provides us with a context within which we can discover that our true nature is best realised by the adoption of a life-plan whose overarching principle is religious. ## 11. Ferguson and the social state One of Blair's colleagues at Edinburgh University was Adam Ferguson (1723-1816). He succeeded David Hume as librarian of the Advocates' Library in Edinburgh and then held in succession two chairs at Edinburgh University, that of natural philosophy (1759-1764) and of pneumatics and moral philosophy (1764-1785). His lectures at Edinburgh were published as The Institutes of Moral Philosophy (1769) and Principles of Moral and Political Science (1792). Ferguson advocates a form of moral science based on the study of human beings as they are and then grounds a normative account of what they ought to be upon this empirical basis. His most influential work is An Essay on the History of Civil Society (1767). There Ferguson attended to one of the main concepts of the Enlightenment, that of human progress, and expressed doubts about whether over the centuries the proportion of human happiness to unhappiness had increased. He believed that each person accommodates himself to the conditions in his own society and the fact that we cannot imagine that we would be contented if we lived in an earlier society does not imply that people in earlier societies were not, more or less, as happy in their own society as we are in ours. As against our unscientific conjectures about how we would have felt in a society profoundly unlike the only one we have ever lived in, Ferguson commends the use of historical records. He talks disparagingly about boundless regions of ignorance in our conjectures about other societies, and among those he has in mind who speak ignorantly about earlier conditions of humanity are Hobbes, Rousseau, and Hume in their discussions of the state of nature and the origins of society. Hobbes and Rousseau in particular had a good deal to say about the pre-social condition of humankind. Ferguson argues, against their theories, that there are no records whatever of a pre-social human condition; and since on the available evidence humankind has always lived in society he concludes that living in society comes naturally to us. Hence the state of nature is a social state and is not antecedent to it. Instead of state of nature and contract theory, Ferguson advocates the use of history, of the accounts of travellers, and the literature and myths of societies to build an account of types of human society. Ferguson's discussion of types of society, ideas of civilisation and civil society has seen him credited with an influential place in the development of sociology. This method is ubiquitous across the thinkers of the Scottish Enlightenment, with a particularly influential discussion of the method to be found in John Millar's The Origin of the Distinction of Ranks (1779). ## 12. Dugald Stewart on history and philosophy One colleague of Blair and Ferguson at Edinburgh University was Dugald Stewart (1753-1828), who was a student first at Edinburgh, and then at Glasgow where his moral philosophy professor was Thomas Reid. Stewart succeeded his father in the chair of mathematics at Edinburgh, and then in 1785 became professor of pneumatic and moral philosophy at Edinburgh when Ferguson resigned the chair. Stewart shared with Ferguson an interest in the kind of historical inquiry that explored how societies operate. In his Account of the Life and Writings of Adam Smith LL.D. Dugald Stewart says of one of Smith's works, the Dissertation on the Origin of Languages (Smith [LRB], pp. 201-26), that "it deserves our attention less, on account of the opinions it contains, than as a specimen of a particular sort of inquiry, which, so far as I know, is entirely of modern origin" (Smith 1795, p. 292). Stewart then spells out the 'particular sort of inquiry' that he has in mind. He notes the lack of direct evidence for the origin of language, of the arts and the sciences, of political union, and so on, and affirms: "In this want of direct evidence, we are under a necessity of supplying the place of fact by conjecture; and when we are unable to ascertain how men have actually conducted themselves upon particular occasions, of considering in what manner they are likely to have proceeded, from the principles of their nature, and the circumstances of their external situation" (Smith 1795, p. 293). For Stewart such enquiries are of practical importance, for by them "a check is given to that indolent philosophy, which refers to a miracle, whatever appearances, both in the natural and moral worlds, it is unable to explain" (Smith 1795, p. 293). Stewart uses the term 'conjectural history' for the sort of history exemplified by Smith's account of the origin of language. Conjectural history works against the illegitimate encroachment of religion into the lives of people who are too quick to reach for God as the solution to a problem when extrapolation from scientifically established principles of human nature would provide a solution satisfying to the intellect. Knowing what we do about human nature, about our intellect and will, our emotions and fundamental beliefs, we ask how people would have behaved in given circumstances. Love and hate, anger and jealousy, joy and fear, do not change much through the generations. Much the same things, speaking generally, have much the same effect first on the emotions and then on behaviour. Dugald Stewart formulates the principle underlying conjectural history: it has "long been received as an incontrovertible logical maxim that the capacities of the human mind have been in all ages the same, and that the diversity of phenomena exhibited by our species is the result merely of the different circumstances in which men are placed" (Stewart 1854-58, vol. 1, p. 69). As regards the credentials of Stewart's 'incontrovertible logical maxim', if the claim that human nature is invariant is an empirical claim, it must be based on observation of our contemporaries and on evidence of people's lives in other places and at other times. Such evidence needs however to be handled with care. The further back we go the more meagre it is, and so the more we need to conjecture to supplement the few general facts available to us. Indeed we can go back so far that we have no facts beyond the generalities that we have worked out in the light of our experience. But to rely on conjecture in order to support the very principle that forms the first premiss in any exercise in conjectural history is to come suspiciously close to arguing in a circle. The incontrovertible logical maxim of Dugald Stewart should probably be accorded at most the status of a well-supported empirical generalisation. Conjectural history is certainly not pure guesswork. We argue on the basis of observed uniformities, and the more experience we have of given uniformities the greater credence we will give to reports that speak of the occurrence of the uniformities, whether they concern dead matter or living people and their institutions. In a famous passage Hume writes: "Whether we consider mankind according to the difference of sexes, ages, governments, conditions, or methods of education; the same uniformity and regular operation of natural principles are discernible. Like causes still produce like effects; in the same manner as in the mutual action of the elements and powers of nature" (Hume [T], p. 401). For Hume the chief point about the similarity between ourselves and our ancestors is that histories greatly contribute to the scientific account of human nature by massively extending our otherwise very limited observational data base. Hume writes: "Mankind are so much the same, in all times and places, that history informs us of nothing new or strange in this particular. Its chief use is only to discover the constant and universal principles of human nature, by showing men in all varieties of circumstances and situations, and furnishing us with materials from which we may form our observations and become acquainted with the regular springs of human action and behaviour. These records of wars, intrigues, factions, and revolutions, are so many collections of experiments, by which the politician or moral philosopher fixes the principles of his science, in the same manner as the physician or natural philosopher becomes acquainted with the nature of plants, minerals, and other external objects, by the experiments which he forms concerning them" (Hume [E], pp. 83-84). On this account of history, it is perhaps the single most important resource for the philosopher seeking to construct a scientific account of human nature. Among the historians produced by eighteenth century Scotland were Turnbull, Hume, Smith, Ferguson and William Robertson. In light of Hume's observation it is not surprising that so much history was written by men prominent for their philosophical writings on human nature.
twardowski	## 1. Life Kazimierz (or Kasimir) Jerzy Skrzypna-Twardowski, Ritter von Ogonczyk was born to Polish parents on October 20, 1866 in Vienna, which was then the capital of the Habsburg Empire. From 1877 to 1885 he attended the Theresian Academy (Theresianum), the secondary school of Vienna's bourgeoise elite. Like many other high-school students at the time, his philosophy textbook was the Philosophische Propedaeutik by Robert Zimmermann, Bolzano's 'favorite pupil' (Herzensjunge). The book covered empirical psychology, logic, and the introduction to philosophy. In 1885 Twardowski enrolled at the University of Vienna. The year after he became a student of Franz Brentano, for whom he felt "the most sincere awe and veneration" and whom he remembered as "uncompromising and relentless in his quest for rigor in formulation, consistency in expression, and precision in the working out of proofs" (Twardowski 1926, 20). In addition to philosophy, Twardowski also studied history, mathematics, physics, and physiology (with Sigmund Exner, the son of Bolzano's correspondent Franz Exner; ibid., 21). While a student, Twardowski was a close friend of Alois Hofler, Christian von Ehrenfels, Josef Klemens Kreibig, and Hans Schmidkunz--the latter initiated regular meetings between younger and older students of Brentano and founded the Vienna Philosophical Society in 1888. Twardowski defended his doctoral dissertation, Idea and Perception--An Epistemological Investigation of Descartes (published in 1892), in the Fall of 1891. Since Brentano had resigned from his chair in 1880, Twardowski's official PhD supervisor was Robert Zimmermann. In 1891-1892 Twardowski spent time as a researcher both in Leipzig, where he followed courses by Wilhelm Wundt and Oswald Kulpe, and in Munich, where he attended the lectures of Carl Stumpf, another pupil of Brentano. Between 1892 and 1895 Twardowski earned a living working for an insurance company and writing for German-language and Polish-language newspapers about philosophy, literature, and music while working on his Habilitation thesis, On the Content and Object of Presentations--A Psychological Investigation (1894), and teaching as Privatdozent at the University of Vienna, where he gave courses in logic, on the immortality of the soul and a practicum on Hume's Enquiry concerning Human Understanding. In 1895 Twardowski, then 29, was appointed as extraordinarius at the University of Lvov (then Lemberg, now Lviv, in Polish Lwow), one of the two Polish-speaking Universities of the Empire. He saw it as his duty to export Brentano's style of philosophizing to Poland and spent most of his time organizing Lvov's philosophical life. He reanimated the Lvov Philosophical Circle and gave lectures aimed at a broader public. He organized his teaching in general 'core courses' in logic, psychology, ethics, and the history of philosophy (given and updated every four years), and placed less emphasis on specialized courses. Further, he inaugurated a philosophical seminar and reading room, where he made his private library available to students, with whom he always maintained close and frequent personal contacts. Relatively many of Twardowski's students were women (among others, Izydora Dambska, Maria Lutman-Kokoszynska, and Seweryna Luszczewska-Romahnowa, all of whom held philosophy chairs later). At a certain point, Twardowski had managed to create three concentric circles of philosophical influence: there was the Philosophical Circle, open to all university departments, the Polish Philosophical Society (1904), open to professional philosophers, and the journal Ruch Filozoficzny, conceived as an organ of promotion of philosophy at large and open to everyone (1911). Twardowski also established a laboratory of psychology in 1907. As he writes, he was waging "a most aggressive propaganda campaign on behalf of philosophy" (Twardowski 1926, 20). The campaign soon resulted in his lectures being moved to the Great Concert Hall of the Lvov Musical Society when the number of students reached two thousand (in the mid-Twenties he lectured in the Apollo movie theater, at 7 a.m. in the summer and 8 a.m. in the winter, without academic quarter). All this amounted to the establishment of a philosophical movement that soon became known as a proper school: first, until the First World War, it was known as the Lvov School; then, when the Russian-speaking University of Warsaw again became Polish-speaking, and Twardowski's students starting getting positions and having their own students there, it was known as the Lvov-Warsaw School (see Lvov-Warsaw School). The name is somewhat inaccurate, for Twardowski's students occupied philosophy chairs in all post-war Polish universities, not only in Lvov and Warsaw. As has often been pointed out, what all students of Twardowski had in common was not a particular set of views, but a rather distinctive attitude to philosophical problems informed by precision and clarity that they inherited from Twardowski's general conception of methodology, and which he valued most highly. According to Jordan, Twardowski led his students > > to undertake painstaking analysis of specific problems which were rich > in conceptual and terminological distinctions, and directed rather to > the clarification than to the solution of the problems involved. > (Jordan 1963, > 7f)[2] > In this, Twardowski's students learned what he had learned from Brentano, namely > > how to strive relentlessly after matter-of-factness, and how to pursue > a method of analysis and investigation that, insofar as that is > possible, guarantees that matter-of-factness. He proved to me by > example that the most difficult problems can be clearly formulated, > and the attempts at their solution no less clearly presented, provided > one is clear within oneself. The emphasis he placed on sharp > conceptual distinctions that did not lapse into fruitless nit-picking > was an important guideline for my own writings. (Twardowski 1926, 20) > Twardowski was convinced that the philosophical way of thinking he advocated, namely precision in thought and writing and rigor in argumentation, was directly beneficial to practical life. It was with the idea of proving exactly this point--by way of his personal example--that he accepted to head various kinds of committees (among others, the University Lectures Series, the Society for Women in Higher Education, the Society for Teachers in High Education, the Federation of Austrian Middle-School Teacher Candidates) and to be Dean twice and Rector three times in a row (though he repeatedly refused to be Minister of Education). All these activities cost him a lot of time: in fact, Twardowski's choice to be most of all an educator and an organizer left him very little time for academic writing. Besides, as he reports, Twardowski wasn't much interested in the publication of his ideas. Thus, since he placed high demands on the clarity and the logical cogency of philosophical work, he set out to publish only when it was required by external circumstances (Twardowski 1926, 30). As a consequence, in his Lvov years, Twardowski published little. He officially retired in 1930. Twardowski died on February 11, 1938. On Twardowski's life and education in Vienna up to 1895 (as well as Twardowski's early activity in Lvov and Vienna's 'war parenthesis' in 1914-1915) see Brozek 2012; Twardowski's heritage and his achievements as an educator and organizer in Lvov, see Czezowski 1948, Ingarden 1948; Ajdukiewicz 1959, Kotarbinski 1959 (in Polish: excerpts in English are to be found in Brozek 2014), Dambska 1978, Wolenski 1989, 1997 and 1998. With special reference to psychology, see Rzepa and Stachowski 1993. ## 2. The Vienna Years: 1891-1895 Twardowski's main publication in the Vienna period is, next to his doctoral dissertation Idea and Perception (1891), his Habilitationsschrift, On the Content and Object of Presentations (1894), Twardowski's most influential work. According to J. N. Findlay, Content and Object is "unquestionably one of the most interesting treatises in the whole range of modern philosophy; it is clear, concentrated, and amazingly rich in ideas" (1963, 8). ### 2.1 On the Content and Object of Presentations (1894): Context, Influence and Historical Background In Content and Object, Twardowski shares a number of Brentano's fundamental theses. Five of them are particularly relevant here. According to the first and most important thesis, the essential characteristic of mental phenomena, and what demarcates them from physical phenomena, is intentionality. We can sum it up as follows: > > Brentano's Thesis: > > > Every mental phenomenon has an object towards which it is directed. > Mental phenomena, also called mental acts, fall into three separate classes (second thesis): presentations (Vorstellungen), judgments, and phenomena of love and hate. In the mental act of presentation, an object is presented; in a judgment, judged; in love and hate, loved or hated.[3] Next (third thesis), mental phenomena are either presentations or are based on presentations. We need to present an object in order to judge it or appreciate it (though we do not need to judge or appreciate an object that we might just present). Importantly, however (fourth thesis), a judgment is not a combination of presentations, but a mental act sui generis that accepts or rejects the object given by the presentation at its basis (see Brentano's Theory of Judgment). In keeping with this idea, all judgments (fifth thesis) can be aptly expressed in the existential form 'A is' (positive judgment) or 'A is not' (negative judgment) (alternatively, 'A exists' or 'A does not exist'). In both cases, the judgment has a so-called 'immanent' object, given by the presentation, which is simply A. Both the notion of an object 'immanent' in a mental act, as well as, in all generality, the term 'object' in Brentano's Thesis, have an ambiguous character. Are the objects of mental acts fully inside us or not? To correctly judge that the aether does not exist means, in Brentanian terms, rejecting the aether which is given to one's consciousness by a presentation. But if the aether is an object immanent in me, fully inside my consciousness, what does it mean, then, that I reject it? Is it some mental entity inside my consciousness that I reject? But how could that possibly be correct? In this case I most certainly have something in my head, and that something exists. So it cannot be correct that I reject it--for, if that is the aether I reject, the judgment that the aether does not exist cannot be true. The aether should be a physical space-filling substance, outside my consciousness: that is what I am rejecting. But if that is what I reject, it seems something must be there for me to be able to affirm that it is not there. This is very puzzling. It was fundamentally due to difficulties of this kind that Brentano's theory of judgment was subjected, from 1889 onwards, to continuous objections by philosophers such as Sigwart and Windelband. Brentano engaged in the debate in his defense, as did his pupils Marty and Hillebrand. It is in this context that Twardowski's Habilitationsschrift was conceived. Twardowski saw that notions such as 'the object of a presentation' and 'immanent object' were ambiguous because in Brentano's writings the object of a presentation was identified with that of content of a presentation (Twardowski 1926, 10). In Content and Object, Twardowski set out to clarify exactly the relationship between the two, with far-reaching implications for Brentano's original position. The distinction between content and object of a presentation was not new in Twardowski's time. It was the most fundamental element of Bolzano 1837; it was later present in works of Bolzanian inspiration (Zimmermann 1853, Kerry 1885-1891; on Zimmermann see Winter 1975, Morscher 1997, Raspa 1999); and it was also mentioned in works of Brentanian inspiration (Hofler and Meinong 1890, Hillebrand 1891, Marty 1884-1895: article 5, 1894). Nevertheless, the distinction was by no means common lore. In particular, before Twardowski, no Brentanian had endorsed the distinction between content and object in a way that offered a basis for solving the problems of Brentano's theory, nor had anyone among the Brentanians devoted any in-depth study to the issue, although the differences between Bolzano's and Brentano's theories are such that reworking the distinction in a Brentanian framework raises philosophical problems that are by no means trivial. A major difference concerns the role of the content of a presentation and its ontological status, since for a Brentanian the content of a presentation can only be something actually existing in one's mind. No Brentanian had thus realised in full the implications that reinterpreting the Bolzanian distinction in a Brentanian key would have. Carrying out this task, and thus helping Brentano's theory out of its most pressing troubles, was Twardowski's original contribution. He drew inspiration from arguments in favor of the content-object distinction present in Bolzano, but he reinterpreted them in a Brentanian framework to sustain conclusions that were opposite to Bolzano's and that were new for the Brentanians. One such conclusion, and a major implication of Twardowski's theory of intentionality, is that there are no objectless presentations, presentations without an object, no matter how strange and improbable that object is[4]. Even presentations of contradictory objects have both content and object. It is this thesis and the conclusions that Twardowski drew from it that have had a major impact on the development of Brentanian theories of intentionality and that opened the way to ontologies as rich as that of Alexius Meinong. On the other hand, this position, together with Twardowski's identification of meaning with psychological content, prompted Husserl's critical reactions and led Husserl to the theory of intentionality set out in the Logical Investigations (1900/01) where Twardowski's distinction between content and object is taken up (on this, see Schuhmann 1993; on Twardowski's influence and 'triggering effect' on Husserl, see Cavallin 1990, especially 28f). Twardowski's ideas were not only influential on the continent. Via G. F. Stout, who published an anonymous review of Content and Object, in Mind, Twardowski's ideas had an influence on Moore and Russell's transition from idealism to analytic philosophy (van der Schaar 1996). ### 2.2 'The Presented' The aim of Content and Object is to distinguish "the presented, in one sense, where it means the content, from the presented in the other sense, where it is used to designate the object" (Twardowski 1894, 4). Its main thesis is that in every mental act a content (Inhalt) and an object (Gegenstand) must be distinguished. This distinction enables Twardowski to clarify that > > Twardowski's Thesis: > > > Every mental phenomenon has a content and an object, and it is > directed towards its object, not towards its content. > On the basis of the distinction between content and object, Twardowski is in turn able to clarify Brentano's notion of 'object immanent (immanentes Objekt) in a presentation' by identifying it with the notion of content, and to clarify Brentano's notion of 'object of a presentation' by identifying it with the notion of object. The distinction between the content and the object of a presentation rests on a psychological or epistemic difference which is, roughly speaking, the mental counterpart of Frege's distinction between sense and reference. The object of a presentation, says Twardowski using Zimmermann's terminology, is that which is presented in a presentation; the content is that through which the object is presented. An important argument Twardowski gives in favor of this distinction--and which strengthens the analogy with Frege's distinction--is that we can present the same object in two different ways by having two presentations with the same object but with different content. Twardowski calls such presentations interchangeable presentations (Wechselvorstellungen). The presentation of the Roman Juvavum and the presentation of the birthplace of Mozart have the same object, Salzburg; however, their content differs. To offer a rough analogy, think of an arrow pointing at an object: the object is what the arrow is pointing at, the content is what in the arrow makes it the arrow it is, that is, its being directed to that object and not to another. The act is just the 'being directed towards' of the arrow. Interchangeable presentations are like two arrows pointing at the same object. Twardowski thinks of the difference between act, content, and object in the following way. An act of presentation is a mental event which takes place in our mind at a certain time. The content is literally inside the mind, and exists dependently on the act, as long as the act does. The object is instead independent of the mental act (1894, 1, 4; SS7, 36), and, generally speaking, not inside one's mind, although in some special cases the object of a presentation might be a mental item. This special case is the case in which the content of some presentation plays the role of the object of another presentation. This is not infrequent, because any time we discuss the content of a presentation, describing its characteristics and its relations to other things, we are presenting it. And for this to be possible, the content must play the role of object in the presentation(s) we are having (what we would call second-order presentations). To use the arrow metaphor once again, second-order presentations, then, are like an arrow directed towards another arrow. The way in which Twardowski relates mind and language makes the distinction between content and object fairly easy to understand. Names are the linguistic counterpart of presentations. By 'names' Twardowski means the categorematic terms of traditional logic ('Barack Obama,' 'The President of the United States,' 'black,' 'man,''he'). A name has three functions: first, a name makes known that in the mind of the person using the name an act of presentation is taking place; secondly, a name means something; thirdly, it names an object. Twardowski takes the meaning of a name to be the content of the presentation that, as the name makes known, is taking place in the speaker (SS3), that is, Twardowski takes meaning to be something mental and individual. This holds, mutatis mutandis, also for judgments. In this sense, in Content and Object the semantic sphere is dependent on the mental sphere. This trait qualifies Twardowski's position as psychologistic (see Psychologism: PA3). Although his position on meaning underwent a development, Twardowski never adhered to a platonistic conception of meaning like Bolzano or Frege did. If all this is intuitive, why are the object and the content of a presentation conflated? Twardowski maintains that the reason why content and object are often identified comes, among others, from a linguistic ambiguity: both the content and the object are said to be 'presented' in a presentation (SS4). Twardowski offers an analysis of the ambiguity of the term 'presented' by appealing to the linguistic distinction between modifying and attributive (or determining) adjectives, and he illustrates it with an analogy between the act of presenting an object and the act of painting a landscape. When a painter paints a landscape, she also paints a painting: so we can say that the painting and the landscape are both painted. But in this situation 'painted landscape' can have two very different meanings. In the first meaning of 'painted', a painted landscape is a landscape; in the second meaning of 'painted' a painted landscape is not a landscape, but a painting (like in Magritte's La trahison des images (1928-9): it is a painted pipe we are looking at, not a pipe). In the first case, 'painted' is used in an attributive sense (the landscape is a portion of nature that happens to be painted by a painter in a painting); in the second case 'painted' is used in a modifying sense (that in which, looking at the painting in a museum, someone may say: this is a landscape!). The painted landscape in the modifying sense is a painting, and thus identical with the painting painted in the attributive sense. Analogously, in an act of presentation, the object can be, like the painted landscape, said to be 'presented' in two senses. The object presented in the modifying sense is identical with the content presented in the attributive sense: it is dependent on the act of presentation, and it is what we mean by 'the object immanent in the act'; the object presented in the attributive sense is the object of the presentation, what happens to be presented in a presentation, and what is independent of the act of presentation. ### 2.3 Nonexistent Objects According to a Brentanian conception of judgment, when we judge, the object is given to us by an act of presentation. Given this fact, Twardowski's analysis of 'presented' in terms of modifying vs. attributive adjectives is fundamental to understand what is exactly 'judged in a judgment.'[5] For it is not only in presentations that we can distinguish act, content and object, but also in judgments. When we pass a judgment, we either accept or reject an object through a content. All judgments have a form which can be linguistically expressed as 'The object A exists' or 'The object A does not exist.' The object of judgment (A) is what is judged in a judgment. This object is the object of the presentation at the basis of the judgment, it is not the content of the presentation; it is the object presented in the attributive sense, not in the modifying sense. The content of a judgment is the existence or non-existence of the object presented. It might seem strange at first to hear Twardowski say that the content of a judgment is the (non-)existence of the object, but Twardowski has something like the following in mind: we judge about the object A that it exists (or that it does not exist). Twardowski's analysis clarifies what is going on when we judge, in a Brentanian theory of judgment, that the aether does not exist. Judging that the aether does not exist means rejecting a physical space-filling substance outside my consciousness. The object of this judgment is not mental. However, this does not mean that there is nothing inside my consciousness: there is a mental content, through which the aether is presented and judged as non-existing. That content, present in me, however, exists; it is the aether itself that does not exist. For a theory of judgment like the one sketched above to work, the object we judge must be the very object we present. Therefore, Twardowski's thesis must be understood in the strong sense that there are no presentations which do not have an object: if a presentation did not have an object, we would not have an object to judge as existing or non-existing. It is for this reason that a crucial part of Content and Object is devoted to defending this claim, and to showing that, despite appearances, every presentation has an object (SS5). This is like saying, coming back to our arrow metaphor, that every arrow points at something. Twardowski's strategy is to show that presentations which are normally deemed by others to be objectless--i.e., arrows that do not point to anything--are not such. There exist no presentations without an object: there are presentations whose object does not exist. Twardowski gives a number of arguments for his position. Of all these arguments, the key one is an argument based on the three functions of names in language. > > If someone uses the expression: 'oblique square,' then he > makes known that there occurs in him an act of presentation. The > content, which belongs to this act, constitutes the meaning of this > name. But this name does not only mean something, it also designates > something, namely, something which combines in itself contradictory > properties and whose existence one denies as soon as one feels > inclined to make a judgment about it. Something is undoubtedly > designated by the name, even though this something does not exist. > (SS5, 23) > On the basis of this reasoning and a linguistic analysis of how 'nothing' functions in language Twardowski rejects Bolzano's claim that 'nothing' is an objectless presentation by showing that 'nothing' is a syncategorematic expression (like 'and,' 'or,' and 'the') not a categorematic one. If an expression is not a categorematic expression, it is not a name; but if an expression is not a name, it does not need to have three functions. To every name there corresponds a presentation and vice versa; if an expression is not a name, then there is no presentation corresponding to it. If there is no presentation corresponding to 'nothing' because 'nothing' is not a name, the question of its object does not arise. This argument has been said to anticipate by 37 years Carnap's analysis of Heidegger's 'The nothing itself nothings.' (Wolenski 1998, 11). The comparison should however not make one think that Twardowski shared Carnap's attitude towards metaphysics (see 2.4 below.) Another argument given by Twardowski for the thesis that every presentation has an object is based on the different ontological status of act, content, and object of a presentation. The act and the content always exist; and, in fact, they always exist together, forming a whole in the mind, a unified mental reality, though the content has a dependent existence on the act.[6] The object may or may not exist. Suppose you have the following presentations: the presentation of Barack Obama, that of a possible object such as the aether, and that of an impossible object, such as a dodecahedron with thirteen sides or a round square. The objects of these presentations (namely Barack Obama, the aether, and the round square) differ greatly, but they are still all objects. Barack Obama is an existing object; the aether and the dodecahedron with thirteen sides are non-existing objects. When we present an object such as a dodecahedron with thirteen sides or a round square, the object, that which is named, is different from the content because the content exists and the object does not. It is the object, not the content, that is rejected in the negative existential judgment 'the round square does not exist,' for it is the object that does not exist; the content 'exists in the truest sense of the word' (SS5, 24). A third argument rests on the difference between being presented and existing (SS5, 24). Those who claim that there are objectless presentations, says Twardowski, confuse 'being presented' with 'existing.' To reinforce his point, Twardowski offers several observations, among others, one resting on the claim that it is the object which is the bearer of contradictory properties, not the content. This claim will be described later by Meinong and Mally as the principle of the independence of Being (existence) from Being-So (bearing properties). When we present an object which does not exist because it is contradictory, i.e. impossible, we need not immediately notice that that object has contradictory qualities: it is possible that we discover this successively by further reasoning. Suppose now that only presentations with possible objects are accepted. Then, Twardowski continues, the presentation of something contradictory would have an object for as long as we did not notice the contradiction; the moment we discovered it, the presentation would cease to have an object. What would then bear the contradictory qualities? Since the content can't be what bears the contradictory qualities, it is the object itself that bears them; but then, this object has to be what is presented. [7] The arguments above ultimately rest on the idea that every name has three functions. This assumption can also be seen, in turn, as resting on an even more basic idea, namely that in all generality 'object' equals 'being capable to take up the role of object in a presentation,' thus an object is anything that can be presented; and anything that can be named by a name (SS3, 12; SS5, 23; SS7, 37). Yet, saying that an object is anything that can be named by a name and that a name always has the function of naming are two sides of the same coin, as are the claim that an object is anything that can be presented by a presentation and the claim that a presentation always has an object. This cluster of correlative assumptions is the core of the theory. Only when this core is accepted can the other claims put forward by Twardowski (namely (1) that 'being presented' is not the same as 'existing' and (2) that objects can possess properties even though they do not exist) become convincing parts of a cogent theory of intentionality. One might note, however, that it is a theory for which there are no non-circular arguments, for the theory is acceptable only to those who are already convinced that there are no empty names, that all categorematic terms, including 'round square,' 'aether,', 'Pegasus,' etc. have the function of naming something, not only the function of being meaningful. As is known, the theory, in Meinong's version, will be the critical target of Russell's 'On Denoting' (1905). ### 2.4 Mereology: Parts of the Content and Parts of the Object Metaphysics is importantly present in Content and Object, and particularly important are the mereological considerations Twardowski offers. According to Ingarden, Content and Object offered "the first consistently constructed theory of objects manifesting a certain theoretical unity since the times of scholasticism and of the 'ontology' of Christian Wolff" (1948, 258). One cannot literally agree with Ingarden considering the Leibnizian metaphysics in Bolzano's Athanasia (1827)--hidden in the two-thousand pages of the Wissenschaftslehre--and Brentano's work in mereology. Nevertheless, if the impact of Content and Object is compared with that of Bolzano's or Brentano's metaphysics, things are different. Content and Object was a fundamental contribution to the renaissance of Aristotelian metaphysics--metaphysics in the sense of a general theory of objects--which led to both Meinong's theory of objects and to Husserl's formal ontology of parts and wholes in the Third Logical Investigation. The story continues, later, with Lesniewski's mereology and Ingarden's ontology. The heritage of Content and Object also includes the fact that Twardowski's pupils had a relation to metaphysics which was vastly different from the approach that was typical, for instance, of the Vienna circle (see Lukasiewicz 1936). In Poland, metaphysics was not rejected as nonsense, but accepted as a respectable area of investigation to be explored using rigorous methods, including axiomatics (see Smith 1988, 315-6; on Twardowski and metaphysics, see Kleszcz 2016) Like anyone in pre-set theoretical times, Twardowski has a very broad notion of (proper) part, covering much more than just the pieces of an object. Twardowski distinguishes material and formal parts of an object. The material parts of an object comprise not only the pieces of an object, but also anything that can be said to be a component of it, such as the series 1, 3, 5 is composed of three numbers (namely, 1, 3, and 5). Among the material parts of an object are also its qualities (Beschaffenheiten) such as extension, weight, color, etc. (1894, SS10, 58). The formal parts of an object are the relations (Beziehungen) obtaining between the object and its material parts (primary formal parts) as well as the relations obtaining among the material parts themselves (secondary formal parts) (1894, SS9, 48 and ff.; SS10, 51 and ff.). In keeping with the tradition, Twardowski calls the matter of an object the sum of its material parts and the form of an object the sum of its formal parts. A special kind of primary formal parts are (relations of) properties (Eigenschaften): these are relations between an object as a whole and one of its material parts, consisting in the whole's having the part at issue (1894, SS10, 56). Since Twardowski accepts that among the parts of an object there are the relations in which that object is in, Twardowski's mereology is, strictly speaking, an atomless mereology: there are no simples (1894, SS12, 74). However, we can speak of atoms if we restrict ourselves to material parts only (for instance, in the case of the number one, we can say it is a simple object only if we consider the (proper) material parts it has, namely zero). The distinction between content and object enables Twardowski to clearly distinguish conceptualizations regarding parts of the object from conceptualizations regarding the parts of the content (as special cases of objects). This, in turn, enables Twardowski to offer sophisticated considerations, leading him among others to fix clearly the notion of the characteristic mark (Merkmal, nota) of an object. Twardowski calls elements the parts of the content of a presentation; he calls characteristic marks the parts of the object of a presentation (SS8, 46-7); the characteristic marks of the object are presented through the elements of the content. The content of a presentation is the collection of the presentations of the characteristic marks of the object of the presentation. The notion of characteristic mark is a relative one because only the parts of an object that happen to be actually presented in the content of a presentation in someone's mind qualify as its characteristics. > > > For example, one can be presented with a table without thinking of the > shape of its legs; in this case, the shape of the table legs is a > (material, metaphysical) constituent (of second order), but not a > characteristic of the table. But if one thinks, while being presented > with the table, of the shape of its legs, then the shape had to be > considered a characteristic of the table. (SS13, 86; Eng. transl. > 81). > > > According to Twardowski, it is not possible to present all the parts of an object in a presentation. Given that the number of parts of an object is boundless, and that we can only present a finite amount of characteristics, the number of the elements of the content is therefore lower than that of the parts of the object (SS12, 78-9; on this point Twardowski is again indebted to Bolzano (Wissenschaftslehre, SS64). It follows that no adequate presentation is possible of any object (SS13, 83). Like Brentano had done, Twardowski distinguishes, with respect to material parts, metaphysical from physical and logical parts on the basis of a notion of dependence and separability.[8] However, differently from Brentano, Twardowski does not construe the notion of dependence in terms of existence of the objects involved. His notion of dependence needs also to be applicable to non-existing objects. Consequently, Twardowski construes the notion of dependence of the parts of an object in terms of modes of presentability of the parts of an object, i.e. as existence of the parts of the content of a presentation (which always exists). The inseparability of a part p1 from a part p2 of an object O is not construed in the sense that it is not possible for p1 to exist without p2 existing, but in the sense that in the content of the presentation of an object O the part that represents p1 cannot exist without the part that represents p2 existing, that is, both must be elements of the content of the presentation of the object O. Two material parts of the content of a presentation of A and of B are mutually separable iff A can be presented without presenting B and vice versa. Mutually separable parts are physical parts. For instance, the parts of the content of the presentations of the pages and the cover of a book are mutually separable. Two material parts of the content of a presentation of A and of B are one-sidedly separable iff A can be presented without B, but not vice versa. Logical parts are one-sidedly separable: for instance, we can have a presentation of color without the presentation of red, but not vice versa. Two material parts of the content of a presentation of A and of B are mutually inseparable (but yet distinguishable) iff they are neither mutually nor one-sidedly separable. Metaphysical parts are mutually inseparable, e.g. you cannot present separately the being colored and the being extended of something colored and extended, although those parts are in abstracto distinguishable in the object. On Twardowski's mereology see Cavallin (1990), Rosiak (1998) and Schaar (2015, 68 and ff.) ### 2.5 The Position of Content and Object in Twardowski's Oeuvre Content and Object should be seen as part of a bigger research aim that Twardowski had set for himself. This included developing a theory of concepts and a theory of judgment. In Content and Object, Twardowski had given a coherent account of how judgments such as 'The aether does not exist' work on the basis of the systematization of the notion of 'object of a presentation.' That systematization however had implications for the theory of judgment which went far beyond the account of judgments such as 'The aether does not exist.' Although Content and Object contains little on the theory of judgment, Twardowski was well aware of those implications. A letter to Meinong is evidence that Twardowski was developing a theory of judgment which continued the work initiated by Content and Object, whose first main outline can be found in Twardowski's manuscripts Logik 1894/5 and Logika 1895/6 (Betti & van der Schaar 2004, Betti 2005). In his 'Self-Portrait' Twardowski also remarks that it was the question of the nature of concepts that brought him to Content and Object. The issue ensued from Twardowski's research on a particular concept in his doctoral dissertation, Descartes' concept of clear and distinct perception. Since concepts are a species of presentations, Twardowski saw that he had first to investigate presentations in general, and thus to dispel the ambiguities that the notion of 'the presented' carried with itself (Twardowski 1926, 10). Work on concepts is to be found in the manuscript Logik 1894/5, where concepts are defined as presentations with well-defined content, in Images and Concepts (1898) and in The Essence of Concepts (1924), where presentations with well-defined content (i.e. fixed by a definition) are called 'logical concepts.' An important role in Twardowski's theory of concepts and definitions is played by the notion of presented judgment, i.e. judgments not really passed, but merely presented (in the modified sense). Twardowski's investigation of the notion of (logical) concept is important to understand his attitude towards objectivity and unicity of meaning, i.e. his relationship to psychologism on the one hand, and the role of the theorizations in Content and Object of two notions: that of the object of a general presentation (which disappears for instance in Logik 1894/5) and that of indirect presentations on the other hand. ## 3. The Lvov Years: 1895-1938 Among the issues that Twardowski only sketched or left open in his Vienna years and investigated later are, next to theories of truth and knowledge (Theory of Knowledge, 1925; see Schaar 2015, Chapter 5.3), the relationship between time and truth ('On the So-called Relative Truths,' 1900) and that between linguistic meaning and the content of mental acts ('Actions and Products,' 1912). Particularly important themes to mention are the relationship between a priori or deductive sciences, a posteriori or inductive sciences and the notion of grounding ('A Priori and A Posteriori Sciences,' 1923), and the relation between philosophy, psychology and physiology ('Psychology vs. Physiology and Philosophy', 1897b). Well worth mentioning are Twardowski's writings on issues of philosophical methodology ('On Clear and Obscure Philosophical Style,' 1919/20; 'Symbolomania and Pragmatophobia,' 1921). He also left a relatively substantial corpus of writings in ethics ('On Ethical Skepticism', 1905-24). Twardowski's publications during his Lvov years are in Polish, but of those writings that Twardowski considered as important academic publications we normally also have German versions. Especially important and influential among these are 'On the So-Called Relative Truths' (1900) and 'Actions and Products' (1912). ### 3.1 On the so-called relative truths (1900) 'On the So-Called Relative Truths' (henceforth: Relative Truths) is a work of fundamental importance for the development of the idea of absolute truth in Poland. Scholars have deemed its impact to have reached as far as Tarski's work on truth (Wolenski & Simons 1989). It certainly influenced the 1910-1913 discussion on future contingents which involved Kotarbinski, Lesniewski, and Lukasiewicz, which served later as metaphysical foundation for Lukasiewicz's three-valued logic (Wolenski 1990). Twardowski's position is distinctive and interesting because it is a non-platonistic theory of absolute truth, though it is difficult to say exactly what place it occupies with respect to Lesniewski's nominalistic or to Tarski's platonistic approach (for these labels see Simons 2003: section 2.) One of the reasons why it was important for Twardowski to defend an anti-relativistic notion of truth was that relativism jeopardizes the possibility of constructing ethics as a science based on principles (on Twardowski's ethics, see Paczkowska-Lagowska 1977). This aspect is important because Twardowski is usually characterized not as a builder of systems, but as championing 'small philosophy.' This is correct, in some sense, but it should not make one mistakenly think that there is no unity sought in Twardowski's thought, or that his works are separate small occasional contributions without overarching ideas in the background governing their development. In Relative Truths, Twardowski opposes Brentano's ideas on the relationship between time and truth, and sides instead with Bolzano's views.[9] By 'truth,' Twardowski means a true judgment and by 'absolute truth' he means a judgment true independently of any circumstance, time, and place. The main claim of Relative Truths is that every truth (falsity) is absolute, i.e. no judgment changes its truth-value relatively to circumstance, time, or place. Although Twardowski's treatment is general, and is not confined to the relation between truth and time, it is his ideas on the latter that were particularly important for later discussions. We can characterize Twardowski's view in this respect as the view that a judgment is true for ever and since ever, that is: > > Omnitemporal truth: > > > For any judgment g, if g is true at a time > t, then it is true also at an arbitrary time > t' past or future with respect to t (the same > applies, mutatis mutandis, to falsity). > Twardowski's aim is to defend this claim against those who argue that truth is relative on the basis of examples of elliptical sentences (such as 'I don't'), sentences with indexicals ('my father is called Vincent'), sentences of general form ('radioactivity is good for you'), and sentences about ethical principles ('it is wrong to speak against one's own convictions'). Those who argue that truth is relative on this basis, says Twardowski, make a mistake. They confuse judgments, which are the real truth-bearers, with the (type) sentences expressing them. But sentences are merely the external expression of judgments, and often they do not express everything one has in mind when judging. For human speech has a purely practical task; it can perfectly serve its communicative purposes and yet be strictly speaking ambiguous or elliptic. Twardowski's main point is that we can disambiguate ambiguous sentences or integrate elliptic ones in such a way that they become the appropriate means of expression of the judgment that they actually and strictly speaking express (Twardowski 1900, 156)--we can make sentences eternal, as we would say in our Post-Quinean age. Once we show that eternalization is possible, the confusion on which relativists rely is dispelled. For instance, if standing in Lvov on the High Castle Hill, I assert that it is raining, > > I do not have in mind just any rain, falling at any place and time, > but I voice a judgment about the rain falling here and now > (Twardowski 1900, 151). > One could object that the true sentence 'it is raining here and now' is relative because it may become false, i.e. be true when uttered in Amsterdam and become false when uttered in the Dry Valley in Antarctica. Twardowski observes that this impression is, again, only due to the ambiguity of the indexicals 'here and now' in the sentence. Take thus the sentence > > (i) It is raining here and now, > when uttered in Lvov, on 1 March 1900, in accordance with the Gregorian calendar, at noon, Central European Time, on the High Castle Hill. That sentence expresses the same judgment as > > (i\) On 1 March 1900, in accordance with the Gregorian calendar, at > noon, Central European Time, it is raining in Lvov, on the High Castle > Hill and in its vicinity. > According to Twardowski, the difference between the two sentences is only in their brevity and their practical use. Take now the sentence > > (ii) It is raining here and now, > when uttered in Krakow on 2 March 1900, in accordance with the Gregorian calendar, at noon, Central European Time, on the Castle Hill. That sentence expresses the same judgment as > > (ii\) On 2 March 1900, in accordance with the Gregorian calendar, at > noon, Central European Time, it is raining in Krakow, on the Castle > Hill and in its vicinity. > It's not that we have in (i) and (ii) one and the same judgment; we have the same sentence, which expresses two different judgments, (i\) and (ii\). Therefore, one cannot argue that the same judgment can turn from true to false. > > > It is apparent that [the judgment expressed by (i) and (i\)], which > [...] asserts that it is raining, is true not only at a particular > time and place, but is always true. (Twardowski 1900, 153) > > > > [...] Only sentences can be said to be relatively true; yet the truth > of a sentence depends on the truth of the judgment expressed by that > sentence; usually a given sentence can express various judgments, some > true, some false, and it is thus relatively true because it expresses > a true judgment only under certain conditions, i.e. if we consider it > as an expression of a judgment which is true (Twardowski 1900, 169). > > > > Notice that Twardowski is not saying that judgments can be integrated or completed: the judgments we formulate in our head, and which are true or false, are complete* and fully unambiguous. It is for this reason that he can think that the procedure of completing sentences as expressions of judgments can be carried out. An important thing to notice in Relative Truths is that Twardowski, differently from what Meinong and Lukasiewicz were to do, did not question the principle of contradiction or the principle of the excluded middle. Twardowski's idea that one can have absolute truth in an all-changing world has been recently taken up by Simons (2003). ### 3.2 Actions and Products (1912) For quite some time Twardowski thought that logic was dependent on psychology [10], and continued to hold the psychologistic idea of meaning as mental content from Content and Object. In his 'Self-Portrait,' Twardowski says that he changed his mind in an anti-psychologistic sense because of Husserl's Logical Investigations (1900/01).[11] 'Actions and Products' (1912) is the first printed work where Twardowski favors an 'Aristotelian' view of ideal meaning, i.e. of meaning in specie, which he associates with that of Husserl's in the Logical Investigations. In marking the line of demarcation between logic and psychology on the basis of the distinction act/product which underlies his theory of meaning, Twardowski writes: > > Indeed, a rigorous demarcation of products from actions has already > contributed enormously to liberating logic from psychological > accretions. (Twardowski 1912, SS45, 132) > Twardowski's mature theory of meaning is connected with the rigorous definition of the distinction between actions and products of actions (while in Relative Truths he speaks of judgments as actions or products). On the basis of grammatical analyses, purporting however to show logically salient differences, Twardowski establishes a basic distinction between physical, psychical (i.e. mental), and psychophysical actions and their products. The analysis is reminiscent in philosophical style of that of 'the presented' in Content and Object, though more general, and a nice example of the method which has become now strongly associated with analytic philosophy. The relationship between an action and what results from it, its product, is exemplified linguistically in the relationship between a verb and the corresponding noun as internal complement (Twardowski 1912, SS1, 10; SS8, 107): > > > > \| \| \| \| > \| --- \| --- \| --- \| > \| \| Act \| Product \| > \| Physical \| running \| run \| > \| Mental \| judging \| judgment \| > \| Psychophysical \| speaking \| speech \| > > > A psychophysical product (e.g., a speech) differs from a mental product (e.g., a judgment) because it is perceptible to senses; it differs from a physical product (e.g., a run) because in the psychophysical action which produces it (speaking) a mental action is also involved, which has bearing on the physical action, and thus on its product (SS10). In some cases, a psychophysical product expresses a mental product; for instance, a sentence is a psychophysical product which expresses a mental product, a judgment. Twardowski points out that the meaning of the noun 'judgment,' like other nouns ('mistake'), is ambiguous between action and product. A judgment in the sense of the action (a judging) is a judgment in the psychological sense, while a judgment in the sense of the product is a judgment in the logical sense (SS14) (a third meaning of 'judgment' is that of 'disposition, aptitude to make correct judgments,' SS15). Twardowski points out that he uses 'judgment' in the sense of 'judgment in the logical sense,' i.e. the product of the action of judging, and he clarifies that what he means now by 'judgment-product' is the content of judgment in Content and Object (SS24, n. 37, 117). Exactly as it was the case there, the judgment exists as long as someone performs the corresponding action of judging; for this reason, it is called a non-enduring product (SS23, 116). Non-enduring products do not exist in actuality separately from the corresponding actions, but only in conjunction with them; we can only analyze them abstractly apart from these actions. On the other hand, enduring products can and do exist in actuality apart from the actions owing to which they arise (Twardowski 1912, n. 41, 116). Enduring products last longer than the action which produces them; they originate from a transformation or a rearrangement of pre-existing physical material in the course of the action: footprints in the sand are enduring products arising from the change of configuration of grains of sand (the material) as a product of the action of walking applied to that material (Twardowski points out that the product is not the grains of sand arranged in some way, but the arrangement itself, SS26). If actions are processes, non-enduring products are events whereas enduring products present themselves as things (SS 27). Among enduring products, we find physical products (such as footprints in the sand) and psychophysical products (such as paintings). A mental product, such as a judgment, is never enduring (SS29), but it can have its expression in an enduring psychophysical product, such as a written sentence. The process of preserving non-enduring products such as judgments in enduring products such as written sentences is a complex one (SS37), and goes in two steps. In step one a spoken sentence is produced which expresses a judgment, in such a way that the judgment is the meaning of the sentence and the sentence is the sign of the judgment. The process goes as follows. A non-enduring mental product, a judgment (together with the action of judging), which is non-perceptible, gives rise--by being its (partial) cause--to a non-enduring psychophysical product, a spoken sentence, which is perceptible. In this case, the spoken sentence is the expression of the judgment (SS30). Now, if the spoken sentence becomes itself a partial cause of another judgment, which (we would say) is a token of the same type of the initial judgment (by partially causing an action of judging which produces that other judgment in another person or at a different time in the same person), then the spoken sentence can also count as the sign of the judgment, and the judgment as the meaning of the expression (SS32, SS34). The condition just sketched in the antecedent is fundamental: without it, a sentence p might very well have been an expression of a judgment j once (j was a partial cause of p), but no meaning is linked with p: for, if p is incomprehensible, namely it is not a partial cause of another judgment j', then nothing can be said to be its meaning (SS31). Step two is to preserve the spoken sentence (a non-enduring psychophysical product) in an enduring psychophysical product s, a written sentence.[12] When a judgment is preserved in this way, it has in the sign an existence called potential. This is because the sign may at any moment cause the formation of an identical or similar judgment (Twardowski 1912, SS34), and it will be able to (partially) cause judgments as long as it lasts. In consequence of this, 'meaning' can therefore also mean > > the capacity to evoke a mental product in the individual on whom a > psychophysical product acts as a sign of that mental product, or, more > briefly, the capacity to bring the corresponding mental product to > awareness. (Twardowski 1912, n. 51, 125.) > Once they are preserved in this two-step way, non-enduring products assume not only the illusory appearance of enduring products, but also of products which are somehow independent from the actions which produce them. The appearance of independence is made stronger by the fact that we do as if one and the same judgment-product existed in all individuals, although many judgment-products are elicited by the written sentence-product. All these many judgments are different from each other, but, insofar as we consider a judgment to be the meaning of a sentence which is its sign > > there must be a group of common attributes in these individual mental > products. And it is precisely these common attributes (in which these > individual products accord) that we ordinarily regard as the meaning > of the psychophysical product, as the content inherent in it, provided > of course that these common attributes correspond to the intent with > which that psychophysical product was utilized as a sign. [...] Thus, > we speak of only a single meaning of a sign--barring cases of > ambiguity--and not of as many meanings as there are mental > products that are aroused or capable of being aroused, by that sign in > the persons on whom it acts. Now, a meaning conceived in this matter > is no longer a concrete mental product, but something at which we > arrive by way of an abstraction performed on concrete products. > (Twardowski 1912, SS39, 128) > To this passage, Twardowski appends a note referring to Husserl's notion of ideal meaning. The relationship between Husserl's and Twardowski's notion of meaning as well as the status of Twardowski's unique meaning has been and remains object of discussion (see Paczkowska-Lagowska 1979, Buczynska-Garewicz 1980, Placek 1996, Brandl 1998, Schaar 2015, 108). These matters should be considered as yet unsettled. The introduction of the notion of the unique meaning of a sentence leads Twardowski to make a further distinction between substitutive (artefacta) and non-substitutive judgments. Substitutive judgments aren't real judgments, but fictitious ones. The latter are in fact the presented judgments of Content and Object; the difference is that in Actions and Products, Twardowski applies the concept of substitutive judgments explicitly to logic: the sentences uttered or written by logicians-at-work are not sentences which express or have as meanings judgments which are really passed by them, but only presented judgments, produced thus by actions of presenting--which are different actions from actual judging acts. This happens for instance when a logician constructs a valid syllogism made up of materially false sentences to give examples of formally valid inferences (SS44, 130). In this case, the logician does not actually judge that all triangles are square, that all squares are round, and that all triangles are round, but she merely presents the corresponding judgments. The sentences 'all triangles are square,' 'all squares are round,' and 'all triangles are round' are not real sentences, but artificial ones, because they are "expressions of artificial products that substitute for actual judgments, namely merely represented judgments" (Ibid.). The meanings of these artificial sentences are artificial judgments, because they are merely presented, not passed. These artificial judgments are the subject-matter of logic (on this point, Twardowski is again indebted to Bolzano). Operating with surrogate sentences such as 'all triangles are square' constitutes the most extreme example of "making mental products independent of the actions owing to which alone they can truly (actually) exist" (SS44, 131). Twardowski's distinction between acts and products is presently being re-discovered in act-based theories of semantic content, within which Twardowski's notion of product is considered an interesting alternative to the notion of proposition (as the mind-independent and language-independent content of assertions, meaning of sentences, primary truthbearer, and object of propositional attitudes). See Moltmann 2014.
twotruths-india	## 1. Abhidharmikas / Sarvastivada (Vaibhasika) In the fourth century, Vasubandhu undertook a comprehensive survey of the Sarvastivada School's thought, and wrote a compendium, Treasury of Knowledge, (Abhidharmakosakarika [AK]; Mngon pa ku 1b-25a) with his own Commentary on the Treasury of Knowledge (Abhidharmakosabhasya [AKB], Mngon pa ku 26b-258a). This commentary not only offers an excellent account of the Sarvastivadin views, including the theory of the two truths, but also offers a sharp critique of many views held by the Sarvastivadins. Vasubandhu based his commentary on the Mahavibhasa (The Great Commentary), as the Sarvastivadins held their philosophical positions according to the teachings of the Mahavibhasa. Consequently, Sarvastivadins are often known as Vaibhasikas. The Sarvastivadin's ontology[2] or the theory of the two truths makes two fundamental claims. 1. the claim that the ultimate reality consists of irreducible spatial units (e.g., atoms of the material category) and irreducible temporal units (e.g., point-instant consciousnesses) of the five basic categories, and 2. the claim that the conventional reality consists of reducible spatial wholes or temporal continua. To put the matter straightforwardly, for the Sarvastivadins, wholes and continua are only conventionally real, whereas the atoms and point-instant consciousness are only ultimately real. ### 1.1 Conventional truth To see how the Sarvastivadins defend these two claims, we shall have a close look at their definitions of the two truths. We will examine conventional truth first. This will provide the argument in support of the second claim. In the Abhidharmakosa Vasubandu defines conventional truth/reality as follows: "An entity, the cognition of which does not arise when it is destroyed and, mentally divided, is conventionally existent like a pot and water. Ultimate existence is otherwise." ([AK] 6.4, Mngon pa khu 7ab) Whatever is, on this definition, designated as "conventionally existent" is taken to be "conventionally real" or conventional truth when the idea or concept of which ceases to arise when it is physically destroyed, by means of a hammer for instance. Or its properties such as shape are stripped away from it by means of subjecting it under analysis, thereby conceptually excluding them. A pot and water are designated as conventionally existent therefore conventionally real for the concept "pot" ceases to exist when it is destroyed physically, and the concept "water" no longer arises when we conceptually exclude from it its shape, colour etc. On the Sarvastivadin definition, for an entity to be real, it does not need to be ultimately real, exclusively. For a thing to be ultimately real is for that thing to be "foundationally existent" (dravya-sat / rdzas yod)[3] in contrast with being "compositely existent" (avayavidravya / rdzas grub). By "foundationally existent" the Sarvastivadin refer to the entity which is fundamentally real, the concept or the cognition of which is not dependent on conceptual construction, hence not conceptually existent (prajnaptisat) nor a composition of the aggregative phenomena. In the case of foundational existent there always remains something irreducible to which the concept of the thing applies, hence it is ultimately real. A simple entity is not reducible to conceptual forms, or conventional designations, nor is it compositely existent entity. We will have lot more to say on this point shortly. Pot and water are not the foundational entities. They are rather composite entities (avayavi-dravya / rdzas grub). By composite entity, we mean an entity or existent which is not fundamental, primary or simple, but is rather a conceptually constructed (prajnaptisat), composition of various properties, and is thus reducible both physically and logically. Hence for the Sarvastivadin, conventional reality (samvrtisatya), composite-existence (avayavi-dravya / rdzas grub), and the lack of intrinsic reality (nihsvabhava) are all equivalents. A conventional reality is therefore characterised as a reducible conventional entity on three grounds: (i) conventional reality is both physically and logically reducible, as it disintegrates when it is subjected to physical destruction and disappears from our minds when its parts are separated from it by logical analysis; (ii) conventional reality borrows its identity from other things including its parts, concepts etc., it does not exist independently in virtue of its intrinsic reality (nihsvabhava), the exclusion of its parts and concepts thus affects and reduces its inherent nature; and (iii) conventional reality is a product of mental constructions, like that of conventionally real wholes, causation, continuum etc, and it does not exists intrinsically. ### 1.2 Ultimate truth The definition of the ultimate reality, as we shall see, offers the Sarvastivadin response to the claim that ultimate reality consists of irreducible atoms and point-instant moments. In glossing the [AK] 6.4 verse his commentary explains that ultimate reality is regarded as ultimately existent, one that is both physically and logically irreducible. Vasubandhu supplies three arguments to support this: (i) ultimate reality is both physically and logically irreducible, as it does not disintegrate when it is subjected to physical destruction and that its identity does not disappear when its parts are separated from it under logical analysis; and (ii) ultimate reality does not borrow its nature from other things including its parts. Rather it exists independently in virtue of its intrinsic reality (svabhava), the exclusion of its parts thus does not affect its inherent nature; and (iii) it is not a product of mental constructions, like that of conventionally real wholes, causation, continuum etc. It exists intrinsically ([AKB] 6.4, Mngon pa khu 214a). Ultimate reality is of two types: the compounded (samskrta) ultimate, and the uncompounded (asamskrta) ultimate. The uncompounded ultimate consists of (a) space (akasa), and (b) nirvana--analytical cessation (pratisamkya-nirodha) and non-analytical cessation (apratisamkhya-nirodha). These three ultimates are uncompounded as each is seen as being causally unconditioned. They are nonspatial concepts. These concepts do not have any physical referent whatsoever. Space is a mere absence of entity. Analytical and nonanalytical cessations are the two forms of nirvana, which is simply freedom from afflictive suffering, or the elimination of afflictive suffering. These concepts are not associated in positing any thing that can be described as remotely physical. They are thus the concepts that are irreducible physically and logically. The compounded ultimate consists of the five aggregates--material aggregate (rupa), feeling aggregate (vedana), perception-aggregate (samjna), dispositional aggregate (samskara), and consciousness-aggregate (vijnana)--since they are causally produced, and the ideas of each aggregate are conceived individually rather than collectively. If the aggregates, the ideas of which are conceived collectively as a whole(s) or a continuum/continua, they could not be ultimately real. The collective concepts of the aggregate as a "whole" or as a "continuum" subject to cessation as they cease to appear to the mind, get excluded from the conceptual framework of the reality of the five aggregates when they are logically analysed. ## 2. Sautrantika The philosophers[4] who championed this view are some of the best-known Indian logicians and epistemologists. Other great names who propogated the tradition include Devendrabuddhi (?), Sakhyabuddhi (?), Vinitadeva (630-700), Dharmottara (750-810), Moksakaragupt (ca. 1100-1200). Dignaga (480-540) and Dharmakirti (600-660) are credited to have founded this school. For the theory of the two truths in the Sautrantika we will need to rely on the following texts: 1. Dignaga's Compendium of Epistemology (Pramanasamuccaya, Tshad ma ce 1b-13a), 2. Dignaga's Auto-commentary on the Compendium of Epistemology (Pramanasamuccayavrtti, Tshad ma ce 14b-85b), 3. Dharmakirti's Verses on Epistemology (Pramanavarttikakarika [PVK]; tshad ma ce 94b-151a), 4. Dharmakirti's Commentary on the Verses of Epistemology (Pramanavarttikavrtti [PVT]; tshad ma ce 261b-365a), 5. Dharmakirti's Ascertainment of Epistemology (Pramanaviniscaya, tshad ma ce 152b-230a), 6. Dharmakirti's Dose of Logical Reasoning (Nyayabindu, tshad ma ce 231b-238a). Broadly, all objects of knowledge are classified into two realities based on the ways in which right-cognition (pramana) engages with its object. They are either directly accessible (pratyaksa), which constitutes objects that are obvious to cognition, or they are directly inaccessible (paroksa), which constitutes objects that are occulted, or obscured from cognition. A directly accessible object is principally known by a direct perceptual right-cognition (pratyasa-pramana), whereas a directly inaccessible object is principally known by an inferential right-cognition (anumana-pramana). ### 2.1 Ultimate truth Of the two types of objects, some are ultimately real while others are only conventionally real, and some are not even conventionally real, they are just unreal, or fictions. In defining ultimate truth in the Sautrantika tradition, we read in Dharmakirti's Verses on Epistemology: "That which is ultimately causally efficient is here an ultimately existent (paramarthasat). Conventionally existent (samvrtisat) is otherwise. They are declared as the definitions of the unique particular (svalaksana) and the universal (samanyalaksana)" (Dharmakirti [PVK] Tshad ma ce 118b). Ultimate truth is, on this definition, a phenomenon (dharma) that is ultimately existent, and ultimately existent are ultimately causally efficient. Phenomenon that is ultimately causally efficient is intrinsically or objectively real, existing in and of itself as a "unique particular" (svalaksana).[5] By "unique particular" Dharmakirti means ultimately real phenomenon--dharma that is self-defined, uniquely individual, objectively real, existing independent of any conceptual fabrication, ultimately causally efficient (artha), a dharma which serves as an objects of direct perception, dharma that presents itself to the cognitions as distinctive/unique individuals. In the Commentary on the Verses of Epistemology, Dharmakirti characterises (Tshad ma ce 274b-279b) all ultimately real unique particulars exist as distinct individuals with their own intrinsic natures. And they satisfy three criteria: 1. They have determinate spatial locations (desaniyata / yul nges pa) of their own, as real things do not have a shared property amongst themselves. The real fire we see is either near or far, or at the left or the right, or at the back or in the front. By contrast, the universal fireness[6], that is, the concept of being a fire, does not occupy a determinate position. 2. Unique particulars are temporally determinate (kalaniyata / dus nges pa or dus ma 'dres pa). They are only momentary instants. They spontaneously go out of existence the moment they have come into existence. This is not the case with the universals. Being purely conceptually constructed, they remain uninfluenced by the dynamism of causal conditions and hence are not affected by time. 3. Unique particulars are ontologically determinate (akaraniyata / ngo bo nges pa / ngo bo ma 'dres pa) as they are causally conditioned; the effects of the aggregation of the causal conditions that have the ability to produce them. When those causal conditions come together at certain points in time, unique particulars come into existence. When those conditions disintegrate and are not replaced by new conditions, unique particulars go out of existence. When the conditions have not yet come together, unique particulars are yet to obtain their ontological status. So "determinate intrinsic natures of the unique particulars," Dharmakirti argues in Commentary on the Verses of Epistemology, "are not accidental or fortuitous since what is not determinate cannot be spatially, temporally and ontologically determinate" (Tshad ma ce 179a). The unique particulars are, Sautrantika claims, ultimately real, and they supply us four arguments to support the claim: (1) Unique particulars are causally efficient phenomena (arthakriyasamartha) ([PVT] Tshad ma ce 179a) because: (a) they have ability to serve pragmatic purpose of life--fulfil the objectives in our life, and (b) ability to produce a variety of cognitive images due to their remoteness or proximity. (Dharmottara's Nyabindutika Tshad ma we 36b-92a) Both of these abilities must be associated exclusively with objects of direct perception (Tshad ma we 45a). (2) Unique particulars present themselves only to a direct perceptual cognition as distinct and uniquely defined individuals because unique particulars are, as Dharmakirti's Nyayabindu points out "the objects whose nearness or remoteness presents the difference of cognitive image" (Tshad ma ce 231a) and that object alone which produces the impression of vividness according to its remoteness or proximity, exists ultimately (DharmottaraTshad ma we 44b-45a). (3) Unique particulars are not denotable by language since they are beyond the full grasp of any conceptual mind (sabdasyavisaya). Although unique particulars are objective references of language and thought, and we may have firm beliefs about them, conceptual mind does not fully grasp their real nature. They are ultimately real, directly cognisable by means of certain perceptions without reliance on other factors (nimitta) such as language and thought. Therefore they must exist. They are the sorts of phenomena whose cognition would not occur if they are not objectively real. In the Sautrantika ontology ultimately real/existent (synonymous) unique particulars are classified into three kinds: 1. momentary instants of matter (rupa), 2. momentary instants of consciousness (vijnana) and 3. momentary instants of the non-associated composite phenomena, which are neither matter nor minds or mental factors (citta-caitta-viprayukta-samskara). The Sautrantika's theory of ultimate truth mirrors its ontology of flux in which unique particulars are viewed as spatially infinitesimal atoms constituting temporally momentary events (ksanika) or as successive flashes of consciousnesses, cognitive events, all devoid of any real continuity as substratum. Unique particulars are ultimately real, although they are not enduring substances (dravyas) inhering in it's qualities (gunas) and actions (karmas) as the Naiyayika-Vaisesika claims. They are rather bundles of events arising and disappearing instantly. Even continuity of things and motion are only successive events closely resembling each other. On this theory ultimate realities are momentary point instants, and Vasubandhu and Dharmakirti both argue that no conditioned phenomenon, therefore, no ultimately real unique particulars, endure more than a single moment--hence they are momentary instants (ksanika). Four closely interrelated arguments provide the defence of the Sautrantika's claim that ultimately real unique particulars are momentary instants. Vasubandhu and Dharmakirti both employ the first two arguments. The third argument is one Dharmakirti specialises in his works. (1) Ultimately real unique particulars are momentary instants because their perishing or destruction is spontaneous to their becoming. This follows because (i) unique particulars are inherently self-destructive (Vasubandhu [AKB], Mngon pa khu 166b-167a), and (ii) their perishing or cessation is intrinsic and does not depend on any other extrinsic causal factors (Dharmakirti, [PVK] Tshad ma ce 102a; [PVT], Tshad ma ce 178ab). (2) The ultimately real unique particulars are momentary instants because they are motionless, and do not move from one temporal or spatial location to another. They perish just where they were born since nothing exists later than its acquisition of existence (Vasubandhu [AKB] Mngon pa khu 166b). (3) The third argument proves momentary instants of unique particulars from the inference of existence (sattvanumana). This is a case of the argument from identity of existence and production (svabhavahetu). All unique particulars which are ultimately existent, are necessarily produced, since only those that are ultimately existent, insofar as Dharmakiriti is concerned, are able to a perform causal function--i.e., to produce effects. And causally efficient unique particulars imply constant change for the renewal and the perishing of their antecedent identities, therefore, they are momentary. Finally (4), unique particulars are ultimately real not only on the ground that they constitute the final ontological status, but also because it forms the basis of the Sautrantika soteriology. The attainment of nirvana -- the ultimate freedom from the afflictions of life--for the Sautrantika, according to Dharmakirti's Vadanyaya, has an immediate perception of the unique particulars as its necessary condition (Tshad ma che 108b-109a). ### 2.2 Conventional truth Dharmakirti defines conventional truth, in his [PVK], as dharma which is "conventionally existent" and he identifies conventional truth with the "universal" (samanya-laksana)[7] just as he identifies ultimate truth with unique particular (Tshad ma ce 118b). When a Sautrantika philosopher describes a certain entity as a universal, he means a conceptual entity not apprehended by virtue of its own being. He means a general property that is conceptually constructed, appearing to be something common amongst all items in a certain class of objects. Unlike Nyaya-Vaisesika view where universals are regarded as objectively real and eternal entities inhering in substances, qualities and particulars, universals, for the Sautrantika are pure conceptual constructs. Sautrantika holds the view known as nominalism or conceptualism--the view that denies universals any independent extramental objective reality existing on their own apart from being mentally constructed. While unique particulars exist independently of linguistic convention, universals have no reality in isolation from linguistic and conceptual conventions. Thus, universals and ultimate reality are mutually exclusive. Universals are therefore only conventionally real, lacking any intrinsic nature, whereas unique particulars are ultimately real, and exist intrinsically. The Sautrantika defends the claim that universals (samanya-laksana) are only conventional reality for the following reasons (tshad ma ce 118b): 1. Universals are domains of inferential cognition since they are exhaustively grasped by conceptual mind by means of language and thought (Dharmakirti, Nyayabindu, tshad ma ce 231a). 2. Universals are objects of the apprehending cognition which arises simply out of having beliefs about the objects without the need to see any real object. 3. Universals are causally inefficient. By "causal inefficiency," the Sautrantika, according to Dharmottara's Nyayabindutika, (Tshad ma we 45ab) means three things: (a) universals are purely conceptually constructed, hence unreal; (b) universals are unable to serve a pragmatic purpose as they do not fulfil life's objectives; and (c) cognitive images produced by universals are independent of the proximity between objects and their cognitions since production of image does not require seeing the object as it is in the case of perception and unique particulars. 4. Universals are products of unifying or mixing of language and their referential objects (unique particulars), and thus appearing them to conceptual minds as generalities, unified wholes, unity, continuity, as phenomena that appear to conceptual mind to having shared properties linking with all items in the same class of objects. 5. Universals, consequently, obscure the individualities of unique particulars from being directly apprehended. This is because, as we have already seen, universals, according to Dharmakirti's [PVK] (Tshad ma ce 97ab) and [PVT] (Tshad ma ce 282ab), are only conventionally real since they are conceptual constructs founded on unifying and putting together the distinct individualities of unique particulars as having one common property being shared by all items in the same class. According to the Sautrantika philosophy, language does not describe reality or unique particulars positively through real universals as suggested by the Naiyayikas. The Sautrantika developed an alternative nominalist theory of universal called the apoha-theory in which language is seen to engage with reality negatively by means of elimination or exclusion of the other (anyapoha / gzhan sel). On this theory, the function of language, specifically naming, is to eliminate object from the class of those objects to which language does not apply. In brief the Sautrantika's theory of the two truths rests on dichotomising objects between unique particulars, which are understood as ultimately reals, dynamic, momentary, causally effective, the objective domain of the direct perception; and universals, which are understood as only conventionally reals, conceptually constructed, static, causally ineffective and the objective domain of the inferential cognition. ## 3. Yogacara The Vaibhasika's realistic theory of the two truths and the Sautrantika's representationalist theory of the two truths both affirm the ultimate reality of physical objects constituted by atoms. The Yogacara rejects physical realism of both the Vaibhasika and the Sautrantika, although it agrees with the Sautrantika's representationalist theory as far as they both affirm representation as the intentional objects in perception and deny in perception a direct access to any external object. Where they part their company is in their response to the questions: what causes representations? Is the contact of senses with physical objects necessary to give rise to representations in perception? The Sautrantika's reply is that external objects cause representations, given that these representations are intentional objects there is indeed a contact between senses and external objects. This affirmative response allows the Sautrantika to affirm reality of external objects. The Yogacarin however replies that "subliminal impressions" (vasanas) from foundational consciousness (alayavijnana) are the causes of the mental representations, and given that these impressions are only internal phenomena acting as intentional objects, the contact between senses and external objects is therefore rejected even conventionally. This allows the Yogacarin to deny even conventional reality of all physical objects, and argue that all conventional realities are our mental representations, mental creations, cognitions etc. The central thesis in the Yogacara philosophy, the theory of the two truths echoes is the assertion that all that is conventionally real is only ideas, representations, images, creations of the mind, and that there is no conventionally real object that exists outside the mind to which it corresponds. These ideas are only objects of any cognition. The whole universe is a mental universe. All physical objects are only fiction, they are unreal even by the conventional standard, similar to a dream, a mirage, a magical illusion, where what we perceive are only products of our mind, without a real external existence. Inspired by the idealistic tendencies of various sutras consisting of important elements of the idealistic doctrines, in the third and the fourth centuries many Indian philosophers developed and systematised a coherent Idealist School. In the beginning of the Vimsatika Vasubandhu treats citta, manas, vijnana, vijnapti as synonymous and uses these terms as the names of the idealistic school. The chief founders were Maitreyanath (ca. 300) and Asanga (315-390), propagated by Vasubandhu (320-380), Dignaga (480-540) Sthiramati (ca. 500), Dharmapala (530-561), Hiuan-tsang (602-664), Dharmakirti (600-660), Santaraksita (ca.725-788) and Kamalasila (ca.740-795). The last two are Yogacara-Madhyamikas in contrast with the earlier figures who are identified as Yogacarins. Like other Buddhist schools, the theory of the two truths captures the central Yogacara doctrines. Maitreyanath asserts in his Verses on the Distinction Between Phenomena and Reality (Dharmadharmatavibhanga-karika, DDVK; Sems tsam phi 50b-53b)--"All this is understood to be condensed in two categories: phenomena (dharma) and reality (dharmata) because they encompass all." (DDVK 2, Sems tsam phi) By "All" Yogacarin means every possible object of knowledge, and they are said to be contained in the two truths since objects are either conventional truth or ultimate truth. Things are either objects of conventional knowledge or objects of ultimate knowledge, and a third category is not admitted. ### 3.1 Conventional Truth Etymologically the term conventional truth covers the sense of what we ordinarily take as commonsensical truths. However, in contrast with naive realism associated with common sense notions of truths, for the Yogacara the term "conventional truth" has somewhat a negative connotation. It exclusively refers to objects of knowledge like forms, sound etc., the mode of existence or mode of being which radically contradicts with the mode of its appearance, and thus they are false, unreal, and deceptive. Forms, sounds, etc., are defined as conventional entities in that they are realities from the perspective of, or by the force of, three forms of convention: 1. fabrication (asatkalpita), 2. consciousness (vijnana), or 3. language, signifier, a convenient designator (sabda). A conventional truth is therefore a truth by virtue of being fabricated by the conceptual mind; or it is truth erroneously apprehended by means of the dualistic consciousness; or it is true concept, meaning, signified and designated by a convenient designator/signifier. Because the Yogacarins admit three conventions, it also admits three categories of conventional truths: 1. fabricated phenomena; 2. mind/consciousness; and 3. language since conventional truths exist due to the force of these three conventions. The first and the last are categories of imaginary phenomena (parikalpita) and the second is dependent phenomena (paratantra). The Yogacara's claim that external objects are not even conventionally real, what is conventionally real are only our impressions, and mental representations is one Vasubandhu closely defends by means of the Yogacara's theory of the three natures (trisvabhava). In his Discernment of the Three Natures (Trisvabhavakarika, or Trisvabhavanirdesa [TSN]; Sems tsam shi 10a-11b), Vasubandhu explains that the Yogacara ontology and phenomenology as consisting of the unity of three natures (svabhava): 1. the dependent or other (paratantra); 2. the imaginary / conceptual (parikalpita); and 3. the perfect / ultimate (parinispanna) ([TSN] 1, Sems tsam shi 10a) The first two account for conventional truth and the latter ultimate truth. We shall consider the import of the three in turn. First, Vasubandhu defines the dependent nature as: (a) one that exists due to being causally conditioned (pratyayadhinavrttitvat), and (b) it is the basis of "what appears" (yat khyati) mistakenly in our cognition as conventionally real, it is the basis for the "unreal conceptual fabrication" (asatkalpa) which is the phenomenological basis of the appearance of reified subjects and objects ([TSN] 2 Sems tsam shi 10a). The implications of the Yogacarin expression "what appears" to describe the dependent nature are therefore twofold: (a) that the things that appear in our cognitions are exclusively the representations, which are the manifest forms of the subliminal impressions, and (b) that the entire web of conventional reality, which presents itself to our cognitions phenomenologically in various ways, is exclusively the appearance of those representations. Apart from those representations, consciousnesses, which appear to be external objects, there is no conventionally real external content which corresponds to what appears. Second, in contrast with the dependent nature which is the basis of "what appears" (yat khyati), the imaginary nature (parikalpita), as Vasubandu defines it, is the mode of appearance "as it appears" (sau yatha khyati) on the ground that its existence is only an "unreal conceptual fabrication" (asatakalpo) ([TSN] 2, Sems tsam chi 10a). The imaginary nature is only an "unreal conceptual construction" because of two reasons: (i) it is the dependent nature--representations--merely dually reified by the mind as an ultimately real subject, or self, or eg, and (ii) the imaginary nature is dualistic reification of beings and objects as existing really and externally, there is no such reality. Third, given the fact that the dependent nature is devoid, or free from this duality, the imaginary nature is a mere superimposition on it. Hence nonduality, the perfect nature (parinispanna) is ultimate reality of the dependent nature. ### 3.2 Ultimate truth In the Commentary on the Sutra of Intent (Arya-samdhinirmocana-sutra, Mdo sde ca 1b-55b) it is stated that "Reality as it is, which is the intentional object of a pure consciousness, is the definition of the perfect nature. This must be so because it is with respect to this that the Victorious Buddha attributed all phenomena as natureless, ultimately." (Mdo sde ca 35b) Vasubandhu's [TSN] defines "the perfect nature (parinispanna) as the eternal nonexistence of 'as it appears' of 'what appears' because it is unalterable." ([TSN] 3, Sems tsam chi 10a) "What appears" is the dependent nature--a series of cognitive events, the representations. "As it appears" is the imaginary nature--the unreal conceptual fabrication of the subject-object duality. The representations, (i.e., the dependent nature) appear in the cognition as if they have in them the subject-object duality, even though the dependent nature is wholly devoid of such subject-object duality. The perfect nature is therefore this eternal nonexistence of the imaginary nature--the duality--in the dependent nature. Vasubandhu defines the perfect nature as the ultimate truth and identifies it with mere-consciousness. "This is the ultimate (paramartha) of the dharmas, and so it is the reality (tathata) too. Because its reality is like this all the time, it is mere consciousness." (Trim 25, Sems tsam shi 3b) Accordingly Sthiramati's Commentary on the Thirty Verses (Trimsikabhasya [TrimB], Sems tsam shi 146b-171b), explains "ultimate" (paramartha) here as referring to "' world-transcending knowledge' (lokottara-nirvikalpa-jnana) in that there is nothing that surpasses it. Since it is the object of [the transcendent knowledge], it is the ultimate. It is even like the space in having the same taste everywhere. It is the perfect nature, which is stainless and unchangable. Therefore, it is known as the 'ultimate.'" ([TrimB], Sems tsam shi 169ab) So, as we can see the dependent and the imaginary natures together explain the Yogacara's conventional reality and the perfect nature explains its conception of the ultimate reality. The first two natures provide an argument for the Yogacara's empirical and practical standpoints (vyavahara), conventional truth and the third nature an argument for its ultimate truth. Even then the dependent nature alone is conventionally real and the perfect nature alone is ultimately real. By contrast, the imaginary nature is unreal and false even by the empirical and practical standards. This is true in spite of the fact the imaginary nature is constitutive of the conventional truth. So, the perfect nature--nondual mind, i.e., emptiness (sunyata) of the subject-object duality--is the ultimate reality of the Yogacara conception. Ultimate truth takes various forms as it is understood within the Yogacarin tradition. As Maitreyanatha states, ultimate truth takes three primary forms--as emptiness it is the ultimate object, as nirvana it is the ultimate attainment, and as nonconceptual knowledge it is the ultimate realization ([MVK], Sems tsam phi 42B). Yogacara Arguments The core argument in support of the only mind thesis is the impossibility of the existence of external objects. Vasubandhu develops this argument in his Vim 1-27 as does Dignaga in his Examination of the Intentional Object (Alambanapariksavrtti, APV 1-8, Tshad ma ce 86a-87b) against the atomists (Naiyayikas-Vaisesika and Abhidharmikas[8]). Against the Yogacara idealist thesis the realist opponents, as Vasubandhu observes, raise three objections:--"If consciousness (does) not (arise) out from an object (i) neither the determination or certainty (niyama) in regard to place (desa) and time (kala), (ii) nor the indetermination or unexclusiveness (aniyama) in regard to the series of (consciousness) (iii) nor the performance of the (specific) function (krtyakriya) are logically possible (yukta)" (Vim 2, Sems tsam shi 3a). Vasubandhu offers his Yogacara reply in the Vim to these realist objections and insists that the idealist position does not face these three problems. The first problem is not an issue for the Yogacara since dreams (svapna) account for the spatio-temporal determination. In dreams, in the absence of an external object, one still has the cognition of a woman, or a man in only determinate / specific time and place, and not arbitrarily and not everywhere and not in any moment. Neither is the second problem of the lack of an intersubjective agreement an issue for the Yogacara. The case of the pretas (hungry-ghosts) account for intersubjective agreement as they look at water they alone, not other beings, see rivers of pus, urine, excrement, and collectively hallucinates demons as the hell guardians. Although pus, urine, excrement and hell guardians are nonexistent externally, due to the maturation of their collective karma, pretas exclusively experience the series of cognitions (vijnapti), other beings do not encounter such experience (Vim 3, Sems tsam shi). Nor is the lack of causal efficacy of the impressions or representations in the consciousness a problem for the Yogacara. As in the case of wet dreams, even without a union of the couple, the emission of semen can occur, and so the representations in the consciousness are causally efficient even without the externally real object. Yogacara's defensive arguments against the realist challenges are quite strong. However unless Yogacara is able to undermine the core realist thesis--i.e., reality of the external objects--and its key supporting argument--the existence of atoms--then the debate could go either way. Therefore Yogacara shows the nonexistence, or unreality of the atoms as the basis of the external object to reject the realist position. Yogacara's impressions-only theory and the Sautrantika's representationalist theory both explain our sensory experience (the spatio-temporal determinacy, intersubjective agreement, and causal efficacy). They agree on what the observables are: mental entities, including mental images but also emotions such as desires. They also agree that karma plays vital role in explaining our experience. The realist theory, though, according to Dignaga's Investigation About the Support of the Cognition (Alambanapariksavr tti [AlamPV]) has to posit the reality of additional physical objects, things that are in principle unobservable, given that all we experience in the cognition are our impressions ([AlamPV], Tshad ma ce 86a). If the realist thesis were correct, then there would be three alternatives for the atoms to act as the intentional objects of cognition. 1. Atoms of the things would be either one in the way the Nyya-Vaieika conceives the whole as something constituted by parts but being single, one, different from the parts that compose it, and having a real existence apart from the existence of the parts. Or 2. things would have to be constituted by multiple atoms i.e., a number or a group of atoms coexisting one besides the other without forming a composite whole as a result of mutual cohesion between the atoms. Or 3. things would have to be atoms grouped together, massed together as a unity among themselves with a tight cohesion. Yogacara contends that these are the only three alternatives in which the reality of the external objects can functioning as the objects of cognition. Of the three alternatives: (1) points to the unity of a thing conceived as a whole; and (2) and (3) point to the multiplicity looking at the things as loose atoms, i.e., composite atoms. Not one of the three possibilities is, on Yogacara's account, admissible as the object of cognition however. The first is inadmissible because nowhere an external object is grasped as a unity, whole, one apart from its parts ([AlamPV] Tshad ma ce 86b). The second is inadmissible because when we see things we find that the atoms are not perceived individually as one by one ([Vim] 13, Sems tsam shi 3b). The third alternative is also rejected because the atoms in this case cannot be proved to exist as an indivisible ([Vim] 14, Sems tsam shi 3b). Further, if there were a simultaneous conjunction of an atom with other six atoms coming from the six directions, the atom would have six parts because that which is the locus of one atom cannot be the locus of another atom. If the locus of one atom were the locus of the six atoms, which were in conjunction with it, then since all the seven atoms would have the same common locus, the whole mass constituted by the seven atoms would be of the size of a single atom, because of the mutual exclusion of the occupants of a locus. Consequently, there would be no visible mass (Vim12, Sems tsam shi 3b). Since unity is an essential characteristic of being the whole, or composite whereas indivisibility an essential characteristic of being an atom when both the unity and individuality of the atoms are rejected, then the whole and indivisible atoms are no longer admissible. Therefore, Vasubandhu sums up the Yogacara objections against the reality of the external sense-objects (ayatanas) as: "An external sense-object is unreal because it cannot be the intentional object of cognition either as (1) a single thing or (2) as a multiple in [isolated] atoms; or (3) as a aggregate because the atom is not proven to exist" (Vim11, Sems tsam shi 3b). The realist insists that the external sense objects are real and that their reality is ascertained by the various means of knowledge (pramanas) of which perception (pratyaksa) is the most important. If the external sense objects are nonexistents, there can be no intentional objects, then cognition would not arise. Since cognitions do arise, there must be external objects as their intentional objects. To this objection, Yogacara employs two arguments to refute the realist claim and to establish the mechanism of cognition, which takes place without the atoms of an external object. The first argument shows that atoms do not satisfy the criterion of being the intentional object, therefore they do not cause the perception. "Perceptual cognition [takes place] just like in dreams and the like [even without an external object]; moreover, when that [cognition] occurs, the external object is not seen; how can it be thought that this is a case of perception?" (Vim16, Sems tsam shi 3b). The second is the time-lag argument, according to which, there is a time-gap between the perceptual judgement we make and the actual perceptual process. When we make a perceptual judgement, at that time we do not perceive the external object as it is in our mental consciousness (manovijnana) that carries out the judgement and since the visual consciousness that perceives the object has already ceased. Hence at the time when the mental consciousness delivers it judgment, the perceptual cognition no longer exists since all things are momentary. Therefore the atoms of an external object is not the intentional object of perceptual cognition, since it has already ceased and does not now exist therefore it is not responsible for the cognition's having the content it has, like the unseen events occurring on the other side of the wall (Vasubandhu's Vimsatika-karikavrtti [VimKV] 16-17). Yogacara therefore concludes that we cannot postulate the reality of an external object through direct perception. However since in perceptual cognition we are directly aware of something, there must be an intentional object of the perceptual cognition. That intentional object of perceptual cognition is, according to the Yogacara, none other than the subliminal impressions (vasanas) passing from their latent state contained in the storehouse consciousness (alayavijnana) to their conscious level. Therefore the impressions are the only things that are conventionally real. Vasubandhu's [TSN] 35-36 inspired by the Buddhist traditional religious beliefs also offers others arguments to defend the idealism of Yogacara: 1. one and the same thing appears differently to beings that are in different states of existence (pretas, men, and gods); 2. the ability of the bodhisattvas and dhyayins (practitioners of meditation) who have attained the power of thinking (cetovasita) to visualise objects at will; 3. the capacity of the yogins who have attained serenity of mind (samatha) and a direct vision of dharmas as they really are (dharma-vipasyana) to perceive things at the very moment of the concentration of mind (manasikara) with their essential characteristics of flux, suffering, nonself, empty; and 4. the power of those who have attained intuitive knowledge (nirvikalpakajnana) which enables them not to block the perception of things. All these arguments based on the facts of experience show that objects do not exist really outside the mind, that they are products of mental creation and that their appearance is entirely mind dependent. Therefore the Yogacara's theory of the two truths concludes that the whole world is a product of mind--it is the collective mental actions (karma) of all beings. All living beings see the same world because of the identical maturation of their karmic consequences. Since the karmic histories of beings are same, there is homogeneity in the way in which the world is experienced and perceived. This is the reason there is an orderly world instead of chaotic and arbitrariness. This is also the reason behind the impressions of the objectivity of the world. ## 4. Madhyamaka After the Buddha the philosopher who broke new ground on the theory of the two truths in the Madhyamaka system is a South Indian monk, Nagarjuna (ca. 100 BCE-100 CE). Amongst his seminal philosophical works delineating the theory are Nagarjuna's * Fundamental Verses on the Middle Way (Mulamadhyamakakarika [MMK], * Seventy Verses on Emptiness (Sunyatasaptati), * Rebutting the Disputes (Vigrahvyavartani[9] and * Sixty Verses on Logical Reasoning (Yuktisastika). Aryadeva's work Catuhsatakasastrakarika (Four Hundred Verses) is also considered as one of the foundational texts delineating Madhyamaka's theory of the two truths. Nagarjuna saw himself as propagating the dharma taught by the Buddha, which he says is precisely based on the theory of the two truths: a truth of mundane conventions and a truth of the ultimate. ([MMK] 24.8, Dbu ma tsa 14b-15a) He saw the theory of the two truths as constituting the Buddha's core teaching and his philosophy. Nagarjuna maintains therefore that those who do not understand the distinction between these two truths would fail to understand the Buddha's teaching ([MMK] 24.9, Dbu ma tsa 15a). This is so, for Nagarjuna, because (1) without relying on the conventional truth, the meaning of the ultimate cannot be explained, and (2) without understanding the meaning of the ultimate, nirvana is not achieved ([MMK] 24.10, Dbu ma tsa 15a). Nagarjuna's theory of the two truths is fundamentally different from all theories of truth in other Indian philosophies. Hindu philosophers of the Nyaya-Vaisesika, Samkya-Yoga, and Mimamsa-Vedanta--all advocate a foundationalism of some kind according to which ultimate reality is taken to be "substantive reality" (drayva) or foundation upon which stands the entire edifice of the conventional ontological structures where the ultimate reality is posited as immutable, fixed, irreducible and independent of any interpretative conventions. That is so, even though the conventional structure that stands upon it constantly changes and transforms. As we saw the Buddhist realism of the Vaibhasika and the representationalism of the Sautrantika both advocate ultimate truth as ultimately real, logically irreducible. The idealism of Yogacara holds nondual mind as the only ultimate reality and the external world as merely conventional truths. On Nagarjuna's Madhyamaka all things including ultimate truth are ultimately unreal, empty (sunya) of any intrinsic nature (svabhava) including the emptiness (sunyata) itself, therefore all are groundless. In this sense a Madhyamika (a proponent of the Madhyamaka thought) is a an advocate of the emptiness (sunyavadin), advocate of the intrinsic unreality (nihsvabhavavadin), groundlessness, essencelessness, or carelessness. Nevertheless to assert that all things are empty of any intrinsic reality, for Nagarjuna, is not to undermine the existential status of things as simply nothing. On the contrary, Nagarjuna argues, to assert that the things are empty of any intrinsic reality is to explain the way things really are as causally conditioned phenomena (pratityasamputpanha). Nagarjuna's central argument to support his radical non-foundationalist theory of the two truths draws upon an understanding of conventional truth as tied to dependently arisen phenomena, and ultimate truth as tied to emptiness of the intrinsic nature. Since the former and the latter are co-constitutive of each other, in that each entials the other, ultimate reality is tied to being that which is conventionally real. Nagarjuna advances important arguments justifying the correlation between conventional truth vis-a-vis dependent arising, and emptiness vis-a-vis ultimate truth. These arguments bring home their epistemological and ontological correlations ([MMK] 24.14; Dbu ma tsa 15a). He argues that wherever applies emptiness as the ultimate reality, there applies the causal efficacy of conventional reality and wherever emptiness does not apply as the ultimate reality, there does not apply the causal efficacy of conventional reality (Vig.71) (Dbu ma tsa 29a). According to Nagarjuna, ultimate reality's being empty of any intrinsic reality affords conventional reality its causal efficacy since being ultimately empty is identical to being causally produced, conventionally. This must be so since, for Nagarjuna, "there is no thing that is not dependently arisen; therefore, there is no such thing that is not empty" ([MMK] 24.19, Dbu ma tsa 15a). Svatantrika / Prasangika and the two truths The theory of the two truths in the Madhyamaka in India took a great resurgence from the fifth century onwards soon after Buddhapalita (ca. 470-540) wrote A Commentary on [Nagarjuna's] Fundamental Verses of the Middle Way (Buddhapalitamulamadhyamakavrtti,[10]Dbu ma tsa 158b-281a). Set forth in this text is a thoroughgoing non-foundationalist philosophic reasoning and method--prasanga arguments or reductio ad absurdum style without relying upon the svatantra, formal probative argument--to elucidate the Madhyamaka metaphysics and epistemology ingrained in theory of the two truths. For this reason, Buddhapalita is often identified later as the founder of the Prasangika Madhyamaka, although elucidation of the theory itself is set out in the works of Candrakirti. Three decades later Bhavaviveka[11] (ca. 500-570) challenged Buddhapalita's interpretation of the two truths, and developed a Madhyamaka account of the two truths that reflects a significant ontological and epistemological shift from Buddhapalita's position, which was later defended in the works of Candrakirti to whom we shall return shortly. Many later commentators ignore the philosophical contents of the debate between the Prasangika and the Svatantrika and claim that their controversy is confined only to pedagogical or methodological issues. There is another view according to which the contents of the debate between the Prasangika and the Svatantrika--Buddhapalita versus Bhavavevika followed by Bhavavevika versus Candrakirti--is essentially philosophic in nature. Underpinning the dialectical or methodological controversy between the two Madhyamaka camps lies a deeper ontological and epistemological divide implied within their theories of the two truths, and this in turn is reflected in the different methodological considerations they each deploy. So on this second view, the variation in the methods used by the two schools of the Madhyamaka are not simple differences in their rhetorical devices or pedagogical tools, they are underlain by more serious philosophical disagreements between the two. According to this view the philosophical differentiations between the Svatantrika and the Prasangika is best contained within the discourse of the two truths. Prasangika's theory of the two truths we leave aside for the time being. First we shall take up the Svatantrika's account. ### 4.1 Svatantrika Madhyamaka Bhavaviveka wrote some of the major Madhyamaka treatises including * Lamp of Wisdom (Prajnapradipamulamadhyamakavrtti* [PPMV], Dbu ma* tsha 45b-259b), * Verses on the Heart of the Middle Way (Madhyamakahrdayakarika [MadhHK], * Blazes of Reasoning: A Commentary on Verses of the Heart of the Middle Way (Tarkajvala). In these texts Bhavavevika rejects the Brahmanical systems of Nyaya-Vaisesika, Samkhya-Yoga, Mimamsa-Vedanta on both metaphysical and epistemological grounds. All the theories of truth and knowledge advanced in these systems from his Madhyamaka point of view are too rigid to be of any significant use. Bhavavevika's critiques of Abhidharmikas--Vaibhasika and Sautrantika--and Yogacara predominantly target the ontological foundationalism which underpins their theories of the ultimate truth. He rejects them on the ground that from an analytic cognitive perspective, which scrutinises the nature of reality, nothing--subject and object--is found to be ultimately real since all things are rationally reduced to spatial parts or temporal moments. Thus he proposes the view that both the subject and the object are conventionally intrinsically real as both are conventional truths, where as both are ultimately intrinsically unreal as both are empty of ultimate reality, hence emptiness alone is the ultimate reality. Although Bhavavevika is said to have founded the Svatantrika Madhyamaka, it is important to note that there exists two different subschools of the Svatantrika tradition: 1. Sautrantika Svatantrika Madhyamaka 2. Yogacara Svatantrika Madhyamaka The two schools of the Svatantrika Madhyamaka take Bhavavevika's intepretation of the two truths, on the whole, to be more cogent than the Prasangika's. Particularly both schools reject ultimate intrinsic reality while positing conventional intrinsic reality. As far as the presentation of ultimate truth is concerned both schools are in agreement--nonself or emptiness alone is the ultimate reality, and the rest--the entire range of dharmas--are ultimately empty of any intrinsic reality. Nevertheless, the two schools differ slightly on the matter related to the theory of conventional truth, specifically concerning reality or unreality of the external or physical objects on the conventional level. The Sautrantika Svatantrika Madhyamaka view held by Bhavavevika himself, and his student Jnanagarbha, affirms intrinsic reality of the conventional truths of both the outer domains and the inner domains and mental faculty along with their six respective consciousnesses. The arguments that affirm the conventional reality of the external objects are the same arguments that Bhavavevika employed to criticise the Yogacara arguments rejecting the reality of external objects. Bhavavevika's [MadhHK] 5.15 points out Yogacara's arguments are contradicted by perception (pratyaksa), tradition (agama) and commonse sense (lokaprasiddha), since they all prove the correctness of the cogniton of material form ([MadhHK] 5.15 Dbu ma dza 20b and Tarkajvala 5.15 Dbu ma dza 204ab). Yogacara's dream argument is also rejected because, according to Bhavavevika, dream-consciousness and so forth have dharmas as their objects (alambana / dmigs pa) ([MadhHK] 5.19 Dbu ma dza 20b). And dharmas are based on intrinsically real conventional objects, because they are the repeated impression of the objects that have been experienced previously, like memory (Tarkajvala 5.19 Dbu ma dza 205a). The Yogacara Svatantrika Madhyamaka view held by Santaraksita and his student Kamalasila affirms conventionally intrinsic reality of the inner domains and rejects the intrinsic reality of the outer domains and it claims the external objects are mere conceptual fictions. #### 4.1.1 Sautrantika Svatantrika Madhyamaka Of the two schools of Svatantraka we shall first take up the theory of the two truths presented in the Sautrantika Svatantrika Madhyamaka. Bhavavevika's position represents the theory of the two truths held by this school which was later promoted by his disciple Jnanagarbha. The cornerstones of the Sautrantika Svatantrika Madhyamaka theory of the two truths are the following theses: 1. That at the level of conventional truth all phenomena are intrinsically real (svabhavatah), because they are established as intrinsic realities from the perspective of the non-analytical cognitions of the naive ordinary beings. Allied to this thesis is Bhavavevika's claim that the Madhyamaka accepts, conventionally, the intrinsic reality of things, for it is the intrinsic reality that defines what is to be conventionally real. 2. That at the level of ultimate truth, all phenomena, with the exception of the emptiness which is ultimately real, are intrinsically unreal (nihsvabhavatah), or ultimately unreal, because they are established as empty of any intrinsic reality from the perspective of analytical exalted cognition (i.e., ultimate cognition of arya-beings). Allied to this thesis is Bhavavevika's claim that Madhyamaka rejects intrinsic reality at the ultimate level since ultimate reality constitutes that which is intrinsically unreal, therefore empty. ##### Conventional truth We shall consider Sautrantika Svatantrika Madhyamaka's defence of the two claims in turn. We will begin first with Bhavavevika's definition of the conventional truth. In the Tarkajvala, he defines conventional truths as an "incontrovertible (phyin ci ma log pa) linguistic convention." (Dbu ma dza 56a) Such conventional truth on Bhavavevika's Svatantrika takes two forms--unique particulars (svalaksana / rang mtshan) and universals (samanyalaksana / spyi mtshan) as the Tarkajvala III.8, explains: Phenomena have dual characteristics (laksana / mtshan nyid), differentiated as being universal or unique. Unique particular (svalaksana / rang mtshan) is a thing's intrinsic reality (svabhavata / rang gi ngo bo), the domain of engagement which is definitively ascertained by a non-conceptual cognition (nirvikalpena jnanena / rnam par rtog pa med pa'i shes pa). Universal (samanyalaksana) is a cognitive domain to be apprehended by an inferential cognition (anumanavijnanena / rjes su dpog pa'i shes pa) which is conceptual (Dbu ma dza 55b). In this passage Bhavavevika essentially summarises some of the key features of conventional truths which are also critical to understanding the differences between the Svatantrika and Prasangika. 1. Bhavavevika accepts, based on this passage, unique particular (svalaksana) as an integral part of the ontological structure of the conventional truth. The Prasangika, by contrast, rejects the concept of unique particular even at the level of conventional truth. 2. Bhavavevika ascribes unique particular an ontological qualification of being "intrinsically real" (svabhavata), or "inherently real" as its given status, explicitly affirming a form of foundationalism of unique particulars at the level of conventional truth, which the Prasangika totally rejects. 3. Bhavavevika endorses the Pramanika's epistemology in which unique particular, which is intrinsically real, conventionally, is considered as the domain of engagement ascertained by a non-conceptual cognition. The universal (samanyalaksana) is considered as the domain to be apprehended by a conceptual inferential cognition. Bhavavevika's defence of the thesis that conventionally things are intrinsically real can be summed up in two arguments. First, as far as things are conventional realities they are also intrinsic or inherent realities, for being intrinsically real is the reason why things are designated as "conventional reality," since from a conventional standpoint--i.e., non-analytical cognitive engagements of the ordinary beings--things do appear to be inherently real. This means that being "intrinsic" is, for Bhavavevika, what makes conventional reality a reality. Bhavavevika's second argument states that as long as a Madhyamika accepts conventional reality it must also accept things as intrinsic in order to avoid the nihilism charge, for conventional reality is defined in terms of its being intrinsically and uniquely real (svalaksana). Denying intrinsic reality of things at the conventional level, would therefore entail the denial of their conventional existence, since it would entail the denial of the defining characteristics of the conventional reality. This follows says Bhavavevika because "The Lord (bhagvan / bcom ldan ldas) has taught the two truths. Based on this [explanatory schema] conventionally things are posited in terms of their intrinsic natures and [unique] particulars ([sva]laksana / mtshan nyid). It is only ultimately that [things are] posited as having no intrinsic reality" ((Dbu ma dza 60a). ##### Ultimate truth Bhavavevika's second thesis is that at the level of ultimate truth, all phenomena are intrinsically unreal (nihsvabhavatah), therefore Madhyamaka rejects intrinsic reality ultimately. Bhavavevika's Tarkajvala 3.26 offers three ways to interpret the compound paramartha "ultimate domain" or "ultimate object" literally. In the first, both the terms "ultimate" (param) and "domain" (artha) stands for the object--emptiness (sunyata)--because it is both the "ultimate" or "final" as well as an "object to be known," "analysed" (pariksaniya / brtag par bya ba) and "understood" (pratipadya / go bar bya ba). In the second compound, paramartha stands for "object of the ultimate" (paramasya artha / dam pa'i don) where "object" refers to emptiness and "ultimate" refers to a non-conceptual exalted cognition (nirvikalpajnana / rnam par mi rtog pa'i ye shes) of the meditative equipoise (samahita / mnyam bzhag). In the third compound paramarth stands for a concordant ultimate (rjes su mthun pa'i shes rab), a cognition that accords with the knowledge of the ultimate since it has the ultimate as its domain. The first two etymological senses of the term paramarthas have in them the sense of emptiness as an objective domain since it is both the final ontic status of things and it is the ultimate object of a non-conceptual exalted cognition. Bhavavevika's third etymological sense of the term paramartha has it identified with cognition that accords with the knowledge of the ultimate truth on the ground that such cognition has emptiness as its object and such knowledge is a means through which one develops a non-conceptual knowledge of ultimate reality. This third sense of paramartha allows Bhavavevika to argue that a cognition of the subsequent attainment (prsthalabdhajnana / rjes thob ye shes) which directly follows from the non-conceptual meditative equipoise, while it is conceptual in character, is nevertheless a concordant ultimate ([PPMV] Dbu ma tsa 228a). So for Bhavavevika then, "the ultimate is of two kinds: one is transcendent (lokuttara / 'jig rten las 'das pa), undefiled (zag pa med pa), free from elaboration (aprapanca / spros med) which needs to be engaged without a deliberative effort. The second one" as he explains in Tarkajvala 3.26 "is elaborative (prapanca / spros pa), hence it can be engaged with a deliberative effort by means of a correct mundane cognitive process (dag pa'i 'jig rten pa'i ye shes) that is sustained by the collective force of virtues (bsod nams) and insights (ye shes)" (Dbu ma dza 60b). Bhavavevika advances several arguments to defend the thesis that ultimately all are intrinsically unreal. The first is the conditionality argument pertaining to the four elements, according to which, all the four elements are ultimately empty of any intrinsic reality, for they all are conditioned by the causal factors appropriated for their becoming and their existence. The second is Bhavavevika's non-foundationalist ontological argument which demonstrates all phenomena, including the atoms, are ultimately non-foundational, for ultimately there is nothing that can be taken as the foundational entity (dravya) or intrinsically real since the ultimate analysis reveals that all phenomena are composed of the atomic particles that are themselves composites. Bhavavevika's third non-foundationalist epistemological argument shows that ultimately the objects are not the domain of cognitions. The argument says that "A visible form is not ultimately apprehended by the visual faculty because it is a composite (bsags) like a sound and because it is a product of the elements like the sound." ([MadhHK] 3.33 Dbu ma dza 5a) The reasons provided to justify the thesis are twofold: (i) being composite and (ii) being a product of the elements. Given that being composites and products equally apply to visible form and sound, these two reasons cannot warrant the validity of the argument justifying the thesis that a visible form is ultimately apprehended by the visual faculty. If they did, they would equally warrant the validity of the argument justifying the thesis that a sound is ultimately apprehended by the visual faculty (Dbu ma dza 65ab). In conclusion: mirroring these two critically important philosophical positions Bhavavevika introduces a type of method or pedagogical device that is unique to the Svatantrika contra to his Prasangika counterpart. #### 4.1.2 Yogacara Svatantrika Madhyamaka In the early eighth century Bhavavevika and Jnanagarbha's theory of the two truths was adopted and elucidated in the works of the latter's student[12] Santaraksita (705-762) and his student Kamalasila (ca. 725-788). Santaraksita wrote: * Verses on the Ornament of the Middle Way (Madhyamakalamkarakarika [MAK], Dbu ma sa 53a-56b), * Commentary on the Verses of the Ornament of the Middle Way (Madhyamakalamkaravrtti, [MAV], Dbu ma sa 56b-84a). In these works he argues for the synthesis of Madhyamaka and Yogacara's theories of the two truths which resulted in the formation of the Yogacara-Svatantrika Madhyamaka--a subschool within the Svatantrika Madhyamaka. Kamalasila, who is a student of Santaraksita, held the same conception of the two truths in his works: * Subcommentary on the Ornament of the Middle Way (Madhyamakalamkarapanjika, Dbu ma sa 84a-133b), * Light of the Middle Way (Madhyamakalokah [MA], Dbu ma sa 133b-244a), * Discussion on the Light of Reality (Tattvaloka-nama-prakarana, Dbu ma sa 244b-273a), * Establishing the Non-intrinsic Reality of All Phenomena (Sarvadharmasvabhavasiddhi, Dbu ma sa 273a-291a), * Stages of Meditation (Bhavanakrama, Dbu ma ki 22a-68b). Unlike Jnanagarbha and Bhavavevika's realistic Sautrantika Svatantrika Madhyamaka which presents conventional truth in agreement with the epistemological realism of the Sautrantika and ultimate truth in agreement with non-foundationalist ontology of the Madhyamaka, Santaraksita and Kamalasila advocate the Yogacara Svatantrika Madhyamaka. They each maintain a basic presentation of conventional truth in agreement with the epistemological idealism of Yogacara and the presentation of ultimate truth in agreement with the ontological non-foundationalism of the Madhyamaka. Therefore Santaraksita concludes: "By relying on the Mind Only (cittamatra / sems tsam) one understands that external phenomena do not exist. And by relying on this [i.e., Madhyamaka] one understands that even [the mind] is thoroughly nonself." ([MAK] 23 Dbu ma sa 56a) "Therefore, one attains a Mahayana [path] when one is able to ride the chariots of the two systems by holding to the reigns of logic." ([MAK] 24 Dbu ma sa 56a) By "the two systems" Santaraksita meant, Yogacara's account of the conventional truth with Madhyamaka's account of the ultimate truth.[13] Underlying this syncretistic view of the two truths lay two fundamental theses that Yogacara Svatantrika Madhyamaka are committed--namely: 1. that there is no real external object, mind is the only thing that is conventionally real, hence Yogacara is right about its presentation of the conventional truth; and 2. that even mind is unreal and empty of any intrinsic nature under ultimate analysis, therefore nothing is ultimately real, hence Madhyamaka is right about its presentation of the ultimate truth. And it is precisely the synthesis of these two theses that defines the characteristic of the theory of the two truths in the Yogacara Svatantrika Madhyamaka. Santarasita and Kamalasila attempt to achieve this distinctive syncretistic characteristic in their works to which we turn our discussion. ##### Ultimate truth We will follow Santaraksita and Kamalasila and first take up the second major thesis--ultimately everything, including the mind, is unreal and empty. Therefore, ultimate truth is the emptiness of any intrinsic reality. According to Kamalasila ultimate truth has three senses: ([MA], Dbu ma sa 233b) 1. reality, the nature of which is characterised as "nonself of person" and "nonself of phenomena," is etymologically both ultimate (param / dam pa) and object (artha / don). 2. transcendent knowledge (lokottaram jnanam), reliable cognition of the ultimate since it is directly engaged with ultimate reality. 3. mundane knowledge, while not in itself ultimate knowledge, but because it is a complementary means of transcendent knowledge, it is also identified as the ultimate. Both Santaraksita and Kamalasila maintains that the emptiness of intrinsic reality is the ultimate truth, and in order to demonstrate that this is so, Kamalasila deploys five forms of arguments--namely: 1. The diamond-sliver argument (vajrakanadiyukti / irdo rje gzegs ma'i gtan tshigs) which shows that all things are empty of intrinsic reality because things are analytically not found to arise--neither from themselves, nor from another nor both nor causelessly ([MA], Dbu ma sa 190a-202b). 2. The argument refuting the arising of existent and non-existent entities (sadasadutpadapratisedhahetu / yod med skye 'gog gi gtan tshigs) which shows that all things are empty of intrinsic reality because their intrinsic reality is not found to arise either from things that exist or from things that do not exist ([MA], Dbu ma sa 202b). 3. The arguments refuting the four modes of arising (catuskotiyutpadapratisedhahetu / mu bzhi skye 'gog gi gtan tshigs) which shows that things are empty of intrinsic reality for the reason that such reality is neither found anaytically in existence (or being) nor in nonexistence (or nonbeing) nor both existence-nonexistence (being-nonbeing) nor neither existence nor nonexistence (neither being nor nonbeing) ([MA], Dbu ma sa 210b). 4. The neither-one-nor many argument (ekanekaviyogahetu / gcig du bral gyi gtan tshigs) which shows all phenomena are empty of intrinsic reality because they all lack the characteristics of being intrinsically one or intrinsically many ([MA], Dbu ma sa 218b). 5. The argument from dependent arising (pratityasamutpadahetu / rten 'brel gyi gtan tshigs) which shows that things are produced from the association of multiple causes and condition, and therefore showing that things do not have any intrinsic reality of their own ([MA], Dbu ma sa 205b). The most well-known of Santaraksita's arguments is the neither-one-nor many argument,[14] which seeks to examine the final (or ultimate) identity (ngo bo la dpyod pa) of all phenomena. In his [MAK] and [MAV] Santaraksita develops this argument in a great detail. The essential feature of the argument is as follows: "The entities that are proclaimed by our own [Buddhist] schools and the other [non-Buddhist] schools, lack intrinsic identity because in reality their identity is neither a (1) singular nor a (2) plural--like a reflection." ([MAK] 1, Dbu ma sa 53a) In this argument: * The subject (rtsod gzhi cos can): the entities that are proclaimed by our own Buddhist schools and the other non-Buddhist schools (from here on 'all entities.) * The property to be proved (sadhyadharma / sgrub bya'i chos) : the lack of intrinsic identity. * What is being proved (sadhya / sgrub bya) : all entities lack the intrinsic identity. * The reason (hetu / rtags) or means of proof (sadhana / sgrub byed) : in reality their identity is neither a singular nor a plural. * Example: like a reflection. According to Santaraksita the validity of the neither-one-nor many argument depends on whether or not it satisfies the triple criteria (trirupalinga / tshul gsum) of formal probative reasoning. The first criterion (paksadharmata / phyogs chos) shows that the reason qualifies the subject (phyogs / paksa or chos can / dharmin), i.e., that the subject must be proven to have the property of the reason. That is, all entities must be shown, in reality, as neither singular nor plural. And therefore this argument consists of two premises: 1. the premise that shows the lack of singularity ("unity") and 2. the premise that shows the lack of plurality (or "many-ness") In [MAK] and [MAV] Santaraksita develops the argument showing the lack of singularity in two parts: (i) the argument that shows the lack of the non-pervasive singularity in (a) permanent phenomena, and (b) in impermanent entities; and (ii) the argument that shows the lack of pervasive singularity by (a) refuting the indivisible atomic particles which are said to have composed the objects, and (b) by refuting the unified consciousness posited by Yogacara. The premise that shows the lack of plurality is based upon the conclusion of the argument that shows the lack of singularity. This is so, argues Santaraksita, because "When we analyse any entity, none is [found] to be a singular. For that for which nothing is singular, there must also be nothing which is plural" ([MAK] 61, Dbu ma sa 55a). This argument of Santaraksita proceeds from understanding that being singular and being plural (manifold) are essentially interdependent--if there are no singular entity, one must accept that there are no plural entity either since the latter is the former assembled. According to Santaraksita the argument from neither-one-nor many satisfies the triple criteria of valid reasoning. It satisfies the first criterion because all instances of the subject, namely the intrinsic identities of those entities asserted by the Buddhist and non-Buddhist opponents, are instances of entities which are neither singular nor plural. The second criterion of a valid reason--the proof of the forward entailment (anvayavyapti / rjes khyab), i.e., the proof that the reason occurs in only "similar instances" (sapaksa / mthun phyogs) where all instances of the reason are the instances of the predicates--is also satisfied because the reason--all phenomena are neither singular nor plural--is an instance of the predicate--phenomena which lack identity. The third criterion of a valid reason--the proof of the counter entailment (vyatirekavyapti / ldog khyab / ldog khyab), or the proof that the reason is completely absent from the dissimilar instances of the predicate (vipaksa / mi mthun phyogs)--hold because there are no instances of the predicate which are not instances of the reason. There are no instances of phenomena, which lacked intrinsic identity, which are not also instances of phenomena, that are neither singular nor plural. Therefore "the counter entailment holds since there are no entities that could be posited as existing in other possible alternatives" ([MAV] 62 Dbu ma sa 69b). Thus the second and the third criteria which are mutually entailing prove the entailment of the neither one-nor many argument. Once the entailment[15] of the argument is accomplished, in Santaraksita's view, all potential loopholes in this argument are addressed. And therefore the argument successfully proves that all phenomena are empty of any intrinsic reality, since they all lack any singular or plural identity. It is clear, then, that the arguments of Santaraksita and Kamalasila have the final thrust of demonstrating the reasons why all phenomena ultimately lack any intrinsic reality, and are therefore geared towards proving the emptiness of intrinsic reality as the ultimate truth, in each argument a specific domain of analysis is used to prove the final ontological position. ##### Conventional Truth Let us now turn to the first thesis and look at the Yogacara-Svatantratika Madhyamaka defence of the theory of conventional truth. As we have seen from the arguments such as neither-one-nor many, Santaraksita and Kamalasila draw the conclusion that entities are ultimately empty of reality, for ultimately they do not exist either as a singular or as a plural. They then move on to explain the way things do exist, and Santaraksita does this with an explicit admission that things do exist conventionally--"Therefore these entities are only conventionally defined. If [someone] accepts them as ultimate, what can I do for that?" ([MAK] 63 Dbu ma sa 55a). Why is it that all entities are defined as only conventionally real? To understand the answer to this question, we need to look at Santaraksita's definition of conventional reality, which consists of three criteria: 1. it strictly appeals to its face-value and cannot be subjected to reasoned analysis (avicaramaniya); 2. it is causally efficient; and 3. it is subjected to arising and disintegration ([MAK] 64, Dbu ma sa 55a). Elsewhere in [MAV] Santaraksita also provides a broader definition of conventionality (samvrti / kun rdzob), and argues that conventionality is not simply a linguistic convention (sabda-vyvahara / sgra'i tha snyad), it also includes what is conventionally real, those that are seen and accepted by the Madhaymikas as dependently coarisen (pratityasamutpanna / rten cing 'brel 'byung), and those which cannot be rationally analysed ([MAV] 65, Dbu ma sa 70a). Kamalasila's definition of conventional truth stresses epistemic error (branta jnana / rnam shes khrul ba) in as much as he does stress on non-erroneous epistemic perspective (abranta jnana / rnam shes ma khrul ba) in his definition of the ultimate truth. So on his definition, the nature of all entities, illusory persons and the like, is classified into two truths by virtue of erroneous (branta / khrula ba) and non-erroneous cognition through which they are each represented. Kamalasila's definition of conventional truth makes explicitly clear the synthesis between Madhayamaka and Yogacara. The Yogacara Svatantrika Madhyamaka claims that the Yogacara is right about the presentation of the conventional truth as mere mental impressions, and that this can be seen from the way in which Santarasita and Kamalasila defend this synthesis textually and rationally through relying heavily on the sutras--Samdhinirmocannasutra, Lankavatarasutra, Saddharmapundarikasutra and so forth--traditionally supporting Yogacara idealism. The claim is made here, and Santaraksita does this explicity, that the Madhyamaka and the idealistic doctrines taught in these scriptures are all "consistent." ([MAV] 91 Dbu ma sa 79a) The works of these two philosophers produce abundant citations from these sutra literatures stressing the significance of the passages like this from the Lankavatarasutra: "Material forms do not exist, one's mind appears to be external." In terms of the argument, Santaraksita argues "that which is cause and effect is mere consciousness only. And whatever is causally established abides in consciousness." ([MAK] 91, Dbu ma sa 56a) Therefore phenomena that appear to be real external objects like a blue-patch are not something that is distinct from the nature of the phenomenological experience of blue-patch, even so, as the material form experienced in dreams is not distinct from the dreaming experience ([MAV] 91 Dbu ma sa 79a). Briefly, then, it is clear the two Svatantriak schools--Sautrantika-Svatantrika-Madhyamak and the Yogacara-Svatantrika Madhyamaka--both uphold the standard Madhyamaka position in terms of treating emptiness as the ultimate truth, although, they vary somewhat in their presentation of conventional truth since the former is realistic and the latter idealistic about the nature of the conventional truth. ### 4.2 Prasangika Madhyamaka Bhavaviveka's Svatantrika Madhyamaka theory of the two truths soon came under attack from Candrakirti (ca. 600-650) who, among other works, wrote the following: * Clear-word Commentary on the Fundamental Verses of the Middle Way (Mulamadhyamakavrttiprasannapada [PP], known simply as Prasannapada, Dbu ma 'a 1b-200a), * An Introduction on Madhyamaka (Madhyamakavatara [M], Dbu ma 'a 201b-219a), * Commentary on Introduction on Madhyamaka (Madhyamakavatarabhasya [MBh], Dbu ma 'a 220b-348a), * Commentary on Four Hundred Verses (Catuusatakatika, Dbu ma ya 30b-239a), * Commentary on Seventy Verses on Emptiness (Sunayatasaptativrtti, Dbu ma tsa 110a-121a), * Commentary on Sixty Verses on Reasoning (Yuktisasthikavrtii, Dbu ma ya 1b-30b), and * Discussion on the Five Aggregates (Pancaskandhaprakarana, Dbu ma ya 239b-266b) In these philosophical works, Candrakirti rejects the theories of the two truths in both Brahmanical and the Buddhist schools of Vaibhasika, Sautrantika, Yogacara and Svatantrika Madhyamaka. His chief reason for doing this is his deep mistrust of the varying degrees of metaphysical and epistemological foundationalism that these theories are committed to. For example Candrakirti rejects Bhavavevika's reading of Nagarjuna's theory of the two truths and explicitly vindicates Buddhapalita's Prasangika approach to understanding Nagarjuna's Madhyamaka. He does this by demonstrating the groundlessness of the fallacies Bhavavevika attributed to Buddhapalita, and thereby rejecting Bhavavevika's admission and imposition upon Nagarjuna's non-foundationalist ontology of a kind of independent formal svatantra argument, which Candrakirti saw as one encumbered by foundationalist metaphysics, and is incompatible with Nagarjuna's ontology. Instead of relying on the two truths theories of other schools, Candrakirti proposes a distinctive non-foundationalist theory of the two truths in the Madhyamaka, and he defends it against his foundationalist opponents. This school later came to be known as the Prasangika Madhyamaka. At the core of the Prasangika's theory of the two truths are these two fundamental theses: 1. Only what is conventionally non-intrinsic is causally effective, for only those phenomena, the conventional nature of which is non-intrinsically real, are subject to conditioned or dependent arising. Conventional reality (or dependently arisen phenomenon), given it is causally effective, is therefore always intrinsically unreal. Hence that which is conventionally (or dependently) coarisen is always conventionally (or dependently) arisen. 2. Only what is ultimately non-intrinsic is causally effective, for only those phenomena, the ultimate nature of which is non-intrinsically real, are subject to conditioned arising. Ultimate truth (or emptiness), given it is causally effective, is therefore intrinsically unreal. Hence ultimate truth is ultimately unreal (or emptiness is always empty). Although these two theses are advanced separately, they are mutually coextensive. There is a compatible relationship between conventional truth and ultimate truth, hence between dependent arising (pratityasamutpada / rten 'brel) and emptiness (sunyata / stong pa nyid), and there is no tension between the two. We shall examine the Prasangika arguments in defence of these theses. #### 4.2.1 Conventional Truth Let us turn to the first thesis. Earlier we saw that both the Vaibhasika and Sautrantika argue that only ultimately intrinsic reality (svabhava) enables things to perform a causal function (arthakriya). The Svatantrika Madhyamaka rejects this, and it instead argues that things are causally efficient because of their conventionally intrinsic reality (svabhava) or unique particularity (svalaksana). The Prasangika Madhyamaka, however, rejects both these positions, and argues only what is conventionally non-intrinsic reality (nihsvabhava) is causally effective, for only those phenomena, the conventional nature of which is non-intrinsic, are subject to conditioned or dependent arising. Conventional reality (here treated as dependently arisen phenomenon), given it is causally effective, is therefore always intrinsically unreal, and hence lacks any intrinsic reality even conventionally. Hence that which is conventionally (or dependently) coarisen is always conventionally (or dependently) arisen and strictly does not arise ultimately. In his etymological analysis of the term "convention" (samvrti / kun rdzob) in [PP] Candrakirti attributes three senses to the term convention: 1. confusion (avidya / mi shes pa) for it obstructs the mundane cognitive processes of the ordinary beings from seeing the reality as it is by way of concealing (samvrti / kun rdzob) its nature; 2. codependent (paraparasambhavana / phan tshun brtan pa) for it is an interdependent phenomenon; and 3. Signifier (samket / rda) or worldly convention (lokavyavahara / 'jig rten tha snyad) in the sense of dependently designated by means of expression and expressed, consciousness and object of consciousness, etc ([PP] 24.8 Dbu ma 'a 163a). Candrakirti's claims that in the case of mundane cognitive processes of the ordinary beings, the first sense of convention eclipses the force of the second and the third senses. As a result, far from understanding things as dependently arisen (the second sense) and dependently designated (the third sense), ordinary beings reify them to be non-dependently arisen, and non-designated. Due to the force of this cognitive confusion, from the perspective of mundane cognitive processes of the ordinary beings, things appear real, and each phenomenon appears intrinsically real in spite of their non-intrinsic nature. Consequently this confusion, according to Candrakirti, defines epistemic practices of the ordinary beings. For this reason, the epistemic norms or the standard set by mundane convention of the ordinary beings, is poles apart from the epistemic norms set by mundane convention of the noble beings (aryas / 'phags pa) whose mundane cognitive processes are not under the sway of afflictive confusion. In order to illustrate this distinction, in his commentary [MBh] 6.28 Candrakirti introduces the concept: mere convention (kun rdzob tsam) of the noble beings to contrast with conventionally real or conventional true (kun rdzob bden pa) of the ordinary beings. Ordinary beings erroneously grasp all conditioned phenomena as intrinsically real, therefore things are conventionally real and thus are categories of conventional truth. For noble beings, however, because they no longer reify them as real, things are perceived as having the nature of being created (bcos ma) and unreal (bden pa ma yin pa) like the reflected image ([MBh] 6.28 Dbu ma 'a 255a). According to Candrakirti's definition of the two truths, all things embody dual nature, for each is constituted by its conventional nature and its ultimate nature. Consequently, all entities, according to this definition, satisfy the criterion of the two truths, for the definition of the two truths is one based on the two natures. The two truths are, therefore, not merely one specific nature mirrored in two different perspectives ([MBh] 6.23 Dbu ma 'a 253a). Conventional nature entails conventional truth and ultimate nature entails ultimate truth, and given each entity is constituted by the two natures, the two truths define what each entity is in its ontological and epistemological terms. Conventional nature is defined as conventional truth because it is the domain of mundane cognitive process, and is readily accessible for ordinary beings, including mundane cognitive process of noble beings. It is a sort of truth, while unreal and illusory in reality, it is yet erroneously and non-analytically taken for granted by mundane cognitive processes of the ordinary beings. Ultimate nature, on the other hand, is defined as ultimate truth because it is domain of the exalted cognitive process which engages with its object analytically, one that is directly accessible for noble beings (aryas) and only inferentially accessible for ordinary beings. It is this sort of truth, both real and non-illusory, that is correctly and directly found out by exalted cognitive processes of noble beings, and by analytic cognitive processes of ordinary beings. Mundane cognitive process that is associated with the definition of conventional truth and therefore allied to the perception of unreal entities is of two kinds ([M] 6.24 Dbu ma 'a 205a): 1. correct cognitive process, and 2. fallacious cognitive process. Correct cognitive process is associated with an acute sense faculty, which is not impaired by any occasional extraneous causes of misperception (see below). Fallacious cognitive process is associated with a defective sense faculty impaired by occasional extraneous causes of misperception. In [PP] Candrakirti introduces us to a similar epistemic distinction: 1. mundane convention (lokasamvrti / 'jig rten gyi kun rdzob) and 2. non-mundane convention (alokasamvrti /'jig rten ma yin pa'i kun rdzo). Candrakirti's key argument behind to support the distinction between two mundane epistemic practices--one mundane convention and the other non-mundane convention--is that the former is, for mundane standard, epistemically reliable whereas the latter is epistemically unreliable. Consequently correct cognitive processes of both ordinary and awakened beings satisfy the epistemic standard of mundane convention, being non-deceptive by mundane standard, thus they set the standard of mundane convention; whereas fallacious cognitive processes of defective sense faculties, of both ordinary and awakened beings, do not satisfy the epistemic standard of mundane convention, and therefore, they are deceptive even by mundane standard. They are thus "non-mundane convention" as Candrakirti characterises them in [PP] 24.8 (Dbu ma 'a 163ab). Just as they are two kinds of sensory faculty--nonerroneous and erroneous--their objects are, according to Candrakirti, of two corresponding kinds: 1. conventionally real and 2. conventionally unreal. First, cognitive processes associated with sense faculties that are unimpaired by extraneous causes of misperception grasp the former objects, and because this kind of object fulfils realistic mundane ontological standard, it is thus real for ordinary beings, (not so for the noble beings the issue to which we turn later). Hence it is only conventionally real. Second, cognitive processes associated with sense faculties that are impaired by extraneous causes of misperception grasp conventionally unreal objects, and because this kind of object does not meet realistic mundane ontological standard, they are thus conventionally unreal. Although Candrakirti recognises that conventional reality satisfies the ontological standard of ordinary beings, he, importantly, argues that it does not satisfy the ontological criterion of the Madhyamaka (here synonymous for noble beings). Just as illusion is partly real although partly unreal, all objects that are considered commonsensically as conventionally real, are unreal. The major difference between the two is that a knowledge of illusion being unreal is available to mundane cognitive processes of ordinary beings where as a knowledge of unreality of conventionally real entities is not available to mundane cognitive processes of ordinary beings. However, for Candrakirti illusory objects are conventionally real and causally efficient, and that the conventionally real entities themselves are illusion-like. Candrakirti develops this argument using the two tier theory of illusion. 1. The argument from the first tier of the theory of illusion demonstrates a causal efficiency of conventionally illusory objects. This, Candrakirti does by identifying conventionally illusory entities with the so-called conventionally real entities. 2. The argument from the second tier of the theory of illusion demonstrates an illusory nature of conventionally real entities. This, Candrakirti does by means of showing that entities that are conventionally real are conventionally empty of intrinsic reality, and by denying conventional reality the ontological status of the so-called intrinsic reality. Candrakirti support his argument from the theory of illusion by means of applying four categories of the conventional entities: 1. Partly unreal entities--illusory objects (e.g. mirage, reflection, echoe etc.); 2. Partly real entities--non-illusory objects (e.g. water, face, sound etc.); 3. Intrinsic reality (e.g. substance, essence, intrinsic nature etc.,) and 4. Conventionally unreal entities (e.g. the rabbit's horn, the sky-flower etc). ([MBh] 6.28 Dbu ma 'a 254b-255a). Candrakirti accepts conventional reality of both entities--partly unreal (i) and partly real (ii)--on the ground that they both appear in mundane cognitive processes who are confused about their ultimate ontological status. Candrakirti rejects conventional reality status of those entities that are supposedly intrinsically real (iii) and conventionally unreal (iv), although for different reasons. Conventional reality of intrinsically real entity is rejected in [MBh] 6.28 on the ground that its existence does not, forever, appear to a mundane cognitive processes. Candrakirti argues that there is a critical epistemic differentiation to be made between the two entities--conventionally real and conventionally illusory. Deceptive, unreal, and dependently arisen nature of conventionally real entities is beyond the grasp of mundane cognitive processes. By contrast, deceptive, unreal and dependently arisen nature of conventionally illusory, partly unreal, entities is grasped by mundane cognitive processes. Therefore illusory objects are regarded as deceptive, and unreal by the standard of mundane knowledge, whereas, conventionally real things are regarded as nondeceptive and real by the standard of mundane knowledge. So the argument from the first tier of the illusory theory shows that mundane cognitive processes of ordinary beings fail to know illusory and unreal nature of conventionally real entities, they instead grasp them to be intrinsically real. The argument from the second tier of the illusory theory offers Candrakirti's reason why this is so. Here, he argues, the presence of the underlying confusion operating beneath mundane cognitive processes of the ordinary beings is the force by which ordinary beings intuitively and erroneously reify the nature of conventional entities. Hence they grasp conventional entities as intrinsically real, although they are in actual fact on a closer analysis, only non-intrinsic, unreal or illusory. Therefore intrinsic reality is not a conventional reality and grasping things to be intrinsically real is only a confused belief, not a knowledge. Hence Candrakirti clearly excludes it from the ontological categories of conventional reality, for the reason that intrinsic reality is a conceptual fiction fabricated by confused minds of the ordinary beings and that it is a conceptual construct thrust upon non-intrinsic entities ([MBh] 6.28 Dbu ma 'a 255a). In addition Candrakirti argues that if things were intrinsically real, even conventionally, as the Svatantrikas take them to be, in Candrakirti's view, things would exist by virtue of their intrinsic reality, at least conventionally. If this were the case then intrinsic reality would become the ultimate nature of things. That would be absurd. For Madhyamikas, things cannot have more than one final mode of existence--if they exist, they must exist qua only conventionally. Consequently, given the mutually exclusive relation of intrinsic reality and emptiness, if intrinsic reality is granted even a conventional reality status, Candrakirti contends, one has to reject emptiness as the ultimate reality ([M] 6.34, DBu ma 'a 205b). But since the Madhyamika asserts emptiness to be the ultimate reality, and given that emptiness and intrinsic reality are mutually exclusive, Madhyamika must reject intrinsic reality in all its forms--conventionally as well as ultimately ([MBh] 6.36, 1994: 118). #### 4.2.2 Ultimate truth We now turn to Candrakirti's second thesis--namely, only what is ultimately non-intrinsic is causally effective, for only those phenomena, an ultimate nature of which is non-intrinsic, are subject to conditioned arising. Ultimate reality (or emptiness), given it is causally effective, is therefore intrinsically unreal. Therefore ultimate reality is ultimately unreal (or put it differently, emptiness is ultimately empty). Candrakirti defines ultimate reality as the nature of things found by the perception of reality ([M] 6.23 Dbu ma 'a 205a). In the commentary Candrakirti expands this definition. "Ultimate is the object, the nature of which is found by particular exalted cognitive processes (yishes) of those who perceive reality. But it does not exist by virtue of its intrinsic objective reality (svarupata / bdag gi ngo bo nyid)." ([MBh] 6.23 Dbu ma 'a 253a) Of the two natures, the object of the perception of reality is the way things really are, and this is, Candrakirti explains, what it means to be ultimate reality ([MBh] 6.23 Dbu ma 'a 253ab). Candrakirti's definition raises a couple of important points here. First, by defining ultimate reality as "the nature of things found by particular exalted cognitive processes (yeshes) of those who perceive reality," he means ultimate reality is not found by any exalted cognitive processes, it must be found by a particular exalted cognitive processes--analytic--that knows things just as they are. Second, by stating that "it does not exist by virtue of its intrinsic objective reality" (svarupata / bdag gi ngo bo nyid), he means ultimate reality is not intrinsically real just as much as conventional reality is not intrinsically real. Third, he says that ultimate reality is the first nature of the two natures of things found by perception of reality. This means ultimate reality is ultimate nature of all conventionally real things. For they all have an ultimate nature representing ultimate status, just as they have conventional nature representing conventional status. Ultimate truth, according to Candrakirti, for the purpose of liberating all sentient beings ([M] 6.179 Dbu ma 'a 213a). 'a 213a) is differentiated by the Buddha into two aspects: 1. nonself of person (pudgalanairatmya) and 2. nonself of phenomena (dharmanairatmya) As we shall see Candrakirti's arguments come from two domains of analysis which he employs to account for his theory of ultimate truth: analysis of personal self (pudgala-atmya / gang zag gi bdag) and analysis of phenomenal self (dharma-atmya / chos kyi bdag). The analysis will demonstrate whether or not conventionally real phenomena and persons are more than what they are conventionally. So we shall consider Candrakirti's arguments under this rubrics: 1. The not-self argumnent: unreality of conventionally real person; 2. The emptiness argument: unreality of conventionally real phenomena; 3. The emptiness is emptiness argument: unreality of ultimate reality. Let us begin with the first. The not-self argumnent demonstrates, as we shall see, unreality of conventionally real person. We shall develop this argument of Candrakirti progressively in three steps: * correlation between personal self and the five aggregates; * arguments against personal self; and * reasons for positing personal self as merely dependently designated. In order to appreciate Candrakirti's not-self argument, or the argument that shows unreality of conventionally real person, we need to understand Candrakirti's purpose of refuting personal self. According to Candrakirti it is important that we look into the question: from where does the assumption of self arise before we go any further. He argues, like all the Buddhist philosophers do, the notion of self or the assumption of self arises in relation to personhood. After all it is a personal identity issue. To analyse the self therefore requires us to look closely among the parts that constitute a person, and given that the conception of self is embedded in personhood which is constituted by the five aggregates, we need to examine its relation to our conception of self. Suffice it here to highlight the important relation that exists between the aggregates and the notion of self. According to Candrakirti, the five aggregates are primary categories upon which we will draw to examine the phenomenal structure of sentient beings, because of the following reasons: because (1) it serves as a phenomenological scheme for examining the nature of human experience, (2) the way in which human beings construct conceptions of the self is based upon different existential experiences within the framework of the five aggregates, (3) they are the basis upon which we develop the sense of self through the view that objectifies the five aggregates which are transient and composites (satkayadrsti / 'jig tshogs la lta ba), either through an appropriation or falsely identifying the aggregates to be self ([MBh] 6.120, Dbu ma 'a 292ab), (4) they are the objective domain of the arising of defilements that cause us suffering since the false view objectifying the aggregates (satkayadrsti / 'jig tshogs la lta ba) to be real self lead to the arising of all "afflictive defilements", and finally (5) the five aggregates are the objective domain of the knowledge of unreality of personal self. For the simple reason that confusion and misconceptions of self arise in relation to the aggregates, thus it is within the same framework that the realisation of nonself must emerge. Therefore, for Candrakirti, it is clear that within the range of the five aggregates exhaust "all" or "everything" that could be considered as the basis for developing the misconception of a reified self and also the basis for correct knowledge of non-self--i.e., reality of person or self. Candrakirti develops his arguments against personal self in a great detail in [M]/[Mbh] chapter six. Suffice it here to consider one such argument called "the argument from the sevenfold analysis of a chariot" (shing rta rnams bdun gyi rigs pa). This argument however is more or less a way of putting all his arguments together. In it, Candrakirti concludes: "the basis of the conception of self, when analysed, is not plausible to be an entity. It is not plausible to be different from the aggregates, nor identity of the aggregates, nor dependent (skandhadhara) on the aggregates. The 'basis' (skandhadhara) at issue is a compound that exposes the implausibility of self as both the container (adhara / rten) and the contained (adheya / brten pa). The self does not own the aggregates. This is established in codependence on the aggregates" ([M] 6.150, Dbu ma 'a 221b). Putting this argument analogically, Candrakirti asserts that the self is, in terms of not being intrinsically real, analogous to a chariot. "We do not accept a chariot to be different from its own parts, nor to be identical [to its parts], nor to be in possession of them, nor is it in the parts, nor are the parts in it, nor is it the mere composite [of its parts], nor is it the shape [or size of the those parts]" ([M] 6.151 Dbu ma 'a 221b). So the structure of the nonself "chariot" argument that demonstrates unreality of a conventionally real person can be briefly stated as follows: * P1 Self is not distinct from the aggregates. * P2 Self is not identical to the aggregates. * P3 Self and the aggregates are not codependent upon each other. * P4 Self and the aggregates do not own each other. * P5 Self is not the shape of the aggregates. * P6 Self is not the collection of the aggregates. * P7 There is nothing more to a person than the five aggregates. * C Therefore a real personal self does not exist. There is no ultimately real self that can be logically proven to exist. If it did exist it has to be found when it is subjected to these sevenfold analyses. But the self is not found either to be different from the aggregates, nor identical to the aggregates, nor to be in possession of them, nor is the self in the aggregates, nor are the aggregates in the self, nor is the self the mere composite of the aggregates; nor is the self the shape or sizes of the aggregates. Since there is no more to a person any more than dependently designated self upon basis of the five aggregates, they exhaust all that there is in a person, Candrakirti therefore concludes that there is no real self anywhere whatsoever to be found in the five aggregates. From this argument, Candrakirti concludes that both ultimately and conventionally, any intrinsic reality of a personal self must remain unproven. Therefore according to any of the sevenfold analysis a personal self is unreal, and empty of any intrinsic reality. What Candrakirti does not conclude from this argument, though, is the implausibility of a conventionally real self for everyday purpose. As matter of fact, Candrakirti argues, the argument from the sevenfold analysis reinforces the idea that only conventionally real self makes sense. For such a self constitutes being dependently designated on the aggregates in as much as a chariot is dependently designated on its parts (M / MBh 6.158). Just as the chariot is conveniently designated in dependence on its parts, and it is referred to in the world as "agent," even so, a conventionally self which is established through dependent designation serves as a conventionally real moral agent. Therefore, although the argument from the sevenfold analyses denies the existence of intrinsically real self or ultimately real self, the argument does not entail a denial of conventionally real self. (M / MBh 6.159). It turns out therefore that a self is merely a convenient designation, or a meaningful label given to the five aggregates, and taken for granted in the everyday purpose as a agent. And it is this nominal self that serves the purpose of moral agent. This nominal or conventional self is comparable to anything that exists conventionally. Candrakirti compares it to a chariot. Second, Candrakirti's argument from emptiness demonstrates the unreality of the conventionally real phenomena by means of appealing to causal processes at work in producing these entities. The understanding here is that the way in which things arise and come into existence definitively informs us about how things actually are. This argument is an extension of Nagarjuna's famous tetralemma argument which states: "neither from itself nor from another nor from both, nor without cause does anything anywhere, ever arise." ([MMK] 1.1) Candrakirti develops this argument in great detail in chapter 6 of [M]/[MBh], we shall only have a quick look at its overall strategy without delving into its details. If entities are intrinsically real there are only two alternative ways for them to arise: either they arise (a) causally or (b) causelessly. If one holds that intrinsically real things arise causally, then there are only three possible ways for them to arise: either they should arise from (1) themselves, or from (2) another or from (3) both. > > > P1 If intrinsic entity arise from itself it would absurdly follow that > cause and effect would be identical and they would exist > simultaneously. In which case, the production of an effect would be > pointless for it would be already in existence. It is thus > unreasonable to assume that something already arisen might arise all > over again. ([M] 6.8) If an entity in existence still requires arising > then an infinite regress would follow ([M] 6. 9-13). > > > > P2 If an intrinsically real entity arises from another, anything could > arise from anything (like a fire from pitch-darkness), given that > cause and effect would be distinct, all causes would be equally > 'another' ([M] 6. 14-21). > > > > P3 If an intrinsic entity arises from both itself and another then > both reductio ad absurdums in P1 and P2 would apply ([M] 6. > 98). > > > > P4 If an intrinsic entity arises causelessly, then anything could > arise from everything ([M] 6. 99-103). > > > > Thus it is not possible for any intrinsic entity to exist because no > such entity could be causally produce either from itself or from > another or both or causelessly. It follows therefore that any entity > that arises causally is not intrinsically real, hence it is > dependently originated entity ([M] 6. 104). > > > Candrakirti therefore insists that entities that are intrinsically unreal are not nonexistent unlike intrinsically real entities which are nonexistent analogous to the rabbit's horn. Even though they are not intrinsically real, hence remain unproduced according to the four analyses, unlike the rabbit's horn, intrinsically unreal entities are objects of mundane cognitive processes, and they arise codependently ([M] 110-114). Hence, in summarising the argument, Candrakirti states: "Entities do not arise causelessly, and they do not arise through causes like God, for example. Nor do they arise out of themselves, nor from another, nor from both. They arise codependently." ([M] 6.114, Dbu ma 'a 209b) So because entities arise codependently, according to Candrakirti, reified concepts such as intrinsic entities cannot stand up under logical analysis. The argument from dependent arising therefore eliminates the web of erroneous views presupposing intrinsic reality. If entities are intrinsically real, then their arising could be through themselves, from another or from both, causelessly. However upon critical analysis the entity as such proves to be nonexistent ([M]/[MBh] 6. 115-116). Third, Candrakirti's argument from the emptiness of emptiness demonstrates the unreality of ultimate reality. The "emptiness-argument" and the "nonself-argument" show that all conventionally real phenomena and persons are empty of intrinsic reality. As we have seen from a critical rational point of view, it is demonstrated that there is no ultimately real or ultimately existent entities or real self of person to be found. The two arguments do not however show that emptiness per se is empty, nor do they show that the selflessness is itself selfless. This raises an important question: does this mean that emptiness is the intrinsic reality of conventionally real entities and selflessness is the intrinsic reality of conventionally real persons? Putting the question differently: could a Madhyamika claim that emptiness and selflessness are ultimately real or are intrinsically real? This question is reasonable. We tend to posit ultimate reality to be something that is timeless, independent, transcendent, nondual etc. So if emptiness is the ultimate reality of all phenomena and if selflessness is the ultimate reality of all persons,could we say that emptiness and selflessness are nonempty, ultimately real or intrinsically real? In order to answer this question Candrakirti maintains that the Buddha reclassifies emptiness and nonself under various aspects--"He elaborated sixteen emptinesses and subsequently he condensed these into four explanations which are adopted by the Mahayana." ([M] 6.180 Dbu ma 'a 213a) We shall consider one example here: Candrakirti argues "The eye is empty of eye[ness] because that is its reality. The ear, nose, tongue, body and mind are also to be explained in this way." ([M] 6.181 Dbu ma 'a 213ab) "Because each of them are non-eternal (ther zugs tu gnas pa ma yin) not subject to disintegration ('jig pa ma yin pa) and their reality is non-intrinsic [the emptiness of] the eye and the other six faculties is accepted as 'internal emptiness' (adhyatmasunyata)"([M] 6.181 Dbu ma 'a 2213b). On Candrakirit's view just as the chariot, self and other conventionally real phenomena are intrinsically unreal, so too is the emptiness and the nonself. Emptiness is also empty of intrinsic reality. Candrakirti explains the emptiness of emptiness as: > > The emptiness of intrinsic reality of things is itself called by the > wise as 'emptiness,' and this emptiness also is considered > to be empty of any intrinsic reality. ([M] 6.185 Dbu > ma 'a 213b) > > > The emptiness of that which is called > 'emptiness' is accepted as 'the emptiness of > emptiness' (sunyatasunyata). It is > explained in this way for the purpose of controverting objectification > of the emptiness as intrinsically real (bhava). ([M] 6.186 > Dbu ma 'a 213b) > > > Also consider Candrakirti's argument against positing emptiness as ultimately real that he develops in his commentary on [MMK] 13.7. If emptiness is ultimately real, emptiness would not be empty of the intrinsic reality--the essence of conventionally real objects. In that case we will have to grant emptiness as existing independently of conventionally real entities, as their underlying substratum. If this is granted the emptiness of a chariot and the empty entity, the chariot itself, would be quite distinct and unrelated. Moreover if the emptiness of the chariot is nonempty, i.e., if it is ultimately real, whereas the chariot itself is empty i.e., ultimately unreal, then, one has to posit two distinct and contradictory verifiable realities even for one conventionally real chariot. But since there is not even the slightest nonempty phenomenon verified in the chariot--not even the slightest bit of entity withstanding critical analysis--emptiness of the chariot, like the chariot itself, is implausible to be ultimately real. Like the chariot itself, the emptiness of the chariot is also ultimately empty. Therefore, while emptiness is the ultimate truth of the conventionally real entities, it is not plausible to posit emptiness to be ultimately real. Similarly, while nonself is the ultimate truth of the personal self, nonself is implausible to be ultimately real. Therefore nonself is not considered as the essence of persons or self. So, to wind up the discussion on the Prasangika's theory of the two truths, just as conventional truth is empty of intrinsic reality, hence ultimately unreal, even so, is ultimate truth empty of intrinsic reality, hence ultimately unreal. It is, therefore, demonstrated that nothing is ultimately real for Candrakirti, hence everything is empty of intrinsic reality. The heart of Candrakirti's two truths theory is the argument that only because they are empty of intrinsic reality can conventionally real entities be causally efficient. Only in the context of a categorical rejection of any foundationalism of intrinsic reality--both conventionally and ultimate, Candrakirti insists, can there be mundane practices rooted in the mutually interdependent character of cognitive processes and objects cognised. For this reason Candrakirti concludes that whomsoever emptiness makes sense, everything makes sense. For whomsoever emptiness makes no sense, dependent arising would not make sense. Hence nothing would make sense ([PP] 24.14, Dbu ma 'a 166b). ## 5. Conclusion To sum up, though this entry provides just an overview of the theory of the two truths in Indian Buddhism discussed overview, it nevertheless offers us enough reasons to believe that there is no single theory of the two truths in Indian Buddhism. As we have seen there are many such competing theories, some of which are highly complex and sophisticated. The essay clearly shows, however, that except for the Prasangika's theory of the two truths, which unconditionally rejects all forms of foundationalism both conventionally and ultimately, all other theories of the two truths, while rejecting some forms of foundationalism, embrace another form of foundationalism. The Sarvastivadin (or Vaibhasika) theory rejects the substance-metaphysics of the Brahmanical schools, yet it claims the irreducible spatial units (e.g., atoms of the material category) and irreducible temporal units (e.g., point-instant consciousnesses) of the five basic categories as ultimate truths, which ground conventional truth, which is comprised of only reducible spatial wholes or temporal continua. Based on the same metaphysical assumption and although with modified definitions, the Sautrantika argues that the unique particulars (svalaksana) which, they say, are ultimately causally efficient, are ultimately real; whereas the universals (samanyalaksana) which are only conceptually constructed, are only conventionally real. Rejecting the Abhidharmika realism, the Yogacara proposes a form of idealism in which which it is argued that only mental impressions are conventionally real and nondual perfect nature is the ultimately real. The Svatantrika Madhyamaka, however, rejects both the Abhidharmika realism and the Yogacara idealism as philosophically incoherent. It argues that things are only intrinsically real, conventionally, for this ensures their causal efficiency, things do not need to be ultimately intrinsically real. Therefore it proposes the theory which states that conventionally all phenomena are intrinsically real (svabhavatah) whereas ultimately all phenomena are intrinsically unreal (nihsvabhavatah). Finally, the Prasangika Madhyamaka rejects all the theories of the two truths including the one advanced by its Madhyamaka counterpart, namely, Svatantrika, on the ground that all the theories are metaphysically too stringent, and they do not provide the ontological malleability necessary for the ontological identity of conventional truth (dependent arising) and ultimate truth (emptiness). It therefore proposes the theory of the two truths in which the notion of intrinsic reality is categorically denied. It argues that only when conventional truth and ultimate truth are both conventionally and ultimately non-intrinsic, can they be causally effective.
twotruths-tibet	## 1. Nyingma Longchen Rabjam sets out the course of the Nyingma theory of the truths and the later philosophers of the school took similar stance without varying much in essence. In Treasure he begins the section on the Madhyamaka theory of the two truths as follows: "The Madhyamaka tradition is the secret and profound teachings of the [Sakya]muni. Although it constitues five ontological categories, the two truths subsume them" (Longchen, 1983: 204f). Nyingma defines conventional truth as consisting of unreal phenomena that appear to be real to the erroneous cognitive processes of the ordinary beings while ultimate truth is reality which transcends any mode of thinking and speech, one that unmistakenly appears to the nonerroneous cognitive processes of the exalted and awakened beings. In other words (1) ultimate truth represents the perspective of epistemically correct and warranted cognitive process of exalted beings ('phags pa); whereas (2) conventional truth represents the perspective of epistemically deceptive and unwarranted cognitive processes of the ordinary beings (Mipham Rinpoche 1993d: 543-544). Let us consider these two definitions turn by turn. Nyingma supports definition (1) with two premises. The first one says that cognitive processes of the exalted ('phags pa) and awakened beings (sang rgyas) are epistemically correct and non-deceptive, because "It is in relation to this cognitive content that the realisation of the ultimate truth is so designated. The object of an exalted cognitive process consists of the way things really are (gshis kyi gnas lugs), phenomena as they really are (chos kyi dbyings) which is undefiled by its very nature (rang bzin dag pa)" (Longchen 1983: 202f). However ultimate truth for Nyingma is not an object per se in the usual sense of the word. It is object only in the metaphorical sense. "From the [ultimate] perspective the meditative equipoise of the realised (sa thob) and awakened beings (sangs rgyas), there exists neither object of knowledge (shes bya) nor knowing cognitive process (shes byed) and so forth, for there is neither object to apprehend nor the subject that does the apprehending. Even the exalted cognitive process (yeshes) as a subject ceases (zhi ba) to operate" (Longchen 1983: 201f). Therefore "At this stage, [Nyingma] accepts the total termination (chad) of all the continua (rgyun) of the cognitive processes ('jug pa) of the mind (sems) and mental factors (sems las byung ba). This exalted cognitive process which is inexpressible beyond words and thoughts (smra bsam brjod du med pa'i yeshes), and thus is designated (btags pa) as a correct and unmistaken cognitive process (yang dag pa'i blo ma khrul ba) as it knows the reality as it is" (Longchen 1983:201f). The second premise comes from Nyingma's transcendent theory. According to this theory ultimate truth constitutes reality and the reality constitutes transcendence of all elaborations. Reality is that which cannot be comprehended by the means of linguistic and conceptual elaboration, as it is utterly beyond the grasp of words and thoughts which merely defile the cognitive states. Given that exalted beings are free of defiled mental states, all forms of thoughts and conceptions are terminated in their realization of the ultimate truth. Ultimate truth transcends all elaborations, thus remains untouched by the philosophical speculations (Longchen, 1983: 203f). "In short the characteristic of [ultimate truth] is that of nirvana, which is profound and peaceful. It is intrinsically unadulterated domain (dbyings). The cognitive process by means of which ultimate truth is realized must therefore be free from all cognitive limitations (sgribs pa), for it is disclosed to the awakened beings (sangs rgyas) in whose exalted cognitive processes appear the objects as they really are without being altered" (Longchen, 1983: 204f). Nyingma's defence of the definition (2) also has two premises. The first one comes from its theory of error, the most commonly used by the Nyingma philosophers. According to this theory, conventional truth is, as matter of fact, an error confined to the ordinary beings (so so skye bo) who are blinded by the dispositions (bag chags) of confusion (ma rig pa). It is argued that under the sway of confusion the ordinary beings falsely and erroneously believe in the reality of entirely unreal entities and the truth of wholly false epistemic instruments, just like the people who mistakenly grasps cataracts and falling hairs to be real objects. Conventional truths are mere errors that appear real to the ordinary beings, but they are in fact no more real than the falling hairs that are reducible into the "modes of apprehensions" (snang tshul) (Mipham Rinpoche 1993c: 3, 1977: 80-81ff, 1993d: 543-544). The second premise comes from its representationalist or elaboration-theory which says that conventional truths constitute merely mental elaborations (spros pa) represented to appear (rnam par snang ba) in the minds of the ordinary beings as if they are realities having the subject-object relation. They are thus deceptive since they are produced through the power of the underlying cognitive confusion or ignorance, and that they do not cohere with any corresponding reality externally. Confusion is samvrti because it conceals (sgrib) the nature, it fabricates all conditioned phenomena to appear as if they are real. Even though conventional truths are to be eventually eradicated (spang bya), the representational images of the conventional reality will nevertheless continue to appear in the minds of even those who are highly realized beings, that is, until they achieved a complete cessation of mind and mental states (Longchen, 1983: 203f). It is argued that when the eye that is affected by cataracts mistakenly sees hairs falling in containers, the healthy eye that is not affected by cataracts does not even perceive the appearances of falling hairs. Likewise, those who are affected by the cataracts of afflictive confusion see things as intrinsically real, hence for them things are conventionally real. Those noble beings who are free from the afflictive confusion and the awakened beings who are free from even the non-afflictive confusion see things as they are (ultimate truth). Just as the person without cataracts does not see the falling hairs, the noble beings and awakened beings do not see any reality of things at all. Mipham Rinpoche also proposes another definition of the two truths which diverges significantly from the one we saw. As we saw Longchen's definition is based on two radically opposed epistemic criteria whereas Mipham's definition, as we shall see, has two radically opposed ontological criteria--one that withstands reasoned analysis and the other that does not withstand reasoned analysis. In his Clearing Dhamchoe's Doubts (Dam chos dogs sel), Mipham says that: "Reality as it is (de bzhin nyid) is established as ultimately real (bden par grub pa). Conventional entities are actually established as unreal, they are subject to deception. Being devoid of such characteristics ultimate is characterized as real, not unreal and not non-deceptive. If this [reality] does not exist," in Mipham's view, "than it would be impossible even for the noble beings (arya / phags pa) to perceive reality. They would, instead, perceive objects that are unreal and deceptive. If that is the case, everyone would be ordinary beings. There would be no one who will attain liberation" (1993a: 602). Mipham anticipates objections to his definition from his Gelug opponents when he writes, "Someone may object: although ultimate truth is real (bden p), its reality is not established ultimately (bden grub), because to be established ultimately is for it to withstands reasoned analysis (rigs pas dpyad bzod)" (Mipham 1993a: 603). The response from Mipham makes it even clearer. "Conventional truth is not ultimately real (bden par grub pa) because it is not able to withstand reasoned analysis, in spite of the fact that those that are conventionally real (yang dag kun rdzob) are nominally real (tha snyad du bden pa) and are the objects of the dualistic cognitive processes" (Mipham 1993a: 603). In contrast: "Reality (dharmata / chos nyid), ultimate truth, is really established (bden par grub pa) on the ground that it is established as the object of nondual exalted cognitive process. Besides, it withstands reasoned analysis, for no logical reasoning whatsoever can undermine it or destroy it. For that reason so long as it does not withstand reasoned analysis, it is not ultimate, it would be conventional entity" (Mipham 1993a: 603). According to Mipham's argument, that an object that cannot withstand reasoned analysis is the kind of object that mundane cognitive processes find dually, and it is not the kind of object found by the exalted cognitive processes that perceive ultimate truth nondually (Mipham 1993a: 603). As we have noticed, Mipham's definition seems to have departed from Longchen's some what significantly. In the final analysis, however, there does not seem to be much variation. Mipham Rinpoche also is committed to the representationalist argument to ascend to nonduality: "In the end," he says, "there are no external objects. It is evident that they appear due to the force of mental impressions ... All texts that supposedly demonstrate the existence of external objects are provisional [descriptions of] their appearances"(Mipham 1977: 159-60ff). Consequently whatever appears to exist externally, according to Nyingma, "is like a horse or an elephant appearing in a dream. When it is subjected to logical analysis, it finally boils down to the interdependent inner predispositions. And this is at the heart of Buddhist philosophy" (Mipham 1977: 159-60ff). Following on from its definition, Nyingma argues that there is no one entity that can be taken as the basis from which divides the two truths; since there is no one entity which is both real (true) and unreal (false). Hence it proposes the two truths division based on the two types of epistemic practices "because it confirms a direct opposition between that which is free of the elaborations and that which is not free of the elaborations, and between what is to be affirmed and what is to be negated excluding the possibility of the third alternative. Thus it ascertains the two" (Longchen, 1983: 205-206ff). Nyingma has two key arguments to support the claim the two truths division is an epistemic. The first argument states, "It is definite that we posit the objects (yul rnams) in dependence upon the subjects (yul can). The subjects are exclusively two types--they are either ultimately (mthar thug pa) fallacious cognitive processes (khrul ba'i blo) or ultimately (mthar thug pa) non-fallacious cognitive processes (ma 'khrula ba'i blo). The fallacious cognitive processes posit [conventional truth]--all samsaric phenomena--whereas as the non-fallacious cognitive processes posit the [ultimate] reality. Therefore it is due to the cognitive processes that we posit the objects in terms of two [truths]" (Longchen, 1983: 206f) The second argument states that ultimate truth is not an objective domain of the cognitive processes with the representational images (dmigs bcas kyi blo'i spyod yul), for the reason that it may be known by means of the exalted cognitive processes (yeshes) with no representational image (dmigs pa med). (Longchen, 1983: 206f) In contrast conventional truth is an objective domain of the cognitive processes with the representational images (dmigs bcas kyi blo'i spyod yul), for the reason that it may be known by means of the cognitive processes having the representational images (dmigs bcas kyi blo'i spyod ul). Nyingma categorically rejects commitment to any philosophical position, for it would entail a commitment to, at least, some forms of elaborations. This is so since "the Prasangika rejects all philosophical systems, and does not accept any self-styled philosophical elaboration (rang las spros pas grub mtha'). (Longchen, 1983: 210ff) The Prasangika therefore presents the theory of the two truths and so forth by merely designating them in accordance with the mundane fabrications (sgro btags) (Longchen 1983: 211f). "Therefore ultimate truth which is transcendent of all elaborations cannot be expressed either as identical to or separate from the conventional truth. They are rather merely different in terms of negating the oneness (Longchen, 1983: 192-3ff, Mipham 1977: 84f). Interestingly Mipham Rinpoche also says that "the two truths constitute a single entity but different conceptual identities (ngo bo gcig la ldog pa tha dad). "This is because," he says, "appearance and emptiness are indistinguishable. This is ascertained through reliable cognitions (tshad ma) by means of which the two truths are investigated. What appears is empty. If emptiness were to exist separately from what appears, the reality of that [apparent] phenomenon would not be empty. Thus the two [truths] are not distinct" (Mipham 1977: 81f). The identity of ultimate truth at issue here, according to Mipham, is that of absolute ultimate (rnam grangs min pa'i don dam ste), rather than provisional ultimate. This is because the kind of ultimate truth under discussion when we are discussing its relation with conventional truth is the ultimate truth "which is beyond the bound of any expression, although it is an object of direct perception" (Mipham 1977: 81f). ## 2. Kagyu According to Karmapa Mikyo Dorje, Atisha and his followers in Tibet, are the authoritative former masters of the Prasangika. It is this line of reading of the Prasangika that Kagyu takes it to be authoritative which the later Kagyu wholeheartedly embraces as the standard position (Mikyo Dorje, 2006: 272-273). Karmapa Mikyo Dorje cliams that it is a characteristic of Kagyu masters and their followers to recognise the ancient Prasangika tradition to be philosophically impeccable ('khrul med) (2006: 272). Unlike the other Tibetan schools, Kagyu rejects the more dominant view that each philosophical school in Indian Buddhism has its own distinctive theory of the two truths. "There are those who advance various theories proclaiming that the Vaibhasika on up to the Prasangika Madhyamaka, for the purpose of distinctive definitions of the two truths, unique rational and unique textual bases. This is true," says Karmapa Mikyo Dorje, an indication "of the failure to understand the logic, and the differentiation between the definition (mtshan nyid) and the defined example (mtshan gzhi)" (Mikyo Dorje 2006: 293). The Indian Buddhist philosophical schools only disagree, according to Kagyu, on the issue concerning the defined example (mtshan gzhi) of the two truths. The instances of ultimate truth in each lower school are, it argues, their philosophical reifications, hence, they are the ones the higher schools logically refute. There is nevertheless no disagreement between the Indian Buddhist schools as far as the definiendum (mtshon bya / laksya) and the definition (mtshan nyid / laksana) of the two truths are concerned. If they disagree on these two issues, there would be no commonly agreed definition, but there is a definition of the ultimate which is common to all schools, even though there is no commonly agreed defined example to illustrate such definition (Mikyo Dorje 2006: 293). Where the Indian Buddhist philosophical schools party their company, Kagyu argues, is their conception of the defined example of the ultimate. For the realists (dngos smra ba) ultimate truth is a foundational entity (rdzas su yod pa) that withstands rational analysis. The Madhyamaka rejects this and argues that if ultimate truth is a foundational entity, it is not possible to attain an awakening to it. If it is possible to attain awakening, then ultimate truth is impossible to be a foundational entity, for the two mutually exclude each other. The foundational entity, like the substance of the Brahmanical metaphysicians, does not allow modification or change to take place whereas the attainment of awakening is an ongoing process of modification and change (Mikyo Dorje 2006: 294). From the position Kagyu takes, it is clear then, that it does not endorse Gelug's position that the Prasangika has its own theory of the two truths which affords a unique framework within which a presentation of the definition and the defined example of the two truths can be set out. (See Yakherds 2021 vol. 1 for the Gelugpa critiques)Kagyu argues that the Prasangika's project is purely for pedagogical and heuristic reasons, "for it seeks to assuage the cognitive errors of its opponents--the non-Buddhists, our own-schools such as Vaibhasika on up to the Svatantrika--who assert philosophical positions based on certain texts" (Mikyo Dorje 2006: 294). The non-Prasangika philosophical schools equate the definition of the ultimate with the defined example, and this is the malady that the Prasangika seeks to allay. Therefore the Prasangika's role, as Kagyu sees it, is to point out the absurdities in the non-Prasangika position and to show that the definition of the ultimate and the defined example are rather two separate issues, and that they cannot be identified with each other (Mikyo Dorje 2006: 294-95). The Prasangika does not, like Nyingma, entertain any position of its own, except for the purpose of a therapeutic reason, and for which "the Prasangika relies on the convention as a framework for things that need to be cultivated and those that are to be abandoned. For this, our own position does not need to base the linguistic conventions on the concepts of definition and the defined example. Instead [our position]," says Kagyu, "conforms to the non-analytical mundane practice in which what needs to be abandoned and what needs to be cultivated are undertaken in agreement with the real-unreal discourses of the conventional [truth]" (Mikyo Dorje 2006: 295). Therefore according to Kagyu, the position of all Indian Buddhist philosophical schools--be it higher or lower--agree in so far as the definitions of the two truths are concerned. (1) "[Ultimate truth] is that which reasoned analysis does not undermine and that which withstands reasoned analysis. Conventional truth is the reverse. (2) Ultimate is the non-deceptive nature and it exists as reality. Conventional [truth] is the reverse. (3) Ultimate is the object of the non-erroneous subject. Conventional [truth] is the object of the erroneous subject" (Mikyo Dorje 2006: 294). Of these three definitions, the first one proposes conventional truth as one that does not withstands reasoned analysis whereas ultimate truth does indeed withstand the reasoned analysis. "Conventional [truth] is a phenomenon that is found to exist from the perspective of the cognitive processes that are either non-analytical (ma dpyad pa) or slightly analytical (cung zad dpyad pa), whereas ultimate [truth] is that which is found either from the perspective of the cognitive process that is thoroughly analytical (shing tu legs par dpyad pa) or meditative equipoise" (Mikyo Dorje 2006: 274). Conventional truth is non-analytical in the sense that ordinary beings assume its reality uncritically, and become fixated to its reifications in virtue of how it appears to their uncritical minds, rather than how it really is from the critical cognitive perspective. When conventional truth is slightly analysed, however, it reveals that it exists merely as collocations of the codependent causes and conditions (Mikyo Dorje 2006: 273). Therefore conventional truth is "one that is devoid of an intrinsic reality, rather it is mere name (ming tsam), mere sign (rda tsam), mere linguistic convention (tha snyad tsam), mere conception (rnam par rtog pa tsam) and mere fabrication (sgro btags pa tsam)--one that merely arises or ceases due to the force of the expressions or the conceptual linguistic convention" (Mikyo Dorje 2006: 273). Ultimate truth is defined, on the other hand, as one that is found by analytical cognitive process or by a meditative equipoise of the exalted beings, for the reason that when it is subjected to analysis, phenomena are fundamentally, by its very nature transcendent of the elaborations (spros pa) and all symbolisms (mtshan ma) (Mikyo Dorje 2006: 273). On the second definition "Conventional truth is that which does not deceive the perspective to which it appears (snang ngor), or that which does not deceive an erroneous perspective ('khrul ngor), or that which is non-deceptive according to the standard of the mundane convention" (Kun Khyen Padkar, 2004: 66). Ultimate truth, on the other hand, is defined as "that which does not deceive a correct perception (yang dag pa'i mthong ba), or that which does not deceive reality as it is (gnas lugs la mi bslu ba), or that which does not deceive an awakened perspective (sang rgyas kyi gzigs ngor)" (Kun Khyen Padkar, 2004: 74). The third definition is based on the two conflicting epistemic practices. Here ultimate is defined as an ultimate object from the perspective of exalted beings (arya / 'phags pa). The ultimate reality, although, is not one that is regarded as cognitively found to exist in and of itself inherently (rang gi bdag nyid). Conventional truth is defined as the cognitive process that sees unreal phenomena (brdzun pa mthong ba) from the perspective of the ordinary beings whose cognitive processes are obscured by the cataracts of confusion. Conventional truth is unreal, for it does not exist in the manner in which it is perceived by the confused cognitive processes of the ordinary beings (Mikyo Dorje 2006: 269-70). Karmapa Mikyo Dorje puts the point in this way. "Take a vase as one entity, for instance. It is a conventional entity, for it is a reality for the ordinary beings, and it is found to be the basis from which arise varieties of hypothetical fabrications. The exalted beings," he argues, "however, do not see any of these [fabrications], anywhere, whatsoever, and this mode of seeing by way of not seeing anything at all is termed as the seeing of the ultimate. Therefore the two truths, in this sense, are not distinct. They are differentiated from the perspective of the cognitive processes that are either erroneous or non-erroneous" (Mikyo Dorje, 2006: 274). Although, like Nyingma and Sakya it is the view of Kagyu to maintain that erroneous cognitive process represents the nature of the ordinary beings, whereas non-erroneous cognitive processes represent the exalted beings. However, unlike Nyingma and Sakya, Kagyu insists that the distinctions between the two truths has nothing to do with the perspective of the exalted beings. Everything, it claims, has to do with how ordinary beings fabricate things erroneously--one more real than the other--and that it has nothing to do with how exalted beings experience things. "Even this distinction is made from the point of view of the cognitive process of the childish beings. Since all things the childish beings perceive are characteristically unreal (brdzun pa) and deceptive (bslu ba), they constitute the conventionality. The exalted beings, however, do not perceive at all anything in the way in which the childish beings perceive and fabricate. So ultimate truth is, according to Kagyu, that which is unseen and unfound by the exalted beings since it is the way things really are. Therefore both the truths are taught for the pedagogical reasons in accord with the perspective of the childish beings, but not because the exalted beings experienced the two truths or that there really exist the two truths (Mikyo Dorje 2006: 274). As for the conventional truth, Kagyu divides it into having two aspects: the conventional in the form of imputed concrete phenomena that are capable to appear concretely as such, and the conventional in terms of merely nominal imputations of abstract entities that are incapable of concrete appearance (Mikyo Dorje, see Yakherds 2021: 153-55). What follows from the above discussion is that Kagyu is monist about truth (Karmapa Mikyo Dorje, 2006: 302) in that it claims only one truth and that truth it equates with transcendent wisdom. Despite minor differences, Nyingma, Kagyu and Sagya all emphasize the synthesis between transcendent wisdom and ultimate truth, arguing that "there is neither separate ultimate truth apart from the transcendent wisdom, nor transcendent wisdom apart from the ultimate truth" (2006: 279). For this reason Kagyu advances the view similar to Nyingma and Sakya that the exceptional quality of awakened knowledge consists in not experiencing anything conventional or empirical from the enlightened perspective, but experiencing everything from the other's--nonenlightened--perspective. (Karmapa Mikyo Dorje, 2006: 141-42ff) Finally, on the relationship of the two truths, Kagyu maintains a position which is consciously ambiguous--that they are expressible neither as identical nor distinct. Responding to an interlocutory question, "Are the two truths identical or distinct?" Karmapa Mikyo Dorje, says, "neither is the case." And he advances three arguments to support this position. First, "From a perspective of the childish beings, [the two truths] are neither identical, for they do not see the ultimate; nor are they distinct, for they do not see them separately" (Karmapa Mikyo Dorje, 2006: 285). Second, "From standpoint of the meditative equipoise of the exalted beings, the two truths are neither identical, for the appearance of the varieties of conventional entities do not appear to them. Nor are they distinct, for they are not perceived as distinct" (Karmapa Mikyo Dorje, 2006: 285). The third is the relativity argument which says, "They are all defined relative to one another--unreality is relative to reality and reality is relative to unreality. Because one is defined relative to the other, one to which it is related (ltos sa) could not be identical to that which it relates (ltos chos). This is because it is contradictory for one thing to be both that which relates (ltos chos) and the related (ltos sa). Nor is it the case that they are distinct because," according to Kagyu's view, "when the related is not established, so is the other [i.e., one that relates] not established. Hence there is no relation. If one insists that relation is still possible, then such a relation would not be relative to another" (Karmapa Mikyo Dorje, 2006: 285-86). So from these three arguments, Kagyu concludes that the two truths are neither expressible as identical nor distinct. It argues that "just as the conceptual images of a golden vase and a silver vase do not become distinct on the account of them not being expressed as identical. Likewise these images do not become identical on the account of them not being expressed as distinct" (Mikyo Dorje, 2006: 286). ## 3. Sakya Sakya's theory of two truths is defended in the works of the succession of Sakya scholars--Sakya Pandita (1182-1251), Rongton Shakya Gyaltsen (Rong ston Sakya rgyal tshan, 1367-1449), the translator Taktsang Lotsawa (Stag tsang Lo tsa ba, 1405-?), Gorampa Sonam Senge (Go rams pa Bsod nams seng ge, 1429-89) and Shakya Chogden (Sakya Mchog ldan, 1428-1509). In particular Gorampa Sonam Senge's works are unanimously recognised as the authoritative representation of the Sakya's position. Sakya agree with Nyingma and Kagyu in maintaining that the distinction between the two truths is merely subjective processes, and that the two truths are reducible to the two conflicting perspectives (Sakya Pandita 1968a: 72d, Rongton Shakya Gyaltsen n.d. 7f, Taktsang Lotsawa n.d.: 27, Shakya Chogden 1975a: 3-4ff, 15f). "Although there are not two truths in terms of the object's ontological mode of being (gnas tshul), the truths are divided into two in terms of [the contrasting perspectives of] the mind that sees the mode of existence and the mind that does not see the mode of existence...This makes perfect sense" (Gorampa 1969a: 374ab). Since it emphasizes the subjective nature of the distinction between the two truths, it proposes "mere mind" (blo tsam) to be the basis of the division (Gorampa 1969a: 374ab). It aruges that "Here in the Madhyamaka system, the object itself cannot be divided into two truths. Conventional truth and ultimate truth are divided in terms of the modes of apprehension (mthong tshul)--in terms of the subject apprehending unreality and the subject apprehending reality; or in terms of mistaken and unmistaken apprehensions ('khrul ma 'khrul); or deluded or undeluded apprehensions (rmongs ma rmongs); or erroneous or nonerroneous apprehensions (phyin ci log ma log); or reliable cognition or unreliable cognitions (tshad ma yin min)" (Gorampa 1969a: 375b). He also adds that: "The position which maintains that the truths are divided in terms of the subjective consciousness is one that all Prasangikas and Svatantrikas of India unanimously accepted because they are posited in terms of the subjective cognitive processes depending on whether it is deluded (rmongs) or nondeluded (ma rmongs), a perception of unreality (brdzun pa thong ba) or a perception of reality (yang dag mthong ba), and mistaken (khrul) or incontrovertible (ma khrul) (1969a: 384c). Sakya's advances two reasons to support its position. First, since the minds of ordinary beings are always deluded, mistaken, and erroneous, they falsely experience conventional truth. Conventional truth is thus posited only in relation to the perspective of the ordinary beings.[2] The wisdom of exalted meditative equipoise is however never mistaken, it is always nondeluded, and nonerroneous, hence exalted beings flawlessly experience ultimate truth. Ultimate truth is thus posited strictly in relation to the cognitive perspective of exalted beings. Second, the view held by Sakya argues for separate cognitive agents corresponding to each of the two truths. Ordinary beings have direct knowledge of conventional truth, but are utterly incapable of knowing ultimate truth. The exalted beings in training have direct knowledge of the ultimate in their meditative equipoise and direct knowledge of the conventional truth in the post meditative states. Fully awakened buddhas, on the other hand, only have access to ultimate truth. They have no access to conventional truth whatsoever from the enlightened perspective, although they may access conventional truth from the perspective of ordinary deluded beings. Etymologically Sakya characterizes ultimate (parama) as a qualification of transcendent exalted cognitive process ('jig rten las 'das pa'i ye shes, lokottarajnana) that belongs to exalted beings with artha as its corresponding object. The sense of ultimate truth (paramarthasatya) grants primacy to the exalted transcendent cognitive process of the noble beings, which supersedes the ontological status of conventional phenomena (Gorampa 1969a: 377d). "There is no realization and realized object, nor is there object and subject" (Gorampa 1969b: 714f, 727-729). As Taktsang Lotsawa puts it: "A wisdom without dual appearance is without any object" (Taktsang n.d.: 305f). Strictly speaking, transcendent wisdom itself becomes the ultimate truth. "Ultimate truth is to be experienced under a total cessation of dualistic appearance through exalted personal wisdom," and further, "Anything that has dualistic appearance, even omniscience,must not be treated as ultimate truth" (Gorampa 1969b: 612-13ff). Therefore conventionality (samvrti, kun rdzob) means primal ignorance (Gorampa, 1969b: 377b, Sakya Pandita 1968a: 72b, Shakya Chogden 1975a: 30f, Rongton Shakya Gyaltsen 1995: 288). In agreement with Nyingma and Kagyu, Sakya treats primal ignorance as to the villain responsible for projecting the entire system of conventional truths. It argues ignorance as constituting the defining characteristics of the conventional truth.[3] (Sakya Pandita 1968a: 72, Rendawa 1995: 121, Shakya Chogden 1975b: 378f, 1975c: 220, Taktsang Lotsawa n.d.: 27f, Rongton Shakya Gyaltsen 1995: 287, n.d: 6-7ff) It formulates the definitions of the two truths in terms of the distinctions between the ignorant experiences of ordinary beings and the enlightened experiences of noble beings during their meditative equipoise. Sakya's definition is based on three reasons. First, each conventionally real phenomenon satisfies only the definition of conventional truth because each phenomenon has only a conventional nature (in contrast to the two nature theory of Gelug), and that ultimate truth has a transcendent ontological status. Second, that each cognitive agent is potentially capable of knowing only one truth exhaustively, being equipped with either the requisite conventionally reliable cognitive process or ultimately reliable cognitive process. Each truth must be verified by a different individual and that access to the two truths is mutually exclusive--a cognitive agent who knows conventional truth cannot know ultimate truth and vice versa except in the case of exalted beings who are not fully enlightened. Third, that each conventional entity has only one nature, namely, its conventional nature, and the so-called ultimate nature must not be associated with any conventionally real phenomenon. If a sprout, for example, actually did possess two natures as proposed in Gelug's theory of the two truths, then, according to Sakya, each nature would have to be ontologically distinct. Since the ontological structure of the sprout cannot be separated into a so-called conventional and ultimate nature, the sprout must possess only one phenomenal nature, i.e., the conventional reality. As this nature is found only under the spell of ignorance, it can be comprehended only under the conventional cognitive processes of ordinary beings, and of unenlightened noble beings in their post meditative equipoise. It is therefore not possible, in Sakya's view, to confine the definition of ultimate truth to the framework of conventional phenomena. Ultimate truth, on the other hand, requires the metaphysical transcendence of conventionality. Unlike conventional reality, it is neither presupposed nor projected by ignorance. Ultimate truth, in Gorampa's words: "is inexpressible through words and is beyond the scope of cognition" (1969a: 370a). The cognition is always conceptual and thus deluded. "Yet ultimate truth is experienced by noble beings in their meditative equipoise, and is free from all conceptual categories. It cannot be expressed through definition, through any defined object, or through anything else" (1969a: 370a). In fact, Sakya goes so far as to combine the definition of ultimate truth with that of intrinsic reality (rang bzhin, svabhava). When an interlocutor ask this question: "What is the nature of the reality of phenomena?" Gorampa replies that reality is transcendent and it has three defining characteristics: namely: "It is not created by causes and conditions, it exists independently of conventions and of other phenomena; and it does not change" (Gorampa, 1969c: 326a). Like Nagarjuna's hypothetical, not real, intrinsic reality, Sakya claims that ultimate truth is ontologically unconditioned, and hence it is not a dependently arisen phenomenon; it is distinct from conventional phenomena in every sense of the word; it is independent of conceptual-linguistic conventions; it is a timeless and unchanging phenomenon. It is clear then, according to Sakya view, any duality ascribed to truth is untenable. Since there is only one truth, it cannot be distinguished any further. Like Nyingma and Kagyu, Sakya holds the view that truth itself is not divisible (Sakya Pandita n.d.: 32ab, Rendawa 1995: 122, Rongton Shakya Gyaltsen, 1995: 287, n.d.: 22f Taktsang Lotsawa n.d.: 263, Shakya Chogden 1975a: 7-8ff, 1975c: 222f). It agrees with Nyingma and Kagyu that the distinction between the two truths is essentially between two conflicting perspectives, rather than any division within truth as such. In Shakya Chogden's words: "Precise enumeration (grangs nges) of the twofold truth all the earlier Tibetans have explained rests on the precise enumeration of the mistaken cognition (blo 'khrul) and unmistaken cognition (blo ma 'khrul). With this underpinning reason, they explained the precise enumeration through the elimination of the third alternative. There is not even a single figure to be found who claims the view comparable with the latter [i.e., Gelug], who asserts a precise enumeration of the twofold truth based on the certification of reliable cognitive processes" (1975a: 9-10ff). Sakya rejects the reality of conventional truth by treating it as a projection of conventional mind--it is the ignorance of ordinary beings. When asked this question: "If this were true, even the mere term conventional truth would be unacceptable, for whatever is conventional is incompatible with truth," Gorampa Replies: "Since [conventional] truth is posited only in relation to a conventional mind, there is no problem. Even so-called real conventionalities (yang dag kun rdzob ces pa yang) are posited as real with respect to a conventional mind" (1969b: 606b). By "conventional mind," Gorampa means the ignorant mind of an ordinary being experiencing the phenomenal world. In other words, conventional truth is described as "truth" only from the perspective of ignorance. It is a truth projected (sgro brtag pa) by it and taken for granted. Sakya equates conventional truth with "the appearances of nonexistent entities like illusions" (Gorampa 1969c: 287c). It follows Sakya Pandita on this point who puts it: "Conventional truths are like reflections of the moon in the water--despite their nonexistence, they appear due to thoughts" (Sakya Pandita 1968a: 72a). According to Sakya Pandita "The defining characteristic of conventional truth constitutes the appearances of the nonexistent objects" (1968a: 72a). In this sense, conventional truths "are things apprehended by the cognition perceiving conventional entities. Those very things are found as nonexistent by the cognition analyzing their mode of existence that is itself posited as the ultimate" (Gorampa 1969a: 377a). Since mere mind is the basis of the division of the two truths wherein ultimate truth--wisdom--alone is seen as satisfying the criterion of truth, so conventional truth--ignorance--cannot properly be taken as truth. Wisdom and ignorance are invariably contradictory, and thus the two truths cannot coexist. Sakya argues, in fact, that conventional truth must be negated in the ascent to ultimate truth. Given wisdom's primacy over ignorance, in the final analysis it is ultimate truth alone that must prevail without its merely conventional counterpart. Conventional truth is an expedient means to achieve ultimate truth, and the Buddha described conventional truth as truth to suit the mentality of ordinary beings (Gorampa 1969a: 370b). The two truths are thus categorized as a means (thabs) and a result (thabs byung). Conventional truth is the means to attain the one and only ultimate truth. According to this view, then, the relationship between the two truths is equivalent to the relationship between the two conflicting perspectives--namely, ignorance and wisdom. The question now arises: How is ignorance related to wisdom? Or conversely, how does wisdom relate to ignorance? It says that the two truths are distinct in the sense that they are incompatible with unity, like entity and without entity. In the ultimate sense, it argues, the two truths transcend identity and difference (Gorampa, 1969a: 376d). The transcendence of identity and difference from the ultimate standpoint is synonymous with the transcendence of identity and difference from the purview of the meditative equipoise of noble beings. However, from the conventional standpoint, it claims that the two truths are distinct in the sense that they are incompatible with their unity. It likens this relationship to the one between entity and without entity (Gorampa, 1969a: 377a). Sakya's claim that the two truths are distinct and incompatible encompasses both ontological and epistemological distinctions. Since what is divided into the two truths is mere mind, it is obvious that there is no single phenomenon that could serve as the objective referent for both. This also means that the two truths must be construed as corresponding to distinct spheres belonging to distinct modes of consciousness: conventional truth corresponds to ignorance and ultimate truth to wisdom. It is thus inappropriate to describe the relationship between the two truths, and their corresponding modes of consciousness, in terms of two ways of perceiving the same entity. Although the two truths can be thought of as two ways of perceiving, one based on ignorance and the other on wisdom, there is no same entity perceived by both. There is nothing common between the two truths, and if they are both ways of perceiving, then they do not perceive the same thing. According to this view, the relationship between conventional truth and ultimate truth is analogous to the relationship between the appearance of falling hairs when vision is impaired by cataracts and the absence of such hairs when vision is unimpaired. Although this is a metaphor, it has a direct application to determining the relationship between the two truths. Conventional truth is like seeing falling hairs as a result of cataracts: both conventional truth and such false seeing are illusory, in the ontological sense that there is nothing to which each corresponds, and in the epistemological sense that there is no true knowledge in either case. Ultimate truth is therefore analogous, ontologically and epistemologically, to the true seeing unimpaired by cataracts and free of the appearance of falling hairs. Just as cataracts give rise to illusory appearances, so ignorance, according to Sakya, gives rise to all conventional truths; wisdom, on the other hand, gives rise to ultimate truth. As each is the result of a different state, there is no common link between them in terms of either an ontological identity or an epistemological or conceptual identity. ## 4. Gelug Gelug's (Dge lugs) theory of the two truths is campioned by Tsongkhapa Lobsang Dragpa (Tsong khapa Blo bzang grags pa, 1357-1419). Tsongkhapa's theory is adopted, expanded and defended in the works of his immediate disciples--Gyaltsab Je (Rgyal tshab Rje, 1364-1432), Khedrub Je (Mkhas grub Rje, 1385-1438), Gendun Drub (Dge 'dun grub,1391-1474)--and other great Gelug thinkers such as Sera Jetsun Chokyi Gyaltsen (Se ra Rje tsun Chos kyi rgyal tshan, 1469-1544), Panchen Sonam Dragpa (Pan chen Bsod nams grags pa,1478-1554), Panchen Lobsang Chokyi Gyaltsen (Pan chen Blo bzang chos kyi rgyal tshan,1567-1662), Jamyang Shepai Dorje ('Jam dbyangs Bzhad ba'i Rdo rje, 1648-1722), Changkya Rolpai Dorje (Lcang skya Rol pa'i rdo rje, 1717-86), Konchog Jigme Wangpo (Kon mchog 'jigs med dbang po, 1728-91), and many others scholars. Gelug maintains that "objects of knowledge" (shes bya) are the basis for dividing the two truths (Tsongkhapa 1984b:176).[4] This means that the two truths relate to "two objects of knowledge," the idea it takes from the statement of the Buddha from Discourse on Meeting of the Father and the Son (Pitaputrasamagama Sutra)[5] By object of knowledge Gelug means an object that is cognizable (blo'i yul du bya rung ba). It must be an object of cognitive processes in general ranging from those of ordinary sentient beings through to those of enlightened beings. This definition attempts to capture any thing knowable in the broadest possible sense. Since the Buddha maintains knowledge of the two truths to be necessary for awakening, the understanding of the two truths must constitute an exhaustive understanding of all objects of knowledge. Gelug's key argument to support this claim comes from its two-nature theory in which it has been argued that every nominally (tha snyad) or conventionally (kun rdzob) given phenomenon possesses dual natures: namely, the nominal (or conventional nature) and the ultimate nature. The conventional nature is unreal and deceptive while the ultimate nature is real and nondeceptive. Since two natures pertain to every phenomenon, the division of the two truths means the division of each entity into two natures. Thus the division of the two truths, "reveals that it makes sense to divide even the nature of a single entity, like a sprout, into dual natures--its conventional and its ultimate natures. It does not however show," as non-Gelug schools have it, "that the one nature of the sprout is itself divided into two truths in relation to ordinary beings (so skye) and to noble beings (aryas)" (Tsongkhapa 1984b: 173, 1992: 406). The relation of the two truths comes down to the way in which the single entity appears to cognitive processes--deceptively and nondeceptively. The two natures correspond to these deceptive or nondeceptive modes of appearance. While they both belong to the same ontological entity, they are epistemically or conceptually mutually exclusive. Take a sprout for instance. If it exists, it necessarily exhibits a dual nature, and yet those two natures cannot be ontologically distinct. The ultimate nature of the sprout cannot be separate from its conventional nature--its color, texture, shape, extension, and so on. As an object of knowledge, the sprout retains its single ontological basis, but it is known through its two natures. These two natures exclude one another so far as knowledge is concerned. The cognitive process that knows the deceptive conventional nature of the sprout does not have direct access to its nondeceptive ultimate nature. Similarly, the cognitive process that apprehends the nondeceptive ultimate nature of the sprout does not have direct access to its deceptive conventional nature. In Newland's words: "A table and its emptiness are a single entity. When an ordinary conventional mind takes a table as its object of observation, it sees a table. When a mind of ultimate analysis searches for the table, it finds the emptiness of the table. Hence, the two truths are posited in relation to a single entity by way of the perspectives of the observing consciousness. This is as close as Ge-luk-bas will come to defining the two truths as perspectives" (1992: 49). Gelug's two-nature theory not only serves as the basic reference point for its exposition of the basis of the division of the two truths, their meanings and definitions, but also serves as the basic ontological reference for its account of the relationship between the two truths. Therefore Gelug proposes the view that the two truths are of single entity with distinct conceptual identities. This view is also founded on the theory of the two natures. But how are the two natures related? Are they identical or distinct? For Gelug argues that there are only two possibilities: either the two natures are identical (ngo bo gcig) or distinct (ngo bo tha dad); there cannot be a third (Tsongkhapa 1984b:176). They are related in terms of being a single entity with distinct conceptual identities--thus they are both the same and different. Since the two natures are the basis of the relationship between the two truths, the relationship between the two truths will reflect the relationship between the two natures. Just as the two natures are of a same entity, ultimate truth and conventional truth are of same ontological status. Gelug argues that the relationship between the two truths, therefore, the two natures, is akin to the relationship between being conditioned and being impermanent (Tsongkhapa 1984b:176). It appropriates this point from Nagarjuna's Bodhicittavivarana, which states: "Reality is not perceived as separate from conventionality. The conventionality is explained to be empty. Empty alone is the conventionality, if one of them does not exist, neither will the other, like being conditioned and being impermanent" (v.67-68) (Nagarjuna 1991:45-45, cited Tsongkhapa, 1984b: 176; Khedrub Je 1992: 364). Commenting on this passage from the Bodhicittavivarana, Tsongkhapa argues that the first four lines (in original Tibetan verses) show that things as they really are, are not ontologically distinct from that of the conventionality. The latter two lines (in the original Tibetan verse) show their relationship such that if one did not exist, neither could the other (med na mi 'bung ba'i 'brel ba). This, in fact, he says, is equivalent to their being constituted by a single-property relationship (bdag cig pa'i 'brel ba). Therefore, like the case of being conditioned and being impermanent, the relation between the two truths is demonstrated as one of a single ontological identity (Tsongkhapa 1984b: 176-77). The way in which the two truths are related is thus analogous to the relationship between being conditioned and being impermanent. They are ontologically identical and mutually entailing. Just as a conditioned state is not a result of impermanence, so emptiness is not a result of the conventional truth (the five aggregates) or the destruction of the five aggregates--Hence in the Vimalakirtinirdesa Sutra it is stated: "Matter itself is void. Voidness does not result from the destruction of matter, but the nature of matter is itself voidness" (Vimalakirti, 1991:74). The same principle applies in the case of consciousness and the emptiness of consciousness, as well as to the rest of the five psychophysical aggregates--the aggregate and its emptiness are not causally related. For the causal relationship would imply either the aggregate is the cause, therefore its emptiness is the result, or the aggregate is the result, and its emptiness the cause. This would imply, according to Gelug's reading, either the aggregate or the emptiness is temporally prior to its counterpart, thus leading to the conlusion that the conventional truth and ultimate truth exist independently of each other. Such a view is for Gelug is completely unacceptable. The ontological identity between being conditioned and being impermanent does not imply identity in all and every respect. Insofar as their epistemic mode is concerned, conditioned and impermanent phenomena are distinct and contrasting. The concept impermanence always presents itself to the cognizing mind as momentary instants, but not as conditioned. Similarly, the concept being-conditioned always presents itself to its cognizing mind as constituted by manifold momentary instants, but not as moments. Thus it does not necessarily follow that the two truths are identical in every respect just because they share a common ontological identity. Where the modes of conceptual appearance are concerned, ultimate nature and conventional nature are distinct. The conceptual mode of appearance of ultimate truth is nondeceptive and consistent with its mode of existence, while that of conceptual mode of conventional truth is deceptive and inconsistent with its mode of existence. Conventional truth is uncritically confirmed by conventionally reliable cognitive process, whereas ultimate truth is critically confirmed by ultimately reliable cognitive process. Hence, just as ultimate truth is inaccessible to the conventionally reliable cognitive process for its uncritical mode of engagement, so, too, is conventional truth inaccessible to ultimately reliable cognitive process for its critical mode of engagement. This is how, in Gelug's view, the truths differ conceptually despite sharing a common ontological entity. In summarizing Gelug's argument, Khedrub Je writes: "The two truths are therefore of the same nature, but different conceptual identities. They have a single-nature relationship such that, if one did not exist, neither could the other, just like being conditioned and impermanent" (1992: 364). The two nature theory also supports Gelug's view that the truth is twofold. Since the two natures of every conventional phenomenon provide the ontological and epistemological foundation for each of the truths, the division of truth into two is entirely appropriate. Both the conventional and the ultimate are actual truths, and since the two natures are mutually interlocking, neither of the two truths has primacy over the other--both have equal status, ontologically, epistemologically, and even soteriologically. Given Gelug's stance on conventional truth as actual truth and its argument for the equal status of the two truths, Gelug must now address the question: How can conventional truth, which is unreal (false) and deceptive, be truth (real) at all? In other words, how can the two truths be of equal status given if conventional truth is unreal (false)? The success of Gelug's reply depends on its ability to maintain harmony between the two truths, or their equal footing.There are several arguments by means of which Gelug defends this. The first, and the most obvious one, is the argument from the two-nature theory. It argues since two truths are grounded in the dual nature of a single phenomenon, then "just as the ultimate reality of the sprout [for instance] is taken as characteristic of the sprout, hence it is described as the sprout's nature, so, too," explains Tsongkhapa, "are the sprout's color, shape, etc., the sprout's characteristics. Therefore they too are its nature" (1992: 406). Since the two natures are ontologically mutually entailing, the sprout's ultimate truth cannot exist ontologically separate from its conventional truth, and vice versa. Neither truth could exist without the other. The most important argument Gelug advances for the unity of the two truths draws upon an understanding of the compatible relation between conventional truth and dependent arising on the one hand, and between ultimate truth and emptiness on the other. For Gelug emptiness and dependent arising are synonymous. The concept of emptiness is incoherent unless it means dependent arising, and equally the concept of dependent arising is incoherent unless it means emptiness of intrinsic reality. Gelug argues, as Tsongkhapa does in the Rten 'brel stod pa (In Praise of Dependent Arising, 1994a), since emptiness means dependent arising, the emptiness of intrinsic reality and the efficacy of action and its agent are not contradictory. If emptiness, however, is misinterpreted as contradictory with dependent arising, then Gelug contends, there would be neither action in empty phenomena, nor empty phenomena in action. But that cannot be the case, for that would entail a rejection of both phenomena and action, a nihilistic view (v.11-12). Since there is no phenomenon other than what is dependently arisen, there is no phenomenon other than what is empty of intrinsic reality (v.15). Understood emptiness in this way, there is indeed no need to say that they are noncontradictory--the utter nonexistence of intrinsic reality and making sense of everything in the light of the principle this arises depending on this (v.18). This is so because despite the fact that whatever is dependently arisen lacks intrinsic reality, and therefore empty, nonetheless its existence is illusion-like (v.27). Sometimes Gelug varies its ontological argument slightly, as does Tsongkhapa in the Lam gtso rnam gsum (The Three Principal Pathways), to stressing the unity of the two truths in terms of their causal efficiency. It argues that empty phenomena is causally efficient is crucial in understanding the inextricable relationship between ultimate truth and conventional truth. The argument takes two forms: (1) appearance avoids realism--the extreme of existence, and (2) emptiness avoids inhilism--the extreme of nonexistence. The former makes sense for Gelug because the appearance arises from causal conditions, and whatever arises from the causal conditions is a non-eternal or non-permanent. When the causal conditions changes any thing that dependently arises from them also changes. Thus the appearance avoids the realism. The latter makes sense because the empty phenomena arises from causal conditions, and whatever arise from the causal conditions is not a non-existent, even though it lacks intrinsic reality. Only when the causal conditions are satisfied do we see arising of the empty phenomena. Hence by understanding that the empty phenomenon itself is causally efficient, the bearer of cause and effect, one is not robbed by the extreme view of nihilism (1985: 252). The argument for the unity of the two truths also takes epistemological form which also rests on the idea that emptiness and dependently arising are unified. The knowledge of empty phenomena is conceptually interlinked with that of dependently arisen phenomena--the latter is, in fact, founded on the former. To the extent that empty phenomena are understood in terms of relational and dependently arisen phenomena, to that extent empty phenomena are always functional and causally effective. The phrase "empty phenomena," although expressed negatively, is not negative in a metaphysical sense--it is not equivalent to no-thingness. Although the empty phenomenon appears to its cognizing consciousness negatively and without any positive affirmation, it is nonetheless equivalent to a relational and dependently arisen phenomenon seen deconstructively. Since seeing phenomena as empty does not violate the inevitable epistemic link with the understanding of phenomena as dependently arisen, and the converse also applies, so the unity between the two truths--understanding things both as empty and as dependently arisen--is still sustained. The unity between the two truths, according to Gelug, does not apply merely to ontological and epistemological issues; it applies equally to soteriology--the practical means to the freedom from suffering. As Jamyang Shepai Dorje argues that undermining either of the two truths would result in a similar downfall--a similar eventual ruin. If, however, they are not undermined, the two are alike insofar as the accomplishment of the two accumulations and the attainment of the two awakened bodies (kayas), and so forth, are concerned. If one undermines conventional truth or denies it's reality, one would succumb to the extreme of nihilism, which would also undermine the fruit and the means by which an awakened physical body (rupakaya) is accomplished. It is therefore not sensible to approach the two truths with bias. Since this relation continues as a means to avoid falling into extremes, and also to accomplish the two accumulations and attain the two awakened bodies (kayas), it is imperative, says Jamyang Shepai Dorje, that the two truths be understood as mutually interrelated (1992: 898-99). One could object Gelug's position as follows: If the two natures are ontologically identical, why is conventional truth unreal and deceptive, while ultimate truth real and nondeceptive? To this Gelugs replies that "nondeceptive is the mode of reality (bden tshul) of the ultimate. That is, ultimate truth does not deceive the world by posing one mode of appearance while existing in another mode" (1992: 411). Ultimate truth is described as ultimate, not because it is absolute or higher than conventional truth, but simply because of its consistent character, therefore non-deceptive--its mode of appearance and its mode of being are the same--in contrast with the inconsistent (therefore deceptive) character of conventional truth. Ultimate truth is nondeceptive for the same reason. The premise follows because to the cognizing consciousness, conventional truth presents itself as inherently or instrinsically real. It appears to be substance, or essence, and therefore it deceives ordinary beings. Insofar as conventional truth presents itself as more than conventional--as inherently real--they deceive the ordinary beings. We take them to be what they are not--to be intrinsically and objectively real. In that sense, they are unreal. "But to the extent that we understand them as dependently arisen, empty, interdependent phenomena," as Garfield explains, "they constitute a conventional truth" (1995: 208). Another objection may be advanced as follows: Gelug's position contradicts the Buddha's teachings since it is not possible to reconcile Gelug's view that there are two actual truths with the Buddha's declaration that nirvana is the only truth.[6] For its answer to this objection Gelug appropriates Candrakirti's Yuktisastikavrtti which states that nirvana is not like conditioned phenomena, which deceives the childish by presenting false appearances. For the existence of nirvana is always consistent with its characteristic of the nonarising nature. Unlike conditioned phenomena, it never appears, even to the childish, as having a nature of arising (skye ba'i ngo bo). Since nirvana is always consistent with the mode of existence of nirvana, it is explained as the noble truth. Yet this explanation is afforded strictly in the terms of worldly conventions (1983: 14-15ff, cited in Tsongkhapa 1992: 312, Khedrub Je 1992: 360). For Gelug the crucial point here, as Candrakirti emphasizes, is that nirvana is the truth strictly in terms of worldly conventions. Gelug recognizes this linguistic convention as a highly significant to the Prasangika system and it insists on conformity with the worldly conventions for the the following reasons. First, just as an illusion, a mirror image, etc., are real in an ordinary sense, despite the fact that they are deceptive and unreal, so, too, conventional phenomena in the Prasangika Madhyamaka sense are conventionally real, and can even be said to constitute truths, despite being recognized by the Madhyamikas themselves as unreal and deceptive. Second, because the concept ultimate truth is also taken from its ordinary convention, nirvana is spoken of as ultimate on the ground of its nondeceptive nature, in the sense that its mode of existence is consistent with its mode of appearance. The nondeceptive nature of the empty phenomenon itself constitutes its reality, and so it is conventionally described as ultimate in the Prasangika system. Thus Gelug asserts that neither of the two truths is more or less significant than the other. Indeed, while the illusion only makes sense as illusion in relation to that which is not illusion, the reflection only makes sense as reflection in relation to that which is reflected. So, too, does the real only make sense as real in relation to the illusion, the thing reflected in relation to its reflection. This also holds in the case of discussions about the ultimate nature of things, such as the being of the sprout--it only makes sense inasmuch as it holds in discussions of ordinary phenomena. The only criterion that determines a thing's truth in the Prasangika Madhyamaka system, according to Gelug, is the causal effectiveness of the thing as opposed to mere heuristic significance. The sprout's empty mode of being and its being as appearance are both truths, insofar as both are causally effective, and thus both functional. The two truths, understood as, respectively, the empty and the dependently arisen characters of phenomena, are on equal footing according to Gelug. Nevertheless these truths have different designations--the sprout's empty mode is always described as "ultimate truth," while the conventional properties, such as color and shape, are described as "conventional truths." The former is accepted as nondeceptive truth while the sprout's conventional properties are accepted as deceptive or false truth, despite common sense dictating that they are true and real. In conclusion Gelug's theory of the two truths is based on one fundamental thesis that each conventionally real phenomenon satisfies the definitions of both truths for each phenomenon, as it sees it, possesses two natures that serve as the basis of the definitions of the two truths. The two truths are conceptual distinctions applied to a particular conventionally real phenomenon, and every conventionally real phenomenon fulfills the criterion of both truths because each phenomenon constitutes these two natures they are not merely one specific nature of a phenomenon mirrored in two different perspectives. As each phenomenon possesses two natures, so each verifying cognitive process has a different nature as its referent, even though there is only one ontological entity and one cognitive agent involved. ## 5. Implications Gelug considers the two natures of each phenomenon as the defining factor of the two truths. It argues that the conventional nature of an entity, as verified by a conventionally reliable cognitive process, determines the defining criterion of conventional truth; the ultimate nature of the same entity, as verified by an ultimately reliable cognitive practice, determines the defining criterion of ultimate truth. Since both truths are ontologically as well as epistemologically interdependent, knowledge of conventionally real entitity as dependently arisen suffices for knowledge of both truths. In contasty non-Gelug schools--Nyingma, Kagyu and Sakya Non-Gelug, as we have seen, rejects Gelug's dual-nature theory, treating each conventional entity as satisfying only the definition of conventional truth and taking the definition of ultimate truth to be ontologically and epistemologically transcendent from conventional truth. They argue, instead, it is through the perspectives of either an ordinary being or an unenlightened exalted being (aryas) that the definition of conventional truth is verified--fully enlightened being (buddhas) do not experience the conventional truth in any respect. Similarly, for non-Gelug, no ordinary being can experience the ultimate truth. Ultimate truth transcends conventional truth, and the knowledge of empirically given phenomena as dependently arisen could not satisfy the criterion of knowing ultimate truth. For Gelug, there is an essential compatibility between between the two truths, for the reason that there is a necessary harmony between dependently arisen and emptiness of intrinsic reality. As dependently arisen, empty phenomena are not constructions of ignorant consciousness, so neither is conventional truth such a construction. Both truths are actual truths that stand on an equal footing. Moreover, according to this view, whosoever knows conventional truth, either directly or inferentially, also knows ultimate truth; whosoever knows ultimate truth, also knows phenomena as dependently arisen, and hence knows them as empty of intrinsic reality. Where there is no knowledge of conventional truth, the converse applies. For non-Gelug, the incommensurability between dependently arisen and emptiness of intrinsic reality also applies to the two truths. Accordingly, whosoever knows conventional truth does not know ultimate truth, and one who knows ultimate truth does not know conventional truth; whosoever knows phenomena as dependently arisen does not know them as empty, whereas whosoever knows phenomena as empty does not know them as dependently arisen. While Gelug thus distances itself from the subjective division of the two truths, Nyingma, Kagyu and Sakya attempt to demonstrate the validity of their view by arguing that perspective provides the primary basis for the division of the two truths. Unlike Gelug, non-Gelug schools hold that the two truths do not have any objective basis. Instead they are entirely reducible to the experiences of the deluded minds of ordinary beings and the experiences of the wisdom of exalted being. According to Gelug, the agent who cognizes the two truths may be one and the same individual. Each agent may have all the requisite cognitive resources that are potentially capable of knowing both truths. Ordinary beings have only conceptual access to ultimate truth, while exalted beings, who are in the process of learning, have direct, but intermittent, access. Awakenened beings, however, invariably have simultaneous access to both truths. The view held by non-Gelug argues for separate cognitive agents corresponding to each of the two truths. Ordinary beings have direct knowledge of conventional truth, but are utterly incapable of knowing ultimate truth. The exalted beings in training directly know ultimate while they are meditative equipoise and conventional truth in post meditative states. Fully awakened buddhas, on the other hand, only have access to ultimate truth. Awakened beings have no access to conventional truth whatsoever from the enlightened perspective, although they may access conventional truth from unenlightened ordinary perspectives.
types-tokens	## 1. The Distinction Between Types and Tokens ### 1.1 What the Distinction Is The distinction between a type and its tokens is an ontological one between a general sort of thing and its particular concrete instances (to put it in an intuitive and preliminary way). So for example consider the number of words in the Gertrude Stein line from her poem Sacred Emily on the page in front of the reader's eyes: > > > Rose is a rose is a rose is a rose. In one sense of 'word' we may count three different words; in another sense we may count ten different words. C. S. Peirce (1931-58, sec. 4.537) called words in the first sense "types" and words in the second sense "tokens". Types are generally said to be abstract and unique; tokens are concrete particulars, composed of ink, pixels of light (or the suitably circumscribed lack thereof) on a computer screen, electronic strings of dots and dashes, smoke signals, hand signals, sound waves, etc. A study of the ratio of written types to spoken types found that there are twice as many word types in written Swedish as in spoken Swedish (Allwood, 1998). If a pediatrician asks how many words the toddler has uttered and is told "three hundred", she might well enquire "word types or word tokens?" because the former answer indicates a prodigy. A headline that reads "From the Andes to Epcot, the Adventures of an 8,000 year old Bean" might elicit "Is that a bean type or a bean token?". ### 1.2 What It Is Not Although the matter is discussed more fully in SS8 below, it should be mentioned here at the outset that the type-token distinction is not the same distinction as that between a type and (what logicians call) its occurrences. Unfortunately, tokens are often explained as the "occurrences" of types, but not all occurrences of types are tokens. To see why, consider this time how many words there are in the Gertrude Stein line itself, the line type, not a token copy of it. Again, the correct answer is either three or ten, but this time it cannot be ten word tokens. The line is an abstract type with no unique spatio-temporal location and therefore cannot consist of particulars, of tokens. But as there are only three word types of which it might consist, what then are we counting ten of? The most apt answer is that (following logicians' usage) it is composed of ten occurrences of word types. See SS8 below, Occurrences, for more details. ## 2. Importance and Applicability of the Distinction ### 2.1 Linguistics It is generally accepted that linguists are interested in types. Some, e.g. Lyons (1977, p. 28), claim the linguist is interested only in types. Whether this is so or not, linguists certainly appear to be heavily committed to types; they "talk as though" there are types. That is, they often quantify over types in their theories and refer to them by means of singular terms. As Quine has emphasized, a theory is committed to the existence of entities of a given sort if and only if they must be counted among the values of the variables in order for the statements in the theory to be true. Linguistics is rife with such quantifications. For example, we are told that an educated person's vocabulary is about 15,000 words, but Shakespeare's was more nearly 30,000. These are types, as are the twenty-six letters of the English alphabet, and its eighteen cardinal vowels. (Obviously the numbers would be much larger if we were counting tokens). Linguists also frequently refer to types by means of singular terms. According to the O.E.D., for example, the noun 'color'' is from early modern English and in addition to its pronunciation [ka' l@r] has two "modern current or most usual spellings" [colour, color], eighteen earlier spellings [collor, collour, coloure, colowr, colowre, colur, colure, cooler, couler, coullor, coullour, coolore, coulor, coulore, coulour, culler, cullor, cullour] and eighteen senses (vol. 2, p. 636). According to Webster's, the word 'schedule' has four current pronunciations: ['ske-(,)ju(@)l], ['ske-j@l] (US), ['she-j@l] (Can) and ['she-(,)dyu(@)l] (Brit) (p. 1044). Thus, linguistics is apparently committed to the existence of these words, which are types. References to types is not limited to letters, vowels and words, but occur extensively in all branches of linguistics. Lexicography discusses, in terms that make it clear that it is types being referred to, nouns, verbs, words, their stems, definitions, forms, pronunciations and origins. Phonetics is committed to consonants, syllables, words and sound segments, the human vocal tract and its parts (the tongue has five). Phonology also organizes sounds but in terms of phonemes, allophones, alternations, utterances, phonological representations, underlying forms, syllables, words, stress-groups, feet and tone groups. Morphology is apparently committed to morphemes, roots, affixes, and so forth, and syntax to sentences, semantic representations, LF representations, among other things. Clearly, just as words and letters and vowels have tokens, so do all of the other items mentioned (nouns, pronunciations, syllables, tone groups and so forth). It is more controversial whether the items studied in semantics (the meanings of signs, their sense relations, etc.) also come in types and tokens, and similarly for pragmatics (including speaker meanings, sentence meanings, implicatures, presuppositions, etc.) It seems to hinge on whether a mental event (token) or part of it could be a meaning, a matter that cannot be gone into here. See Davis (2003) for a view according to which concepts and thoughts--varieties of meaning--come in types and tokens. It is notable that when one of the above types is defined, it is defined in terms of other types. So for example, sentences might be (partly) defined in terms of words, and words in terms of phonemes. The universal and largely unscrutinized reliance of linguistics on the type-token relationship and related distinctions like that of langue to parole, and competence to performance, is the subject of Hutton's cautionary book (1990). ### 2.2 Philosophy Obviously then, types play an important role in philosophy of language, linguistics and, with its emphasis on expressions, logic. Especially noteworthy is the debate concerning the relation between the meaning of a sentence type and the speaker's meaning in using a token (a relation that figures prominently in Grice 1969). But the type-token distinction also functions significantly in other branches of philosophy as well. In philosophy of mind, it yields two versions of the identity theory of mind (each of which is criticized in Kripke 1972). The type version of the identity theory (defended by Smart (1959) and Place (1956) among others) identifies types of mental events/states/processes with types of physical events/states/processes. It says that just as lightning turned out to be electrical discharge, so pain might turn out to be c-fiber stimulation, and consciousness might turn out to be brain waves of 40 cycles per second. On this type view, thinking and feeling are certain types of neurological processes, so absent those processes, there can be no thinking. The token identity theory (defended by Kim (1966) and Davidson (1980) among others) maintains that every token mental event is some token physical event or other, but it denies that a type match-up must be expected. So for example, even if pain in humans turns out to be c-fiber stimulation, there may be other life forms that lack c-fibers but have pains too. And even if consciousness in humans turns out to be a brain waves that occur 40 times per second, perhaps androids have consciousness even if they lack such brain waves. In aesthetics, it is generally necessary to distinguish works of art themselves (types) from their physical incarnations (tokens). (See, for example, Wollheim 1968, Wolterstorff 1980 and Davies 2001.) This is not the case with respect to oil paintings like the Mona Lisa where there is and perhaps can be only one token, but seems to be the case for many other works of art. There can be more than one token of a sculpture made from a mold, more than one elegant building made from a blueprint, more than one copy of a film, and more than one performance of a musical work. Beethoven wrote nine symphonies, but although he conducted the first performance of Symphony No. 9, he never heard the Ninth, whereas the rest of us have all heard it, that is, we have all heard tokens of it. In ethics, actions are said to be right/wrong--but is it action types or only action tokens? There is a dispute about this. Most ethicists from Mill (1979) to Ross (1988) hold that the hallmark of ethical conduct is universalizability, so that a particular action is right/wrong only if it is right/wrong for anyone else in similar circumstances--in other words, only if it is the right/wrong type of action. If certain types of actions are right and others wrong, then there may be general indefeasible ethical principles (however complicated they may be to state, and whether they can be stated at all). But some ethicists hold that there are no general ethical principles that hold come what may--that there is always some circumstance in which such principles would prescribe the wrong outcome--and such ethicists may go on to add that only particular (token) actions are right/wrong, not types of actions. See, for example, Murdoch 1970 and Dancy 2004. ### 2.3 Science and Everyday Discourse Outside of philosophy and linguistics, scientists often quantify over types in their theories and refer to them by means of singular terms. When, for example, we read that the "Spirit Bear" is a rare white bear that lives in rain forests along the British Columbia coast, we know that no particular bear is rare, but rather a type of bear. When we are told that these Kermode bears "have a mutation in the gene for the melanocortin 1 receptor" (The Washington Post 9/24/01 A16) we know that it is not a token mutation, token gene and token receptor being referred to, but a type. It is even more evident that a type is being referred to when it is claimed that "all men carry the same Y chromosome.... This one and only Y has the same sequence of DNA units in every man alive except for the occasional mutation that has cropped up every thousand years" (The New York Times, Nicholas Wade 5/27/03). Similarly, to say the ivory-billed woodpecker is not extinct is to be referring to a type, a species. (The status of species will be discussed in greater detail in SS4 below.) The preceding paragraph contains singular terms that (apparently) refer to types. An even more telling commitment to types are the frequent quantifications over them. Mayr (1970, p. 233), for example, tells us that "there are about 28,500 subspecies of birds in a total of 8,600 species, an average of 3.3 subspecies per species...79 species of swallows have an average of 2.6 subspecies, while 70 species of cuckoo shrikes average 4.6 subspecies". Although these examples come from biology, physics (or any other science) would provide many examples too. It was claimed (in the sixties), for example, that "there are thirty particles, yet all but the electron, neutrino, photon, graviton, and proton are unstable." Artifactual types (the Volvo 850 GLT, the Dell Latitude D610 laptop) easily lend themselves to reference also. In chess we are told that accepting the Queen's Gambit with 2...dc has been known since 1512, but Black must be careful in this opening--the pawn snatch is too risky. Type-talk is ubiquitous. ## 3. Types and Universals Are types universals? They have usually been so conceived, and with good reason. But the matter is controversial. It depends in part on what a universal is. (See the entry on properties.) Universals, in contrast to particulars, have been characterized as having instances, being repeatable, being abstract, being acausal, lacking a spatio-temporal location and being predicable of things. Whether universals have all these characteristics cannot be resolved here. The point is that types seem to have some, but not all, of these characteristics. As should be clear from the preceding discussion, types have or are capable of having instances, of being exemplified; they are repeatable. To many, this is enough to count as universals. With respect to being abstract and lacking a spatio-temporal location, types are also akin to universals--that is, they are if universals are. On certain views of types and universals, types, unlike their instances, are abstract and lack a spatio-temporal location. On other views, types and universals are in their instances and hence are neither abstract nor acausal; far from lacking a spatio-temporal location, they usually have many. (For more details, see SS5 below, The Relation between Types and Tokens.) So far, then, types appear to be a species of universal, and most metaphysicians would so classify them. (Although a few would not. Zemach (1992), for example, holds that there are no universals, but there are types, which are repeatable particulars--the cat may be in many different places at the same time.) When it comes to being predicable, however, most types diverge from such classic examples of universals as the property of being white or the relation of being east of. They seem not to be predicable, or at least not as obviously so as the classic examples of universals. That is, if the hallmark of a universal is to answer to a predicate or open sentence such as being white answers to 'is white', then most types do not resemble universals, as they more readily answer to singular terms. This is amply illustrated by the type talk exhibited in SS2 above. It is also underscored by the observation that it is more natural to say of a token of a word--'infinity', say--that it is a token of the word 'infinity' than that it is an 'infinity'. That is to say, types seem to be objects, like numbers and sets, rather than properties or relations; it's just that they are not concrete particulars but are general objects--abstract objects, on some views. If, then, we follow Gottlob Frege (1977) in classifying all objects as being the sort of thing referred to by singular terms, and all properties as the sort of thing referred to by predicates, then types would be objects. Hence they would not fall into the same category as the classic examples of universals such as being white and being east of, and thus perhaps should not be considered universals at all. (Although perhaps all this shows is that they are not akin to properties, but are their own kind of universal.) A general exception to what has just been claimed about how we refer to types (with singular terms) might be inferred from the fact that we do more often say of an animal that it is a tiger, rather than that it is a member of the species Felis Tigris. This raises the question as to whether the species Felis Tigris is just the property of being a tiger, and if it isn't, then what the relation between these two items is. Wollheim (1968, p. 76) insightfully puts the point that types seem to be objects as that the relationship between a type and its tokens is "more intimate" than that between (a classic example of) a property and its instances because "for much of the time we think and talk of the type as though it were itself a kind of token, though a peculiarly important or pre-eminent one". He (1968, p. 77) mentions two other differences worth noting between types and the classic examples of universals. One is that although types and the classic examples of properties often satisfy the same predicates, there are many more predicates shared between a type and its tokens than between a classic example of a property and its instances. (Beethoven's Symphony No. 9 is in the same key, has the same number of measures, same number of notes, etc. as a great many of its tokens.) Second, he argues that predicates true of tokens in virtue of being tokens of the type are therefore true of the type (Old Glory is rectangular) but this is never the case with classic properties (being white is not white.) These considerations may not suffice to show that types aren't universals, but they do point to a difference between types and the classic examples of properties. ## 4. What is a Type? The question permits answers at varying levels of generality. At its most general, it is a request for a theory of types, the way set theory answers the question "what is a set?" A general answer would tell us what sort of thing a type--any type--is. For example, is it sui generis, or a universal, or perhaps the set of its tokens, or a function from worlds to sets, or a kind, or, as Peirce maintained, a law? These options are discussed in SS4.1. At a more specific level, "what is a type?" is a request for a theory that would shed some light on the identity conditions for specific types of types, not necessarily all of them. It would yield an account of what a word (or a symphony, a species, a disease, etc.) is. This is in many ways a more difficult thing to do. To see just how difficult it is to give the identity conditions for an individual type, SS4.2 considers what a word is, both because words are our paradigm of types, since the type-token distinction is generally illustrated by means of words, and because doing so will show that some of the most common assumptions made about types and their tokens are not correct. It will also illuminate some of the things we want from a theory of types. ### 4.1 Some General Answers As mentioned in the previous paragraph, one way a theory of types might answer the question "what is a type?" is the way set theory answers the question "what is a set?" If types are universals, as most thinkers assume, then there are as many theories of types as there are theories of universals. Some axiomatic theories include Zalta 1983 and Jubien 1988. Since theories of universals are discussed at length in this Encyclopedia elsewhere, they will not be repeated here. (See properties.) However it might be said that types are not universals for the reasons mentioned in SS3 above, where it was urged that types might be neither properties nor relations but objects, and there is an absolute difference between objects and properties. Identifying types as universals would appear to fly in the face of that consideration. #### 4.1.1 A Set It might appear that types are better construed as sets (assuming sets themselves are not universals). The natural thought is that a type is the set of its tokens, which is how Quine sometimes (1987, p. 218) construes a type. After all, a species is often said to be "the class of its members". There are two serious problems with this construal. One is that many types have no tokens and yet they are different types. For example, there are a lot of very long sentences that have no tokens. So if a type were just the set of its tokens, these distinct sentences would be wrongly classified as identical, because each would be identical to the null set. Another closely related problem also stems from the fact that sets, or classes, are defined extensionally, in terms of their members. The set of natural numbers without the number 17 is a distinct set from the set of natural numbers. One way to put this is that classes have their members essentially. Not so the species homo sapiens, the word 'the', nor Beethoven's Symphony No. 9. The set of specimens of homo sapiens without George W. Bush is a different set from the set of specimens of homo sapiens with him, but the species would be the same even if George W. Bush did not exist. That is, it is false that had George W. Bush never existed, the species homo sapiens would not have existed. The same species might have had different members; it does not depend for its existence on the existence of all its members as sets do. Better, then, but still in keeping with an extensional set-theoretic approach, would be to identify a type as a function from worlds to sets of objects in that world. It is difficult to see any motivation for this move that would not also motivate identifying properties as such functions and then we are left with the question of whether types are universals, discussed in SS3. #### 4.1.2 A Kind The example of homo sapiens suggests that perhaps a type is a kind, where a kind is not a set (for the reasons mentioned two paragraphs above). Of course, this raises the question of what a kind is; Wolterstorff (1970) adopts the kind view of types and identifies kinds as universals. In Wolterstorff 1980, he takes being an example of as undefined and uses it to define kinds--so that, for example, a possible kind is one such that it is possible there is an example of it. Norm kinds he then defines as kinds "such that it is possible for them to have properly formed and also possible for them to have improperly formed examples" (p. 56). He identifies both species and artworks as norm-kinds. Bromberger (1992a) also views the tokens of a type as a quasi-natural kind relative to appropriate projectible ("What is its freezing point?" e.g.) and individuating questions ("Where was it on June 13th, 2005?"). However, he doesn't identify the type as the kind itself, since to do so does not do justice to the semantic facts mentioned in SS2 above, that types are largely referred to by singular terms. Instead he views the type as what he calls the archetype of the kind, defined as something that models all the tokens of a kind with respect to projectible questions but not something that admits of answers to individuating questions. Thus for Bromberger the type is not the kind itself, but models all the tokens of the kind. We shall see some difficulties for this view in SS5 below. #### 4.1.3 A Law It wouldn't do to ignore what the coiner of the type-token distinction had to say about types. Unfortunately it cannot be adequately unpacked without an in-depth explication of Peirce's semiotics, which cannot be embarked upon here. (See the entries on Charles Sanders Peirce and Peirce's theory of signs.) Peirce said types "do not exist", yet they are "definitely Significant Forms" that "determine things that do exist" (4.423). A type, or "legisign" as he also calls it, "has a definite identity, though usually admitting a great variety of appearances. Thus, & and and the sound are all one word" (8.334). Elsewhere he tells us that a type is "a general law that is a sign. This law is usually established by men. Every conventional sign is a legisign. It is not a single object, but a general type which...shall be significant. ...[E]very Legisign requires Sinsigns" (2.246). Sinsigns are tokens. (It should be mentioned that for Peirce there is actually a trichotomy among types, tokens and tones, or qualisigns, which are "the mere quality of appearance" (8.334).) Thus types have a definite identity as signs, are general laws established by men, but they do not exist. Perhaps all Peirce meant by saying they do not exist was that they are "not individual things" (6.334), that they are, instead what he calls "generals"--not to be confused with universals. What he might have meant by a "general law" is uncertain. Stebbing (1935, p.15) suggests "a rule in accordance with which tokens ... could be so used as to have meaning attached to them". Greenlee (1973, p. 137) suggests that for Peirce a type is "a habit controlling a specific way of responding interpretatively." Perhaps, then, types are of a psychological nature. Obviously two people can have the same habit, so habits also come in types and tokens. Presumably, types are then habit types. This account may be plausible for words, but it is not plausible for sentences, because there are sentences that have no tokens because if Ph,Ps are sentences, then so is (Ph & Ps) and it is clear that for Peirce "every [type] requires [tokens]" (2.246). And it is much less plausible for non-linguistic types, like types of beetles, some of which have yet to be discovered. ### 4.2 What is a Word? No general theory of types can tell us what we often want to know when we ask: what is a species, a symphony, a word, a poem, or a disease? Such questions are just as difficult to answer as what a type is in general. Even if types were sets, for example, set theory by itself will not answer the burning question of which set a species is. One would then have to go to biology and philosophy of biology to find out whether a species is (i) "a set of individuals closely resembling each other" as Darwin (1859, p. 52) would have it, (ii) a set of "individuals sharing the same genetic code" as Putnam (1975, p. 239) would have it, (iii) a set of "interbreeding natural populations that are reproductively isolated from other such groups", as Mayr (1970, p. 12) would have it, or (iv) a set comprising "a lineage evolving separately from others and with its own unitary evolutionary role and tendencies", as Simpson (1961, p. 153) would have it. Similarly, if there is a question of copyright infringement, one had best look to industry standards and aesthetics for what a film or a song is, and not set theory. In general, questions such as "what is a poem, a phoneme, a disease, a flag,....?" are to be pursued in conjunction with a specific discipline, and not within the confines of a general theory of types. It is largely up to linguistics and the philosophy of it, e.g., to determine the identity conditions for phonemes, allophones, cardinal vowels, LF representations, tone groups and all the other linguistic types mentioned in SS2 above. #### 4.2.1 Identity Conditions for Words It's instructive to consider what our paradigm of a type is--a word. It will reveal how complicated the identity conditions are for even one specific type, and help to dispel the idea that tokens are to types as cookies are to cookie cutters. It will also show what we desire from a theory of types, by exhibiting the facts that a theory of types has to accommodate. We illustrated the type-token distinction by appealing to words, so presumably we think we know at least roughly what a word type is. Unfortunately, everyone seems to think they know, but there is massive disagreement on the matter in philosophy. However, as urged in the preceding paragraph, it is crucial to rely on linguistics when we consider what a word is. When we do we find that there are different linguistic criteria for what a word is, and a good deal of the disagreement can be chalked up to this fact. McArthur 1992's The Oxford Companion to the English Language (pp. 1120-1) lists eight: orthographic, phonological, morphological, lexical, grammatical, onomastic, lexicographical and statistical-- but adds that more can be demarcated. There are different types of types of words. However deserving of attention each of these is, it will be useful to focus on just one, and I will do so in what follows. There is an important and very common use of the word 'word' that lexicographers and the rest of us use frequently. It is, roughly, the sort of thing that merits a dictionary entry. (Roughly, because some entries in the dictionary, e.g.,'il-,' '-ile,' and 'metric system,' are not words, and some words, e.g. many proper names, do not get a dictionary entry.) This notion was at play in our opening remarks in SS2 about Shakespeare's vocabulary containing 30,000 words, and the twenty spellings and eighteen senses of the noun 'color'/'colour', the verb 'color'/'colour', and four current pronunciations of the noun 'schedule'. These examples show (in this ordinary sense of 'word') that the same word can be written or spoken, can have more than one correct spelling, can have more than one correct spelling at the same time, can have more than one sense at the same time and can have more than one correct pronunciation at the same time. It also shows that different words can have the same correct spelling and pronunciation; further obvious examples would show that different words can have the same sense--e.g. English 'red' and French 'rouge'. A theory of types, or of word types, that cannot accommodate this notion of a word is worthless. In what follows, I shall use 'word' in this sense. #### 4.2.2 What a Theory of Words Might Tell Us Ideally, a theory of words and their tokens should tell us not only (i) what a word is (in the sense indicated), but (ii) how a word is to be individuated, (iii) whether there is anything all and only tokens of a particular word have in common (other than being tokens of that word); (iv) how we know about words; (v) what the relation is between words and their tokens; (vi) what makes a token a token of one word rather than another; (vii) how word tokens are to be individuated; and (viii) what makes us think a particular is a token of one word rather than another. These questions are distinct, although they are apt to run together because the answer to one may give rise to answers to others. For example, if we say in answer to (iii), that all tokens of a certain word (say, 'cat') have something in common besides being tokens of that word--they are all spelled 'c'-'a'-'t', for example--then we may be inclined to say to (vi) that spelling makes a word token a token of 'cat' rather than some other type; and to (vii) that word tokens of 'cat' are to be individuated on the basis of their being spelled 'c'-'a'-'t'; and to (viii) that we think something is a token of 'cat' when we see that it is spelled 'c'-'a'-'t'; and to (ii) that the word 'cat' itself is to be individuated by its spelling; and to (i) that a word type is a sequence of letters--e.g. the word 'cat' just is the sequence of letters <'c', 'a', 't'>; and to (iv) that we know about a particular word, about what properties it has, by perceiving its tokens: it has all the properties that every one of its tokens has (except for properties types cannot have, e.g., being concrete). #### 4.2.3 Orthography The advantage of starting with (iii) is that if there is some nontrivial property that all tokens of a word (in the sense indicated) have in common, then perhaps we can use it to individuate the tokens, and also to get a handle on what the type is like and on how we know what the type is like. Unfortunately, it is not spelling, contrary to what many philosophers seem to think. Stebbing, e.g., considers the inscribed word a shape. But not even the linguist's much narrower notion of an orthographic word ('a visual sign with a space around it') requires a canonical spelling. We have seen that not all tokens of 'color' have the same spelling, even when they are spelled correctly, which they sometimes are not. Not all tokens are spelled at all--spoken tokens are not. Moreover, two words can have the same spelling, as the noun 'color' and verb 'color' prove, or to take a different example, the noun 'down' from German meaning "the fine soft covering of fowls" and the different noun 'down' from Celtic meaning "open expanse of elevated land". (They are not the same word with two senses, but different words with different etymologies.) Notice that even if, contrary to fact, all tokens of a word had the same spelling and we concluded that the word type itself just is the sequence of letter types that compose it, we would have analyzed word types in terms of letter types, but since we are wondering what types are in the first place, we would still need an account of what letters are since they are types too. Providing one is surprisingly difficult. Letters of the alphabet such as the letter 'A' are not just shapes, for example, contrary to what is implicit in Stebbing 1935 and more explicit in Goodman and Quine 1947, because Braille and Morse code tokens of the letter 'A' cannot be said to have "the same shape", and even standard inscriptions of the letter 'A' do not have the same shape--in either a Euclidean or a topological sense--as these examples obtained from a few minutes in the library illustrate: > > ![letter A forms](fig-a.jpg) > Moreover, the letter 'A' has a long history and many of its earlier "forms" would not be recognizable to the modern reader. #### 4.2.4 Phonology If we switch instead to a phonemic analysis of words, as being more fundamental, similar problems arise. Not all tokens of a word are composed of the same phonemes, because some tokens are inscriptions. But even ignoring inscriptions, the example of the two 'down''s shows that neither can we identify a word with a sequence of phonemes. This particular difficulty might be avoided if we identify a word with a phonemic analysis paired with a sense. But this is too strong; we saw earlier that the noun 'color' has eighteen senses. Moreover, 'schedule' has more than one phonemic analysis. A phoneme itself is a type with tokens, and so we'd also need an account of what a phoneme is, and what its tokens have in common (if anything). Saying what a phoneme is promises to be at least as hard as saying what a letter is. Phonology, the study of phonemes, is distinct from phonetics, the scientific study of speech production. Phonetics is concerned with the physical properties of sounds produced and is not language relative. Phonemes, on the other hand, are language relative: two phonetically distinct speech tokens may be classified as tokens of the same phoneme relative to one language, or as tokens of different phonemes relative to another language. Phonemes are theoretical entities, and abstract ones at that: they are sometimes said to be sets of features. The bottom line is that the phonological word is not the lexicographical one either. It might be thought that we started at too abstract a level--that if we think there is a hierarchy of types of words, we started "too high" on the hierarchy and we should start lower on the hierarchy. That is, that we should first gather together those tokens that are phonetically (and perhaps semantically) identical on the grounds that this is a perfectly good notion of a word. But notice: this would mean that different dialects of the same language would have far fewer "words" in common than one would have supposed, and it would misclassify many words because, for example, according to Fudge (1990, p. 39) a Cockney 'know' is like the Queen's 'now'; the Queen's 'know' is like Scottish 'now'; and a Yorkshire 'know' is like the Queen's 'gnaw'. Worse, even within the very same idiolect it would distinguish as different "words" what one would have thought were the same word. For example, the word 'extraordinary' is variously pronounced with six, five, four, three or even two syllables by speakers of British English. According to Fudge (1990, p. 40) it ranges "for most British English speakers from the hyper-careful ['ekstr@'?o:dIn@rI] through the fairly careful [Ik'stro:dnrI] to the very colloquial ['stro:nrI]." That is, the very same person may use any of five pronunciations for what should be considered the same word. Only an absolute diehard of this "bottom-up" approach would insist on distinguishing as different words representations for the same idiolectal word. Not only would a phonologist take this as excessively complicated, but the representation types themselves can receive realizations that are acoustically very different (for the small child and the man may speak the same idiolect). Fudge (1990, p. 31) assures us that "It is very rare for two repetitions of an utterance to be exactly identical, even when spoken by the same person." Pretty soon, each word token would have to count as tokens of different "words". The example of ['stro:nri] demonstrates that there may be no phonetic signal in a token for every phoneme that is supposed to compose the word: it is "missing" several syllables! This is also demonstrated by reflection on casual speech: [jeet?] for 'did you eat?'. No wonder, then, that many phoneticians have given up on the attempt to reduce phonological types to acoustic/articulatory types. (See Bromberger and Halle 1986). Even the physicalist Bjorn Lindblom (1986, p. 495) concedes that "for a given language there seems to be no unique set of acoustic properties that will always be present in the production of a given unit (feature, phoneme, syllable) and that will reliably be found in all conceivable contexts." However, the final nail in the coffin for the suggestion according to which all tokens of the same word have the "same sound" is that words can be mispronounced. As Kaplan (1990) has argued, a word can suffer extreme mispronunciation and still be (a token of) the same word. He asks us to imagine a test subject, who faithfully repeats each word she is told. After a time, we put filters on her that radically alter the results of her efforts. Nonetheless, we would say she is saying the word she hears. Kaplan concludes that differences in sound between tokens of the same word can be just about as great as we would like. Notice that in such circumstances intention--what word the test subject intended to produce--is key. This suggests that perhaps what all tokens of a word, say, 'color', have in common is that they were produced as the result of an intention to produce a token of the word 'color'. Unfortunately, counterexamples are not hard to manufacture. (A clear phonemic example of 'supercalifragilisticexpealadocious' in English would probably not count as a token of 'color', for example.) Counterexamples aside, it would be putting the cart before the horse to try to explain what the word 'color' is by appealing to the intention to produce a token of the word 'color'. It would be like trying to explain what a fork is by appealing to the intention to produce a fork. Intentions are important in helping to identify which type a token is a token of--question (viii)--but will not help us with what the type is--question (i)--and so I shall ignore them in what follows. #### 4.2.5 Conclusion The upshot of all this is that there is no nontrivial, interesting, "natural", projectible property that all tokens of a word have in common, other than being tokens of that word (in the sense of 'word' indicated). Tokens are all the same word, but they are not all the same. That is, the answer to (iii) is no. What then, of the other questions, (i)-(viii)? They become more difficult to answer. Wetzel (2002) attempts to answer them. The primary conclusion of Wetzel 2002 is that words are theoretical entities, postulated by and justified by linguistic theory. Words, in the sense indicated, are individuated by a number of variables, including orthography, phonology, etymology, grammatical function and sense(s). As for their tokens, they are apt to have some but not all of the properties of the type. And, as the story from Kaplan shows, tokens may even be quite deformed. These considerations impact significantly on the relation between types and their tokens, discussed in the next section, SS5. ## 5. The Relation between Types and their Tokens The relation between types and their tokens is obviously dependent on what a type is. If it is a set, as Quine (1987, p. 217) maintains, the relation is class membership. If it is a kind, as Wolterstorff maintains, the relation is kind-membership. If it is a law, as Peirce maintains, it is something else again, perhaps being used in accordance with. (Although Peirce also says a token is an "instance" of a type and that the token signifies the type.) Nonetheless, it has often been taken to be the relation of instantiation, or exemplification; a token is an instance of a type; it exemplifies the type. (Not that every instance of a type is a token--e.g. capital 'A', small 'A', and all the other types of 'A's tokened in the display in SS4.2 above may be said to be instances of the letter 'A'.) As with other universals, there are two versions of this relation, Platonic and Aristotelian. Although the two versions of property realism are discussed at length under this encyclopedia's entry for properties, a few remarks about the type versions are in order. According to Platonic versions of type realism, e.g., Bromberger 1989, Hale 1990, Katz 1981 and Wetzel 2002, the type is an abstract object, not located anywhere in space-time, although its tokens are. This version appears to give rise to serious epistemological problems--we don't see or hear the type, it isn't located anywhere in space-time, so how do spatiotemporal creatures such as ourselves know it exists, or what properties it has? Admittedly, we see and hear tokens, but how are they a guide to what the type is like? One answer, given for example, by Wolterstorff, for whom we saw in SS4 that a type is a norm-kind, is that ordinary induction from tokens would give us knowledge of types, at least in the case of instantiated types. Bromberger (1992a, p.176) claims that linguists "often impute properties to types after observing and judging some of their tokens and seem to do this in a principled way" and calls the principle that licenses this inference the Platonic Relationship Principle. More specifically, he proposes (1992a) that the type, as the archetype of the quasi-natural kind which comprises the tokens, has just those projectible properties that all the tokens have. He has in mind properties such as the same underlying phonological structure, for words, and the same boiling point, for elements. However, as we saw in SS4, generally there are no such properties had by all and only tokens of a type, at least in the case of words--not same phonological structure, nor same sense nor same spelling. Not all tokens have any such natural projectible property (except for the property of being tokens of the same type). It should be clear from SS4 that the cookie cutter model--the idea of the type as just a perceptible pattern for what all the tokens look like--does not work. Goodman (1972, pp. 437-8) follows Peirce in using the word 'replica' to apply to all tokens of the same type, (although Peirce seemed to think they were replicas of the type, whereas Goodman, being a nominalist, cannot think this) but not all tokens resemble each other in any ordinary sense beyond being tokens of the same type (although of course some do). Goodman himself is clear about this, for he notes there that "Similarity, ever ready to solve philosophical problems and overcome obstacles, is a pretender, an impostor, a quack.... Similarity does not pick out inscriptions that are 'tokens of a common type', or replicas of each other. Only our addiction to similarity deludes us into accepting similarity as the basis for grouping inscriptions into the several letters, words, and so forth." But others, e.g. Stebbing (1935, p. 6) and Hardie (1936), claim that all spoken tokens are more or less similar to each other. Because they are not, Wetzel (2002) and (2008) proposes that since the only property all the tokens of a type generally share is being tokens of the type, one of the primary justifications for positing word types is that being a token of the word 'color,' say, is the glue that binds the considerable variety of space-time particulars together. The type is thus a very important theoretical object, whose function is to unify all the tokens as being "of the same type"; in accordance with the Platonic Relationship Principle, the type has properties based on the properties of some of its tokens, but in a complex way--in addition, the tokens have some of their properties in virtue of what properties the type has. In Aristotelian versions of exemplification, such as Wollheim 1968 and Armstrong 1978, the type has no independent existence apart from its tokens. It is "in" each and every one of its tokens, and so can be seen or heard just as the tokens can be. This avoids the epistemological problem mentioned in the preceding paragraph, but makes it hard to explain how some types--such as very long sentences--can have no tokens. As against instantiation of any sort, Stebbing (1935, p. 9) argues that a token is not an instance of a type, because "the type is a logical construction out of tokens having similarity or conventional association [as the inscribed word with the spoken]. It follows that the type is logically dependent upon the tokens, in the sense that it is logically impossible to mention the type-word without using a token, and further, the meaning of the type has to be defined by reference to the tokens." These claims are quite controversial. For example, it is clear one can refer to a type word without using a token of it--one can say "the word a token of which is the first word on the page". Another alternative to exemplification is representation. According to Szabo (1999, p. 150), types are abstract particulars, as with Platonic realism, but tokens represent their types, just as "paintings, photographs, maps, numerals, hand gestures, traffic signs and horn signals" represent, or "stand for" their representata. A word token of 'horse' represents the word 'horse', which in turn represents horses. Just as a correct map of the planet can provide us with knowledge of the planet, so too a token can provide us with knowledge of properties of the type, thus addressing the epistemological problem. The representation view gives rise to a problem however, for it turns out that what we have been calling word tokens aren't words at all on this view, anymore than a map of a planet is a planet, and this runs contrary to our usual thinking. ## 6. What is a Token? It might seem that tokens are less problematic than types, being spatio-temporal particulars. But certain complications should be noted. (We continue to use linguistic examples, but the remarks hold true for tokens generally). * As Kaplan (1990, p. 97) pointed out, a token of a word could be suitably circumscribed empty space (e.g., in a piece of cardboard after letters have been cut out of it). * It would seem that a token might also be a mental particular--because a poem might be composed prior to its being read or written down--although there is disagreement over whether such a mental particular is to count as a token. * It will generally be the case that some tokens of the type resemble in their outward physical appearance some other tokens of the type, but as we saw in SS4-5, it is generally not the case that all the tokens of the same type resemble one another in appearance. (Recall the example of the letter 'A' in SS4.) * Even being similar in appearance (say by sound or spelling) to a canonical exemplar token of the type is not enough to make a physical object/event a token of that type. The phonetic sequence [Ah 'key ess 'oon ah 'may sah] is the same phonetic (type) spoken in Spanish or Yiddish. Yet if a Spanish speaker uses it to say a table goes here, she has not, in addition, said a cow eats without a knife in Yiddish. She has not said anything in Yiddish, however phonetically similar what she said might be to a sentence token of Yiddish. So her token is a token in Spanish, not Yiddish. Meaningful tokens are tokens in a language. * Along the same lines, being a physical object/event that is similar in appearance (say by sound or spelling) to a canonical exemplar token of the type is not even enough to make the physical object/event meaningful, because, as Putnam (1981, pp. 1-2) among others has argued, the token must have been appropriately intentionally produced. * If it is not already clear from the foregoing, it should be noted that a physical object that is a token of a type is not one intrinsically--merely by being a certain sequence of shaped ink marks, say. Just as being a brother is not an intrinsic property of brothers because one is a brother only relative to another person, so a token is a token only relative to a type, and to a language, perhaps to an orientation, and perhaps, as Hugly and Sayward (1981 p. 186) have argued, to a tokening system (e.g., Morse Code), which they define as "a set of instructions that tell how, for any given expression, a speaker of the language could construct a perceptible particular that tokens that expression given that the speaker had enough physical and mental resources...". ## 7. Do Types Exist? ### 7.1 Universals Since types are usually thought to be universals, the debate over whether they exist is as longstanding as the debate over universals, and debaters fall into the same camps. Realists say they do, as we saw in SS5 which surveyed several varieties of realism. Realism's traditional opponents were nominalists and conceptualists. Nominalists, who renounce universals and abstract objects, say they don't. (See, e.g., Goodman and Quine 1947, Quine 1953, Goodman 1977 and Bromberger 1992b). Conceptualists argued that there are no general things such as the species homo sapiens; there are only general ideas--that is, ideas that apply to more than one thing. Applied to words, the thesis would be that words are not abstract objects "out there", but objects in the mind. Their existence then, would be contingent on having been thought of. While this contingency may have a good deal of plausibility in the case of linguistic items, by itself conceptualism is just a stopgap measure on the issue of types and tokens generally. For ideas themselves are also either types or tokens (as evidenced by the fact that two people sometimes have the same idea). So either the conceptualist is proposing that types are idea types--which would be a species of realism--or she is proposing that there are no types, only mental idea particulars in particular persons, which is a version of nominalism. Conceptualism, therefore, will be ignored in what follows. ### 7.2 Realism Realism is the most natural view in the debate with nominalism, because as we saw in some detail in SS2 type talk is ubiquitous. That is, we talk as though there are types in philosophy, science and everyday life. To say that we talk as though there are types is not to invoke the traditional argument for universals, which is that a sunset and a rose are both red, so they have something in common; and this something can only be the property of being red; so properties exist. Quine (1953) convinced many at mid-century that this traditional argument for universals, which relies on predicates referring to something, fails. He objected that "the rose is red because the rose partakes of redness" is uninformative--we are no better off in terms of explanatory power with such extra objects as redness than we are without them; perhaps a rose's being red and a sunset's being red are just brute facts. Rather, to say we talk as though there are types is, as we saw in SS2, to appeal to the fact that in our theories we frequently use singular terms for types, and we quantify over them. As we saw, Frege emphasized that singular term usage is in indicator of objecthood, and Quine stressed that we are ontologically committed to that over which we quantify. Such considerations led Quine himself (1987, p. 217) to hold that expression types such as 'red' exist, even while he denied that redness does. Since at least on the face of it we are committed to types in many fields of inquiry, therefore, it is incumbent upon the nominalist to "analyze them away". (Or to maintain that all theories which appear to refer to types are false--but this is a pretty radical approach, which will be ignored below.) ### 7.3 Nominalism Realism is not without its problems, as was noted in SS5 above. Also favoring nominalism is Occam's principle which would have us prefer theories with fewer sorts of entities, other things being equal. The main problem for nominalists is to account for our apparent theoretical commitment to types, which, whatever types are, are not spatio-temporal particulars (according to nearly everyone). Traditional nominalists argued (as their name implies) that there are no general things, there are only general words, and such words simply apply to more than one thing. But this is not a solution to the current problem, presupposing as it does that there are word types--types are the problem. (Attempts to avoid this are given by Goodman and by Sellars; see below.) So-called class nominalists hold that a word type is just the class, or set, of its tokens. But this is unsatisfactory because, first, as we saw in SS5, classes are ill-suited for the job, since classes have their membership and their cardinality necessarily, but how many tokens a type has is usually a contingent matter. (For the same reason, mereological sums of tokens are unsuited for the job of types, as they have their parts essentially.) And second, classes are abstract objects too, so it is hard to see how this is really a form of nominalism about abstract objects at all. Initially more promising is the nominalistic claim that the surface grammar of type talk is misleading, that talk of types is just shorthand for talk of tokens and is thus harmless. To say 'The horse is a mammal' is just to say 'All horses are mammals'; to say 'The horse is a four-legged animal' is to say, as Frege himself (1977) suggested, 'All properly constituted horses are four-legged animals'. The idea is to "analyze away" apparent references to types by offering translations that are apparently type-free, and otherwise nominalistically acceptable. The problem is how to do this for each and every reference to or quantification over a type/types. Given the ubiquity of such references/quantifications, only the procurement of some sort of systematic procedure could assure us it can be done. However, chances of formulating a systematic procedure appear slight, in view of the following obstacles that would have to be overcome. * If the "translation" is not to turn out to be trivially true because there are no tokens, it must be the case that all types have tokens. We saw in SS5 there is reason to deny this, because it would falsify the law of syntax that says that if Ph,Ps are sentences, then so is (Ph & Ps), since there are only finitely many sentence tokens. * The examples of 'The horse is a mammal' and 'The horse is four-legged' suggest that 'The Type is P' is to be analyzed as 'All tokens, or all normal tokens, are P'. But this won't work. The noun 'colour' is spelled 'color' (among other ways) , but not all properly formed tokens of it are. 'Colour' also means of the face or skin (among other things), but not only do not all tokens of it mean of the face or skin, probably most do not, and the average one does not. * These semantic facts show that 'The Type is P' cannot be analyzed as 'Either all tokens are P, all properly formed tokens are P, most tokens are P or the average token is P'. Nor is there any point in searching for some other magic statistical formula that would express exactly how many tokens have to be P for the Type to be P. This is because some of the properties of the Type are collective properties--e.g., when it is said that "The grizzly bear, Ursus Horribilis, had at one time a U.S. range of most of the West, and numbered 10,000 is California alone, but today its range is Montana, Wyoming and Idaho and it numbers less than 1000." Having a range of most of the West is true of no individual bear. * It is very difficult even to find a nominalistic paraphrase for "Old Glory had twenty-eight stars in 1846 but now has fifty", as Wetzel (2008) argues. * So far we have only been considering analyzing away singular terms. Quantifications are indeed a nightmare. Here is Quine and Goodman's (1947 p. 180) nominalized version of "There are more cats than dogs": "Every individual that contains a bit of each cat is bigger than some individual that contains a bit of each dog" where a bit "is an object that is just as big as the smallest animal among all cats and dogs". Ingenious though their strategy is, it offers no help for nominalizing "Of 20,481 species examined, two-thirds were secure, seven percent were critically imperiled, and fifteen percent were vulnerable". #### 7.3.1 Goodman's Nominalism I have been writing as though it were easy to pick out and quantify over tokens of a type without referring to the type. Sometimes it is: 'The horse...'-- 'All horses...'. But this will not generally be the case. Consider the noun 'color'. The natural way to pick out its tokens is "all tokens of the noun 'color'", but obviously this will not do as a nominalistic paraphrase, for it contains a reference to a type. Goodman (1977) suggests a systematic way of paraphrasing and a general procedure for substituting nominalistic paraphrases for linguistic type-ridden sentences. In 1977 (p. 262) he claims that "any " 'Paris' consists of five letters" is short for "Every 'Paris'-inscription consists of five letter-inscriptions" ". The idea seems to be to replace singular terms by predicates, which nominalists such as Goodman think carry no ontological commitment. So instead of "an inscription of 'color'" write "a 'color'-inscription." It seems pretty clear, however, that, based on the rules of quote-names for words, this predicate still retains a singular term for a type, the word 'color' just as clearly as "a George W. Bush appearance" still contains a reference to Bush. That this is so is even more obvious in the case of "an 'a cat is on the mat'-inscription". So the question is how we might identify these tokens grammatically but without referring to the noun 'color' itself and still say something true and (in some appropriate sense) equivalent. Is there perhaps some other way of quantifying over tokens of a word, without referring to the word or to any other type? The fact that tokens are all "the same type" suggests they are all "the same" somehow, which begets the idea that the type must embody certain similar features that all and only its tokens have, such as spelling, or sense, or phonological structure or some combination of them. This is a beguiling idea, until one tries to find such a feature, or features, amid the large variety of its tokens--even the well-formed tokens. As we saw in SS4, there is no such feature (consider again 'color' and 'schedule'). They demonstrate that e.g. neither same spelling, same sense, nor same pronunciation prevail. In any event, such possible defining features involve reference to types: letter types in the spellings, phoneme types in the pronunciation. These too would have to be "analyzed away" in terms of quantifications over particulars. It might be thought that Sellars solved this problem by appealing to the notion of a linguistic role, which Loux (1998, p. 79) defines as: two word (tokens) have the same linguistic role if they "function in the same way as responses to perceptual input; they enter into the same inference patterns; and they play the same role in guiding behavior". It is dubious whether this notion can be unpacked without referring to abstract objects (same inference patterns?), but in any event it cannot be used to pick out all tokens of a word, as we have been using the word 'word'. The reason is that 'red' and 'rouge' are different words in our sense, but their tokens play the same linguistic role for Sellars. But even if expressions like "a 'color'-inscription" did not contain singular terms, Goodman's suggestion that whatever is true of the type is true of all the tokens has two fatal defects. First, it would turn truths into falsehoods. 'Paris' consists of five letters, as does 'color', but not every 'color'-inscription consists of just five letter-inscriptions since some are spelled 'c'-'o'-'l'-'o'-'u'-'r'. (As for denying that these are inscriptions of the word 'color' see "Orthography" in SS4 above.) It doesn't even work for the word 'Paris', as it has the forms 'Pareiss' and 'Parrys' also. Second, Goodman's technique of replacing singular terms by predicates only works if we are employing a realist semantics. That is, the key to his program is that for every word in the dictionary and every sentence in the language there corresponds a unique predicate that is true of just the tokens of that word/that sentence. (Without such predicates, true statements apparently referring to types would be short for nothing at all.) But this is only plausible on a realist semantics. If 'predicate' meant 'predicate token' there would not even be enough predicates for every word in the dictionary, much less every sentence in the language. That is, it is certainly false that for every word a in the dictionary, there is an actual predicate token of the form 'is an a-inscription'. For more details on the problems with Goodman's strategy, see Wetzel (2000). #### 7.3.2 Sellars's nominalism Sellars's suggestion is rather similar to Goodman's, and faces similar difficulties, but is sufficiently different to be worth mentioning. Loux (1998, pp. 75-83), for one, argues that Sellars (1963) achieves the best nominalist account available by overcoming critical objections to Carnap's "metalinguistic" account as follows. Carnap (1959, pp. 284-314) had suggested that all claims involving apparent reference to abstract objects, such as "Red is a color", are systematically to be understood as metalinguistic claims about the word involved ("The word 'red' is a color predicate"). There are two obvious objections to Carnap's suggestion. The first is that we still have word types being referred to; the second is that translation of "Red is a color" and "The word 'red' is a color predicate" into French, say, will not result in sentences that are equivalent, since the latter ("Le mot 'red' est un predicat de coleur") will still refer to an English word, but the former will not. To counter the first objection, Sellars (1963, pp. 632-33) claims "the word 'red'" is a distributive singular term, which typically consists of 'the' followed by a common noun that purports to name a kind--e.g., 'the lion' in "The lion is tawny"--and that use of it results in a generic claim that is about token lions, not the type--"All lions are tawny"--since "The K is f" is logically equivalent to "All Ks are f". Note that this is similar to Goodman's suggestion and as such is subject to the same criticism as is Goodman's: the two sentence forms said to be logically equivalent are not. See the second bullet point three paragraphs above. Nor is there some other simple and straightforward logical equivalence that does the job; see the third and fourth bullet points above. At best this is a suggestion that some of the sentences in question--the ones that do not contain "collective" predicates like 'extinct'--are logically equivalent to generic claims--e.g. "Lions are tawny"--which are capable of being true while admitting of exceptions. However, that there is a uniform semantics (even a very complex one) for generic claims, one moreover that "analyzes away" kind talk, is a strong empirical claim about language, one that does not receive strong support from current efforts to analyze generic claims. (See Carlson and Pelletier 1995.) To counter the second objection (about non-equivalent translations) Sellars suggests we introduce a new sort of quotation device, dot quotation, that would permit translation of the quoted word. Then, e.g. "The word red is a color predicate" translates as "Le mot rouge est un predicat de coleur". The justification is that 'red' and 'rouge' are functionally equivalent--they play the same linguistic role. As mentioned above, this means that they have the same causes (of perceptual stimuli, e.g.) and effects (conduct, e.g.) and function similarly in inferential transitions. Again, whether Sellars can unpack the notion of "same linguistic role" without appealing to any universals remains an open question. Less open is the question of how successful such an approach is to dealing with talk of words, e.g., "The word 'red' consists of three letters". If we apply Sellars' analysis systematically, this becomes ""The word 'red'" consists of a three-lettered predicate" which is just false; if instead we reckon that "the word 'red'" is already a distributive singular term and apply his analysis to it, getting "The word red is a three-lettered predicate", this then translates incorrectly as "Le mot rouge est un predicat trois-en lettres". If we stick instead with regular quotation, which resists translation, then we cannot appeal to "same linguistic role" to say what class of things "the word 'red'" distributes over. In any event, Sellars's account also faces the difficulty noted above with Goodman's account: it is only plausible on a realist semantics, because if 'predicate' meant 'predicate token' there would not even be enough predicates for every word in the dictionary, much less every sentence in the language. (For other problems with Sellars's account, see Loux 1978: 81-85.) ## 8. Occurrences As mentioned in SS1 above, there is a related distinction that needs to be mentioned in connection with the type-token distinction. It is that between a thing, or type of thing, and (what is best called) an occurrence of it--where an occurrence is not necessarily a token. The reason the reader was asked above to count the words in Stein's line in front of the reader's eyes, was to ensure that tokens would be counted. Tokens are concrete particulars; whether objects or events they have a unique spatio-temporal location. Had the reader been asked to count the words in Stein's line itself, the reader might still have correctly answered either 'three' or 'ten'. There are exactly three word types, but although there are ten word tokens in a token copy of the line, there aren't any tokens at all in the line itself. The line itself is an abstract type, as is the poem in which it first appeared. Nor are there ten word types in the line, because as we just said it contains only the three word types, 'a,' 'is' and 'rose,' each of which is unique. So what are there ten of? Occurrences of words, as logicians say: three occurrences of the word (type) 'a,' three of 'is' and four of 'rose'. Or, to put it in a more ontologically neutral fashion: the word 'a' occurs three times in the line, 'is' three times and 'rose' four times. Similarly, the variable 'x' occurs three times in the formula '[?]x (Ax & Bx)'. Now this may seem impossible; how can one and the same thing occur more than once without there being two tokens of it? Simons (1982) concludes that it can't. Wetzel (1993) argues that it is useful to distinguish objects from occurrences of them. For example, in the sequence of numbers <0,1,0,1> the very same number, the number one, occurs twice, yet its first occurrence is distinct from its second. The notion of an occurrence of x in y involves not only x and y, but also how x is situated in y. Even a concrete object can occur more than once in a sequence--the same person occurs twice in the sequence of New Jersey million dollar lottery winners, remarkably enough. If we think of an expression as a sequence, then the air of mystery over how the same identical thing can occur twice vanishes. Does this mean that, in addition to word types and word tokens, word occurrences must also be recognized? Not necessarily; we can unpack the notion of an occurrence using "occurs in" if we have the notion of a sequence available; see Wetzel 1993 for details. The need to distinguish tokens of types from occurrences of types arises, not just in linguistics, but whenever types of things have other types of things occurring in them. There are 10,000 (or so) notes in Beethoven's Sonate Pathetique, but there are only 88 notes (types) the piano can produce. There are supposed to be fifty stars (types) in the current Old Glory (type), but the five-pointed star (type) is unique. And what could it mean to say that the very same atom (type), hydrogen, "occurs four times" in the methane molecule? Again, the perplexing thing is how the very same thing can "occur" more than once, without there being more than one token of it. Armstrong (1986), Lewis (1986a,b) and Forrest (1986) called such types "structural universals", which were the subject of a debate among the three. Armstrong and Forrest defended Armstrong's (1978) view of universals against Lewis, who delineated seven different views of structural universals compatible with Armstrong's theory, and found all of them wanting. Basically, Lewis (1986a) assumed that there are two sorts of ways structural universals might be constructed of simpler universals: set-theoretically and mereologically. He argued that set-theoretical constructions resulted in ersatz universals, not universals worthy of the name, and that the various mereological constructions just resulted in a heap of simpler universals, where there could not be two of the simpler universals. Wetzel (2008) argues that there is a conception of a structural universal, the "occurrence conception"--which is an extension of the occurrence conception of expressions mentioned above--that escapes Lewis's objections.
type-theory	## 1. Paradoxes and Russell's Type Theories The theory of types was introduced by Russell in order to cope with some contradictions he found in his account of set theory and was introduced in "Appendix B: The Doctrine of Types" of Russell 1903. This contradiction was obtained by analysing a theorem of Cantor that no mapping \[ F:X \rightarrow \Pow(X) \] (where $\Pow(X)$ is the class of subclasses of a class $X)$ can be surjective; that is, $F$ cannot be such that every member $b$ of $\Pow(X)$ is equal to $F(a)$ for some element $a$ of $X$. This can be phrased "intuitively" as the fact that there are more subsets of $X$ than elements of $X$. The proof of this fact is so simple and basic that it is worthwhile giving it here. Consider the following subset of $X$: \[ A = \{ x \in X \mid x \not\in F(x)\}. \] This subset cannot be in the range of $F$. For if $A = F(a)$, for some $a$, then \[\begin{align} a \in F(a) &\text{ iff } a \in A \\ &\text{ iff } a \not\in F(a) \end{align}\] which is a contradiction. Some remarks are in order. First, the proof does not use the law of excluded middle and is thus valid intuitionistically. Second, the method that is used, called diagonalisation was already present in the work of du Bois-Reymond for building real functions growing faster than any function in a given sequence of functions. Russell analysed what happens if we apply this theorem to the case where A is the class of all classes, admitting that there is such a class. He was then lead to consider the special class of classes that do not belong to themselves \[\tag{\} R = \{ w \mid w \not\in w \}. \] We then have \[ R \in R \text{ iff } R \not\in R. \] It seems indeed that Cantor was already aware of the fact that the class of all sets cannot be considered itself to be a set. Russell communicated this problem to Frege, and his letter, together with Frege's answer appear in van Heijenoort 1967. It is important to realise that the formulation (\) does not apply as it is to Frege's system. As Frege himself wrote in his reply to Russell, the expression "a predicate is predicated of itself" is not exact. Frege had a distinction between predicates (concepts) and objects. A (first-order) predicate applies to an object but it cannot have a predicate as argument. The exact formulation of the paradox in Frege's system uses the notion of the extension of a predicate $P$, which we designate as $\varepsilon P$. The extension of a predicate is itself an object. The important axiom V is: \[\tag{Axiom V} \varepsilon P = \varepsilon Q \equiv \forall x [P(x) \equiv Q(x)] \] This axiom asserts that the extension of $P$ is identical to the extension of $Q$ if and only if $P$ and $Q$ are materially equivalent. We can then translate Russell's paradox (\) in Frege's system by defining the predicate \[ R(x) \text{ iff } \exists P[x = \varepsilon P \wedge \neg P(x)] \] It can then been checked, using Axiom V in a crucial way, that \[ R(\varepsilon R) \equiv \neg R(\varepsilon R) \] and we have a contradiction as well. (Notice that for defining the predicate $R$, we have used an impredicative* existential quantification on predicates. It can be shown that the predicative version of Frege's system is consistent (see Heck 1996 and for further refinements Ferreira 2002). It is clear from this account that an idea of types was already present in Frege's work: there we find a distinction between objects, predicates (or concepts), predicates of predicates, etc. (This point is stressed in Quine 1940.) This hierarchy is called the "extensional hierarchy" by Russell (1959), and its necessity was recognised by Russell as a consequence of his paradox. ## 2. Simple Type Theory and the $\lambda$-Calculus As we saw above, the distinction: objects, predicates, predicate of predicates, etc., seems enough to block Russell's paradox (and this was recognised by Chwistek and Ramsey). We first describe the type structure as it is in Principia and later in this section we present the elegant formulation due to Church 1940 based on $\lambda$-calculus. The types can be defined as 1. $i$ is the type of individuals 2. $(\,)$ is the type of propositions 3. if $A\_1 ,\ldots ,A\_n$ are types then $(A\_1 ,\ldots ,A\_n)$ is the type of $n$-ary relations over objects of respective types $A\_1 ,\ldots ,A\_n$ For instance, the type of binary relations over individuals is $(i, i)$, the type of binary connectives is $((\,),(\,))$, the type of quantifiers over individuals is $((i))$. For forming propositions we use this type structure: thus $R(a\_1 ,\ldots ,a\_n)$ is a proposition if $R$ is of type $(A\_1 ,\ldots ,A\_n)$ and $a\_i$ is of type $A\_i$ for $i = 1,\ldots ,n$. This restriction makes it impossible to form a proposition of the form $P(P)$: the type of $P$ should be of the form $(A)$, and $P$ can only be applied to arguments of type $A$, and thus cannot be applied to itself since $A$ is not the same as $(A)$. However simple type theory is not predicative: we can define an object $Q(x, y)$ of type $(i, i)$ by \[ \forall P[P(x) \supset P(y)] \] Assume that we have two individuals $a$ and $b$ such that $Q(a, b)$ holds. We can define $P(x)$ to be $Q(x, a)$. It is then clear that $P(a)$ holds, since it is $Q(a, a)$. Hence $P(b)$ holds as well. We have proved, in an impredicative way, that $Q(a, b)$ implies $Q(b, a)$. Alternative simpler formulations, which retain only the notion of classes, classes of classes, etc., were formulated by Godel and Tarski. Actually this simpler version was used by Godel in his 1931 paper on formally undecidable propositions. The discovery of the undecidable propositions may have been motivated by a heuristic argument that it is unlikely that one can extend the completeness theorem of first-order logic to type theory (see the end of his Lecture at Konigsberg 1930 in Godel Collected Work, Volume III, Awodey and Carus 2001 and Goldfarb 2005). Tarski had a version of the definability theorem expressed in type theory (see Hodges 2008). See Schiemer and Reck 2013. We have objects of type 0, for individuals, objects of type 1, for classes of individuals, objects of type 2, for classes of classes of individuals, and so on. Functions of two or more arguments, like relations, need not be included among primitive objects since one can define relations to be classes of ordered pairs, and ordered pairs to be classes of classes. For example, the ordered pair of individuals a, b can be defined to be $\{\{a\},\{a,b\}\}$ where $\{x,y\}$ denotes the class whose sole elements are $x$ and $y$. (Wiener 1914 had suggested a similar reduction of relations to classes.) In this system, all propositions have the form $a(b)$, where $a$ is a sign of type $n+1$ and $b$ a sign of type $n$. Thus this system is built on the concept of an arbitrary class or subset of objects of a given domain and on the fact that the collection of all subsets of the given domain can form a new domain of the next type. Starting from a given domain of individuals, this process is then iterated. As emphasised for instance in Scott 1993, in set theory this process of forming subsets is iterated into the transfinite. In these versions of type theory, as in set theory, functions are not primitive objects, but are represented as functional relation. The addition function for instance is represented as a ternary relation by an object of type $(i,i,i)$. An elegant formulation of the simple type theory which extends it by introducing functions as primitive objects was given by Church in 1940. It uses the $\lambda$-calculus notation (Barendregt 1997). Since such a formulation is important in computer science, for the connection with category theory, and for Martin-Lof type theory, we describe it in some detail. In this formulation, predicates are seen as a special kind of functions (propositional functions), an idea that goes back to Frege (see for instance Quine 1940). Furthermore, the notion of function is seen as more primitive than the notion of predicates and relations, and a function is not defined anymore as a special kind of relation. (Oppenheimer and Zalta 2011 presents some arguments against such a primitive representation of functions.) The types of this system are defined inductively as follows 1. there are two basic types $i$ (the type of individuals) and $o$ (the type of propositions) 2. if $A, B$ are types then $A \rightarrow B$, the type of functions from $A$ to $B$, is a type We can form in this way the types: \| \| \| \| --- \| --- \| \| $i\rightarrow o$ \| (type of predicates) \| \| $(i\rightarrow o) \rightarrow o$ \| (type of predicates of predicates) \| which correspond to the types $(i)$ and $((i))$ but also the new types \| \| \| \| --- \| --- \| \| $i\rightarrow i$ \| (type of functions) \| \| $(i\rightarrow i) \rightarrow i$ \| (type of functionals) \| It is convenient to write \[ A\_1 ,\ldots ,A\_n \rightarrow B \] for \[ A\_1 \rightarrow(A\_2 \rightarrow \ldots \rightarrow(A\_n \rightarrow B)) \] In this way \[ A\_1 ,\ldots ,A\_n \rightarrow o \] corresponds to the type $(A\_1 ,\ldots ,A\_n)$. First-order logic considers only types of the form \| \| \| \| \| --- \| --- \| --- \| \| $i,\ldots ,i \rightarrow i$ \| (type of function symbols), \| and \| \| $i,\ldots ,i \rightarrow o$ \| (type of predicate, relation symbols) \| \| Notice that \[ A \rightarrow B \rightarrow C \] stands for \[ A \rightarrow(B\rightarrow C) \] (association to the right). For the terms of this logic, we shall not follow Church's account but a slight variation of it, due to Curry (who had similar ideas before Church's paper appeared) and which is presented in detail in R. Hindley's book on type theory. Like Church, we use $\lambda$-calculus, which provides a general notation for functions \[ M ::= x \mid M M \mid \lambda x.M \] Here we have used the so-called BNF notation, very convenient in computing science. This gives a syntactic specification of the $\lambda$-terms which, when expanded, means: * every variable is a function symbol; * every juxtaposition of two function symbols is a function symbol; * every $\lambda x.M$ is a function symbol; * there are no other function symbols. The notation for function application $M N$ is a little different than the mathematical notation, which would be $M(N)$. In general, \[ M\_1 M\_2 M\_3 \] stands for \[ (M\_1 M\_2) M\_3 \] (association to the left). The term $\lambda x.M$ represents the function which to $N$ associates $M[x:=N$]. This notation is so convenient that one wonders why it is not widely used in mathematics. The main equation of $\lambda$-calculus is then $(\beta$-conversion) \[ (\lambda x.M) N = M[x:=N] \] which expresses the meaning of $\lambda x.M$ as a function. We have used $M[x:=N$] as a notation for the value of the expression that results when $N$ is substituted for the variable $x$ in the matrix $M$. One usually sees this equation as a rewrite rule $(\beta$-reduction) \[ (\lambda x.M) N \rightarrow M[x:=N] \] In untyped lambda calculus, it may be that such rewriting does not terminate. The canonical example is given by the term $\Delta = \lambda x.x x$ and the application \[ \Delta \Delta \rightarrow \Delta \Delta \] (Notice the similarity with Russell's paradox.) The idea of Curry is then to look at types as predicates over lambda terms, writing $M:A$ to express that $M$ satisfies the predicate/type $A$. The meaning of \[ N:A\rightarrow B \] is then \[ \forall M, \text{ if } M:A \text{ then } N M:B \] which justifies the following rules \[ \frac{N:A\rightarrow B M:A}{N M:B} \] \[ \frac{M:B [x:A]}{\lambda x.M:A \rightarrow B} \] In general one works with judgements of the form \[ x\_1 :A\_1,...,x\_n :A\_n \vdash M:A \] where $x\_1,..., x\_n$ are distinct variables, and $M$ is a term having all free variables among $x\_1,..., x\_n$. In order to be able to get Church's system, one adds some constants in order to form propositions. Typically \| \| \| \| --- \| --- \| \| not: \| $o\rightarrow o$ \| \| imply: \| $o\rightarrow o\rightarrow o$ \| \| and: \| $o\rightarrow o\rightarrow o$ \| \| forall: \| $(A\rightarrow o) \rightarrow o$ \| The term \[ \lambda x. \neg(x x) \] represents the predicate of predicates that do not apply to themselves. This term does not have a type however, that is, it is not possible to find $A$ such that \[ \lambda x. \neg(x x) : (A\rightarrow o) \rightarrow o \] which is the formal expression of the fact that Russell's paradox cannot be expressed. Leibniz equality \[ Q: i \rightarrow i \rightarrow o \] will be defined as \[ Q = \lambda x . \lambda y. \forall(\lambda P.\imply(P x) (P y)) \] One usually writes $\forall x[M$] instead of $\forall(\lambda x.M)$, and the definition of $Q$ can then be rewritten as \[ Q = \lambda x.\lambda y.\forall P[\imply (P x) (P y)] \] This example again illustrates that we can formulate impredicative definitions in simple type theory. The use of $\lambda$-terms and $\beta$-reduction is most convenient for representing the complex substitution rules that are needed in simple type theory. For instance, if we want to substitute the predicate $\lambda x.Q a x$ for $P$ in the proposition \[ \imply (P a) (P b) \] we get \[ \imply ((\lambda x.Q a x) a) ((\lambda x.Q a x) b) \] and, using $\beta$-reduction, \[ \imply (Q a a) (Q a b) \] In summary, simple type theory forbids self-application but not the circularity present in impredicative definitions. The $\lambda$-calculus formalism also allows for a clearer analysis of Russell's paradox. We can see it as the definition of the predicate \[ R x = \neg(x x) \] If we think of $\beta$-reduction as the process of unfolding a definition, we see that there is a problem already with understanding the definition of R R \[ R R \rightarrow \neg(R R) \rightarrow \neg(\neg(R R)) \rightarrow \ldots \] In some sense, we have a non-wellfounded definition, which is as problematic as a contradiction (a proposition equivalent to its negation). One important theorem, the normalisation theorem, says that this cannot happen with simple types: if we have $M:A$ then $M$ is normalisable in a strong way (any sequence of reductions starting from $M$ terminates). For more information on this topic, we refer to the entry on Church's simple type theory. ## 3. Ramified Hierarchy and Impredicative Principles Russell introduced another hierarchy, that was not motivated by any formal paradoxes expressed in a formal system, but rather by the fear of "circularity" and by informal paradoxes similar to the paradox of the liar. If a man says "I am lying", then we have a situation reminiscent of Russell's paradox: a proposition which is equivalent to its own negation. Another informal such paradoxical situation is obtained if we define an integer to be the "least integer not definable in less than 100 words". In order to avoid such informal paradoxes, Russell thought it necessary to introduce another kind of hierarchy, the so-called "ramified hierarchy". The need for such a hierarchy is hinted in Appendix B of Russell 1903. Interestingly this is connected there to the question of the identity of equivalent propositions and of the logical product of a class of propositions. A thorough discussion can be found in Chapter 10 of Russell 1959. Since this notion of ramified hierarchy has been extremely influential in logic and especially proof theory, we describe it in some details. In order to further motivate this hierarchy, here is one example due to Russell. If we say Napoleon was Corsican. we do not refer in this sentence to any assemblage of properties. The property "to be Corsican" is said to be predicative. If we say on the other hand Napoleon had all the qualities of a great general we are referring to a totality of qualities. The property "to have all qualities of a great general" is said to be impredicative. Another example, also coming from Russell, shows how impredicative properties can potentially lead to problems reminiscent of the liar paradox. Suppose that we suggest the definition A typical Englishman is one who possesses all the properties possessed by a majority of Englishmen. It is clear that most Englishmen do not possess all the properties that most Englishmen possess. Therefore, a typical Englishman, according to this definition, should be untypical. The problem, according to Russell, is that the word "typical" has been defined by a reference to all properties and has been treated as itself a property. (It is remarkable that similar problems arise when defining the notion of random numbers, cf. Martin-Lof "Notes on constructive mathematics" (1970).) Russell introduced the ramified hierarchy in order to deal with the apparent circularity of such impredicative definitions. One should make a distinction between the first-order properties, like being Corsican, that do not refer to the totality of properties, and consider that the second-order properties refer only to the totality of first-order properties. One can then introduce third-order properties, that can refer to the totality of second-order property, and so on. This clearly eliminates all circularities connected to impredicative definitions. At about the same time, Poincare carried out a similar analysis. He stressed the importance of "predicative" classifications, and of not defining elements of a class using a quantification over this class (Poincare 1909). Poincare used the following example. Assume that we have a collection with elements 0, 1 and an operation + satisfying \[\begin{align} x+0 &= 0 \\ x+(y+1) &= (x+y)+1 \end{align}\] Let us say that a property is inductive if it holds of 0 and holds for $x+1$ if it holds for $x$. An impredicative, and potentially "dangerous", definition would be to define an element to be a number if it satisfies all inductive properties. It is then easy to show that this property "to be a number" is itself inductive. Indeed, 0 is a number since it satisfies all inductive properties, and if $x$ satisfies all inductive properties then so does $x+1$. Similarly it is easy to show that $x+y$ is a number if $x,y$ are numbers. Indeed the property $Q(z)$ that $x+z$ is a number is inductive: $Q$(0) holds since $x+0=x$ and if $x+z$ is a number then so is $x+(z+1) = (x+z)+1$. This whole argument however is circular since the property "to be a number" is not predicative and should be treated with suspicion. Instead, one should introduce a ramified hierarchy of properties and numbers. At the beginning, one has only first-order inductive properties, which do not refer in their definitions to a totality of properties, and one defines the numbers of order 1 to be the elements satisfying all first-order inductive properties. One can next consider the second-order inductive properties, that can refer to the collection of first-order properties, and the numbers of order 2, that are the elements satisfying the inductive properties of order 2. One can then similarly consider numbers of order 3, and so on. Poincare emphasizes the fact that a number of order 2 is a fortiori a number of order 1, and more generally, a number of order $n+1$ is a fortiori a number of order $n$. We have thus a sequence of more and more restricted properties: inductive properties of order 1, 2, ... and a sequence of more and more restricted collections of objects: numbers of order 1, 2, ... Also, the property "to be a number of order $n$" is itself an inductive property of order $n+1$. It does not seem possible to prove that $x+y$ is a number of order $n$ if $x,y$ are numbers of order $n$. On the other hand, it is possible to show that if $x$ is a number of order $n+1$, and $y$ a number of order $n+1$ then $x+y$ is a number of order $n$. Indeed, the property $P(z)$ that "$x+z$ is a number of order $n$" is an inductive property of order $n+1: P$(0) holds since $x+0 = x$ is a number of order $n+1$, and hence of order $n$, and if $P(z)$ holds, that is if $x+z$ is a number of order $n$, then so is $x+(z+1) = (x+z)+1$, and so $P(z+1)$ holds. Since $y$ is a number of order $n+1$, and $P(z)$ is an inductive property of order $n+1, P(y)$ holds and so $x+y$ is a number of order $n$. This example illustrates well the complexities introduced by the ramified hierarchy. The complexities are further amplified if one, like Russell as for Frege, defines even basic objects like natural numbers as classes of classes. For instance the number 2 is defined as the class of all classes of individuals having exactly two elements. We again obtain natural numbers of different orders in the ramified hierarchy. Besides Russell himself, and despite all these complications, Chwistek tried to develop arithmetic in a ramified way, and the interest of such an analysis was stressed by Skolem. See Burgess and Hazen 1998 for a recent development. Another mathematical example, often given, of an impredicative definition is the definition of least upper bound of a bounded class of real numbers. If we identify a real with the set of rationals that are less than this real, we see that this least upper bound can be defined as the union of all elements in this class. Let us identify subsets of the rationals as predicates. For example, for rational numbers $q, P(q)$ holds iff $q$ is a member of the subset identified as $P$. Now, we define the predicate $L\_C$ (a subset of the rationals) to be the least upper bound of class $C$ as: \[ \forall q[L\_C (q) \leftrightarrow \exists P[C(P) \wedge P(q)]] \] which is impredicative: we have defined a predicate $L$ by an existential quantification over all predicates. In the ramified hierarchy, if $C$ is a class of first-order classes of rationals, then $L$ will be a second-order class of rationals. One obtains then not one notion or real numbers, but real numbers of different orders 1, 2, ... The least upper bound of a collection of reals of order 1 will then be at least of order 2 in general. As we saw earlier, yet another example of an impredicative definition is given by Leibniz' definition of equality. For Leibniz, the predicate "is equal to $a$" is true for $b$ iff $b$ satisfies all the predicates satisfied by $a$. How should one deal with the complications introduced by the ramified hierarchy? Russell showed, in the introduction to the second edition to Principia Mathematica, that these complications can be avoided in some cases. He even thought, in Appendix B of the second edition of Principia Mathematica, that the ramified hierarchy of natural numbers of order 1,2,... collapses at order 5 for defining the reflexive transitive closure of a relation. However, Godel later found a problem in his argument, and indeed, it was shown by Myhill 1974 that this hierarchy actually does not collapse at any finite level. A similar problem, discussed by Russell in the introduction to the second edition to Principia Mathematica arises in the proof of Cantor's theorem that there cannot be any injective functions from the collection of all predicates to the collection of all objects (the version of Russell's paradox in Frege's system that we presented in the introduction). Can this be done in a ramified hierarchy? Russell doubted that this could be done within a ramified hierarchy of predicates and this was indeed confirmed indeed later (see Chwistek 1926, Fitch 1939 and Heck 1996). Because of these problems, Russell and Whitehead introduced in the first edition of Principia Mathematica the following reducibility axiom: the hierarchy of predicates, first-order, second-order, etc., collapses at level 1. This means that for any predicate of any order, there is a predicate of the first-order level which is equivalent to it. In the case of equality for instance, we postulate a first-order relation "$a=b$" which is equivalent to "$a$ satisfies all properties that $b$ satisfies". The motivation for this axiom was purely pragmatic. Without it, all basic mathematical notions, like real or natural numbers are stratified into different orders. Also, despite the apparent circularity of impredicative definitions, the axiom of reducibility does not seem to lead to inconsistencies. As noticed however first by Chwistek, and later by Ramsey, in the presence of the axiom of reducibility, there is actually no point in introducing the ramified hierarchy at all! It is much simpler to accept impredicative definitions from the start. The simple "extensional" hierarchy of individuals, classes, classes of classes, ... is then enough. We get in this way the simpler systems formalised in Godel 1931 or Church 1940 that were presented above. The axiom of reducibility draws attention to the problematic status of impredicative definitions. To quote Weyl 1946, the axiom of reducibility "is a bold, an almost fantastic axiom; there is little justification for it in the real world in which we live, and none at all in the evidence on which our mind bases its constructions"! So far, no contradictions have been found using the reducibility axiom. However, as we shall see below, proof-theoretic investigations confirm the extreme strength of such a principle. The idea of the ramified hierarchy has been extremely important in mathematical logic. Russell considered only the finite iteration of the hierarchy: first-order, second-order, etc., but from the beginning, the possibility of extending the ramification transfinitely was considered. Poincare (1909) mentions the work of Koenig in this direction. For the example above of numbers of different order, he also defines a number to be inductive of order $\omega$ if it is inductive of all finite orders. He then points out that x+y is inductive of order $\omega$ if both $x$ and $y$ are. This shows that the introduction of transfinite orders can in some case play the role of the axiom of reducibility. Such transfinite extension of the ramified hierarchy was analysed further by Godel who noticed the key fact that the following form of the reducibility axiom is actually provable: when one extends the ramified hierarchy of properties over the natural numbers into the transfinite this hierarchy collapses at level $\omega\_1$, the least uncountable ordinal (Godel 1995, and Prawitz 1970). Furthermore, while at all levels $\lt \omega\_1$, the collection of predicates is countable, the collection of predicates at level $\omega\_1$ is of cardinality $\omega\_1$. This fact was a strong motivation behind Godel's model of constructible sets. In this model the collection of all subsets of the set of natural numbers (represented by predicates) is of cardinality $\omega\_1$ and is similar to the ramified hierarchy. This model satisfies in this way the Continuum Hypothesis, and gives a relative consistency proof of this axiom. (The motivation of Godel was originally only to build a model where the collection of all subsets of natural numbers is well-ordered.) The ramified hierarchy has been also the source of much work in proof theory. After the discovery by Gentzen that the consistency of Arithmetic could be proved by transfinite induction (over decidable predicates) along the ordinal $\varepsilon\_0$, the natural question was to find the corresponding ordinal for the different levels of the ramified hierarchy. Schutte (1960) found that for the first level of the ramified hierarchy, that is if we extend arithmetic by quantifying only over first-order properties, we get a system of ordinal strength $\varepsilon\_{\varepsilon\_0}$. For the second level we get the ordinal strength $\varepsilon\_{\varepsilon\_{ \varepsilon\_0}}$, etc. We recall that $\varepsilon\_{\alpha}$ denotes the $\alpha$-th $\varepsilon$-ordinal number, an $\varepsilon$-ordinal number being an ordinal $\beta$ such that $\omega^{\beta} = \beta$, see Schutte (1960). Godel stressed the fact that his approach to the problem of the continuum hypothesis was not constructive, since it needs the uncountable ordinal $\omega\_1$, and it was natural to study the ramified hierarchy along constructive ordinals. After preliminary works of Lorenzen and Wang, Schutte analysed what happens if we proceed in the following more constructive way. Since arithmetic has for ordinal strength $\varepsilon\_0$ we consider first the iteration of the ramified hierarchy up to $\varepsilon\_0$. Schutte computed the ordinal strength of the resulting system and found an ordinal strength $u(1)\gt \varepsilon\_0$. We iterate then ramified hierarchy up to this ordinal $u(1)$ and get a system of ordinal strength $u(2)\gt u(1)$, etc. Each $u(k)$ can be computed in terms of the so-called Veblen hierarchy: $u(k+1)$ is $\phi\_{u(k)}(0)$. The limit of this process gives an ordinal called $\Gamma\_0$: if we iterate the ramified hierarchy up to the ordinal $\Gamma\_0$ we get a system of ordinal strength $\Gamma\_0$. Such an ordinal was obtained independently about the same time by S. Feferman. It has been claimed that $\Gamma\_0$ plays for predicative systems a role similar to $\varepsilon\_0$ for Arithmetic. Recent proof-theoretical works however are concerned with systems having bigger proof-theoretical ordinals that can be considered predicative (see for instance Palmgren 1995). Besides these proof theoretic investigations related to the ramified hierarchy, much work has been devoted in proof theory to analysing the consistency of the axiom of reducibility, or, equivalently, the consistency of impredicative definitions. Following Gentzen's analysis of the cut-elimination property in the sequent calculus, Takeuti found an elegant sequent formulation of simple type theory (without ramification) and made the bold conjecture that cut-elimination should hold for this system. This conjecture seemed at first extremely dubious given the circularity of impredicative quantification, which is well reflected in this formalism. The rule for quantifications is indeed \[ \frac{\Gamma \vdash \forall X[A(X)]}{\Gamma \vdash A[X:=T]} \] where $T$ is any term predicate, which may itself involve a quantification over all predicates. Thus the formula $A[X:=T]$ may be itself much more complex than the formula $A(X)$. One early result is that cut-elimination for Takeuti's impredicative system implies in a finitary way the consistency of second-order Arithmetic. (One shows that this implies the consistency of a suitable form of infinity axiom, see Andrews 2002.) Following work by Schutte (1960b), it was later shown by W. Tait and D. Prawitz that indeed the cut-elimination property holds (the proof of this has to use a stronger proof theoretic principle, as it should be according to the incompleteness theorem.) What is important here is that these studies have revealed the extreme power of impredicative quantification or, equivalently, the extreme power of the axiom of reducibility. This confirms in some way the intuitions of Poincare and Russell. The proof-theoretic strength of second-order Arithmetic is way above all ramified extensions of Arithmetic considered by Schutte. On the other hand, despite the circularity of impredicative definitions, which is made so explicit in Takeuti's calculus, no paradoxes have been found yet in second-order Arithmetic. Another research direction in proof theory has been to understand how much of impredicative quantification can be explained from principles that are available in intuitionistic mathematics. The strongest such principles are strong forms of inductive definitions. With such principles, one can explain a limited form of an impredicative quantification, called $\Pi\_{1}^1$-comprehension, where one uses only one level of impredicative quantification over predicates. Interestingly, almost all known uses of impredicative quantifications: Leibniz equality, least upper bound, etc., can be done with $\Pi\_{1}^1$-comprehension. This reduction of $\Pi\_{1}^1$-comprehension was first achieved by Takeuti in a quite indirect way, and was later simplified by Buchholz and Schutte using the so-called $\Omega$-rule. It can be seen as a constructive explanation of some restricted, but nontrivial, uses of impredicative definitions. ## 4. Type Theory/Set Theory Type theory can be used as a foundation for mathematics, and indeed, it was presented as such by Russell in his 1908 paper, which appeared the same year as Zermelo's paper, presenting set theory as a foundation for mathematics. It is clear intuitively how we can explain type theory in set theory: a type is simply interpreted as a set, and function types $A \rightarrow B$ can be explained using the set theoretic notion of function (as a functional relation, i.e. a set of pairs of elements). The type $A \rightarrow o$ corresponds to the powerset operation. The other direction is more interesting. How can we explain the notion of sets in terms of types? There is an elegant solution, due to A. Miquel, which complements previous works by P. Aczel (1978) and which has also the advantage of explaining non necessarily well-founded sets a la Finsler. One simply interprets a set as a pointed graph (where the arrow in the graph represents the membership relation). This is very conveniently represented in type theory, a pointed graph being simply given by a type A and a pair of elements \[ a:A, R:A \rightarrow A \rightarrow o \] We can then define in type theory when two such sets $A, a, R$ and $B, b, S$ are equal: this is the case iff there is a bisimulation $T$ between $A$ and $B$ such that $Tab$ holds. A bisimulation is a relation \[ T:A\rightarrow B\rightarrow o \] such that whenever $Txy$ and $Rxu$ hold, there exists $v$ such that $Tuv$ and $Syv$ hold, and whenever $Txy$ and $Ryv$ hold, there exists $u$ such that $Tuv$ and $Rxu$ hold. We can then define the membership relation: the set represented $B, b, S$ is a member of the set represented by $A, a, R$ iff there exists $a\_1$ such that $Ra\_1a$ and $A, a\_1, R$ and $B, b, S$ are bisimilar. It can then be checked that all the usual axioms of set theory extensionality, power set, union, comprehension over bounded formulae (and even antifoundation, so that the membership relation does not need to be well-founded) hold in this simple model. (A bounded formula is a formula where all quantifications are of the form $\forall x \in a\ldots$ or $\exists x \in a\ldots$). In this way it can been shown that Church's simple type theory is equiconsistent with the bounded version of Zermelo's set theory. ## 5. Type Theory/Category Theory There are deep connections between type theory and category theory. We limit ourselves to presenting two applications of type theory to category theory: the constructions of the free cartesian closed category and of the free topos (see the entry on category theory for an explanation of "cartesian closed" and "topos"). For building the free cartesian closed category, we extend simple type theory with the type 1 (unit type) and the product type $A \times B$, for $A, B$ types. The terms are extended by adding pairing operations and projections and a special element of type 1. As in Lambek and Scott 1986, one can then define a notion of typed conversions between terms, and show that this relation is decidable. One can then show (Lambek and Scott 1986) that the category with types as objects and as morphisms from $A$ to $B$ the set of closed terms of type $A \rightarrow B$ (with conversion as equality) is the free cartesian closed category. This can be used to show that equality between arrows in this category is decidable. The theory of types of Church can also be used to build the free topos. For this we take as objects pairs $A,E$ with $A$ type and $E$ a partial equivalence relation, that is a closed term $E:A \rightarrow A \rightarrow o$ which is symmetric and transitive. We take as morphisms between $A, E$ and $B, F$ the relations $R:A\rightarrow B\rightarrow o$ that are functional that is such that for any $a:A$ satisfying $E a a$ there exists one, and only one (modulo $F)$ element $b$ of $B$ such that $F b b$ and $R a b$. For the subobject classifier we take the pair $o, E$ with $E:o\rightarrow o\rightarrow o$ defined as \[ E M N = \text{ and } (\imply\, M N) (\imply\,N M) \] One can then show that this category forms a topos, indeed the free topos. It should be noted that the type theory in Lambek and Scott 1986 uses a variation of type theory, introduced by Henkin and refined by P. Andrews (2002) which is to have an extensional equality as the only logical connective, i.e. a polymorphic constant \[ \text{eq} : A \rightarrow A \rightarrow o \] and to define all logical connectives from this connective and constants $T, F : o$. For instance, one defines \[ \forall P =\_{df} \text{eq} (\lambda x.T) P \] The equality at type $o$ is logical equivalence. One advantage of the intensional formulation is that it allows for a direct notation of proofs based on $\lambda$-calculus (Martin-Lof 1971 and Coquand 1986). ## 6. Extensions of Type System, Polymorphism, Paradoxes We have seen the analogy between the operation A $\rightarrow$ A $\rightarrow$ o on types and the powerset operation on sets. In set theory, the powerset operation can be iterated transfinitely along the cumulative hierarchy. It is then natural to look for analogous transfinite versions of type theory. One such extension of Church's simple type theory is obtained by adding universes (Martin-Lof 1970). Adding a universe is a reflection process: we add a type $U$ whose objects are the types considered so far. For Church's simple type theory we will have \[ o:U, i:U \text{ and } A\rightarrow B:U \text{ if } A:U, B:U \] and, furthermore, $A$ is a type if $A:U$. We can then consider types such as \[ (A:U) \rightarrow A \rightarrow A \] and functions such as \[ \text{id} = \lambda A.\lambda x.x : (A:U) \rightarrow A \rightarrow A \] The function id takes as argument a "small" type $A:U$ and an element $x$ of type $A$, and outputs an element of type $A$. More generally if $T(A)$ is a type under the assumption $A:U$, one can form the dependent type \[ (A:U) \rightarrow T(A) \] That $M$ is of this type means that $M A:T(A)$ whenever $A:U$. We get in this way extensions of type theory whose strength is similar to the one of Zermelo's set theory (Miquel 2001). More powerful form of universes are considered in (Palmgren 1998). Miquel (2003) presents a version of type theory of strength equivalent to the one of Zermelo-Fraenkel. One very strong form of universe is obtained by adding the axiom $U:U$. This was suggested by P. Martin-Lof in 1970. J.Y. Girard showed that the resulting type theory is inconsistent as a logical system (Girard 1972). Though it seems at first that one could directly reproduce Russell's paradox using a set of all sets, such a direct paradox is actually not possible due to the difference between sets and types. Indeed the derivation of a contradiction in such a system is subtle and has been rather indirect (though, as noticed in Miquel 2001, it can now be reduced to Russell's paradox by representing sets as pointed graphs). J.Y. Girard first obtained his paradox for a weaker system. This paradox was refined later (Coquand 1994 and Hurkens 1995). (The notion of pure type system, introduced in Barendregt 1992, is convenient for getting a sharp formulation of these paradoxes.) Instead of the axiom $U:U$ one assumes only \[ (A:U) \rightarrow T(A) : U \] if $T(A) : U [A:U]$. Notice the circularity, indeed of the same kind as the one that is rejected by the ramified hierarchy: we define an element of type $U$ by quantifying over all elements of $U$. For instance the type \[ (A:U) \rightarrow A \rightarrow A:U \] will be the type of the polymorphic identity function. Despite this circularity, J.Y. Girard was able to show normalisation for type systems with this form of polymorphism. However, the extension of Church's simple type theory with polymorphism is inconsistent as a logical system, i.e. all propositions (terms of type o) are provable. J.Y. Girard's motivation for considering a type system with polymorphism was to extend Godel's Dialectica (Godel 1958) interpretation to second-order arithmetic. He proved normalisation using the reducibility method, that had been introduced by Tait (1967) while analysing Godel 1958. It is quite remarkable that the circularity inherent in impredicativity does not result in non-normalisable terms. (Girard's argument was then used to show that cut-elimination terminates in Takeuti's sequent calculus presented above.) A similar system was introduced independently by J. Reynolds (1974) while analysing the notion of polymorphism in computer science. Martin-Lof's introduction of a type of all types comes from the identification of the concept of propositions and types, suggested by the work of Curry and Howard. It is worth recalling here his three motivating points: 1. Russell's definition of types as ranges of significance of propositional functions 2. the fact that one needs to quantify over all propositions (impredicativity of simple type theory) 3. identification of proposition and types Given (1) and (2) we should have a type of propositions (as in simple type theory), and given (3) this should also be the type of all types. Girard's paradox shows that one cannot have (1),(2) and (3) simultaneously. Martin-Lof's choice was to take away (2), restricting type theory to be predicative (and, indeed, the notion of universe appeared first in type theory as a predicative version of the type of all types). The alternative choice of taking away (3) is discussed in Coquand 1986. ## 7. Univalent Foundations The connections between type theory, set theory and category theory gets a new light through the work on Univalent Foundations (Voevodsky 2015) and the Axiom of Univalence. This involves in an essential way the extension of type theory described in the previous section, in particular dependent types, the view of propositions as types, and the notion of universe of types. These development are also relevant for discussing the notion of structure, the importance of which was for instance emphasized in Russell 1959. Martin-Lof 1975 [1973] introduced a new basic type $\mathbf{Id}\_A (a,b)$, if $a$ and $b$ are in the type $A$, which can be thought as the type of equality proofs of the element $a$ and $b$. An important feature of this new type is that it can be iterated, so that we can consider the type $\mathbf{Id}\_{\mathbf{Id}\_A (a,b)}(p,q)$ if $p$ and $q$ are of type $\mathbf{Id}\_A (a,b)$. If we think of a type as a special kind of set, it is natural to conjecture that such a type of equality proofs is always inhabited for any two equality proofs $p$ and $q$. Indeed, intuitively, there seems to be at most an equality proof between two elements $a$ and $b$. Surprisingly, Hofmann and Streicher 1996 designed a model of dependent type theory where this is not valid, that is a model where they can be different proofs that two elements are equal. In this model, a type is interpreted by a groupoid and the type $\mathbf{Id}\_A (a,b)$ by the set of isomorphisms between $a$ and $b$, set which may have more than one element. The existence of this model has the consequence that it cannot be proved in general in type theory that an equality type has at most one element. This groupoid interpretation has been generalized in the following way, which gives an intuitive interpretation of the identity type. A type is interpreted by a topological space, up to homotopy, and a type $\mathbf{Id}\_A (a,b)$ is interpreted by the type of paths connecting $a$ and $b$. (See Awodey et al. 2013 and [HoTT 2013, Other Internet Resources].) Voevodsky 2015 introduced the following stratification of types. (This stratification was motivated in part by this interpretation of a type as a topological space, but can be understood directly without reference to this interpretation.) We say that a type $A$ is a proposition if we have $\mathbf{Id}\_A (a,b)$ for any element $a$ and $b$ of $A$ (this means that the type $A$ has at most one element). We say that a type $A$ is a set if the type $\mathbf{Id}\_A (a,b)$ is a proposition for any element $a$ and $b$ of $A$. We say that a type $A$ is a groupoid if the type $\mathbf{Id}\_A (a,b)$ is a set for any element $a$ and $b$ of $A$. The justification of this terminology is that it can be shown, only using the rules of type theory, that any such type can indeed be seen as a groupoid in the usual categorical sense, where the objects are the elements of this type, the set of morphisms between $a$ and $b$ being represented by the set $\mathbf{Id}\_A (a,b)$. The composition is the proof of transitivity of equality, and the identity morphism is the proof of reflexivity of equality. The fact that each morphism has an inverse corresponds to the fact that identity is a symmetric relation. This stratification can then be extended and we can define when a type is a 2-groupoid, 3-groupoid and so on. In this view, type theory appears as a vast generalization of set theory, since a set is a particular kind of type. Voevodsky 2015 introduces also a notion of equivalence between types, notion which generalizes in an uniform way the notions of logical equivalence between propositions, bijection between sets, categorical equivalence between groupoids, and so on. We say that a map $f:A\rightarrow B$ is an equivalence if, for any element $b$ in $B$ the type of pairs $a,p$ where $p$ is of type $\mathbf{Id}\_B (f a,b)$, is a proposition and is inhabited. This expresses in a strong way that an element in $B$ is the image of exactly one element in $A$, and if $A$ and $B$ are sets, we recover the usual notion of bijection between sets. (In general if $f:A\rightarrow B$ is an equivalence, then we have a map $B\rightarrow A$, which can be thought of as the inverse of $f$.) It can be shown for instance that the identity map is always an equivalence. Let $\text{Equiv}(A,B)$ be the type of pairs $f,p$ where $f:A\rightarrow B$ and $p$ is a proof that $f$ is an equivalence. Using the fact that the identity map is an equivalence we have an element of $\text{Equiv}(A,A)$ for any type $A$. This implies that we have a map \[ \mathbf{Id}\_U (A,B)\rightarrow \text{Equiv}(A,B) \] and the Axiom of Univalence states that this map is an equivalence. In particular, we have the implication \[ \text{Equiv}(A,B)\rightarrow \mathbf{Id}\_U (A,B) \] and so if there is an equivalence between two small types then these types are equal. This Axiom can be seen as a strong form of the extensionality principle. It indeed generalizes the Axiom of propositional extensionality mentioned by Church 1940, which states that two logically equivalent propositions are equal. Surprisingly, it also implies the Axiom of function extensionality, Axiom 10 in Church 1940, which states that two pointwise equal functions are equal (Voevodsky 2015). It also directly implies that two isomorphic sets are equal, that two categorically equivalent groupoids are equal, and so one. This can be used to give a formulation of the notion of transport of structures (Bourbaki 1957) along equivalences. For instance, let $M A$ be the type of monoid structures on the set $A$: this is the type of tuples $m, e, p$ where $m$ is a binary operation on $A$ and $e$ an element of $A$ and $p$ a proof that these elements satisfy the usual monoid laws. The rule of substitution of equal by equal takes the form \[ \mathbf{Id}\_U (A,B)\rightarrow M A\rightarrow M B \] If there is a bijection between $A$ and $B$ they are equal by the Axiom of Univalence, and we can use this implication to transport any monoid structure of $A$ in a monoid structure of $B$. Both Russell 1919 and Russell 1959 stress the importance of the notion of structure. For instance, in Chapter VI of Russell 1919, it is noticed that two similar relations essentially have the same properties, and hence have the same "structure". (The notion of "similarity" of relations was introduced in Russell 1901.) We can also use this framework to refine Russell's discussions on the notion of structure. For instance, let Monoid be the type of pairs $A,p$ where $p$ is an element of $M A$. Two such pairs $A,p$ and $B,q$ are isomorphic if there exists a bijection $f$ from $A$ to $B$ such that $q$ is equal to the transport of structure of $p$ along $f$. A consequence of the Axiom of Univalence is that two isomorphic elements of the type Monoid are equal, and hence shares the same properties. Notice that such a general transport of properties is not possible when structures are formulated in a set theoretic framework. Indeed, in a set theoretic framework, it is possible to formulate properties using the membership relations, for instance the property that the carrier set of the structure contains the natural number $0$, property that is not preserved in general by isomorphisms. Intuitively, the set theoretical description of a structure is not abstract enough since we can talk about the way this structure is built up. This difference between set theory and type theory is yet another illustration of the characterization by J.Reynolds 1983 of a type structure as a "syntactical discipline for enforcing level of abstraction".
type-theory-church	## 1. Syntax ### 1.1 Fundamental Ideas We start with an informal description of the fundamental ideas underlying the syntax of Church's formulation of type theory. All entities have types, and if a and b are types, the type of functions from elements of type b to elements of type a is written as $(\alpha \beta)$. (This notation was introduced by Church, but some authors write $(\beta \rightarrow \alpha)$ instead of $(\alpha \beta)$. See, for example, Section 2 of the entry on type theory.) As noted by Schonfinkel (1924), functions of more than one argument can be represented in terms of functions of one argument when the values of these functions can themselves be functions. For example, if f is a function of two arguments, for each element x of the left domain of f there is a function g (depending on x) such that $gy = fxy$ for each element y of the right domain of f. We may now write $g = fx$, and regard f as a function of a single argument, whose value for any argument x in its domain is a function $fx$, whose value for any argument y in its domain is fxy. For a more explicit example, consider the function + which carries any pair of natural numbers to their sum. We may denote this function by $+\_{((\sigma \sigma)\sigma)}$, where $\sigma$ is the type of natural numbers. Given any number x, $[+\_{((\sigma \sigma)\sigma)}x]$ is the function which, when applied to any number y, gives the value $[[+\_{((\sigma \sigma)\sigma)}x]y]$, which is ordinarily abbreviated as $x + y$. Thus $[+\_{((\sigma \sigma)\sigma)}x]$ is the function of one argument which adds x to any number. When we think of $+\_{((\sigma \sigma)\sigma)}$ as a function of one argument, we see that it maps any number x to the function $[+\_{((\sigma \sigma)\sigma)}x]$. More generally, if f is a function which maps n-tuples $\langle w\_{\beta},x\_{\gamma},\ldots ,y\_{\delta},z\_{\tau}\rangle$ of elements of types $\beta$, $\gamma$,..., $\delta$ ,$\tau$, respectively, to elements of type a, we may assign to f the type $((\ldots((\alpha \tau)\delta)\ldots \gamma)\beta)$. It is customary to use the convention of association to the left to omit parentheses, and write this type symbol simply as $(\alpha \tau \delta \ldots \gamma \beta)$. A set or property can be represented by a function (often called characteristic function) which maps elements to truth values, so that an element is in the set, or has the property, in question iff the function representing the set or property maps that element to truth. When a statement is asserted, the speaker means that it is true, so that $s x$ means that $s x$ is true, which also expresses the assertions that s maps x to truth and that $x \in s$. In other words, $x \in s$ iff $s x$. We take ${o}$ as the type symbol denoting the type of truth values, so we may speak of any function of type $({o}\alpha)$ as a set of elements of type a. A function of type $(({o}\alpha)\beta)$ is a binary relation between elements of type b and elements of type a. For example, if $\sigma$ is the type of the natural numbers, and $<$ is the order relation between natural numbers, $<$ has type $({o}\sigma \sigma)$, and for all natural numbers x and $y, {<}x y$ (which we ordinarily write as $x < y)$ has the value truth iff x is less than y. Of course, $<$ can also be regarded as the function which maps each natural number x to the set $<x$ of all natural numbers y such that x is less than y. Thus sets, properties, and relations may be regarded as particular kinds of functions. Church's type type theory is thus a logic of functions, and, in this sense, it is in the tradition of the work of Frege's Begriffsschrift. The opposite approach would be to reduce functions to relations, which was the approach taken by Whitehead and Russell (1927a) in the Principia Mathematica. Expressions which denote elements of type a are called well-formed formulas (wffs) of type a. Thus, statements of type theory are wffs of type ${o}$. If $\bA\_{\alpha}$ is a wff of type a in which $\bu\_{\alpha \beta}$ is not free, the function (associated with) $\bu\_{\alpha \beta}$ such that $\forall \bv\_{\beta}[\bu\_{\alpha \beta}\bv\_{\beta} = \bA\_{\alpha}]$ is denoted by $[\lambda \bv\_{\beta}\bA\_{\alpha}]$. Thus $\lambda \bv\_{\beta}$ is a variable-binder, like $\forall \bv\_{\beta}$ or $\exists \bv\_{\beta}$ (but with a quite different meaning, of course); l is known as an abstraction operator. $[\lambda \bv\_{\beta}\bA\_{\alpha}]$ denotes the function whose value on any argument $\bv\_{\beta}$ is $\bA\_{\alpha}$, where $\bv\_{\beta}$ may occur free in $\bA\_{\alpha}$. For example, $[\lambda n\_{\sigma}[4\cdot n\_{\sigma}+3]]$ denotes the function whose value on any natural number n is $4\cdot n+3$. Hence, when we apply this function to the number 5 we obtain $[\lambda n\_{\sigma}[4\cdot n\_{\sigma}+3]]5 = 4\cdot 5+3 = 23$. We use $\textsf{Sub}(\bB,\bv,\bA)$ as a notation for the result of substituting $\bB$ for $\bv$ in $\bA$, and $\textsf{SubFree}(\bB,\bv,\bA)$ as a notation for the result of substituting $\bB$ for all free occurrences of $\bv$ in $\bA$. The process of replacing $[\lambda \bv\_{\beta}\bA\_{\alpha}]\bB\_{\beta}$ by $\textsf{SubFree}(\bB\_{\beta},\bv\_{\beta},\bA\_{\alpha})$ (or vice-versa) is known as b-conversion, which is one form of l-conversion. Of course, when $\bA\_{{o}}$ is a wff of type ${o}$, $[\lambda \bv\_{\beta}\bA\_{{o}}]$ denotes the set of all elements $\bv\_{\beta}$ (of type $\beta)$ of which $\bA\_{{o}}$ is true; this set may also be denoted by $\{\bv\_{\beta}\|\bA\_{{o}}\}$. For example, $[\lambda x\ x<y]$ denotes the set of x such that x is less than y (as well as that property which a number x has if it is less than y). In familiar set-theoretic notation, \[[\lambda \bv\_{\beta} \bA\_{{o}}]\bB\_{\beta} = \textsf{SubFree}(\bB\_{\beta},\bv\_{\beta},\bA\_{{o}})\] would be written \[\bB\_{\beta} \in \{\bv\_{\beta}\|\bA\_{{o}}\} \equiv \textsf{SubFree}(\bB\_{\beta},\bv\_{\beta},\bA\_{{o}}).\] (By the Axiom of Extensionality for truth values, when $\bC\_{{o}}$ and $\bD\_{{o}}$ are of type ${o}, \bC\_{{o}} \equiv \bD\_{{o}}$ is equivalent to $\bC\_{{o}} = \bD\_{{o}}$.) Propositional connectives and quantifiers can be assigned types and can be denoted by constants of these types. The negation function maps truth values to truth values, so it has type $({o}{o})$. Similarly, disjunction and conjunction (etc.) are binary functions from truth values to truth values, so they have type $({o}{o}{o})$. The statement $\forall \bx\_{\alpha}\bA\_{{o}}$ is true iff the set $[\lambda \bx\_{\alpha}\bA\_{{o}}]$ contains all elements of type a. A constant $\Pi\_{{o}({o}\alpha)}$ can be introduced (for each type symbol $\alpha)$ to denote a property of sets: a set $s\_{{o}\alpha}$ has the property $\Pi\_{{o}({o}\alpha)}$ iff $s\_{{o}\alpha}$ contains all elements of type a. With this interpretation \[ \forall s\_{{o}\alpha}\left[\Pi\_{{o}({o}\alpha)}s\_{{o}\alpha} \equiv \forall x\_{\alpha}\left[s\_{{o}\alpha}x\_{\alpha}\right]\right] \] should be true, as well as \[ \Pi\_{{o}({o}\alpha)}[\lambda \bx\_{\alpha}\bA\_{{o}}] \equiv \forall \bx\_{\alpha}[[\lambda \bx\_{\alpha}\bA\_{{o}}]\bx\_{\alpha}] \label{eqPi} \] for any wff $\bA\_{{o}}$ and variable $\bx\_{\alpha}$. Since by l-conversion we have \[ [\lambda \bx\_{\alpha}\bA\_{{o}}]\bx\_{\alpha} \equiv \bA\_{{o}} \] this equation can be written more simply as \[ \Pi\_{{o}({o}\alpha)}[\lambda \bx\_{\alpha}\bA\_{{o}}] \equiv \forall \bx\_{\alpha}\bA\_{{o}} \] Thus, $\forall \bx\_{\alpha}$ can be defined in terms of $\Pi\_{{o}({o}\alpha)}$, and l is the only variable-binder that is needed. ### 1.2 Formulas (as certain terms) Before we state the definition of a "formula", a word of caution is in order. The reader may be accustomed to thinking of a formula as an expression which plays the role of an assertion in a formal language, and of a term as an expression which designates an object. Church's terminology is somewhat different, and provides a uniform way of discussing expressions of many different types. What we call well-formed formula of type a ($\text{wff}\_{\alpha}$) below would in more standard terminology be called term of type a, and then only certain terms, namely those with type ${o}$, would be called formulas. Anyhow, in this entry we have decided to stay with Church's original terminology. Another remark concerns the use of some specific mathematical notation. In what follows, the entry distinguishes between the symbols $\imath$, $\iota\_{(\alpha({o}\alpha))}$, and $\atoi$. The first is the symbol used for the type of individuals; the second is the symbol used for a logical constant (see Section 1.2.1 below); the third is the symbol used as a variable-binding operator that represents the definite description "the" (see Section 1.3.4). The reader should not confuse them and check to see that the browser is displaying these symbols correctly. #### 1.2.1 Definitions Type symbols are defined inductively as follows: * $\imath$ is a type symbol (denoting the type of individuals). There may also be additional primitive type symbols which are used in formalizing disciplines where it is natural to have several sorts of individuals. * ${o}$ is a type symbol (denoting the type of truth values). * If a and b are type symbols, then $(\alpha \beta)$ is a type symbol (denoting the type of functions from elements of type b to elements of type a). The primitive symbols are the following: * Improper symbols: [, ], $\lambda$ * For each type symbol a, a denumerable list of variables of type $\alpha : a\_{\alpha}, b\_{\alpha}, c\_{\alpha}\ldots$ * Logical constants: $\nsim\_{({o}{o})}$, $\lor\_{(({o}{o}){o})}$, $\Pi\_{({o}({o}\alpha))}$ and $\iota\_{(\alpha({o}\alpha))}$ (for each type symbol a). The types of these constants are indicated by their subscripts. The symbol $\nsim\_{({o}{o})}$ denotes negation, $\lor\_{(({o}{o}){o})}$ denotes disjunction, and $\Pi\_{({o}({o}\alpha))}$ is used in defining the universal quantifier as discussed above. $\iota\_{(\alpha({o}\alpha))}$ serves either as a description or selection operator as discussed in Section 1.3.4 and Section 1.3.5 below. * In addition, there may be other constants of various types, which will be called nonlogical constants or parameters. Each choice of parameters determines a particular formulation of the system of type theory. Parameters are typically used as names for particular entities in the discipline being formalized. A formula is a finite sequence of primitive symbols. Certain formulas are called well-formed formulas (wffs). We write $\textrm{wff}\_{\alpha}$ as an abbreviation for wff of type a, and define this concept inductively as follows: * A primitive variable or constant of type a is a wff$\_{\alpha}$. * If $\bA\_{\alpha \beta}$ and $\bB\_{\beta}$ are wffs of the indicated types, then $[\bA\_{\alpha \beta}\bB\_{\beta}]$ is a wff$\_{\alpha}$. * If $\bx\_{\beta}$ is a variable of type b and $\bA\_{\alpha}$ is a wff, then $[\lambda \bx\_{\beta}\bA\_{\alpha}]$ is a wff$\_{(\alpha \beta)}$. Note, for example, that by (a) $\nsim\_{({o}{o})}$ is a wff$\_{({o}{o})}$, so by (b) if $\bA\_{{o}}$ is a wff$\_{{o}}$, then $[\nsim\_{({o}{o})}\bA\_{{o}}]$ is a wff$\_{{o}}$. Usually, the latter wff will simply be written as $\nsim \bA$. It is often convenient to avoid parentheses, brackets and type symbols, and use conventions for omitting them. For formulas we use the convention of association to the right, and we may write $\lor\_{((oo)o)}\bA\_{{o}} \bB\_{{o}}$ instead of $[[\lor\_{((oo)o)}\bA\_{{o}}] \bB\_{{o}}]$. For types the corresponding convention is analogous. ##### Abbreviations: * $\bA\_{{o}} \lor \bB\_{{o}}$ stands for $\lor\_{ooo}\bA\_{{o}} \bB\_{{o}}$. * $\bA\_{{o}} \supset \bB\_{{o}}$ stands for $[\nsim\_{{o}{o}}\bA\_{{o}}] \lor \bB\_{{o}}$. * $\forall \bx\_{\alpha}\bA\_{{o}}$ stands for $\Pi\_{{o}({o}\alpha)} [\lambda \bx\_{\alpha}\bA\_{{o}}]$. * Other propositional connectives, and the existential quantifier, are defined in familiar ways. In particular, \[ {\bA\_{{o}} \equiv \bB\_{{o}} \quad \text{stands for} \quad [\bA\_{{o}} \supset \bB\_{{o}}] \land [\bB\_{{o}} \supset \bA\_{{o}}]} \] * $\sfQ\_{{o}\alpha \alpha}$ stands for $\lambda $x$\_{\alpha}\lambda $y$\_{\alpha}\forall f\_{{o}\alpha}{[}f\_{{o}\alpha} x\_{\alpha} \supset f\_{{o}\alpha}y\_{\alpha}{]}$. * $\bA\_{\alpha} = \bB\_{\alpha}$ stands for $\sfQ\_{{o}\alpha \alpha}\bA\_{\alpha}\bB\_{\alpha}$. The last definition is known as the Leibnizian definition of equality. It asserts that x and y are the same if y has every property that x has. Actually, Leibniz called his definition "the identity of indiscernibles" and gave it in the form of a biconditional: x and y are the same if x and y have exactly the same properties. It is not difficult to show that these two forms of the definition are logically equivalent. #### 1.2.2 Examples We now provide a few examples to illustrate how various assertions and concepts can be expressed in Church's type theory. Example 1 To express the assertion that "Napoleon is charismatic" we introduce constants $\const{Charismatic}\_{{o}\imath}$ and $\const{Napoleon}\_{\imath}$, with the types indicated by their subscripts and the obvious meanings, and assert the wff \[\const{Charismatic}\_{{o}\imath} \const{Napoleon}\_{\imath}\] If we wish to express the assertion that "Napoleon has all the properties of a great general", we might consider interpreting this to mean that "Napoleon has all the properties of some great general", but it seems more appropriate to interpret this statement as meaning that "Napoleon has all the properties which all great generals have". If the constant $\const{GreatGeneral}\_{{o}\imath}$ is added to the formal language, this can be expressed by the wff \[\forall p\_{{o}\imath}[\forall g\_{\imath}{[}\const{GreatGeneral}\_{{o}\imath}g\_{\imath} \supset p\_{{o}\imath}g\_{\imath}] \supset p\_{{o}\imath} \const{Napoleon}\_{\imath}]\] As an example of such a property, we note that the sentence "Napoleon's soldiers admire him" can be expressed in a similar way by the wff \[\forall x\_{\imath}{[}\const{Soldier}\_{{o}\imath}x\_{\imath} \land \const{CommanderOf}\_{\imath\imath}x\_{\imath} = \const{Napoleon}\_{\imath} \supset \const{Admires}\_{{o}\imath\imath}x\_{\imath}\const{Napoleon}\_{\imath}]\] By l-conversion, this is equivalent to \[[\lambda n\_{\imath}\forall x\_{\imath}{[}\const{Soldier}\_{{o}\imath}x\_{\imath} \land \const{CommanderOf}\_{\imath\imath}x\_{\imath} = n\_{\imath} \supset \const{Admires}\_{{o}\imath\imath}x\_{\imath}n\_{\imath}]]\, \const{Napoleon}\_{\imath}\] This statement asserts that one of the properties which Napoleon has is that of being admired by his soldiers. The property itself is expressed by the wff \[\lambda n\_{\imath}\forall x\_{\imath}{[}\const{Soldier}\_{{o}\imath}x\_{\imath} \land \const{CommanderOf}\_{\imath\imath}x\_{\imath} = n\_{\imath} \supset \const{Admires}\_{{o}\imath\imath}x\_{\imath}n\_{\imath}]\] Example 2 We illustrate some potential applications of type theory with the following fable. > > > A rich and somewhat eccentric lady named Sheila has an ostrich and a > cheetah as pets, and she wishes to take them from her hotel to her > remote and almost inaccessible farm. Various portions of the trip may > involve using elevators, boxcars, airplanes, trucks, very small boats, > donkey carts, suspension bridges, etc., and she and the pets will not > always be together. She knows that she must not permit the ostrich and > the cheetah to be together when she is not with them. > > > We consider how certain aspects of this problem can be formalized so that Sheila can use an automated reasoning system to help analyze the possibilities. There will be a set Moments of instants or intervals of time during the trip. She will start the trip at the location $\const{Hotel}$ and moment $\const{Start}$, and end it at the location $\const{Farm}$ and moment $\const{Finish}$. Moments will have type $\tau$, and locations will have type $\varrho$. A state will have type $\sigma$ and will specify the location of Sheila, the ostrich, and the cheetah at a given moment. A plan will specify where the entities will be at each moment according to this plan. It will be a function from moments to states, and will have type $(\sigma \tau)$. The exact representation of states need not concern us, but there will be functions from states to locations called $\const{LocationOfSheila}$, $\const{LocationOfOstrich}$, and $\const{LocationOfCheetah}$ which provide the indicated information. Thus, $\const{LocationOfSheila}\_{\varrho \sigma}[p\_{\sigma \tau}t\_{\tau}]$ will be the location of Sheila according to plan $p\_{\sigma \tau}$ at moment $t\_{\tau}$. The set $\const{Proposals}\_{{o}(\sigma \tau)}$ is the set of plans Sheila is considering. We define a plan p to be acceptable if, according to that plan, the group starts at the hotel, finishes at the farm, and whenever the ostrich and the cheetah are together, Sheila is there too. Formally, we define $\const{Acceptable}\_{{o}(\sigma \tau)}$ as \[\begin{multline} \lambda p\_{\sigma \tau} \left[ \begin{aligned} & \const{LocationOfSheila}\_{\varrho \sigma} [p\_{\sigma \tau}\const{Start}\_{\tau}] = \const{Hotel}\_{\varrho} \,\land\\ & \const{LocationOfOstrich}\_{\varrho \sigma}[p\_{\sigma \tau} \const{Start}\_{\tau}] = \const{Hotel}\_{\varrho} \,\land\\ & \const{LocationOfCheetah}\_{\varrho \sigma}[p\_{\sigma \tau} \const{Start}\_{\tau}] = \const{Hotel}\_{\varrho} \,\land\\ & \const{LocationOfSheila}\_{\varrho \sigma}[p\_{\sigma \tau} \const{Finish}\_{\tau}] = \const{Farm}\_{\varrho} \,\land\\ & \const{LocationOfOstrich}\_{\varrho \sigma}[p\_{\sigma \tau} \const{Finish}\_{\tau}] = \const{Farm}\_{\varrho} \,\land\\ & \const{LocationOfCheetah}\_{\varrho \sigma}[p\_{\sigma \tau} \const{Finish}\_{\tau}] = \const{Farm}\_{\varrho} \,\land\\ & \forall t\_{\tau} \left [ \begin{split} & \const{Moments}\_{{o}\tau}t\_{\tau} \\ & \supset \left [ \begin{split} & \left[ \begin{split} & \const{LocationOfOstrich}\_{\varrho \sigma}[p\_{\sigma \tau}t\_{\tau}] \\ & = \const{LocationOfCheetah}\_{\varrho \sigma}[p\_{\sigma \tau}t\_{\tau}] \end{split} \right] \\ & \supset \left[ \begin{split} & \const{LocationOfSheila}\_{\varrho \sigma}[p\_{\sigma \tau}t\_{\tau}] \\ & = \const{LocationOfCheetah}\_{\varrho\sigma}[p\_{\sigma\tau}t\_{\tau}] \end{split} \right] \end{split} \right] \end{split} \right] \end{aligned} \right] \end{multline}\] We can express the assertion that Sheila has a way to accomplish her objective with the formula \[\exists p\_{\sigma \tau}[\const{Proposals}\_{{o}(\sigma \tau)}p\_{\sigma \tau} \land \const{Acceptable}\_{{o}(\sigma \tau)}p\_{\sigma \tau}]\] Example 3 We now provide a mathematical example. Mathematical ideas can be expressed in type theory without introducing any new constants. An iterate of a function f from a set to itself is a function which applies f one or more times. For example, if $g(x) = f(f(f(x)))$, then g is an iterate of f. $[\text{ITERATE+}\_{{o}(\imath\imath)(\imath\imath)}f\_{\imath\imath}g\_{\imath\imath}]$ means that $g\_{\imath\imath}$ is an iterate of $f\_{\imath\imath}$. $\text{ITERATE+}\_{{o}(\imath\imath)(\imath\imath)}$ is defined (inductively) as \[ \lambda f\_{\imath\imath}\lambda g\_{\imath\imath}\forall p\_{{o}(\imath\imath)} \left [p\_{{o}(\imath\imath)}f\_{\imath\imath} \land \forall j\_{\imath\imath} \left [p\_{{o}(\imath\imath)}j\_{\imath\imath} \supset p\_{{o}(\imath\imath)} \left [\lambda x\_{\imath}f\_{\imath\imath} \left [j\_{\imath\imath}x\_{\imath} \right] \right] \right] \supset p\_{{o}(\imath\imath)}g\_{\imath\imath} \right] \] Thus, g is an iterate of f if g is in every set p of functions which contains f and which contains the function $\lambda x\_{\imath}f\_{\imath\imath}[j\_{\imath\imath}x\_{\imath}]$ (i.e., f composed with j) whenever it contains j. A fixed point of f is an element y such that $f(y) = y$. It can be proved that if some iterate of a function f has a unique fixed point, then f itself has a fixed point. This theorem can be expressed by the wff \[\begin{aligned} \forall f\_{\imath\imath} \left [ \begin{split} \exists g\_{\imath\imath} & \left [ \begin{split} &\text{ITERATE+}\_{{o}(\imath\imath)(\imath\imath)} f\_{\imath\imath} g\_{\imath\imath} \\ &{} \land \exists x\_{\imath} \left [ \begin{split} & g\_{\imath\imath}x\_{\imath} = x\_{\imath} \\ &{} \land \forall z\_{\imath} \left [g\_{\imath\imath} z\_{\imath} = z\_{\imath} \supset z\_{\imath} = x\_{\imath} \right] \end{split} \right] \end{split} \right]\\ & \supset { } \exists y\_{\imath} [f\_{\imath\imath}y\_{\imath} = y\_{\imath}] \end{split} \right]. \end{aligned}\] See Andrews et al. 1996, for a discussion of how this theorem, which is called THM15B, can be proved automatically. Example 4 An example from philosophy is Godel's variant of the ontological argument for the existence of God. This example illustrates two interesting aspects: * Church's type theory can be employed as a meta-logic to concisely embed expressive other logics such as the higher-order modal logic assumed by Godel. By exploiting the possible world semantics of this target logic, its syntactic elements are defined in such a way that the infrastructure of the meta-logic are reused as much as possible. In this technique, called shallow semantical embedding, the modal operator $\Box$, for example, is simply identified with (taken as syntactic sugar for) the l-formula \[\lambda \varphi\_{{o}\imath} \lambda w\_{\imath} \forall v\_{\imath} [R\_{{o}\imath\imath} w\_{\imath} v\_{\imath} \supset \varphi\_{{o}\imath} v\_{\imath}]\] where $R\_{{o}\imath\imath}$ denotes the accessibility relation associated with $\Box$ and type ${\imath}$ is identified with possible worlds. Moreover, since $\forall x\_{\alpha} [\bA\_{{o}\alpha} x\_{\alpha}]$ is shorthand in Church's type theory for $\Pi\_{{o}({o}\alpha)} [\lambda x\_{\alpha} [\bA\_{{o}\alpha} x\_{\alpha}]]$, the modal formula \[\Box \forall x \bP x\] is represented as \[\Box \Pi' [\lambda x\_{\alpha} \lambda w\_{\imath} [\bP\_{{o}\imath\alpha} x\_{\alpha} w\_{\imath}]]\] where $\Pi'$ stands for the l-term \[\lambda \Phi\_{{o}\imath\alpha} \lambda w\_{\imath} \forall x\_{\alpha} [\Phi\_{{o}\imath\alpha} x\_{\alpha} w\_{\imath}]\] and the $\Box$ gets resolved as described above. The above choice of $\Pi'$ realizes a possibilist notion of quantification. By introducing a binary "existence" predicate in the metalogic and by utilizing this predicate as an additional guard in the definition of $\Pi'$ an actualist notion of quantification can be obtained. Expressing that an embedded modal formula $\varphi\_{{o}\imath}$ is globally valid is captured by the formula $\forall x\_{\imath} [\varphi\_{{o}\imath} x\_{\imath}]$. Local validity (and also actuality) can be modeled as $\varphi\_{{o}\imath} n\_{\imath}$, where $n\_{\imath}$ is a nominal (constant symbol in the meta-logic) denoting a particular possible world. * The above technique can be exploited for a natural encoding and automated assessment of Godel's ontological argument in higher-order modal logic, which unfolds into formulas in Church's type theory such that higher-order theorem provers can be applied. Further details are presented in Section 6 (Logic and Philosophy) of the SEP entry on automated reasoning and also in Section 5.2; moreover, see Benzmuller & Woltzenlogel-Paleo 2014 and Benzmuller 2019. Example 5 Suppose we omit the use of type symbols in the definitions of wffs. Then we can write the formula $\lambda x\nsim[xx]$, which we shall call $\textrm{R}$. It can be regarded as denoting the set of all sets x such that x is not in x. We may then consider the formula $[\textrm{R R}]$, which expresses the assertion that $\textrm{R}$ is in itself. We can clearly prove $[\textrm{R R}] \equiv [[\lambda x\nsim [xx]] \textrm{R}]$, so by l-conversion we can derive $[\textrm{R R}] \equiv\, \nsim[\textrm{R R}]$, which is a contradiction. This is Russell's paradox. Russell's discovery of this paradox (Russell 1903, 101-107) played a crucial role in the development of type theory. Of course, when type symbols are present, $\textrm{R}$ is not well-formed, and the contradiction cannot be derived. ### 1.3 Axioms and Rules of Inference #### 1.3.1 Rules of Inference 1. Alphabetic Change of Bound Variables $(\alpha$-conversion). To replace any well-formed part $\lambda \bx\_{\beta}\bA\_{\alpha}$ of a wff by $\lambda \by\_{\beta} \textsf{Sub}(\by\_{\beta},\bx\_{\beta},\bA\_{\alpha})$, provided that $\by\_{\beta}$ does not occur in $\bA\_{\alpha}$ and $\bx\_{\beta}$ is not bound in $\bA\_{\alpha}$. 2. b-contraction. To replace any well-formed part $[\lambda \bx\_{\alpha}\bB\_{\beta}] \bA\_{\alpha}$ of a wff by $\textsf{Sub}(\bA\_{\alpha},\bx\_{\alpha},\bB\_{\beta})$, provided that the bound variables of $\bB\_{\beta}$ are distinct both from $\bx\_{\alpha}$ and from the free variables of $\bA\_{\alpha}$. 3. b-expansion. To infer $\bC$ from $\bD$ if $\bD$ can be inferred from $\bC$ by a single application of b-contraction. 4. Substitution. From $\bF\_{({o}\alpha)}\bx\_{\alpha}$, to infer $\bF\_{({o}\alpha)}\bA\_{\alpha}$, provided that $\bx\_{\alpha}$ is not a free variable of $\bF\_{({o}\alpha)}$. 5. Modus Ponens. From $[\bA\_{{o}} \supset \bB\_{{o}}]$ and $\bA\_{{o}}$, to infer $\bB\_{{o}}$. 6. Generalization. From $\bF\_{({o}\alpha)}\bx\_{\alpha}$ to infer $\Pi\_{{o}({o}\alpha)}\bF\_{({o}\alpha)}$, provided that $\bx\_{\alpha}$ is not a free variable of $\bF\_{({o}\alpha)}$. #### 1.3.2 Elementary Type Theory We start by listing the axioms for what we shall call elementary type theory. \[\begin{align} [p\_{{o}} \lor p\_{{o}}] & \supset p\_{{o}} \tag{E1}\\ p\_{{o}} & \supset [p\_{{o}} \lor q\_{{o}}] \tag{E2}\\ [p\_{{o}} \lor q\_{{o}}] & \supset [q\_{{o}} \lor p\_{{o}}] \tag{E3}\\ [p\_{{o}} \supset q\_{{o}}] & \supset [[r\_{{o}} \lor p\_{{o}}] \supset [r\_{{o}} \lor q\_{{o}}]] \tag{E4}\\ \Pi\_{{o}({o}\alpha)}f\_{({o}\alpha)} & \supset f\_{({o}\alpha)}x\_{\alpha} \tag{E$5^{\alpha}$} \\ \forall x\_{\alpha}[p\_{{o}} \lor f\_{({o}\alpha)}x\_{\alpha}] & \supset \left[p\_{{o}} \lor \Pi\_{{o}({o}\alpha)}f\_{({o}\alpha)}\right] \tag{E$6^{\alpha}$} \end{align}\] The theorems of elementary type theory are those theorems which can be derived, using the rules of inference, from Axioms $(\rE1){-}(\rE6^{\alpha})$ (for all type symbols $\alpha)$. We shall sometimes refer to elementary type theory as $\cT$. It embodies the logic of propositional connectives, quantifiers, and l-conversion in the context of type theory. To illustrate the rules and axioms introduced above, we give a short and trivial proof in $\cT$. Following each wff of the proof, we indicate how it was inferred. (The proof is actually quite inefficient, since line 3 is not used later, and line 7 can be derived directly from line 5 without using line 6. The additional proof lines have been inserted to illustrate some relevant aspects. For the sake of readability, many brackets have been deleted from the formulas in this proof. The diligent reader should be able to restore them.) \[\begin{alignat}{2} \forall x\_{\imath}\left[p\_{{o}} \lor f\_{{o}\imath}x\_{\imath}\right] \supset\left[p\_{{o}} \lor \Pi\_{{o}({o}\imath)} f\_{{o}\imath}\right] && \text{Axiom $6^{\imath}$} \tag\{1.}\\ \bigg[\lambda f\_{{o}\imath}\bigg[\forall x\_{\imath}[p\_{{o}} \lor f\_{{o}\imath}x\_{\imath}] \supset \bigg[p\_{{o}} \lor \Pi\_{{o}({o}\imath)}f\_{{o}\imath}\bigg]\bigg]\bigg] f\_{{o}\imath} && \text{b-expansion: 1} \tag\{2.}\\ \Pi\_{{o}({o}({o}\imath))}\bigg[\lambda f\_{{o}\imath}\bigg[\forall x\_{\imath}[p\_{{o}} \lor f\_{{o}\imath}x\_{\imath}] \supset \bigg[p\_{{o}} \lor \Pi\_{{o}({o}\imath)}f\_{{o}\imath}\bigg]\bigg]\bigg] && \text{Generalization: 2} \tag\{3.}\\ \bigg[\lambda f\_{{o}\imath}\bigg[\forall x\_{\imath}[p\_{{o}}\lor f\_{{o}\imath}x\_{\imath}] \supset \bigg[p\_{{o}} \lor \Pi\_{{o}({o}\imath)}f\_{{o}\imath}\bigg]\bigg]\bigg] [\lambda x\_{\imath}r\_{{o}\imath}x\_{\imath}] && \text{Substitution: 2} \tag\{4.}\\ \forall x\_{\imath}[p\_{{o}} \lor [\lambda x\_{\imath}r\_{{o}\imath}x\_{\imath}]x\_{\imath}] \supset \left[p\_{{o}} \lor \Pi\_{{o}({o}\imath)}\left[\lambda x\_{\imath}r\_{{o}\imath}x\_{\imath}\right]\right] && \text{b-contraction: 4} \tag\{5.}\\ \forall x\_{\imath}[p\_{{o}} \lor [\lambda y\_{\imath} r\_{{o}\imath} y\_{\imath}] x\_{\imath}] \supset \left[p\_{{o}} \lor \Pi\_{{o}({o}\imath)}\left[\lambda x\_{\imath}r\_{{o}\imath}x\_{\imath}\right]\right] && \text{a-conversion: 5} \tag\{6.}\\ \forall x\_{\imath}\left[p\_{{o}} \lor r\_{{o}\imath}x\_{\imath}\right] \supset \left[p\_{{o}} \lor \Pi\_{{o}({o}\imath)}\left[\lambda x\_{\imath}r\_{{o}\imath}x\_{\imath}\right]\right] && \text{b-contraction: 6} \tag\{7.} \end{alignat}\] Note that line 3 can be written as \[ \forall f\_{{o}\imath} [\forall x\_{\imath} [p\_{{o}} \lor f\_{{o}\imath}x\_{\imath}] \supset [p\_{{o}} \lor [\forall x\_{\imath} f\_{{o}\imath} x\_{\imath}] ]] \tag\{$3'.$} \] and line 7 can be written as \[ \forall x\_{\imath}[p\_{{o}} \lor r\_{{o}\imath}x\_{\imath}] \supset [p\_{{o}} \lor \forall x\_{\imath}r\_{{o}\imath}x\_{\imath}] \tag\{$7'.$} \] We have thus derived a well known law of quantification theory. We illustrate one possible interpretation of the line $7'$ wff (which is closely related to Axiom E6) by considering a situation in which a rancher puts some horses in a corral and leaves for the night. Later, he cannot remember whether he closed the gate to the corral. While reflecting on the situation, he comes to a conclusion which can be expressed on line $7'$ if we take the horses to be the elements of type $\imath$, interpret $p\_{{o}}$ to mean "the gate was closed", and interpret $r\_{{o}\imath}$ so that $r\_{{o}\imath}x\_{\imath}$ asserts "$x\_{\imath}$ left the corral". With this interpretation, line $7'$ says > > > If it is true of every horse that the gate was closed or that the > horse left the corral, then the gate was closed or every horse left > the corral. > > > To the axioms listed above we add the axioms below to obtain Church's type theory. #### 1.3.3 Axioms of Extensionality The axioms of boolean and functional extensionality are the following: \[\begin{align} [x\_{{o}} \equiv y\_{{o}}] & \supset x\_{{o}} = y\_{{o}} \tag{$7^{o}$} \\ \forall x\_{\beta}[f\_{\alpha \beta}x\_{\beta} = g\_{\alpha \beta}x\_{\beta}] & \supset f\_{\alpha \beta} = g\_{\alpha \beta}\tag{$7^{\alpha \beta}$} \end{align}\] Church did not include Axiom $7^{{o}}$ in his list of axioms in Church 1940, but he mentioned the possibility of including it. Henkin did include it in Henkin 1950. #### 1.3.4 Descriptions The expression \[\exists\_1\bx\_{\alpha}\bA\_{{o}}\] stands for \[[\lambda p\_{{o}\alpha}\exists y\_{\alpha}[p\_{{o}\alpha}y\_{\alpha} \land \forall z\_{\alpha}[p\_{{o}\alpha}z\_{\alpha} \supset z\_{\alpha} = y\_{\alpha}]]]\, [\lambda \bx\_{\alpha}\bA\_{{o}}]\] For example, \[\exists\_1 x\_{\alpha}P\_{{o}\alpha}x\_{\alpha}\] stands for \[[\lambda p\_{{o}\alpha}\exists y\_{\alpha}[p\_{{o}\alpha}y\_{\alpha} \land \forall z\_{\alpha}[p\_{{o}\alpha}z\_{\alpha} \supset z\_{\alpha} = y\_{\alpha}]]]\, [\lambda x\_{\alpha}P\_{{o}\alpha}x\_{\alpha}]\] By l-conversion, this is equivalent to \[\exists y\_{\alpha}[[\lambda x\_{\alpha}P\_{{o}\alpha}x\_{\alpha}]y\_{\alpha} \land \forall z\_{\alpha}[[\lambda x\_{\alpha}P\_{{o}\alpha}x\_{\alpha}] z\_{\alpha} \supset z\_{\alpha} = y\_{\alpha}]]\] which reduces by l-conversion to \[\exists y\_{\alpha}[P\_{{o}\alpha}y\_{\alpha} \land \forall z\_{\alpha}[P\_{{o}\alpha}z\_{\alpha} \supset z\_{\alpha} = y\_{\alpha}]]\] This asserts that there is a unique element which has the property $P\_{{o}\alpha}$. From this example we can see that in general, $\exists\_1\bx\_{\alpha}\bA\_{{o}}$ expresses the assertion that "there is a unique $\bx\_{\alpha}$ such that $\bA\_{{o}}$". When there is a unique such element $\bx\_{\alpha}$, it is convenient to have the notation $\atoi\bx\_{\alpha}\bA\_{{o}}$ to represent the expression "the $\bx\_{\alpha}$ such that $\bA\_{{o}}$". Russell showed in Whitehead & Russell 1927b how to provide contextual definitions for such notations in his formulation of type theory. In Church's type theory $\atoi\bx\_{\alpha}\bA\_{{o}}$ is defined as $\iota\_{\alpha({o}\alpha)}[\lambda \bx\_{\alpha}\bA\_{{o}}]$. Thus, $\atoi$ behaves like a variable-binding operator, but it is defined in terms of l with the aid of the logical constant $\iota\_{\alpha({o}\alpha)}$. Thus, l is still the only variable-binding operator that is needed. Since $\bA\_{{o}}$ describes $\bx\_{\alpha}, \iota\_{\alpha({o}\alpha)}$ is called a description operator. Associated with this notation is the following: ##### Axiom of Descriptions: \[ \exists\_1 x\_{\alpha}[p\_{{o}\alpha}x\_{\alpha}] \supset p\_{{o}\alpha} [\iota\_{\alpha({o}\alpha)}p\_{{o}\alpha}] \tag{$8^{\alpha}$} \] This says that when the set $p\_{{o}\alpha}$ has a unique member, then $\iota\_{\alpha({o}\alpha)}p\_{{o}\alpha}$ is in $p\_{{o}\alpha}$, and therefore is that unique member. Thus, this axiom asserts that $\iota\_{\alpha({o}\alpha)}$ maps one-element sets to their unique members. If from certain hypotheses one can prove \[\exists\_1\bx\_{\alpha}\bA\_{{o}}\] then by using Axiom $8^{\alpha}$ one can derive \[[\lambda \bx\_{\alpha}\bA\_{{o}}] [\iota\_{\alpha({o}\alpha)}[\lambda \bx\_{\alpha}\bA\_{{o}}]]\] which can also be written as \[[\lambda \bx\_{\alpha}\bA\_{{o}}] {[\atoi\bx\_{\alpha}\bA\_{{o}}]}\] We illustrate the usefulness of the description operator with a small example. Suppose we have formalized the theory of real numbers, and our theory has constants $1\_{\varrho}$ and $\times\_{\varrho \varrho \varrho}$ to represent the number 1 and the multiplication function, respectively. (Here $\varrho$ is the type of real numbers.) To represent the multiplicative inverse function, we can define the wff $\textrm{INV}\_{\varrho \varrho}$ as \[{\lambda z\_{\varrho} \atoi x\_{\varrho} [\times\_{\varrho \varrho \varrho}z\_{\varrho}x\_{\varrho} = 1\_{\varrho}]}\] Of course, in traditional mathematical notation we would not write the type symbols, and we would write $\times\_{\varrho \varrho \varrho}z\_{\varrho}x\_{\varrho}$ as $z \times x$ and write $\textrm{INV}\_{\varrho \varrho}z$ as $z^{-1}$. Thus $z^{-1}$ is defined to be that x* such that $z \times x = 1$. When Z is provably not 0, we will be able to prove $\exists\_1 x\_{\varrho}[\times\_{\varrho \varrho \varrho} \textrm{Z x}\_{\varrho} = 1\_{\varrho}]$ and $Z \times Z^{-1} = 1$, but if we cannot establish that Z is not 0, nothing significant about $Z^{-1}$ will be provable. #### 1.3.5 Axiom of Choice The Axiom of Choice can be expressed as follows in Church's type theory: \[ \exists x\_{\alpha}p\_{{o}\alpha}x\_{\alpha} \supset p\_{{o}\alpha}[\iota\_{\alpha({o}\alpha)}p\_{{o}\alpha}] \tag{$9^{\alpha}$} \] $(9^{\alpha})$ says that the choice function $\iota\_{\alpha({o}\alpha)}$ chooses from every nonempty set $p\_{{o}\alpha}$ an element, designated as $\iota\_{\alpha({o}\alpha)}p\_{{o}\alpha}$, of that set. When this form of the Axiom of Choice is included in the list of axioms, $\iota\_{\alpha({o}\alpha)}$ is called a selection operator instead of a description operator, and $\atoi\bx\_{\alpha} \bA\_{{o}}$ means "an $\bx\_{\alpha}$ such that $\bA\_{{o}}$" when there is some such element $\bx\_{\alpha}$. These selection operators have the same meaning as Hilbert's $\epsilon$-operator (Hilbert 1928). However, we here provide one such operator for each type a. It is natural to call $\atoi$ a definite description operator in contexts where $\atoi\bx\_{\alpha}\bA\_{{o}}$ means "the $\bx\_{\alpha}$ such that $\bA\_{{o}}$", and to call it an indefinite description operator in contexts where $\atoi\bx\_{\alpha}\bA\_{{o}}$ means "an $\bx\_{\alpha}$ such that $\bA\_{{o}}$". Clearly the Axiom of Choice implies the Axiom of Descriptions, but sometimes formulations of type theory are used which include the Axiom of Descriptions, but not the Axiom of Choice. Another formulation of the Axiom of Choice simply asserts the existence of a choice function without explicitly naming it: \[ \exists j\_{\alpha ({o}\alpha)}\forall p\_{{o}\alpha}[\exists x\_{\alpha}p\_{{o}\alpha}x\_{\alpha} \supset p\_{{o}\alpha}[j\_{\alpha({o}\alpha)}p\_{{o}\alpha}]] \tag{$\text{AC}^{\alpha}$} \] Normally when one assumes the Axiom of Choice in type theory, one assumes it as an axiom schema, and asserts AC$^{\alpha}$ for each type symbol a. A similar remark applies to the axioms for extensionality and description. However, modern proof systems for Church's type theory, which are, e.g., based on resolution, do in fact avoid the addition of such axiom schemata for reasons as further explained in Sections 3.4 and 4 below. They work with more constrained, goal-directed proof rules instead. Before proceeding, we need to introduce some terminology. $\cQ\_0$ is an alternative formulation of Church's type theory which will be described in Section 1.4 and is equivalent to the system described above using Axioms (1)-(8). A type symbol is propositional if the only symbols which occur in it are ${o}$ and parentheses. Yasuhara (1975) defined the relation "$\ge$" between types as the reflexive transitive closure of the minimal relation such that $(\alpha \beta) \ge \alpha$ and $(\alpha \beta) \ge \beta$. He established that: * If $\alpha \ge \beta$, then $\cQ\_0 \vdash$ AC$^{\alpha} \supset$ AC$^{\beta}$. * Given a set S of types, none of which is propositional, there is a model of $\cQ\_0$ in which AC$^{\alpha}$ fails if and only if $\alpha \ge \beta$ for some b in S. The existence of a choice functions for "higher" types thus entails the existence of choice functions for "lower" types, the opposite is generally not the case though. Buchi (1953) has shown that while the schemas expressing the Axiom of Choice and Zorn's Lemma can be derived from each other, the relationships between the particular types involved are complex. #### 1.3.6 Axioms of Infinity One can define the natural numbers (and therefore other basic mathematical structures such as the real and complex numbers) in type theory, but to prove that they have the required properties (such as Peano's Postulates), one needs an Axiom of Infinity. There are many viable possibilities for such an axiom, such as those discussed in Church 1940, section 57 of Church 1956, and section 60 of Andrews 2002. ### 1.4 A Formulation Based on Equality In Section 1.2.1, $\nsim\_{({o}{o})}, \lor\_{(({o}{o}){o})}$, and the $\Pi\_{({o}({o}\alpha))}$'s were taken as primitive constants, and the wffs $\sfQ\_{{o}\alpha \alpha}$ which denote equality relations at type a were defined in terms of these. We now present an alternative formulation $\cQ\_0$ of Church's type theory in which there are primitive constants $\sfQ\_{{o}\alpha \alpha}$ denoting equality, and $\nsim\_{({o}{o})}, \lor\_{(({o}{o}){o})}$, and the $\Pi\_{({o}({o}\alpha))}$'s are defined in terms of the $\sfQ\_{{o}\alpha \alpha}$'s. Tarski (1923) noted that in the context of higher-order logic, one can define propositional connectives in terms of logical equivalence and quantifiers. Quine (1956) showed how both quantifiers and connectives can be defined in terms of equality and the abstraction operator l in the context of Church's type theory. Henkin (1963) rediscovered these definitions, and developed a formulation of Church's type theory based on equality in which he restricted attention to propositional types. Andrews (1963) simplified the axioms for this system. $\cQ\_0$ is based on these ideas, and can be shown to be equivalent to a formulation of Church's type theory using Axioms (1)-(8) of the preceding sections. This section thus provides an alternative to the material in the preceding Sections 1.2.1-1.3.4. More details about $\cQ\_0$ can be found in Andrews 2002. #### 1.4.1 Definitions * Type symbols, improper symbols, and variables of $\cQ\_0$ are defined as in Section 1.2.1. * The logical constants of $\cQ\_0$ are $\sfQ\_{(({o}\alpha)\alpha)}$ and $\iota\_{(\imath({o}\imath))}$ (for each type symbol a). * Wffs of $\cQ\_0$ are defined as in Section 1.2.1. ##### Abbreviations: * $\bA\_{\alpha} = \bB\_{\alpha}$ stands for $\sfQ\_{{o}\alpha \alpha}\bA\_{\alpha}\bB\_{\alpha}$ * $\bA\_{{o}} \equiv \bB\_{{o}}$ stands for $\sfQ\_{{o}{o}{o}}$A$\_{{o}}$B$\_{{o}}$ * $T\_{{o}}$ stands for $\sfQ\_{{o}{o}{o}} = \sfQ\_{{o}{o}{o}}$ * $F\_{{o}}$ stands for $[\lambda x\_{{o}}T\_{{o}}] = [\lambda x\_{{o}}x\_{{o}}]$ * $\Pi\_{{o}({o}\alpha)}$ stands for $\sfQ\_{{o}({o}\alpha)({o}\alpha)}[\lambda x\_{\alpha}T\_{{o}}]$ * $\forall \bx\_{\alpha}\bA$ stands for $\Pi\_{{o}({o}\alpha)}[\lambda \bx\_{\alpha}\bA]$ * $\land\_{{o}{o}{o}}$ stands for $\lambda x\_{{o}}\lambda y\_{{o}}[[\lambda g\_{{o}{o}{o}}[g\_{{o}{o}{o}}T\_{{o}}T\_{{o}}]] = [\lambda g\_{{o}{o}{o}}[g\_{{o}{o}{o}}x\_{{o}}y\_{{o}}]]]$ * $\bA\_{{o}} \land \bB\_{{o}}$ stands for $\land\_{{o}{o}{o}} \bA\_{{o}} \bB\_{{o}}$ * $\nsim\_{{o}{o}}$ stands for $\sfQ\_{{o}{o}{o}}F\_{{o}}$ $T\_{{o}}$ denotes truth. The meaning of $\Pi\_{{o}({o}\alpha)}$ was discussed in Section 1.1. To see that this definition of $\Pi\_{{o}({o}\alpha)}$ is appropriate, note that $\lambda x\_{\alpha}T$ denotes the set of all elements of type a, and that $\Pi\_{{o}({o}\alpha)}s\_{{o}\alpha}$ stands for $\sfQ\_{{o}({o}\alpha)({o}\alpha)}[\lambda x\_{\alpha}T] s\_{{o}\alpha}$, respectively for $[\lambda x\_{\alpha}T] = s\_{{o}\alpha}$. Therefore $\Pi\_{{o}({o}\alpha)}s\_{{o}\alpha}$ asserts that $s\_{{o}\alpha}$ is the set of all elements of type a, so $s\_{{o}\alpha}$ contains all elements of type a. It can be seen that $F\_{{o}}$ can also be written as $\forall x\_{{o}}x\_{{o}}$, which asserts that everything is true. This is false, so $F\_{{o}}$ denotes falsehood. The expression $\lambda g\_{{o}{o}{o}}[g\_{{o}{o}{o}}x\_{{o}}y\_{{o}}]$ can be used to represent the ordered pair $\langle x\_{{o}},y\_{{o}}\rangle$, and the conjunction $x\_{{o}} \land y\_{{o}}$ is true iff $x\_{{o}}$ and $y\_{{o}}$ are both true, i.e., iff $\langle T\_{{o}},T\_{{o}}\rangle = \langle x\_{{o}},y\_{{o}}\rangle$. Hence $x\_{{o}} \land y\_{{o}}$ can be expressed by the formula $[\lambda g\_{{o}{o}{o}}[g\_{{o}{o}{o}}T\_{{o}}T\_{{o}}]] = [\lambda g\_{{o}{o}{o}}[g\_{{o}{o}{o}}x\_{{o}}y\_{{o}}]]$. Other propositional connectives, such as $\lor, \supset$, and the existential quantifier $\exists$, are easily defined. By using $\iota\_{(\imath({o}\imath))}$, one can define description operators $\iota\_{\alpha({o}\alpha)}$ for all types a. #### 1.4.2 Axioms and Rules of Inference $\cQ\_0$ has a single rule of inference. > > Rule R: From $\bC$ and $\bA\_{\alpha} = > \bB\_{\alpha}$, to infer the result of replacing one occurrence of > $\bA\_{\alpha}$ in $\bC$ by an occurrence of $\bB\_{\alpha}$, > provided that the occurrence of $\bA\_{\alpha}$ in $\bC$ is not (an > occurrence of a variable) immediately preceded by l. The axioms for $\cQ\_0$ are the following: \[\begin{align} [g\_{{o}{o}}T\_{{o}} \land g\_{{o}{o}}F\_{{o}}] &= \forall x\_{{o}}[g\_{{o}{o}}x\_{{o}}] \tag{Q1}\\ [x\_{\alpha} = y\_{\alpha}] & \supset [h\_{{o}\alpha}x\_{\alpha} = h\_{{o}\alpha}y\_{\alpha}] \tag{Q$2^{\alpha}$}\\ [f\_{\alpha \beta} = g\_{\alpha \beta}] & = \forall x\_{\beta}[f\_{\alpha \beta}x\_{\beta} = g\_{\alpha \beta}x\_{\beta}] \tag{Q$3^{\alpha \beta}$}\\ [\lambda \bx\_{\alpha}\bB\_{\beta}]\bA\_{\alpha} & = \textsf{SubFree}(\bA\_{\alpha},\bx\_{\alpha},\bB\_{\beta}), \tag{Q4}\\ &\quad \text{ provided } \bA\_{\alpha} \text{ is free for } \bx \text{ in } \bB\_{\beta}\\ \iota\_{\imath({o}\imath)}[\sfQ\_{{o}\imath\imath}y\_{\imath}] &= y\_{\imath} \tag{Q5} \end{align}\] The additional condition in axiom $(\rQ4)$ ensures that the substitution does not lead to free variables of A being bound in the result of the substitution. ## 2. Semantics It is natural to compare the semantics of type theory with the semantics of first-order logic, where the theorems are precisely the wffs which are valid in all interpretations. From an intuitive point of view, the natural interpretations of type theory are standard models, which are defined below. However, it is a consequence of Godel's Incompleteness Theorem (Godel 1931) that axioms (1)-(9) do not suffice to derive all wffs which are valid in all standard models, and there is no consistent recursively axiomatized extension of these axioms which suffices for this purpose. Nevertheless, experience shows that these axioms are sufficient for most purposes, and Leon Henkin considered the problem of clarifying in what sense they are complete. The definitions and theorem below constitute Henkin's (1950) solution to this problem, which is often referred to as general semantics or Henkin semantics. A frame is a collection $\{\cD\_{\alpha}\}\_{\alpha}$ of nonempty domains (sets) $\cD\_{\alpha}$, one for each type symbol a, such that $\cD\_{{o}} = \{\sfT,\sfF\}$ (where $\sfT$ represents truth and $\sfF$ represents falsehood), and $\cD\_{\alpha \beta}$ is some collection of functions mapping $\cD\_{\beta}$ into $\cD\_{\alpha}$. The members of $\cD\_{\imath}$ are called individuals. An interpretation $\langle \{\cD\_{\alpha}\}\_{\alpha}, \frI\rangle$ consists of a frame and a function $\frI$ which maps each constant C of type a to an appropriate element of $\cD\_{\alpha}$, which is called the denotation of C. The logical constants are given their standard denotations. An assignment of values in the frame $\{\cD\_{\alpha}\}\_{\alpha}$ to variables is a function $\phi$ such that $\phi \bx\_{\alpha} \in \cD\_{\alpha}$ for each variable $\bx\_{\alpha}$. (Notation: The assignment $\phi[a/x]$ maps variable x to value a and it is identical with $\phi$ for all other variable symbols different from x.) An interpretation $\cM = \langle \{\cD\_{\alpha}\}\_{\alpha}, \frI\rangle$ is a general model (aka Henkin model) iff there is a binary function $\cV$ such that $\cV\_{\phi}\bA\_{\alpha} \in \cD\_{\alpha}$ for each assignment $\phi$ and wff $\bA\_{\alpha}$, and the following conditions are satisfied for all assignments and all wffs: * $\cV\_{\phi}\bx\_{\alpha} = \phi \bx\_{\alpha}$ for each variable $\bx\_{\alpha}$. * $\cV\_{\phi}A\_{\alpha} = \frI A\_{\alpha}$ if $A\_{\alpha}$ is a primitive constant. * $\cV\_{\phi}[\bA\_{\alpha \beta}\bB\_{\beta}] = (\cV\_{\phi}\bA\_{\alpha \beta})(\cV\_{\phi}\bB\_{\beta})$ (the value of a function $\cV\_{\phi}\bA\_{\alpha \beta}$ at the argument $\cV\_{\phi}\bB\_{\beta})$. * $\cV\_{\phi}[\lambda \bx\_{\alpha}\bB\_{\beta}] =$ that function from $\cD\_{\alpha}$ into $\cD\_{\beta}$ whose value for each argument $z \in \cD\_{\alpha}$ is $\cV\_{\psi}\bB\_{\beta}$, where $\psi$ is that assignment such that $\psi \bx\_{\alpha} = z$ and $\psi \by\_{\beta} = \phi \by\_{\beta}$ if $\by\_{\beta} \ne \bx\_{\alpha}$. If an interpretation $\cM$ is a general model, the function $\cV$ is uniquely determined. $\cV\_{\phi}\bA\_{\alpha}$ is called the value of $\bA\_{\alpha}$ in $\cM$ with respect to $\phi$. One can easily show that the following statements hold in all general models $\cM$ for all assignments $\phi$ and all wffs $\bA$ and $\bB$: * $\cV\_{\phi} T\_{{o}} = \sfT$ and $\cV\_{\phi} F\_{{o}} = \sfF$ * $\cV\_{\phi} [\nsim\_{{o}{o}} \bA\_{{o}}] = \sfT$ iff $\cV\_{\phi} \bA\_{{o}} = \sfF$ * $\cV\_{\phi} [ \bA\_{{o}} \lor \bB\_{{o}} ] = \sfT$ iff $\cV\_{\phi} \bA\_{{o}} = \sfT$ or $\cV\_{\phi} \bB\_{{o}} = \sfT$ * $\cV\_{\phi} [ \bA\_{{o}} \land \bB\_{{o}} ] = \sfT$ iff $\cV\_{\phi} \bA\_{{o}} = \sfT$ and $\cV\_{\phi} \bB\_{{o}} = \sfT$ * $\cV\_{\phi} [ \bA\_{{o}} \supset \bB\_{{o}} ] = \sfT$ iff $\cV\_{\phi} \bA\_{{o}} = \sfF$ or $\cV\_{\phi} \bB\_{{o}} = \sfT$ * $\cV\_{\phi} [ \bA\_{{o}} \equiv \bB\_{{o}} ] = \sfT$ iff $\cV\_{\phi} \bA\_{{o}} = \cV\_{\phi} \bB\_{{o}}$ * $\cV\_{\phi} [\forall \bx\_{\alpha}\bA] = \sfT$ iff $\cV\_{\phi[a/x]} \bA= \sfT$ for all $a \in \cD\_{\alpha}$ * $\cV\_{\phi} [\exists \bx\_{\alpha}\bA] = \sfT$ iff there exists an $a \in \cD\_{\alpha}$ such that $\cV\_{\phi[a/x]} \bA= \sfT$ The semantics of general models is thus as expected. However, there is a subtlety to note regarding the following condition for arbitrary types a: * [equality] $\cV\_{\phi} [ \bA\_{\alpha} = \bB\_{\alpha} ] = \sfT$ iff $\cV\_{\phi} \bA\_{\alpha} = \cV\_{\phi} \bB\_{\alpha}$ When the definitions of Section 1.2.1 are employed, where equality has been defined in terms of Leibniz' principle, then this statement is not implied for all types a. It only holds if we additionally require that the domains $\cD\_{{o}\alpha}$ contain all the unit sets of objects of type a, or, alternatively, that the domains $\cD\_{{o}\alpha\alpha}$ contain the respective identity relations on objects of type a (which entails the former). The need for this additional requirement, which is not included in the original work of Henkin (1950), has been demonstrated in Andrews 1972a. When instead the alternative definitions of Section 1.4 are employed, then this requirement is obviously met due to the presence of the logical constants $\sfQ\_{{o}\alpha \alpha}$ in the signature, which by definition denote the respective identity relations on the objects of type a and therefore trivially ensure their existence in each general model $\cM$. It is therefore a natural option to always assume primitive equality constants (for each type a) in a concrete choice of base system for Church's type theory, just as realized in Andrews' system $\cQ\_0$. An interpretation $\langle \{\cD\_{\alpha}\}\_{\alpha}, \frI\rangle$ is a standard model iff for all a and $\beta , \cD\_{\alpha \beta}$ is the set of all functions from $\cD\_{\beta}$ into $\cD\_{\alpha}$. Clearly a standard model is a general model. We say that a wff $\bA$ is valid in a model $\cM$ iff $\cV\_{\phi}\bA = \sfT$ for every assignment $\phi$ into $\cM$. A model for a set $\cH$ of wffs is a model in which each wff of $\cH$ is valid. A wff $\bA$ is valid in the general [standard] sense iff $\bA$ is valid in every general [standard] model. Clearly a wff which is valid in the general sense is valid in the standard sense, but the converse of this statement is false. Completeness and Soundness Theorem (Henkin 1950): A wff is a theorem if and only if it is valid in the general sense. Not all frames belong to interpretations, and not all interpretations are general models. In order to be a general model, an interpretation must have a frame satisfying certain closure conditions which are discussed further in Andrews 1972b. Basically, in a general model every wff must have a value with respect to each assignment. A model is said to be finite iff its domain of individuals is finite. Every finite model for $\cQ\_0$ is standard (Andrews 2002, Theorem 5404), but every set of sentences of $\cQ\_0$ which has infinite models also has nonstandard models (Andrews2002, Theorem 5506). An understanding of the distinction between standard and nonstandard models can clarify many phenomena. For example, it can be shown that there is a model $\cM = \langle \{\cD\_{\alpha}\}\_{\alpha}, \frI\rangle$ in which $\cD\_{\imath}$ is infinite, and all the domains $\cD\_{\alpha}$ are countable. Thus $\cD\_{\imath}$ and $\cD\_{{o}\imath}$ are both countably infinite, so there must be a bijection h between them. However, Cantor's Theorem (which is provable in type theory and therefore valid in all models) says that $\cD\_{\imath}$ has more subsets than members. This seemingly paradoxical situation is called Skolem's Paradox. It can be resolved by looking carefully at Cantor's Theorem, i.e., $\nsim \exists g\_{{o}\imath\imath}\forall f\_{{o}\imath}\exists j\_{\imath}[g\_{{o}\imath\imath}j\_{\imath} = f\_{{o}\imath}]$, and considering what it means in a model. The theorem says that there is no function $g \in \cD\_{{o}\imath\imath}$ from $\cD\_{\imath}$ into $\cD\_{{o}\imath}$ which has every set $f\_{{o}\imath} \in \cD\_{{o}\imath}$ in its range. The usual interpretation of the statement is that $\cD\_{{o}\imath}$ is bigger (in cardinality) than $\cD\_{\imath}$. However, what it actually means in this model is that h cannot be in $\cD\_{{o}\imath\imath}$. Of course, $\cM$ must be nonstandard. While the Axiom of Choice is presumably true in all standard models, there is a nonstandard model for $\cQ\_0$ in which AC$^{\imath}$ is false (Andrews 1972b). Thus, AC$^{\imath}$ is not provable in $\cQ\_0$. Thus far, investigations of model theory for Church's type theory have been far less extensive than for first-order logic. Nevertheless, there has been some work on methods of constructing nonstandard models of type theory and models in which various forms of extensionality fail, models for theories with arbitrary (possibly incomplete) sets of logical constants, and on developing general methods of establishing completeness of various systems of axioms with respect to various classes of models. Relevant papers include Andrews 1971, 1972a,b, and Henkin 1975. Further related work can be found in Benzmuller et al. 2004, Brown 2004, 2007, and Muskens 2007. ## 3. Metatheory ### 3.1 Lambda-Conversion The first three rules of inference in Section 1.3.1 are called rules of l-conversion. If $\bD$ and $\bE$ are wffs, we write $\bD \conv \bE$ to indicate that $\bD$ can be converted to $\bE$ by applications of these rules. This is an equivalence relation between wffs. A wff $\bD$ is in b-normal form iff it has no well-formed parts of the form $[[\lambda \bx\_{\alpha}\bB\_{\beta}]\bA\_{\alpha}]$. Every wff is convertible to one in b-normal form. Indeed, every sequence of contractions (applications of rule 2, combined as necessary with alphabetic changes of bound variables) of a wff is finite; obviously, if such a sequence cannot be extended, it terminates with a wff in b-normal form. (This is called the strong normalization theorem.) By the Church-Rosser Theorem, this wff in b-normal form is unique modulo alphabetic changes of bound variables. For each wff $\bA$ we denote by ${\downarrow}\bA$ the first wff (in some enumeration) in b-normal form such that $\bA \conv {\downarrow} \bA$. Then $\bD \conv \bE$ if and only if ${\downarrow} \bD = {\downarrow} \bE$. By using the Axiom of Extensionality one can obtain the following derived rule of inference: > > > $\eta$-Contraction. Replace a well-formed part $[\lambda > \by\_{\beta}[\bB\_{\alpha \beta}\by\_{\beta}]]$ of a wff by > $\bB\_{\alpha \beta}$, provided $\by\_{\beta}$ does not occur free > in $\bB\_{\alpha \beta}$. > > > This rule and its inverse (which is called $\eta$-Expansion) are sometimes used as additional rules of l-conversion. See Church 1941, Stenlund 1972, Barendregt 1984, and Barendregt et al. 2013 for more information about l-conversion. It is worth mentioning (again) that l-abstraction replaces the need for comprehension axioms in Church's type theory. ### 3.2 Higher-Order Unification The challenges in higher-order unification are outlined very briefly. More details on the topic are given in Dowek 2001; its utilization in higher-order theorem provers is also discussed in Benzmuller & Miller 2014. Definition. A higher-order unifier for a pair $\langle \bA,\bB\rangle$ of wffs is a substitution $\theta$ for free occurrences of variables such that $\theta \bA$ and $\theta \bB$ have the same b-normal form. A higher-order unifier for a set of pairs of wffs is a unifier for each of the pairs in the set. Higher-order unification differs from first-order unification (Baader & Snyder 2001) in a number of important respects. In particular: 1. Even when a unifier for a pair of wffs exists, there may be no most general unifier (Gould 1966). 2. Higher-order unification is undecidable (Huet 1973b), even in the "second-order" case (Goldfarb 1981). However, an algorithm has been devised (Huet 1975, Jensen & Pietrzykowski 1976), called pre-unification, which will find a unifier for a set of pairs of wffs if one exists. The pre-unifiers computed by Huet's procedure are substitutions that can reduce the original unification problem to one involving only so called flex-flex unification pairs. Flex-flex pairs have variable head symbols in both terms to be unified and they are known to always have a solution. The concrete computation of these solutions can thus be postponed or omitted. Pre-unification is utilized in all the resolution based theorem provers mentioned in Section 4. Pattern unification refers a small subset of unification problems, first studied by Miller 1991, whose identification has been important for the construction of practical systems. In a pattern unification problem every occurrence of an existentially quantified variable is applied to a list of arguments that are all distinct variables bound by either a l-binder or a universal quantifier in the scope of the existential quantifier. Thus, existentially quantified variables cannot be applied to general terms but a very restricted set of bound variables. Pattern unification, like first-order unification, is decidable and most general unifiers exist for solvable problems. This is why pattern unification is preferably employed (when applicable) in some state-of-the-art theorem provers for Church's type theory. ### 3.3 A Unifying Principle The Unifying Principle was introduced in Smullyan 1963 (see also Smullyan 1995) as a tool for deriving a number of basic metatheorems about first-order logic in a uniform way. The principle was extended to elementary type theory by Andrews (1971) and to extensional type theory, that is, Henkin's general semantics without description or choice, by Benzmuller, Brown and Kohlhase (2004). We outline these extensions in some more detail below. #### 3.3.1 Elementary Type Theory The Unifying Principle was extended to elementary type theory (the system $\cT$ of Section 1.3.2) in Andrews 1971 by applying ideas in Takahashi 1967. This Unifying Principle for $\cT$ has been used to establish cut-elimination for $\cT$ in Andrews 1971 and completeness proofs for various systems of type theory in Huet 1973a, Kohlhase 1995, and Miller 1983. We first give a definition and then state the principle. Definition. A property $\Gamma$ of finite sets of wffs$\_{{o}}$ is an abstract consistency property iff for all finite sets $\cS$ of wffs$\_{{o}}$, the following properties hold (for all wffs A, B): 1. If $\Gamma(\cS)$, then there is no atom $\bA$ such that $\bA \in \cS$ and $[\nsim \bA] \in \cS$. 2. If $\Gamma(\cS \cup \{\bA\})$, then $\Gamma(\cS \cup {\downarrow} \bA\})$. 3. If $\Gamma(\cS \cup \{\nsim \nsim \bA\})$, then $\Gamma(\cS \cup \{\bA\})$. 4. If $\Gamma(\cS \cup \{\bA \lor \bB\})$, then $\Gamma(\cS \cup \{\bA\})$ or $\Gamma(\cS \cup \{\bB\})$. 5. If $\Gamma(\cS \cup \{\nsim[\bA \lor \bB]\})$, then $\Gamma(\cS \cup \{\nsim \bA,\nsim \bB\})$. 6. If $\Gamma(\cS \cup \{\Pi\_{{o}({o}\alpha)}\bA\_{{o}\alpha}\})$, then $\Gamma(\cS \cup \{\Pi\_{{o}({o}\alpha)}\bA\_{{o}\alpha}, \bA\_{{o}\alpha}\bB\_{\alpha}\})$ for each wff $\bB\_{\alpha}$. 7. If $\Gamma(\cS \cup \{\nsim \Pi\_{{o}({o}\alpha)}\bA\_{{o}\alpha}\})$, then $\Gamma(\cS \cup \{\nsim \bA\_{{o}\alpha}\bc\_{\alpha}\})$, for any variable or parameter $\bc\_{\alpha}$ which does not occur free in $\bA\_{{o}\alpha}$ or any wff in $\cS$. Note that consistency is an abstract consistency property. Unifying Principle for $\cT$. If $\Gamma$ is an abstract consistency property and $\Gamma(\cS)$, then $\cS$ is consistent in $\cT$. Here is a typical application of the Unifying Principle. Suppose there is a procedure $\cM$ which can be used to refute sets of sentences, and we wish to show it is complete for $\cT$. For any set of sentences, let $\Gamma(\cS)$ mean that $\cS$ is not refutable by $\cM$, and show that $\Gamma$ is an abstract consistency property. Now suppose that $\bA$ is a theorem of $\cT$. Then $\{\nsim \bA\}$ is inconsistent in $\cT$, so by the Unifying Principle not $\Gamma(\{\nsim \bA\})$, so $\{\nsim \bA\}$ is refutable by $\cM$. #### 3.3.2 Extensional Type Theory Extensions of the above Unifying principle towards Church's type theory with general semantics were studied since the mid nineties. A primary motivation was to support (refutational) completeness investigations for the proof calculi underlying the emerging higher-order automated theorem provers (see Section 4 below). The initial interest was on a fragment of Church's type theory, called extensional type theory, that includes the extensionality axioms, but excludes $\iota\_{(\alpha({o}\alpha))}$ and the axioms for it (description and choice were largely neglected in the automated theorem provers at the time). Analogous to before, a distinction has been made between extensional type theory with defined equality (as in Section 1.2.1, where equality is defined via Leibniz' principle) and extensional type theory with primitive equality (e.g., system $\cQ\_0$ as in Section 1.4, or, alternatively, a system based on logical constants $\nsim\_{({o}{o})}, \lor\_{(({o}{o}){o})}$, and the $\Pi\_{({o}({o}\alpha))}$'s as in Section 1.2.1, but with additional primitive logical constants $=\_{{o}\alpha\alpha}$ added). A first attempt towards a Unifying Principle for extensional type theory with primitive equality is presented in Kohlhase 1993. The conditions given there, which are still incomplete[1], were subsequently modified and complemented as follows: * To obtain a Unifying Principle for extensional type theory with defined equality, Benzmuller & Kohlhase 1997 added the following conditions for boolean extensionality, functional extensionality and saturation to the above conditions 1.-7. for $\cT$ (their presentation has been adapted here; for technical reasons, they also employ a slightly stronger variant for condition 2. based on b-conversion rather than b-normalization): 8. If $\Gamma(\cS \cup \{ \bA\_{{o}} = \bB\_{{o}} \})$, then $\Gamma(\cS \cup \{ \bA\_{{o}}, \bB\_{{o}} \})$ or $\Gamma(\cS \cup \{ \nsim \bA\_{{o}}, \nsim \bB\_{{o}} \})$ 9. If $\Gamma(\cS \cup \{ \bA\_{\alpha\beta} = \bB\_{\alpha\beta} \})$, then $\Gamma(\cS \cup \{ \bA\_{\alpha\beta} \bc\_\beta = \bB\_{\alpha\beta} \bc\_\beta \})$ for any parameter $\bc\_{\beta}$ which does not occur free in $\cS$. 10. $\Gamma(\cS \cup \{ \bA\_{{o}} \})$ or $\Gamma(\cS \cup \{ \nsim \bA\_{{o}} \})$ The saturation condition 10. was required to properly establish the principle. However, since this condition is related to the proof theoretic notion of cut-elimination, it limits the utility of the principle in completeness proofs for machine-oriented calculi. The principle was nevertheless used in Benzmuller & Kohlhase 1998a and Benzmuller 1999a,b to obtain a completeness proof for a system of extensional higher-order resolution. The principle was also applied in Kohlhase 1998 to study completeness for a related extensional higher-order tableau calculus,[2] in which the extensionality rules for Leibniz equality were adapted from Benzmuller & Kohlhase 1998a, respectively Benzmuller 1997. * Different options for achieving a Unifying Principle for extensional type theory with primitive equality are presented in Benzmuller 1999a (in this work primitive logical constants $=\_{{o}\alpha\alpha}$ were used in addition to $\nsim\_{({o}{o})}, \lor\_{(({o}{o}){o})}$, and the $\Pi\_{({o}({o}\alpha))}$'s; such a redundant choice of logical constants is not rare in higher-order theorem provers). One option is to introduce a reflexivity and substitutivity condition. An alternative is to combine a reflexivity condition with a condition connecting primitive with defined equality, so that the substitutivity condition follows. Note that introducing a defined notion of equality based on the Leibniz principle is, of course, still possible in this context (defined equality is denoted in the remainder of this section by $\doteq$ to properly distinguish it from primitive equality $=$): 8. Not $\Gamma(\cS \cup \{ \nsim [\bA\_{\alpha} = \bA\_{\alpha}] \})$ 9. If $\Gamma(\cS \cup \{ \bA\_{\alpha} = \bA\_{\alpha} \})$, then $\Gamma(\cS \cup \{ \bA\_{\alpha} \doteq \bA\_{\alpha} \})$ 10. $\Gamma(\cS \cup \{ \bA\_{{o}} \})$ or $\Gamma(\cS \cup \{ \nsim \bA\_{{o}} \})$ The saturation condition 10. still has to be added independent of which option is considered. The principle was applied in Benzmuller 1999a,b to prove completeness for the extensional higher-order RUE-resolution[3] calculus underlying the higher-order automated theorem prover LEO and its successor LEO-II. * In Benzmuller et al. 2004 the principle is presented in a very general way which allows for various possibilities concerning the treatment of extensionality and equality in the range between elementary type theory and extensional type theory. The principle is applied to obtain completeness proofs for an associated range of natural deduction calculi. The saturation condition is still used in this work. * Based on insights from Brown's (2004, 2007) thesis, a solution for replacing the undesirable saturation condition by two weaker conditions is presented in Benzmuller, Brown, and Kohlhase 2009; this work also further studies the relation between saturation and cut-elimination. The two weaker conditions, termed mating and decomposition, are easier to demonstrate than saturation in completeness proofs for machine-oriented calculi. They are (omitting some type information in the second one and abusing notation): 1. If $\Gamma(\cS \cup \{ \nsim \bA\_{{o}}, \bB\_{{o}} \})$ for atoms $\bA\_{{o}}$ and $\bB\_{{o}}$, then $\Gamma(\cS \cup \{ \nsim [\bA\_{{o}} \doteq \bB\_{{o}}] \})$ 2. If $\Gamma(\cS \cup \{ \nsim [h \overline{\bA^n\_{\alpha^n}} \doteq h \overline{\bB^n\_{\alpha^n}} ] \})$, where head symbol $h\_{\beta\overline{\alpha^n}}$ is a parameter, then there is an $i\ (1 \leq i \leq n)$ such that $\Gamma(\cS \cup \{ \nsim [\bA^i\_{\alpha^i} \doteq \bB^i\_{\alpha^i}] \})$. The modified principle is applied in Benzmuller et al. 2009 to show completeness for a sequent calculus for extensional type theory with defined equality. * A further extended Unifying Principle for extensional type theory with primitive equality is presented and used in Backes & Brown 2011 to prove the completeness of a tableau calculus for type theory which incorporates the axiom of choice. * A closely related and further simplified principle has also been presented and studied in Steen 2018, where it was applied for showing completeness of the paramodulation calculus that is underlying the theorem prover Leo-III (Steen & Benzmuller 2018, 2021). ### 3.4 Cut-Elimination and Cut-Simulation Cut-elimination proofs (see also the SEP entry on proof theory) for Church's type theory, which are often closely related to such proofs (Takahashi 1967, 1970; Prawitz 1968; Mints 1999) for other formulations of type theory, may be found in Andrews 1971, Dowek & Werner 2003, and Brown 2004. In Benzmuller et al. 2009 it is shown how certain wffs$\_{{o}}$, such as axioms of extensionality, descriptions, choice (see Sections 1.3.3 to 1.3.5), and induction, can be used to justify cuts in cut-free sequent calculi for elementary type theory. Moreover, the notions of cut-simulation and cut-strong axioms are introduced in this work, and the need for omitting defined equality and for eliminating cut-strong axioms such as extensionality, description, choice and induction in machine-oriented calculi (e.g., by replacing them with more constrained, goal-directed rules) in order to reduce cut-simulation effects are discussed as a major challenge for higher-order automated theorem proving. In other words, including cut-strong axioms in a machine-oriented proof calculus for Church's type theory is essentially as bad as including a cut rule, since the cut rule can be mimicked by them. ### 3.5 Expansion Proofs An expansion proof is a generalization of the notion of a Herbrand expansion of a theorem of first-order logic; it provides a very elegant, concise, and nonredundant representation of the relationship between the theorem and a tautology which can be obtained from it by appropriate instantiations of quantifiers and which underlies various proofs of the theorem. Miller (1987) proved that a wff $\bA$ is a theorem of elementary type theory if and only if $\bA$ has an expansion proof. In Brown 2004 and 2007, this concept is generalized to that of an extensional expansion proof to obtain an analogous theorem involving type theory with extensionality. ### 3.6 The Decision Problem Since type theory includes first-order logic, it is no surprise that most systems of type theory are undecidable. However, one may look for solvable special cases of the decision problem. For example, the system $\cQ\_{0}^1$ obtained by adding to $\cQ\_0$ the additional axiom $\forall x\_{\imath}\forall y\_{\imath}[x\_{\imath}=y\_{\imath}]$ is decidable. Although the system $\cT$ of elementary type theory is analogous to first-order logic in certain respects, it is a considerably more complex language, and special cases of the decision problem for provability in $\cT$ seem rather intractable for the most part. Information about some very special cases of this decision problem may be found in Andrews 1974, and we now summarize this. A wff of the form $\exists \bx^1 \ldots \exists \bx^n [\bA=\bB]$ is a theorem of $\cT$ iff there is a substitution $\theta$ such that $\theta \bA \conv \theta \bB$. In particular, $\vdash \bA=\bB$ iff $\bA \conv \bB$, which solves the decision problem for wffs of the form $[\bA=\bB]$. Naturally, the circumstance that only trivial equality formulas are provable in $\cT$ changes drastically when axioms of extensionality are added to $\cT$. $\vdash \exists \bx\_{\beta}[\bA=\bB]$ iff there is a wff $\bE\_{\beta}$ such that $\vdash[\lambda \bx\_{\beta}[\bA=\bB]]\bE\_{\beta}$, but the decision problem for the class of wffs of the form $\exists \bx\_{\beta}[\bA=\bB]$ is unsolvable. A wff of the form $\forall \bx^1 \ldots \forall \bx^n\bC$, where $\bC$ is quantifier-free, is provable in $\cT$ iff ${\downarrow} \bC$ is tautologous. On the other hand, the decision problem for wffs of the form $\exists \bz\bC$, where $\bC$ is quantifier-free, is unsolvable. (By contrast, the corresponding decision problem in first-order logic with function symbols is known to be solvable (Maslov 1967).) Since irrelevant or vacuous quantifiers can always be introduced, this shows that the only solvable classes of wffs of $\cT$ in prenex normal form defined solely by the structure of the prefix are those in which no existential quantifiers occur. ## 4. Automation ### 4.1 Machine-Oriented Proof Calculi The development, respectively improvement, of machine-oriented proof calculi for Church's type theory has made good progress in recent years; see, e.g., the references on superposition based calculi given below. The challenges are obviously much bigger than in first-order logic. The practically way more expressive nature of the term-language of Church's type theory causes a larger, bushier and more difficult to traverse proof search space than in first-order logic. Moreover, remember that unification, which constitutes a very important control and filter mechanism in first-order theorem proving, is undecidable (in general) in type theory; see Section 3.2. On the positive side, however, there is a chance to find significantly shorter proofs than in first-order logic. This is well illustrated with a small, concrete example in Boolos 1987, for which a fully automated proof seems now in reach (Benzmuller et al. 2023). Clearly, further progress is needed to further leverage the practical relevance of existing calculi for Church's type theory and their implementations (see Section 4.3). The challenges include * an appropriate handling of the impredicative nature of Church's type theory (some form of blind guessing cannot generally be avoided in a complete proof procedure, but must be intelligently guided), * the elimination/reduction of cut-simulation effects (see Section 3.4) caused by defined equality or cut-strong axioms (e.g., extensionality, description, choice, induction) in the search space, * the general undecidability of unification, rendering it a rather problematic filter mechanism for controlling proof search, * the invention of suitable heuristics for traversing the search space, * the provision of suitable term-orderings and their effective exploitation in term rewriting procedures, * and efficient data structures in combination with strong technical support for essential operations such l-conversion, substitution and rewriting. It is planned that future editions of this article further elaborate on machine-oriented proof calculi for Church's type theory. For the time being, however, we provide only a selection of historical and more recent references for the interested reader (see also Section 5 below): * Sequent calculi: Schutte 1960; Takahashi 1970; Takeuti 1987; Mints 1999; Brown 2004, 2007; Benzmuller et al. 2009. * Mating method: Andrews 1981; Bibel 1981; Bishop 1999. * Resolution calculi: Andrews 1971; Huet 1973a; Jensen & Pietrzykowski 1976; Benzmuller 1997, 1999a; Benzmuller & Kohlhase 1998a. * Tableau method: Kohlhase[4] 1995, 1998; Brown & Smolka 2010; Backes & Brown 2011. * Paramodulation calculi: Benzmuller 1999a,b; Steen 2018, Steen & Benzmuller 2018, 2021. * Superposition calculi: Bentkamp et al. 2018, Bentkamp et al. 2021, Bentkamp et al. 2023a,c * Combinator-based superposition calculi: Bhayat & Reger 2020 ### 4.2 Proof Assistants Early computer systems for proving theorems of Church's type theory (or extensions of it) include HOL (Gordon 1988; Gordon & Melham 1993), TPS (Andrews et al. 1996; Andrews & Brown 2006), Isabelle (Paulson 1988, 1990), PVS (Owre et al. 1996; Shankar 2001), IMPS (Farmer et al. 1993), HOL Light (Harrison 1996), OMEGA (Siekmann et al. 2006), and lClam (Richardson et al. 1998). Prominent proof assistents that support more powerful dependent type theory include Coq (Bertot & Casteran 2004) and the recent Lean system (de Moura & Ullrich 2021). See Other Internet References section below for links to further info on these and other provers mentioned later. Most of the early systems mentioned above focused (at least initially) on interactive proof. However, the TPS project in particular had been working since the mid-1980s on the integration of ND-style interactive proof and automated theorem proving using the mating method, and had investigated proof transformation techniques between the two. This research was further intensified in the 1990s, when other projects investigated various solutions such as proof planning and bridges to external theorem provers using so-called hammer tools. The hammers developed at that time were early precursors of todays very successful hammer tools such as Sledgehammer, HolyHammer and related systems (Blanchette et al. 2013, Kaliszyk & Urban 2015 and Czaika & Kaliszyk 2018). While, triggered by the mentioned dynamics, some first initial progress was made in the late 80s and during the 90s with regards to the automation of Church's type theory, the resource investments and achievements at the time were lacking much behind those seen, for example, in first-order theorem proving. Good progress was fostered only later, in particular, through the development of a commonly supported syntax for Church's type theory, called TPTP THF (Sutcliffe & Benzmuller 2010, Sutcliffe 2022), and the inclusion, from 2009 onwards, of a TPTP THF division in the yearly CASC competitions (kind of world championships for automated theorem proving; see Sutcliffe 2016 for further details). These competitions, in combination with further factors, triggered an increasing interest in the full automation of Church's type theory. Particularly successful was recently the exploration of equality based theorem proving using the superposition approach and the adaptation of SMT techniques to HOL. ### 4.3 Automated Theorem Provers A selection of theorem provers for Church's type theory is presented. The focus is on systems that have successfully participated in TPTP THF CASC competitions in the past, and the order of presentation is motivated by their first-time CASC participation. The latest editions of most mentioned systems can be accessed online via the SystemOnTPTP infrastructure (Sutcliffe 2017). Nearly all mentioned systems produce verifiable proof certificates in the TPTP TSTP syntax. The TPS prover (Andrews et al. 1996, Andrews & Brown 2006) can be used to prove theorems of elementary type theory or extensional type theory automatically, interactively, or semi-automatically. When searching for a proof automatically, TPS first searches for an expansion proof (Miller 1987) or an extensional expansion proof (Brown 2004, 2007) of the theorem. Part of this process involves searching for acceptable matings (Andrews 1981, Bishop 1999). The behavior of TPS is controlled by sets of flags, also called modes. A simple scheduling mechanism is employed in the latest versions of TPS to sequentially run a about fifty modes for a limited amount of time. TPS was the winner of the first THF CASC competition in 2009. The LEO-II prover (Benzmuller et al. 2015) is the successor of LEO (Benzmuller & Kohlhase 1998b, which was hardwired with the OMEGA proof assistant (LEO stands for Logical Engine of OMEGA). The provers are based on the RUE-resolution calculi developed in Benzmuller 1999a,b. LEO was the first prover to implement calculus rules for extensionality to avoid cut-simulation effects. LEO-II inherits and adapts them, and provides additional calculus rules for description and choice. The prover, which internally collaborates with first-order provers (preferably E) and SAT solvers, has pioneered cooperative higher-order/first-order proof automation. Since the prover is often too weak to find a refutation among the steadily growing set of clauses on its own, some of the clauses in LEO-II's search space attain a special status: they are first-order clauses modulo the application of an appropriate transformation function. Therefore, LEO-II progressively launches time limited calls with these clauses to a first-order theorem prover, and when the first-order prover reports a refutation, LEO-II also terminates. Parts of these ideas were already implemented in the predecessor LEO. Communication between LEO-II and the cooperating first-order theorem provers uses the TPTP language and standards. LEO-II was the winner of the second THF CASC competition in 2010. The Satallax prover (Brown 2012) is based on a complete ground tableau calculus for Church's type theory with choice (Backes & Brown 2011). An initial tableau branch is formed from the assumptions of a conjecture and negation of its conclusion. From that point on, Satallax tries to determine unsatisfiability or satisfiability of this branch. Satallax progressively generates higher-order formulas and corresponding propositional clauses. Satallax uses the SAT solver MiniSat as an engine to test the current set of propositional clauses for unsatisfiability. If the clauses are unsatisfiable, the original branch is unsatisfiable. Satallax provides calculus rules for extensionality, description and choice. If there are no quantifiers at function types, the generation of higher-order formulas and corresponding clauses may terminate. In that case, if MiniSat reports the final set of clauses as satisfiable, then the original set of higher-order formulas is satisfiable (by a standard model in which all types are interpreted as finite sets). Satallax was the winner of the THF CASC competition in 2011 and from 2013 to 2019 The Isabelle/HOL system (Nipkow, Wenzel, & Paulson 2002) has originally been designed as an interactive prover. However, in order to ease user interaction several automatic proof tactics have been added over the years. By appropriately scheduling a subset of these proof tactics, some of which are quite powerful, Isabelle/HOL has since about 2011 been turned also into an automatic theorem prover for TPTP THF (and other TPTP syntax formats), that can be run from a command shell like other provers. The most powerful proof tactics that are scheduled by Isabelle/HOL include the Sledgehammer tool (Blanchette et al. 2013), which invokes a sequence of external first-order and higher-order theorem provers, the model finder Nitpick (Blanchette & Nipkow 2010), the equational reasoner simp, the untyped tableau prover blast, the simplifier and classical reasoners auto, force, and fast, and the best-first search procedure best. In contrast to all other automated theorem provers mentioned above, the TPTP incarnation of Isabelle/HOL does not output proof certificates. Isabelle/HOL was the winner of the THF CASC competition in 2012. The agsyHOL prover is based on a generic lazy narrowing proof search algorithm. Backtracking is employed and a comparably small search state is maintained. The prover outputs proof terms in sequent style which can be verified in the Agda system. coqATP implements (the non-inductive) part of the calculus of constructions (Bertot & Casteran 2004). The system outputs proof terms which are accepted as proofs (after the addition of a few definitions) by the Coq proof assistant. The prover employs axioms for functional extensionality, choice, and excluded middle. Boolean extensionality is not supported. In addition to axioms, a small library of basic lemmas is employed. The Leo-III prover implements a paramodulation calculus for Church's type theory (Steen 2018). The system, which is a descendant of LEO and LEO-II, provides calculus rules for extensionality, description and choice. The system has put an emphasis on the implementation of an efficient set of underlying data structures, on simplification routines and on heuristic rewriting. In the tradition of its predecessors, Leo-III cooperates with first-order reasoning tools using translations to many-sorted first-order logic. The prover accepts every common TPTP syntax dialect and is thus very widely applicable. Recently, the prover has also been extended to natively supports almost every normal higher-order modal logic (Steen et al. 2023). CVC4, CVC5 and Verit (Barbosa et al. 2019) are SMT-based automated provers with built-in support for many theories, including linear arithmetic, arrays, bit vectors, data types, finite sets and strings. These provers have been extended to support (fragments of) Church's type theory. The Zipperposition prover (Bentkamp et al. 2018, Vukmirovic et al. 2022) focuses on the effective implementation of superposition calculi for Church's type theory. It comes with Logtk, a supporting library for manipulating terms, formulas, clauses, etc. in Church's type theory, and supports relevant extensions such as datatypes, recursive functions, and arithmetic. Zipperposition was the winner of the THF CASC competition in 2020, 2021 and 2022. Vampire (Bhayat & Reger 2020), which has dominated the TPTP competitions in first-order logic for more than two decades, now also supports the automation of Church's type theory using a combinator-based superposition calculus. Vampire was the winner of the THF CASC competition in 2023. The theorem prover E (Vukmirovic et al. 2023), another prominent and leading first-order automated theorem prover, has been extended for Church's type theory using a variant of the supperposition approach that is also implemented in Zipperposition. Lash (Brown and Kaliszyk 2022) is a theorem prover for Church's type theory created as a fork of Satallax. Duper is a superposition-based theorem prover for dependent type theory, deigned to prove theorems in the proof assistant Lean. In recent years, there has thus been a significant shift from first-order to higher-order automated theorem proving. ### 4.4 (Counter-)Model Finding Support for finding finite models or countermodels for formulas of Church's type theory was implemented already in the tableau-based prover HOT (Konrad 1998). Restricted (counter-)model finding capabilities are also implemented in the provers Satallax, LEO-II and LEO-III. The most advanced (finite) model finding support is currently realized in the systems Nitpick, Nunchaku and Refute. These tools have been integrated with the Isabelle proof assistant. Nitpick is also available as a standalone tool that accepts TPTP THF syntax. The systems are particularly valuable for exposing errors and misconceptions in problem encodings, and for revealing bugs in the THF theorem provers. ## 5. Applications In addition to its elegance, Church's type theory also has many practical advantages. This aspect is emphasised, for example, in the recent textbook by Farmer (2023), which deals with various relevant and important further aspects that could not be covered in this article. Because of its good practical expressiveness, the range of applications of Church's type theory is very wide, and only a few examples can be given below (a good source for further information on practical examples are the proceedings of the conferences on Interactive Theorem Proving). ### 5.1 Semantics of Natural Language Church's type theory plays an important role in the study of the formal semantics of natural language. Pioneering work on this was done by Richard Montague. See his papers "English as a formal language", "Universal grammar", and "The proper treatment of quantification in ordinary English", which are reprinted in Montague 1974. A crucial component of Montague's analysis of natural language is the definition of a tensed intensional logic (Montague 1974: 256), which is an enhancement of Church's type theory. Montague Grammar had a huge impact, and has since been developed in many further directions, not least in Typelogical/Categorical Grammar. Further related work on intensional and higher-order modal logic is presented in Gallin 1975 and Muskens 2006. ### 5.2 Mathematics and Computer Science Proof assistants based on Church's Type Theory, including Isabelle/HOL, HOL Light, HOL4, and PVS, have been successfully utilized in a broad range of application in computer science and mathematics. Applications in computer science include the verification of hardware, software and security protocols. A prominent example is the L4.verified project in which Isabelle/HOL was used to formally prove that the seL4 operating system kernel implements an abstract, mathematical model specifying of what the kernel is supposed to do (Klein et al. 2018). In mathematics proof assistants have been applied for the development of libraries mathematical theories and the verification of challenge theorems. An early example is the mathematical library that was developed since the eighties in the TPS project. A exemplary list of theorems that were proved automatically with TPS is given in Andrews et al. 1996. A very prominent recent example is Hales Flyspeck in which HOL Light was employed to develop a formal proof for Kepler's conjecture (Hales et al. 2017). An example that strongly exploits automation support in Isabelle/HOL with Sledgehammer and Nitpick is presented in Benzmuller & Scott 2019. In this work different axiom systems for category theory were explored and compared. A solid overview of past and ongoing formalization projects can be obtained by consulting appropriate sources such as Isabelle's Archive of Formal Proofs, the Journal of Formalized Reasoning, or the THF entries in Sutcliffe's TPTP problem library. Relevant further information and a discussion of implications and further work can also be found in Bentkamp et al. 2023b and in Bayer et al. 2024. Further improving proof automation within these proof assistants--based on proof hammering tools or on other forms of prover integration--is relevant for minimizing interaction effort in future applications. ### 5.3 Computational Metaphysics and Artificial Intelligence Church's type theory is a classical logic, but topical applications in philosophy and artificial intelligence often require expressive non-classical logics. In order to support such applications with reasoning tools for Church's type theory, the shallow semantical embedding technique (see also Section 1.2.2) has been developed that generalizes and extends the ideas underlying the well known standard translation of modal logics to first-order logic. The technique was applied for the assessment of modern variants of the ontological argument with a range of higher-order theorem provers, including LEO-II, Satallax, Nitpick and Isabelle/HOL. In the course of experiments, LEO-II detected an inconsistency in the premises of Godel's argument, while the provers succeeded in automatically proving Scott's emendation of it and to confirm the consistency of the emended premises. More details on this work are presented in a related SEP entry on automated reasoning (see Section 4.6 on Logic and Philosophy). The semantical embedding approach has been adapted and further extended for a range of other non-classical logics and related applications. In philosophy this includes the encoding and formal assessment of ambitious ethical and metaphysical theories, and in artificial intelligence this includes the mechanization of deontic logics and normative reasoning to control AI systems (Benzmuller et al. 2020) as well as an automatic proof of the muddy children puzzle (see Appendix B of dynamic epistemic logic), which is a famous puzzle in epistemic reasoning, respectively dynamic epistemic reasoning. ### 5.4. Metalogical Studies Church's type theory is also well suited to support metalogical studies, including the encoding of other logical formalisms and formalized soundness and completeness studies (see, e.g., Schlichtkrull et al. 2020, Halkjaer et al. 2023, and various further pointers given in Benzmuller 2019). In particular, Church's type theory can be studied formally in itself, as shown, for example, by Kumar et al. (2016), Schlichtkrull (2023), and Diaz (2023).
type-theory-intuitionistic	## 1. Overview We begin with a bird's eye view of some important aspects of intuitionistic type theory. Readers who are unfamiliar with the theory may prefer to skip it on a first reading. The origins of intuitionistic type theory are Brouwer's intuitionism and Russell's type theory. Like Church's classical simple theory of types it is based on the lambda calculus with types, but differs from it in that it is based on the propositions-as-types principle, discovered by Curry (1958) for propositional logic and extended to predicate logic by Howard (1980) and de Bruijn (1970). This extension was made possible by the introduction of indexed families of types (dependent types) for representing the predicates of predicate logic. In this way all logical connectives and quantifiers can be interpreted by type formers. In intuitionistic type theory further types are added, such as a type of natural numbers, a type of small types (a universe) and a type of well-founded trees. The resulting theory contains intuitionistic number theory (Heyting arithmetic) and much more. The theory is formulated in natural deduction where the rules for each type former are classified as formation, introduction, elimination, and equality rules. These rules exhibit certain symmerties between the introduction and elimination rules following Gentzen's and Prawitz' treatment of natural deduction, as explained in the entry on proof-theoretic semantics. The elements of propositions, when interpreted as types, are called proof-objects. When proof-objects are added to the natural deduction calculus it becomes a typed lambda calculus with dependent types, which extends Church's original typed lambda calculus. The equality rules are computation rules for the terms of this calculus. Each function definable in the theory is total and computable. Intuitionistic type theory is thus a typed functional programming language with the unusual property that all programs terminate. Intuitionistic type theory is not only a formal logical system but also provides a comprehensive philosophical framework for intuitionism. It is an interpreted language, where the distinction between the demonstration of a judgment and the proof of a proposition plays a fundamental role (Sundholm 2012). The framework clarifies the Brouwer-Heyting-Kolmogorov interpretation of intuitionistic logic and extends it to the more general setting of intuitionistic type theory. In doing so it provides a general conception not only of what a constructive proof is, but also of what a constructive mathematical object is. The meaning of the judgments of intuitionistic type theory is explained in terms of computations of the canonical forms of types and terms. These informal, intuitive meaning explanations are "pre-mathematical" and should be contrasted to formal mathematical models developed inside a standard mathematical framework such as set theory. This meaning theory also justifies a variety of inductive, recursive, and inductive-recursive definitions. Although proof-theoretically strong notions can be justified, such as analogues of certain large cardinals, the system is considered predicative. Impredicative definitions of the kind found in higher-order logic, intuitionistic set theory, and topos theory are not part of the theory. Neither is Markov's principle, and thus the theory is distinct from Russian constructivism. An alternative formal logical system for predicative constructive mathematics is Myhill and Aczel's constructive Zermelo-Fraenkel set theory (CZF). This theory, which is based on intuitionistic first-order predicate logic and weakens some of the axioms of classical Zermelo-Fraenkel Set Theory, has a natural interpretation in intuitionistic type theory. Martin-Lof's meaning explanations thus also indirectly form a basis for CZF. Variants of intuitionistic type theory underlie several widely used proof assistants, including NuPRL, Coq, and Agda. These proof assistants are computer systems that have been used for formalizing large and complex proofs of mathematical theorems, such as the Four Colour Theorem in graph theory and the Feit-Thompson Theorem in finite group theory. They have also been used to prove the correctness of a realistic C compiler (Leroy 2009) and other computer software. Philosophically and practically, intuitionistic type theory is a foundational framework where constructive mathematics and computer programming are, in a deep sense, the same. This point has been emphasized by (Gonthier 2008) in the paper in which he describes his proof of the Four Colour Theorem: > > The approach that proved successful for this proof was to turn > almost every mathematical concept into a data structure or a program > in the Coq system, thereby converting the entire enterprise into one > of program verification. > > > ## 2. Propositions as Types ### 2.1 Intuitionistic Type Theory: a New Way of Looking at Logic? Intuitionistic type theory offers a new way of analyzing logic, mainly through its introduction of explicit proof objects. This provides a direct computational interpretation of logic, since there are computation rules for proof objects. As regards expressive power, intuitionistic type theory may be considered as an extension of first-order logic, much as higher order logic, but predicative. #### 2.1.1 A Type Theory Russell developed type theory in response to his discovery of a paradox in naive set theory. In his ramified type theory mathematical objects are classified according to their types: the type of propositions, the type of objects, the type of properties of objects, etc. When Church developed his simple theory of types on the basis of a typed version of his lambda calculus he added the rule that there is a type of functions between any two types of the theory. Intuitionistic type theory further extends the simply typed lambda calculus with dependent types, that is, indexed families of types. An example is the family of types of $n$-tuples indexed by $n$. Types have been widely used in programming for a long time. Early high-level programming languages introduced types of integers and floating point numbers. Modern programming languages often have rich type systems with many constructs for forming new types. Intuitionistic type theory is a functional programming language where the type system is so rich that practically any conceivable property of a program can be expressed as a type. Types can thus be used as specifications of the task of a program. #### 2.1.2 An intuitionstic logic with proof-objects Brouwer's analysis of logic led him to an intuitionistic logic which rejects the law of excluded middle and the law of double negation. These laws are not valid in intuitionistic type theory. Thus it does not contain classical (Peano) arithmetic but only intuitionistic (Heyting) arithmetic. (It is another matter that Peano arithmetic can be interpreted in Heyting arithmetic by the double negation interpretation, see the entry on intuitionistic logic.) Consider a theorem of intuitionistic arithmetic, such as the division theorem \[\forall m, n. m > 0 \supset \exists q, r. mq + r = n \wedge m > r \] A formal proof (in the usual sense) of this theorem is a sequence (or tree) of formulas, where the last (root) formula is the theorem and each formula in the sequence is either an axiom (a leaf) or the result of applying an inference rule to some earlier (higher) formulas. When the division theorem is proved in intuitionistic type theory, we do not only build a formal proof in the usual sense but also a construction (or proof-object) "$\divi$" which witnesses the truth of the theorem. We write \[\divi : \forall m, n {:} \N.\, m > 0 \supset \exists q, r {:} \N.\, mq + r = n \wedge m > r \] to express that $\divi$ is a proof-object for the division theorem, that is, an element of the type representing the division theorem. When propositions are represented as types, the $\forall$-quantifier is identified with the dependent function space former (or general cartesian product) $\Pi$, the $\exists$-quantifier with the dependent pairs type former (or general disjoint sum) $\Sigma$, conjunction $\wedge$ with cartesian product $ \times $, the identity relation = with the type former $\I$ of proof-objects of identities, and the greater than relation $>$ with the type former $\GT$ of proof-objects of greater-than statements. Using "type-notation" we thus write \[ \divi : \Pi m, n {:} \N.\, \GT(m,0)\rightarrow \Sigma q, r {:} \N.\, \I(\N,mq + r,n) \times \GT(m,r) \] to express that the proof object "$\divi$" is a function which maps two numbers $m$ and $n$ and a proof-object $p$ witnessing that $m > 0$ to a quadruple $(q,(r,(s,t)))$, where $q$ is the quotient and $r$ is the remainder obtained when dividing $n$ by $m$. The third component $s$ is a proof-object witnessing the fact that $mq + r = n$ and the fourth component $t$ is a proof object witnessing $m > r $. Crucially, $\divi$ is not only a function in the classical sense; it is also a function in the intuitionistic sense, that is, a program which computes the output $(q,(r,(s,t)))$ when given $m$, $n$, $p$ as inputs. This program is a term in a lambda calculus with special constants, that is, a program in a functional programming language. #### 2.1.3 An extension of first-order predicate logic Intuitionistic type theory can be considered as an extension of first-order logic, much as higher order logic is an extension of first order logic. In higher order logic we find some individual domains which can be interpreted as any sets we like. If there are relational constants in the signature these can be interpreted as any relations between the sets interpreting the individual domains. On top of that we can quantify over relations, and over relations of relations, etc. We can think of higher order logic as first-order logic equipped with a way of introducing new domains of quantification: if $S\_1, \ldots, S\_n$ are domains of quantification then $(S\_1,\ldots,S\_n)$ is a new domain of quantification consisting of all the n-ary relations between the domains $S\_1,\ldots,S\_n$. Higher order logic has a straightforward set-theoretic interpretation where $(S\_1,\ldots,S\_n)$ is interpreted as the power set $P(A\_1 \times \cdots \times A\_n)$ where $A\_i$ is the interpretation of $S\_i$, for $i=1,\ldots,n$. This is the kind of higher order logic or simple theory of types that Ramsey, Church and others introduced. Intuitionistic type theory can be viewed in a similar way, only here the possibilities for introducing domains of quantification are richer, one can use $\Sigma, \Pi, +, \I$ to construct new ones from old. (Section 3.1; Martin-Lof 1998 [1972]). Intuitionistic type theory has a straightforward set-theoretic interpretation as well, where $\Sigma$, $\Pi$ etc are interpreted as the corresponding set-theoretic constructions; see below. We can add to intuitionistic type theory unspecified individual domains just as in HOL. These are interpreted as sets as for HOL. Now we exhibit a difference from HOL: in intuitionistic type theory we can introduce unspecified family symbols. We can introduce $T$ as a family of types over the individual domain $S$: \[T(x)\; {\rm type} \;(x{:}S).\] If $S$ is interpreted as $A$, $T$ can be interpreted as any family of sets indexed by $A$. As a non-mathematical example, we can render the binary relation loves between members of an individual domain of people as follows. Introduce the binary family Loves over the domain People \[{\rm Loves}(x,y)\; {\rm type}\; (x{:}{\rm People}, y{:}{\rm People}).\] The interpretation can be any family of sets $B\_{x,y}$ ($x{:}A$, $y{:}A$). How does this cover the standard notion of relation? Suppose we have a binary relation $R$ on $A$ in the familiar set-theoretic sense. We can make a binary family corresponding to this as follows \[ B\_{x,y} = \begin{cases} \{0\} &\text{if } R(x,y) \text{ holds} \\ \varnothing &\text{if } R(x,y) \text{ is false.} \end{cases}\] Now clearly $B\_{x,y}$ is nonempty if and only if $R(x,y)$ holds. (We could have chosen any other element from our set theoretic universe than 0 to indicate truth.) Thus from any relation we can construct a family whose truth of $x,y$ is equivalent to $B\_{x,y}$ being non-empty. Note that this interpretation does not care what the proof for $R(x,y)$ is, just that it holds. Recall that intuitionistic type theory interprets propositions as types, so $p{:} {\rm Loves}({\rm John}, {\rm Mary})$ means that ${\rm Loves}({\rm John}, {\rm Mary})$ is true. The interpretation of relations as families allows for keeping track of proofs or evidence that $R(x,y)$ holds, but we may also chose to ignore it. In Montague semantics, higher order logic is used to give semantics of natural language (and examples as above). Ranta (1994) introduced the idea to instead employ intuitionistic type theory to better capture sentence structure with the help of dependent types. In contrast, how would the mathematical relation $>$ between natural numbers be handled in intuitionistic type theory? First of all we need a type of numbers $\N$. We could in principle introduce an unspecified individual domain $\N$, and then add axioms just as we do in first-order logic when we set up the axiom system for Peano arithmetic. However this would not give us the desirable computational interpretation. So as explained below we lay down introduction rules for constructing new natural numbers in $\N$ and elimination and computation rules for defining functions on $\N$ (by recursion). The standard order relation $>$ should satisfy \[\mbox{$x > y$ iff there exists $z{:} \N$ such that $y+z+1 = x$}. \] The right hand is rendered as $\Sigma z{:}\N.\, \I(\N,y+z+1,x)$ in intuitionistic type theory, and we take this as definition of relation $>$. ($+$ is defined by recursive equations, $\I$ is the identity type construction). Now all the properties of $>$ are determined by the mentioned introduction and elimination and computation rules for $\N$. #### 2.1.4 A logic with several forms of judgment The type system of intuitionistic type theory is very expressive. As a consequence the well-formedness of a type is no longer a simple matter of parsing, it is something which needs to be proved. Well-formedness of a type is one form of judgment of intuitionistic type theory. Well-typedness of a term with respect to a type is another. Furthermore, there are equality judgments for types and terms. This is yet another way in which intuitionistic type theory differs from ordinary first order logic with its focus on the sole judgment expressing the truth of a proposition. #### 2.1.5 Semantics While a standard presentation of first-order logic would follow Tarski in defining the notion of model, intuitionistic type theory follows the tradition of Brouwerian meaning theory as further developed by Heyting and Kolmogorov, the so called BHK-interpretation of logic. The key point is that the proof of an implication $A \supset B $ is a method that transforms a proof of $A$ to a proof of $B$. In intuitionistic type theory this method is formally represented by the program $f {:} A \supset B$ or $f {:} A \rightarrow B$: the type of proofs of an implication $A \supset B$ is the type of functions which maps proofs of $A$ to proofs of $B$. Moreover, whereas Tarski semantics is usually presented meta-mathematically, and assumes set theory, Martin-Lof's meaning theory of intuitionistic type theory should be understood directly and "pre-mathematically", that is, without assuming a meta-language such as set theory. #### 2.1.6 A functional programming language Readers with a background in the lambda calculus and functional programming can get an alternative first approximation of intuitionistic type theory by thinking about it as a typed functional programming language in the style of Haskell or one of the dialects of ML. However, it differs from these in two crucial aspects: (i) it has dependent types (see below) and (ii) all typable programs terminate. (Note that intuitionistic type theory has influenced recent extensions of Haskell with generalized algebraic datatypes which sometimes can play a similar role as inductively defined dependent types.) ### 2.2 The Curry-Howard Correspondence As already mentioned, the principle that > > a proposition is the type of its proofs. > > > is fundamental to intuitionistic type theory. This principle is also known as the Curry-Howard correspondence or even Curry-Howard isomorphism. Curry discovered a correspondence between the implicational fragment of intuitionistic logic and the simply typed lambda-calculus. Howard extended this correspondence to first-order predicate logic. In intuitionistic type theory this correspondence becomes an identification of proposition and types, which has been extended to include quantification over higher types and more. ### 2.3 Sets of Proof-Objects So what are these proof-objects like? They should not be thought of as logical derivations, but rather as some (structured) symbolic evidence that something is true. Another term for such evidence is "truth-maker". It is instructive, as a somewhat crude first approximation, to replace types by ordinary sets in this correspondence. Define a set $\E\_{m,n}$, depending on $m, n \in {{\mathbb N}}$, by: \[\E\_{m,n} = \left\{\begin{array}{ll} \{0\} & \mbox{if $m = n$}\\ \varnothing & \mbox{if $m \ne n$.} \end{array} \right.\] Then $\E\_{m,n}$ is nonempty exactly when $m=n$. The set $\E\_{m,n}$ corresponds to the proposition $m=n$, and the number $0$ is a proof-object (truth-maker) inhabiting the sets $\E\_{m,m}$. Consider the proposition that $m$ is an even number expressed as the formula $\exists n \in {{\mathbb N}}. m= 2n$. We can build a set of proof-objects corresponding to this formula by using the general set-theoretic sum operation. Suppose that $A\_n$ ($n\in {{\mathbb N}}$) is a family of sets. Then its disjoint sum is given by the set of pairs \[ (\Sigma n \in {{\mathbb N}})A\_n = \{ (n,a) : n \in {{\mathbb N}}, a \in A\_n\}.\] If we apply this construction to the family $A\_n = \E\_{m,2n}$ we see that $(\Sigma n \in {{\mathbb N}})\E\_{m,2n}$ is nonempty exactly when there is an $n\in {{\mathbb N}}$ with $m=2n$. Using the general set-theoretic product operation $(\Pi n \in {{\mathbb N}})A\_n$ we can similarly obtain a set corresponding to a universally quantified proposition. ### 2.4 Dependent Types In intuitionistic type theory there are primitive type formers $\Sigma$ and $\Pi$ for general sums and products, and $\I$ for identity types, analogous to the set-theoretic constructions described above. The identity type $\I(\N,m,n)$ corresponding to the set $\E\_{m,n}$ is an example of a dependent type since it depends on $m$ and $n$. It is also called an indexed family of types since it is a family of types indexed by $m$ and $n$. Similarly, we can form the general disjoint sum $\Sigma x {:} A.\, B$ and the general cartesian product $\Pi x {:} A.\, B$ of such a family of types $B$ indexed by $x {:} A$, corresponding to the set theoretic sum and product operations above. Dependent types can also be defined by primitive recursion. An example is the type of $n$-tuples $A^n$ of elements of type $A$ and indexed by $n {:} N$ defined by the equations \[\begin{align\} A^0 &= 1\\ A^{n+1} &= A \times A^n \end{align\}\] where $1$ is a one element type and $\times$ denotes the cartesian product of two types. We note that dependent types introduce computation in types: the defining rules above are computation rules. For example, the result of computing $A^3$ is $A \times (A \times (A \times 1))$. ### 2.5 Propositions as Types in Intuitionistic Type Theory With propositions as types, predicates become dependent types. For example, the predicate $\mathrm{Prime}(x)$ becomes the type of proofs that $x$ is prime. This type depends on $x$. Similarly, $x < y$ is the type of proofs that $x$ is less than $y$. According to the Curry-Howard interpretation of propositions as types, the logical constants are interpreted as type formers: \[\begin{align\} \bot &= \varnothing\\ \top &= 1\\ A \vee B &= A + B\\ A \wedge B &= A \times B\\ A \supset B &= A \rightarrow B\\ \exists x {:} A.\, B &= \Sigma x {:} A.\, B\\ \forall x {:} A.\, B &= \Pi x {:} A.\, B \end{align\}\] where $\Sigma x {:} A.\, B$ is the disjoint sum of the $A$-indexed family of types $B$ and $\Pi x {:} A.\, B$ is its cartesian product. The canonical elements of $\Sigma x {:} A.\, B$ are pairs $(a,b)$ such that $a {:} A$ and $b {:} B[x:=a]$ (the type obtained by substituting all free occurrences of $x$ in $B$ by $a$). The elements of $\Pi x {:} A.\, B$ are (computable) functions $f$ such that $f\,a {:} B[x:=a]$, whenever $a {:} A$. For example, consider the proposition \[\begin{equation} \forall m {:} \N.\, \exists n {:} \N.\, m \lt n \wedge \mathrm{Prime}(n) \tag{1}\label{prop1} \end{equation}\] expressing that there are arbitrarily large primes. Under the Curry-Howard interpretation this becomes the type $\Pi m {:} \N.\, \Sigma n {:} \N.\, m \lt n \times \mathrm{Prime}(n)$ of functions which map a number $m$ to a triple $(n,(p,q))$, where $n$ is a number, $p$ is a proof that $m \lt n$ and $q$ is a proof that $n$ is prime. This is the proofs as programs principle: a constructive proof that there are arbitrarily large primes becomes a program which given any number produces a larger prime together with proofs that it indeed is larger and indeed is prime. Note that the proof which derives a contradiction from the assumption that there is a largest prime is not constructive, since it does not explicitly give a way to compute an even larger prime. To turn this proof into a constructive one we have to show explicitly how to construct the larger prime. (Since proposition (\ref{prop1}) above is a $\Pi^0\_2$-formula we can for example use Friedman's A-translation to turn such a proof in classical arithmetic into a proof in intuitionistic arithmetic and thus into a proof in intuitionistic type theory.) ## 3. Basic Intuitionistic Type Theory We now present a core version of intuitionistic type theory, closely related to the first version of the theory presented by Martin-Lof in 1972 (Martin-Lof 1998 [1972]). In addition to the type formers needed for the Curry-Howard interpretation of typed intuitionistic predicate logic listed above, we have two types: the type $\N$ of natural numbers and the type $\U$ of small types. The resulting theory can be shown to contain intuitionistic number theory $\HA$ (Heyting arithmetic), Godel's System $\T$ of primitive recursive functions of higher type, and the theory $\HA^\omega$ of Heyting arithmetic of higher type. This core intuitionistic type theory is not only the original one, but perhaps the minimal version which exhibits the essential features of the theory. Later extensions with primitive identity types, well-founded tree types, universe hierarchies, and general notions of inductive and inductive-recursive definitions have increased the proof-theoretic strength of the theory and also made it more convenient for programming and formalization of mathematics. For example, with the addition of well-founded trees we can interpret the Constructive Zermelo-Fraenkel Set Theory $\CZF$ of Aczel (1978 [1977]). However, we will wait until the next section to describe those extensions. ### 3.1 Judgments In Martin-Lof (1996) a general philosophy of logic is presented where the traditional notion of judgment is expanded and given a central position. A judgment is no longer just an affirmation or denial of a proposition, but a general act of knowledge. When reasoning mathematically we make judgments about mathematical objects. One form of judgment is to state that some mathematical statement is true. Another form of judgment is to state that something is a mathematical object, for example a set. The logical rules give methods for producing correct judgments from earlier judgments. The judgments obtained by such rules can be presented in tree form \[ \begin{prooftree} \AxiomC{$J\_1$} \AxiomC{$J\_2$} \RightLabel{$r\_1$} \BinaryInfC{$J\_3$} \AxiomC{$J\_4$} \RightLabel{$r\_5$} \UnaryInfC{$J\_5$} \AxiomC{$J\_6$} \RightLabel{$r\_3$} \BinaryInfC{$J\_7$} \RightLabel{$r\_4$} \BinaryInfC{$J\_8$} \end{prooftree}\] or in sequential form * (1) $J\_1 \quad\text{ axiom} $ * (2) $J\_2 \quad\text{ axiom} $ * (3) $J\_3 \quad\text{ by rule \(r\_1$ from (1) and (2)} \) * (4) $J\_4 \quad\text{ axiom} $ * (5) $J\_5 \quad\text{ by rule \(r\_2$ from (4)} \) * (6) $J\_6 \quad\text{ axiom} $ * (7) $J\_7 \quad\text{ by rule \(r\_3$ from(5) and (6)} \) * (8) $J\_8 \quad\text{ by rule \(r\_4$ from (3) and (7)} \) The latter form is common in mathematical arguments. Such a sequence or tree formed by logical rules from axioms is a derivation or demonstration of a judgment. First-order reasoning may be presented using a single kind of judgment: > > the proposition $B$ is true under the hypothesis that the > propositions $A\_1, \ldots, A\_n$ are all true. > > > We write this hypothetical judgment as a so-called Gentzen sequent \[A\_1, \ldots, A\_n {\vdash}B.\] Note that this is a single judgment that should not be confused with the derivation of the judgment ${\vdash}B$ from the judgments ${\vdash}A\_1, \ldots, {\vdash}A\_n$. When $n=0$, then the categorical judgment $ {\vdash}B$ states that $B$ is true without any assumptions. With sequent notation the familiar rule for conjunctive introduction becomes \[\begin{prooftree} \AxiomC{$A\_1, \ldots,A\_n {\vdash}B$} \AxiomC{$A\_1, \ldots, A\_n {\vdash}C$} \RightLabel{$(\land I)$.} \BinaryInfC{$A\_1, \ldots, A\_n {\vdash}B \land C$} \end{prooftree}\] ### 3.2 Judgment Forms Martin-Lof type theory has four basic forms of judgments and is a considerably more complicated system than first-order logic. One reason is that more information is carried around in the derivations due to the identification of propositions and types. Another reason is that the syntax is more involved. For instance, the well-formed formulas (types) have to be generated simultaneously with the provably true formulas (inhabited types). The four forms of categorical judgment are * $\vdash A \; {\rm type}$, meaning that $A$ is a well-formed type, * $\vdash a {:} A$, meaning that $a$ has type $A$, * $\vdash A = A'$, meaning that $A$ and $A'$ are equal types, * $\vdash a = a' {:} A$, meaning that $a$ and $a'$ are equal elements of type $A$. In general, a judgment is hypothetical, that is, it is made in a context $\Gamma$, that is, a list $x\_1 {:} A\_1, \ldots, x\_n {:} A\_n$ of variables which may occur free in the judgment together with their respective types. Note that the types in a context can depend on variables of earlier types. For example, $A\_n$ can depend on $x\_1 {:} A\_1, \ldots, x\_{n-1} {:} A\_{n-1}$. The four forms of hypothetical judgments are * $\Gamma \vdash A \; {\rm type}$, meaning that $A$ is a well-formed type in the context $\Gamma$, * $\Gamma \vdash a {:} A$, meaning that $a$ has type $A$ in context $\Gamma$, * $\Gamma \vdash A = A'$, meaning that $A$ and $A'$ are equal types in the context $\Gamma$, * $\Gamma \vdash a = a' {:} A$, meaning that $a$ and $a'$ are equal elements of type $A$ in the context $\Gamma$. Under the proposition as types interpretation \[\tag{2}\label{analytic} \vdash a {:} A \] can be understood as the judgment that $a$ is a proof-object for the proposition $A$. When suppressing this object we get a judgment corresponding to the one in ordinary first-order logic (see above): \[\tag{3}\label{synthetic} \vdash A\; {\rm true}. \] Remark 3.1. Martin-Lof (1994) argues that Kant's analytic judgment a priori and synthetic judgment a priori can be exemplified, in the realm of logic, by ([analytic]) and ([synthetic]) respectively. In the analytic judgment ([analytic]) everything that is needed to make the judgment evident is explicit. For its synthetic version ([synthetic]) a possibly complicated proof construction $a$ needs to be provided to make it evident. This understanding of analyticity and syntheticity has the surprising consequence that "the logical laws in their usual formulation are all synthetic." Martin-Lof (1994: 95). His analysis further gives: > " [...] the logic of analytic judgments, > that is, the logic for deriving judgments of the two analytic forms, > is complete and decidable, whereas the logic of synthetic judgments is > incomplete and undecidable, as was shown by Godel." > Martin-Lof (1994: 97). > > > The decidability of the two analytic judgments ($\vdash a{:}A$ and $\vdash a=b{:}A$) hinges on the metamathematical properties of type theory: strong normalization and decidable type checking. Sometimes also the following forms are explicitly considered to be judgments of the theory: * $\Gamma \; {\rm context}$, meaning that $\Gamma$ is a well-formed context. * $\Gamma = \Gamma'$, meaning that $\Gamma$ and $\Gamma'$ are equal contexts. Below we shall abbreviate the judgment $\Gamma \vdash A \; {\rm type}$ as $\Gamma \vdash A$ and $\Gamma \; {\rm context}$ as $\Gamma \vdash.$ ### 3.3 Inference Rules When stating the rules we will use the letter $\Gamma$ as a meta-variable ranging over contexts, $A,B,\ldots$ as meta-variables ranging over types, and $a,b,c,d,e,f,\ldots$ as meta-variables ranging over terms. The first group of inference rules are general rules including rules of assumption, substitution, and context formation. There are also rules which express that equalities are equivalence relations. There are numerous such rules, and we only show the particularly important rule of type equality which is crucial for computation in types: \[\frac{\Gamma \vdash a {:} A\hspace{2em}\Gamma \vdash A = B} {\Gamma \vdash a {:} B}\] The remaining rules are specific to the type formers. These are classified as formation, introduction, elimination, and equality rules. ### 3.4 Intuitionistic Predicate Logic We only give the rules for $\Pi$. There are analogous rules for the other type formers corresponding to the logical constants of typed predicate logic. In the following $B[x := a]$ means the term obtained by substituting the term $a$ for each free occurrence of the variable $x$ in $B$ (avoiding variable capture). $\Pi$-formation. \[\frac{\Gamma \vdash A\hspace{2em} \Gamma, x {:} A \vdash B} {\Gamma \vdash \Pi x {:} A. B}\] $\Pi$-introduction. \[\frac{\Gamma, x {:} A \vdash b {:} B} {\Gamma \vdash \lambda x. b {:} \Pi x {:} A. B}\] $\Pi$-elimination. \[\frac {\Gamma \vdash f {:} \Pi x {:} A.B\hspace{2em}\Gamma \vdash a {:} A} {\Gamma \vdash f\,a {:} B[x := a]}\] $\Pi$-equality. \[\frac {\Gamma, x {:} A \vdash b {:} B\hspace{2em}\Gamma \vdash a {:} A} {\Gamma \vdash (\lambda x.b)\,a = b[x := a] {:} B[x := a]}\] This is the rule of $\beta$-conversion. We may also add the rule of $\eta$-conversion: \[\frac {\Gamma \vdash f {:} \Pi x {:} A. B} {\Gamma \vdash \lambda x. f\,x = f {:} \Pi x {:} A. B}.\] Furthermore, there are congruence rules expressing that operations introduced by the formation, introduction, and elimination rules preserve equality. For example, the congruence rule for $\Pi$ is \[\frac{\Gamma \vdash A = A'\hspace{2em} \Gamma, x {:} A \vdash B=B'} {\Gamma \vdash \Pi x {:} A. B = \Pi x {:} A'. B'}.\] ### 3.5 Natural Numbers As in Peano arithmetic the natural numbers are generated by 0 and the successor operation $\s$. The elimination rule states that these are the only possible ways to generate a natural number. We write $f(c) = \R(c,d,xy.e)$ for the function which is defined by primitive recursion on the natural number $c$ with base case $d$ and step function $xy.e$ (or alternatively $\lambda xy.e$) which maps the value $y$ for the previous number $x {:} \N$ to the value for $\s(x)$. Note that $\R$ is a new variable-binding operator: the variables $x$ and $y$ become bound in $e$. $\N$-formation. \[\Gamma \vdash \N\] $\N$-introduction. \[\Gamma \vdash 0 {:} \N \hspace{2em} \frac{\Gamma \vdash a {:} \N} {\Gamma \vdash s(a) {:} \N}\] $\N$-elimination. \[\frac{ \Gamma, x {:} \N \vdash C \hspace{1em} \Gamma \vdash c {:} \N \hspace{1em} \Gamma \vdash d {:} C[x := 0] \hspace{1em} \Gamma, y {:} \N, z {:} C[x := y] \vdash e {:} C[x := s(y)] } { \Gamma \vdash \R(c,d,yz.e) {:} C[x := c] }\] $\N$-equality (under appropriate premises). \[\begin{align\} \R(0,d,yz.e) &= d {:} C[x := 0]\\ \R(s(a),d,yz.e) &= e[y := a, z := \R(a,d,yz.e)] {:} C[x := s(a)] \end{align\}\] The rule of $\N$-elimination simultaneously expresses the type of a function defined by primitive recursion and, under the Curry-Howard interpretation, the rule of mathematical induction: we prove the property $C$ of a natural number $x$ by induction on $x$. Godel's System $\T$ is essentially intuitionistic type theory with only the type formers $\N$ and $A \rightarrow B$ (the type of functions from $A$ to $B$, which is the special case of $(\Pi x {:} A)B$ where $B$ does not depend on $x {:} A$). Since there are no dependent types in System $\T$ the rules can be simplified. ### 3.6 The Universe of Small Types Martin-Lof's first version of type theory (Martin-Lof 1971a) had an axiom stating that there is a type of all types. This was proved inconsistent by Girard who found that the Burali-Forti paradox could be encoded in this theory. To overcome this pathological impredicativity, but still retain some of its expressivity, Martin-Lof introduced in 1972 a universe $\U$ of small types closed under all type formers of the theory, except that it does not contain itself (Martin-Lof 1998 [1972]). The rules are: $\U$-formation. \[\Gamma \vdash \U\] $\U$-introduction. \[\Gamma \vdash \varnothing {:} \U \hspace{3em} \Gamma \vdash 1 {:} \U\] \[\frac{\Gamma \vdash A {:} \U\hspace{2em} \Gamma \vdash B {:} \U} {\Gamma \vdash A + B {:} \U} \hspace{3em} \frac{\Gamma \vdash A {:} \U\hspace{2em} \Gamma \vdash B {:} \U} {\Gamma \vdash A \times B {:} \U}\] \[\frac{\Gamma \vdash A {:} \U\hspace{2em} \Gamma \vdash B {:} \U} {\Gamma \vdash A \rightarrow B {:} \U}\] \[\frac{\Gamma \vdash A {:} U\hspace{2em} \Gamma, x {:} A \vdash B {:} \U} {\Gamma \vdash \Sigma x {:} A.\, B {:} \U} \hspace{3em} \frac{\Gamma \vdash A {:} \U\hspace{2em} \Gamma, x {:} A \vdash B {:} \U} {\Gamma \vdash \Pi x {:} A.\, B {:} \U}\] \[\Gamma \vdash \N {:} \U\] $\U$-elimination. \[\frac{\Gamma \vdash A {:} \U} {\Gamma \vdash A}\] Since $\U$ is a type, we can use $\N$-elimination to define small types by primitive recursion. For example, if $A : \U$, we can define the type of $n$-tuples of elements in $A$ as follows: \[A^n = \R(n,1,xy.A \times y) {:} \U\] This type-theoretic universe $\U$ is analogous to a Grothendieck universe in set theory which is a set of sets closed under all the ways sets can be constructed in Zermelo-Fraenkel set theory. The existence of a Grothendieck universe cannot be proved from the usual axioms of Zermelo-Fraenkel set theory but needs a new axiom. In Martin-Lof (1975) the universe is extended to a countable hierarchy of universes \[\U\_0 : \U\_1 : \U\_2 : \cdots .\] In this way each type has a type, not only each small type. ### 3.7 Propositional Identity Above, we introduced the equality judgment \[\tag{4}\label{defeq} \Gamma \vdash a = a' {:} A.\] This is usually called a "definitional equality" because it can be decided by normalizing the terms $a$ and $a'$ and checking whether the normal forms are identical. However, this equality is a judgment and not a proposition (type) and we thus cannot prove such judgmental equalities by induction. For this reason we need to introduce propositional identity types. For example, the identity type for natural numbers $\I(\N,m,n)$ can be defined by $\U$-valued primitive recursion. We can then express and prove the Peano axioms. Moreover, extensional equality of ufnctions can be defined by \[\I(\N\rightarrow \N,f,f') = \Pi x {:} \N. \I(\N,f\,x,f'\,x).\] ### 3.8 The Axiom of Choice is a Theorem The following form of the axiom of choice is an immediate consequence of the BHK-interpretation of the intuitionistic quantifiers, and is easily proved in intuitionistic type theory: \[(\Pi x {:} A. \Sigma y {:} B. C) \rightarrow \Sigma f {:} (\Pi x {:} A. B). C[y := f\,x]\] The reason is that $\Pi x {:} A. \Sigma y {:} B. C$ is the type of functions which map elements $x {:} A$ to pairs $(y,z)$ with $y {:} B$ and $z {:} C$. The choice function $f$ is obtained by returning the first component $y {:} B$ of this pair. It is perhaps surprising that intuitionistic type theory directly validates an axiom of choice, since this axiom is often considered problematic from a constructive point of view. A possible explanation for this state of affairs is that the above is an axiom of choice for types, and that types are not in general appropriate constructive approximations of sets in the classical sense. For example, we can represent a real number as a Cauchy sequence in intuitionistic type theory, but the set of real numbers is not the type of Cauchy sequences, but the type of Cauchy sequences up to equiconvergence. More generally, a set in Bishop's constructive mathematics is represented by a type (commonly called "preset") together with an equivalence relation. If $A$ and $B$ are equipped with equivalence relations, there is of course no guarantee that the choice function, $f$ above, is extensional in the sense that it maps equivalent element to equivalent elements. This is the failure of the extensional axiom of choice, see Martin-Lof (2009) for an analysis. ## 4. Extensions ### 4.1 The Logical Framework The above completes the description of a core version of intuitionistic type theory close to that of (Martin-Lof 1998 [1972]). In 1986 Martin-Lof proposed a reformulation of intuitionistic type theory; see Nordstrom, Peterson and Smith (1990) for an exposition. The purpose was to give a more compact formulation, where $\lambda$ and $\Pi$ are the only variable binding operations. It is nowadays considered the main version of the theory. It is also the basis for the Agda proof assistant. The 1986 theory has two parts: * the theory of types (the logical framework); * the theory of sets (small types). Remark 4.1. Note that the word "set" in the logical framework does not coincide with the way it is used in Bishop's constructive mathematics. To avoid this confusion, types together with equivalence relations are usually called "setoids" or "extensional sets" in intuitionistic type theory. The logical framework has only two type formers: $\Pi x {:} A. B$ (usually written $(x {:} A)B$ or $(x {:} A) \rightarrow B$ in the logical framework formulation) and $\U$ (usually called $\Set$). The rules for $\Pi x{:} A. B$ ($(x {:} A) \rightarrow B$) are the same as given above (including $\eta$-conversion). The rules for $\U$ ($\Set$) are also the same, except that the logical framework only stipulates closure under $\Pi$-type formation. The other small type formers ("set formers") are introduced in the theory of sets. In the logical framework formulation each formation, introduction, and elimination rule can be expressed as the typing of a new constant. For example, the rules for natural numbers become \[\begin{align\} \N &: \Set,\\ 0 &: \N,\\ \s &: \N \rightarrow \N,\\ \R &: (C {:} \N \rightarrow \Set) \rightarrow C\,0 \rightarrow (( x {:} \N) \rightarrow C\,x \rightarrow C\,(\s\,x)) \rightarrow (c {:} \N) \rightarrow C\,c. \end{align\}\] where we have omitted the common context $\Gamma$, since the types of these constants are closed. Note that the recursion operator $R$ has a first argument $C {:} \N \rightarrow \Set$ unlike in the original formulation. Moreover, the equality rules can be expressed as equations \[\begin{align\} \R\, C\, d\, e\, 0 &= d {:} C\,0\\ \R\, C\, d\, e\, (\s\, a) &= e\, a\, (\R\, C\, d\, e\, a) {:} C\,(\s\,a) \end{align\}\] under suitable assumptions. In the sequel we will present several extensions of type theory. To keep the presentation uniform we will however not use the logical framework presentation of type theory, but will use the same notation as in section 2. ### 4.2 A General Identity Type Former As we mentioned above, identity on natural numbers can be defined by primitive recursion. Identity relations on other types can also be defined in the basic version of intuitionistic type theory presented in section 2. However, Martin-Lof (1975) extended intuitionistic type theory with a uniform primitive identity type former $\I$ for all types. The rules for $\I$ express that the identity relation is inductively generated by the proof of reflexivity, a canonicial constant called $\r$. (Note that $\r$ was coded by the number 0 in the introductory presentation of proof-objects in 2.3. The elimination rule for the identity type is a generalization of identity elimination in predicate logic and introduces an elimination constant $\J$. We here show the formulation due to Paulin-Mohring (1993) rather than the original formulation of Martin-Lof (1975). The inference rules are the following. $\I$-formation. \[\frac{\Gamma \vdash A \hspace{1em} \Gamma \vdash a {:} A \hspace{1em} \Gamma \vdash a' {:} A} {\Gamma \vdash \I(A,a,a')}\] $\I$-introduction. \[\frac{\Gamma \vdash A \hspace{1em} \Gamma \vdash a {:} A} {\Gamma \vdash \r {:} \I(A,a,a)}\] $\I$-elimination. \[\frac{ \Gamma, x {:} A, y {:} \I(A,a,x) \vdash C \hspace{1em} \Gamma \vdash b {:} A \hspace{1em} \Gamma \vdash c {:} \I(A,a,b) \hspace{1em} \Gamma \vdash d {:} C[x := a, y := r]} { \Gamma \vdash \J(c,d) {:} C[x := b, y:= c]}\] $\I$-equality (under appropriate assumptions). \[\begin{align\} \J(r,d) &= d \end{align\}\] Note that if $C$ only depends on $x : A$ and not on the proof $y : \I(A,a,x)$ (and we also suppress proof objects) in the rule of $\I$-elimination we recover the rule of identity elimination in predicate logic. By constructing a model of type theory where types are interpreted as groupoids (categories where all arrows are isomorphisms) Hofmann and Streicher (1998) showed that it cannot be proved in intuitionistic type theory that all proofs of $I(A,a,b)$ are identical. This may seem as an incompleteness of the theory and Streicher suggested a new axiom $\K$ from which it follows that all proofs of $\I(A,a,b)$ are identical to $\r$. The $\I$-type is often called the intensional identity type, since it does not satisfy the principle of function extensionality. Intuitionistic type theory with the intensional identity type is also often called intensional intuitionistic type theory to distinguish it from extensional intuitionistic type theory which will be presented in section 7.1. ### 4.3 Well-Founded Trees A type of well-founded trees of the form $\W x {:} A. B$ was introduced in Martin-Lof 1982 (and in a more restricted form by Scott 1970). Elements of $\W x {:} A. B$ are trees of varying and arbitrary branching: varying, because the branching type $B$ is indexed by $x {:} A$ and arbitrary because $B$ can be arbitrary. The type is given by a generalized inductive definition since the well-founded trees may be infinitely branching. We can think of $\W x{:}A. B$ as the free term algebra, where each $a {:} A$ represents a term constructor $\sup\,a$ with (possibly infinite) arity $B[x := a]$. $\W$-formation. \[\frac{\Gamma \vdash A\hspace{2em} \Gamma, x {:} A \vdash B} {\Gamma \vdash \W x {:} A. B}\] $\W$-introduction. \[\frac{\Gamma \vdash a {:} A \hspace{2em} \Gamma, y {:} B[x:=a] \vdash b : Wx{:}A. B} {\Gamma \vdash \sup(a, y.b) : \W x {:} A. B}\] We omit the rules of $\W$-elimination and $\W$-equality. Adding well-founded trees to intuitionistic type theory increases its proof-theoretic strength significantly (Setzer (1998)). ### 4.4 Iterative Sets and CZF An important application of well-founded trees is Aczel's (1978) construction of a type-theoretic model of Constructive Zermelo Fraenkel Set Theory. To this end he defines the type of iterative sets as \[\V = \W x {:} \U. x.\] Let $A {:} \U$ be a small type, and $x {:} A\vdash M$ be an indexed family of iterative sets. Then $\sup(A,x.M)$, or with a more suggestive notation $\{ M\mid x {:} A\}$, is an iterative set. To paraphrase: an iterative set is a family of iterative sets indexed by a small type. Note that an iterative set is a data-structure in the sense of functional programming: a possibly infinitely branching well-founded tree. Different trees may represent the same set. We therefore need to define a notion of extensional equality between iterative sets which disregards repetition and order of elements. This definition is formally similar to the definition of bisimulation of processes in process algebra. The type $\V$ up to extensional equality can be viewed as a constructive type-theoretic model of the cumulative hierarchy, see the entry on set theory: constructive and intuitionistic ZF for further information about CZF. ### 4.5 Inductive Definitions The notion of an inductive definition is fundamental in intuitionistic type theory. It is a primitive notion and not, as in set theory, a derived notion where an inductively defined set is defined impredicatively as the smallest set closed under some rules. However, in intuitionistic type theory inductive definitions are considered predicative: they are viewed as being built up from below. The inductive definability of types is inherent in the meaning explanations of intuitionistic type theory which we shall discuss in the next section. In fact, intuitionistic type theory can be described briefly as a theory of inductive, recursive, and inductive-recursive definitions based on a framework of lambda calculus with dependent types. We have already seen the type of natural numbers and the type of well-founded trees as examples of types given by inductive definitions; the natural numbers is an example of an ordinary finitary inductive definition and the well-founded trees of a generalized possibly infinitary inductive definition. The introduction rules describe how elements of these types are inductively generated and the elimination and equality rules describe how functions from these types can be defined by structural recursion on the way these elements are generated. According to the propositions as types principle, the elimination rules are simultaneously rules for proof by structural induction on the way the elements are generated. The type formers $0, 1, +, \times, \rightarrow, \Sigma,$ and $\Pi$ which interpret the logical constants for intuitionistic predicate logic are examples of degenerate inductive definitions. Even the identity type (in intensional intuitionistic type theory) is inductively generated; it is the type of proofs generated by the reflexivity axiom. Its elimination rule expresses proof by pattern matching on the proof of reflexivity. The common structure of the rules of the type formers can be captured by a general schema for inductive definitions (Dybjer 1991). This general schema has many useful instances, for example, the type $\List(A)$ of lists with elements of type $A$ has the following introduction rules: \[\Gamma \vdash \nil {:} \List(A) \hspace{3em} \frac{\Gamma \vdash a {:} A\hspace{2em}\Gamma \vdash as {:} \List(A)} {\Gamma \vdash \cons(a,as) {:} \List(A)}\] Other useful instances are types of binary trees and other trees such as the infinitely branching trees of the Brouwer ordinals of the second and higher number classes. The general schema does not only cover inductively defined types, but also inductively defined families of types, such as the identity relation. The above mentioned type $A^n$ of $n$-tuples of type $A$ was defined above by primitive recursion on $n$. It can also be defined as an inductive family with the following introduction rules \[\Gamma \vdash \nil {:} A^0 \hspace{3em} \frac{\Gamma \vdash a {:} A\hspace{2em}\Gamma \vdash as {:} A^n} {\Gamma \vdash \cons(a,as) {:} A^{\s(n)}}\] The schema for inductive types and families is a type-theoretic generalization of a schema for iterated inductive definitions in predicate logic (formulated in natural deduction) presented by Martin-Lof (1971b). This paper immediately preceded Martin-Lof's first version of intuitionistic type theory. It is both conceptually and technically a forerunner to the development of the theory. It is an essential feature of proof assistants such as Agda and Coq that it enables users to define their own inductive types and families by listing their introduction rules (the types of their constructors). This is much like in typed functional programming languages such as Haskell and the different dialects of ML. However, unlike in these programming languages the schema for inductive definitions in intuitionistic type theory enforces a restriction amounting to well-foundedness of the elements of the defined types. ### 4.6 Inductive-Recursive Definitions We already mentioned that there are two main definition principles in intuitionistic type theory: the inductive definition of types (sets) and the (primitive, structural) definition of functions by recursion on the way the elements of such types are inductively generated. Usually, the inductive definition of a set comes first: the formation and introduction rules make no reference to the elimination rule. However, there are definitions in intuitionistic type theory for which this is not the case and we simultaneously inductively generate a type and a function from that type defined by structural recursion. Such definitions are simultaneously inductive-recursive. The first example of such an inductive-recursive definition is an alternative formulation a la Tarski of the universe of small types. Above we presented the universe formulated a la Russell, where there is no notational distinction between the element $A {:} \U$ and the corresponding type $A$. For a universe a la Tarski there is such a distinction, for example, between the element $\hat{\N} {:} \U$ and the corresponding type $\N$. The element $\hat{\N}$ is called the code for $\N$. The elimination rule for the universe a la Tarski is: \[\frac{\Gamma \vdash a {:} \U} {\Gamma \vdash \T(a)}\] This expresses that there is a function $\T$ which maps a code $a$ to its corresponding type $T(a)$. The equality rules define this correspondence. For example, \[\T(\hat{\N}) = \N.\] We see that $\U$ is inductively generated with one introduction rule for each small type former, and $\T$ is defined by recursion on these small type formers. The simultaneous inductive-recursive nature of this definition becomes apparent in the rules for $\Pi$ for example. The introduction rule is \[\frac{\Gamma \vdash a {:} \U\hspace{2em} \Gamma, x {:} \T(a) \vdash b {:} \U} {\Gamma \vdash \hat{\Pi} x {:} a. b {:} \U}\] and the corresponding equality rule is \[\T(\hat{\Pi}x {:} a. b) = \Pi x {:} \T(a). \T(b)\] Note that the introduction rule for $\U$ refers to $\T$, and hence that $\U$ and $\T$ must be defined simultaneously. There are a number of other universe constructions which are defined inductive-recursively: universe hierarchies, superuniverses (Palmgren 1998; Rathjen, Griffor, and Palmgren 1998), and Mahlo universes (Setzer 2000). These universes are analogues of certain large cardinals in set theory: inaccessible, hyperinaccessible, and Mahlo cardinals. Other examples of inductive-recursive definitions include an informal definition of computability predicates used by Martin-Lof in an early normalization proof of intuitionistic type theory (Martin-Lof 1998 [1972]). There are also many natural examples of "small" inductive-recursive definitions, where the recursively defined (decoding) function returns an element of a type rather than a type. A large class of inductive-recursive definitions, including the above, can be captured by a general schema (Dybjer 2000) which extends the schema for inductive definitions mentioned above. As shown by Setzer, intuitionistic type theory with this class of inductive-recursive definitions is very strong proof-theoretically (Dybjer and Setzer 2003). However, as proposed in recent unpublished work by Setzer, it is possible to increase the strength of the theory even further and define universes such as an autonomous Mahlo universe which are analogues of even larger cardinals. ## 5. Meaning Explanations The consistency of intuitionistic type theory relative to set theory can be proved by model constructions. Perhaps the simplest method is an interpretation whereby each type-theoretic concept is given its corresponding set-theoretic meaning, as outlined in section 2.3. For example the type of functions $A \rightarrow B$ is interpreted as the set of all functions in the set-theoretic sense between the set denoted by $A$ and the set denoted by $B$. To interpret $\U$ we need a set-theoretic universe which is closed under all (set-theoretic analogues of) the type constructors. Such a universe can be proved to exist if we assume the existence of an inaccessible cardinal $\kappa$ and interpret $\U$ by $V\_\kappa$ in the cumulative hierarchy. Alternatives are realizability models, and for intensional type theory, a model of terms in normal forms. The latter can also be used for proving decidability of the judgments of the theory. Mathematical models only prove consistency relative to classical set theory (or whatever other meta-theory we are using). Is it possible to be convinced about the consistency of the theory in a more direct way, so called simple minded consistency (Martin-Lof 1984)? In fact, is there a way to explain what it means for a judgment to be correct in a direct pre-mathematical way? And given that we know what the judgments mean can we then be convinced that the inference rules of the theory are valid? An answer to this problem was proposed by Martin-Lof in 1979 in the paper "Constructive Mathematics and Computer Programming" (Martin-Lof 1982) and elaborated later on in numerous lectures and notes, see for example, Martin-Lof (1984, 1987). These meaning explanations for intuitionistic type theory are also referred to as the direct semantics, intuitive semantics, informal semantics, standard semantics, or the syntactico-semantical approach to meaning theory. This meaning theory follows the Wittgensteinian meaning-as-use tradition. The meaning is based on rules for building objects (introduction rules) of types and computation rules (elimination rules) for computing with these objects. A difference from much of the Wittgensteinian tradition is that also higher order types like $\N \rightarrow \N$ are given meaning using rules. To explain the meaning of a judgment we must first know how the terms in the judgment are computed to canonical form. Then the formation rules explain how correct canonical types are built and the introduction rules explain how correct canonical objects of such canonical types are built. We quote (Martin-Lof 1982): > > A canonical type $A$ is defined by prescribing how a canonical > object of type $A$ is formed as well as how two equal canonical > objects of type $A$ are formed. There is no limitation on this > prescription except that the relation of equality which it defines > between canonical objects of type $A$ must be reflexive, symmetric > and transitive. In other words, a canonical type is equipped with an equivalence relation on the canonical objects. Below we shall give a simplified form of the meaning explanations, where this equivalence relation is extensional identity of objects. In spite of the pre-mathematical nature of this meaning theory, its technical aspects can be captured as a mathematical model construction similar to Kleene's realizability interpretation of intuitionistic logic, see the next section. The realizers here are the terms of type theory rather than the number realizers used by Kleene. ### 5.1 Computation to Canonical Form The meaning of a judgment is explained in terms of the computation of the types and terms in the judgment. These computations stop when a canonical form is reached. By canonical form we mean a term where the outermost form is a constructor (introduction form). These are the canonical forms used in lazy functional programming (for example in the Haskell language). For the purpose of illustration we consider meaning explanations only for three type formers: $\N, \Pi x {:} A.B$, and $\U$. The context free grammar for the terms of this fragment of Intuitionistic Type Theory is as follows: \[ a :: = 0 \mid \s(a) \mid \lambda x.a \mid \N \mid \Pi x{:}a.a \mid \U \mid \R(a,a,xx.a) \mid a\,a . \] The canonical terms are generated by the following grammar: \[v :: = 0 \mid \s(a) \mid \lambda x.a \mid \N \mid \Pi x{:}a.a \mid \U ,\] where $a$ ranges over arbitrary, not necessarily canonical, terms. Note that $\s(a)$ is canonical even if $a$ is not. To explain how terms are computed to canonical form, we introduce the relation $a \Rightarrow v$ between closed terms $a$ and canonical forms (values) $v$ given by the following computation rules: \[ \frac{c \Rightarrow 0\hspace{1em}d \Rightarrow v}{\R(c,d,xy.e)\Rightarrow v} \hspace{2em} \frac{c \Rightarrow \s(a)\hspace{1em}e[x := d,y := \R(a,d,xy.e)]\Rightarrow v}{\R(c,d,xy.e)\Rightarrow v} \] \[ \frac{f\Rightarrow \lambda x.b\hspace{1em}b[x := a]\Rightarrow v}{f\,a \Rightarrow v} \] in addition to the rule \[v \Rightarrow v\] stating that a canonical term has itself as value. ### 5.2 The Meaning of Categorical Judgments A categorical judgment is a judgment where the context is empty and there are no free variables. The meaning of the categorical judgment $\vdash A$ is that $A$ has a canonical type as value. In our fragment this means that either of the following holds: * $A \Rightarrow \N$, * $A \Rightarrow \U$, * $A \Rightarrow \Pi x {:} B. C$ and furthermore that $\vdash B$ and $x {:} B \vdash C$. The meaning of the categorical judgment $\vdash a {:} A$ is that $a$ has a canonical term of the canonical type of $A$ as value. In our fragment this means that either of the following holds: * $A \Rightarrow \N$ and either $a \Rightarrow 0$ or $a \Rightarrow \s(b)$ and $\vdash b {:} \N$, * $A \Rightarrow \U$ and either $a \Rightarrow \N$ or $a \Rightarrow \Pi x {:} b. c$ where furthermore $\vdash b {:} \U$ and $x {:} b \vdash c {:} \U$, * $A \Rightarrow \Pi x {:} B. C$ and $a \Rightarrow \lambda x.c$ and $x {:} B \vdash c {:} C$. The meaning of the categorical judgment $\vdash A = A'$ is that $A$ and $A'$ have the same canonical types as values. In our fragment this means that either of the following holds: * $A \Rightarrow \N$ and $A' \Rightarrow \N$, * $A \Rightarrow \U$ and $A' \Rightarrow \U$, * $A \Rightarrow \Pi x {:} B. C$ and $A' \Rightarrow \Pi x {:} B'. C'$ and furthermore that $\vdash B = B'$ and $x {:} B \vdash C = C'$. The meaning of the categorical judgment $\vdash a = a' {:} A$ is explained in a similar way. It is a tacit assumption of the meaning explanations that the repeated computations of canonical forms is well-founded. For example, a natural number is the result of finitely many computations of the successor function $\s$ ended by $0$. A computation which results in infinitely many computations of $\s$ is not a natural number in intuitionistic type theory. (However, there are extensions of type theory, for example, partial type theory, and non-standard type theory, where such infinite computations can occur, see section 7.3. To justify the rules of such theories the present meaning explanations do not suffice.) ### 5.3 The Meaning of Hypothetical Judgments According to Martin-Lof (1982) the meaning of a hypothetical judgment is reduced to the meaning of the categorical judgments by substituting the closed terms of appropriate types for the free variables. For example, the meaning of \[x\_1 {:} A\_1, \ldots, x\_n {:} A\_n \vdash a {:} A\] is that the categorical judgment \[\vdash a[x\_1 := a\_1, \ldots , x\_n := a\_n] : A[x\_1 := a\_1, \ldots , x\_n := a\_n]\] is valid whenever the categorical judgments \[\vdash a\_1 {:} A\_1, \ldots , \vdash a\_n[x\_1 := a\_1, \ldots , x\_{n-1} := a\_{n-1}] {:} A\_n[x\_1 := a\_1, \ldots , x\_{n-1} := a\_{n-1}]\] are valid. ## 6. Mathematical Models ### 6.1 Categorical Models #### 6.1.1 Hyperdoctrines Curry's correspondence between propositions and types was extended to predicate logic in the late 1960s by Howard (1980) and de Bruijn (1970). At around the same time Lawvere developed related ideas in categorical logic. In particular he proposed the notion of a hyperdoctrine (Lawvere 1970) as a categorical model of (typed) predicate logic. A hyperdoctrine is an indexed category $P {:} T^{op} \rightarrow \mathbf{Cat}$, where $T$ is a category where the objects represent types and the arrows represent terms. If $A$ is a type then the fibre $P(A)$ is a category of propositions depending on a variable $x {:} A$. The arrows in this category are proofs $Q \vdash R$ and can be thought of as proof-objects. Moreover, since we have an indexed category, for each arrow $t$ from $A$ to $B$, there is a reindexing functor $P(B) \rightarrow P(A)$ representing substitution of $t$ for a variable $y {:} B$. The category $P(A)$ is assumed to be cartesian closed and conjunction and implications are modelled by products and exponentials in this category. The quantifiers $\exists$ and $\forall$ are modelled by the left and right adjoints of the reindexing functor. Moreover, Lawvere added further structure to hyperdoctrines to model identity propositions (as left adjoints to a diagonal functor) and a comprehension schema. #### 6.1.2 Contextual categories, categories with attributes, and categories with families Lawvere's definition of hyperdoctrines preceded intuitionistic type theory but did not go all the way to identifying propositions and types. Nevertheless Lawvere influenced Scott's (1970) work on constructive validity, a somewhat preliminary precursor of intuitionistic type theory. After Martin-Lof (1998 [1972]) had presented a more definite formulation of the theory, the first work on categorical models was presented by Cartmell in 1978 with his notions of category with attributes and contextual category (Cartmell 1986). However, we will not define these structures here but instead the closely related categories with families (Dybjer 1996) which are formulated so that they directly model a variable-free version of a formulation of intuitionistic type theory with explicit substitutions (Martin-Lof 1995). A category with families is a functor $T {:} C^{op} \rightarrow \mathbf{Fam}$, where $\mathbf{Fam}$ is the category of families of sets. The category $C$ is the category of contexts and substitutions. If $\Gamma$ is an object of $C$ (a context), then $T(\Gamma)$ is the family of terms of type $A$ which depend on variables in $\Gamma$. If $\gamma$ is an arrow in $C$ representing a substitution, then the arrow part of the functor represents substitution of $\gamma$ in types and terms. A category with families also has a terminal object and a notion of context comprehension, reminiscent of Lawvere's comprehension in hyperdoctrines. The terminal object captures the rules for empty contexts and empty substitutions. Context comprehension captures the rules for extending contexts and substitutions, and has projections capturing weakening and assumption of the last variable. Categories with families are algebraic structures which model the general rules of dependent type theory, those which come before the rules for specific type formers, such as $\Pi$, $\Sigma$, identity types, universes, etc. In order to model specific type-former corresponding extra structure needs to be added. #### 6.1.3 Locally cartesian closed categories From a categorical perspective the above-mentioned structures may appear somewhat special and ad hoc. A more regular structure which gives rise to models of intuitionistic type theory are the locally cartesian closed categories. These are categories with a terminal object, where each slice category is cartesian closed. It can be shown that the pullback functor has a left and a right adjoint, representing $\Sigma$- and $\Pi$-types, respectively. Locally cartesian closed categories correspond to intuitionistic type theory with extensional identity types and $\Sigma$ and $\Pi$-types (Seely 1984, Clairambault and Dybjer 2014). It should be remarked that the correspondence with intuitionistic type theory is somewhat indirect, since a coherence problem, in the sense of category theory, needs to be solved. The problem is that in locally cartesian closed categories type substituion is represented by pullbacks, but these are only defined up to isomorphism, see Curien 1993 and Hofmann 1994. ### 6.2 Set-Theoretic Model Intuitionistic type theory is a possible framework for constructive mathematics in Bishop's sense. Such constructive mathematics is compatible with classical mathematics: a constructive proof in Bishop's sense can directly be understood as a proof in classical logic. A formal way to understand this is by constructing a set-theoretic model of intuitionistic type theory, where each concept of type theory is interpreted as the corresponding concept in Zermelo-Fraenkel Set Theory. For example, a type is interpreted as a set, and the type of functions in $A \rightarrow B$ is interpreted as the set of all functions in the set-theoretic sense from the set representing $A$ to the set representing $B$. The type of natural numbers is interpreted as the set of natural numbers. The interpretations of identity types, and $\Sigma$ and $\Pi$-types were already discussed in the introduction. And as already mentioned, to interpret the type-theoretic universe we need an inaccessible cardinal. #### 6.2.1 Model in CZF It can be shown that the interpretation outlined above can be carried out in Aczel's constructive set theory CZF. Hence it does not depend on classical logic or impredicative features of set theory. ### 6.3 Realizability Models The set-theoretic model can be criticized on the grounds that it models the type of functions as the set of all set-theoretic functions, in spite of the fact that a function in type theory is always computable, whereas a set-theoretic function may not be. To remedy this problem one can instead construct a realizability model whereby one starts with a set of realizers. One can here follow Kleene's numerical realizability closely where functions are realized by codes for Turing machines. Or alternatively, one can let realizers be terms in a lambda calculus or combinatory logic possibly extended with appropriate constants. Types are then represented by sets of realizers, or often as partial equivalence relations on the set of realizers. A partial equivalence relation is a convenient way to represent a type with a notion of "equality" on it. There are many variations on the theme of realizability model. Some such models tacitly assume set theory as the metatheory (Aczel 1980, Beeson 1985), whereas others explictly assume a constructive metatheory (Smith 1984). Realizability models are also models of the extensional version of intuitionistic type theory (Martin-Lof 1982) which will be presented in section 7.1 below. ### 6.4 Model of Normal Forms and Type-Checking In intuitionistic type theory each type and each well-typed term has a normal form. A consequence of this normal form property is that all the judgments are decidable: for example, given a correct context $\Gamma$, a correct type $A$ and a possibly ill-typed term $a$, there is an algorithm for deciding whether $\Gamma \vdash a {:} A$. This type-checking algorithm is the key component of proof-assistants for Intensional Type Theory, such as Agda. The correctness of the normal form property can be expressed as a model of normal forms, where each context, type, and term are interpreted as their respective normal forms. ## 7. Variants of the Theory ### 7.1 Extensional Type Theory In extensional intuitionistic type theory (Martin-Lof 1982) the rules of $\I$-elimination and $\I$-equality for the general identity type are replaced by the following two rules: \[\frac{\Gamma\vdash c {:} \I(A,a,a')} {\Gamma \vdash a=a' {:} A} \hspace{3em} \frac{\Gamma\vdash c{:}I(A,a,a')} {\Gamma\vdash c = \r {:} \I(A,a,a')}\] The first causes the distinction between propositional and judgmental equality to disappear. The second forces identity proofs to be unique. Unlike the rules for the intensional identity type former, the rules for extensional identity types do not fit into the schema for inductively defined types mentioned above. These rules are however justified by the meaning explanations in Martin-Lof (1982). This is because the categorical judgment \[\vdash c {:} \I(A,a,a')\] is valid iff $c \Rightarrow \r$ and the judgment $\vdash a = a' {:} A$ is valid. However, these rules make it possible to define terms without normal forms. Since the type-checking algorithm relies on the computation of normal forms of types, it no longer works for extensional type theory, see (Castellan, Clairambault, and Dybjer 2015). On the other hand, certain constructions which are not available in intensional type theory are possible in extensional type theory. For example, function extensionality \[(\Pi x {:} A. \I(B,f\,x,f'\,x)) \rightarrow \I(\Pi x{:}A.B,f,f')\] is a theorem. Another example is that $\W$-types can be used for encoding other inductively defined types in Extensional Type Theory. For example, the Brouwer ordinals of the second and higher number classes can be defined as special instances of the $\W$-type (Martin-Lof 1984). More generally, it can be shown that all inductively defined types which are given by a strictly positive type operator can be represented as instances of well-founded trees (Dybjer 1997). ### 7.2 Univalent Foundations and Homotopy Type Theory Univalent foundations refer to Voevodsky's programme for a new foundation of mathematics based on intuitionistic type theory and employing ideas from homotopy theory. Here every type $A$ is considered as a space, and the identity type $\I(A,a,b)$ is the space of paths from point $a$ to point $b$ in $A$. Iterated identity types represent higher homotopies, e.g. \[\I(\I(A,a,b),f,g)\] is the space of homotopies between $f$ and $g$. The notion of ordinary set can be thought of as a discrete space $A$ where all paths in $\I(A,a,b)$ are trivial loops. The origin of these ideas was the remarkable discovery by (Hofmann and Streicher 1998) that the axioms of intensional type theory do not force all proofs of an identity to be equal, that is, not all paths need to be trivial. This was shown by a model construction where each type is interpreted as a groupoid. Further connections between identity types and notions from homotopy theory and higher categories were subsequently discovered by (Awodey and Warren 2009), (Lumsdaine 2010), and (van den Berg and Garner 2011). Voevodsky realized that the whole intensional intuitionistic type theory could be modelled by a well-known category studied in homotopy theory, namely the Kan simplicial sets. Inspired by this model he introduced the univalence axiom. For a universe $\U$ of small types, this axiom states that the substitution map associated with the $J$-operator \[\I(\U,a,b) \longrightarrow \T(a) \cong \T(b)\] is an equivalence. Equivalence ($\cong$) here refers to a general notion of equivalence of higher dimensional objects, as in the sequence equal elements, isomorphic sets, equivalent groupoids, biequivalent bigroupoids, etc. The univalence axiom expresses that "everything is preserved by equivalence", thereby realizing the informal categorical slogan that all categorical constructions are preserved by isomorphism, and its generalization, that all constructions of categories are preserved by equivalence of categories, etc. The axiom of univalence was originally justified by Voevodsky's simplical set model. This model is however not constructive and (Bezem, Coquand and Huber 2014 [2013]) has more recently proposed a model in Kan cubical sets. Although univalent foundations concern preservation of mathematical structure in general, strongly inspired by category theory, applications within homotopy theory are particularly actively investigated. Intensional type theory extended with the univalence axiom and so called higher inductive types is therefore also called "homotopy type theory". We refer to the entry on type theory for further details. ### 7.3 Partial and Non-Standard Type Theory Intuitionistic type theory is not intended to model Brouwer's notion of free choice sequence, although lawlike choice sequences can be modelled as functions from $\N$. However, there are extensions of the theory which incorporate such choice sequences: namely partial type theory and non-standard type theory (Martin-Lof 1990). The types in partial type theory can be interpreted as Scott domains (Martin-Lof 1986, Palmgren and Stoltenberg-Hansen 1990, Palmgren 1991). In this way a type $\N$ which contains an infinite number $\infty$ can be interpreted. However, in partial type theory all types are inhabited by a least element $\bot$, and thus the propositions as types principle is not maintained. Non-standard type theory incorporates non-standard elements, such as an infinite number $\infty {:} \N$ without inhabiting all types. ### 7.4 Impredicative Type Theory The inconsistent version of intuitionistic type theory of Martin-Lof (1971a) was based on the strongly impredicative axiom that there is a type of all types. However, (Coquand and Huet 1988) showed with their calculus of constructions, that there is a powerful impredicative but consistent version of type theory. In this theory the universe $\U$ (usually called ${\bf Prop}$ in this theory) is closed under the following formation rule for cartesian product of families of types: \[\frac{\Gamma \vdash A \hspace{2em} \Gamma, x {:} A \vdash B {:} \U} {\Gamma \vdash \Pi x {:} A. B {:} \U}\] This rule is more general than the rule for constructing small cartesian products of families of small types in intuitionistic type theory, since we can now quantify over arbitrary types $A$, including $\U$, and not just small types. We say that $\U$ is impredicative since we can construct a new element of it by quantifying over all elements, even the element which is constructed. The motivation for this theory was that inductively defined types and families of types become definable in terms of impredicative quantification. For example, the type of natural numbers can be defined as the type of Church numerals: \[\N = \Pi X {:} \U. X \rightarrow (X \rightarrow X) \rightarrow X {:} \U\] This is an impredicative definition, since it is a small type which is constructed by quantification over all small types. Similarly we can define an identity type by impredicative quantification: \[\I(A,a,a')= \Pi X {:} A \rightarrow \U. X\,a \rightarrow X\,a' {:} \U\] This is Leibniz' definition of equality: $a$ and $a'$ are equal iff they satisfy the same properties (ranged over by $X$). Unlike in intuitionistic type theory, the function type in impredicative type cannot be interpreted set-theoretically in a straightfoward way, see (Reynolds 1984). ### 7.5 Proof Assistants In 1979 Martin-Lof wrote the paper "Constructive Mathematics and Computer Programming" where he explained that intuitionistic type theory is a programming language which can also be used as a formal foundation for constructive mathematics. Shortly after that, interactive proof systems which help the user to derive valid judgments in the theory, so called proof assistants, were developed. One of the first systems was the NuPrl system (PRL Group 1986), which is based on an extensional type theory similar to (Martin-Lof 1982). Systems based on versions of intensional type theory go back to the type-checker for the impredicative calculus of constructions which was written around 1984 by Coquand and Huet. This led to the Coq system, which is based on the calculus of inductive constructions (Paulin-Mohring 1993), a theory which extends the calculus of construction with primitive inductive types and families. The encodings of the pure calculus of constructions were found to be inconvenient, since the full elimination rules could not be derived and instead had to be postulated. We also remark that the calculus of inductive constructions has a subsystem, the predicative calculus of inductive constructions, which follows the principles of Martin-Lof's intuitionistic type theory. Agda is another proof assistant which is based on the logical framework formulation of intuitionistic type theory, but adds numerous features inspired by practical programming languages (Norell 2008). It is an intensional theory with decidable judgments and a type-checker similar to Coq's. However, in contrast to Coq it is based on Martin-Lof's predicative intuitionistic type theory. There are several other systems based either on the calculus of constructions (Lego, Matita, Lean) or on intuitionistic type theory (Epigram, Idris); see (Pollack 1994; Asperti et al. 2011; de Moura et al. 2015; McBride and McKinna 2004; Brady 2011).
type-theory-intuitionistic	## 1. Overview We begin with a bird's eye view of some important aspects of intuitionistic type theory. Readers who are unfamiliar with the theory may prefer to skip it on a first reading. The origins of intuitionistic type theory are Brouwer's intuitionism and Russell's type theory. Like Church's classical simple theory of types it is based on the lambda calculus with types, but differs from it in that it is based on the propositions-as-types principle, discovered by Curry (1958) for propositional logic and extended to predicate logic by Howard (1980) and de Bruijn (1970). This extension was made possible by the introduction of indexed families of types (dependent types) for representing the predicates of predicate logic. In this way all logical connectives and quantifiers can be interpreted by type formers. In intuitionistic type theory further types are added, such as a type of natural numbers, a type of small types (a universe) and a type of well-founded trees. The resulting theory contains intuitionistic number theory (Heyting arithmetic) and much more. The theory is formulated in natural deduction where the rules for each type former are classified as formation, introduction, elimination, and equality rules. These rules exhibit certain symmerties between the introduction and elimination rules following Gentzen's and Prawitz' treatment of natural deduction, as explained in the entry on proof-theoretic semantics. The elements of propositions, when interpreted as types, are called proof-objects. When proof-objects are added to the natural deduction calculus it becomes a typed lambda calculus with dependent types, which extends Church's original typed lambda calculus. The equality rules are computation rules for the terms of this calculus. Each function definable in the theory is total and computable. Intuitionistic type theory is thus a typed functional programming language with the unusual property that all programs terminate. Intuitionistic type theory is not only a formal logical system but also provides a comprehensive philosophical framework for intuitionism. It is an interpreted language, where the distinction between the demonstration of a judgment and the proof of a proposition plays a fundamental role (Sundholm 2012). The framework clarifies the Brouwer-Heyting-Kolmogorov interpretation of intuitionistic logic and extends it to the more general setting of intuitionistic type theory. In doing so it provides a general conception not only of what a constructive proof is, but also of what a constructive mathematical object is. The meaning of the judgments of intuitionistic type theory is explained in terms of computations of the canonical forms of types and terms. These informal, intuitive meaning explanations are "pre-mathematical" and should be contrasted to formal mathematical models developed inside a standard mathematical framework such as set theory. This meaning theory also justifies a variety of inductive, recursive, and inductive-recursive definitions. Although proof-theoretically strong notions can be justified, such as analogues of certain large cardinals, the system is considered predicative. Impredicative definitions of the kind found in higher-order logic, intuitionistic set theory, and topos theory are not part of the theory. Neither is Markov's principle, and thus the theory is distinct from Russian constructivism. An alternative formal logical system for predicative constructive mathematics is Myhill and Aczel's constructive Zermelo-Fraenkel set theory (CZF). This theory, which is based on intuitionistic first-order predicate logic and weakens some of the axioms of classical Zermelo-Fraenkel Set Theory, has a natural interpretation in intuitionistic type theory. Martin-Lof's meaning explanations thus also indirectly form a basis for CZF. Variants of intuitionistic type theory underlie several widely used proof assistants, including NuPRL, Coq, and Agda. These proof assistants are computer systems that have been used for formalizing large and complex proofs of mathematical theorems, such as the Four Colour Theorem in graph theory and the Feit-Thompson Theorem in finite group theory. They have also been used to prove the correctness of a realistic C compiler (Leroy 2009) and other computer software. Philosophically and practically, intuitionistic type theory is a foundational framework where constructive mathematics and computer programming are, in a deep sense, the same. This point has been emphasized by (Gonthier 2008) in the paper in which he describes his proof of the Four Colour Theorem: > > The approach that proved successful for this proof was to turn > almost every mathematical concept into a data structure or a program > in the Coq system, thereby converting the entire enterprise into one > of program verification. > > > ## 2. Propositions as Types ### 2.1 Intuitionistic Type Theory: a New Way of Looking at Logic? Intuitionistic type theory offers a new way of analyzing logic, mainly through its introduction of explicit proof objects. This provides a direct computational interpretation of logic, since there are computation rules for proof objects. As regards expressive power, intuitionistic type theory may be considered as an extension of first-order logic, much as higher order logic, but predicative. #### 2.1.1 A Type Theory Russell developed type theory in response to his discovery of a paradox in naive set theory. In his ramified type theory mathematical objects are classified according to their types: the type of propositions, the type of objects, the type of properties of objects, etc. When Church developed his simple theory of types on the basis of a typed version of his lambda calculus he added the rule that there is a type of functions between any two types of the theory. Intuitionistic type theory further extends the simply typed lambda calculus with dependent types, that is, indexed families of types. An example is the family of types of $n$-tuples indexed by $n$. Types have been widely used in programming for a long time. Early high-level programming languages introduced types of integers and floating point numbers. Modern programming languages often have rich type systems with many constructs for forming new types. Intuitionistic type theory is a functional programming language where the type system is so rich that practically any conceivable property of a program can be expressed as a type. Types can thus be used as specifications of the task of a program. #### 2.1.2 An intuitionstic logic with proof-objects Brouwer's analysis of logic led him to an intuitionistic logic which rejects the law of excluded middle and the law of double negation. These laws are not valid in intuitionistic type theory. Thus it does not contain classical (Peano) arithmetic but only intuitionistic (Heyting) arithmetic. (It is another matter that Peano arithmetic can be interpreted in Heyting arithmetic by the double negation interpretation, see the entry on intuitionistic logic.) Consider a theorem of intuitionistic arithmetic, such as the division theorem \[\forall m, n. m > 0 \supset \exists q, r. mq + r = n \wedge m > r \] A formal proof (in the usual sense) of this theorem is a sequence (or tree) of formulas, where the last (root) formula is the theorem and each formula in the sequence is either an axiom (a leaf) or the result of applying an inference rule to some earlier (higher) formulas. When the division theorem is proved in intuitionistic type theory, we do not only build a formal proof in the usual sense but also a construction (or proof-object) "$\divi$" which witnesses the truth of the theorem. We write \[\divi : \forall m, n {:} \N.\, m > 0 \supset \exists q, r {:} \N.\, mq + r = n \wedge m > r \] to express that $\divi$ is a proof-object for the division theorem, that is, an element of the type representing the division theorem. When propositions are represented as types, the $\forall$-quantifier is identified with the dependent function space former (or general cartesian product) $\Pi$, the $\exists$-quantifier with the dependent pairs type former (or general disjoint sum) $\Sigma$, conjunction $\wedge$ with cartesian product $ \times $, the identity relation = with the type former $\I$ of proof-objects of identities, and the greater than relation $>$ with the type former $\GT$ of proof-objects of greater-than statements. Using "type-notation" we thus write \[ \divi : \Pi m, n {:} \N.\, \GT(m,0)\rightarrow \Sigma q, r {:} \N.\, \I(\N,mq + r,n) \times \GT(m,r) \] to express that the proof object "$\divi$" is a function which maps two numbers $m$ and $n$ and a proof-object $p$ witnessing that $m > 0$ to a quadruple $(q,(r,(s,t)))$, where $q$ is the quotient and $r$ is the remainder obtained when dividing $n$ by $m$. The third component $s$ is a proof-object witnessing the fact that $mq + r = n$ and the fourth component $t$ is a proof object witnessing $m > r $. Crucially, $\divi$ is not only a function in the classical sense; it is also a function in the intuitionistic sense, that is, a program which computes the output $(q,(r,(s,t)))$ when given $m$, $n$, $p$ as inputs. This program is a term in a lambda calculus with special constants, that is, a program in a functional programming language. #### 2.1.3 An extension of first-order predicate logic Intuitionistic type theory can be considered as an extension of first-order logic, much as higher order logic is an extension of first order logic. In higher order logic we find some individual domains which can be interpreted as any sets we like. If there are relational constants in the signature these can be interpreted as any relations between the sets interpreting the individual domains. On top of that we can quantify over relations, and over relations of relations, etc. We can think of higher order logic as first-order logic equipped with a way of introducing new domains of quantification: if $S\_1, \ldots, S\_n$ are domains of quantification then $(S\_1,\ldots,S\_n)$ is a new domain of quantification consisting of all the n-ary relations between the domains $S\_1,\ldots,S\_n$. Higher order logic has a straightforward set-theoretic interpretation where $(S\_1,\ldots,S\_n)$ is interpreted as the power set $P(A\_1 \times \cdots \times A\_n)$ where $A\_i$ is the interpretation of $S\_i$, for $i=1,\ldots,n$. This is the kind of higher order logic or simple theory of types that Ramsey, Church and others introduced. Intuitionistic type theory can be viewed in a similar way, only here the possibilities for introducing domains of quantification are richer, one can use $\Sigma, \Pi, +, \I$ to construct new ones from old. (Section 3.1; Martin-Lof 1998 [1972]). Intuitionistic type theory has a straightforward set-theoretic interpretation as well, where $\Sigma$, $\Pi$ etc are interpreted as the corresponding set-theoretic constructions; see below. We can add to intuitionistic type theory unspecified individual domains just as in HOL. These are interpreted as sets as for HOL. Now we exhibit a difference from HOL: in intuitionistic type theory we can introduce unspecified family symbols. We can introduce $T$ as a family of types over the individual domain $S$: \[T(x)\; {\rm type} \;(x{:}S).\] If $S$ is interpreted as $A$, $T$ can be interpreted as any family of sets indexed by $A$. As a non-mathematical example, we can render the binary relation loves between members of an individual domain of people as follows. Introduce the binary family Loves over the domain People \[{\rm Loves}(x,y)\; {\rm type}\; (x{:}{\rm People}, y{:}{\rm People}).\] The interpretation can be any family of sets $B\_{x,y}$ ($x{:}A$, $y{:}A$). How does this cover the standard notion of relation? Suppose we have a binary relation $R$ on $A$ in the familiar set-theoretic sense. We can make a binary family corresponding to this as follows \[ B\_{x,y} = \begin{cases} \{0\} &\text{if } R(x,y) \text{ holds} \\ \varnothing &\text{if } R(x,y) \text{ is false.} \end{cases}\] Now clearly $B\_{x,y}$ is nonempty if and only if $R(x,y)$ holds. (We could have chosen any other element from our set theoretic universe than 0 to indicate truth.) Thus from any relation we can construct a family whose truth of $x,y$ is equivalent to $B\_{x,y}$ being non-empty. Note that this interpretation does not care what the proof for $R(x,y)$ is, just that it holds. Recall that intuitionistic type theory interprets propositions as types, so $p{:} {\rm Loves}({\rm John}, {\rm Mary})$ means that ${\rm Loves}({\rm John}, {\rm Mary})$ is true. The interpretation of relations as families allows for keeping track of proofs or evidence that $R(x,y)$ holds, but we may also chose to ignore it. In Montague semantics, higher order logic is used to give semantics of natural language (and examples as above). Ranta (1994) introduced the idea to instead employ intuitionistic type theory to better capture sentence structure with the help of dependent types. In contrast, how would the mathematical relation $>$ between natural numbers be handled in intuitionistic type theory? First of all we need a type of numbers $\N$. We could in principle introduce an unspecified individual domain $\N$, and then add axioms just as we do in first-order logic when we set up the axiom system for Peano arithmetic. However this would not give us the desirable computational interpretation. So as explained below we lay down introduction rules for constructing new natural numbers in $\N$ and elimination and computation rules for defining functions on $\N$ (by recursion). The standard order relation $>$ should satisfy \[\mbox{$x > y$ iff there exists $z{:} \N$ such that $y+z+1 = x$}. \] The right hand is rendered as $\Sigma z{:}\N.\, \I(\N,y+z+1,x)$ in intuitionistic type theory, and we take this as definition of relation $>$. ($+$ is defined by recursive equations, $\I$ is the identity type construction). Now all the properties of $>$ are determined by the mentioned introduction and elimination and computation rules for $\N$. #### 2.1.4 A logic with several forms of judgment The type system of intuitionistic type theory is very expressive. As a consequence the well-formedness of a type is no longer a simple matter of parsing, it is something which needs to be proved. Well-formedness of a type is one form of judgment of intuitionistic type theory. Well-typedness of a term with respect to a type is another. Furthermore, there are equality judgments for types and terms. This is yet another way in which intuitionistic type theory differs from ordinary first order logic with its focus on the sole judgment expressing the truth of a proposition. #### 2.1.5 Semantics While a standard presentation of first-order logic would follow Tarski in defining the notion of model, intuitionistic type theory follows the tradition of Brouwerian meaning theory as further developed by Heyting and Kolmogorov, the so called BHK-interpretation of logic. The key point is that the proof of an implication $A \supset B $ is a method that transforms a proof of $A$ to a proof of $B$. In intuitionistic type theory this method is formally represented by the program $f {:} A \supset B$ or $f {:} A \rightarrow B$: the type of proofs of an implication $A \supset B$ is the type of functions which maps proofs of $A$ to proofs of $B$. Moreover, whereas Tarski semantics is usually presented meta-mathematically, and assumes set theory, Martin-Lof's meaning theory of intuitionistic type theory should be understood directly and "pre-mathematically", that is, without assuming a meta-language such as set theory. #### 2.1.6 A functional programming language Readers with a background in the lambda calculus and functional programming can get an alternative first approximation of intuitionistic type theory by thinking about it as a typed functional programming language in the style of Haskell or one of the dialects of ML. However, it differs from these in two crucial aspects: (i) it has dependent types (see below) and (ii) all typable programs terminate. (Note that intuitionistic type theory has influenced recent extensions of Haskell with generalized algebraic datatypes which sometimes can play a similar role as inductively defined dependent types.) ### 2.2 The Curry-Howard Correspondence As already mentioned, the principle that > > a proposition is the type of its proofs. > > > is fundamental to intuitionistic type theory. This principle is also known as the Curry-Howard correspondence or even Curry-Howard isomorphism. Curry discovered a correspondence between the implicational fragment of intuitionistic logic and the simply typed lambda-calculus. Howard extended this correspondence to first-order predicate logic. In intuitionistic type theory this correspondence becomes an identification of proposition and types, which has been extended to include quantification over higher types and more. ### 2.3 Sets of Proof-Objects So what are these proof-objects like? They should not be thought of as logical derivations, but rather as some (structured) symbolic evidence that something is true. Another term for such evidence is "truth-maker". It is instructive, as a somewhat crude first approximation, to replace types by ordinary sets in this correspondence. Define a set $\E\_{m,n}$, depending on $m, n \in {{\mathbb N}}$, by: \[\E\_{m,n} = \left\{\begin{array}{ll} \{0\} & \mbox{if $m = n$}\\ \varnothing & \mbox{if $m \ne n$.} \end{array} \right.\] Then $\E\_{m,n}$ is nonempty exactly when $m=n$. The set $\E\_{m,n}$ corresponds to the proposition $m=n$, and the number $0$ is a proof-object (truth-maker) inhabiting the sets $\E\_{m,m}$. Consider the proposition that $m$ is an even number expressed as the formula $\exists n \in {{\mathbb N}}. m= 2n$. We can build a set of proof-objects corresponding to this formula by using the general set-theoretic sum operation. Suppose that $A\_n$ ($n\in {{\mathbb N}}$) is a family of sets. Then its disjoint sum is given by the set of pairs \[ (\Sigma n \in {{\mathbb N}})A\_n = \{ (n,a) : n \in {{\mathbb N}}, a \in A\_n\}.\] If we apply this construction to the family $A\_n = \E\_{m,2n}$ we see that $(\Sigma n \in {{\mathbb N}})\E\_{m,2n}$ is nonempty exactly when there is an $n\in {{\mathbb N}}$ with $m=2n$. Using the general set-theoretic product operation $(\Pi n \in {{\mathbb N}})A\_n$ we can similarly obtain a set corresponding to a universally quantified proposition. ### 2.4 Dependent Types In intuitionistic type theory there are primitive type formers $\Sigma$ and $\Pi$ for general sums and products, and $\I$ for identity types, analogous to the set-theoretic constructions described above. The identity type $\I(\N,m,n)$ corresponding to the set $\E\_{m,n}$ is an example of a dependent type since it depends on $m$ and $n$. It is also called an indexed family of types since it is a family of types indexed by $m$ and $n$. Similarly, we can form the general disjoint sum $\Sigma x {:} A.\, B$ and the general cartesian product $\Pi x {:} A.\, B$ of such a family of types $B$ indexed by $x {:} A$, corresponding to the set theoretic sum and product operations above. Dependent types can also be defined by primitive recursion. An example is the type of $n$-tuples $A^n$ of elements of type $A$ and indexed by $n {:} N$ defined by the equations \[\begin{align\} A^0 &= 1\\ A^{n+1} &= A \times A^n \end{align\}\] where $1$ is a one element type and $\times$ denotes the cartesian product of two types. We note that dependent types introduce computation in types: the defining rules above are computation rules. For example, the result of computing $A^3$ is $A \times (A \times (A \times 1))$. ### 2.5 Propositions as Types in Intuitionistic Type Theory With propositions as types, predicates become dependent types. For example, the predicate $\mathrm{Prime}(x)$ becomes the type of proofs that $x$ is prime. This type depends on $x$. Similarly, $x < y$ is the type of proofs that $x$ is less than $y$. According to the Curry-Howard interpretation of propositions as types, the logical constants are interpreted as type formers: \[\begin{align\} \bot &= \varnothing\\ \top &= 1\\ A \vee B &= A + B\\ A \wedge B &= A \times B\\ A \supset B &= A \rightarrow B\\ \exists x {:} A.\, B &= \Sigma x {:} A.\, B\\ \forall x {:} A.\, B &= \Pi x {:} A.\, B \end{align\}\] where $\Sigma x {:} A.\, B$ is the disjoint sum of the $A$-indexed family of types $B$ and $\Pi x {:} A.\, B$ is its cartesian product. The canonical elements of $\Sigma x {:} A.\, B$ are pairs $(a,b)$ such that $a {:} A$ and $b {:} B[x:=a]$ (the type obtained by substituting all free occurrences of $x$ in $B$ by $a$). The elements of $\Pi x {:} A.\, B$ are (computable) functions $f$ such that $f\,a {:} B[x:=a]$, whenever $a {:} A$. For example, consider the proposition \[\begin{equation} \forall m {:} \N.\, \exists n {:} \N.\, m \lt n \wedge \mathrm{Prime}(n) \tag{1}\label{prop1} \end{equation}\] expressing that there are arbitrarily large primes. Under the Curry-Howard interpretation this becomes the type $\Pi m {:} \N.\, \Sigma n {:} \N.\, m \lt n \times \mathrm{Prime}(n)$ of functions which map a number $m$ to a triple $(n,(p,q))$, where $n$ is a number, $p$ is a proof that $m \lt n$ and $q$ is a proof that $n$ is prime. This is the proofs as programs principle: a constructive proof that there are arbitrarily large primes becomes a program which given any number produces a larger prime together with proofs that it indeed is larger and indeed is prime. Note that the proof which derives a contradiction from the assumption that there is a largest prime is not constructive, since it does not explicitly give a way to compute an even larger prime. To turn this proof into a constructive one we have to show explicitly how to construct the larger prime. (Since proposition (\ref{prop1}) above is a $\Pi^0\_2$-formula we can for example use Friedman's A-translation to turn such a proof in classical arithmetic into a proof in intuitionistic arithmetic and thus into a proof in intuitionistic type theory.) ## 3. Basic Intuitionistic Type Theory We now present a core version of intuitionistic type theory, closely related to the first version of the theory presented by Martin-Lof in 1972 (Martin-Lof 1998 [1972]). In addition to the type formers needed for the Curry-Howard interpretation of typed intuitionistic predicate logic listed above, we have two types: the type $\N$ of natural numbers and the type $\U$ of small types. The resulting theory can be shown to contain intuitionistic number theory $\HA$ (Heyting arithmetic), Godel's System $\T$ of primitive recursive functions of higher type, and the theory $\HA^\omega$ of Heyting arithmetic of higher type. This core intuitionistic type theory is not only the original one, but perhaps the minimal version which exhibits the essential features of the theory. Later extensions with primitive identity types, well-founded tree types, universe hierarchies, and general notions of inductive and inductive-recursive definitions have increased the proof-theoretic strength of the theory and also made it more convenient for programming and formalization of mathematics. For example, with the addition of well-founded trees we can interpret the Constructive Zermelo-Fraenkel Set Theory $\CZF$ of Aczel (1978 [1977]). However, we will wait until the next section to describe those extensions. ### 3.1 Judgments In Martin-Lof (1996) a general philosophy of logic is presented where the traditional notion of judgment is expanded and given a central position. A judgment is no longer just an affirmation or denial of a proposition, but a general act of knowledge. When reasoning mathematically we make judgments about mathematical objects. One form of judgment is to state that some mathematical statement is true. Another form of judgment is to state that something is a mathematical object, for example a set. The logical rules give methods for producing correct judgments from earlier judgments. The judgments obtained by such rules can be presented in tree form \[ \begin{prooftree} \AxiomC{$J\_1$} \AxiomC{$J\_2$} \RightLabel{$r\_1$} \BinaryInfC{$J\_3$} \AxiomC{$J\_4$} \RightLabel{$r\_5$} \UnaryInfC{$J\_5$} \AxiomC{$J\_6$} \RightLabel{$r\_3$} \BinaryInfC{$J\_7$} \RightLabel{$r\_4$} \BinaryInfC{$J\_8$} \end{prooftree}\] or in sequential form * (1) $J\_1 \quad\text{ axiom} $ * (2) $J\_2 \quad\text{ axiom} $ * (3) $J\_3 \quad\text{ by rule \(r\_1$ from (1) and (2)} \) * (4) $J\_4 \quad\text{ axiom} $ * (5) $J\_5 \quad\text{ by rule \(r\_2$ from (4)} \) * (6) $J\_6 \quad\text{ axiom} $ * (7) $J\_7 \quad\text{ by rule \(r\_3$ from(5) and (6)} \) * (8) $J\_8 \quad\text{ by rule \(r\_4$ from (3) and (7)} \) The latter form is common in mathematical arguments. Such a sequence or tree formed by logical rules from axioms is a derivation or demonstration of a judgment. First-order reasoning may be presented using a single kind of judgment: > > the proposition $B$ is true under the hypothesis that the > propositions $A\_1, \ldots, A\_n$ are all true. > > > We write this hypothetical judgment as a so-called Gentzen sequent \[A\_1, \ldots, A\_n {\vdash}B.\] Note that this is a single judgment that should not be confused with the derivation of the judgment ${\vdash}B$ from the judgments ${\vdash}A\_1, \ldots, {\vdash}A\_n$. When $n=0$, then the categorical judgment $ {\vdash}B$ states that $B$ is true without any assumptions. With sequent notation the familiar rule for conjunctive introduction becomes \[\begin{prooftree} \AxiomC{$A\_1, \ldots,A\_n {\vdash}B$} \AxiomC{$A\_1, \ldots, A\_n {\vdash}C$} \RightLabel{$(\land I)$.} \BinaryInfC{$A\_1, \ldots, A\_n {\vdash}B \land C$} \end{prooftree}\] ### 3.2 Judgment Forms Martin-Lof type theory has four basic forms of judgments and is a considerably more complicated system than first-order logic. One reason is that more information is carried around in the derivations due to the identification of propositions and types. Another reason is that the syntax is more involved. For instance, the well-formed formulas (types) have to be generated simultaneously with the provably true formulas (inhabited types). The four forms of categorical judgment are * $\vdash A \; {\rm type}$, meaning that $A$ is a well-formed type, * $\vdash a {:} A$, meaning that $a$ has type $A$, * $\vdash A = A'$, meaning that $A$ and $A'$ are equal types, * $\vdash a = a' {:} A$, meaning that $a$ and $a'$ are equal elements of type $A$. In general, a judgment is hypothetical, that is, it is made in a context $\Gamma$, that is, a list $x\_1 {:} A\_1, \ldots, x\_n {:} A\_n$ of variables which may occur free in the judgment together with their respective types. Note that the types in a context can depend on variables of earlier types. For example, $A\_n$ can depend on $x\_1 {:} A\_1, \ldots, x\_{n-1} {:} A\_{n-1}$. The four forms of hypothetical judgments are * $\Gamma \vdash A \; {\rm type}$, meaning that $A$ is a well-formed type in the context $\Gamma$, * $\Gamma \vdash a {:} A$, meaning that $a$ has type $A$ in context $\Gamma$, * $\Gamma \vdash A = A'$, meaning that $A$ and $A'$ are equal types in the context $\Gamma$, * $\Gamma \vdash a = a' {:} A$, meaning that $a$ and $a'$ are equal elements of type $A$ in the context $\Gamma$. Under the proposition as types interpretation \[\tag{2}\label{analytic} \vdash a {:} A \] can be understood as the judgment that $a$ is a proof-object for the proposition $A$. When suppressing this object we get a judgment corresponding to the one in ordinary first-order logic (see above): \[\tag{3}\label{synthetic} \vdash A\; {\rm true}. \] Remark 3.1. Martin-Lof (1994) argues that Kant's analytic judgment a priori and synthetic judgment a priori can be exemplified, in the realm of logic, by ([analytic]) and ([synthetic]) respectively. In the analytic judgment ([analytic]) everything that is needed to make the judgment evident is explicit. For its synthetic version ([synthetic]) a possibly complicated proof construction $a$ needs to be provided to make it evident. This understanding of analyticity and syntheticity has the surprising consequence that "the logical laws in their usual formulation are all synthetic." Martin-Lof (1994: 95). His analysis further gives: > " [...] the logic of analytic judgments, > that is, the logic for deriving judgments of the two analytic forms, > is complete and decidable, whereas the logic of synthetic judgments is > incomplete and undecidable, as was shown by Godel." > Martin-Lof (1994: 97). > > > The decidability of the two analytic judgments ($\vdash a{:}A$ and $\vdash a=b{:}A$) hinges on the metamathematical properties of type theory: strong normalization and decidable type checking. Sometimes also the following forms are explicitly considered to be judgments of the theory: * $\Gamma \; {\rm context}$, meaning that $\Gamma$ is a well-formed context. * $\Gamma = \Gamma'$, meaning that $\Gamma$ and $\Gamma'$ are equal contexts. Below we shall abbreviate the judgment $\Gamma \vdash A \; {\rm type}$ as $\Gamma \vdash A$ and $\Gamma \; {\rm context}$ as $\Gamma \vdash.$ ### 3.3 Inference Rules When stating the rules we will use the letter $\Gamma$ as a meta-variable ranging over contexts, $A,B,\ldots$ as meta-variables ranging over types, and $a,b,c,d,e,f,\ldots$ as meta-variables ranging over terms. The first group of inference rules are general rules including rules of assumption, substitution, and context formation. There are also rules which express that equalities are equivalence relations. There are numerous such rules, and we only show the particularly important rule of type equality which is crucial for computation in types: \[\frac{\Gamma \vdash a {:} A\hspace{2em}\Gamma \vdash A = B} {\Gamma \vdash a {:} B}\] The remaining rules are specific to the type formers. These are classified as formation, introduction, elimination, and equality rules. ### 3.4 Intuitionistic Predicate Logic We only give the rules for $\Pi$. There are analogous rules for the other type formers corresponding to the logical constants of typed predicate logic. In the following $B[x := a]$ means the term obtained by substituting the term $a$ for each free occurrence of the variable $x$ in $B$ (avoiding variable capture). $\Pi$-formation. \[\frac{\Gamma \vdash A\hspace{2em} \Gamma, x {:} A \vdash B} {\Gamma \vdash \Pi x {:} A. B}\] $\Pi$-introduction. \[\frac{\Gamma, x {:} A \vdash b {:} B} {\Gamma \vdash \lambda x. b {:} \Pi x {:} A. B}\] $\Pi$-elimination. \[\frac {\Gamma \vdash f {:} \Pi x {:} A.B\hspace{2em}\Gamma \vdash a {:} A} {\Gamma \vdash f\,a {:} B[x := a]}\] $\Pi$-equality. \[\frac {\Gamma, x {:} A \vdash b {:} B\hspace{2em}\Gamma \vdash a {:} A} {\Gamma \vdash (\lambda x.b)\,a = b[x := a] {:} B[x := a]}\] This is the rule of $\beta$-conversion. We may also add the rule of $\eta$-conversion: \[\frac {\Gamma \vdash f {:} \Pi x {:} A. B} {\Gamma \vdash \lambda x. f\,x = f {:} \Pi x {:} A. B}.\] Furthermore, there are congruence rules expressing that operations introduced by the formation, introduction, and elimination rules preserve equality. For example, the congruence rule for $\Pi$ is \[\frac{\Gamma \vdash A = A'\hspace{2em} \Gamma, x {:} A \vdash B=B'} {\Gamma \vdash \Pi x {:} A. B = \Pi x {:} A'. B'}.\] ### 3.5 Natural Numbers As in Peano arithmetic the natural numbers are generated by 0 and the successor operation $\s$. The elimination rule states that these are the only possible ways to generate a natural number. We write $f(c) = \R(c,d,xy.e)$ for the function which is defined by primitive recursion on the natural number $c$ with base case $d$ and step function $xy.e$ (or alternatively $\lambda xy.e$) which maps the value $y$ for the previous number $x {:} \N$ to the value for $\s(x)$. Note that $\R$ is a new variable-binding operator: the variables $x$ and $y$ become bound in $e$. $\N$-formation. \[\Gamma \vdash \N\] $\N$-introduction. \[\Gamma \vdash 0 {:} \N \hspace{2em} \frac{\Gamma \vdash a {:} \N} {\Gamma \vdash s(a) {:} \N}\] $\N$-elimination. \[\frac{ \Gamma, x {:} \N \vdash C \hspace{1em} \Gamma \vdash c {:} \N \hspace{1em} \Gamma \vdash d {:} C[x := 0] \hspace{1em} \Gamma, y {:} \N, z {:} C[x := y] \vdash e {:} C[x := s(y)] } { \Gamma \vdash \R(c,d,yz.e) {:} C[x := c] }\] $\N$-equality (under appropriate premises). \[\begin{align\} \R(0,d,yz.e) &= d {:} C[x := 0]\\ \R(s(a),d,yz.e) &= e[y := a, z := \R(a,d,yz.e)] {:} C[x := s(a)] \end{align\}\] The rule of $\N$-elimination simultaneously expresses the type of a function defined by primitive recursion and, under the Curry-Howard interpretation, the rule of mathematical induction: we prove the property $C$ of a natural number $x$ by induction on $x$. Godel's System $\T$ is essentially intuitionistic type theory with only the type formers $\N$ and $A \rightarrow B$ (the type of functions from $A$ to $B$, which is the special case of $(\Pi x {:} A)B$ where $B$ does not depend on $x {:} A$). Since there are no dependent types in System $\T$ the rules can be simplified. ### 3.6 The Universe of Small Types Martin-Lof's first version of type theory (Martin-Lof 1971a) had an axiom stating that there is a type of all types. This was proved inconsistent by Girard who found that the Burali-Forti paradox could be encoded in this theory. To overcome this pathological impredicativity, but still retain some of its expressivity, Martin-Lof introduced in 1972 a universe $\U$ of small types closed under all type formers of the theory, except that it does not contain itself (Martin-Lof 1998 [1972]). The rules are: $\U$-formation. \[\Gamma \vdash \U\] $\U$-introduction. \[\Gamma \vdash \varnothing {:} \U \hspace{3em} \Gamma \vdash 1 {:} \U\] \[\frac{\Gamma \vdash A {:} \U\hspace{2em} \Gamma \vdash B {:} \U} {\Gamma \vdash A + B {:} \U} \hspace{3em} \frac{\Gamma \vdash A {:} \U\hspace{2em} \Gamma \vdash B {:} \U} {\Gamma \vdash A \times B {:} \U}\] \[\frac{\Gamma \vdash A {:} \U\hspace{2em} \Gamma \vdash B {:} \U} {\Gamma \vdash A \rightarrow B {:} \U}\] \[\frac{\Gamma \vdash A {:} U\hspace{2em} \Gamma, x {:} A \vdash B {:} \U} {\Gamma \vdash \Sigma x {:} A.\, B {:} \U} \hspace{3em} \frac{\Gamma \vdash A {:} \U\hspace{2em} \Gamma, x {:} A \vdash B {:} \U} {\Gamma \vdash \Pi x {:} A.\, B {:} \U}\] \[\Gamma \vdash \N {:} \U\] $\U$-elimination. \[\frac{\Gamma \vdash A {:} \U} {\Gamma \vdash A}\] Since $\U$ is a type, we can use $\N$-elimination to define small types by primitive recursion. For example, if $A : \U$, we can define the type of $n$-tuples of elements in $A$ as follows: \[A^n = \R(n,1,xy.A \times y) {:} \U\] This type-theoretic universe $\U$ is analogous to a Grothendieck universe in set theory which is a set of sets closed under all the ways sets can be constructed in Zermelo-Fraenkel set theory. The existence of a Grothendieck universe cannot be proved from the usual axioms of Zermelo-Fraenkel set theory but needs a new axiom. In Martin-Lof (1975) the universe is extended to a countable hierarchy of universes \[\U\_0 : \U\_1 : \U\_2 : \cdots .\] In this way each type has a type, not only each small type. ### 3.7 Propositional Identity Above, we introduced the equality judgment \[\tag{4}\label{defeq} \Gamma \vdash a = a' {:} A.\] This is usually called a "definitional equality" because it can be decided by normalizing the terms $a$ and $a'$ and checking whether the normal forms are identical. However, this equality is a judgment and not a proposition (type) and we thus cannot prove such judgmental equalities by induction. For this reason we need to introduce propositional identity types. For example, the identity type for natural numbers $\I(\N,m,n)$ can be defined by $\U$-valued primitive recursion. We can then express and prove the Peano axioms. Moreover, extensional equality of ufnctions can be defined by \[\I(\N\rightarrow \N,f,f') = \Pi x {:} \N. \I(\N,f\,x,f'\,x).\] ### 3.8 The Axiom of Choice is a Theorem The following form of the axiom of choice is an immediate consequence of the BHK-interpretation of the intuitionistic quantifiers, and is easily proved in intuitionistic type theory: \[(\Pi x {:} A. \Sigma y {:} B. C) \rightarrow \Sigma f {:} (\Pi x {:} A. B). C[y := f\,x]\] The reason is that $\Pi x {:} A. \Sigma y {:} B. C$ is the type of functions which map elements $x {:} A$ to pairs $(y,z)$ with $y {:} B$ and $z {:} C$. The choice function $f$ is obtained by returning the first component $y {:} B$ of this pair. It is perhaps surprising that intuitionistic type theory directly validates an axiom of choice, since this axiom is often considered problematic from a constructive point of view. A possible explanation for this state of affairs is that the above is an axiom of choice for types, and that types are not in general appropriate constructive approximations of sets in the classical sense. For example, we can represent a real number as a Cauchy sequence in intuitionistic type theory, but the set of real numbers is not the type of Cauchy sequences, but the type of Cauchy sequences up to equiconvergence. More generally, a set in Bishop's constructive mathematics is represented by a type (commonly called "preset") together with an equivalence relation. If $A$ and $B$ are equipped with equivalence relations, there is of course no guarantee that the choice function, $f$ above, is extensional in the sense that it maps equivalent element to equivalent elements. This is the failure of the extensional axiom of choice, see Martin-Lof (2009) for an analysis. ## 4. Extensions ### 4.1 The Logical Framework The above completes the description of a core version of intuitionistic type theory close to that of (Martin-Lof 1998 [1972]). In 1986 Martin-Lof proposed a reformulation of intuitionistic type theory; see Nordstrom, Peterson and Smith (1990) for an exposition. The purpose was to give a more compact formulation, where $\lambda$ and $\Pi$ are the only variable binding operations. It is nowadays considered the main version of the theory. It is also the basis for the Agda proof assistant. The 1986 theory has two parts: * the theory of types (the logical framework); * the theory of sets (small types). Remark 4.1. Note that the word "set" in the logical framework does not coincide with the way it is used in Bishop's constructive mathematics. To avoid this confusion, types together with equivalence relations are usually called "setoids" or "extensional sets" in intuitionistic type theory. The logical framework has only two type formers: $\Pi x {:} A. B$ (usually written $(x {:} A)B$ or $(x {:} A) \rightarrow B$ in the logical framework formulation) and $\U$ (usually called $\Set$). The rules for $\Pi x{:} A. B$ ($(x {:} A) \rightarrow B$) are the same as given above (including $\eta$-conversion). The rules for $\U$ ($\Set$) are also the same, except that the logical framework only stipulates closure under $\Pi$-type formation. The other small type formers ("set formers") are introduced in the theory of sets. In the logical framework formulation each formation, introduction, and elimination rule can be expressed as the typing of a new constant. For example, the rules for natural numbers become \[\begin{align\} \N &: \Set,\\ 0 &: \N,\\ \s &: \N \rightarrow \N,\\ \R &: (C {:} \N \rightarrow \Set) \rightarrow C\,0 \rightarrow (( x {:} \N) \rightarrow C\,x \rightarrow C\,(\s\,x)) \rightarrow (c {:} \N) \rightarrow C\,c. \end{align\}\] where we have omitted the common context $\Gamma$, since the types of these constants are closed. Note that the recursion operator $R$ has a first argument $C {:} \N \rightarrow \Set$ unlike in the original formulation. Moreover, the equality rules can be expressed as equations \[\begin{align\} \R\, C\, d\, e\, 0 &= d {:} C\,0\\ \R\, C\, d\, e\, (\s\, a) &= e\, a\, (\R\, C\, d\, e\, a) {:} C\,(\s\,a) \end{align\}\] under suitable assumptions. In the sequel we will present several extensions of type theory. To keep the presentation uniform we will however not use the logical framework presentation of type theory, but will use the same notation as in section 2. ### 4.2 A General Identity Type Former As we mentioned above, identity on natural numbers can be defined by primitive recursion. Identity relations on other types can also be defined in the basic version of intuitionistic type theory presented in section 2. However, Martin-Lof (1975) extended intuitionistic type theory with a uniform primitive identity type former $\I$ for all types. The rules for $\I$ express that the identity relation is inductively generated by the proof of reflexivity, a canonicial constant called $\r$. (Note that $\r$ was coded by the number 0 in the introductory presentation of proof-objects in 2.3. The elimination rule for the identity type is a generalization of identity elimination in predicate logic and introduces an elimination constant $\J$. We here show the formulation due to Paulin-Mohring (1993) rather than the original formulation of Martin-Lof (1975). The inference rules are the following. $\I$-formation. \[\frac{\Gamma \vdash A \hspace{1em} \Gamma \vdash a {:} A \hspace{1em} \Gamma \vdash a' {:} A} {\Gamma \vdash \I(A,a,a')}\] $\I$-introduction. \[\frac{\Gamma \vdash A \hspace{1em} \Gamma \vdash a {:} A} {\Gamma \vdash \r {:} \I(A,a,a)}\] $\I$-elimination. \[\frac{ \Gamma, x {:} A, y {:} \I(A,a,x) \vdash C \hspace{1em} \Gamma \vdash b {:} A \hspace{1em} \Gamma \vdash c {:} \I(A,a,b) \hspace{1em} \Gamma \vdash d {:} C[x := a, y := r]} { \Gamma \vdash \J(c,d) {:} C[x := b, y:= c]}\] $\I$-equality (under appropriate assumptions). \[\begin{align\} \J(r,d) &= d \end{align\}\] Note that if $C$ only depends on $x : A$ and not on the proof $y : \I(A,a,x)$ (and we also suppress proof objects) in the rule of $\I$-elimination we recover the rule of identity elimination in predicate logic. By constructing a model of type theory where types are interpreted as groupoids (categories where all arrows are isomorphisms) Hofmann and Streicher (1998) showed that it cannot be proved in intuitionistic type theory that all proofs of $I(A,a,b)$ are identical. This may seem as an incompleteness of the theory and Streicher suggested a new axiom $\K$ from which it follows that all proofs of $\I(A,a,b)$ are identical to $\r$. The $\I$-type is often called the intensional identity type, since it does not satisfy the principle of function extensionality. Intuitionistic type theory with the intensional identity type is also often called intensional intuitionistic type theory to distinguish it from extensional intuitionistic type theory which will be presented in section 7.1. ### 4.3 Well-Founded Trees A type of well-founded trees of the form $\W x {:} A. B$ was introduced in Martin-Lof 1982 (and in a more restricted form by Scott 1970). Elements of $\W x {:} A. B$ are trees of varying and arbitrary branching: varying, because the branching type $B$ is indexed by $x {:} A$ and arbitrary because $B$ can be arbitrary. The type is given by a generalized inductive definition since the well-founded trees may be infinitely branching. We can think of $\W x{:}A. B$ as the free term algebra, where each $a {:} A$ represents a term constructor $\sup\,a$ with (possibly infinite) arity $B[x := a]$. $\W$-formation. \[\frac{\Gamma \vdash A\hspace{2em} \Gamma, x {:} A \vdash B} {\Gamma \vdash \W x {:} A. B}\] $\W$-introduction. \[\frac{\Gamma \vdash a {:} A \hspace{2em} \Gamma, y {:} B[x:=a] \vdash b : Wx{:}A. B} {\Gamma \vdash \sup(a, y.b) : \W x {:} A. B}\] We omit the rules of $\W$-elimination and $\W$-equality. Adding well-founded trees to intuitionistic type theory increases its proof-theoretic strength significantly (Setzer (1998)). ### 4.4 Iterative Sets and CZF An important application of well-founded trees is Aczel's (1978) construction of a type-theoretic model of Constructive Zermelo Fraenkel Set Theory. To this end he defines the type of iterative sets as \[\V = \W x {:} \U. x.\] Let $A {:} \U$ be a small type, and $x {:} A\vdash M$ be an indexed family of iterative sets. Then $\sup(A,x.M)$, or with a more suggestive notation $\{ M\mid x {:} A\}$, is an iterative set. To paraphrase: an iterative set is a family of iterative sets indexed by a small type. Note that an iterative set is a data-structure in the sense of functional programming: a possibly infinitely branching well-founded tree. Different trees may represent the same set. We therefore need to define a notion of extensional equality between iterative sets which disregards repetition and order of elements. This definition is formally similar to the definition of bisimulation of processes in process algebra. The type $\V$ up to extensional equality can be viewed as a constructive type-theoretic model of the cumulative hierarchy, see the entry on set theory: constructive and intuitionistic ZF for further information about CZF. ### 4.5 Inductive Definitions The notion of an inductive definition is fundamental in intuitionistic type theory. It is a primitive notion and not, as in set theory, a derived notion where an inductively defined set is defined impredicatively as the smallest set closed under some rules. However, in intuitionistic type theory inductive definitions are considered predicative: they are viewed as being built up from below. The inductive definability of types is inherent in the meaning explanations of intuitionistic type theory which we shall discuss in the next section. In fact, intuitionistic type theory can be described briefly as a theory of inductive, recursive, and inductive-recursive definitions based on a framework of lambda calculus with dependent types. We have already seen the type of natural numbers and the type of well-founded trees as examples of types given by inductive definitions; the natural numbers is an example of an ordinary finitary inductive definition and the well-founded trees of a generalized possibly infinitary inductive definition. The introduction rules describe how elements of these types are inductively generated and the elimination and equality rules describe how functions from these types can be defined by structural recursion on the way these elements are generated. According to the propositions as types principle, the elimination rules are simultaneously rules for proof by structural induction on the way the elements are generated. The type formers $0, 1, +, \times, \rightarrow, \Sigma,$ and $\Pi$ which interpret the logical constants for intuitionistic predicate logic are examples of degenerate inductive definitions. Even the identity type (in intensional intuitionistic type theory) is inductively generated; it is the type of proofs generated by the reflexivity axiom. Its elimination rule expresses proof by pattern matching on the proof of reflexivity. The common structure of the rules of the type formers can be captured by a general schema for inductive definitions (Dybjer 1991). This general schema has many useful instances, for example, the type $\List(A)$ of lists with elements of type $A$ has the following introduction rules: \[\Gamma \vdash \nil {:} \List(A) \hspace{3em} \frac{\Gamma \vdash a {:} A\hspace{2em}\Gamma \vdash as {:} \List(A)} {\Gamma \vdash \cons(a,as) {:} \List(A)}\] Other useful instances are types of binary trees and other trees such as the infinitely branching trees of the Brouwer ordinals of the second and higher number classes. The general schema does not only cover inductively defined types, but also inductively defined families of types, such as the identity relation. The above mentioned type $A^n$ of $n$-tuples of type $A$ was defined above by primitive recursion on $n$. It can also be defined as an inductive family with the following introduction rules \[\Gamma \vdash \nil {:} A^0 \hspace{3em} \frac{\Gamma \vdash a {:} A\hspace{2em}\Gamma \vdash as {:} A^n} {\Gamma \vdash \cons(a,as) {:} A^{\s(n)}}\] The schema for inductive types and families is a type-theoretic generalization of a schema for iterated inductive definitions in predicate logic (formulated in natural deduction) presented by Martin-Lof (1971b). This paper immediately preceded Martin-Lof's first version of intuitionistic type theory. It is both conceptually and technically a forerunner to the development of the theory. It is an essential feature of proof assistants such as Agda and Coq that it enables users to define their own inductive types and families by listing their introduction rules (the types of their constructors). This is much like in typed functional programming languages such as Haskell and the different dialects of ML. However, unlike in these programming languages the schema for inductive definitions in intuitionistic type theory enforces a restriction amounting to well-foundedness of the elements of the defined types. ### 4.6 Inductive-Recursive Definitions We already mentioned that there are two main definition principles in intuitionistic type theory: the inductive definition of types (sets) and the (primitive, structural) definition of functions by recursion on the way the elements of such types are inductively generated. Usually, the inductive definition of a set comes first: the formation and introduction rules make no reference to the elimination rule. However, there are definitions in intuitionistic type theory for which this is not the case and we simultaneously inductively generate a type and a function from that type defined by structural recursion. Such definitions are simultaneously inductive-recursive. The first example of such an inductive-recursive definition is an alternative formulation a la Tarski of the universe of small types. Above we presented the universe formulated a la Russell, where there is no notational distinction between the element $A {:} \U$ and the corresponding type $A$. For a universe a la Tarski there is such a distinction, for example, between the element $\hat{\N} {:} \U$ and the corresponding type $\N$. The element $\hat{\N}$ is called the code for $\N$. The elimination rule for the universe a la Tarski is: \[\frac{\Gamma \vdash a {:} \U} {\Gamma \vdash \T(a)}\] This expresses that there is a function $\T$ which maps a code $a$ to its corresponding type $T(a)$. The equality rules define this correspondence. For example, \[\T(\hat{\N}) = \N.\] We see that $\U$ is inductively generated with one introduction rule for each small type former, and $\T$ is defined by recursion on these small type formers. The simultaneous inductive-recursive nature of this definition becomes apparent in the rules for $\Pi$ for example. The introduction rule is \[\frac{\Gamma \vdash a {:} \U\hspace{2em} \Gamma, x {:} \T(a) \vdash b {:} \U} {\Gamma \vdash \hat{\Pi} x {:} a. b {:} \U}\] and the corresponding equality rule is \[\T(\hat{\Pi}x {:} a. b) = \Pi x {:} \T(a). \T(b)\] Note that the introduction rule for $\U$ refers to $\T$, and hence that $\U$ and $\T$ must be defined simultaneously. There are a number of other universe constructions which are defined inductive-recursively: universe hierarchies, superuniverses (Palmgren 1998; Rathjen, Griffor, and Palmgren 1998), and Mahlo universes (Setzer 2000). These universes are analogues of certain large cardinals in set theory: inaccessible, hyperinaccessible, and Mahlo cardinals. Other examples of inductive-recursive definitions include an informal definition of computability predicates used by Martin-Lof in an early normalization proof of intuitionistic type theory (Martin-Lof 1998 [1972]). There are also many natural examples of "small" inductive-recursive definitions, where the recursively defined (decoding) function returns an element of a type rather than a type. A large class of inductive-recursive definitions, including the above, can be captured by a general schema (Dybjer 2000) which extends the schema for inductive definitions mentioned above. As shown by Setzer, intuitionistic type theory with this class of inductive-recursive definitions is very strong proof-theoretically (Dybjer and Setzer 2003). However, as proposed in recent unpublished work by Setzer, it is possible to increase the strength of the theory even further and define universes such as an autonomous Mahlo universe which are analogues of even larger cardinals. ## 5. Meaning Explanations The consistency of intuitionistic type theory relative to set theory can be proved by model constructions. Perhaps the simplest method is an interpretation whereby each type-theoretic concept is given its corresponding set-theoretic meaning, as outlined in section 2.3. For example the type of functions $A \rightarrow B$ is interpreted as the set of all functions in the set-theoretic sense between the set denoted by $A$ and the set denoted by $B$. To interpret $\U$ we need a set-theoretic universe which is closed under all (set-theoretic analogues of) the type constructors. Such a universe can be proved to exist if we assume the existence of an inaccessible cardinal $\kappa$ and interpret $\U$ by $V\_\kappa$ in the cumulative hierarchy. Alternatives are realizability models, and for intensional type theory, a model of terms in normal forms. The latter can also be used for proving decidability of the judgments of the theory. Mathematical models only prove consistency relative to classical set theory (or whatever other meta-theory we are using). Is it possible to be convinced about the consistency of the theory in a more direct way, so called simple minded consistency (Martin-Lof 1984)? In fact, is there a way to explain what it means for a judgment to be correct in a direct pre-mathematical way? And given that we know what the judgments mean can we then be convinced that the inference rules of the theory are valid? An answer to this problem was proposed by Martin-Lof in 1979 in the paper "Constructive Mathematics and Computer Programming" (Martin-Lof 1982) and elaborated later on in numerous lectures and notes, see for example, Martin-Lof (1984, 1987). These meaning explanations for intuitionistic type theory are also referred to as the direct semantics, intuitive semantics, informal semantics, standard semantics, or the syntactico-semantical approach to meaning theory. This meaning theory follows the Wittgensteinian meaning-as-use tradition. The meaning is based on rules for building objects (introduction rules) of types and computation rules (elimination rules) for computing with these objects. A difference from much of the Wittgensteinian tradition is that also higher order types like $\N \rightarrow \N$ are given meaning using rules. To explain the meaning of a judgment we must first know how the terms in the judgment are computed to canonical form. Then the formation rules explain how correct canonical types are built and the introduction rules explain how correct canonical objects of such canonical types are built. We quote (Martin-Lof 1982): > > A canonical type $A$ is defined by prescribing how a canonical > object of type $A$ is formed as well as how two equal canonical > objects of type $A$ are formed. There is no limitation on this > prescription except that the relation of equality which it defines > between canonical objects of type $A$ must be reflexive, symmetric > and transitive. In other words, a canonical type is equipped with an equivalence relation on the canonical objects. Below we shall give a simplified form of the meaning explanations, where this equivalence relation is extensional identity of objects. In spite of the pre-mathematical nature of this meaning theory, its technical aspects can be captured as a mathematical model construction similar to Kleene's realizability interpretation of intuitionistic logic, see the next section. The realizers here are the terms of type theory rather than the number realizers used by Kleene. ### 5.1 Computation to Canonical Form The meaning of a judgment is explained in terms of the computation of the types and terms in the judgment. These computations stop when a canonical form is reached. By canonical form we mean a term where the outermost form is a constructor (introduction form). These are the canonical forms used in lazy functional programming (for example in the Haskell language). For the purpose of illustration we consider meaning explanations only for three type formers: $\N, \Pi x {:} A.B$, and $\U$. The context free grammar for the terms of this fragment of Intuitionistic Type Theory is as follows: \[ a :: = 0 \mid \s(a) \mid \lambda x.a \mid \N \mid \Pi x{:}a.a \mid \U \mid \R(a,a,xx.a) \mid a\,a . \] The canonical terms are generated by the following grammar: \[v :: = 0 \mid \s(a) \mid \lambda x.a \mid \N \mid \Pi x{:}a.a \mid \U ,\] where $a$ ranges over arbitrary, not necessarily canonical, terms. Note that $\s(a)$ is canonical even if $a$ is not. To explain how terms are computed to canonical form, we introduce the relation $a \Rightarrow v$ between closed terms $a$ and canonical forms (values) $v$ given by the following computation rules: \[ \frac{c \Rightarrow 0\hspace{1em}d \Rightarrow v}{\R(c,d,xy.e)\Rightarrow v} \hspace{2em} \frac{c \Rightarrow \s(a)\hspace{1em}e[x := d,y := \R(a,d,xy.e)]\Rightarrow v}{\R(c,d,xy.e)\Rightarrow v} \] \[ \frac{f\Rightarrow \lambda x.b\hspace{1em}b[x := a]\Rightarrow v}{f\,a \Rightarrow v} \] in addition to the rule \[v \Rightarrow v\] stating that a canonical term has itself as value. ### 5.2 The Meaning of Categorical Judgments A categorical judgment is a judgment where the context is empty and there are no free variables. The meaning of the categorical judgment $\vdash A$ is that $A$ has a canonical type as value. In our fragment this means that either of the following holds: * $A \Rightarrow \N$, * $A \Rightarrow \U$, * $A \Rightarrow \Pi x {:} B. C$ and furthermore that $\vdash B$ and $x {:} B \vdash C$. The meaning of the categorical judgment $\vdash a {:} A$ is that $a$ has a canonical term of the canonical type of $A$ as value. In our fragment this means that either of the following holds: * $A \Rightarrow \N$ and either $a \Rightarrow 0$ or $a \Rightarrow \s(b)$ and $\vdash b {:} \N$, * $A \Rightarrow \U$ and either $a \Rightarrow \N$ or $a \Rightarrow \Pi x {:} b. c$ where furthermore $\vdash b {:} \U$ and $x {:} b \vdash c {:} \U$, * $A \Rightarrow \Pi x {:} B. C$ and $a \Rightarrow \lambda x.c$ and $x {:} B \vdash c {:} C$. The meaning of the categorical judgment $\vdash A = A'$ is that $A$ and $A'$ have the same canonical types as values. In our fragment this means that either of the following holds: * $A \Rightarrow \N$ and $A' \Rightarrow \N$, * $A \Rightarrow \U$ and $A' \Rightarrow \U$, * $A \Rightarrow \Pi x {:} B. C$ and $A' \Rightarrow \Pi x {:} B'. C'$ and furthermore that $\vdash B = B'$ and $x {:} B \vdash C = C'$. The meaning of the categorical judgment $\vdash a = a' {:} A$ is explained in a similar way. It is a tacit assumption of the meaning explanations that the repeated computations of canonical forms is well-founded. For example, a natural number is the result of finitely many computations of the successor function $\s$ ended by $0$. A computation which results in infinitely many computations of $\s$ is not a natural number in intuitionistic type theory. (However, there are extensions of type theory, for example, partial type theory, and non-standard type theory, where such infinite computations can occur, see section 7.3. To justify the rules of such theories the present meaning explanations do not suffice.) ### 5.3 The Meaning of Hypothetical Judgments According to Martin-Lof (1982) the meaning of a hypothetical judgment is reduced to the meaning of the categorical judgments by substituting the closed terms of appropriate types for the free variables. For example, the meaning of \[x\_1 {:} A\_1, \ldots, x\_n {:} A\_n \vdash a {:} A\] is that the categorical judgment \[\vdash a[x\_1 := a\_1, \ldots , x\_n := a\_n] : A[x\_1 := a\_1, \ldots , x\_n := a\_n]\] is valid whenever the categorical judgments \[\vdash a\_1 {:} A\_1, \ldots , \vdash a\_n[x\_1 := a\_1, \ldots , x\_{n-1} := a\_{n-1}] {:} A\_n[x\_1 := a\_1, \ldots , x\_{n-1} := a\_{n-1}]\] are valid. ## 6. Mathematical Models ### 6.1 Categorical Models #### 6.1.1 Hyperdoctrines Curry's correspondence between propositions and types was extended to predicate logic in the late 1960s by Howard (1980) and de Bruijn (1970). At around the same time Lawvere developed related ideas in categorical logic. In particular he proposed the notion of a hyperdoctrine (Lawvere 1970) as a categorical model of (typed) predicate logic. A hyperdoctrine is an indexed category $P {:} T^{op} \rightarrow \mathbf{Cat}$, where $T$ is a category where the objects represent types and the arrows represent terms. If $A$ is a type then the fibre $P(A)$ is a category of propositions depending on a variable $x {:} A$. The arrows in this category are proofs $Q \vdash R$ and can be thought of as proof-objects. Moreover, since we have an indexed category, for each arrow $t$ from $A$ to $B$, there is a reindexing functor $P(B) \rightarrow P(A)$ representing substitution of $t$ for a variable $y {:} B$. The category $P(A)$ is assumed to be cartesian closed and conjunction and implications are modelled by products and exponentials in this category. The quantifiers $\exists$ and $\forall$ are modelled by the left and right adjoints of the reindexing functor. Moreover, Lawvere added further structure to hyperdoctrines to model identity propositions (as left adjoints to a diagonal functor) and a comprehension schema. #### 6.1.2 Contextual categories, categories with attributes, and categories with families Lawvere's definition of hyperdoctrines preceded intuitionistic type theory but did not go all the way to identifying propositions and types. Nevertheless Lawvere influenced Scott's (1970) work on constructive validity, a somewhat preliminary precursor of intuitionistic type theory. After Martin-Lof (1998 [1972]) had presented a more definite formulation of the theory, the first work on categorical models was presented by Cartmell in 1978 with his notions of category with attributes and contextual category (Cartmell 1986). However, we will not define these structures here but instead the closely related categories with families (Dybjer 1996) which are formulated so that they directly model a variable-free version of a formulation of intuitionistic type theory with explicit substitutions (Martin-Lof 1995). A category with families is a functor $T {:} C^{op} \rightarrow \mathbf{Fam}$, where $\mathbf{Fam}$ is the category of families of sets. The category $C$ is the category of contexts and substitutions. If $\Gamma$ is an object of $C$ (a context), then $T(\Gamma)$ is the family of terms of type $A$ which depend on variables in $\Gamma$. If $\gamma$ is an arrow in $C$ representing a substitution, then the arrow part of the functor represents substitution of $\gamma$ in types and terms. A category with families also has a terminal object and a notion of context comprehension, reminiscent of Lawvere's comprehension in hyperdoctrines. The terminal object captures the rules for empty contexts and empty substitutions. Context comprehension captures the rules for extending contexts and substitutions, and has projections capturing weakening and assumption of the last variable. Categories with families are algebraic structures which model the general rules of dependent type theory, those which come before the rules for specific type formers, such as $\Pi$, $\Sigma$, identity types, universes, etc. In order to model specific type-former corresponding extra structure needs to be added. #### 6.1.3 Locally cartesian closed categories From a categorical perspective the above-mentioned structures may appear somewhat special and ad hoc. A more regular structure which gives rise to models of intuitionistic type theory are the locally cartesian closed categories. These are categories with a terminal object, where each slice category is cartesian closed. It can be shown that the pullback functor has a left and a right adjoint, representing $\Sigma$- and $\Pi$-types, respectively. Locally cartesian closed categories correspond to intuitionistic type theory with extensional identity types and $\Sigma$ and $\Pi$-types (Seely 1984, Clairambault and Dybjer 2014). It should be remarked that the correspondence with intuitionistic type theory is somewhat indirect, since a coherence problem, in the sense of category theory, needs to be solved. The problem is that in locally cartesian closed categories type substituion is represented by pullbacks, but these are only defined up to isomorphism, see Curien 1993 and Hofmann 1994. ### 6.2 Set-Theoretic Model Intuitionistic type theory is a possible framework for constructive mathematics in Bishop's sense. Such constructive mathematics is compatible with classical mathematics: a constructive proof in Bishop's sense can directly be understood as a proof in classical logic. A formal way to understand this is by constructing a set-theoretic model of intuitionistic type theory, where each concept of type theory is interpreted as the corresponding concept in Zermelo-Fraenkel Set Theory. For example, a type is interpreted as a set, and the type of functions in $A \rightarrow B$ is interpreted as the set of all functions in the set-theoretic sense from the set representing $A$ to the set representing $B$. The type of natural numbers is interpreted as the set of natural numbers. The interpretations of identity types, and $\Sigma$ and $\Pi$-types were already discussed in the introduction. And as already mentioned, to interpret the type-theoretic universe we need an inaccessible cardinal. #### 6.2.1 Model in CZF It can be shown that the interpretation outlined above can be carried out in Aczel's constructive set theory CZF. Hence it does not depend on classical logic or impredicative features of set theory. ### 6.3 Realizability Models The set-theoretic model can be criticized on the grounds that it models the type of functions as the set of all set-theoretic functions, in spite of the fact that a function in type theory is always computable, whereas a set-theoretic function may not be. To remedy this problem one can instead construct a realizability model whereby one starts with a set of realizers. One can here follow Kleene's numerical realizability closely where functions are realized by codes for Turing machines. Or alternatively, one can let realizers be terms in a lambda calculus or combinatory logic possibly extended with appropriate constants. Types are then represented by sets of realizers, or often as partial equivalence relations on the set of realizers. A partial equivalence relation is a convenient way to represent a type with a notion of "equality" on it. There are many variations on the theme of realizability model. Some such models tacitly assume set theory as the metatheory (Aczel 1980, Beeson 1985), whereas others explictly assume a constructive metatheory (Smith 1984). Realizability models are also models of the extensional version of intuitionistic type theory (Martin-Lof 1982) which will be presented in section 7.1 below. ### 6.4 Model of Normal Forms and Type-Checking In intuitionistic type theory each type and each well-typed term has a normal form. A consequence of this normal form property is that all the judgments are decidable: for example, given a correct context $\Gamma$, a correct type $A$ and a possibly ill-typed term $a$, there is an algorithm for deciding whether $\Gamma \vdash a {:} A$. This type-checking algorithm is the key component of proof-assistants for Intensional Type Theory, such as Agda. The correctness of the normal form property can be expressed as a model of normal forms, where each context, type, and term are interpreted as their respective normal forms. ## 7. Variants of the Theory ### 7.1 Extensional Type Theory In extensional intuitionistic type theory (Martin-Lof 1982) the rules of $\I$-elimination and $\I$-equality for the general identity type are replaced by the following two rules: \[\frac{\Gamma\vdash c {:} \I(A,a,a')} {\Gamma \vdash a=a' {:} A} \hspace{3em} \frac{\Gamma\vdash c{:}I(A,a,a')} {\Gamma\vdash c = \r {:} \I(A,a,a')}\] The first causes the distinction between propositional and judgmental equality to disappear. The second forces identity proofs to be unique. Unlike the rules for the intensional identity type former, the rules for extensional identity types do not fit into the schema for inductively defined types mentioned above. These rules are however justified by the meaning explanations in Martin-Lof (1982). This is because the categorical judgment \[\vdash c {:} \I(A,a,a')\] is valid iff $c \Rightarrow \r$ and the judgment $\vdash a = a' {:} A$ is valid. However, these rules make it possible to define terms without normal forms. Since the type-checking algorithm relies on the computation of normal forms of types, it no longer works for extensional type theory, see (Castellan, Clairambault, and Dybjer 2015). On the other hand, certain constructions which are not available in intensional type theory are possible in extensional type theory. For example, function extensionality \[(\Pi x {:} A. \I(B,f\,x,f'\,x)) \rightarrow \I(\Pi x{:}A.B,f,f')\] is a theorem. Another example is that $\W$-types can be used for encoding other inductively defined types in Extensional Type Theory. For example, the Brouwer ordinals of the second and higher number classes can be defined as special instances of the $\W$-type (Martin-Lof 1984). More generally, it can be shown that all inductively defined types which are given by a strictly positive type operator can be represented as instances of well-founded trees (Dybjer 1997). ### 7.2 Univalent Foundations and Homotopy Type Theory Univalent foundations refer to Voevodsky's programme for a new foundation of mathematics based on intuitionistic type theory and employing ideas from homotopy theory. Here every type $A$ is considered as a space, and the identity type $\I(A,a,b)$ is the space of paths from point $a$ to point $b$ in $A$. Iterated identity types represent higher homotopies, e.g. \[\I(\I(A,a,b),f,g)\] is the space of homotopies between $f$ and $g$. The notion of ordinary set can be thought of as a discrete space $A$ where all paths in $\I(A,a,b)$ are trivial loops. The origin of these ideas was the remarkable discovery by (Hofmann and Streicher 1998) that the axioms of intensional type theory do not force all proofs of an identity to be equal, that is, not all paths need to be trivial. This was shown by a model construction where each type is interpreted as a groupoid. Further connections between identity types and notions from homotopy theory and higher categories were subsequently discovered by (Awodey and Warren 2009), (Lumsdaine 2010), and (van den Berg and Garner 2011). Voevodsky realized that the whole intensional intuitionistic type theory could be modelled by a well-known category studied in homotopy theory, namely the Kan simplicial sets. Inspired by this model he introduced the univalence axiom. For a universe $\U$ of small types, this axiom states that the substitution map associated with the $J$-operator \[\I(\U,a,b) \longrightarrow \T(a) \cong \T(b)\] is an equivalence. Equivalence ($\cong$) here refers to a general notion of equivalence of higher dimensional objects, as in the sequence equal elements, isomorphic sets, equivalent groupoids, biequivalent bigroupoids, etc. The univalence axiom expresses that "everything is preserved by equivalence", thereby realizing the informal categorical slogan that all categorical constructions are preserved by isomorphism, and its generalization, that all constructions of categories are preserved by equivalence of categories, etc. The axiom of univalence was originally justified by Voevodsky's simplical set model. This model is however not constructive and (Bezem, Coquand and Huber 2014 [2013]) has more recently proposed a model in Kan cubical sets. Although univalent foundations concern preservation of mathematical structure in general, strongly inspired by category theory, applications within homotopy theory are particularly actively investigated. Intensional type theory extended with the univalence axiom and so called higher inductive types is therefore also called "homotopy type theory". We refer to the entry on type theory for further details. ### 7.3 Partial and Non-Standard Type Theory Intuitionistic type theory is not intended to model Brouwer's notion of free choice sequence, although lawlike choice sequences can be modelled as functions from $\N$. However, there are extensions of the theory which incorporate such choice sequences: namely partial type theory and non-standard type theory (Martin-Lof 1990). The types in partial type theory can be interpreted as Scott domains (Martin-Lof 1986, Palmgren and Stoltenberg-Hansen 1990, Palmgren 1991). In this way a type $\N$ which contains an infinite number $\infty$ can be interpreted. However, in partial type theory all types are inhabited by a least element $\bot$, and thus the propositions as types principle is not maintained. Non-standard type theory incorporates non-standard elements, such as an infinite number $\infty {:} \N$ without inhabiting all types. ### 7.4 Impredicative Type Theory The inconsistent version of intuitionistic type theory of Martin-Lof (1971a) was based on the strongly impredicative axiom that there is a type of all types. However, (Coquand and Huet 1988) showed with their calculus of constructions, that there is a powerful impredicative but consistent version of type theory. In this theory the universe $\U$ (usually called ${\bf Prop}$ in this theory) is closed under the following formation rule for cartesian product of families of types: \[\frac{\Gamma \vdash A \hspace{2em} \Gamma, x {:} A \vdash B {:} \U} {\Gamma \vdash \Pi x {:} A. B {:} \U}\] This rule is more general than the rule for constructing small cartesian products of families of small types in intuitionistic type theory, since we can now quantify over arbitrary types $A$, including $\U$, and not just small types. We say that $\U$ is impredicative since we can construct a new element of it by quantifying over all elements, even the element which is constructed. The motivation for this theory was that inductively defined types and families of types become definable in terms of impredicative quantification. For example, the type of natural numbers can be defined as the type of Church numerals: \[\N = \Pi X {:} \U. X \rightarrow (X \rightarrow X) \rightarrow X {:} \U\] This is an impredicative definition, since it is a small type which is constructed by quantification over all small types. Similarly we can define an identity type by impredicative quantification: \[\I(A,a,a')= \Pi X {:} A \rightarrow \U. X\,a \rightarrow X\,a' {:} \U\] This is Leibniz' definition of equality: $a$ and $a'$ are equal iff they satisfy the same properties (ranged over by $X$). Unlike in intuitionistic type theory, the function type in impredicative type cannot be interpreted set-theoretically in a straightfoward way, see (Reynolds 1984). ### 7.5 Proof Assistants In 1979 Martin-Lof wrote the paper "Constructive Mathematics and Computer Programming" where he explained that intuitionistic type theory is a programming language which can also be used as a formal foundation for constructive mathematics. Shortly after that, interactive proof systems which help the user to derive valid judgments in the theory, so called proof assistants, were developed. One of the first systems was the NuPrl system (PRL Group 1986), which is based on an extensional type theory similar to (Martin-Lof 1982). Systems based on versions of intensional type theory go back to the type-checker for the impredicative calculus of constructions which was written around 1984 by Coquand and Huet. This led to the Coq system, which is based on the calculus of inductive constructions (Paulin-Mohring 1993), a theory which extends the calculus of construction with primitive inductive types and families. The encodings of the pure calculus of constructions were found to be inconvenient, since the full elimination rules could not be derived and instead had to be postulated. We also remark that the calculus of inductive constructions has a subsystem, the predicative calculus of inductive constructions, which follows the principles of Martin-Lof's intuitionistic type theory. Agda is another proof assistant which is based on the logical framework formulation of intuitionistic type theory, but adds numerous features inspired by practical programming languages (Norell 2008). It is an intensional theory with decidable judgments and a type-checker similar to Coq's. However, in contrast to Coq it is based on Martin-Lof's predicative intuitionistic type theory. There are several other systems based either on the calculus of constructions (Lego, Matita, Lean) or on intuitionistic type theory (Epigram, Idris); see (Pollack 1994; Asperti et al. 2011; de Moura et al. 2015; McBride and McKinna 2004; Brady 2011).
god-ultimates	## 1. Conceptual Foundations and Motivations ### 1.1 Definition of "ultimate" Brahman, the Dao, emptiness, God, the One, Reasonableness--there, in alphabetical order, are names of the central subjects of concern in what are commonly parsed as some of the world's religions, philosophies and quasi-religious-philosophies.[1] They are all names for what is ultimate, at least on some uses of the names (for instance, "God" is not always taken to be ultimate, more soon). But what is it to be ultimate, in this sense? To answer in terms of its use, the term "ultimacy", meaning the state or nature of being ultimate, has Brahman, the Dao, emptiness etc. as instances. To answer semantically, with a meaning, is difficult for at least two reasons. First, though there is abundant precedent in the literature for collecting these subjects as ideas of ultimacy,[2] doing so presupposes they have some shared characteristic or family resemblance that makes them count as ultimate. But is there a shared core idea of being ultimate? "Particularists" among others argue no: the diverse range of cultural and historical contexts from which these subjects come, coupled with how hard it is to talk across such contexts, makes them all "separate cultural islands" (Hedges 2014: 206; see also Berthrong 2001: 237-239, 255-256).[3] The second concern is related, and not far from one Tomoko Masuzawa (2005) among others has raised about religion: even if we found a substantive account of ultimacy visible in multiple traditions, such an account necessarily will be borne from a cultural conceptual context. Thus, far from delivering the notions at work in other traditions, such an account actually risks de-forming them. Regarding the first concern, John H. Berthrong, among others, is far more optimistic than the particularists that concepts not only can be shared across cultures but in fact are > > > already comparative, having been generated by the interactions of > people, texts, rituals, cultural sensibilities and the vagaries of > history and local customs. (2001: 238) > > > Other theorists explore factors that could detail or add to Berthrong's list--e.g., trade and conquests (Gayatriprana 2020), shared human evolutionary biology (Wildman 2017), and the evolution of moral development (Wright 2010).[4] Still, most take the second concern about enculturation to stick and thus to temper the optimism: there is both a shared humanity and real cultural difference to own in reaching a global idea of ultimacy. Raimon Panikkar says it well: > > > Brahman is certainly not the one true and living God of the > Abrahamic traditions. Nor can it be said that Shang-ti or > kami are the same as brahman. And yet they are not > totally unrelated. (Panikkar 1987 [2005: 2254]) > > > The Cross-Cultural Comparative Religious Ideas Project, run by Robert C. Neville and Wesley Wildman from 1995-99, balanced the overlap and difference when they concluded that an account of ultimacy should be a "properly vague" category: it needs enough shared content to count as a category, but enough vagueness to cover the disparate instances generally taken to be ultimates.[5] There are multiple contenders for such vague categories, including, e.g., Paul Tillich's "object of ultimate concern" (1957a, e.g., 10-11), John Hick's "the Real" (1989: 11ff); Keith Ward's "the Transcendent" (1998); and the Project's own proposal as "that which is most important to religious life because of the nature of reality" (Neville & Wildman 2001, see 151ff for an explanation of each part of the phrase). Informed by the Project's finding, this entry will "vague-ify" the content of John Schellenberg's account of what is ultimate to use as a cross-cultural core idea of ultimacy and as an organizing principle. To paraphrase, Schellenberg takes being ultimate to mean being that which is (1) the most real, (2) the most valuable, and (3) the source of deepest fulfillment among all that is or perhaps could be. More carefully, and in Schellenberg's words, being ultimate requires being ultimate in three ways: (1) metaphysically ultimate, i.e., the "most fundamental fact" about the nature of things (2016: 168), (2) axiologically ultimate, i.e., that which "has unsurpassably great value" (2009: 31), meaning greatness along all its categories of being, and (3) soteriologically ultimate, i.e., "the source of an ultimate good (salvific)" (2009: xii, also 2005: 15), meaning being the source of salvation or liberation of the kind practitioners of the world's religions and philosophies ardently seek (e.g., nirvana, communing with God, moksha, ascent to the One, etc.), whether these all amount to the same salvation (e.g., Hick 1989: Ch. 14) or constitute radically distinct types of salvations (Heim 1995: Ch. 5). Schellenberg's choice of these three terms is insightful: most extant takes on ultimacy per se and on Brahman, God and the Dao in particular are variations on a theme of Schellenberg, as scrutiny of even the brief definitions above as well as the models in Section 2 will bear out.[6] Schellenberg's account of what it takes to be ultimate is already somewhat vague: he recommends no further precisification of his three terms in order to stay open about what counts as ultimate as our knowledge grows (2009: 31). This entry will loosen his account further by placing a disjunction between its terms instead of a conjunction. That is, Schellenberg takes metaphysical, axiological and soteriological ultimacy to be severally necessary and jointly sufficient for something to count as ultimate; he requires "triple ultimacy", as James Elliott calls it, for ultimacy per se (2017: 103-04). Those who agree in principle include, e.g., Clooney 2001 and Rubenstein 2019,[7] as well as, e.g., Aquinas, Leibniz and Samuel Clarke, who in fact argue that, given the entailments between the terms, there is triple ultimacy or none at all.[8] Others take double or even single ultimacy to be not only possible but also sufficient for being ultimate. For example, Neville defines ultimacy in strictly metaphysical terms as "the ontological act of creation" (2013: 1); Elliott and Paul Draper each take soteriological ultimacy to be sufficient for ultimacy (Elliott 2017: 105-109; Draper 2019: 161); and John Bacon takes his understanding of God as "<Creator, Good>" to be metaphysically and soteriologically ultimate though a "let-down axiologically" (2013: 548). Moreover, requiring triple ultimacy stops some of the paradigmatic models of Brahman, God and the Dao from counting as ultimacy, as Section 2 will demonstrate (see also Diller 2013b). Thus, this entry will take some combinations of the three types of ultimacy to be sufficient for being ultimate, without settling which, provided nothing else in a system has more. Note that replacing Schellenberg's conjunction with a disjunction makes the field of ultimates a family resemblance class. Even disjunctivized, the Schellenbergian view adopted here is clearly an inheritance from Abrahamic perfect being theology,[9] so two cautions about scope. First, when we look outward and find that some non-Abrahamic traditions have ultimacy in the sense here--as we will in Section 2 for Hinduism and Daoism--we should acknowledge that this result comes framed from the outside. Second, not all non-Abrahamic traditions will have ultimates in the sense just adopted. For instance, to offer just one example, Barbara Mann suggests that in non-colonized interpretations of Native American spiritualities there is nothing that is ultimate in Schellenberg's sense (2010: 33-36).[10] So ultimacy as just defined may be widespread in the world's religions and spiritualities, but it is not universal. Finally, three points of clarification about terminology. First, talking about ultimacy does not entail that anything ultimate exists. The concern that it might is related to "the problem of singular negative existential statements", to which Frege and Russell offered solutions that in turn have been enshrined in predicate logic, though not without complaint (for more see Kripke 2013 and the SEP articles on existence and on nonexistent objects). Second, the term "ultimate reality" could be taken to mean how reality ultimately is--i.e., that which is metaphysically ultimate, perhaps a part of every complete ontology--instead of meaning the richer sense of "ultimacy" at issue here, which is not a part of every complete ontology.[11] To avoid confusion, this entry will reserve the term "ultimate reality" for metaphysical ultimacy per se and use "ultimate" and variants for the combinations of the Schellenbergian disjunction.[12] Thus framed, the distinction leaves open this central question: is ultimate reality ultimate? Lastly, regarding the choice of the term "ultimate" and its variants, there is a syntactic parallel to the semantic issue above: we need a sufficiently vague kind of speech to cover the diverse ontological kinds implicit in accounts of ultimacy, including concrete or abstract particular things (e.g., God or Brahman on some views); states of being (e.g., Existence-Consciousness-Bliss for Brahman, see Section 2.1); properties (e.g., everything is empty on Buddhism or divinely intentional for Karl Pfeifer, see Section 2.2); actions and events that things perform or undergo (D. Cooper models God as "a verb" as in "God-ing", 1997: 70; cf. Bishop & Perszyk 2017); and grounds of being that are meant to be category-less (e.g., as in "the creative source of the categories themselves", Vallicella 2006 [2019], see also, e.g., Tillich in Section 2.2, the Dao in Section 2.3). Though no term is quite right given the diversity, this entry uses the adjective form of being "ultimate" as primary, to describe x as metaphysically, axiologically or soteriologically ultimate per above, whatever x's ontological kind (cf. with the more familiar term from Abrahamic monotheism of "divine"). It also uses "ultimacy" as a noun for the nature or state of being ultimate (cf. "divinity"), the nouns "the ultimate", "an ultimate" or "ultimates" to function flexibly both as a mass noun (such as "water" or "butter") for the property or uncountable substance of being ultimate and as a count noun for things, events and grounds of being (cf. "God" or "gods"); and "to ultimize" as the verb form, if ever we need it.[13] ### 1.2 Definition of "model" A "model" of what is ultimate is a way it can be conceived. In general, a model is a representation of a target phenomenon for some purpose. For instance, R. Axelrod developed a computational model that represented a target of two "agents" caught in an iterated prisoner's dilemma, with the purpose of solving the dilemma.[14] The term "model" in religious and philosophical contexts is related to but not identical with its use in scientific contexts, in ways Ian Barbour (1974) foundationally traced. Though there is some disagreement between the fields on whether models describe their target or not and what the point of modeling is, importantly, in both contexts, the term "model" (1) connotes that its target is somehow out of reach--not able to be directly examined--and, perhaps by force of this, (2) encodes a conceptual distance between the target and the model. In particular, the model is not a copy of the target but rather chooses revealing aspects of the target to relay by leaving out or distorting other aspects.[15] Think of a model of a city that is by design not to scale, precisely so viewers can see the relationships between the city's streets, buildings and neighborhoods. Models are thus simultaneously epistemically instructive and humble. Thus understood, "model" captures well the ways people have thought about what is ultimate. Taking a model's target to be obscured per (1) and the model itself to be fallible per (2) is not only apt but also crucial for thinking about what is ultimate given our necessarily limited knowledge of it (see Section 1.4). Moreover, among the choices on the linguistic menu, "model" is a middle way. It is more specific than "idea" understood in the Cartesian sense as "whatever is immediately perceived by the mind",[16] since a model is the more particular kind of idea just relayed. At the same time, "model" is general enough to cover the very diverse kinds of extant linguistic accounts of what is ultimate in the literature, including, e.g., "concepts" understood in the classic sense as necessary and sufficient conditions for being ultimate, such as Anselm's idea of God as "that than which no greater can be conceived;" "conceptions" which are "more particular fillers" for more general concepts, such as Richard Swinburne's verdict about what it would take to be that than which nothing greater can be conceived;[17] sustained "metaphors" such as Ramanuja's Brahman as the "Soul" of the cosmos (see Section 2.1) or Sallie McFague's "God as Mother, Lover and Friend" (1987); and "indexical signs" that point to an indeterminate ultimate visible in Neville (2013). This entry will call all these linguistic types "models" of what is ultimate, given that they each in their own way represent or aim at a target of what is ultimate for various purposes, at least as much as language can (more in Section 1.4).[18] As Section 1.5 will detail, some models of what is ultimate nest, e.g., Shankara's idea of Brahman is a species of Vedantic ideas of Brahman which in turn are species of Hindu ideas of Brahman. To simplify, this entry will use the term "model" for ideas of what is ultimate at all levels, and take a model's target to be "the ultimate" if the modeler thinks what is ultimate is single or uncountable (as in models of e.g., Brahman, God and the Dao) or "ultimates" if multiple (as in polytheistic or perhaps communotheistic models). ### 1.3 Motivations Understanding the scope of the work on modeling what is ultimate is central in multiple ways for assessing claims about its existence. First, the meaning we have in mind for "x" can decide our take on whether x exists. For example, in the case of God, some are convinced by arguments from suffering that there is no God, but such arguments, even if they succeed, generally apply only to God conceived as an omnipotent, omniscient and omnibenevolent ("OOO") person and as written do not apply to God conceived differently ("generally" since there are notable exceptions, e.g., Bishop 2007 and Bishop & Perszyk 2016). Further, the act of abandoning one model while being aware that there are alternative models can set up an exploration into the field of models of the ultimate or ultimates. For example, if one decides there is no OOO personal God, one might ask: are there any other models of God genuinely worth the name "God" without these properties? If not, are there perhaps any non-theistic (non-God) models of what is ultimate without these properties? In other words, for those who have a limited sense of what it takes to be ultimate (e.g., what is ultimate is a OOO God or nothing at all), a search of extant models can open up the range of options on their menu and position them to see whether any are philosophically palatable. In fact, such a search may be required in order to settle the question of whether anything ultimate exists, because it is invalid to conclude that there is nothing ultimate on the basis of arguments against specific models of ultimacy if there could still be genuine ultimacy of another kind (see Diller 2016: 123-124). In addition to their pivotal role in deciding questions of existence, notions of what is ultimate profoundly impact questions about religious diversity, such as whether multiple religions can have simultaneously true core beliefs. For example, on "plenitude of being" models--where the ultimate is infinitely full of incommensurable content--multiple religions are taken to be grasping different aspects of it accurately. So multiple religions can have true beliefs about what is ultimate, even if each has only a part of the truth. Finally, models of what is ultimate are also intimately bound up with philosophical questions of meaning, e.g., in meaning's "Cosmic" and "Ultimate" senses as Rivka Weinberg explains them, though she decides that even if there were something ultimate, it would not finally deliver meaning in either sense (2021). ### 1.4 Challenges Though some model what is ultimate, others object to the entire project of modeling it. There are three main kinds of concerns in the literature: about the existence of ultimacy, about the coherence of the very idea of ultimacy, and about whether talk and knowledge of ultimacy is humanly possible, even assuming the idea is coherent. The first concern is motivational: for those who think there is nothing ultimate, there seems no point worrying about how it has been conceived. However, as indicated in Section 1.3, it is invalid to conclude that there is nothing ultimate without having some sense of the range of what it might be. So even those who think there is nothing ultimate are thrown into the project of modeling it--at least enough to license the conclusion that this work is not worth doing. To offer just one example of the second challenge, Stephen Maitzen (2017) argues against the concept of ultimacy on multiple fronts, including, e.g., that nothing can be metaphysically ultimate in Schellenberg's sense because it would need to be simultaneously a se and concrete so that it can be metaphysically independent and explain concrete things, respectively. But no concrete thing can explain itself because (to shorten the argument considerably) even the necessity of such a thing is not identical to itself so it is not explaining itself (2017: 53). If Maitzen is right, nothing can be metaphysically ultimate in Schellenberg's sense, and thus not metaphysically, axiologically and soteriologically ultimate at once. The third kind of concern is more widespread--that it makes no sense for us to model what is ultimate because it is beyond human language (ineffable) or beyond human cognitive grasp (cognitively unknowable), or both. Why think this? To combine a few common arguments: what is ultimate (1) goes beyond the world, (2) is in a class by itself, and (3) is infinite, while our predicates are (1') suited to describe things in the world, (2') classify things with other things, and (3') are limited (see, e.g., Wildman 2013: 770; Seeskin 2013: 794-795). Thus, for any P, a statement of the form "ultimacy is P" seems necessarily false, if it is meaningful at all. Ineffability and unknowability are related: if we can say nothing true about ultimacy, then we can know nothing about it--at least nothing that can be said in words. That last hesitation--"at least not in words"--leaves room for, e.g., embodied ways of knowing by way of religious, mystical or spiritual experiences which are reported in the world's religious traditions and more generally (whether such experiences actually happen or not, see, e.g., James 1902 [1961]). Still, the combined arguments above and the gut intuition behind them represent an enormous challenge to the whole enterprise of modeling what is ultimate in words. There are those who opt for silence in the face of these arguments, and thus understandably but alas "literally disappear from the conversation" (Wildman 2013: 768, "alas" because they are missed). One main move of those who do keep talking is to distinguish what is ultimate as it is in itself, which they concede we can never talk about or know, from it as it affects our experience, which they think we can talk about and know. So some distinguish, e.g., the absolute from the relative Dao (Daodejing, chapter 1); the Godhead from God as revealed (Meister Eckhart, e.g., Sermon 97; Panikkar 1987 [2005: 2254]); the noumenal from the phenomenal Real (Hick 1989) etc. and talk or make knowledge claims only about the latter (see also, e.g., Paul Hedges 2020 [Other Internet Resources]).[19] Some also claim to make true statements about what is ultimate by restricting those statements to certain kinds of claims about it. One tactic is to talk about how ultimacy is related to us or to other parts of the natural (or non-natural) world instead of talking about how it is in itself, i.e., to talk about its extrinsic vs. intrinsic properties. For example, Maimonides suggests that one way to make "ultimacy is P" true is to make P an "attribute of action", i.e., an "action that he who is described has performed, such as Zayd carpentered this door..." (The Guide of the Perplexed, I.52-3, italics added), an attribute which says nothing about Zayd's intrinsic properties save that he has what it takes to carpenter this door. An analogue of this for ultimacy is, e.g., the Dao generated being, again an attribute which indicates only that the Dao, whatever it is or is not, can and has generated being. Other possible ways of speaking truly about ultimacy include famously the via negativa ("God is not P", Shaffer 2013: 783), the via eminentia ("God is better than P"), the way of analogy ("God is perfectly P", Copleston 1952: 351 on Aquinas, e.g., Summa Theologica I, 13 and Summa contra Gentiles I, 30, see also Kennedy 2013: 158-159), the way of super-eminence ("God is beyond P or not P", Pseudo-Dionysius, Shaffer 2013: 786ff) and--though this seems doable in theory only--equivocal predication ("God is P\" where P\ is a predicate outside of human language, Shaffer 2013: 783). What does all this really allow us to say, know, and do philosophically, though? More than one might think, says Neville: though ineffability might seem to stop metaphysics, it actually tells us how to do the metaphysics, e.g., "the dao...can be discussed mainly by negations and indirections" (2008: 43), or in these other ways. ### 1.5 Philosophical categories of ultimacy For those who decide to model what is ultimate in the face of or informed by these challenges, there are several common categories, or "model types" as Philip Clayton once called them, which distinguish kinds of models of what is ultimate from each other. Knowing these categories can organize what might be an otherwise haphazard array of models, in something like the way knowing what an oak, maple and birch are can help sort the sights on a walk through a forest. The categories of ultimacy are best grasped by framing them with a question. For example, Hartshorne and Reese (1953) categorized models of God in particular as the logically possible variations on a theme of five questions, whose positive answers get symbolized by ETCKW: * E: Is God eternal? * T: Is God temporal? * C: Is God conscious? * K: Does God know the world? * W: Does God include the world? To use models that will be discussed in Section 2, Shankara's pantheism is ECKW since he takes Brahman to be an "Eternal Consciousness, Knowing and including the World" but which is atemporal and never changes; Hartshorne and Reese's own panentheism is ETCKW, etc. (see their 1953 [2000: 17] and Viney 2013 for more examples). Wildman (2017) has his own set of questions and entire system built out of them (more in Section 3), which adds another important question that ETCKW leaves out or perhaps merely implies: Is what is ultimate personal/anthropomorphic--i.e., is it aware and does it have intentions--or is it impersonal/non-anthropomorphic? One might also look for functional categories: is what is ultimate the efficient, material or final cause of the universe?, does it intervene in it?, does it provoke religious experience?, and more. Another question that may be asked about a particular model is: How many ultimates does the model conceive ultimacy to be? Though the very idea of plural ultimates sounds contradictory--wouldn't one thing need to beat out all competitors in order to satisfy the superlatives "most fundamental" fact, "highest" value, "deepest" source of fulfillment?--those who model "ultimate multiplicity" see a tie for first place. Some models of Zoroaster's view, for instance, take there to be two ultimates, the good Ahura Mazda and the evil Angra Mainyu engaged in fundamental battle (though such models will have to explain why Ahura Mazda is not the true ultimate given his apparent axiological edge on Angra Mainyu). John Cobb and David Ray Griffin at least some of the time take there to be three distinct ultimates: the Supreme Being experienced in theistic experiences, Being Itself (or emptiness) experienced in acosmic experiences, and the Cosmos experienced in cosmic experiences (Griffin 2005: 47-49).[20] Monica Coleman models what is ultimate in traditional Yoruba religion as a "communotheism", in which "the Divine is a community of gods who are fundamentally related to each other but ontologically equal", including Olodumare and the 401 orisa (2013: 345-349). All the models we will look at in Section 2 will be "ultimate unities" although there will be some diversity in the unity for the panentheisms in particular. George Mavrodes also cautions us to be wary of the unity/multiplicity distinction: > > > there are monotheisms that seem to include an element of > multiplicity--e.g. Christianity with its puzzling idea of the > divine trinity--and views of divine multiplicity--such as > the African religions [have]--that seem to posit some sort of > unity composed of a large number of individual divine entities. (2013: > 660) > > > Probably the most frequently-used categories to sort models from each other are the logically exhaustive answers to the question: How does the ultimate or ultimates relate to the world? Though the answers (and question actually) are commonly framed with the word "God", as the theos root in some of the category names below belies, their descriptions at least are framed here with "ultimacy" to include non-theistic ultimates, too:[21] * Monism (literally, one-ism): There exists just one thing or one kind of thing (the "One" or "Unity"), depending on how the monism is read: either ultimacy is identical to the world, or all there is is ultimacy, or all there is is the world. Pantheism is a species of monism in which the one thing or kind of thing is God, or as Linda Mercadante once said in conversation, it is a monism in which the emphasis is on the divinity of the world instead of on the worldliness of the divine. Baruch Spinoza's view of "God, or Nature" (Deus sive natura) is often deemed a pantheism,[22] though some experts deem it a panentheism (1677, Part IV.4; see, e.g., Edwin Curley 2013). * Panentheism (all-in-God-ism): The world is a proper part of ultimacy. I.e., though the world is in ultimacy, ultimacy is more than the world. Panentheism is a wide tent since there is ambiguity in the "in", but R. T. Mullins, for example, offers one disambiguated panentheism: take space and time to be attributes of God, and "affirm that the universe is literally in God because the universe is spatially and temporally located in God" (2016: 343). * Merotheism (part-God-ism): Ultimacy is a proper part of the world. I.e., though ultimacy is in the world, the world is more than ultimacy. This type is rare, but see, e.g., Alexander 1920 and Draper 2019 (more soon). * Dualism (two-ism): Ultimacy is not the world and the world is not ultimacy, so there are two fundamentally different things or kinds of things, depending on how the underlying ontology is read. Dualism includes both the view that ultimacy and the world are disjoint (they share no parts) and the view that they bear a relation of proper overlap (they overlap in part, but not in whole).[23] The various models of God, Brahman and the Dao in Section 2 taken together will instantiate all four of these categories. ## 2. Models of Brahman, God, and the Dao This section turns to multiple models of three ultimates extant in living world traditions: Brahman in the Hindu traditions indigenous in India, God in the Abrahamic traditions (Judaism, Christianity and Islam) indigenous in the Middle East and the Arabian Peninsula, and the Dao in the Daoist and Ru (Confucian) traditions indigenous in China. The sections here are ordered historically, with the idea of Brahman surfacing first in the Rig Veda ca. 1000-1500 BCE, the idea of the Dao during the Warring States period ca. fifth century BCE, and the idea of God in the Jewish tradition shifting from a henotheistic to monotheistic deity somewhere in between. A main point of looking at these models is to grasp some ways people across the globe have conceived and are conceiving of what is most real, most valuable or most fulfilling to them--knowledge worth having for its own sake. Looking at the models also deepens understanding of ultimacy in general, by seeing how it plays out in the specifics. Finally, studying the models is a window into how they relate. Specifically, as we go, look for (1) the wide intra-traditional disagreement about how to model what is ultimate (e.g., there are Hindu monisms, panentheisms and dualisms) and related (2) the significant inter-traditional agreement about it (e.g., there are Hindu, Christian, and Daoist panentheisms). The amount of disagreement within a tradition makes talk about "the idea of Brahman" or "the idea of God" ambiguous, and crucially so in some contexts, as indicated in Section 1.2. The amount of agreement between traditions creates strange bedfellows across the landscape of models--strange because the modelers disagree about their religious or philosophical tradition, but bedfellows all the same because they agree about which philosophical categories from Section 1.5 best suit what is ultimate (see Diller 2013a). The existence of such agreement at the very heart of diverse traditions is an important fact. One perplexing issue before we encounter the models is the range of their connection to a lived religious tradition. The "religious models" purport to provide an understanding of reality as it is believed (or held by faith) to be in a particular tradition--as evidenced by their efforts to capture as much as they can of a tradition's key texts, figures, practices, symbols etc.--while "philosophical models" do not do this. For example, the models of the Dao in Section 2.3 seem to be religious models given the regular citations of the Daodejing and the Zhuangzi and liturgical practices in explication and defense of the models. Some philosophical models include Aristotle's Unmoved Mover or divine Nous (Physics Bk. VIII, Metaphysics Bk. XII), Plato's Demiurge in the Timaeus (29-30) or Plotinus' One in the Enneads. None of these modelers turn to traditional sources to explain or defend their views; indeed, it is not even clear what traditional sources would be germane. Perhaps there are intermediate "quasi-religio-philosophical models", e.g., Spinoza 1677 or Hegel 1832, which have some salient connection with a religious tradition, either because of the model's status as a revision of a traditional model or because they share some key features of ultimacy within a religious tradition. The puzzle: are all these modelers modeling the same thing? This question may be just a rehash of the "God of Abraham Isaac and Jacob" vs. "God of the philosophers" issue (see Section 2.2). The answer to it is yes in the vague-categorial sense that both philosophical and religious models are talking about that which is fundamentally real, valuable or liberating, but no in the sense that philosophical models are about what is ultimate per se while the religious models are about what is ultimate as it appears in their tradition. In any case, nota bene: Section 2.3 (Models of God) houses several recent philosophical models because they use the term "God" to name the ultimate they are modeling. Perhaps they are misnamed and should be renamed models of "the ultimate" and housed in an entirely different section. Or perhaps they are rightly named "God" and God is not dead as Nietzsche had it, but is becoming increasingly unhinged from the lived religious traditions. ### 2.1 Models of Brahman Though the term "Hinduism" and its variants was originally a foreign imposition during the British Raj in India, they have become widely used in the public realm to refer to forms of religiosity and spirituality that had their start in the Indus River valley at least since 2000 BCE and have endured with great internal diversification ever since.[24] The idea of Brahman was birthed in the Rig Veda (ca. 1000-1500 BCE), was defined in the Upanishads (ca. 500 BCE), came to full flower in the Bhagavad-Gita (ca. 200 BCE) and the Brahmasutras (400-450 CE), and has been refined in commentaries about them for over the past millennium, first by Adi Shankara (788-820 BCE) and then by many others after him which constitute the Vedanta system of Hindu philosophy (Vedanta = "end of the Vedas").[25] Vedanta has been dominant among the traditional six Hindu philosophical systems for at least the last 500 years.[26] All the Vedanta schools agree on three things about Brahman, or ultimacy (here is this entry's first, very general model of an ultimate). First, Brahman's nature is essentially sat-chid-ananda, "Existence-Consciousness-Bliss", which means that the metaphysical rock bottom of reality is--surprise, given the phenomenal mess--a blissful consciousness. Second, most of the schools take Brahman to be what Jeffery D. Long calls a "theocosm": (1) God and (2) a cosmos, meaning a universe/multiverse of all natural forms.[27] The original Sanskrit in the Vedantic texts is conceptually precise and better than what we have in the God-world relations in Section 1.5 because there is a name for God and the world put together: the theocosm gets called "Brahman", God is "Ishvara" (or "Narayana" or "Krishna" etc.), and the cosmos is "samsara", and the whole thing and each part are eternal.[28] Third, the Vedanta schools all agree that their view about Brahman is associated with (1) an epistemological license in direct experience, either in texts heard by spiritual adepts (sruti) or in proponents' own firsthand spiritual experience (for much more, see Phillips 2019), and (2) a life expression that lives out such experience and the view so deeply that it is hard to know which came first, the life expression or the metaphysical commitments that enable it.[29] What the Vedanta schools disagree about is the kind of link between God and the cosmos in the theocosm. These disagreements spread them out in a range. On one end is the Dvaita (dvaita = "two") Vedanta school, which is dualistic in its ultimate-world relation and not far from the classical monotheistic views (model 2 in this entry). It reads the theocosm to be two really distinct things, viz., God and the cosmos, with no organic link between them. Though there is no creation ex nihilo in Dvaita or any other Vedanta school, Ishvara (God) is the Divine creator and sustainer of the distinct cosmos and ensouls it and resides within it. The ultimate is the two eternally co-existing, like an eternal dweller eternally content to live in an eternal house It built out of something other than Itself. Of the four traditional Hindu yogas, Dvaita is a deeply bhakti-oriented system (bhakti = "devotion"), generally practicing devotion to Vishnu/Krishna. On the other end of the range is the Advaita Vedanta school (advaita = not two) which is well-known in the West but not as dominant in India. It reads the theocosm/Brahman as one/non-dual, and takes God and the cosmos not to be really distinct, i.e., God = the cosmos. The paradigmatic filling out of this view, often and perhaps mistakenly attributed to Shankara, is pantheistic and world-denying.[30] Specifically (here is model 3), such classical Advaitans hold that since reality is one and the cosmos is many, the cosmos cannot be real. It looks as if there is a cosmos filled with many things, but the cosmos is merely an appearance (maya), and taking it to be reality is like taking a rope at dusk to be a snake (Shankara's famous metaphor, see, e.g., Tapasyananda 1990: 34). With the cosmos out of the picture, the theocosm is just the "theo" part; to use the metaphor above, the ultimate is all dweller, no house. That move makes the terms "Brahman" and "God" interchangeable, and on it, God-Brahman is generally read as impersonal. The epistemological license in direct experience for this view is samadhi, a state of spiritual absorption in which all dualities vanish and one experiences just infinite bliss. Since the classical Advaitan view denies the world, Anantanand Rambachan, a contemporary Advaitan, takes its best life expression to be that of the renunciant (sannyasi) who lives out that denial (2006: 69). In the middle range between Dvaita and Advaita are a dozen subtly distinct schools of mainstream Vedanta, often called the "Bhakti schools" given their emphasis on devotion (the distinctions between them make a dozen distinct models here). These schools are all panentheistic and world-affirming. Their best representative is Ramanuja (ca. 1017-1157), the first comprehensive critic of Shankara, who synthesized Advaita with bhakti in a system called Vishishtadvaita (meaning "non-duality of the qualified whole", model 4 for this entry).[31] Like classical Advaitans, Ramanuja takes the theocosm/Brahman to be non-dual because it is all God, but unlike them, he takes the cosmos and the many things within it to be real and distinct from Brahman. How can the ultimate be one but have parts? Ramanuja's panentheistic answer: the cosmic parts form an "inseparable and integral union" (aprthaksiddhi) with Brahman, like the union of body and soul (sarira and sariri): Brahman is the cosmos' soul, and the cosmos is Brahman's body--a body which, to use Ramanuja's illustration from a quote from the Upanishads, > > > "is born in, sustained by, and is dissolved in > Brahman"....[just] as pearls strung on a thread...are > held as a unity without impairing their manifoldness. (Tapasyananda > 1990: 6) > > > So for Ramanuja, Brahman builds a house not out of something else as in Dvaita but out of Brahmanself, and eternally dwells in this eternal Brahman-house which (here is the mark of panentheism) depends on Brahman for its existence but not vice versa, i.e., (Brahman - the cosmos) but not (the cosmos - Brahman). Ramanuja also takes Brahman to be not impersonal but the Divine Person, known under different sacred names, e.g., "Vishnu", "Narayana", "Isvara", etc. The moves to a real cosmos and divine personality have Ramanuja deciding that Brahman's essential nature as sat-chid-ananda is filled out with countless extrinsic "auspicious properties" ("qualities manifested in him in relation to finite beings"), among them some of the classical perfections such as omnipotence, omniscience and immutability, as well as compassion, generosity, lordship, creative power, and splendor (1990: 36-37).The experiential epistemological licenses for Vishishtadvaita are the combination of samadhi (which justifies the non-dual piece) and darsan (visual contact with the divine through the eyes of images, which justifies the qualification to non-duality). Its best life expression is the practice of bhakti yoga, supported by the reality and personality of both devotee and devoted in Ramanuja's metaphysics, and exemplified by the Alvars' passionate devotion to Vishnu from the second to eighth century CE (Tapasyananda 1990: 33). In the midst of this centuries-long dispute between the Dvaitans, Advaitans and Bhakti schools, a key modern figure arose: Ramakrishna (1836-86), sometimes called "the Great Reconciler" because he attempted to integrate all the schools into one pluralistic, non-sectarian approach. His major insight (birthing model 5 for this entry) is that "God is infinite, and the paths to God are infinite", and the schools are among these infinite paths (Maharaj 2018: frontispiece). Ramakrishna came by this view firsthand when, after being in a state of samadhi for six months, which he said was "like reaching the roof of a house by leaving the steps behind",[32] he had a divine command to come down and stay in a state he called "vijnana" (intimate knowledge), during which he could see the roof but also see that "the steps are made of the same material as the roof (brick, lime, brick dust)". In other words, in vijnana he saw that Brahman is both non-dual and dual, and thus began to affirm the "spiritual core" of both schools, and eventually of all the Hindu schools and Christian and Muslim ones, too (Long 2020: 166). He decided they all must be contacting different "aspects" of one and the same reality, and that this reality thus must be deeply, indeed infinitely complex in order to make such diverse experiences possible (Maharaj 2018: Chapter 1, part 3, tenets 1 and 3; see also interpretive principle 4 and K422/G423).[33] Because he affirms the core of all schools, Ramakrishna is hard to classify, but his view is probably best read as panentheistic and world-affirming since he says over and over again that "all is Brahman" and that Brahman "has become everything", i.e., Brahman - the cosmos but not vice versa (Long 2020, especially 163). Contemporary Vedanta is alive and well. To offer just three examples: Long recently merged Ramakrishna's thought with Whitehead's to develop a Hindu process theology that offers an ultimate unity behind the God-world relations that was missing in both Whitehead's and Griffin and Cobb's interpretations (see Section 2.2 and Long 2013, our model 6). Ayon Maharaj has systematized Ramakrishna's thought (no small task!) and put it into conversation with major Western philosophers to advance global philosophical work on the problem of evil, religious pluralism and mystical experience (2018). Finally, Rambachan, a present-day Advaitan, moves Advaita closer to Vishishtadvaita when he asserts that "not-two is not one" (2015): non-dualism is not pantheistic but rather panentheistic because Brahman is the cosmos' material and efficient cause;[34] Brahman intentionally self-multiplies to make the cosmos.[35] The cosmos is thus not maya but rather a finite "celebrative expression of Brahman's fullness" (Rambachan 2006: 79). This reading (model 7 here) opens up Advaitic theology to help heal a variety of human problems from low self-esteem to the caste system because there is real power in re-seeing people as infinite-conscious-bliss, so worthy of respect (Rambachan 2015). Much of the explicit discussion among the schools is debate over metaphysical ultimacy in general and over the Brahman-God-cosmos relation in particular. Still, there is an implicit drumbeat in all the schools that one can find ultimate fulfillment in Brahman, whether that is in a state of samadhi, vijnana or devotion (e.g., the "profound and mutual sharing in the life of God...creates unsurpassed bliss [for the devotee]", Long 2013: 364). So Brahman seems to be soteriologically as well as metaphysically ultimate. What is less clear is whether Brahman is axiologically ultimate, especially in the moral category of being. There are suggestive phrases sprinkled through the texts in the affirmative, but as Alan Watts says: > > > reason and the moral sense rebel at pantheistic monism which must > reduce all things to a flat uniformity and assert that even the most > diabolical things are precisely God, thus destroying all values. > (Hartshorne & Reese 1953 [2000: 325], quoting Watts 1947) > > > Though this sentiment is softened by the panentheistic readings in several of the schools since the world can take the fall, in the end, all the schools affirm that Brahman is everything, in some sense. And if Brahman is everything, and not everything seems to be good, then Brahman seems not to be good, at least not full stop. There is no question that Brahman is still the ultimate in Vedantic schools even if Brahman is not axiologically ultimate; as indicated at the start, metaphysical and soteriological ultimacy are sufficient for ultimacy when nothing else in a system has all three marks. But it is jarring and a real contribution to global thought about ultimacy to consider dropping axiological ultimacy from the trio: what is most deeply real may not be all good, though contact with it may still manage to fulfill us deeply all the same. ### 2.2 Models of God As in Vedanta, the extant models of God disagree about the ultimate-world relation and those disagreements spread them out in a range, with dualisms on one end and monisms on the other, and panentheisms and--for the first time in this entry--merotheisms in between. The idea of God seems to be nothing if not flexible. Even the relatively common view that God is by definition a personal ultimate--an ultimate that is conscious and self-aware--has been on the move for millennia and is hotly debated today. The most venerable model of God that is often read dualistically is known as "perfect being theology", which bears traces of its origin in its name (this is model 8--a general model, species coming). The idea fully grown, as we have it today, defines God as that which is perfect (whether personal or not), where perfection is typically taken to entail being unsurpassable in power, knowledge, and goodness, and several models add being immutable, impassible, a se, eternal, simple and necessary in some sense. Most perfect being theologians take God to have created the universe out of nothing (ex nihilo), and that view can be taken to entail dualism for a variety of reasons. To offer one, as Brian Davies says, "God makes things to be, but not out of anything" (italics his), including not out of Godself, so the cosmos is entirely fresh stuff--a second kind of stuff, distinct from and radically dependent on God (2004: 3).[36] Perfect being theology was birthed during the Hellenistic era from the fusing of the Jewish idea of a single God that acts in history (the theos in "perfect being theology") with the Greek philosophical idea of perfect ultimacy ("perfect being").[37] From the very start, there were conceptual tensions in the combination: how can the God who led us out of Egypt, who hears our prayers and who intervenes in the world as the Jews say (Cohen 1987: 44) also be immutable, impassible and a se as the Greeks say (e.g., Guthrie 1965: 26-39, 272-279; Guthrie 1981: 254-263)? This question is sometimes framed: how can "the God of Abraham, Isaac and Jacob" be the "God of the philosophers?" Even after perfect being theology had passed for centuries from Judaism to Christianity to Islam--with an important handoff in the midst by Anselm who amped up the Greek perfection by taking God to be that than which no greater can be conceived--the great medieval theologians in all three faiths were still hitting up against the tensions and finding ways to tamp them down. For instance, on the issue of anthropomorphic descriptions of God in the Bible and the Quran, both Maimonides and Aquinas read them as negations and said that God "is not a body" (Dorff 2013: 113; Kennedy 2013: 158) and both Ibn Rushd (Averroes) and al-Ghazali parted with "theologians who took all these descriptions literally" because "beings that have bodily form...have characteristics incompatible with a perfect being" (Hasan 2013: 142). The tensions continued into the modern era and are still felt in our time. Perhaps as early as 1644, perfect being theology split into two camps over them (see Davies 2004: chapter 1, and Page 2019). Both camps take God to be absolutely perfect, but disagree over what it takes to be perfect: "classical theists" deny or weaken God's personhood to save the Greek perfections such as impassibility, immutability and simplicity, while "theistic personalists" (a species of "neoclassical theists") conversely deny or weaken the Greek perfections to save God's personhood. "Open theists" (model 9 in this entry), for example, are theistic personalists: they call for new readings of, e.g., omnipotence and omniscience and drop immutability and impassibility to comport with God's desire "to be in an ongoing, dynamic relationship with us" (Basinger 2013: 264-268, see also, e.g., Clark 1992; Pinnock et al. 1994; Sanders 1998).[38] Other neoclassical theists aim merely to resolve inconsistencies among the perfections, as in Nagasawa's Maximal God Theism (2008, 2017; model 10). In addition to its old challenges, perfect being theology also hit new ones in the modern era from advances in science. When it met Newtonian mechanics (and more) during the Enlightenment, the combination spawned "deism", the idea that God set the initial conditions of the universe and then left it to play out on its own (model 11). Deism is a dualism because it assumes God can leave the world behind and thus is neither "in" it as in panentheism nor identical with it as in pantheism. Picture all these theistic dualisms as close to Dvaita Vedanta's image of the eternal builder building a house out of something different from itself and dwelling in it as it pleases, but make the house not necessarily eternal (it may have had a start and may end), and for classical theism, give the builder all the perfections; for neoclassical theisms, give it a few less and perhaps have it throw better parties in the house; and for deism, have the builder abandon the house altogether once it is built and leave it to its own devices, like an "absentee landlord" (Mitchell 2008: 169). On the other end of the spectrum from these varieties of theistic dualism, we find pantheism, the species of monism that takes the One to be God (a general model, 13). All monisms face a problem of unity: how are the many things in the world integrated enough to call them One? But pantheisms face an additional problem of divinity: even if all is truly One, does the One have what it takes to be God? Here we will focus on two contemporary pantheisms, both in Buckareff & Nagasawa (2016): what we might call a "one-thing" pantheism by Peter Forrest (2016) (a specific pantheism, model 14), where the One is a count noun (as in "a walrus is sleeping over there"), so the cosmos as a whole is One thing, and a "one-stuff" pantheism by Karl Pfeifer (2016) where the One is a mass term (as in "that little lamb is made of butter"), so everything in the cosmos is made of the same kind of One-stuff (model 15). Unlike the Advaita pantheists who take the universe to be a mere appearance, Forrest and Pfeifer definitely take the universe to exist. So for them (and pantheists like them), the One will have to be identical to the universe, and the work is to show how the universe can be identical to God. In other words, this is not all builder no house as in Advaita; the builder is the house, and the builder-house is special enough to call it "God". Forrest's main move to effect this is to take the universe to be a conscious self, by way of a "properly anthropocentric" non-reductive physicalism: just as our brain processes correlate with our mental states, so also the universe's physical processes correlate with universal mental states, which on the model involve a unity of consciousness and thus a sense of self. Forrest has a strong reply to the problem of unity here: the One is an integrated Self precisely because of what emerges from the processes of the many. But is the Self conscious in high enough ways to meet the problem of divinity, to count as God? Though Forrest does not argue like this, the resources for nascent perfections are here, such as omnipotence (the Self has all the power in the universe), omniscience (It could know the entire universe by biofeedback), good will for all (since to hurt any part of the universe is to hurt Itself) etc.--enough in theory to count as God in the classical or at least neoclassical sense. For Pfeifer's view, instead of picturing the universe as a person, picture it as an "intentional field", like an electromagnetic field except spread physical dispositional states across space instead of magnetic forces and electrons. Because those physical dispositional states have the same extension as intentional states,[39] intentional states are effectively spread everywhere too. That spreading means there is a kind of "panintentionalism", and if the intentions are divine enough, then a panGodism, i.e., pantheism. So if Forrest is right, the universe is God itself, and if Pfeifer is right, the universe is made of God-stuff, this field of divine intentional states--both strong thoughts. How plausible, though? On the plus side, Forrest's view is an instance of "cosmopsychism" ("the cosmos as a whole is phenomenal", i.e., the Cosmos as a whole has conscious states) and Pfeifer's an instance of "panpsychism" ("everything in the cosmos is phenomenal", every particular in the cosmos is conscious), both of which are receiving growing attention in philosophy of mind since, e.g., cosmopsychism may solve physicalism's problem of strong emergence and panpsychism's combination problem at once (Nagasawa 2019, for more on these views and their link with Hinduism and Buddhism see, e.g., Shani 2015, Albahari 2019, Mathews 2019). However, even if either the cosmo- or panpsychic aspect of Forrest's or Pfeifer's views turns out to be true, the divine part seems doubtful for a reason Pfeifer enunciates: the kind of intentionality the universe would have within it (on Pfeifer's view) or that would supervene on it (in Forrest's view) seems likely to be at best the consciousness of an animal, or a comatose or schizoid human, etc.--not even close to the kind of consciousness that would make it count as God (see Pfeifer's footnote on 2016: 49). In the middle, between the theistic dualisms and the pantheisms, stand the merotheisms and the panentheisms (two general kinds of models, 16 and 17, again, species coming). As indicated above, the merotheisms are rare, the "odd bird" idea that God is in the world, but the world goes beyond God. Though the term "merotheism" was coined only recently by Paul Draper for his own view (2019: 160), merotheisms have been around well before, for example, in divine emergence theories such as Samuel Alexander's (1920, a specific model 18) on which the world is metaphysically ultimate and God arises in it.[40] So on the metaphor, the house comes first, then God grows within it. Alexander, for instance, thinks the rock-bottom reality is space-time, and that when "patterns" or "groupings" of it become complex enough, matter comes to evolve in it, then life, then mind and then deity (257). The universe now is at mind, so we are waiting for deity to emerge, not from small "groupings" of things as with the other levels, but from the universe as a whole. Because things can think only about the things below themselves in the hierarchy, we cannot know what deity will be like when it comes (Thomas 2016: 258)--a nice way to explain why God is ineffable and unknowable, albeit one that gives no (other) content to say why Alexander's "deity" should count as God. In contrast to the emergence merotheisms, Draper (2019) offers a sheer "meros" one (model 19), in which nature, instead of growing God, always has God as one proper part. Specifically, nature is composed of two parts which are both metaphysically ultimate: fundamental matter, and fundamental mind. So what there is not only all the familiar material stuff but also one and only one immaterial mind, i.e., God--"the single subject of all phenomenally conscious experiences", located in and coextensive with space (2019: 163). Assuming that minds are the source of value, this one mind is the fundamental "source of all the value there is", and hence is axiologically ultimate (2019: 163). Interestingly, just like prisms immersed in sunlight naturally diffract the electromagnetic spectrum (2019: 167, originally from William James), our brains, which are with everything else immersed in this omnipresent universal mind, naturally diffract what we might think of as the divine spectrum--displaying aspects of the universal consciousness by generating one of its "multiple streams", "making use" of it for our own ends, tuning in to it in mystical experiences, etc. (2019: 163, 170). So brains don't produce consciousness--they tap into it--and God doesn't make the universe or emerge in it--God is the mental part of it that gives it value, and gives us a hope for a form of life after death because the consciousness that runs through our brains and that we mistakenly call our own continues to live on after the brain dies as the aspect of the enduring universal consciousness it always was. This hope secures some soteriological ultimacy: though it makes sense to mourn our deaths, we should "not despair" (2019: 170) since, if we ally ourselves with our consciousnesses, we are even after death still what we always were, an aspect of the mental fundamental reality, as Shankara and Ramakrishna and others would tell us. Panentheistic models of God (on which the world is in God but God goes beyond the world) have been popular for millennia, to the point that John Cooper calls them "the other God of the Philosophers" in the title of his book on panentheism (Cooper 2006, a general model, #20 in this entry). There are literally too many panentheistic models of God to count, from a star-studded list of historical thinkers including Plato, Pseudo-Dionysius, Ibn Arabi, Meister Eckhart, Nicholas of Cusa, Kant, Hegel, Peirce and more, with a resurgence in the last decade owing at least in part to Yujin Nagasawa and Andrei Buckareff's Pantheism and Panentheism Project (2017-19 [see Other Internet Resources]). Though some complain that the "in" in panentheism is so ambiguous it is not obviously a single view (see Gasser 2019), Chad Meister suggests that the recent appeal of panentheism is a direct result of (1) some of the neoclassical revisions to the idea of God (more immanent, more passible, etc.) which can be explained by the world's being in God, as well as (2) the advent of emergentist theories in science which make room not only for the emergence merotheisms sketched above (on which God emerges from the world) but also for their converse, the emergence panentheisms (on which the world emerges from God), among other reasons (Meister 2017: section 4). Hartshorne's process theology is a great example of the first impulse Meister identifies (so it is a specific panentheism, model 21). Hartshorne's process view begins with Whitehead's metaphysics from Process and Reality--with the idea that the world is dynamic, not static, and indeed that the fundamental units are events, "actual occasions", not substances, which > > > do not endure through a tiny bit of time unchanged but [take] a tiny > bit of time to become...concrete ("concresence", Cobb > & Griffin 1976: 15) > > > and which are thus dynamic all the way down. Hartshorne then places this dynamic world of events in God, by taking a page from Ramanuja's book and saying that all of it--this "totality of individuals as a physical or spatial whole is God's body, the Soul of which is God" (Hartshorne 1984: 94, quoted in Meister 2017: section 5, italics added)--a move which cements his view as a panentheism, since the world is literally in God, but God, as Soul of the world, goes beyond the world. The practical pay dirt of the view is that, in the same way we feel our bodies, so also God as the Soul of the world feels the world--feels every last "drop of experience" as Whitehead says, every last bit of change happening in every last actual occasion. Moreover, just as we respond to what we feel in our bodies, so also God responds to each felt occasion, and in that instant does two things: runs through a catalog of all possible next occasions, next moves as it were, and then "lures the world forward" with suggestions for the best next moves to actualize in the next occasion. The world can "listen" or not to these suggestions as the next occasion concresces, and then God will regroup again, moment after moment after moment. This is the dynamic process of perfecting--from the world to God back to the world again--which gives process theology its name, and makes it a kind of "becoming-perfect-being" theology. John Bishop and Ken Perszyk (2016, 2017) propose a panentheism they call a "euteleological conception of divinity" (model 22 here), on which (1) divinity is the property or activity of being the supreme good ("eu" in "euteleological") and (2) realizing this property or activity is the point ("telos" or final cause) of the universe. In addition--inspired by an unusual kind of efficient causation called "axiarchism" on which final causes can function as efficient ones, an idea visible at least since Plato and having something of a revival in the last couple decades--Bishop and Perszyk (3) take concrete realizations of the supreme good to be the efficient cause of the universe.[41] Thus, on (2) and (3), these realizations of the good are both the efficient and final cause of the universe, both alpha and omega. This model is, as its authors say, "prone to be met with incomprehension or blank incredulity" (2017: 613): how can effects in the universe be the cause the universe, and thus their own causes? Though Bishop and Perszyk do not answer this question in 2016 or 2017, they do point out the eerie precedent in the Christian tradition, the model's home context, for efficient causes to double as final ones: Jesus is both the source and offspring of David, both "root and flower;" Mary "gives birth to her own creator;" the Divine word is both "without which was not anything made that was made" and "born late in time" (2017: 614). They also identify the supreme good in Christianity as perfect love, take Jesus to have instantiated it in his person and time and again in relationships, and take us to do so too when we "love one another as he has loved us" (2017: 613). Note such concrete occasions of love are per (1) literally divinity dotting (and hopefully eventually overrunning) the universe, and that they deserve to count as divinity because they are triply ultimate: metaphysically since they are the efficient and final cause of the universe; axiologically since they are the best of things, and soteriologically since they are deeply fulfilling (to quote the Beatles, "all you need is love"). Whether or not the axiarchism at its heart is a strike against euteleological theism, an enormous point in its favor is how profoundly it addresses the problem of evil: it makes God the force in nature that defuses evil instead of intending it.[42] This section would be incomplete without at least mentioning Tillich's "ground of being theology" in closing (model 23). His view is not filed into the range of God-world relations above because it is famously difficult to categorize: Christopher Demuth Rodkey (2013) says Tillich has been read variously as a panentheist, deist (i.e., dualist), and pantheist, and that it is in fact best to characterize him as none of the above but rather as an "ecstatic naturalist", where the Power of Being delivers the naturalism (since this Power is "the power in every thing that has power") and the Depth of Being delivers the ecstasy (persons experience this Power of Being ecstatically, as holy). This interpretation tracks Tillich's method of correlative theology in Systematic Theology I and II: ecstasy is a "state of mind" which is "an exact correlate" to the "state of reality" of the power of being which animates and transcends the finite world (see, e.g., Tillich 1957b: 13). So for Tillich, God is the power or energy that animates the world which, when truly encountered, provokes ecstatic response. This view is spare enough that it is not obvious how someone might work up an "ultimate concern" about God, another of Tillich's central ideas mentioned at the start of this entry (1957a, e.g., 10-11). Tillich will have to hypothesize that the ecstasy provoked is, for believers, strong enough to rouse such a concern. These are, then, several models of God, sorted mainly by how they see the relationship between God and the world. Is the God that is modeled in each of these ways metaphysically, axiologically and soteriologically ultimate, in Schellenberg's terms? Interestingly, the answers differ dramatically for each model. To offer just two examples, on classical theism we get a yes, yes, yes: God as single-handed origin of the universe, making everything out of nothing, is metaphysically the fundamental fact; and, in Anselm's hands, God as the greatest not only actual but also possible being in every category of being, is as axiologically ultimate as anything can be; and in Aquinas' idea, God as our very telos, the point of our being, is soteriologically ultimate as well. In contrast, God on Alexander's view gets a no, maybe, maybe. Alexander's deity is not metaphysically the most fundamental fact in any of the ways collected in the models seen so far: it is neither the efficient cause of the universe (as in the dualisms), nor its material cause (as in the pantheisms and some panentheisms) nor its final cause (as in Bishop and Perszyk).[43] Alexander also cannot say if deity will be axiologically or soteriologically ultimate when it arrives, since deity is by definition unknown for him. Thus, God as modeled in some ways is ultimate and in others is not. ### 2.3 Models of the Dao The idea of the Dao (Way, Path, Guide) emerged during the Warring States period in China (fifth to second centuries BCE), when the reigning idea of Tian (Heaven) as a kind of personal god or God started to shatter along with the rest of the imperial structures of the Zhou Dynasty. Chinese thinkers faced their version of the problem of evil: "Why is Tian letting this chaos persist?" and added "Where is the dao to harmony?" (Perkins 2019; Miller 2003: 37). An extended debate arose among different schools of thought arguing for different answers (Zurn 2018: 300ff), including two schools that have endured: the early Ru (Confucian) thinkers who said the dao could be brought back into the human world by reestablishing right social relationships and customs, and the early or proto-Daoists[44] who found a new focus in the dao in the impersonal, consistent patterns of the non-human natural world. The Daodejing (ca. sixth to fourth centuries BCE, hereafter "DDJ") is the earliest Daoist text that reads these natural patterns as evidence of a single force or principle of all that there is--as a single metaphysical ultimate--and "tentatively", as Perkins (2019) says well, names this ultimate "the dao" or in some translations "the Dao" or "Dao". Though this entry will focus mainly on the Daoist tradition and use the word "Dao" (hereafter not italicized) to refer to it, the res in question runs under other important names and concepts in both the Daoist and Ru traditions, including Taiji (Great Ultimate or Grand One), Xuan Tian (Dark Heaven), Zhen (Truth or noumenal Reality) and conjoined with Tian as Tiandao in Ruism. Gradually, the early Daoist thinkers took the Dao to have multiple functional roles--metaphysically, as the cosmos' origin, its pattern or structure (ti), its functioning (yong); and soteriologically as a guide through the cosmos for humans, as Robin Wang says (DDJ ch. 25, Wang 2012: 47). Combining the Dao's role as the origin of all things with its undeniable unitariness threw Daoist thinkers into the question of how the One became Many, and thus into a focus on cosmogony. The Daoist cosmogonists generally agreed, and agree now, on at least six things about the Dao (the last general model this entry will showcase)--though there is substantial diversity in interpretations of each which help constitute various thinkers' models of the Dao. First, in a seeming nod to the consistent patterns of the universe that encouraged postulation of the Dao in the first place, the Dao is taken to be immanent in everything. As the Zhuangzi says, > > > There's no place [the Dao] doesn't exist....It is in > the panic grass....in the tiles and shards...in the piss and > shit!..."Complete", "universal", > "all-inclusive"--all point to a single reality. > (Zhuangzi, sec. 22, Watson translation) > > > Second, because it is capable of singlehandedly originating everything, the Dao is taken to be necessarily ziran, meaning "self-so" or "spontaneous", which is read as entailing something like the kind of necessity and aseity of being causa sui in the Thomist tradition (Perkins 2019). Wang explains the entailment in her explication of a famous passage ("Human beings follow earth, earth follows heaven, heaven follows dao, and dao follows ziran"): the Dao's following ziran arrests the regress because "following" spontaneity is the opposite of following since spontaneity is making it up yourself on the fly (DDJ, ch. 25; Wang 2012: 51). What is the nature of a ziran generator of all things, then? Zhuangzi answers in his inimitable way: "what things things is not itself a thing" (ch. 11, see Schipper 1982 [1993: 115]). In other words, the third commonly held claim is that the Dao is no thing, nothing, nonbeing (wu). Bin Song (2018) helpfully disambiguates several readings of nonbeing, including as (a) sheer nothingness, a great vacuum "before" time and things; or (b) abstract forms not yet made concrete, e.g., Zhu-Xi's "pattern-principles" or Wang's "patterns and processes of interrelatedness" (2018: 48) or, instead of nothingness or abstractness, (c) concrete no-thingness, i.e., a totally undetermined whole of being, stuff without form. Song and Poul Andersen favor option (c), translating a key phrase in DDJ 21 that describes the Dao as a "complete blend" and as having "murky indistinctness", respectively (Song 2018: 224-230, Andersen 2019: 131-132). Another line in that chapter also tells against the Dao's being sheer nothingness: > > > yet within it is a substance, within it is an essence, quite genuine, > within it, something that can be tested. (DDJ, Lau > translation) > > > On any of these readings of nonbeing, it is clear why the Dao is taken to be impersonal: the Dao is not only not anthropomorphic; it is not even thingmorphic. It is also clear why it is taken to be ineffable: it is not just because its being is beyond us; it is also because it is not a being at all, and most uses of words (to talk like Zhuangzi) thing it. So we find Daoist texts using the tricks of the ineffability trade to talk about the Dao, including famously, e.g., a use of the via negativa in the opening line of the DDJ: "the way that can be spoken of is not the constant way..." Also, in an expression that is perhaps less about the Dao's ineffability and more about the futility of finding it intellectually, there is Zhuangzi's dynamic semper negativa of "continuous self-negation" or "unsaying", visible also in the Buddhist tradition and in Tillich millennia after and oceans away: > > > There is being, there is no-being, there is not yet beginning to be > no-being, there is not yet beginning to be not yet beginning to be > no-being. (Zhuangzi, sec. 2; on Tillich, Rodkey 2013: > 491-493) > > > The fourth and fifth widely held views of the Dao are both about how nonbeing generates being, namely with wu wei ("non-action"), and in stages. Andersen describes wu wei more fully: "the Way does not cause [things] to come into being but provides a gap that allows things to emerge" (2019: 131). To reveal wu wei, Daoist literature frequently uses images of the female and infants, e.g., twice over in DDJ 10, where wu wei is likened first to "keeping the role of the female" who with no apparent (anyway) action naturally nurtures the fetus and then, to use Andersen's words, provides a gap to allow it to emerge in birth; and second to "being as supple as a babe" who is the epitome of the wu wei ruler since a baby does not do anything but gets everyone else to act to please it (Zurn in conversation; see also Erkes and Ho-Shang-Kung1945: 128). The Dao is also taken to generate in stages, and actually in four of them (DDJ 40, 42; Perkins 2019; Robson 2015: 1483; Wang 2012: 48; etc.). There are various readings of the sequence, but one prominent view takes 0 as nonbeing, the Dao; 1 as unity, Being; 2 as duality, yinyang;[45] and 3 as multiplicity, namely heaven, earth and human beings. In a prescient recognition of how natural (heaven and earth) and social (human) constructions combine to make reality, 3 burgeons into 4, i.e., the 10,000 or myriad things that comprise the universe. Crucially, 4 returns to 0--to use the feminine imagery, returns to the womb where everything is possible and everything develops--and then the sequence repeats (Pregadio 2016 [2020: sec. 4], Wang 2012, 51 citing the Huainanzi). This world-to-Dao-and-back cycle is reminiscent of the occasion-to-God-and-back cycle in process theology, though on a grand vs. momentary scale.[46] Interestingly, in general, Daoists read the sequence as strictly cyclical (so 4 returns to 0, i.e., 0, 1, 2, 3, 4, 0, 1, ...) while Ruists read it as an "endless advance to novelty" so that we never step into the same cycle twice (i.e., 0, 1, ..., 0', 1', ..., 0'', 1'', ... etc.) (Song 2018: 230-232). Moreover, there is an open question whether the sequence is temporal or ontological (Song 2018: 225-226) which is sometimes crystallized into a debate about whether 0 is temporally (and thus ontologically) prior to 1 or merely ontologically so. If the move from 0 to 1 is temporal, then 0 happens before 1 and they are really distinct; as Andersen argues: > > > the One is a product of the Way, not the Way itself... [because] > identity and stability as a thing in the world depends on being one. > (Andersen 2019: 180) > > > If, on the other hand, 0 is only ontologically prior to 1, then nonbeing and being are always co-existing as two eternal aspects of what there is, with being still depending for its existence on non-being like a burning eternal candle depends for its existence on its eternal flame. The ontological-only reading seems licensed by DDJ 1 which talks about "the nameless [who] was the beginning of heaven and earth" and "the named [who] was the mother of the myriad creatures....these two are the same but diverge in name as they issue forth" (italics mine, for more see Pregadio 2016 [2020: sec. 4.1])--assuming that the passage implies that they diverge in name "only". The dispute about how nonbeing and being relate is vexing enough that it is a relief, actually, when DDJ 81 throws up its hands and says: "their coexistence is a mystery"! Grasping all this together--that the Dao is the origin of all, immanent in all, ziran, essentially nonbeing, generating forms of being by wu wei, generating stage by stage until we reach the roiling boil of being in the myriad things--we are able to catch the last and perhaps deepest thought of all about the Dao: that it is generative by nature, or as Neville says: "nonbeing is simply fecund from the perspective of being" (2008: 3). This idea hails back to the I Ching, which takes "the foundation of the changes" to be sheng sheng, literally meaning "life life" or "generating generating", sometimes glossed with the phrase shengsheng buxi "generating generating never ceasing!"--held as the highest metaphysical principle by Ruists (especially neo-Ruists) and not far from Spinoza's natura naturans (Gao Heng 1998: 388 cited in Perkins 2019). Thus, at the bottom of it all, there is just endless, unformed, spontaneous life stuff, which is generating increasingly formed spontaneous life stuff which, because it is shot through with the Dao, in turn generates even more, even further formed, spontaneous life stuff, and we are off and running to the 10,000 things, until, as yang wanes to yin, life life goes back to 0 (death death?), and all is still until it waxes to yang again. In spite of the cycle that always sends it back to yin, it is fitting to call the Dao "life life" because it always waxes to life again; the life force is irrepressible, no matter how many times it temporarily dies. Though the Daoist literature does not use these philosophical distinctions much, and though occasionally it is read there as a monism in passing, the standard model of the Dao just recounted is best understood as an impersonal panentheism--as are the classical theisms and Bishop and Perszyk's model of God (Section 2.2). The Dao-world relation has the asymmetry that defines panentheism: all the forms of being at 1-4 (or at least 2-4, if Being is read as identical to the Dao) depend for their existence on the Dao, but the Dao does not depend on them, or anything, for its existence.[47] In addition, if we take the theme of immanence in a full-throated way so that really there is nowhere down the line of the stages that the Dao is not, then the Dao is both the efficient and material cause of the universe, as we saw in the Bhakti Vedanta views. So the view of the Dao traced here is close to the idea of Brahman as the builder of a house out of Brahmanself, who eternally dwells in this eternal Brahman-house which depends on Brahman for its existence but not vice versa. But "builder" is too intentional for the Dao, and, at least on the temporal reading of the sequence, there is no eternal house and thus no eternal dwelling in it. So try this: think of the Dao as an eternal seed for a house. The seed sprouts naturally in stages into a house filled with 10,000 things which depends on the seed but not vice versa, until the house dies back into the seed, which then lies dormant, pregnant with being, until it sprouts into being again, and so on, eternally. After all that has been said, the Dao is clearly metaphysically ultimate in Schellenberg's sense: it is "the most fundamental fact about the nature of things" (2016: 168). The Dao is soteriologically ultimate as well, but, as with Brahman, it is not clearly axiologically so. Regarding axiology, first, if we understand axiology as greatness along all the categories of being, we can see immediately that the Dao, at least understood as 0, has greatness along no category of being since it is not being at all (for more see Kohn 2001: 18).[48] Moreover, if we narrow the idea of axiology just to the moral category of being, the Dao is still not axiologically ultimate. In multiple texts, the Dao is taken to be neither good nor bad; it is taken to be what is. Famous among them is DDJ, ch. 5 ("Heaven and earth are not kind, they treat the ten thousand beings as straw dogs", see also chs. 18, 62). Interpreters take these passages to mean that the Dao is either amoral--as nonbeing, not the kind of thing that has moral interests in the first place; or "value-contrarian" (Hansen 2020); or, to add a thought, perhaps "ananthropocentric", having moral concerns that are not human-centered (see Mulgan 2015 and 2017). Wang Bi's commentary on the straw dogs passage suggests that these amoral or anti-moral moments spring from the Dao's and the sage's wu wei: > > > The one who is kind necessarily creates and erects, impacts and > transforms. He shows mercy and acts. If someone creates and erects, > impacts and transforms things, then these things lose their true > reality. If he shows mercy and acts, then these things are not > entirely there. (quoted in Andersen 2019: 130) > > > In other words, acting wu wei actually requires not being kind in the usual sense. But if Wang Bi is right, maybe there is an axiology after all to treating the myriad things like straw dogs: being kind in the traditional sense may not be being kind deeply since it destroys a thing's power to be itself. Regarding soteriology, it is agreed that a--or even the--central goal of Daoist practices such as inner alchemy, t'ai chi, etc. is to return to the Dao (fandao, huandao, DDJ 16 and 40; Andersen 2019: 126), and specifically to return to nonbeing, which is the Dao at its most creative, powerful and sublime, on the crest of becoming being (Song 2018: 234-5). If we can return to 0, we embody this power and sublimity in human form and, as Andersen says, also do our part to return the cosmos to the start for a new beginning (2019: 123). There are specific rituals Daoists do in communal contexts to return. In one of the important rites in Daoist liturgy called bugang ("walking along the guideline"), a Daoist high priest walks through the ritual space, with the audience making their own occasional movements too, to embody a complete motion of return with a successful arrival back to 0 by the end of the rite, when > > > the forces that animated the universe at the beginning of time may > once again be channeled into the community on behalf of which the > ritual is performed. (2019: 118-123) > > > Practitioners outside of ritual contexts also try to return by inwardly cultivating the skill of acting as the Dao does when it generates being: with wu wei. Miller reminds us that wu wei is not some loose form of letting go, but is rather a specific "spiritual technology" of intervening very gently at the right time, in the right place--as Neville says, with "a subtle infinitesimal dose" when there is a rare "opening for spontaneity [in the otherwise hard-to-beat] inertial forces of the Dao" (Miller 2003: 140; Neville 2008: 47-51). Andersen's take on these efforts is haunting: > > > An accomplished Daoist resides in the gap between being and nonbeing. > The fundamental truth of Daoism is in this gap, in the Way and its > manifestation as true and real. (2019: 130) > > > This thought suggests an answer to the puzzle that surfaced about Brahman--about why contact with a metaphysical ultimate that is not axiologically so might still be fulfilling to us. At least for those seeking awareness of Brahman and harmony with the Dao, our whole desire looks like it is to be in the presence of what is true and real (Zhen), whether what is true and real is bad or good or neither or both. We are fans of unvarnished reality. ## 3. Responses to the Diversity of Models of What is Ultimate After surveying these many models of Brahman, God, and the Dao, and recognizing that they are just a small sample of the range of options for modeling Brahman, God, and the Dao, which are in turn a small sample of the range of ultimates that could be modeled, one may wonder: What should one do with all this information? People respond in various ways after grasping the diversity of the models. Some abandon the models altogether, either exhausted by their complexity (embodied in Watt's wonderful phrase "the which than which there is no whicher", 1972: 110), or convinced by their number and inconsistency that some models logically must be making a mistake, and it will be very hard to tell which ones. In other words, one response to the diversity is to decide more deeply that the nature of what is ultimate is indeed beyond us, if there is anything ultimate at all, so it is not worth thinking about it. In sharp contrast, others actually embrace the diversity of models as part of the path to understanding what is ultimate. The comparative theologians, for example, study novel models in order to carry fresh insights from them back to their own tradition and re-see their own models more deeply (see Clooney 2010 for an introduction and, e.g., Feldmeier 2019 for the method applied to Buddhist and Christian models of the ultimate). An emerging movement, Theology Without Walls (TWW), draws on the models to understand the nature of a globally shared ultimate, one to which all the models may be intending to refer, reading the body of models as data and their number and inconsistencies as an interpretive challenge instead of a deal breaker (see, e.g., Martin 2020). Ramakrishna offers one such interpretation in the TWW spirit: he decides there is no need to choose between the models because each is a finite start on the "infinite paths to an infinite reality" (see Section 2.1)--each is news about an ultimate whose nature is so full that we actually need all the models to help us see it. Both TWW and Ramakrishna will have to explain how it is possible for many or all the models to deliver news of what is ultimate given their inconsistency, e.g., by relying on perspectivalism, a phenomenal/noumenal distinction, the models' incommensurability, etc. (see Ruhmkorff 2013 for a survey of options). Others fall somewhere between abandoning and embracing the plurality of models by recommending that we hold the models loosely somehow, that we attenuate our commitment to them. Kierkegaard for instance tells us to move our focus from the content of our model to our orientation to what we are attempting to model: it is better to pray to a false God truly than to a true God falsely (paraphrase of 1846, Part Two, Chapter II).[49] Similarly, J.R. Hustwit cautions us to "balance engagement with non-attachment" to models to avoid ego-reinforcement and more (2013: 1003-1007)--not far perhaps from Zhuangzi's and Tillich's semper negativa of holding and letting go model after model, a view which converts the pile of models into grist for the mill of a spiritual practice. For his part, Schellenberg suggests that instead of having faith in a specific model of, e.g., Brahman, God or the Dao, we do better to have it in the more general thing (res) that underlies them all--the axiological, soteriological, and metaphysical ultimate which has been the organizing principle of this entry. One advantage of reading ultimacy in Schellenberg's way is that the general ultimate is more likely to exist than any of the particular ultimates it covers, since it exists if any of them do.[50] A second advantage is that Schellenberg's general ultimate is by design the core, the overlooked "heart" of the many models--the same thing that the particular models it covers are about, just at a higher level of description. So faith in Schellenberg's ultimate permits a "faith without details" (2009: ch. 2) in many of the world's religions and philosophies at once. For those who, after this long journey through the landscape of models and now these responses to them, still hope to discover which model is philosophically the best of them all, know that Wildman (2017) has a plan for "think[ing] our way through the morass" (2017: viii). In brief: 1. identify the models worth your time; 2. place them in a "reverent competition" that scores them on "comparative criteria" (2017: viii-ix, ff.), then 3. adopt the winner, at least provisionally, since the entire inquiry is "fallibilist" (2017: 161 and elsewhere). Wildman demonstrates his plan by following it himself (2017). Interestingly, he frames his options for step 1 in terms of "entire systems of thought" comprised of combinations of models of what is ultimate (which he calls "U types") plus "ontological cosmologies" ("C types")--an idea which may really do a better job of identifying our choices than the models per se do, given their fuller capture of an entire worldview. His U types include agential models on which the ultimate is personal, ground of being models on which it is impersonal, and "subordinate deity" models such as process theology on which it is "disjoint:" one or more personal deities operate in an impersonal ultimacy (2017: 13, 165, 182). His C types include supernaturalism, which involves disembodied agency; naturalism, which does not; and monism. Combining the U and C types produces nine U + C views, and to live out step 1 of his plan, he chooses his top three to place in competition: supernaturalist theistic personalism (God as a personal perfect being), naturalist ground of being (think, e.g., the panentheistic impersonal Dao), and Whitehead's or Hartshorne's process theism. For step (2), he then subjects these three views to his comparative criteria, which include coherence, ability to handle the problems of evil and the One and the Many, fit with the sciences, and most importantly non-anthropomorphism, his main criterion since he takes anthropomorphism to result from misapplying human cognitive structures that were naturally selected for mere survival purposes to ideas of ultimacy (2017: 217). When Wildman ran his competitors against these criteria, the naturalist ground of being system won. But it is obviously up to each of us interested in such a project to run our own competitions on the models we take to be worth our time, with comparative criteria we think make a model truth-conducive, in order to light on the most philosophically satisfying model of what is ultimate that we can. That model would be the one to then subject to the best arguments for and against the existence of God and other ultimates, to discover in a fully researched and now clarified way, whether there is anything ultimate.
umar-khayyam	## 1. The Formative Period Abu'l Fath 'Umar ibn Ibrahim Khayyam, commonly known as Umar Khayyam, is the best known Iranian poet-scientist in the West. He was born in the district of Shadyakh of Nayshabur (originally "Nayshapur") in the province of Khorasan sometime around 439 AH/1048 CE,[1] and died there between 515 and 520 AH/1124 and 1129 CE.[2] The word "Khayyam," means "tent maker," and thus, it is likely that his father Ibrahim or forefathers were tent makers. Khayyam is said to have been quiet, reserved, and humble. His reluctance to accept students drew criticism from opponents, who claimed that he was impatient, bad tempered, and uninterested in sharing his knowledge. Given the radical nature of his views in the Ruba'iyyat, he may merely have wished to remain intellectually inconspicuous. > > The secrets which my book of love has bred, > > Cannot be told for fear of loss of head; > > Since none is fit to learn, or cares to know, > > 'Tis better all my thoughts remain unsaid. > > (Ruba'iyyat, Tirtha 1941, 266) > > > Khayyam's reference to Ibn Sina as "his teacher" has led some to speculate that he actually studied with Ibn Sina. Although this is incorrect, several traditional biographers indicate that Umar Khayyam may have studied with Bahmanyar, an outstanding student of Ibn Sina.[3] Following a number of journeys to Herat, Ray, and Isfahan (the latter being the capital of the Seljuqs) in search of libraries and in pursuit of astronomical calculations, Khayyam's declining health caused him to return to Nayshabur, where he died in the district of Shadyakh. ## 2. The Philosophical Works and Thoughts of Umar Khayyam Khayyam wrote little, but his works--some fourteen treatises identified to date--were remarkable. They can be categorized primarily in three genres: mathematics, philosophy, and poetry. His philosophical works which have been edited and published recently are (we have not translated risalah (treatise) in the titles): 1. [Lucid Discourse] "A Translation of Ibn Sina's (Avicenna's) Lucid Discourse" (Khutbah al-gharra'). 2. [On Being and Obligation] "On Being and Obligation" (Risalah fi'l-kawn wa'l-taklif). 3. [The Necessity of Contrariety] "The Response to Three Problems: The Necessity of Contrariety in the World, Predeterminism and Persistence" (Al-jawab 'an thalath masa'il: Darurat al-tadad fi'l-'alam wa'l-jabr wa'l-baqa'). 4. [The Light of the Intellect] "The Light of the Intellect on the Subject of Universal Knowledge" (Risalah al-diya' al-'aqli fi mawdu' al-'ilm al-kulli). 5. [Principles of Existence] "On the Knowledge of the Universal Principles of Existence" (Risalah dar 'ilm kulliyat-i wujud). 6. [On Existence] "On Existence" (Risalah fi'l-wujud). 7. [A Response] "A Response to Three Problems" (Risalah jawaban li thalath masa'il). (3 and 7 are distinct works.) Except the first work mentioned above which is a free translation and commentary on a discourse by Ibn Sina, the other six philosophical treatises represent Khayyam's own independent philosophical views. It is noteworthy that Khayyam's philosophical treatises were written in the Peripatetic tradition at a time when philosophy in general and rationalism in particular were under attack by orthodox Muslim jurists--so much that Khayyam had to defend himself against the charge of "being a philosopher." > > "A philosopher I am," my enemies falsely say, > > But God knows I am not what they say; > > While in this sorrow-laden nook, I reside > > Need to know who I am, and why Here stay > > (translation by Aminrazavi) > > > Khayyam identifies the main types of inquiry in "philosophy" along the Peripatetic line: "The essential and real inquiries that are discussed in philosophy are three inquiries, [first], 'is it?'...second, 'what is it?'...third, 'why is it?'" (On Being and Obligation, Malik (ed.) 1998, 335). While these are standard Aristotelian questions, for Khayyam they have a wider range of philosophical implications, especially with regard to the following topics: 1. The existence of God, His attributes and knowledge 2. Hierarchy of Existents and the problem of multiplicity 3. Eschatology 4. Theodicy 5. Predeterminism and free will 6. Subjects and predicates 7. Existence and essence ### 2.1 The existence of God, His attributes and knowledge In accordance with the Avicennan tradition, Khayyam refers to God as the "Necessary Existence" (or as it is more common in English translation, "Necessary Existent") and "that which cannot be conceived unless being existent," and "the one whose existence is from its essence, from the intellect's [point of view]" (On Existence, Jamshid Nijad Awwal (ed.), 112) and then offers cosmological, teleological, and ontological arguments for His existence.[4] The most fundamental principle about the necessary existent is its absolute oneness. Khayyam discusses issues such as unity, necessity, causality, and the impossibility of a chain of causes and effects continuing ad infinitum. In his translation of Avicenna's treatise, as Malik (ed.) (1998, 307) has rightly pointed out, Khayyam adds short remarks and explications to clarify and support Avicenna's view after each sub-section. He completely follows Avicenna's view on God's existence and His attributes. God, the necessary existent, is neither a substance nor an accident (Principles of Existence, Malik (ed.) 1998, 388). There is no motion in Him, and thus He is not in time. He has no intention (qasd), end or goal (gharad), since having an end or a goal indicates not having something [desirable], and this in turn implies imperfection (Lucid Discourse, Malik (ed.) 1998, 313-6). Among other topics pertaining to God which Khayyam discusses are God's knowledge of universals and particulars, the absolutely simple essence of the Necessary Existent, and that "all His attributes, with regard to Its essence, are [rational] considerations (i'itbarat)" (ibid). We will return to this last point, on rational considerations, below. ### 2.2 Hierarchy of existents and the Problem of Unity and Multiplicity For Khayyam, one of the most complex philosophical problems is to account for the hierarchy of existents and the manner in which they are ranked in terms of their nobility. In On Being and Obligation, Khayyam asserts: > > What remains from among the most important and difficult problems [to solve] is the difference among the existents in their degree of nobility.... Perhaps I, and my teacher, the master of all who have proceeded before him, Avicenna, have thoughtfully reflected upon this problem, and consequently have reached the point of convincing our selves [of its truth]. This conviction is either because of the weakness of our souls, which are convinced by something inwardly faulty and apparently fancy, or because of the strength in that view, which makes us convince ourselves [of its truth], and we will shortly express part of that view symbolically. (On Being and Obligation, Malik (ed.) 1998, 338) > > > How to read the last sentence of the above quote may be subject to scholarly debate. Particularly, Khayyam's formulation may suggest that he is not fully satisfied with the Neoplatonic view he is going to present. Nonetheless, in his treatise "On the Knowledge of the Universal Principles of Existence" (Principles of Existence, Malik (ed.) 1998, 381-3), as well as some other works, Khayyam adopts the Neoplatonic scheme of emanation and offers an analysis of a number of traditional philosophical themes within this context. Particularly, he follows the theory of hierarchical intellects and souls associated with the celestial spheres, as interpreted in the Islamic philosophical "emanationist" tradition, to explain creation and celestial motions (ibid, 382). ### 2.3 Eschatology Khayyam has been accused of believing in the transmigration of the soul and even corporeal resurrection in this world. This is partially due to some of the inauthentic quatrains attributed to him. Khayyam's philosophical treatises indicate that he did believe in life after death, and in this regard his views were in line with traditional Islamic eschatological doctrine. Khayyam the poet, however, plays with the notion of life after death in a variety of ways. First, he casts doubt on the very existence of a life beyond our earthly existence; second, he says that based on our very experience in this world, all things seem to perish and not return. Some of his poems play with the idea of the transmigration of the soul (tanasukh). This can be taken symbolically, rather than literally; in numerous poems he tells us that we turn to dust and it is from our dust that other living beings rise. Khayyam's comments regarding the possibility of life after death may well have been an indirect criticism of the orthodox jurists who spoke of the intricacies of heaven and hell with certainty.[5] Khayyam's ideas about the human soul and its persistence are scarce and scattered in his philosophical writings. Toward the end of The Necessity of Contrariety (Naji Isfahani (ed.) 2000, 169-70), he explains the notion of persistence as existence through time, and rejects the view that the concept of existence itself implies duration (in time). It seems to follow that if an entity exists atemporally, its ontological status cannot be described as "persisting." In his translation of Avicenna's treatise, Khayyam is explicit that "when it [the rational soul] is separated from the matter, it becomes like the angels, in simplicity and apprehension of the intelligible [meanings], till immortal persistence inevitably accompanies it" (Lucid Discourse, Malik (ed.) 1998, 317). Describing the fate of the rational soul as "immortally persisting" suggests that for Khayyam, like Avicenna, the rational soul is an incorporeal entity with temporal existence. ### 2.4 Theodicy (The Problem of Evil) The problem of theodicy, which Khayyam handles both philosophically and poetically, is one of the most prevalent themes in his quatrains, yet his approach differs in each medium. It is an irony that while in his philosophy Khayyam offers a rational explanation for the existence of evil, in his Ruba'iyyat he strongly criticizes the presence of evil and finds no justification for it fully satisfying. One may argue that such an inconsistency bears witness to the fact that the philosophical treatises and the Ruba'iyyat are not authored by the same person. While this remains a possibility, it is also reasonable that these seemingly contradictory works might belong to the same person. The discrepancy speaks to the human condition that despite our rationalization of the problem of evil, on a practical and emotional level, we remain fundamentally bewildered by the unnecessary presence of so much pain and suffering. Qadi Abu Nasr, a statesman and scholar from Shiraz, posed the following question to Khayyam: > > It is therefore implied that the Necessary Existent is the cause of the occurrence of evil, contrariety and corruption in the world. This is not worthy of the Divine status. So how can we resolve this problem and the conflict, so evil will not be attributed to the Necessary Existent? (The Necessity of Contrariety, Malik (ed.), 1998, 362-3) > > > The first sentence of the above quote is in fact the conclusion of a short argument, described on the previous page. Here is a slightly revised formulation of that argument: 1. Evil and contrary events occur in this world. 2. The evil and contrary events that occur in this world are either necessary existents or contingent existents. 3. They cannot be necessary existents (because, this implies that multiple things are necessary existents, but it has been demonstrated that there is one and only one Necessary Existent, which is absolutely simple and one in all respects). So, 4. The evil and contrary events that occur in this world are contingent existents. But 5. The Necessary Existent is the ultimate cause of all contingent existents. Thus, 6. The Necessary Existent is the ultimate cause of the occurrence of evil and contrary events. 7. If A is the cause of B and B is the cause of C, then A is the cause of C (Khayyam has formulated this premise explicitly in his response to the question). Therefore, 8. The Necessary Existent is (not only the ultimate cause but) the cause of the occurrence of evil and contrary events. In The Necessity of Contrariety, Khayyam offers an extended argument to exonerate God from being morally blameworthy for the creation of evil (the expression "morally blameworthy" is ours). Some have understood his response in terms of associating evil with non-existence or absence. Accordingly, God has created the essences of all contingent beings, which are good in and of themselves since any being, ontologically speaking, is better than non-being. Evil therefore represents an absence, a non-being for which God cannot be blamed (Nasr, The Poet Scientist Khayyam as Philosopher; Aminrazavi 2007, 172-5).[6] A closer reading of the text, however, may suggest a different and novel solution. Evil is sometimes introduced as nonexistence (On Existence, Jamshid Nijad Awwal (ed.) 2000, 118) and sometimes as following from nonexistence (ibid. and The Necessity of Contrariety, Naji Isfahani 2000, 168), which itself follows from the contrary relations that hold between contingent objects or events. Many instances of evil, according to Khayyam, do exist in reality. Khayyam's main point is not that evil acts and events are absences or negative facts, as it were. Though he talks about nonexistence and associates evil with absence, the contribution of his argument is not to conceptualize all instances of evil in terms of nonexistence or merely mental existence. Rather, evil is characterized as the concomitant of the contrary relations that obtain between contingent existents, which themselves are essentially created by God. In this formulation, two notions call for clarification. First, the notion of "concomitant" (lawazim) as a "necessary non-essential predicable": this notion is well known in the Avicennan school and borrowed from the Aristotelian notion of per se accidents (kath' hauto sembebekos). The second notion is the distinction between essential and accidental causation. This is also rooted in an Aristotelian theory of causation and Avicenna's reconstruction of that. Evil, as the concomitant of the contrary relations that hold between contingent existents, is only accidentally created by God. Otherwise put, God essentially causes the ("quiddative," this is our term) contingent beings and only accidentally causes contrary relations, and, as a result, the evil that follows from the co-existence of those essences. Furthermore, Khayyam claims, the problem in this case is whether God is the essential cause of the occurrence of evil. This can be interpreted as a value claim: causing something is not worthy of Divine status only if God essentially and intentionally causes it. (Khayyam immediately modifies this claim, explaining that the First, properly speaking, has no "intention" (qasd) but there is eternal guardianship ('inaya).) Since God does not essentially and intentionally cause evil (in fact, the existence of some contrary relations is a necessary consequence of creating (quiddative) contingent existents), it follows that God accidentally causes evil and this is not morally problematic. In the appendix to his response to the first question, Khayyam raises another objection: Given that evil is accidentally created by God and assuming that God is omniscient, "Why did God essentially create things, i.e. (quiddative) contingent existents, if He knew their existence would imply contrary relations, nonexistence, and eventually evil?" Khayyam's response is based on three premises: (1) the existence of evil can be justified if the benefits following from essentially creating (quiddative) contingent beings massively outweigh the harms following from accidentally creating contrary relations (and eventually evil). Furthermore, (2) withholding an act whose benefits massively outweigh the harms following it, given an appropriate principle of proportionality, would imply massive evil. And (3), in fact the benefits following from essentially creating (quiddative) contingent beings massively outweigh the harms following from accidentally creating contrary relations (and eventually evil). These premises are intended to justify God's essential creation of contingent beings despite His knowledge of the ensuing evil. ### 2.5 Determinism and Free Will Both his Western and Eastern expositors consider Khayyam to be a predeterminist (jabri). However, his views on the subject matter are far more complex. Some (Nasr, The Poet Scientist Khayyam as Philosopher; Aminrazavi 2007, 175) have interpreted "On Being and Obligation" as a treatise on the problem of determinism and free will. According to Aminrazavi (2007, 175), Khayyam uses the term taklif (translated by Nasr (ibid) and Aminrazavi (ibid) as "necessity" as well) to denote "determinism." There is a short paragraph in The Necessity of Contrariety that Nasr and Aminrazavi interpret as indicating that Khayyam is inclined to "predeterminism," or "determinism" as they use the term, provided it is not taken to its extreme: > > As to his question [i.e., Qadi Nasawi's question] concerning which of the two groups [pre-determinists or free will theorists] are closer to truth, I say prima facie and at the first sight, perhaps the pre-determinists are closer to truth, provided they do not enter into their nonsensical and absurd [claims], for in this case they verily depart far from truth. (The Necessity of Contrariety, Naji Isfahani (ed.) 2000, 169) > > > Aminrazavi (2007, 177-80), then, identifies and explains three types of "determinism" in Khayyam's view as follows: 1. Universal-cosmic 2. Socio-economic 3. Ontological In the universal and cosmic sense, our presence in this world and our entry and exit is predetermined, a condition that Khayyam bemoans throughout his Ruba'iyyat. The universal-cosmic determinism is the cause of our bewilderment and existential anxiety. Khayyam expresses this when he says: > > With Earth's first Clay They did the Last Man knead, > > And there of the Last Harvest sow'd the Seed: > > And the first Morning of Creation wrote > > What the Last Dawn of Reckoning shall read. > > (Ruba'iyyat, FitzGerald 1859, 41) > > > The second sense of determinism is Socio-economic, which is rarely addressed by Muslim philosophers. Khayyam incidentally refers to this notion, for example, in: > > God created the human species such that it is not possible for it to survive and reach perfection unless it is through reciprocity, assistance, and help. Until food, clothes, and a home that are the essentials of life are not prepared, the possibility of the attainment of perfection does not exist. (On Being and Obligation, Hashemipour (ed.) 2000, 143). > > > Finally there is "ontological determinism," which relies on a Neoplatonic scheme of emanation which Khayyam considers to be "among the most significant and complex of all questions," since "the order of the world is in accordance to how the wisdom of God decreed it" (On Being and Obligation, Hashemipour (ed.) 2000, 145). He continues, "obligation (taklif) is a command issued from God Most High, so people may attain those perfections that lead them to happiness" (ibid, 143). This Greek concept of happiness, restated by Farabi as "For every being is made to achieve the ultimate perfection it is susceptible of achieving according to its specific place in the order of being," (al-Farabi 1973, 224) implies that our ontological status or capacity is (at least to some extent) pre-determined. Here, we have translated taklif as "obligation" (not "necessity") and jabr as "predeterminism" (not "determinism"). This may be justified as follows: "Determinism" as we use it today is considered to be compatible with free will (from the compatibilist point of view) but jabr, both in its historical context and in contemporary usage, is not considered to be compatible with free will. Furthermore, we are not sure how to interpret the short quote in response to the second question in The Necessity of Contrariety (Naji Isfahani (ed.) 2000, 169). It can be interpreted as a formulation of Khayyam's dissatisfaction with both predeterminism and the free will view. This is because he says that predeterminism is closer to truth "prima facie and at the first sight" and then suggests that if the view is expanded, or fully applied, it can lead to nonsensical consequences. It is noteworthy that in the same passage he does not attempt to defend the free will view either. The original term for what has been translated as the "free will" view here, is qadariyya (Danish namah-yi Khayyami, Malik (ed.) 1998, 344, note 2). The term is associated with different views and problems, one of which is whether humans are the creators (khaliq) of their sinful acts. The term "creator" is important in this context, particularly because it is not necessarily synonymous with "cause" and more importantly, because some, particularly in the early Islamic Kalam tradition, held the view that the only real creator is God. So, the context of this discussion is not just the metaphysics of causation, as we use the term. Finally, the above considerations make reconstructing Khayyam's view on this matter and mapping it on the contemporary views on "determinism" more difficult than it might initially appear. Khayyam has the concept of "necessitation" (wujub) and discusses that in his treatises on existence [references 5 and 6 above] at length. Thus, the three notions of "determinism" introduced by Aminrazavi may require further investigation. ### 2.6 Subjects, Predicates, and Attributes In a complex discussion, Khayyam presents his views on the relationship between the subject, predicate, and attributes using a mixture of original insight and Aristotelian precedent. Dividing the attributes (al-awsaf) into two categories, essential and accidental, he discusses both categories and their subdivisions, such as concomitant accidental attributes vs. detachable accidental attributes (and the latter, in turn, is divided into merely detachable in estimation vs. detachable in estimation and in existence) in details (On Existence, Jamshid Nijad Awwal (ed.), 101-2). Developing the same line of argument in The Necessity of Contrariety (Naji Isfahani (ed.) 2000, 164), Khayyam proposes the following characterization of an "essential attribute": > > [An attribute (wasf)] is essential if [1] it is not possible to conceive the subject (al-mawsuf) [of the attribution] except by conceiving that as having the attribute in the first place, and it is required for it [2] to belong to the subject [of the attribution] with no [extra] cause, like animality [as an essential attribute] of human, and [3 it is required for it] to be prior to the subject [of the attribution] essentially (bi al-dat), that is, it [i.e. the essential attribute] is the cause of the subject [of the attribution] and not its effect, like animal [as an essential attribute] of human and [also] rational [as an essential attribute of human] (The Necessity of Contrariety, Naji Isfahani 2000, 164; emphasis is ours). > > > (We have translated wasf as attribute and mawsuf as the subject of attribution.) Khayyam's explication includes three conditions for an essential attribute (in relation to the subject of attribution): (1) conceptual priority, (2) no causal dependence (or posteriority), and (3) causal priority. First, conceptual priority indicates that Khayyam understands "essential attributes" not merely as metaphysically essential properties; they should satisfy some epistemological condition as well. Second, "no causal dependence" distinguishes essential attributes from all other attributes that do not follow from the essence of something (and thus their attribution to the subject requires a cause distinct from the subject). This is a strong, and well-known, metaphysical condition in the Islamic Aristotelian tradition. Nonetheless, "concomitants" may satisfy "no causal dependence" requirement as well, because their attribution to the subject does not require any extra cause. Third, causal priority is supposed to exclude concomitants. So, no concomitant, even though necessarily predicable of its subject, is an essential attribute in the sense under discussion. According to Aminrazavi's interpretation of the above quote, "essential attributes are those which are not possible to conceive of without the preconception of these a priori (badawi) attributes, such as 'animality which is an essential attribute of man'" (Aminrazavi 2007, 183). The term badawi is not in the original Arabic text, in fact it can only be found in a Persian translation of The Necessity of Contrariety (Naji Isfahani 2000, 171). Moreover, badawi can be translated in many ways and "a priori" is not its most straightforward translation, if a proper translation in the first place. Finally, the above observations suggest that Khayyam's notion of "essential attributes" only implies conceptual priority (not a prioricity) of the essential attribute(s) to its subject. Conceiving animal is prior to conceiving human; this does not imply that the attribute animal is a priori (in its Kantian sense). If one could find further pieces of evidence in Khayyam's philosophy, with regard to a priori concepts or innate ideas, in its Cartesian sense, one might attempt to bolster Aminrazavi's claim. At present, we are not aware of any such evidence or argument. Khayyam also divides (essential and accidental) attributes into considerational (i'tibari) and existential (wujudi) (i'tibari has also been translated in the literature of Islamic/Arabic philosophy as item of consideration, abstract, secondary, fictional, intentional, mental, and conceptual; see, for example, Online Dictionary of Arabic Philosophical Terms. In this context, these translations may have misleading connotations. Thus, we suggest considerational, and use it in contrast with existential). Khayyam attempts to provide different criteria for this distinction: > > And the existential [attribute] is like the attribute black [attributed to a] material object (jism), when [in fact] it is black. Thus, being black is an existential attribute, that is, it is an additional meaning to the essence [to which] black [is attributed], as [it is] existing in re (mawjudun fi al-a'yan). Thus, if being black is an existential attribute (wasf), then black is [called] an existential description (sifa). (On Existence, Jamshid Nijad Awwal (ed.) 2000, 102) > > > Accordingly, an existential attribute is something that exists in re as an additional attribute of its subject. This point, i.e. being additional to the subject as it exists in re, is crucial to Khayyam's formulation of existential attributes since considerational attributes are only additional to their subject in intellectu (or, more specifically, in the intellect or in the estimative faculty). Khayyam finds existential attributes easily understandable and then quickly moves to introduce considerational attributes:[7] > > If the intellect intellects a meaning (ma'na) and details that intelligible [meaning] in an intelligible manner and considers its status/conditions (al-ahwal), if that meaning happens/occurs to some simple, not multiple, [meaning] like all existing accidents in re (fi al-a'yan), and it [i.e., the simple meaning] happens to have some attributes, then [the intellect] knows that all those attributes belong to that [simple meaning] on the basis of consideration, not on the basis of the existence in re. (On Existence, Jamshid Nijad Awwal (ed.) 2000, 103) > > > A considerational attribute is an intelligible meaning whose subject of attribution is another intelligible meaning that is simple, not multiple, in re and has multiple attributes. These attributes of the simple meaning are considerational attributes. This is because simple things, with no multiplicity in re, are not hylomorphically composite, that is, they are not composed of matter and form. So, their attributes are not grounded on real multiplicity or composition. And according to the Peripatetic philosophical school, accidents (in the sense of the nine Aristotelian categories) are simple in re, with no matter or form. If black is an accident (quality), being a color is a considerational attribute of that. Likewise, if two is an accident (quantity), being half of four is a considerational attribute of that as well. Let us explain these two examples in more detail. Considerational attributes, as well as existential ones, come in two varieties: essential and accidental. Khayyam illustrates both cases. An example of an essential considerational attribute is being a color as an attribute of black. Conceiving being a color is conceptually prior to conceiving black and is the cause of black, that is, being a color is essentially prior to black (or white). Furthermore, an existential attribute of a subject is additional to its essence. But black itself is an accident and, here Khayyam seems to assume that, no accident can be the subject (in the metaphysical sense) of another accident. Finally, being a color and black cannot be accidents of the same subject because if this were the case, then black could come apart from being a color, at least in estimation, but this is impossible. So, being a color is an essential considerational attribute of black (On Existence, Jamshid Nijad Awwal (ed.) 2000, 103-4). An example of an accidental considerational attribute is being half of four as an attribute of two. Conceiving being half of four is not conceptually prior to conceiving two, nor is being half of four the cause of two. So, being half of four is not an essential attribute of two. Hence, being half of four is an accidental attribute of two. Furthermore, an existential attribute of a subject is additional to its essence. If being half of four were something additional (za'id) to two (in re), then two would have infinite meanings added to its essence (in re). But this is impossible, according to Khayyam. Therefore, being half of four is not an accidental existential attribute of two. Therefore, being half of four is an accidental considerational attribute of two (On Existence, Jamshid Nijad Awwal (ed.) 2000, 102-3). (This example suggests that two, as an accident, is a simple meaning, with no matter or form, even though it is true of two entities.) By way of analogy, and in some respects, the problem of considerational vs. existential attributes, as Khayyam discusses, is like the problem of non-natural vs. natural properties in contemporary analytic philosophy. This semantic division plays a significant metaphysical role in Khayyam's metaphysics, to which we shall turn next. ### 2.7 Existence (wujud) and Essence (mahiyyah) Khayyam offers a series of arguments for the thesis that "existence is a 'considerational' attribute" (On Existence, Jamshid Nijad Awwal (ed.) 2000, 106; The Light of the Intellect, Nadvi (ed.) 1933 [2010, 347-9]). In section seventeen of On Existence, entitled "Existence is an added meaning (ma'na) to the intelligible essence," he writes, "And it is as if [the soul/intellect] encounters existence in all things, by way of accidents, and there is no doubt that existence is a meaning added to the intelligible essence" (On Existence, Jamshid Nijad Awwal (ed.) 2000, 111). By "by way of accidents" Khayyam should mean as a form of accident, and "accident" should be used here in the sense of predicable, not one of the nine Aristotelian categories of accidents. The "additionality" should also mean "being additional in intellectu," not in re. By relying on reductio ad absurdum, he concludes that if existence were to be an existential attribute, it would have to exist prior to itself, which is impossible. Khayyam states "essence (dat) was non-existent and then became existent." He goes on to argue that "essence does not need existence [before coming to existence] or a relation to existence because the essence prior to existing was non-existing (ma'dum), thus how can something need something else prior to its existence?" (On Existence, Jamshid Nijad Awwal (ed.) 2000, 110). Towards the end of this treatise he uses the Neoplatonic scheme of emanation to explain the origin of essences and states: "Therefore, it became clear that all essences (dawat) and quiddities (mahiyyat) emanate from the essence of the First Exalted Origin, in an orderly fashion, may glory be upon Him" (On Existence, Jamshid Nijad Awwal (ed.) 2000, 118). In Aminrazavi and Van Brummelen (2017) earlier edition of this entry, as well as in Nasr (The Poet Scientist Khayyam as Philosopher) and Aminrazavi 2007, 168-71), it is suggested that "Clearly Khayyam supports the principality of essence" (i.e., the second view) among the following three views: 1. The existence of an existent is the same as its essence. This view is attributed to Abu'l-Hasan Ash'ari, Abu'l-Hasan Basri, and some of the other Ash'arite theologians. 2. Commonly known as the principality of essence (asalat al-mahiyyah), this view maintains that essence is primary and existence is added to it. Many philosophers such as Abu Hashim Juba'i and later Suhrawardi and Mir Damad came to advocate this view. 3. Commonly known as the principality of existence (asalat al-wujud), this view maintains that existence is primary and essence is then added (Aminrazavi 2007, 168). Nasr (The Poet Scientist Khayyam as Philosopher) who prefers the same reading, that is "Khayyam asserts that for each existent, it is the quiddity that is principal while wujud is a conceptual (i'tibari) quality," immediately qualifies his view by adding that "the distinction between the principality of wujud (asalat al-wujud) and the principality of mahiyyah (asalat al-mahiyyah) goes back to the School of Isfahan and especially Mulla Sadra," and Khayyam "does not use the term asalat al-mahiyyah as was done by Mulla Sadra." We find Nasr's comment on the notion of "principality" important and take it to indicate that there may be room to argue that not only the term "principality of quiddity" but also the corresponding concept is absent from Khayyam's philosophical treatises. What one finds in Khayyam's work is a series of arguments for the thesis that existence is a considerational (or "secondary" as translated by Aminrazavi) attribute. If one further grants the conditional claim that if existence is a considerational attribute, then quiddity is principal (in the sense at stake in the debate over the principality of existence vs. the principality of quiddity in the post Isfahan school). Thus, one might derive the conclusion that Khayyam supports the principality of quiddity. The latter conditional presumption, however, is in need of a sound argument, and as far we can see, no such argument has yet been provided. Aminrazavi and Van Brummelen (2017) then explain that "a more careful reading reveals an interesting twist: namely, that Khayyam's understanding of how essences came to be casts doubt on his belief in the principality of essence" (Note: we have translated al-mahiyyah as "quiddity," not "essence"). This comment may strengthen our hypothesis that Khayyam does not support the principality of quiddity in the first place, at least in the sense rejected by Mulla Sadra and his followers. Perhaps Khayyam does not offer an argument against it either (if he does not work with the notion of principality in question). Aminrazavi and Van Brummelen (2017) then continue: "Towards the end of the Risalah fi'l-wujud he uses the Neoplatonic scheme of emanation to explain the origin of essences [...] Khayyam replaces essence with existence here and the question is whether he equates them and thereby deviates from his teacher Ibn Sina. [...] It appears that Khayyam equates existence and essence as having emanated from God in an orderly fashion, but there is no explanation of how essence becomes primary and existence secondary. In fact, if existence did not exist how could essences come to be?" After discussing all these "problems," they conclude: "Although the distinction between the principality of wujud (asalat al-wujud) and the principality of mahiyyah (asalat al-mahiyyah) can be found among early Muslim philosophers, the subject matter became particularly significant in later Islamic philosophy, especially through the School of Isfahan and the work of its most outstanding figure, Mulla Sadra. This is important for our discussion since Umar Khayyam may simply have presented the arguments for and against the priority and posterity of essence and existence without attaching much significance to their philosophical consequences, as was the case in later Islamic philosophy" (Aminrazavi and Van Brummelen 2017). The above concerns and the ensuing questions are based on the assumption that Khayyam supports the principality of quiddity, which in turn stems from casting a post Isfahan school conceptual framework on Khayyam's philosophy. This latter interpretational strategy, however, requires proper doctrinal and textual justification. In the absence of such evidence, Khayyam's view that existence is a considerational attribute does not seem to imply the principality of quiddity and hence may not be subject to the above criticisms. In The Light of the Intellect, Khayyam offers three reasons why existence is not added to essence in re. A summary of his reasons goes as follows: 1. Existence cannot be added to essence in re; otherwise an infinite regress will follow, which is untenable. 2. Existence is not added to essence in re; otherwise essence should have existed prior to its existence, and this is absurd. 3. With regard to the Necessary Existence, existence clearly is not added to essence in re, for dualism would follow. Some have also read The Light of the Intellect as providing an argument for the principality of essence (or quiddity); we believe such readings face problems similar to the ones discussed above. This treatise reconfirms the view that Khayyam believes that existence, oneness, and many other key metaphysical and scientific attributes (e.g., color), are considerational, not existential, attributes. Khayyam's philosophical works are the least studied aspects of his thought, and were not even available in published form until a few years ago. They permit a fresh look at overall Khayyamian thought and prove indispensable to an understanding of his Ruba'iyyat. In his philosophical works, Khayyam writes as a Muslim philosopher and treats a variety of traditional philosophical problems in a careful manner; but in his Ruba'iyyat, our Muslim philosopher morphs into an agnostic Epicurean, or so it seems. A detailed study of Khayyam's philosophical works reveals several explanations for this dichotomy, the most likely of which is the conflict between pure and practical reasoning. Whereas such questions as theodicy, the existence of God, soul and the possibility of life after death may be argued for philosophically, such arguments hardly seem relevant to the human condition, given our daily share of suffering. It is in light of the distinction between "is" and "ought," the "ideal" and the "actual," that discrepancies between Khayyam's Ruba'iyyat and his philosophical views should be understood. Khayyam's Ruba'iyyat are the works of a sober philosopher and not those of a hedonistic poet. Whereas Khayyam the philosopher-mathematician justifies theism based on the existing order in the universe, Khayyam the poet, for whom suffering in the world remains insoluble, does not talk about theism, or any type of eschatological doctrine, as a solution to the problem of the meaning of human existence. ## 3. The Ruba'iyyat (Quatrains) > > Here with a Loaf of Bread beneath the Bough, > > A Flask of Wine, a Book of Verse--and Thou > > Beside me singing in the Wilderness-- > > And Wilderness is Paradise enow. > > (Ruba'iyyat, FitzGerald 1859, 30; 2009, 21) > > > Although Umar Khayyam's Ruba'iyyat have been admired in the Persian speaking world for many centuries, they have only been known in the West widely since the mid 19th century, when Edward FitzGerald rendered the Ruba'iyyat into English. The word Ruba'i (plural: Ruba'iyyat), meaning "quatrain," comes from the word al-Rabi', the number four in Arabic. It refers to a poetic form which consists of a four-lined stanza and two hemistiches for a total of four parts. Taranah (snatch) and dobaiti (two-liner) are very similar to quatrains; they differ from each other in terms of rhyme scheme and themes or message. They all have a short and simple form that provides a type of "poetic punch line." The overwhelming majority of the literary works on the Ruba'iyyat have been devoted to the monumental task of determining the authentic Ruba'iyyat from the inauthentic ones. In our current discussion, we shall bypass that controversy and rely on the most authoritative Ruba'iyyat in order to provide a commentary on Khayyam's critique of the fundamental tenets of organized religion(s). The salient features of his critique address the following: 1. Impermanence and the quest for the meaning of life 2. Theodicy 3. The here and now 4. Epistemology 5. Eschatology 6. Determinism and free will 7. Philosophical wisdom ### 3.1 Impermanence and the quest for the meaning of life The Ruba'iyyat's overarching theme is the temporality of human existence and the suffering that one endures during a seemingly senseless existence. Clearly, such a view based on his observation of the world around him is in sharp contrast with the Islamic view presented in the Quran: "I (Allah) have not created the celestial bodies and the earth in vain" (Quran, 38:27). Umar Khayyam was caught between the rationalistic tradition of the Peripatetics deeply entrenched in the Islamic religious universe and his own failure to find any meaning or purpose in human existence on a more immediate and experiential level. Khayyam the poet criticizes the meaninglessness of life whereas Khayyam the philosopher remains loyal to the Islamic Peripatetic tradition which adheres to a theocentric world view. Using the imagery of a kuzah ("jug") and clay throughout the Ruba'iyyat, Khayyam alludes to the temporality of life and its senselessness:[8] > > I saw the potter in the market yesterday > > Pounding and pounding upon a piece of clay > > "Behold," said the clay to the potter > > Treat me gently for once like you, now I am clay > > (translation by Aminrazavi) > > > The above verse, for example, may be read as suggesting that Khayyam does not see a profound meaning in human existence; his existential anxiety is compounded by the fact that we are subject to our daily share of suffering, a concept that runs contrary to that of the all merciful and compassionate God of Islam. ### 3.2 Theodicy and Justice The problem of suffering has an ominous presence in the Ruba'iyyat, which contains both Epicurean and Stoic themes. On theodicy, Khayyam remarks: > > In what life yields in this Two-door monastery > > Your share in the pain of heart and death will tarry > > The one who does not bear a child is happy > > And he not born of a mother, merry > > (translation by Aminrazavi) > > > And also: > > Life is dark and maze-like, it is > > Suffering cast upon us and comfort in abyss > > Praise the Lord for all the means of evil > > Ask none other than He for malice > > (translation by Aminrazavi) > > > (It is an irony that while Khayyam complains about theodicy and human suffering throughout his Ruba'iyyat, in his philosophical works he offers a treatise almost entirely devoted to a philosophical justification of the problem of evil.) It might seem that theodicy as a theological and philosophical problem in Islam never received the attention it did in Western intellectual tradition; however, a series of problems associated with evil have been discussed under different names in this tradition. In the discussion of Tawhid, absolute oneness and unity of God in Islam, the problem of existence of evil, particularly the manner in which evil is created by God, is discussed (see section 2.4 above). In the debate over God's attribute of justice, between the Asharites and the Mu'tazilites, some aspects of the problem of evil, e.g., whether our morally wrong actions are ultimately created by God or attributed to him, are carefully examined as well. Also, in the discussion of relative and negative (privative) attributes, the question of the nature of evil is raised again. Khayyam's poem seems to acknowledge both the reality and malicious nature of evil and the ensuing suffering, on the one hand, and God's role in allowing evil and his "answerability," on the other hand. This makes questioning God on the existence of evil not illegitimate: "Ask none other than He for malice." ### 3.3 Here and Now For Khayyam the poet, "the tale of the seventy-two nations," by which he refers to organized branches of Islam, and of religions in general, and traditional metaphysical beliefs (mainly Platonic and neo-Platonic), are merely flight of fancy, for the human condition, which he describes as a "sorrow laden nest," at least in part contradicts many such beliefs. The art of living in the present, a theme dealt with in Sufi literature, is a type of wisdom that must be acquired, since living for the hereafter and heavenly rewards is conventional wisdom, and more suitable for the masses. On this point, Khayyam asserts: > > Today is thine to spend, but not to-morrow, > > Counting on morrow breedeth naught but sorrow; > > Oh! Squander not this breath that heaven hath lent thee, > > Nor make too sure another breath to borrow > > (Whinfield 2001, 30; modified by Aminrazavi) > > > And also: > > What matters if I feast, or have to fast? > > What if my days in joy or grief are cast? > > Fill me with Thee, O Guide! I cannot ken > > If breath I draw returns or fails at last. > > (Whinfield 2001, 144) > > > Khayyam's emphasis on living in the present, or as Sufis say, "Sufi is the son of time," along with his use of other Sufi metaphors such as wine, intoxication and love making, have been interpreted by some scholars as merely mystical allegories.[9] Although a mystical interpretation of the Ruba'iyyat has been advocated by some, it remains the view of a minority of scholars. The complexity of the world according to Khayyam the mathematician-astronomer necessitates the existence of a creator and sustainer of the universe; and yet on a more immediate and existential level, he finds no reason or meaning for human existence. He seems to want to hold both the idea of "a necessary existence" and what necessarily follows from it (as a divine character is always present and sometimes critically addressed in his quatrains) and the immediate intuition of the presence of apparently unexplainable suffering, evil, and meaninglessness in this world. This leads to the theme of doubt and bewilderment. A practical response to this existential crisis opens room for a number of Epicurean and Stoic themes in Khayyam's view, such as being "the son of time," or the art of living in presence. ### 3.4 Doubt and Bewilderment Humans, Khayyam tells us, are thrown into an existence they cannot make sense of: > > The sphere upon which mortals come and go, > > Has no end nor beginning that we know; > > And none there is to tell us in plain truth: > > Whence do we come and whither do we go. > > (Whinfield 2001, 132) > > > Again, the vivid and apparent inconsistency between a seemingly senseless existence and a complex and orderly world leads to existential and philosophical doubt and bewilderment. The tension between Khayyam's philosophical writings in which he embraces the Islamic Peripatetic philosophical tradition and his Ruba'iyyat where he expresses his profound skepticism stems from this paradox. In his Ruba'iyyat Khayyam embraces, or comes very close to embracing, humanism and agnosticism, leaving the individual human being disoriented, anxious and bewildered; whereas in his philosophical writings he operates within a theistic world where the most fundamental questions, like the existence of God and explanation of evil, find (relatively) decisive solutions. Lack of certainty with regard to religious truth leaves the individual in an epistemologically suspended state where one has to live in the here and now, irrespective of the question of truth: > > Since neither truth nor certitude is at hand > > Do not waste your life in doubt for a fairyland > > O let us not refuse the goblet of wine > > For, sober or drunken, in ignorance we stand > > (translation by Aminrazavi) > > > This skepticism towards organized religions runs deep in Khayyam's quatrains; he explicitly questions the orthodoxy and how the final disclosure of ultimate truth might be surprising to all "believers," and perhaps "nonbelievers" (note: your reward is neither here nor there): > > Alike for those who for To-day prepare, > > And those that after a To-morrow stare, > > A Muezzin from the Tower of Darkness cries > > "Fools! your Reward is neither Here nor There!" > > (Ruba'iyyat, FitzGerald 2009, 28) > > > ### 3.5 Eschatology The Ruba'iyyat casts doubt on Islamic eschatological and soteriological views. Once again the tension between Khayyam's poetic and philosophical modes of thought surfaces; experientially there is evidence to conclude that death is the end: > > Behind the curtain none has found his way > > None came to know the secret as we could say > > And each repeats the dirge his fancy taught > > Which has no sense-but never ends the lay > > (Whinfield 2001, 229) > > > In the Ruba'iyyat, Khayyam portrays the universe as a beautiful ode which reads "from dust we come and to dust we return," and "every brick is made from the skull of a man." While Khayyam does not explicitly deny the existence of life after death, perhaps for political reasons and fear of being labeled a heretic, there are subtle references throughout his Ruba'iyyat that the hereafter should be taken with a grain of salt. In contrast, in his philosophical writings we see him argue for the incorporeality of the soul, which paves the path for the existence of life after death. The irreconcilable conflict between Khayyam's observation that death is the inevitable end for all beings, and his philosophical reflections in favor of the possibility of the existence of life after death, remains an insoluble riddle.[10] ### 3.6 Free Will, Determinism and Predestination Khayyam is known as a determinist in both the East and the West, and deterministic themes can be seen in much of the Ruba'iyyat. But if we read his Ruba'iyyat together with his philosophical writings, the picture that emerges may be more rightly called "soft determinism." One of Khayyam's best known quatrains in which determinism is clearly conveyed asserts: > > The Moving Finger writes; and, having writ, > > Moves on: nor all your Piety nor Wit > > Shall lure it back to cancel half a Line, > > Nor all your Tears wash out a Word of it > > (Ruba'iyyat, FitzGerald 1859, 20) > > > Again, in his philosophical treatise The Necessity of Contrariety, Khayyam adheres to three types of determinism. On a universal or cosmic level, our birth is determined in the sense that we had no choice in this matter. Ontologically speaking, our essence and our place on the overall hierarchy of beings appears also to be predetermined. However, the third category of determinism, socio-economic determinism, is man-made and thus changeable. Our attempt to face this determined end of our life, however, will again lead us to nowhere but bewilderment: > > At first they brought me perplexed in this way > > Amazement still enhances day by day > > We all alike are tasked to go but Oh! > > Why are we brought and sent? This none can say > > (Ruba'iyyat, Tirtha 1941, 18) > > > Thus a reading of the Ruba'iyyat in conjunction with Khayyam's philosophical reflections bring forward a more sophisticated view of free will and determinism, indicating that Khayyam believed in free will within a form of cosmic determinism and did not support a full-blown predeterminism. ### 3.7 Philosophical Wisdom Khayyam uses the concept of "wine and intoxication" throughout his Ruba'iyyat in three distinct ways: 1. The intoxicant wine 2. The mystical wine 3. The wine of wisdom The pedestrian use of wine in the Ruba'iyyat, devoid of any intellectual significance, emphasizes the need to forget our daily suffering. The mystical allusions to wine pertain to a type of intoxication which stands opposed to discursive thought. The esoteric use of wine and drinking, which has a long history in Persian Sufi literature, refers to the state of ecstasy in which one is intoxicated with Divine love. Those supporting the Sufi interpretation of Ruba'iyyat rely on this literary genre. While Khayyam was not a Sufi in the traditional sense of the word, he includes the mystical use of wine among his allusions. Khayyam's use of wine in the profound sense in his Ruba'iyyat is a type of Sophia that provides a sage with philosophical wisdom, allowing one to come to terms with the temporality of life and to live in the here and now: > > Those imprisoned by the intellect's need to decipher > > Humbled; knowing being from non-being, they proffer > > Seek ignorance and drink the juice of the grape > > Those fools acting as wise, scoffer. > > (translation by Aminrazavi) > > > Khirad is the type of wisdom that brings about a rapprochement between the poetic and discursive modes of thought, one that sees the fundamental irony in what appears to be a senseless human existence within an orderly and complex physical universe. For Khayyam the mathematician-astronomer, the universe cannot be the result of a random chance; on the other hand, Khayyam the poet fails to find any purpose for an individual human existence in this orderly universe. > > As Spring and Fall make their appointed turn, > > The leaves of life one aft another turn; > > Drink wine and brood not--as the Sage has said: > > "Life's cares are poison, wine the cure in turn." > > (Ruba'iyyat, Sa'idi 1994, 58) > > > ## 4. Khayyam the Mathematician and Scientist > > And the part of philosophy known as mathematics is the easiest of its parts to grasp both as to conception and as to assent. As to the numerical part of it, it is a very evident matter. And as to the geometrical, likewise hardly anything of it will be unknown to someone who has an unimpaired constitution, a sharp view, an excellent intuition. (Commentary on the Difficulties of Certain Postulates of Euclid's Work, Rashed and Vahabzadeh, 2000, 217-218). > > > In several respects Khayyam's mathematical writings are similar to his texts in other genres: they are relatively few in number, but deal with well-chosen topics and carry deep implications. Some of his mathematics relate in passing to philosophical matters (in particular, reasoning from postulates and definitions), but his most significant work deals with issues internal to mathematics and in particular the intersection between geometry and algebra. Khayyam's mathematical works with recent editions and publications are: 1. "Treatise on the division of a quadrant of a circle" (Risala fi taqsim rub' al-da'ira) in Hakim Omare Khayyam as an Algebraist, Arabic text and Persian translation in Mossaheb, 1960, 59-74, 251-291; English translation and critical edition in Rashed and Vahabzadeh, 2000, 165-180. 2. "Treatise on Algebra" (Risala fi'l-barahin 'ala masa'il al-jabr wa'l-muqabala) in Hakim Omare Khayyam as an Algebraist, Persian translation in Mossaheb, 1960, 159-250; English translation and critical edition in Rashed and Vahabzadeh, 2000, 111-164. 3. "Commentary on the Difficulties of Certain Postulates of Euclid's Work" (Sharh ma ashkal min musadirat al-uqlidis) in Danish namah-yi Khayyami, Malik (ed.) 1377 (A.H.s.), 71-112; English translation and critical edition in Rashed and Vahabzadeh, 2000, 217-255. 4. "Problems of Arithmetic" (Mushkilat al-hisab), (only the title page is extant at the University of Leiden library collection Cod. or. 199, but Khayyam references the work in his Algebra). ### 4.1 Solutions of Cubic Equations Khayyam seems to have been attracted to cubic equations originally through his consideration of the following geometric problem: in a quadrant of a circle, drop a perpendicular from some point on the circumference to one of the radii so that the ratio of the perpendicular to the radius is equal to the ratio of the two parts of the radius on which the perpendicular falls. In the short "Treatise on the division of a quadrant of a circle," Khayyam leads us from one case of this problem to the equation $x^3 + 200x = 20x^2 + 2000$.[11] Khayyam states that an exact solution is not possible and provides an approximation. Khayyam also generates a direct geometric solution: he uses the numbers in the equation to determine intersecting curves of two conic sections (a circle and a hyperbola) and demonstrates that the solution $x$ is equal to the length of a particular line segment in the diagram. Solving algebraic problems using geometric tools was not new; in the case of quadratic equations these geometric methods date back as far as the Greeks and even to the Babylonians. Khayyam's predecessors such as al-Khwarizmi (early 9th century) and Thabit ibn Qurra (late 9th century) already had solved quadratic equations using the straightedge and compass geometry of Euclid's Elements. Since negative numbers were avoided in Arabic mathematics, Muslim mathematicians needed to solve several different types of quadratic equations with positive coefficients: for instance, $x^2 = mx + n$ was fundamentally different from $x^2 + mx = n$. For cubic equations, fourteen distinct types of equations cannot be reduced to a linear or quadratic form after dividing by $x$ or $x^2$. In his "Treatise on Algebra"[12] Khayyam notes that four of these fourteen types had been solved and says that al-Khazin (ca. 900-971) had solved a problem from Archimedes' treatise On the Sphere and Cylinder that al-Mahani (fl. ca. 860) had previously converted into a cubic equation and solved using conic sections. Khayyam also states that al-Layth (fl. 10th century) had not treated these cubic forms exhaustively. In the Algebra, Khayyam claims to be the first to deal systematically with all fourteen types of cubic equations. He solves each one in turn, again through the use of intersecting conic sections. Khayyam also considers circumstances when certain cubic equations have multiple solutions. Although he does not handle this topic perfectly, his effort nonetheless exceeds previous efforts. In an algebra where powers of $x$ corresponded to geometrical dimensions, the solution of cubic equations was the apex of the discipline. Nevertheless, even here Khayyam advanced algebra by considering its unknowns as dimension-free abstractions of continuous quantities.[13] A geometric solution to a polynomial equation may seem peculiar to modern eyes, but the study of cubic equations (and indeed much of medieval algebra) was motivated by geometric problems. Khayyam was explicitly aware that the algebraic solution of the cubic equation remained to be solved. He never produced such a solution, nor did anyone else until Scipione del Ferro, Niccolo Fontana Tartaglia, and Gerolamo Cardano in the mid-16th century. Khayyam's work intersecting algebra and geometry is a precursor to the analytic geometry popularized in Descartes' La Geometrie published in 1637. Descartes' La Geometrie refines and generalizes Khayyam's methods and forms a bridge from the medieval mathematics of Khayyam to the modern mathematics of Newton and Leibniz (Rashed and Vahabzadeh, 2000, 20-29). ### 4.2 The Parallel Postulate and the Theory of Ratios The process of reasoning from postulates and definitions has been basic to mathematics at least since the time of Euclid. Islamic geometers were well versed in this art, but also spent some effort examining the logical foundations of the method. They were unafraid to revise and improve upon Euclid's starting points, and they rebuilt Euclid's Elements from the foundation in several ways. Khayyam's "Commentary on the Difficulties of Certain Postulates of Euclid's Work " [14] deals with the two most important issues in this context, the parallel postulate and the definition of equality of ratios. Euclid's fifth "parallel" postulate states that if a line falls on two given lines such that the two interior angles add up to less than two right angles, then the given lines must meet on that side. This statement is logically equivalent to several more easily understood assertions, such as: there is exactly one parallel to a given line that passes through a given point not on the given line; or, the angles of a triangle sum to two right angles. It has been known since the early 19th century that non-Euclidean geometries violate these properties; indeed, it is not yet known whether the space in which we live satisfies them. The parallel postulate, however, was not subject to doubt at Khayyam's time, so it is more appropriate to think of Islamic efforts in this area as part of the tradition of improving upon Euclid rather than as the origin of non-Euclidean geometry. Khayyam's reconstruction of Euclid is one of the better ones: he does not try to prove the parallel postulate. Rather, he replaces it with two statements, which he attributes to Aristotle,[15] that are both simpler and more self-evident: two lines that converge must intersect, and two lines that converge can never diverge in the direction of convergence. Khayyam then replaces Euclid's 29th proposition, the first in which the parallel postulate is used, with a new sequence of eight propositions. Khayyam's insertion amounts to determining that the so-called Saccheri quadrilateral (one with two equal sides perpendicular to the base) is in fact a rectangle. As Giovanni Girolamo Saccheri published this quadrilateral in his book Euclides ab omni naevo vindicatus in 1733, it is also known as the Khayyam-Saccheri quadrilateral. Khayyam believed his approach was an improvement on that of his predecessor Ibn al-Haytham (ca. 965-1040) because his method does not rely on the concept of motion, which Khayyam said should be excluded from geometry. Apparently Nasir al-Din al-Tusi agreed, since he followed Khayyam's path in the next century (Rashed and Vahabzadeh, 2000, 186). Book II of "Commentary on the Difficulties of Certain Postulates of Euclid's Work" takes up the question of the proper definition of ratio. This topic is obscure to the modern reader, but it was fundamental to Greek and medieval mathematics. If the quantities joined in a ratio are whole numbers, then the definition of their ratio poses no difficulty. If the quantities are geometric magnitudes, the situation is more complicated because the two line segments might be incommensurable (in modern terms, their ratio corresponds to an irrational number). Euclid, following Eudoxus, asserts that $A/B = C/D$ when, for any magnitudes $x$ and $y$, the magnitudes $xA$ and $xC$ are both (i) greater than, (ii) equal to, or (iii) less than the magnitudes $yB$ and $yD$ respectively. There is little wonder that Khayyam and others were unhappy with this definition, for while it is clearly true, it does not get at the heart of what it means for ratios to be equal. An alternate approach, which may have existed in ancient Greece but certainly existed in the 9th century, is the "anthyphairetic" definition (Hogendijk 2002). The Euclidean algorithm is an iterative process that is used to find the greatest common divisor of a pair of numbers. The algorithm may be applied equally well to find the greatest common measure of two geometric magnitudes, but the algorithm will never terminate if the ratio between the two magnitudes is irrational. A sequence of divisions within the algorithm results in a "continued fraction" that corresponds to the ratio between the original two quantities. Khayyam, following several earlier Islamic mathematicians, defines the equality of $A/B$ and $C/D$ according to whether their continued fractions are equal. One may wonder why the proponents of the anthyphairetic definition felt that it was more natural than Euclid's approach. There is no doubt, however, that it was preferred; Khayyam even refers to the anthyphairetic definition as the "true" nature of proportionality. Part of the explanation might be simply that the Euclidean algorithm applied to geometric quantities was much more familiar to medieval mathematicians than to us. Khayyam's preference is due to the fact that the anthyphairetic definition allows a ratio to be considered on its own, rather than always in equality to some other ratio. Khayyam's achievement in this topic was not to invent a new definition, but rather to demonstrate that each of the existing definitions logically implies the other. Thus Islamic mathematicians could continue to use ratio theorems from the Elements without having to prove them again according to the anthyphairetic definition. Book III continues the discussion of ratios; Khayyam sets himself the task of demonstrating the seemingly innocuous proposition $A/C = (A/B) (B/C)$, a fact which is used in the Elements but never proved. During this process Khayyam sets an arbitrary fixed magnitude to serve as a unit, to which he relates all other magnitudes of the same kind. This arrangement allows Khayyam to incorporate both numbers and geometric magnitudes within the same system. Thus Khayyam thinks of irrational magnitudes as numbers themselves, which effectively defines the set of "real numbers" that we take for granted today which is composed of rational numbers together with irrational numbers. This revolutionary step to consider irrational numbers as mathematical entities in their own right was one of the most significant advances of conception to occur between ancient Greek and modern mathematics (Youschkevitch and Rosenfeld 1973, 327). ### 4.3 Root Calculations and the Binomial Theorem We know that Khayyam wrote a treatise, now lost, titled "Problems of Arithmetic," involving the determination of $n$-th roots (Youschkevitch and Rosenfeld 1973, 325-326). In his Algebra Khayyam writes that methods for calculating square and cube roots come from India, and that he is the first to extend these methods to the determination of roots of any order (Rashed and Vahabzadeh 2000, 6). Even more interestingly, Khayyam states that he has demonstrated the validity of his methods using proofs that "are only numerical demonstrations based on the arithmetical Books of the Elements" (Rashed and Vahabzadeh 2000, 116-117). If both of these statements are true, then it is hard to avoid the conclusion that Khayyam had used the binomial theorem \[(a + b)^n = a^n + na^{n-1}b + \cdots + nab^{n-1}+ b^n\] in his method, improving on and extending Abu Bakr al-Karaji's (ca. 953-1029) work on binomial expansions. "Pascal's" celebrated arithmetic triangle is a triangular arrangement of the binomial expansion terms' coefficients. Scholars such as Chavoshi advocate renaming Pascal's triangle as "al-Kharaji-Khayyam's" triangle, given how European mathematicians adopted the same method of extracting roots as Khayyam's (Chavoshi 1380 A.H.s., 262-273). The history of Pascal's triangle appears to originate in India with Acharya Pingala's (ca. 200 BCE) study of meter and sound combinations in Sanskrit poetry. The Indian commentator Halayudha wrote a commentary on Pingala's Chandahsutra at the end of the 10th century CE giving the first appearance of a triangular arrangement of the binomial coefficients, solely as a means of finding the number of combinations of short and long syllable patterns. Meanwhile in China, Jia Xian (ca. 1050) had also discovered the method of finding $n$-th roots with Pascal's triangle, as relayed by the 13th century mathematician Yang Hui. However, Jia Xian's work is not extant. Thus, we rely on two lost manuscripts, over approximately the same time period, to determine the Islamic and Chinese origins of Pascal's triangle as a means for finding $n$-th roots (Joseph 2011, 247, 270-281, 354-355). ### 4.4 Astronomy and Other Works Khayyam moved to Isfahan in 1074 to help establish a new observatory under the patronage of Jalal al-Din Malik-shah, the Seljuk sultan, and his vizier, Nizam al-Mulk. Undoubtedly, Khayyam played a major role in the creation of the Maliki or Jalali calendar, the observatory's most significant project. Khayyam succeeded in devising a calendar more accurate than the current Western Gregorian calendar, needing correction of one day every 5,000 years compared to every 3,333 years respectively. In addition to the calendar, the Isfahan observatory produced the Zij Malik-shahi (of which only a fragment of its star catalogue survives), apparently one of the more important astronomical handbooks (Youschkevitch and Rosenfeld 1973, 324). Remaining works attributed to Khayyam in other fields are: 1. "Treatise on the Difficulties of the 'Book of Music' (by Euclid)" (Sharh al-mushkil min kitab al-musiqi). Only one chapter titled "Discussion on Genera Contained in a Fourth" (Al-Qawl 'ala ajnas allati bi'l-arba'a) is extant. For a critical edition see Chavoshi, "Omar al-Khayyam wa'l-musiqi al-nadariyyah," translated into English by M.M. 'Abd al-Jalil, Farhang, vol. 1, 14 (1378 A.H.s.): 29-32 and 203-214. 2. "The Balance of Wisdoms" (Mizan al-hikmah), "Two Treatises on the Level Balance" (Fi'l-qustas al-mustaqim), and "On the Art of Determination of Gold and Silver in a Body Consisting of Them" (Risalah fi'l- ihtiyaj lima 'rifat miqdari al-dhahab wa'l-fidah fi jism murakkab minha) in Danish namah-yi Khayyami, Malik (ed.) 1377 A.H.s., 293-300, and Persian trans., 301-304. 3. "On discovering the truth of Noruz" (Risalah dar kashf haqiqat Noruz) in Danish namah-yi Khayyami, Malik (ed.) 1377 A.H.s., 424-438. Book III of "Commentary on the Difficulties of Certain Postulates of Euclid's Work" shows Khayyam discussing how compounding ratios in music composition are numerical. In this discussion, Khayyam refers to his earlier commentary of Euclid's "Book of Music" (Rashed and Vahabzadeh, 2000, 251-252). However, only one chapter "Discussion on Genera Contained in a Fourth" of Khayyam's commentary discussing tetrachords (a series of four notes separated by three intervals) survives (Aminrazavi 2007, 198-199). Khayyam's treatise "On the Art of Determination of Gold and Silver in a Body Consisting of Them" appears in the book Mizan al-hikmah written by his student Abd'l-Rahman Khazeni (fl. 1115-1130). Khazeni attributes this article, as well as the "Two Treatises on the Level Balance," to Khayyam. Techniques for determining the proportions of gold and silver in a compound date back to Archimedes, and Khayyam's contribution appears to be the precision of his method along with the intricate designs of his level balance (Hall 1981, 341-343). Another treatise titled "On discovering the truth of Noruz" is apocryphal, where only one of its three parts is believed to have been written by Khayyam. Khayyam allegedly writes of the history and mythology of the Persian new year at the beginning of the spring equinox, although the style and errors of the text lend doubt to Khayyam's authorship (Aminrazavi 2007, 38). As with his philosophical and mathematical writings, Khayyam's texts in other disciplines appear to have been taken seriously by his contemporaries and later scholars. ## 5. Khayyam in the West ### 5.1 Orientalism and the European Khayyam The earliest extant translation of the Ruba'iyyat was produced by Thomas Hyde in the 1760s when his translation of a single quatrain appeared in the Veterum Persarum et Parthorum et Medorum Religionis. It was not until the 19th century, however, that the Western world and literary circles discovered Umar Khayyam in all his richness. The voyage of the Ruba'iyyat to the West began when Sir Gore Ouseley, the British ambassador to Iran, presented his collection to the Bodleian Library at Oxford University upon his return to England. In the 1840s Professor Edward Byles Cowell of Oxford University discovered a copy of the Ruba'iyyat of Khayyam and translated several of the Ruba'iyyat. Amazed by their profundity, he shared them with Edward FitzGerald, who took an immediate interest and published the first edition of his own translation in 1859. Four versions of FitzGerald's Ruba'iyyat were published over his lifetime as new quatrains were discovered. Realizing the free nature of his work in his first translation, FitzGerald chose the word rendered to appear on the title page in later editions instead of "translation" (Lange 1968). ### 5.2 The Impact of Khayyam on Western Literary and Philosophical Circles While the connection between the Pre-Raphaelites and Umar Khayyam should not be exaggerated, the relationship that Algernon Charles Swinburne, George Meredith, and Dante G. Rossetti shared with Edward FitzGerald and their mutual admiration of Khayyam cannot be ignored. The salient themes of the Ruba'iyyat became popular among the Pre-Raphaelites and their circle (Lange 1968). Khayyam's popularity led to the formation of the "Omar Khayyam Club of London" (Conway 1893, 305) in 1892, which attracted a number of literary figures and intellectuals. The success of the Club soon led to the simultaneous formation of the Omar Khayyam Clubs of Germany and America. In America, Umar Khayyam was well received in the New England area where his poetry was propagated by the official members of the Omar Khayyam Club of America. The academic community discovered Khayyam's mathematical writings and poetry in the 1880s, when his scholarly articles and translations of his works were published. Some, such as William Edward Story, praised Umar as a mathematician and compared his views with those of Johannes Kepler, Gottfried Wilhelm Leibniz, and Isaac Newton, while others drew their inspiration from his literary tradition and called themselves "Umarians." This new literary movement soon attracted such figures as Mark Twain, who composed forty-five burlesque versions of FitzGerald's quatrains and integrated them with two of FitzGerald's stanzas entitled AGE-A Ruba'iyat (Twain, 1983, 14). The movement also drew the attention of T.S. Eliot's grandfather William Greenleaf Eliot (1811-1887), two of T.S. Eliot's cousins, and T.S. Eliot himself. Umar Khayyam's Ruba'iyyat seems to have elicited two distinct responses among many of his followers in general and the Eliot family in particular: admiration for a rational theology on the one hand, and concern with the rise of skepticism and moral decay in America on the other. Among other figures influenced by the Ruba'iyyat of Umar Khayyam were certain members of the New England School of Transcendentalism, including Henry Wadsworth Longfellow, Ralph Waldo Emerson, and Henry David Thoreau (Aminrazavi 2013; for a complete discussion on Umar Khayyam in the West see Aminrazavi 2007, 204-278). ## 6. Conclusion In the foregoing discussion, we have seen that Umar Khayyam was a philosopher-sage (hakim) and a spiritual-pragmatist whose Ruba'iyyat should be seen as a philosophical commentary on the human condition. The salient features of Umar Khayyam's pioneering work in various branches of mathematics were also discussed. Khayyam's mathematical genius not only produced the most accurate calendar to date, but the issues he treated remained pertinent up until the modern period. For Khayyam, there are two discourses, each of which pertains to one dimension of human existence: philosophical and poetic. Philosophically, Khayyam defended rationalism against the rise of orthodoxy and made an attempt to revive the spirit of rationalism which was so prevalent in the first four centuries in Islam. Poetically, Khayyam represents a voice of protest against what he regards to be a fundamentally unjust world. Many people found in him a voice they needed to hear, and centuries after he had died, his works became a venue for those who were experiencing the same trials and tribulations as Khayyam had.
qt-uncertainty	## 1. Introduction The uncertainty principle is certainly one of the most famous aspects of quantum mechanics. It has often been regarded as the most distinctive feature in which quantum mechanics differs from classical theories of the physical world. Roughly speaking, the uncertainty principle (for position and momentum) states that one cannot assign exact simultaneous values to the position and momentum of a physical system. Rather, these quantities can only be determined with some characteristic "uncertainties" that cannot become arbitrarily small simultaneously. But what is the exact meaning of this principle, and indeed, is it really a principle of quantum mechanics? (In his original work, Heisenberg only speaks of uncertainty relations.) And, in particular, what does it mean to say that a quantity is determined only up to some uncertainty? These are the main questions we will explore in the following, focusing on the views of Heisenberg and Bohr. The notion of "uncertainty" occurs in several different meanings in the physical literature. It may refer to a lack of knowledge of a quantity by an observer, or to the experimental inaccuracy with which a quantity is measured, or to some ambiguity in the definition of a quantity, or to a statistical spread in an ensemble of similarly prepared systems. Also, several different names are used for such uncertainties: inaccuracy, spread, imprecision, indefiniteness, indeterminateness, indeterminacy, latitude, etc. As we shall see, even Heisenberg and Bohr did not decide on a single terminology for quantum mechanical uncertainties. Forestalling a discussion about which name is the most appropriate one in quantum mechanics, we use the name "uncertainty principle" simply because it is the most common one in the literature. ## 2. Heisenberg ### 2.1 Heisenberg's road to the uncertainty relations Heisenberg introduced his famous relations in an article of 1927, entitled Ueber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. A (partial) translation of this title is: "On the anschaulich content of quantum theoretical kinematics and mechanics". Here, the term anschaulich is particularly notable. Apparently, it is one of those German words that defy an unambiguous translation into other languages. Heisenberg's title is translated as "On the physical content ..." by Wheeler and Zurek (1983). His collected works (Heisenberg 1984) translate it as "On the perceptible content ...", while Cassidy's biography of Heisenberg (Cassidy 1992), refers to the paper as "On the perceptual content ...". Literally, the closest translation of the term anschaulich is "visualizable". But, as in most languages, words that make reference to vision are not always intended literally. Seeing is widely used as a metaphor for understanding, especially for immediate understanding. Hence, anschaulich also means "intelligible" or "intuitive".[1] Why was this issue of the Anschaulichkeit of quantum mechanics such a prominent concern to Heisenberg? This question has already been considered by a number of commentators (Jammer 1974; Miller 1982; de Regt 1997; Beller 1999). For the answer, it turns out, we must go back a little in time. In 1925 Heisenberg had developed the first coherent mathematical formalism for quantum theory (Heisenberg 1925). His leading idea was that only those quantities that are in principle observable should play a role in the theory, and that all attempts to form a picture of what goes on inside the atom should be avoided. In atomic physics the observational data were obtained from spectroscopy and associated with atomic transitions. Thus, Heisenberg was led to consider the "transition quantities" as the basic ingredients of the theory. Max Born, later that year, realized that the transition quantities obeyed the rules of matrix calculus, a branch of mathematics that was not so well-known then as it is now. In a famous series of papers Heisenberg, Born and Jordan developed this idea into the matrix mechanics version of quantum theory. Formally, matrix mechanics remains close to classical mechanics. The central idea is that all physical quantities must be represented by infinite self-adjoint matrices (later identified with operators on a Hilbert space). It is postulated that the matrices $\bQ$ and $\bP$ representing the canonical position and momentum variables of a particle satisfy the so-called canonical commutation rule \[\tag{1} \bQ\bP - \bP\bQ = i\hslash\] where $\hslash = h/2\pi$, $h$ denotes Planck's constant, and boldface type is used to represent matrices (or operators). The new theory scored spectacular empirical success by encompassing nearly all spectroscopic data known at the time, especially after the concept of the electron spin was included in the theoretical framework. It came as a big surprise, therefore, when one year later, Erwin Schrodinger presented an alternative theory, that became known as wave mechanics. Schrodinger assumed that an electron in an atom could be represented as an oscillating charge cloud, evolving continuously in space and time according to a wave equation. The discrete frequencies in the atomic spectra were not due to discontinuous transitions (quantum jumps) as in matrix mechanics, but to a resonance phenomenon. Schrodinger also showed that the two theories were equivalent.[2] Even so, the two approaches differed greatly in interpretation and spirit. Whereas Heisenberg eschewed the use of visualizable pictures, and accepted discontinuous transitions as a primitive notion, Schrodinger claimed as an advantage of his theory that it was anschaulich. In Schrodinger's vocabulary, this meant that the theory represented the observational data by means of continuously evolving causal processes in space and time. He considered this condition of Anschaulichkeit to be an essential requirement on any acceptable physical theory. Schrodinger was not alone in appreciating this aspect of his theory. Many other leading physicists were attracted to wave mechanics for the same reason. For a while, in 1926, before it emerged that wave mechanics had serious problems of its own, Schrodinger's approach seemed to gather more support in the physics community than matrix mechanics. Understandably, Heisenberg was unhappy about this development. In a letter of 8 June 1926 to Pauli he confessed that "The more I think about the physical part of Schrodinger's theory, the more disgusting I find it", and: "What Schrodinger writes about the Anschaulichkeit of his theory, ... I consider Mist" (Pauli 1979: 328). Again, this last German term is translated differently by various commentators: as "junk" (Miller 1982) "rubbish" (Beller 1999) "crap" (Cassidy 1992), "poppycock" (Bacciagaluppi & Valentini 2009) and perhaps more literally, as "bullshit" (Moore 1989; de Regt 1997). Nevertheless, in published writings, Heisenberg voiced a more balanced opinion. In a paper in Die Naturwissenschaften (1926) he summarized the peculiar situation that the simultaneous development of two competing theories had brought about. Although he argued that Schrodinger's interpretation was untenable, he admitted that matrix mechanics did not provide the Anschaulichkeit which made wave mechanics so attractive. He concluded: > > > to obtain a contradiction-free anschaulich interpretation, we > still lack some essential feature in our image of the structure of > matter. > > > The purpose of his 1927 paper was to provide exactly this lacking feature. ### 2.2 Heisenberg's argument Let us now look at the argument that led Heisenberg to his uncertainty relations. He started by redefining the notion of Anschaulichkeit. Whereas Schrodinger associated this term with the provision of a causal space-time picture of the phenomena, Heisenberg, by contrast, declared: > > > We believe we have gained anschaulich understanding of a > physical theory, if in all simple cases, we can grasp the experimental > consequences qualitatively and see that the theory does not lead to > any contradictions. Heisenberg 1927: 172) > > > His goal was, of course, to show that, in this new sense of the word, matrix mechanics could lay the same claim to Anschaulichkeit as wave mechanics. To do this, he adopted an operational assumption: terms like "the position of a particle" have meaning only if one specifies a suitable experiment by which "the position of a particle" can be measured. We will call this assumption the "measurement=meaning principle". In general, there is no lack of such experiments, even in the domain of atomic physics. However, experiments are never completely accurate. We should be prepared to accept, therefore, that in general the meaning of these quantities is also determined only up to some characteristic inaccuracy. As an example, he considered the measurement of the position of an electron by a microscope. The accuracy of such a measurement is limited by the wave length of the light illuminating the electron. Thus, it is possible, in principle, to make such a position measurement as accurate as one wishes, by using light of a very short wave length, e.g., $\gamma$-rays. But for $\gamma$-rays, the Compton effect cannot be ignored: the interaction of the electron and the illuminating light should then be considered as a collision of at least one photon with the electron. In such a collision, the electron suffers a recoil which disturbs its momentum. Moreover, the shorter the wave length, the larger is this change in momentum. Thus, at the moment when the position of the particle is accurately known, Heisenberg argued, its momentum cannot be accurately known: > > > At the instant of time when the position is determined, that is, at > the instant when the photon is scattered by the electron, the electron > undergoes a discontinuous change in momentum. This change is the > greater the smaller the wavelength of the light employed, i.e., the > more exact the determination of the position. At the instant at which > the position of the electron is known, its momentum therefore can be > known only up to magnitudes which correspond to that discontinuous > change; thus, the more precisely the position is determined, the less > precisely the momentum is known, and conversely. (Heisenberg 1927: > 174-5) > > > This is the first formulation of the uncertainty principle. In its present form it is an epistemological principle, since it limits what we can know about the electron. From "elementary formulae of the Compton effect" Heisenberg estimated the "imprecisions" to be of the order \[\tag{2} \delta p\delta q \sim h\] He continued: "In this circumstance we see the direct anschaulich content of the relation $\boldsymbol{QP} - \boldsymbol{PQ} = i\hslash$." He went on to consider other experiments, designed to measure other physical quantities and obtained analogous relations for time and energy: \[\tag{3} \delta t \delta E \sim h\] and action $J$ and angle $w$ \[\tag{4} \delta w \delta J \sim h\] which he saw as corresponding to the "well-known" relations \[\tag{5} \boldsymbol{tE} - \boldsymbol{Et} = i\hslash \text{ or } \boldsymbol{wJ} - \boldsymbol{Jw} = i\hslash\] However, these generalisations are not as straightforward as Heisenberg suggested. In particular, the status of the time variable in his several illustrations of relation (3) is not at all clear (Hilgevoord 2005; see also Section 2.5). Heisenberg summarized his findings in a general conclusion: all concepts used in classical mechanics are also well-defined in the realm of atomic processes. But, as a pure fact of experience (rein erfahrungsgemass), experiments that serve to provide such a definition for one quantity are subject to particular indeterminacies, obeying relations (2)-(4) which prohibit them from providing a simultaneous definition of two canonically conjugate quantities. Note that in this formulation the emphasis has slightly shifted: he now speaks of a limit on the definition of concepts, i.e., not merely on what we can know, but what we can meaningfully say about a particle. Of course, this stronger formulation follows by application of the above measurement=meaning principle: if there are, as Heisenberg claims, no experiments that allow a simultaneous precise measurement of two conjugate quantities, then these quantities are also not simultaneously well-defined. Heisenberg's paper has an interesting "Addition in proof" mentioning critical remarks by Bohr, who saw the paper only after it had been sent to the publisher. Among other things, Bohr pointed out that in the microscope experiment it is not the change of the momentum of the electron that is important, but rather the circumstance that this change cannot be precisely determined in the same experiment. An improved version of the argument, responding to this objection, is given in Heisenberg's Chicago lectures of 1930. Here (Heisenberg 1930: 16), it is assumed that the electron is illuminated by light of wavelength $\lambda$ and that the scattered light enters a microscope with aperture angle $\varepsilon$. According to the laws of classical optics, the accuracy of the microscope depends on both the wave length and the aperture angle; Abbe's criterium for its "resolving power", i.e., the size of the smallest discernable details, gives \[\tag{6} \delta q \sim \frac{\lambda}{\sin \varepsilon}.\] On the other hand, the direction of a scattered photon, when it enters the microscope, is unknown within the angle $\varepsilon$, rendering the momentum change of the electron uncertain by an amount \[\tag{7} \delta p \sim \frac{h \sin \varepsilon}{\lambda}\] leading again to the result (2). Let us now analyse Heisenberg's argument in more detail. Note that, even in this improved version, Heisenberg's argument is incomplete. According to Heisenberg's "measurement=meaning principle", one must also specify, in the given context, what the meaning is of the phrase "momentum of the electron", in order to make sense of the claim that this momentum is changed by the position measurement. A solution to this problem can again be found in the Chicago lectures (Heisenberg 1930: 15). Here, he assumes that initially the momentum of the electron is precisely known, e.g., it has been measured in a previous experiment with an inaccuracy $\delta p\_{i}$, which may be arbitrarily small. Then, its position is measured with inaccuracy $\delta q$, and after this, its final momentum is measured with an inaccuracy $\delta p\_{f}$. All three measurements can be performed with arbitrary precision. Thus, the three quantities $\delta p\_{i}, \delta q$, and $\delta p\_{f}$ can be made as small as one wishes. If we assume further that the initial momentum has not changed until the position measurement, we can speak of a definite momentum until the time of the position measurement. Moreover we can give operational meaning to the idea that the momentum is changed during the position measurement: the outcome of the second momentum measurement (say $p\_{f}$ will generally differ from the initial value $p\_{i}$. In fact, one can also show that this change is discontinuous, by varying the time between the three measurements. Let us try to see, adopting this more elaborate set-up, if we can complete Heisenberg's argument. We have now been able to give empirical meaning to the "change of momentum" of the electron, $p\_{f} - p\_{i}$. Heisenberg's argument claims that the order of magnitude of this change is at least inversely proportional to the inaccuracy of the position measurement: \[\tag{8} \abs{p\_{f} - p\_{i}} \delta q \sim h\] However, can we now draw the conclusion that the momentum is only imprecisely defined? Certainly not. Before the position measurement, its value was $p\_{i}$, after the measurement it is $p\_{f}$. One might, perhaps, claim that the value at the very instant of the position measurement is not yet defined, but we could simply settle this by a convention, e.g., we might assign the mean value $(p\_{i} + p\_{f})/2$ to the momentum at this instant. But then, the momentum is precisely determined at all instants, and Heisenberg's formulation of the uncertainty principle no longer follows. The above attempt of completing Heisenberg's argument thus overshoots its mark. A solution to this problem can again be found in the Chicago Lectures. Heisenberg admits that position and momentum can be known exactly. He writes: > > > If the velocity of the electron is at first known, and the position > then exactly measured, the position of the electron for times previous > to the position measurement may be calculated. For these past times, > $\delta p\delta q$ is smaller than the usual bound. (Heisenberg > 1930: 15) > > > Indeed, Heisenberg says: "the uncertainty relation does not hold for the past". Apparently, when Heisenberg refers to the uncertainty or imprecision of a quantity, he means that the value of this quantity cannot be given beforehand. In the sequence of measurements we have considered above, the uncertainty in the momentum after the measurement of position has occurred, refers to the idea that the value of the momentum is not fixed just before the final momentum measurement takes place. Once this measurement is performed, and reveals a value $p\_{f}$, the uncertainty relation no longer holds; these values then belong to the past. Clearly, then, Heisenberg is concerned with unpredictability: the point is not that the momentum of a particle changes, due to a position measurement, but rather that it changes by an unpredictable amount. It is, however always possible to measure, and hence define, the size of this change in a subsequent measurement of the final momentum with arbitrary precision. Although Heisenberg admits that we can consistently attribute values of momentum and position to an electron in the past, he sees little merit in such talk. He points out that these values can never be used as initial conditions in a prediction about the future behavior of the electron, or subjected to experimental verification. Whether or not we grant them physical reality is, as he puts it, a matter of personal taste. Heisenberg's own taste is, of course, to deny their physical reality. For example, he writes, > > > I believe that one can formulate the emergence of the classical > "path" of a particle succinctly as follows: the > "path" comes into being only because we observe it. > (Heisenberg 1927: 185) > > > Apparently, in his view, a measurement does not only serve to give meaning to a quantity, it creates a particular value for this quantity. This may be called the "measurement=creation" principle. It is an ontological principle, for it states what is physically real. This then leads to the following picture. First we measure the momentum of the electron very accurately. By "measurement= meaning", this entails that the term "the momentum of the particle" is now well-defined. Moreover, by the "measurement=creation" principle, we may say that this momentum is physically real. Next, the position is measured with inaccuracy $\delta q$. At this instant, the position of the particle becomes well-defined and, again, one can regard this as a physically real attribute of the particle. However, the momentum has now changed by an amount that is unpredictable by an order of magnitude $\abs{p\_{f} - p\_{i}} \sim h/\delta q$. The meaning and validity of this claim can be verified by a subsequent momentum measurement. The question is then what status we shall assign to the momentum of the electron just before its final measurement. Is it real? According to Heisenberg it is not. Before the final measurement, the best we can attribute to the electron is some unsharp, or fuzzy momentum. These terms are meant here in an ontological sense, characterizing a real attribute of the electron. ### 2.3 The interpretation of Heisenberg's uncertainty relations Heisenberg's relations were soon considered to be a cornerstone of the Copenhagen interpretation of quantum mechanics. Just a few months later, Kennard (1927) already called them the "essential core" of the new theory. Taken together with Heisenberg's contention that they provide the intuitive content of the theory and their prominent role in later discussions on the Copenhagen interpretation, a dominant view emerged in which the uncertainty relations were regarded as a fundamental principle of the theory. The interpretation of these relations has often been debated. Do Heisenberg's relations express restrictions on the experiments we can perform on quantum systems, and, therefore, restrictions on the information we can gather about such systems; or do they express restrictions on the meaning of the concepts we use to describe quantum systems? Or else, are they restrictions of an ontological nature, i.e., do they assert that a quantum system simply does not possess a definite value for its position and momentum at the same time? The difference between these interpretations is partly reflected in the various names by which the relations are known, e.g., as "inaccuracy relations", or: "uncertainty", "indeterminacy" or "unsharpness relations". The debate between these views has been addressed by many authors, but it has never been settled completely. Let it suffice here to make only two general observations. First, it is clear that in Heisenberg's own view all the above questions stand or fall together. Indeed, we have seen that he adopted an operational "measurement=meaning" principle according to which the meaningfulness of a physical quantity was equivalent to the existence of an experiment purporting to measure that quantity. Similarly, his "measurement=creation" principle allowed him to attribute physical reality to such quantities. Hence, Heisenberg's discussions moved rather freely and quickly from talk about experimental inaccuracies to epistemological or ontological issues and back again. However, ontological questions seemed to be of somewhat less interest to him. For example, there is a passage (Heisenberg 1927: 197), where he discusses the idea that, behind our observational data, there might still exist a hidden reality in which quantum systems have definite values for position and momentum, unaffected by the uncertainty relations. He emphatically dismisses this conception as an unfruitful and meaningless speculation, because, as he says, the aim of physics is only to describe observable data. Similarly, in the Chicago Lectures, he warns against the fact that the human language permits the utterance of statements which have no empirical content, but nevertheless produce a picture in our imagination. He notes, > > > One should be especially careful in using the words > "reality", "actually", etc., since these words > very often lead to statements of the type just mentioned. (Heisenberg > 1930: 11) > > > So, Heisenberg also endorsed an interpretation of his relations as rejecting a reality in which particles have simultaneous definite values for position and momentum. The second observation is that although for Heisenberg experimental, informational, epistemological and ontological formulations of his relations were, so to say, just different sides of the same coin, this is not so for those who do not share his operational principles or his view on the task of physics. Alternative points of view, in which e.g., the ontological reading of the uncertainty relations is denied, are therefore still viable. The statement, often found in the literature of the thirties, that Heisenberg had proved the impossibility of associating a definite position and momentum to a particle is certainly wrong. But the precise meaning one can coherently attach to Heisenberg's relations depends rather heavily on the interpretation one favors for quantum mechanics as a whole. And because no agreement has been reached on this latter issue, one cannot expect agreement on the meaning of the uncertainty relations either. ### 2.4 Uncertainty relations or uncertainty principle? Let us now move to another question about Heisenberg's relations: do they express a principle of quantum theory? Probably the first influential author to call these relations a "principle" was Eddington, who, in his Gifford Lectures of 1928 referred to them as the "Principle of Indeterminacy". In the English literature the name uncertainty principle became most common. It is used both by Condon and Robertson in 1929, and also in the English version of Heisenberg's Chicago Lectures (Heisenberg 1930), although, remarkably, nowhere in the original German version of the same book (see also Cassidy 1998). Indeed, Heisenberg never seems to have endorsed the name "principle" for his relations. His favourite terminology was "inaccuracy relations" (Ungenauigkeitsrelationen) or "indeterminacy relations" (Unbestimmtheitsrelationen). We know only one passage, in Heisenberg's own Gifford lectures, delivered in 1955-56 (Heisenberg 1958: 43), where he mentioned that his relations "are usually called relations of uncertainty or principle of indeterminacy". But this can well be read as his yielding to common practice rather than his own preference. But does the relation (2) qualify as a principle of quantum mechanics? Several authors, foremost Karl Popper (1967), have contested this view. Popper argued that the uncertainty relations cannot be granted the status of a principle on the grounds that they are derivable from the theory, whereas one cannot obtain the theory from the uncertainty relations. (The argument being that one can never derive any equation, say, the Schrodinger equation, or the commutation relation (1), from an inequality.) Popper's argument is, of course, correct but we think it misses the point. There are many statements in physical theories which are called principles even though they are in fact derivable from other statements in the theory in question. A more appropriate departing point for this issue is not the question of logical priority but rather Einstein's distinction between "constructive theories" and "principle theories". Einstein proposed this famous classification in Einstein 1919. Constructive theories are theories which postulate the existence of simple entities behind the phenomena. They endeavour to reconstruct the phenomena by framing hypotheses about these entities. Principle theories, on the other hand, start from empirical principles, i.e., general statements of empirical regularities, employing no or only a bare minimum of theoretical terms. The purpose is to build up the theory from such principles. That is, one aims to show how these empirical principles provide sufficient conditions for the introduction of further theoretical concepts and structure. The prime example of a theory of principle is thermodynamics. Here the role of the empirical principles is played by the statements of the impossibility of various kinds of perpetual motion machines. These are regarded as expressions of brute empirical fact, providing the appropriate conditions for the introduction of the concepts of energy and entropy and their properties. (There is a lot to be said about the tenability of this view, but that is not our topic here.) Now obviously, once the formal thermodynamic theory is built, one can also derive the impossibility of the various kinds of perpetual motion. (They would violate the laws of energy conservation and entropy increase.) But this derivation should not misguide one into thinking that they were no principles of the theory after all. The point is just that empirical principles are statements that do not rely on the theoretical concepts (in this case entropy and energy) for their meaning. They are interpretable independently of these concepts and, further, their validity on the empirical level still provides the physical content of the theory. A similar example is provided by special relativity, another theory of principle, which Einstein deliberately designed after the ideal of thermodynamics. Here, the empirical principles are the light postulate and the relativity principle. Again, once we have built up the modern theoretical formalism of the theory (Minkowski space-time), it is straightforward to prove the validity of these principles. But again this does not count as an argument for claiming that they were no principles after all. So the question whether the term "principle" is justified for Heisenberg's relations, should, in our view, be understood as the question whether they are conceived of as empirical principles. One can easily show that this idea was never far from Heisenberg's intentions. We have already seen that Heisenberg presented the relations as the result of a "pure fact of experience". A few months after his 1927 paper, he wrote a popular paper "Uber die Grundprincipien der Quantenmechanik" ("On the fundamental principles of quantum mechanics") where he made the point even more clearly. Here Heisenberg described his recent break-through in the interpretation of the theory as follows: "It seems to be a general law of nature that we cannot determine position and velocity simultaneously with arbitrary accuracy". Now actually, and in spite of its title, the paper does not identify or discuss any "fundamental principle" of quantum mechanics. So, it must have seemed obvious to his readers that he intended to claim that the uncertainty relation was a fundamental principle, forced upon us as an empirical law of nature, rather than a result derived from the formalism of the theory. This reading of Heisenberg's intentions is corroborated by the fact that, even in his 1927 paper, applications of his relation frequently present the conclusion as a matter of principle. For example, he says "In a stationary state of an atom its phase is in principle indeterminate" (Heisenberg 1927: 177, [emphasis added]). Similarly, in a paper of 1928, he described the content of his relations as: > > > It has turned out that it is in principle impossible to know, > to measure the position and velocity of a piece of matter with > arbitrary accuracy. (Heisenberg 1984: 26, [emphasis added]) > > > So, although Heisenberg did not originate the tradition of calling his relations a principle, it is not implausible to attribute the view to him that the uncertainty relations represent an empirical principle that could serve as a foundation of quantum mechanics. In fact, his 1927 paper expressed this desire explicitly: > > > Surely, one would like to be able to deduce the quantitative laws of > quantum mechanics directly from their anschaulich > foundations, that is, essentially, relation > [(2)]. > (ibid: 196) > > > This is not to say that Heisenberg was successful in reaching this goal, or that he did not express other opinions on other occasions. Let us conclude this section with three remarks. First, if the uncertainty relation is to serve as an empirical principle, one might well ask what its direct empirical support is. In Heisenberg's analysis, no such support is mentioned. His arguments concerned thought experiments in which the validity of the theory, at least at a rudimentary level, is implicitly taken for granted. Jammer (1974: 82) conducted a literature search for high precision experiments that could seriously test the uncertainty relations and concluded they were still scarce in 1974. Real experimental support for the uncertainty relations in experiments in which the inaccuracies are close to the quantum limit have come about only more recently (see Kaiser, Werner, and George 1983; Uffink 1985; Nairz, Andt, and Zeilinger 2002). A second point is the question whether the theoretical structure or the quantitative laws of quantum theory can indeed be derived on the basis of the uncertainty principle, as Heisenberg wished. Serious attempts to build up quantum theory as a full-fledged Theory of Principle on the basis of the uncertainty principle have never been carried out. Indeed, the most Heisenberg could and did claim in this respect was that the uncertainty relations created "room" (Heisenberg 1927: 180) or "freedom" (Heisenberg 1931: 43) for the introduction of some non-classical mode of description of experimental data, not that they uniquely lead to the formalism of quantum mechanics. A serious proposal to approach quantum mechanics as a theory of principle was provided more recently by Bub (2000) and Chiribella & Spekkens (2016). But, remarkably, this proposal does not use the uncertainty relation as one of its fundamental principles. Third, it is remarkable that in his later years Heisenberg put a somewhat different gloss on his relations. In his autobiography Der Teil und das Ganze of 1969 he described how he had found his relations inspired by a remark by Einstein that "it is the theory which decides what one can observe"--thus giving precedence to theory above experience, rather than the other way around. Some years later he even admitted that his famous discussions of thought experiments were actually trivial since > > > ... if the process of observation itself is subject to the laws > of quantum theory, it must be possible to represent its result in the > mathematical scheme of this theory. (Heisenberg 1975: 6) > > > ### 2.5 Mathematical elaboration When Heisenberg introduced his relation, his argument was based only on qualitative examples. He did not provide a general, exact derivation of his relations.[3] Indeed, he did not even give a definition of the uncertainties $\delta q$, etc., occurring in these relations. Of course, this was consistent with the announced goal of that paper, i.e., to provide some qualitative understanding of quantum mechanics for simple experiments. The first mathematically exact formulation of the uncertainty relations is due to Kennard. He proved in 1927 the theorem that for all normalized state vectors $\ket{\psi}$ the following inequality holds: \[\tag{9} \Delta\_{\psi}\bP \Delta\_{\psi}\bQ \ge \hslash/2 \] Here, $\Delta\_{\psi}\bP$ and $\Delta\_{\psi}\bQ$ are standard deviations of position and momentum in the state vector $\ket{\psi}$, i.e., \[\tag{10} \begin{align\} (\Delta\_{\psi}\bP)^2 &= \expval{\bP^2}\_{\psi} - \expval{\bP}\_{\psi}^2 \\ (\Delta\_{\psi}\bQ)^2 &= \expval{\bQ^2}\_{\psi} - \expval{\bQ}\_{\psi}^2 \end{align\}\] where $\expval{\cdot}\_{\psi} = \expvalexp{\cdot}{\psi}$ denotes the expectation value in state $\ket{\psi}$. Equivalently we can use the wave function $\psi(q)$ and its Fourier transform: \[\begin{align\} \tag{11} \psi(q) &= \braket{q}{\psi} \\ \notag \tilde{\psi}(p) & = \braket{p}{\psi} =\frac{1}{\sqrt{2\pi \hbar} }\int \! \! dq\, e^{-ipq/\hbar} \psi(q) \end{align\}\] to write \[\begin{align\} (\Delta\_\psi {\bQ})^2 & = \! \int\!\! dq\, \abs{\psi(q)}^2 q^2 - \left(\int \!\!dq \, \abs{\psi(q)}^2 q \right)^2 \\ (\Delta\_\psi {\bP})^2 & = \! \int \!\!dp \, \abs{\tilde{\psi}(p)}^2 p^2 - \left(\int\!\!dp \, \abs{\tilde{\psi}(p)}^2 p \right)^2 \end{align\}\] The inequality (9) was generalized by Robertson (1929) who proved that for all observables (self-adjoint operators) $\bA$ and $\bB$: \[\tag{12} \Delta \_{\psi}\bA \Delta\_{\psi}\bB \ge \frac{1}{2} \abs{\expval{[\bA,\bB]}\_{\psi}} \] where $[\bA,\bB] := \bA\bB - \bB\bA$ denotes the commutator. Since the above inequalities (9) and (12) have the virtue of being exact, in contrast to Heisenberg's original semi-quantitative formulation, it is tempting to regard them as the exact counterpart of Heisenberg's relations (2)-(4). Indeed, such was Heisenberg's own view. In his Chicago Lectures (Heisenberg 1930: 15-19), he presented Kennard's derivation of relation (9) and claimed that "this proof does not differ at all in mathematical content" from his semi-quantitative argument, the only difference being that now "the proof is carried through exactly". But it may be useful to point out that both in status and intended role there is a difference between Kennard's inequality and Heisenberg's previous formulation (2). The inequalities discussed here are not statements of empirical fact, but theorems of the quantum mechanical formalism. As such, they presuppose the validity of this formalism, and in particular the commutation relation (1), rather than elucidating its intuitive content or to create "room" or "freedom" for the validity of this formalism. At best, one should see the above inequalities as showing that the formalism is consistent with Heisenberg's empirical principle. This situation is similar to that arising in other theories of principle where, as noted in Section 2.4, one often finds that, next to an empirical principle, the formalism also provides a corresponding theorem. And similarly, this situation should not, by itself, cast doubt on the question whether Heisenberg's relation can be regarded as a principle of quantum mechanics. There is a second notable difference between (2) and (9). Heisenberg did not give a general definition for the "uncertainties" $\delta p$ and $\delta q$. The most definite remark he made about them was that they could be taken as "something like the mean error". In the discussions of thought experiments, he and Bohr would always quantify uncertainties on a case-to-case basis by choosing some parameters which happened to be relevant to the experiment at hand. By contrast, the inequalities (9) and (12) employ a single specific expression as a measure for "uncertainty": the standard deviation. At the time, this choice was not unnatural, given that this expression is well-known and widely used in error theory and the description of statistical fluctuations. However, there was very little or no discussion of whether this choice was appropriate for a general formulation of the uncertainty relations. A standard deviation reflects the spread or expected fluctuations in a series of measurements of an observable in a given state. It is not at all easy to connect this idea with the concept of the "inaccuracy" of a measurement, such as the resolving power of a microscope. In fact, even though Heisenberg had taken Kennard's inequality as the precise formulation of the uncertainty relation, he and Bohr never relied on standard deviations in their many discussions of thought experiments, and indeed, it has been shown (Uffink and Hilgevoord 1985; Hilgevoord and Uffink 1988) that these discussions cannot be framed in terms of standard deviations. Another problem with the above elaboration is that the "well-known" relations (5) are actually false if energy $\boldsymbol{E}$ and action $\boldsymbol{J}$ are to be positive operators (Jordan 1927). In that case, self-adjoint operators $\boldsymbol{t}$ and $\boldsymbol{w}$ do not exist and inequalities analogous to (9) cannot be derived. Also, these inequalities do not hold for angle and angular momentum (Uffink 1990). These obstacles have led to a quite extensive literature on time-energy and angle-action uncertainty relations (Busch 1990; Hilgevoord 1996, 1998, 2005; Muga et al. 2002; Hilgevoord and Atkinson 2011; Pashby 2015). ## 3. Bohr In spite of the fact that Heisenberg's and Bohr's views on quantum mechanics are often lumped together as (part of) "the Copenhagen interpretation", there is considerable difference between their views on the uncertainty relations. ### 3.1 From wave-particle duality to complementarity Long before the development of modern quantum mechanics, Bohr had been particularly concerned with the problem of particle-wave duality, i.e., the problem that experimental evidence on the behaviour of both light and matter seemed to demand a wave picture in some cases, and a particle picture in others. Yet these pictures are mutually exclusive. Whereas a particle is always localized, the very definition of the notions of wavelength and frequency requires an extension in space and in time. Moreover, the classical particle picture is incompatible with the characteristic phenomenon of interference. His long struggle with wave-particle duality had prepared him for a radical step when the dispute between matrix and wave mechanics broke out in 1926-27. For the main contestants, Heisenberg and Schrodinger, the issue at stake was which view could claim to provide a single coherent and universal framework for the description of the observational data. The choice was, essentially between a description in terms of continuously evolving waves, or else one of particles undergoing discontinuous quantum jumps. By contrast, Bohr insisted that elements from both views were equally valid and equally needed for an exhaustive description of the data. His way out of the contradiction was to renounce the idea that the pictures refer, in a literal one-to-one correspondence, to physical reality. Instead, the applicability of these pictures was to become dependent on the experimental context. This is the gist of the viewpoint he called "complementarity". Bohr first conceived the general outline of his complementarity argument in early 1927, during a skiing holiday in Norway, at the same time when Heisenberg wrote his uncertainty paper. When he returned to Copenhagen and found Heisenberg's manuscript, they got into an intense discussion. On the one hand, Bohr was quite enthusiastic about Heisenberg's ideas which seemed to fit wonderfully with his own thinking. Indeed, in his subsequent work, Bohr always presented the uncertainty relations as the symbolic expression of his complementarity viewpoint. On the other hand, he criticized Heisenberg severely for his suggestion that these relations were due to discontinuous changes occurring during a measurement process. Rather, Bohr argued, their proper derivation should start from the indispensability of both particle and wave concepts. He pointed out that the uncertainties in the experiment did not exclusively arise from the discontinuities but also from the fact that in the experiment we need to take into account both the particle theory and the wave theory. It is not so much the unknown disturbance which renders the momentum of the electron uncertain but rather the fact that the position and the momentum of the electron cannot be simultaneously defined in this experiment (see the "Addition in Proof" to Heisenberg's paper). We shall not go too deeply into the matter of Bohr's interpretation of quantum mechanics since we are mostly interested in Bohr's view on the uncertainty principle. For a more detailed discussion of the former we refer to Scheibe (1973), Folse (1985), Honner (1987) and Murdoch (1987). It may be useful, however, to sketch some of the main points. Central in Bohr's considerations is the language we use in physics. No matter how abstract and subtle the concepts of modern physics may be, they are essentially an extension of our ordinary language and a means to communicate the results of our experiments. These results, obtained under well-defined experimental circumstances, are what Bohr calls the "phenomena". A phenomenon is "the comprehension of the effects observed under given experimental conditions" (Bohr 1939: 24), it is the resultant of a physical object, a measuring apparatus and the interaction between them in a concrete experimental situation. The essential difference between classical and quantum physics is that in quantum physics the interaction between the object and the apparatus cannot be made arbitrarily small; the interaction must at least comprise one quantum. This is expressed by Bohr's quantum postulate: > > > [... the] essence [of the formulation of the quantum theory] may > be expressed in the so-called quantum postulate, which attributes to > any atomic process an essential discontinuity or rather individuality, > completely foreign to classical theories and symbolized by > Planck's quantum of action. (Bohr 1928: 580) > > > A phenomenon, therefore, is an indivisible whole and the result of a measurement cannot be considered as an autonomous manifestation of the object itself independently of the measurement context. The quantum postulate forces upon us a new way of describing physical phenomena: > > > In this situation, we are faced with the necessity of a radical > revision of the foundation for the description and explanation of > physical phenomena. Here, it must above all be recognized that, > however far quantum effects transcend the scope of classical physical > analysis, the account of the experimental arrangement and the record > of the observations must always be expressed in common language > supplemented with the terminology of classical physics. (Bohr 1948: > 313) > > > This is what Scheibe (1973) has called the "buffer postulate" because it prevents the quantum from penetrating into the classical description: A phenomenon must always be described in classical terms; Planck's constant does not occur in this description. Together, the two postulates induce the following reasoning. In every phenomenon the interaction between the object and the apparatus comprises at least one quantum. But the description of the phenomenon must use classical notions in which the quantum of action does not occur. Hence, the interaction cannot be analysed in this description. On the other hand, the classical character of the description allows us to speak in terms of the object itself. Instead of saying: "the interaction between a particle and a photographic plate has resulted in a black spot in a certain place on the plate", we are allowed to forgo mentioning the apparatus and say: "the particle has been found in this place". The experimental context, rather than changing or disturbing pre-existing properties of the object, defines what can meaningfully be said about the object. Because the interaction between object and apparatus is left out in our description of the phenomenon, we do not get the whole picture. Yet, any attempt to extend our description by performing the measurement of a different observable quantity of the object, or indeed, on the measurement apparatus, produces a new phenomenon and we are again confronted with the same situation. Because of the unanalyzable interaction in both measurements, the two descriptions cannot, generally, be united into a single picture. They are what Bohr calls complementary descriptions: > > > [the quantum of action]...forces us to adopt a new mode of > description designated as complementary in the sense that any given > application of classical concepts precludes the simultaneous use of > other classical concepts which in a different connection are equally > necessary for the elucidation of the phenomena. (Bohr 1929: 10) > > > The most important example of complementary descriptions is provided by the measurements of the position and momentum of an object. If one wants to measure the position of the object relative to a given spatial frame of reference, the measuring instrument must be rigidly fixed to the bodies which define the frame of reference. But this implies the impossibility of investigating the exchange of momentum between the object and the instrument and we are cut off from obtaining any information about the momentum of the object. If, on the other hand, one wants to measure the momentum of an object the measuring instrument must be able to move relative to the spatial reference frame. Bohr here assumes that a momentum measurement involves the registration of the recoil of some movable part of the instrument and the use of the law of momentum conservation. The looseness of the part of the instrument with which the object interacts entails that the instrument cannot serve to accurately determine the position of the object. Since a measuring instrument cannot be rigidly fixed to the spatial reference frame and, at the same time, be movable relative to it, the experiments which serve to precisely determine the position and the momentum of an object are mutually exclusive. Of course, in itself, this is not at all typical for quantum mechanics. But, because the interaction between object and instrument during the measurement can neither be neglected nor determined the two measurements cannot be combined. This means that in the description of the object one must choose between the assignment of a precise position or of a precise momentum. Similar considerations hold with respect to the measurement of time and energy. Just as the spatial coordinate system must be fixed by means of solid bodies so must the time coordinate be fixed by means of unperturbed, synchronised clocks. But it is precisely this requirement which prevents one from taking into account of the exchange of energy with the instrument if this is to serve its purpose. Conversely, any conclusion about the object based on the conservation of energy prevents following its development in time. The conclusion is that in quantum mechanics we are confronted with a complementarity between two descriptions which are united in the classical mode of description: the space-time description (or coordination) of a process and the description based on the applicability of the dynamical conservation laws. The quantum forces us to give up the classical mode of description (also called the "causal" mode of description by Bohr[4]: it is impossible to form a classical picture of what is going on when radiation interacts with matter as, e.g., in the Compton effect. > > > Any arrangement suited to study the exchange of energy and momentum > between the electron and the photon must involve a latitude in the > space-time description sufficient for the definition of wave-number > and frequency which enter in the relation [$E = h\nu$ and $p = > h\sigma$]. Conversely, any attempt of locating the collision between > the photon and the electron more accurately would, on account of the > unavoidable interaction with the fixed scales and clocks defining the > space-time reference frame, exclude all closer account as regards the > balance of momentum and energy. (Bohr 1949: 210) > > > A causal description of the process cannot be attained; we have to content ourselves with complementary descriptions. "The viewpoint of complementarity may be regarded", according to Bohr, "as a rational generalization of the very ideal of causality". In addition to complementary descriptions Bohr also talks about complementary phenomena and complementary quantities. Position and momentum, as well as time and energy, are complementary quantities.[5] We have seen that Bohr's approach to quantum theory puts heavy emphasis on the language used to communicate experimental observations, which, in his opinion, must always remain classical. By comparison, he seemed to put little value on arguments starting from the mathematical formalism of quantum theory. This informal approach is typical of all of Bohr's discussions on the meaning of quantum mechanics. One might say that for Bohr the conceptual clarification of the situation has primary importance while the formalism is only a symbolic representation of this situation. This is remarkable since, finally, it is the formalism which needs to be interpreted. This neglect of the formalism is one of the reasons why it is so difficult to get a clear understanding of Bohr's interpretation of quantum mechanics and why it has aroused so much controversy. We close this section by citing from an article of 1948 to show how Bohr conceived the role of the formalism of quantum mechanics: > > > The entire formalism is to be considered as a tool for deriving > predictions, of definite or statistical character, as regards > information obtainable under experimental conditions described in > classical terms and specified by means of parameters entering into the > algebraic or differential equations of which the matrices or the > wave-functions, respectively, are solutions. These symbols themselves, > as is indicated already by the use of imaginary numbers, are not > susceptible to pictorial interpretation; and even derived real > functions like densities and currents are only to be regarded as > expressing the probabilities for the occurrence of individual events > observable under well-defined experimental conditions. (Bohr 1948: > 314) > > > ### 3.2 Bohr's view on the uncertainty relations In his Como lecture, published in 1928, Bohr gave his own version of a derivation of the uncertainty relations between position and momentum and between time and energy. He started from the relations \[\tag{13} E = h\nu \text{ and } p = h/\lambda\] which connect the notions of energy $E$ and momentum $p$ from the particle picture with those of frequency $\nu$ and wavelength $\lambda$ from the wave picture. He noticed that a wave packet of limited extension in space and time can only be built up by the superposition of a number of elementary waves with a large range of wave numbers and frequencies. Denoting the spatial and temporal extensions of the wave packet by $\Delta x$ and $\Delta t$, and the extensions in the wave number $\sigma := 1/\lambda$ and frequency by $\Delta \sigma$ and $\Delta \nu$, it follows from Fourier analysis that in the most favorable case $\Delta x \Delta \sigma \approx \Delta t \Delta \nu \approx 1$, and, using (13), one obtains the relations \[\tag{14} \Delta t \Delta E \approx \Delta x \Delta p \approx h\] Note that $\Delta x, \Delta \sigma$, etc., are not standard deviations but unspecified measures of the size of a wave packet. (The original text has equality signs instead of approximate equality signs, but, since Bohr does not define the spreads exactly the use of approximate equality signs seems more in line with his intentions. Moreover, Bohr himself used approximate equality signs in later presentations.) These equations determine, according to Bohr: > > > the highest possible accuracy in the definition of the energy and > momentum of the individuals associated with the wave field. (Bohr > 1928: 571). > He noted, > > > This circumstance may be regarded as a simple symbolic expression of > the complementary nature of the space-time description and the claims > of causality. > (ibid).[6] > > > > We note a few points about Bohr's view on the uncertainty relations. First of all, Bohr does not refer to discontinuous changes in the relevant quantities during the measurement process. Rather, he emphasizes the possibility of defining these quantities. This view is markedly different from Heisenberg's view. A draft version of the Como lecture is even more explicit on the difference between Bohr and Heisenberg: > > > These reciprocal uncertainty relations were given in a recent paper of > Heisenberg as the expression of the statistical element which, due to > the feature of discontinuity implied in the quantum postulate, > characterizes any interpretation of observations by means of classical > concepts. It must be remembered, however, that the uncertainty in > question is not simply a consequence of a discontinuous change of > energy and momentum say during an interaction between radiation and > material particles employed in measuring the space-time coordinates of > the individuals. According to the above considerations the question is > rather that of the impossibility of defining rigorously such a change > when the space-time coordination of the individuals is also > considered. (Bohr 1985: 93) > > > Indeed, Bohr not only rejected Heisenberg's argument that these relations are due to discontinuous disturbances implied by the act of measuring, but also his view that the measurement process creates a definite result: > > > The unaccustomed features of the situation with which we are > confronted in quantum theory necessitate the greatest caution as > regard all questions of terminology. Speaking, as it is often done of > disturbing a phenomenon by observation, or even of creating physical > attributes to objects by measuring processes is liable to be > confusing, since all such sentences imply a departure from conventions > of basic language which even though it can be practical for the sake > of brevity, can never be unambiguous. (Bohr 1939: 24) > > > Nor did he approve of an epistemological formulation or one in terms of experimental inaccuracies: > > > [...] a sentence like "we cannot know both the momentum and > the position of an atomic object" raises at once questions as to > the physical reality of two such attributes of the object, which can > be answered only by referring to the mutual exclusive conditions for > an unambiguous use of space-time concepts, on the one hand, and > dynamical conservation laws on the other hand. (Bohr 1948: 315; also > Bohr 1949: 211) > > > > It would in particular not be out of place in this connection to warn > against a misunderstanding likely to arise when one tries to express > the content of Heisenberg's well-known indeterminacy relation by > such a statement as "the position and momentum of a particle > cannot simultaneously be measured with arbitrary accuracy". > According to such a formulation it would appear as though we had to do > with some arbitrary renunciation of the measurement of either the one > or the other of two well-defined attributes of the object, which would > not preclude the possibility of a future theory taking both attributes > into account on the lines of the classical physics. (Bohr 1937: > 292) > > > Instead, Bohr always stressed that the uncertainty relations are first and foremost an expression of complementarity. This may seem odd since complementarity is a dichotomic relation between two types of description whereas the uncertainty relations allow for intermediate situations between two extremes. They "express" the dichotomy in the sense that if we take the energy and momentum to be perfectly well-defined, symbolically $\Delta E = \Delta p$ = 0, the position and time variables are completely undefined, $\Delta x = \Delta t = \infty$, and vice versa. But they also allow intermediate situations in which the mentioned uncertainties are all non-zero and finite. This more positive aspect of the uncertainty relation is mentioned in the Como lecture: > > > At the same time, however, the general character of this relation > makes it possible to a certain extent to reconcile the conservation > laws with the space-time coordination of observations, the idea of a > coincidence of well-defined events in space-time points being replaced > by that of unsharply defined individuals within space-time regions. > (Bohr 1928: 571) > > > However, Bohr never followed up on this suggestion that we might be able to strike a compromise between the two mutually exclusive modes of description in terms of unsharply defined quantities. Indeed, an attempt to do so, would take the formalism of quantum theory more seriously than the concepts of classical language, and this step Bohr refused to take. Instead, in his later writings he would be content with stating that the uncertainty relations simply defy an unambiguous interpretation in classical terms: > > > These so-called indeterminacy relations explicitly bear out the > limitation of causal analysis, but it is important to recognize that > no unambiguous interpretation of such a relation can be given in words > suited to describe a situation in which physical attributes are > objectified in a classical way. (Bohr 1948: 315) > > > Finally, on a more formal level, we note that Bohr's derivation does not rely on the commutation relations (1) and (5), but on Fourier analysis. These two approaches are equivalent as far as the relationship between position and momentum is concerned, but this is not so for time and energy since most physical systems do not have a time operator. Indeed, in his discussion with Einstein (Bohr 1949), Bohr considered time as a simple classical variable. This even holds for his famous discussion of the "clock-in-the-box" thought-experiment where the time, as defined by the clock in the box, is treated from the point of view of classical general relativity. Thus, in an approach based on commutation relations, the position-momentum and time-energy uncertainty relations are not on equal footing, which is contrary to Bohr's approach in terms of Fourier analysis. For more details see (Hilgevoord 1996 and 1998). ## 4. The Minimal Interpretation In the previous two sections we have seen how both Heisenberg and Bohr attributed a far-reaching status to the uncertainty relations. They both argued that these relations place fundamental limits on the applicability of the usual classical concepts. Moreover, they both believed that these limitations were inevitable and forced upon us. However, we have also seen that they reached such conclusions by starting from radical and controversial assumptions. This entails, of course, that their radical conclusions remain unconvincing for those who reject these assumptions. Indeed, the operationalist-positivist viewpoint adopted by these authors has long since lost its appeal among philosophers of physics. So the question may be asked what alternative views of the uncertainty relations are still viable. Of course, this problem is intimately connected with that of the interpretation of the wave function, and hence of quantum mechanics as a whole. Since there is no consensus about the latter, one cannot expect consensus about the interpretation of the uncertainty relations either. Here we only describe a point of view, which we call the "minimal interpretation", that seems to be shared by both the adherents of the Copenhagen interpretation and of other views. In quantum mechanics a system is supposed to be described by its wave function, also called its quantum state or state vector. Given the state vector $\ket{\psi}$, one can derive probability distributions for all the physical quantities pertaining to the system, usually called its observables, such as its position, momentum, angular momentum, energy, etc. The operational meaning of these probability distributions is that they correspond to the distribution of the values obtained for these quantities in a long series of repetitions of the measurement. More precisely, one imagines a great number of copies of the system under consideration, all prepared in the same way. On each copy the momentum, say, is measured. Generally, the outcomes of these measurements differ and a distribution of outcomes is obtained. The theoretical momentum distribution derived from the quantum state is supposed to coincide with the hypothetical distribution of outcomes obtained in an infinite series of repetitions of the momentum measurement. The same holds, mutatis mutandis, for all the other physical quantities pertaining to the system. Note that no simultaneous measurements of two or more quantities are required in defining the operational meaning of the probability distributions. The uncertainty relations discussed above can be considered as statements about the spreads of the probability distributions of the several physical quantities arising from the same state. For example, the uncertainty relation between the position and momentum of a system may be understood as the statement that the position and momentum distributions cannot both be arbitrarily narrow--in some sense of the word "narrow"--in any quantum state. Inequality (9) is an example of such a relation in which the standard deviation is employed as a measure of spread. From this characterization of uncertainty relations follows that a more detailed interpretation of the quantum state than the one given in the previous paragraph is not required to study uncertainty relations as such. In particular, a further ontological or linguistic interpretation of the notion of uncertainty, as limits on the applicability of our concepts given by Heisenberg or Bohr, need not be supposed. Of course, this minimal interpretation leaves the question open whether it makes sense to attribute precise values of position and momentum to an individual system. Some interpretations of quantum mechanics, e.g., those of Heisenberg and Bohr, deny this; while others, e.g., the interpretation of de Broglie and Bohm insist that each individual system has a definite position and momentum (see the entry on Bohmian mechanics). The only requirement is that, as an empirical fact, it is not possible to prepare pure ensembles in which all systems have the same values for these quantities, or ensembles in which the spreads are smaller than allowed by quantum theory. Although interpretations of quantum mechanics, in which each system has a definite value for its position and momentum are still viable, this is not to say that they are without strange features of their own; they do not imply a return to classical physics. We end with a few remarks on this minimal interpretation. First, it may be noted that the minimal interpretation of the uncertainty relations is little more than filling in the empirical meaning of inequality (9). As such, this view shares many of the limitations we have noted above about this inequality. Indeed, it is not straightforward to relate the spread in a statistical distribution of measurement results with the inaccuracy of this measurement, such as, e.g., the resolving power of a microscope, or of a disturbance of the system by the measurement. Moreover, the minimal interpretation does not address the question whether one can make simultaneous accurate measurements of position and momentum. As a matter of fact, one can show that the standard formalism of quantum mechanics does not allow such simultaneous measurements. But this is not a consequence of relation (9). Rather, it follows from the fact that this formalism simply does not contain any observable that would accomplish such a task. The extension of this formalism that allows observables to be represented by positive-operator-valued measures or POVM's, does allow the formal introduction of observables describing joint measurements (see also section 6.1). But even here, for the case of position and momentum, one finds that such measurements have to be "unsharp", which entails that they cannot be regarded as simultaneous accurate measurements. If one feels that statements about inaccuracy of measurement, or the possibility of simultaneous measurements, belong to any satisfactory formulation of the uncertainty principle, one will need to look for other formulations of the uncertainty principle. Some candidates for such formulations will be discussed in Section 6. First, however, we will look at formulations of the uncertainty principle that stay firmly within the minimal interpretation, and differ from (9) only by using measures of uncertainty other than the standard deviation. ## 5. Alternative measures of uncertainty While the standard deviation is the most well-known quantitative measure for uncertainty or the spread in the probability distribution, it is not the only one, and indeed it has distinctive drawbacks that other such measures may lack. For example, in the definition of the standard deviations (11) one can see that that the probability density function $\abs{\psi(q)}^2$ is weighed by a quadratic factor $q^2$ that puts increasing emphasis on its tails. Therefore, the value of $\Delta\_\psi \bQ$ will depend predominantly at how this density behaves at the tails: if these falls off very quickly, e.g., like a Gaussian, it will be small, but if the tails drop off only slowly the standard deviation may be very large, even when most probability is concentrated in a small interval. The upshot of this objection is that having a lower bound on the product of the standard deviations of position and momentum, as the Heisenberg-Kennard uncertainty relation (9) gives, does not by itself rule out a state where both the probability densities for position and momentum are extremely concentrated, in the sense of having more than $(1- \epsilon)$ of their probability concentrated in a a region of size smaller than $\delta$, for any choice of $\epsilon, \delta >0$. This means, in our view, that relation (9) actually fails to express what most physicists would take to be the very core idea of the uncertainty principle. One way to deal with this objection is to consider alternative measures to quantify the spread or uncertainty associated with a probability density. Here we discuss two such proposals. ### 5.1 Landau-Pollak uncertainty relations The most straightforward alternative is to pick some value $\alpha$ close to one, say $\alpha = 0.9$, and ask for the width of the smallest interval that supports the fraction $\alpha$ of the total probability distribution in position and similarly for momentum: \[\begin{align\} \tag{15} W\_{\alpha}(\bQ, \psi) &:= \inf\_{\abs{I}} \left\{ I: \int\_I {\abs{\psi(q)}}^2 dq \geq \alpha \right\} \\ \notag W\_{\beta}(\bP,\psi) &:= \inf\_I \left\{\int\_I \abs{\tilde\psi(p)}^2 dp \geq \beta \right\} \end{align\}\] In a previous work (Uffink and Hilgevoord 1985) we called such measures bulk widths, because they indicate how concentrated the "bulk" (i.e., fraction $\alpha$ or $\beta$) of the probability distribution is. Landau and Pollak (1961) obtained an uncertainty relation in terms of these bulk widths. \[\begin{align\} \tag{16} W\_\alpha (\bQ, \psi) W\_\beta (\bP, \psi) &\geq 2\pi \hbar \left( \alpha \beta - \sqrt{(1-\alpha)(1-\beta)} \right)^2 \\ \notag &\mbox{if } \alpha+ \beta \geq 1/2 \end{align\}\label{LP}\] This Landau-Pollak inequality shows that if the choices of $\alpha, \beta$ are not too low, there is a state-independent lower bound on the product of the bulk widths of the position and momentum distribution for any quantum state. Note that bulk widths are not so sensitive to the behavior of the tails of the distributions and, therefore, the Landau-Pollak inequality is immune to the objection above.Thus, this inequality expresses constraints on quantum mechanical states not contained in relation (9). Further, by the well-known Bienayme-Chebyshev inequality, one has \[\begin{align\} \tag{17} W\_\alpha (\bQ,\psi) &\leq \frac{2}{\sqrt {1- \alpha}} \Delta\_\psi \bQ \\ \notag W\_\beta (\bP, \psi) &\leq \frac{2}{\sqrt {1- \beta}} \Delta\_\psi \bP \end{align\}\] so that inequality (16) implies (by choosing $\alpha,\beta$ optimal) that $ \Delta\_\psi \bQ \Delta\_\psi \bP \geq 0.12 \hbar $. This, obviously, is not the best lower bound for the product of standard deviations, but the important point is here that the Landau-Pollak inequality (16) in terms of bulk widths implies the existence of a lower bound on the product of standard deviations, while conversely, the Heisenberg-Kennard equality (9) does not imply any bound on the product of bulk widths. A generalization of this approach to non-commuting observables in a finite-dimensional Hilbert space is discussed in Uffink 1990. ### 5.2 Entropic uncertainty relations Another approach to express the uncertainty principle is to use entropic measures of uncertainty. The foremost example of these is the Shannon entropy, which for the position and momentum distribution of a given state vector $\ket{\psi}$ may be defined as: \[\begin{align\} \tag{18} H(\bQ, \psi) &:= -\int \abs{\psi(q)}^2 \ln \abs{\psi(q)}^2 dq \\ \notag H(\bP, \psi) &:= -\int \abs{\tilde{\psi}(p)}^2 \ln \abs{\tilde{\psi}(p)}^2 dp \end{align\}\] One can then show (see Beckner 1975; Bialinicki-Birula and Micielski 1975) that \[\tag{19} H(\bQ, \psi) + H(\bP,\psi) \geq \ln (e \pi \hbar) \] A nice feature of this entropic uncertainty relation is that it provides a strict improvement of the Heisenberg-Kennard relation. That is to say, one can show (independently of quantum theory) that for any probability density function $p(x)$ \[\tag{20} -\int\! p(x) \ln p(x) dx \leq \ln (\sqrt{2 \pi e} \Delta x )\] Applying this to the inequality (19) we get: \[\tag{21} \frac{\hslash}{2} \leq(2\pi e)^{-1} \exp (H(\bQ, \psi) + H(\bP,\psi)) \leq \Delta\_\psi \bQ\Delta\_\psi \bP \] showing that the entropic uncertainty relation implies the Heisenberg-Kennard uncertainty relation. A drawback of this relation is that it does not completely evade the objection mentioned above, (i.e., these entropic measures of uncertainty can become as large as one pleases while $1-\epsilon$ of the probability in the distribution is concentrated on a very small interval), but the examples needed to show this are admittedly more far-fetched. For non-commuting observables in a $n$-dimensional Hilbert space, one can similarly define an entropic uncertainty in the probability distribution $\abs{\braket{a\_i}{\psi}}^2$ for a given state $\ket{\psi}$ and a complete set of eigenstates $\ket{a\_i}$, $ (i= 1, \ldots n)$, of the observable $\bA$: \[\tag{22} H(\bA ,\psi) := -\sum\_{i=1}^n \abs{\braket{a\_i}{\psi}}^2 \ln \abs{\braket{a\_i}{\psi}}^2 \] and $H(\bB,\psi)$ similarly in terms of the probability distribution $\abs{\braket{b\_j}{\psi}}^2$ for a complete set of eigenstates $\ket{b\_j}$, ($j =1, \ldots, n$) of observable $\bB$. Then we obtain the uncertainty relation (Maassen and Uffink 1988): \[\tag{23} H( bA, \psi) + H(\bB, \psi) \geq 2 \ln \max\_{i,j} \abs{\braket{a\_i}{b\_j}}, \] which was further generalized and improved by (Frank and Lieb 2012). The most important advantage of these relations is that, in contrast to Robertson's inequality (12), the lower bound is a positive constant, independent of the state. ## 6. Uncertainty relations for inaccuracy and disturbance Both the standard deviation and the alternative measures of uncertainty considered in the previous subsection (and many more that we have not mentioned!) are designed to indicate the width or spread of a single given probability distribution. Applied to quantum mechanics, where the probability distributions for position and momentum are obtained from a given quantum state vector, one can use them to formulate uncertainty relations that characterize the spread in those distribution for any given state. The resulting inequalities then express limitations on what state-preparations quantum mechanics allows. They are thus expressions of what may be called a preparation uncertainty principle: > > > In quantum mechanics, it is impossible to prepare any system in a > state $\ket{\psi}$ such that its position and momentum > are both precisely predictable, in the sense of having both the > expected spread in a measurement of position and the expected spread > in a momentum measurement arbitrarily small. > > > The relations (9, 16, 19) all belong to this category; the mere difference being that they employ different measures of spread: viz. the standard deviation, the bulk width or the Shannon entropy. Note that in this formulation, there is no reference to simultaneous or joint measurements, nor to any notion of accuracy like the resolving power of the measurement instrument, nor to the issue of how much the system in the state that is being measured is disturbed by this measurement. This section is devoted to attempts that go beyond the mold of this preparation uncertain principle. ### 6.1 The recent debate on error-disturbance relations We have seen that in 1927 Heisenberg argued that the measurement of (say) position must necessarily disturb the conjugate variable (i.e., momentum) by an amount that is inversely proportional to the inaccuracy of measurement of the former. We have also seen that this idea was not maintained in the Kennard's uncertainty relation (9), a relation that was embraced by Heisenberg (1930) and most textbooks. A rather natural question thus arises whether there are further inequalities in quantum mechanics that would address Heisenberg's original thinking more directly, i.e., that do deal with how much one variable is disturbed by the accurate measurement of another. That is, we will look at attempts that would establish a claim which may be called a measurement uncertainty principle. > > > In quantum mechanics, there is no measurement procedure by which one > can accurately measure the position of a system without disturbing it > momentum, in the sense that some measure of inaccuracy in position and > some measure of the disturbance of momentum of the system by the > measurement cannot both be arbitrarily small. > > > This formulation of the uncertainty principle has always remained controversial. Uncertainty relations that would express this alleged principle are often called "error-disturbance" relations or "noise-disturbance" relations We will look at two recent proposals to search for such relations: Ozawa (2003) and Busch, Lahti, and Werner (2013). In Ozawa's approach, we assume that a system $\cal S$ of interest in state $\ket{\psi}$ is coupled to a measurement device $\cal M$ in state $\ket{\chi}$, and their interaction is governed by a unitary operator $U$. On the Hilbert space of the joint system the observable $\bQ$ of the system $\cal S$ we are interested in is represented by \[\tag{24} \bQ\_{\rm in} = \bQ \otimes \mathbb{1}\] The measurement interaction will allow us to perform an (inaccurate) measurement of this quantity by reading off a pointer observable $\boldsymbol{Q'}$ of the measurement device after the interaction. Hence this inaccurate observable may be represented as \[\tag{25} \bQ'\_{\rm out} = U^\dagger( \mathbb{1} \otimes \bQ') U\] The measure of noise in the measurement of $\bQ$ is then chosen as: \[\tag{26} \epsilon\_\psi(\bQ) := \expval{(\bQ'\_{\rm out} - \bQ\_{\rm in})^2}\_{\psi \otimes \chi}^{1/2}\] A comparison of the initial momentum $\bP\_{\rm in} = \bP \otimes \mathbb{1}$ and the final momentum after the measurement $\bP\_{\rm out} = U^\dagger (\bP \otimes \mathbb{1})U$ is made by choosing a measure of the disturbance of $\bP$ by the measurement procedure: \[\tag{27} \eta\_\psi(\bP):= \expval{(\bP\_{\rm in} - \bP\_{\rm out})^2}\_{\psi\otimes\chi}^{1/2} \] Ozawa obtained an inequality involving those two measures, which, however, is more involved than previous uncertainty relations. For our purposes, however, the important point is that Ozawa showed that the product $\epsilon\_\psi (\bQ) \eta\_\psi (\bP)$ has no positive lower bound. His conclusion from this was that Heisenberg's noise-disturbance relation is violated. Yet, whether Ozawa's result indeed succeeds in formulating Heisenberg's qualitative discussion of disturbance and accuracy in the microscope example has come under dispute. See Busch, Lahti and Werner (2013, and 2014 (Other Internet Resources)), and Ozawa (2013, Other Internet Resources). An objection raised in this dispute is that a quantity like $\expval{(\bQ'\_{\rm out} - \bQ\_{\rm in})^2}^{1/2}$ tells us very little about how good the observable ${\bQ'}\_{\rm out}$ can stand in as an inaccurate measurement of $\bQ\_{\rm in}$. The main point to observe here is that these operators generally do not commute, and that measurements of $\bQ'\_{\rm out}$, of $\bQ\_{\rm in}$ and of their difference will require altogether three different measurement contexts. To require that $\epsilon\_\psi(\bQ)$ vanishes, for example, means only that the state prepared belongs to the linear subspace corresponding to the zero eigenvalue of the operator $\bQ'\_{\rm out} - {\bQ}\_{\rm in}$, and therefore that $\expval{\bQ'\_{\rm out}}\_\psi = \expval{\bQ\_{\rm in}}\_\psi$, but this does not preclude that the probability distribution of $\bQ\_{\rm out}$ in state $\psi$ might be wildly different from that of $\bQ\_{\rm in}$. But then no one would think of $\bQ\_{\rm out}$ as an accurate measurement of $\bQ\_{\rm in}$ so that the definition of $\epsilon\_\psi(\bQ)$ does not express what it is supposed to express. A similar objection can also be raised against $\eta\_\psi (\bP)$. Another observation is that Ozawa's conclusion that there is no lower bound for his error-disturbance product for is not at all surprising. That is, even without probing the system by a measurement apparatus, one can show that such a lower bound does not exist. If the initial state of a system is prepared at time $t=0$ as a Gaussian quasi-monochromatic wave packet with $\expval{\bQ\_0}\_\psi =0$ and evolves freely, we can use a time-of-flight measurement to learn about its later position. Ehrenfest's theorem tells us: $\expval{\bQ\_t}\_\psi = \frac{t}{m} \expval{\bP}\_\psi$. Hence, as an approximative measurement of the position $\bQ\_t$, one could propose the observable $\bQ'\_t = \frac{t}{m}\bP$. It is known that under the stated conditions (and with $m$ and $t$ large) this approximation holds very well, i.e., we do not only have $\expval{\bQ'\_t -\bQ\_t}\_\psi =0$, but also $\expval{(\bQ'\_t -\bQ\_t)^2} \approx 0$, as nearly as we please. But since $\bQ'\_t$ is just the momentum multiplied by a constant, its measurement will obviously not disturb the momentum of the system. In other words, for this example, one has $\epsilon\_\psi (\bQ)$ as small as we please with zero disturbance of the momentum. Therefore, any hopes that there could be a positive lower bound for the product $\epsilon\_\psi (\bQ) \eta\_\psi (\bP)$ seem to be dashed, even with the simplest of measurement schemes, i.e. a free evolution. Ozawa's results do not show that Heisenberg's analysis of the microscope argument was wrong. Rather, they throw doubt on the appropriateness of the definitions he used to formalize Heisenberg's informal argument. An entirely different analysis of the problem of substantiating a measurement uncertainty relation was offered by Busch, Lahti, and Werner (2013). These authors consider a measurement device $\cal M$ that makes a joint unsharp measurement of both position and momentum. To describe such joint unsharp measurements, they employ the extended modern formalism that characterizes obervables not by self-adjoint operators but by positive-operator-valued measures (POVM's). In the present case, this means that the measurement procedure is characterized by a collection of positive operators, $M(p,q)$, where the pair $p,q$ represent the outcome variables of the measurement, with \[\tag{28} M(p,q) \geq 0, \iint \! dp dq \, M(p,q) =\mathbb{1} .\] The two marginals of this POVM, \[\tag{29} \begin{align\} M\_1(q) &= \int\! dp M(p,q)\\ M\_2(p) &= \int\! dq M(p,q) \end{align\} \] are also POVM's in their own right and represent the unsharp position $Q'$ and unsharp momentum $P'$ observables respectively. (Note that these do not refer to a self-adjoint operator!) For a system prepared in a state $\ket{\psi}$, the joint probability density of obtaining outcomes $(p,q)$ in the joint unsharp measurement (28) is then \[\tag{30} \rho(p,q) := \expvalexp{M(p,q)}{\psi},\] while the marginals of this joint probability distribution give the distributions for $Q'$ and $P'$. \[\begin{align\} \tag{31} \mu'(q) &: = \int \! dp \, \rho(p,q) = \expvalexp{M\_1(q)}{\psi} \\ \notag \nu'(p) &:= \int \! dp \, \rho(p,q) = \expvalexp{M\_2(q)}{\psi} \end{align\}\] Since a joint sharp measurement of position and momentum is impossible in quantum mechanics, these marginal distributions (31) obtained from $M$ will differ from that of ideal measurements of $\bQ$ and of $\bP$ on the system of interest in state $\ket{\psi}$. However, one can indicate how much these marginals deviate from separate exact position and momentum measurements on the state $\ket{\psi}$ by a pairwise comparison of (31) to the exact distributions \[\begin{align\} \tag{32} \mu(q) &:= \abs{\braket{q}{\psi}}^2 \\ \notag \nu(p) &:= \abs{\braket{p}{\psi}}^2 \end{align\}\] In order to do so, BLW propose a distance function $D$ between probability distributions, such that $D(\mu, \mu')$ tells us how close the marginal position distribution $\mu'(q)$ for the unsharp position $Q'$ is to the exact distribution $\mu(q)$ in a sharp position measurement, and likewise, $D(\nu ,\nu')$ tells us how close the marginal momentum distribution $\nu'(p)$ for $P'$ is to the exact momentum distribution $\nu(p)$. The distance they chose is the Wasserstein-2 distance, a.k.a. (a variation on) the earth-movers distance. Definition (Wasserstein-2 distance) Let $\mu(x)$ and $\mu'(y)$ be any two probability distributions on the real line, and $\gamma(x,y)$ any joint probability distribution that has $\mu'$ and $\mu$ as its marginals. Then: \[\tag{33} D(\mu, \mu') := \inf\_\gamma \left(\iint (x-y)^2 \gamma (x,y) dx dy \right)^{1/2}\] Applying this definition to the case at hand, i.e. pairwise to the quantum mechanical distributions $\mu'(q)$ and $\mu(q)$ and to $\nu'(p)$ and $\nu(p)$ in (31) and (32), BLW's final step is to take a supremum over all possible input states $\ket{\psi}$ to obtain \[\tag{34} \begin{align\} \Delta(Q, Q') & = \sup\_{\ket{\psi}} D(\mu, \mu') \\ \Delta(P, P') & = \sup\_{\ket{\psi}} D(\nu, \nu') \end{align\} \] From these definitions, they obtain \[\tag{35}\Delta(Q, Q') \Delta (P,P') \geq \frac{\hbar}{2}\] Arguing that $\Delta(Q, Q')$ provides a sensible measure for the inaccuracy or noise about position, and $\Delta(P, P')$ for the disturbance of momentum by any such joint unsharp measurement, the authors conclude, in contrast to Ozawa's analysis, that an error-disturbance uncertainty relation does hold, which they take as "a remarkable vindication of Heisenberg's intuitions" in the microscope thought experiment. In comparison of the two, there are a few positive remarks to make about the Busch-Lahti-Werner (BLW) approach. First of all, by focusing on the distance (33) this approach is comparing entire probability distributions rather than just the expectations of operator differences. When this distance is very small, one is justified to conclude that the distribution has changed very little under the measurement procedure. This brings us closer to the conclusion that the error or disturbance introduced is small. Secondly, by introducing a supremum over all states to obtain $\Delta( Q, Q')$, it follows that when this latter expression is small, the measured distribution $\mu'$ differs only little from the exact distribution $\mu$ whatever the state of the system is. As the authors argue, this means that $\Delta(Q,Q')$ can be seen as a figure-of-merit of the measurement device alone, and in this sense analogous to the resolving power of a microscope. But we also think there is an undesirable feature of the BLW approach. This is due to the supremum over states appearing twice, both in $\Delta(Q,Q')$ and in $\Delta(P,P')$. This feature, we argue, deprives their result from practical applicability. To elucidate: In concrete applications, one would prepare a system in some state (not exactly known) and perform a given joint measurement $M$ of $Q'$ and $P'$. If it is given that, say, $\Delta(Q,Q')$ is very small, one can safely infer that $Q$ has been measured with small inaccuracy, since this guarantees that the measured position distribution differs very little from what an exact position measurement would give, regardless of the state of the system. Now, one would like to be able to infer that in this case the disturbance of the momentum $P$ from $P'$ must be considerable for the state prepared. But the BLW only gives us: \[ \Delta(P, P') = \sup\_{\ket{\psi}} D(\nu, \nu') \geq \frac{\hbar}{2 \Delta(Q, Q')} \] and this does not imply anything for the state in question! Thus, the BLW uncertainty relation does not rule out that for some states it might be possible to perform a joint measurement in which both $D(\mu, \mu')$ and $D(\nu, \nu')$ are very small, and in this sense have negligibe error and disturbance. It seems premature to say that this vindicates Heisenberg's intuitions. Summing up, we emphasize that there is no contradiction between the BLW analysis and the Ozawa analysis: where Ozawa claims that the product of two quantities might for some states be less than the usual limit, BLW show that product of different quantities will satisfy this limit. The dispute is not about mathematically validity, but about how reasonable these quantities are to capture Heisenberg's qualitative considerations. The present authors feel that, in this dispute, Ozawa's analysis fail to be convincing. On the other hand, we also think that the BLW uncertainty relation is not satisfactory. Also, we would like to remark that both protagonists employ measures that are akin to standard deviations in being very sensitive to the tail behavior of probability distributions, and thus face a similar objection as raised in section 5. The final word in this dispute on whether a measurement uncertainty principle holds has not been reached, in our view.
scientific-underdetermination	## 1. A First Look: Duhem, Quine, and the Problems of Underdetermination The scope of the epistemic challenge arising from underdetermination is not limited only to scientific contexts, as is perhaps most readily seen in classical skeptical attacks on our knowledge more generally. Rene Descartes ([1640] 1996) famously sought to doubt any and all of his beliefs which could possibly be doubted by supposing that there might be an all-powerful Evil Demon who sought to deceive him. Descartes' challenge appeals to a form of underdetermination: he notes that all our sensory experiences would be just the same if they were caused by this Evil Demon rather than an external world of rocks and trees. Likewise, Nelson Goodman's (1955) "New Riddle of Induction" turns on the idea that the evidence we now have could equally well be taken to support inductive generalizations quite different from those we usually take them to support, with radically different consequences for the course of future events.[1] Nonetheless, underdetermination has been thought to arise in scientific contexts in a variety of distinctive and important ways that do not simply recreate such radically skeptical possibilities. The traditional locus classicus for underdetermination in science is the work of Pierre Duhem, a French physicist as well as historian and philosopher of science who lived at the turn of the 20th Century. In The Aim and Structure of Physical Theory, Duhem formulated various problems of scientific underdetermination in an especially perspicuous and compelling way, although he himself argued that these problems posed serious challenges only to our efforts to confirm theories in physics. In the middle of the 20th Century, W. V. O. Quine suggested that such challenges applied not only to the confirmation of all types of scientific theories, but to all knowledge claims whatsoever. His incorporation and further development of these problems as part of a general account of human knowledge was one of the most significant developments of 20th Century epistemology. But neither Duhem nor Quine was careful to systematically distinguish a number of fundamentally distinct lines of thinking about underdetermination found in their work. Perhaps the most important division is between what we might call holist and contrastive forms of underdetermination. Holist underdetermination (Section 2 below) arises whenever our inability to test hypotheses in isolation leaves us underdetermined in our response to a failed prediction or some other piece of disconfirming evidence. That is, because hypotheses have empirical implications or consequences only when conjoined with other hypotheses and/or background beliefs about the world, a failed prediction or falsified empirical consequence typically leaves open to us the possibility of blaming and abandoning one of these background beliefs and/or 'auxiliary' hypotheses rather than the hypothesis we set out to test in the first place. But contrastive underdetermination (Section 3 below) involves the quite different possibility that for any body of evidence confirming a theory, there might well be other theories that are also well confirmed by that very same body of evidence. Moreover, claims of underdetermination of either of these two fundamental varieties can vary in strength and character in any number of ways: one might, for example, suggest that the choice between two theories or two ways of revising our beliefs is transiently underdetermined simply by the evidence we happen to have at present, or instead permanently underdetermined by all possible evidence. Indeed, the variety of forms of underdetermination that confront scientific inquiry, and the causes and consequences claimed for these different varieties, are sufficiently heterogeneous that attempts to address "the" problem of underdetermination for scientific theories have often engendered considerable confusion and argumentation at cross-purposes.[2] Moreover, such differences in the character and strength of various claims of underdetermination turn out to be crucial for resolving the significance of the issue. For example, in some recently influential discussions of science it has become commonplace for scholars in a wide variety of academic disciplines to make casual appeal to claims of underdetermination (especially of the holist variety) to support the idea that something besides evidence must step in to do the further work of determining beliefs and/or changes of belief in scientific contexts. Perhaps most prominent among these are adherents of the sociology of scientific knowledge (SSK) movement and some feminist science critics who have argued that it is typically the sociopolitical interests and/or pursuit of power and influence by scientists themselves which play a crucial and even decisive role in determining which beliefs are actually abandoned or retained in response to conflicting evidence. As we will see in Section 2.2, however, Larry Laudan has argued that such claims depend upon simple equivocation between comparatively weak or trivial forms of underdetermination and the far stronger varieties from which they draw radical conclusions about the limited reach of evidence and rationality in science. In the sections that follow we will seek to clearly characterize and distinguish the various forms of both holist and contrastive underdetermination that have been suggested to arise in scientific contexts (noting some important connections between them along the way), assess the strength and significance of the heterogeneous argumentative considerations offered in support of and against them, and consider just which forms of underdetermination pose genuinely consequential challenges for scientific inquiry. ## 2. Holist Underdetermination and Challenges to Scientific Rationality ### 2.1 Holist Underdetermination: The Very Idea Duhem's original case for holist underdetermination is, perhaps unsurprisingly, intimately bound up with his arguments for confirmational holism: the claim that theories or hypotheses can only be subjected to empirical testing in groups or collections, never in isolation. The idea here is that a single scientific hypothesis does not by itself carry any implications about what we should expect to observe in nature; rather, we can derive empirical consequences from an hypothesis only when it is conjoined with many other beliefs and hypotheses, including background assumptions about the world, beliefs about how measuring instruments operate, further hypotheses about the interactions between objects in the original hypothesis' field of study and the surrounding environment, etc. For this reason, Duhem argues, when an empirical prediction is falsified, we do not know whether the fault lies with the hypothesis we originally sought to test or with one of the many other beliefs and hypotheses that were also needed and used to generate the failed prediction: > > A physicist decides to demonstrate the inaccuracy of a proposition; in > order to deduce from this proposition the prediction of a phenomenon > and institute the experiment which is to show whether this phenomenon > is or is not produced, in order to interpret the results of this > experiment and establish that the predicted phenomenon is not > produced, he does not confine himself to making use of the proposition > in question; he makes use also of a whole group of theories accepted > by him as beyond dispute. The prediction of the phenomenon, whose > nonproduction is to cut off debate, does not derive from the > proposition challenged if taken by itself, but from the proposition at > issue joined to that whole group of theories; if the predicted > phenomenon is not produced, the only thing the experiment teaches us > is that among the propositions used to predict the phenomenon and to > establish whether it would be produced, there is at least one error; > but where this error lies is just what it does not tell us. ([1914] > 1954, 185) > Duhem supports this claim with examples from physical theory, including one designed to illustrate a celebrated further consequence he draws from it. Holist underdetermination ensures, Duhem argues, that there cannot be any such thing as a "crucial experiment" (experimentum crucis): a single experiment whose outcome is predicted differently by two competing theories and which therefore serves to definitively confirm one and refute the other. For example, in a famous scientific episode intended to resolve the ongoing heated battle between partisans of the theory that light consists of a stream of particles moving at extremely high speed (the particle or "emission" theory of light) and defenders of the view that light consists instead of waves propagated through a mechanical medium (the wave theory), the physicist Foucault designed an apparatus to test the two theories' competing claims about the speed of transmission of light in different media: the particle theory implied that light would travel faster in water than in air, while the wave theory implied that the reverse was true. Although the outcome of the experiment was taken to show that light travels faster in air than in water,[3] Duhem argues that this is far from a refutation of the hypothesis of emission: > > in fact, what the experiment declares stained with error is the whole > group of propositions accepted by Newton, and after him by Laplace and > Biot, that is, the whole theory from which we deduce the relation > between the index of refraction and the velocity of light in various > media. But in condemning this system as a whole by declaring it > stained with error, the experiment does not tell us where the error > lies. Is it in the fundamental hypothesis that light consists in > projectiles thrown out with great speed by luminous bodies? Is it in > some other assumption concerning the actions experienced by light > corpuscles due to the media in which they move? We know nothing about > that. It would be rash to believe, as Arago seems to have thought, > that Foucault's experiment condemns once and for all the very > hypothesis of emission, i.e., the assimilation of a ray of light to a > swarm of projectiles. If physicists had attached some value to this > task, they would undoubtedly have succeeded in founding on this > assumption a system of optics that would agree with Foucault's > experiment. ([1914] 1954, p. 187) > From this and similar examples, Duhem drew the quite general conclusion that our response to the experimental or observational falsification of a theory is always underdetermined in this way. When the world does not live up to our theory-grounded expectations, we must give up something, but because no hypothesis is ever tested in isolation, no experiment ever tells us precisely which belief it is that we must revise or give up as mistaken: > > In sum, the physicist can never subject an isolated hypothesis to > experimental test, but only a whole group of hypotheses; when the > experiment is in disagreement with his predictions, what he learns is > that at least one of the hypotheses constituting this group is > unacceptable and ought to be modified; but the experiment does not > designate which one should be changed. ([1914] 1954, 187) > The predicament Duhem here identifies is no mere rainy day puzzle for philosophers of science, but a methodological challenge that consistently arises in the course of scientific practice itself. It is simply not true that for practical purposes and in concrete contexts there is always just a single revision of our beliefs in response to disconfirming evidence that is obviously correct, most promising, or even most sensible to pursue. To cite a classic example, when Newton's celestial mechanics failed to correctly predict the orbit of Uranus, scientists at the time did not simply abandon the theory but protected it from refutation by instead challenging the background assumption that the solar system contained only seven planets. This strategy bore fruit, notwithstanding the falsity of Newton's theory: by calculating the location of a hypothetical eighth planet influencing the orbit of Uranus, the astronomers Adams and Leverrier were eventually led to discover Neptune in 1846. But the very same strategy failed when used to try to explain the advance of the perihelion in Mercury's orbit by postulating the existence of "Vulcan", an additional planet located between Mercury and the sun, and this phenomenon would resist satisfactory explanation until the arrival of Einstein's theory of general relativity. So it seems that Duhem was right to suggest not only that hypotheses must be tested as a group or a collection, but also that it is by no means a foregone conclusion which member of such a collection should be abandoned or revised in response to a failed empirical test or false implication. Indeed, this very example illustrates why Duhem's own rather hopeful appeal to the 'good sense' of scientists themselves in deciding when a given hypothesis ought to be abandoned promises very little if any relief from the general predicament of holist underdetermination. As noted above, Duhem thought that the sort of underdetermination he had described presented a challenge only for theoretical physics, but subsequent thinking in the philosophy of science has tended to the opinion that the predicament Duhem described applies to theoretical testing in all fields of scientific inquiry. We cannot, for example, test an hypothesis about the phenotypic effects of a particular gene without presupposing a host of further beliefs about what genes are, how they work, how we can identify them, what other genes are doing, and so on. In the middle of the 20th Century, W. V. O. Quine would incorporate confirmational holism and its associated concerns about underdetermination into an extraordinarily influential account of knowledge in general. As part of his famous (1951) critique of the widely accepted distinction between truths that are analytic (true by definition, or as a matter of logic or language alone) and those that are synthetic (true in virtue of some contingent fact about the way the world is), Quine argued that all of the beliefs we hold at any given time are linked in an interconnected web, which encounters our sensory experience only at its periphery: > > The totality of our so-called knowledge or beliefs, from the most > casual matters of geography and history to the profoundest laws of > atomic physics or even of pure mathematics and logic, is a man-made > fabric which impinges on experience only along the edges. Or, to > change the figure, total science is like a field of force whose > boundary conditions are experience. A conflict with experience at the > periphery occasions readjustments in the interior of the field. But > the total field is so underdetermined by its boundary conditions, > experience, that there is much latitude of choice as to what > statements to reevaluate in the light of any single contrary > experience. No particular experiences are linked with any particular > statements in the interior of the field, except indirectly through > considerations of equilibrium affecting the field as a whole. (1951, > 42-3) > One consequence of this general picture of human knowledge is that all of our beliefs are tested against experience only as a corporate body--or as Quine sometimes puts it, "The unit of empirical significance is the whole of science" (1951, p. 42).[4] A mismatch between what the web as a whole leads us to expect and the sensory experiences we actually receive will occasion some revision in our beliefs, but which revision we should make to bring the web as a whole back into conformity with our experiences is radically underdetermined by those experiences themselves. To use Quine's example, if we find our belief that there are brick houses on Elm Street to be in conflict with our immediate sense experience, we might revise our beliefs about the houses on Elm Street, but we might equally well modify instead our beliefs about the appearance of brick, our present location, or innumerable other beliefs constituting the interconnected web. In a pinch, we might even decide that our present sensory experiences are simply hallucinations! Quine's point was not that any of these are particularly likely or reasonable responses to recalcitrant experiences (indeed, an important part of his account is the explanation of why they are not), but instead that they would serve equally well to bring the web of belief as a whole in line with our experience. And if the belief that there are brick houses on Elm Street were sufficiently important to us, Quine insisted, it would be possible for us to preserve it "come what may" (in the way of empirical evidence), by making sufficiently radical adjustments elsewhere in the web of belief. It is in principle open to us, Quine argued, to revise even beliefs about logic, mathematics, or the meanings of our terms in response to recalcitrant experience; it might seem a tempting solution to certain persistent difficulties in quantum mechanics, for example, to reject classical logic's law of the excluded middle (allowing physical particles to both have and not have some determinate classical physical property like position or momentum at a given time). The only test of a belief, Quine argued, is whether it fits into a web of connected beliefs that accords well with our experience on the whole. And because this leaves any and all beliefs in that web at least potentially subject to revision on the basis of our ongoing sense experience or empirical evidence, he insisted, there simply are no beliefs that are analytic in the originally supposed sense of immune to revision in light of experience, or true no matter what the world is like. Quine recognized, of course, that many of the logically possible ways of revising our beliefs in response to recalcitrant experiences that remain open to us nonetheless strike us as ad hoc, perfectly ridiculous, or worse. He argues (1955) that our actual revisions of the web of belief seek to maximize the theoretical "virtues" of simplicity, familiarity, scope, and fecundity, along with conformity to experience, and elsewhere suggests that we typically seek to resolve conflicts between the web of our beliefs and our sensory experiences in accordance with a principle of "conservatism", that is, by making the smallest possible number of changes to the least central beliefs we can that will suffice to reconcile the web with experience. That is, Quine recognized that when we encounter recalcitrant experience we are not usually at a loss to decide which of our beliefs to revise in response to it, but he claimed that this is simply because we are strongly disposed as a matter of fundamental psychology to prefer whatever revision requires the most minimal mutilation of the existing web of beliefs and/or maximizes virtues that he explicitly recognizes as pragmatic in character. Indeed, it would seem that on Quine's view the very notion of a belief being more central or peripheral or in lesser or greater "proximity" to sense experience should be cashed out simply as a measure of our willingness to revise it in response to recalcitrant experience. That is, it would seem that what it means for one belief to be located "closer" to the sensory periphery of the web than another is simply that we are more likely to revise the first than the second if doing so would enable us to bring the web as a whole into conformity with otherwise recalcitrant sense experience. Thus, Quine saw the traditional distinction between analytic and synthetic beliefs as simply registering the endpoints of a psychological continuum ordering our beliefs according to the ease and likelihood with which we are prepared to revise them in order to reconcile the web as a whole with our sense experience as a whole. ### 2.2 Challenging the Rationality of Science It is perhaps unsurprising that such holist underdetermination has been taken to pose a threat to the fundamental rationality of the scientific enterprise. The claim that the empirical evidence alone underdetermines our response to failed predictions or recalcitrant experience might even seem to invite the suggestion that what systematically steps into the breach to do the further work of singling out just one or a few candidate responses to disconfirming evidence is something irrational or at least arational in character. Imre Lakatos and Paul Feyerabend each suggested that because of underdetermination, the difference between empirically successful and unsuccessful theories or research programs is largely a function of the differences in talent, creativity, resolve, and resources of those who advocate them. And at least since the influential work of Thomas Kuhn, one important line of thinking about science has held that it is ultimately the social and political interests (in a suitably broad sense) of scientists themselves which serve to determine their responses to disconfirming evidence and therefore the further empirical, methodological, and other commitments of any given scientist or scientific community. Mary Hesse suggests that Quinean underdetermination showed why certain "non-logical" and "extra-empirical" considerations must play a role in theory choice, and claims that "it is only a short step from this philosophy of science to the suggestion that adoption of such criteria, that can be seen to be different for different groups and at different periods, should be explicable by social rather than logical factors" (1980, 33). Perhaps the most prominent modern day defenders of this line of thinking are those scholars in the sociology of scientific knowledge (SSK) movement and in feminist science studies who argue that it is typically the career interests, political affiliations, intellectual allegiances, gender biases, and/or pursuit of power and influence by scientists themselves which play a crucial or even decisive role in determining precisely which beliefs are abandoned or retained when faced with conflicting evidence (classic works in SSK include Bloor 1991, Collins 1992, and Shapin and Schaffer 1985; in feminist science studies, see Longino, 1990, 2002, and for a recent review, Nelson 2022). The shared argumentative schema here is one on which holist underdetermination ensures that the evidence alone cannot do the work of picking out a unique response to failed predictions or recalcitrant experience, thus something else must step in to do the job, and sociologists of scientific knowledge, feminist critics of science, and other interest-driven theorists of science each have their favored suggestions close to hand. (For useful further discussion, see Okasha 2000. Note that historians of science have also appealed to underdetermination in presenting "counterfactual histories" exploring the ways in which important historical developments in science might have turned out quite differently than they actually did; see, for example, Radick 2023.) In response to this line of argument, Larry Laudan (1990) argues that the significance of such underdetermination has been greatly exaggerated. Underdetermination actually comes in a wide variety of strengths, he insists, depending on precisely what is being asserted about the character, the availability, and (most importantly) the rational defensibility of the various competing hypotheses or ways of revising our beliefs that the evidence supposedly leaves us free to accept. Laudan usefully distinguishes a number of different dimensions along which claims of underdetermination vary in strength, and he goes on to insist that those who attribute dramatic significance to the thesis that our scientific theories are underdetermined by the evidence defend only the weaker versions of that thesis, yet draw dire consequences and shocking morals regarding the character and status of the scientific enterprise from much stronger versions. He suggests, for instance, that Quine's famous claim that any hypothesis can be preserved "come what may" in the way of evidence can be defended simply as a description of what it is psychologically possible for human beings to do, but Laudan insists that in this form the thesis is simply bereft of interesting or important consequences for epistemology-- the study of knowledge. Along this dimension of variation, the strong version of the thesis asserts that it is always normatively or rationally defensible to retain any hypothesis in the light of any evidence whatsoever, but this latter, stronger version of the claim, Laudan suggests, is one for which no convincing evidence or argument has ever been offered. More generally, Lauden argues, arguments for underdetermination turn on implausibly treating all logically possible responses to the evidence as equally justified or rationally defensible. For example, Laudan suggests that we might reasonably hold the resources of deductive logic to be insufficient to single out just one acceptable response to disconfirming evidence, but not that deductive logic plus the sorts of ampliative principles of good reasoning typically deployed in scientific contexts are insufficient to do so. Similarly, defenders of underdetermination might assert the nonuniqueness claim that for any given theory or web of beliefs, either there is at least one alternative that can also be reconciled with the available evidence, or the much stronger claim that all of the contraries of any given theory can be reconciled with the available evidence equally well. And the claim of such "reconciliation" itself disguises a wide range of further alternative possibilities: that our theories can be made logically compatible with any amount of disconfirming evidence (perhaps by the simple expedient of removing any claim(s) with which the evidence is in conflict), that any theory may be reformulated or revised so as to entail any piece of previously disconfirming evidence, or so as to explain previously disconfirming evidence, or that any theory can be made to be as well supported empirically by any collection of evidence as any other theory. And in all of these respects, Laudan claims, partisans have defended only the weaker forms of underdetermination while founding their further claims about and conceptions of the scientific enterprise on versions much stronger than those they have managed or even attempted to defend. Laudan is certainly right to distinguish these various versions of holist underdetermination, and he is equally right to suggest that many of the thinkers he confronts have derived grand morals concerning the scientific enterprise from much stronger versions of underdetermination than they are able to defend, but the underlying situation is somewhat more complex than he suggests. Laudan's overarching claim is that champions of holist underdetermination show only that a wide variety of responses to disconfirming evidence are logically possible (or even just psychologically possible), rather than that these are all rationally defensible or equally well-supported by the evidence. But his straightforward appeal to further epistemic resources like ampliative principles of belief revision that are supposed to help narrow the merely logical possibilities down to those which are reasonable or rationally defensible is itself problematic, at least as part of any attempt to respond to Quine. This is because on Quine's holist picture of knowledge such further ampliative principles governing legitimate belief revision are themselves simply part of the web of our beliefs, and are therefore open to revision in response to recalcitrant experience as well. Indeed, this is true even for the principles of deductive logic and the (consequent) demand for particular forms of logical consistency between parts of the web itself! So while it is true that the ampliative principles we currently embrace do not leave all logically or even psychologically possible responses to the evidence open to us (or leave us free to preserve any hypothesis "come what may"), our continued adherence to these very principles, rather than being willing to revise the web of belief so as to abandon them, is part of the phenomenon to which Quine is using underdetermination to draw our attention, and so cannot be taken for granted without begging the question. Put another way, Quine does not simply ignore the further principles that function to ensure that we revise the web of belief in one way rather than others, but it follows from his account that such principles are themselves part of the web and therefore candidates for revision in our efforts to bring the web of beliefs into conformity (by the resulting web's own lights) with sensory experience. This recognition makes clear why it will be extremely difficult to say how the shift to an alternative web of belief (with alternative ampliative or even deductive principles of belief revision) should or even can be evaluated for its rational defensibility. Each proposed revision is likely to be maximally rational by the lights of the principles it itself sanctions.[5] Of course we can rightly say that many candidate revisions would violate our presently accepted ampliative principles of rational belief revision, but the preference we have for those rather than the alternatives is itself simply generated by their position in the web of belief we have inherited, and the role that they themselves play in guiding the revisions we are inclined to make to that web in light of ongoing experience. Thus, if we accept Quine's general picture of knowledge, it becomes quite difficult to disentangle normative from descriptive issues, or questions about the psychology of human belief revision from questions about the justifiability or rational defensibility of such revisions. It is in part for this reason that Quine famously suggests (1969, 82; see also p 75-76) that epistemology itself "falls into place as a chapter of psychology and hence of natural science." His point is not that epistemology should simply be abandoned in favor of psychology, but instead that there is ultimately no way to draw a meaningful distinction between the two. (James Woodward, in comments on an earlier draft of this entry, pointed out that this makes it all the harder to assess the significance of Quinean underdetermination in light of Laudan's complaint or even know the rules for doing so, but in an important way this difficulty was Quine's point all along!) Quine's claim is that "[e]ach man is given a scientific heritage plus a continuing barrage of sensory stimulation; and the considerations which guide him in warping his scientific heritage to fit his continuing sensory promptings are, where rational, pragmatic" (1951, 46), but the role of these pragmatic considerations or principles in selecting just one of the many possible revisions of the web of belief in response to recalcitrant experience is not to be contrasted with those same principles having rational or epistemic justification. Far from conflicting with or even being orthogonal to the search for truth and our efforts to render our beliefs maximally responsive to the evidence, Quine insists, revising our beliefs in accordance with such pragmatic principles "at bottom, is what evidence is" (1955, 251). Whether or not this strongly naturalistic conception of epistemology can ultimately be defended, it is misleading for Laudan to suggest that the thesis of underdetermination becomes trivial or obviously insupportable the moment we inquire into the rational defensibility rather than the mere logical or psychological possibility of alternative revisions to the holist's web of belief. In fact, there is an important connection between this lacuna in Laudan's discussion and the further uses made of the thesis of underdetermination by sociologists of scientific knowledge, feminist epistemologists, and other vocal champions of holist underdetermination. When faced with the invocation of further ampliative standards or principles that supposedly rule out some responses to disconfirmation as irrational or unreasonable, these thinkers typically respond by insisting that the embrace of such further standards or principles (or perhaps their application to particular cases) is itself underdetermined, historically contingent, and/or subject to ongoing social negotiation. For this reason, they suggest, such appeals (and their success or failure in convincing the members of a given community) should be explained by reference to the same broadly social and political interests that they claim are at the root of theory choice and belief change in science more generally (see, e.g., Shapin and Schaffer, 1985). On both accounts, then, our response to recalcitrant evidence or a failed prediction is constrained in important ways by features of the existing web of beliefs. But for Quine, the continuing force of these constraints is ultimately imposed by the fundamental principles of human psychology (such as our preference for minimal mutilation of the web, or the pragmatic virtues of simplicity, fecundity, etc.), while for constructivist theorists of science such as Shapin and Schaffer, the continuing force of any such constraints is limited only by the ongoing negotiated agreement of the communities of scientists who respect them. As this last contrast makes clear, recognizing the limitations of Laudan's critique of Quine and the fact that we cannot dismiss holist underdetermination with any straightforward appeal to ampliative principles of good reasoning by itself does nothing to establish the further positive claims about belief revision advanced by interest-driven theorists of science. Conceding that theory choice or belief revision in science is underdetermined by the evidence in just the ways that Duhem and/or Quine suggested leaves entirely open whether it is instead the (suitably broad) social or political interests of scientists themselves that do the further work of singling out the particular beliefs or responses to falsifying evidence that any particular scientist or scientific community will actually adopt or find compelling. Even many of those philosophers of science who are most strongly convinced of the general significance of various forms of underdetermination remain deeply skeptical of this latter thesis and thoroughly unconvinced by the empirical evidence that has been offered in support of it (usually in the form of case studies of particular historical episodes in science). Appeals to underdetermination have also loomed large in recent philosophical debates concerning the place of values in science, with a number of authors arguing that the underdetermination of theory by data is among the central reasons that values (or "non-epistemic" values) do and perhaps must play a central role in scientific inquiry. Feminist philosophers of science have sometimes suggested that it is such underdetermination which creates room not only for unwarranted androcentric values or assumptions to play central roles in the embrace of particular theoretical possibilities, but also for the critical and alternative approaches favored by feminists themselves (e.g. Nelson 2022). But appeals to underdetermination also feature prominently in more general arguments against the possibility or desirability of value-free science. Perhaps most influentially, Helen Longino's "contextual empiricism" suggests that a wide variety of non-epistemic values play important roles in determining our scientific beliefs in part because underdetermination prevents data or evidence alone from doing so. For this and other reasons she concludes that objectivity in science is therefore best served by a diverse set of participants who bring a variety of different values or value-laden assumptions to the enterprise (Longino 1990, 2002). ## 3. Contrastive Underdetermination, Empirical Equivalents, and Unconceived Alternatives ### 3.1 Contrastive Underdetermination: Back to Duhem Although it is also a form of underdetermination, what we described in Section 1 above as contrastive underdetermination raises fundamentally different issues from the holist variety considered in Section 2 (Bonk 2008 raises many of these issues). John Stuart Mill articulated the challenge of contrastive underdetermination with impressive clarity in A System of Logic, where he writes: > > Most thinkers of any degree of sobriety allow, that an hypothesis...is > not to be received as probably true because it accounts for all the > known phenomena, since this is a condition sometimes fulfilled > tolerably well by two conflicting hypotheses...while there are > probably a thousand more which are equally possible, but which, for > want of anything analogous in our experience, our minds are unfitted > to conceive. ([1867] 1900, 328) > This same concern is also evident in Duhem's original writings concerning so-called crucial experiments, where he seeks to show that even when we explicitly suspend any concerns about holist underdetermination, the contrastive variety remains an obstacle to our discovery of truth in theoretical science: > > But let us admit for a moment that in each of these systems > [concerning the nature of light] everything is compelled to be > necessary by strict logic, except a single hypothesis; consequently, > let us admit that the facts, in condemning one of the two systems, > condemn once and for all the single doubtful assumption it contains. > Does it follow that we can find in the 'crucial > experiment' an irrefutable procedure for transforming one of the > two hypotheses before us into a demonstrated truth? Between two > contradictory theorems of geometry there is no room for a third > judgment; if one is false, the other is necessarily true. Do two > hypotheses in physics ever constitute such a strict dilemma? Shall we > ever dare to assert that no other hypothesis is imaginable? Light may > be a swarm of projectiles, or it may be a vibratory motion whose waves > are propagated in a medium; is it forbidden to be anything else at > all? ([1914] 1954, 189) > Contrastive underdetermination is so-called because it questions the ability of the evidence to confirm any given hypothesis against alternatives, and the central focus of discussion in this connection (equally often regarded as "the" problem of underdetermination) concerns the character of the supposed alternatives. Of course the two problems are not entirely disconnected, because it is open to us to consider alternative possible modifications of the web of beliefs as alternative theories between which the empirical evidence alone is powerless to decide. But we have already seen that one need not think of the alternative responses to recalcitrant experience as competing theoretical alternatives to appreciate the character of the holist's challenge, and we will see that one need not embrace any version of holism about confirmation to appreciate the quite distinct problem that the available evidence might support more than one theoretical alternative. It is perhaps most useful here to think of holist underdetermination as starting from a particular theory or body of beliefs and claiming that our revision of those beliefs in response to new evidence may be underdetermined, while contrastive underdetermination instead starts from a given body of evidence and claims that more than one theory may be well-supported by that evidence. Part of what has contributed to the conflation of these two problems is the holist presuppositions of those who originally made them famous. After all, on Quine's view, we simply revise the web of belief in response to recalcitrant experience, and so the suggestion that there are multiple possible revisions of the web available in response to any particular evidential finding just is the claim that there are in fact many different "theories" (i.e. candidate webs of belief) that are equally well-supported by any given body of data.[6] But if we give up such extreme holist views of evidence, meaning, and/or confirmation, the two problems take on very different identities, with very different considerations in favor of taking them seriously, very different consequences, and very different candidate solutions. Notice, for instance, that even if we somehow knew that no other hypothesis on a given subject was well-confirmed by a given body of data, that would not tell us where to place the blame or which of our beliefs to give up if the remaining hypothesis in conjunction with others subsequently resulted in a failed empirical prediction. And as Duhem suggests in the passage cited above, even if we supposed that we somehow knew exactly which of our hypotheses to blame in response to a failed empirical prediction, this would not help us to decide whether or not there are other hypotheses available that are also well-confirmed by the data we actually have. One way to see why not is to consider an analogy that champions of contrastive underdetermination have sometimes used to support their case. If we consider any finite group of data points, an elementary proof reveals that there are an infinite number of distinct mathematical functions describing different curves that will pass through all of them. As we add further data to our initial set we will eliminate functions describing curves which no longer capture all of the data points in the new, larger set, but no matter how much data we accumulate, there will always be an infinite number of functions remaining that define curves including all the data points in the new set and which would therefore seem to be equally well supported by the empirical evidence. No finite amount of data will ever be able to narrow the possibilities down to just a single function or indeed, any finite number of candidate functions, from which the distribution of data points we have might have been generated. Each new data point we gather eliminates an infinite number of curves that previously fit all the data (so the problem here is not the holist's challenge that we do not know which beliefs to give up in response to failed predictions or disconfirming evidence), but also leaves an infinite number still in contention. ### 3.2 Empirically Equivalent Theories Of course, generating and testing fundamental scientific hypotheses is rarely if ever a matter of finding curves that fit collections of data points, so nothing follows directly from this mathematical analogy for the significance of contrastive underdetermination in most scientific contexts. But Bas van Fraassen has offered an extremely influential line of argument intended to show that such contrastive underdetermination is a serious concern for scientific theorizing more generally. In The Scientific Image (1980), van Fraassen uses a now-classic example to illustrate the possibility that even our best scientific theories might have empirical equivalents: that is, alternative theories making the very same empirical predictions, and which therefore cannot be better or worse supported by any possible body of evidence. Consider Newton's cosmology, with its laws of motion and gravitational attraction. As Newton himself realized, exactly the same predictions are made by the theory whether we assume that the entire universe is at rest or assume instead that it is moving with some constant velocity in any given direction: from our position within it, we have no way to detect constant, absolute motion by the universe as a whole. Thus, van Fraassen argues, we are here faced with empirically equivalent scientific theories: Newtonian mechanics and gravitation conjoined either with the fundamental assumption that the universe is at absolute rest (as Newton himself believed), or with any one of an infinite variety of alternative assumptions about the constant velocity with which the universe is moving in some particular direction. All of these theories make all and only the same empirical predictions, so no evidence will ever permit us to decide between them on empirical grounds.[7] Van Fraassen is widely (though mistakenly) regarded as holding that the prospect of contrastive underdetermination grounded in such empirical equivalents demands that we restrict our epistemic ambitions for the scientific enterprise itself. His constructive empiricism holds that the aim of science is not to find true theories, but only theories that are empirically adequate: that is, theories whose claims about observable phenomena are all true. Since the empirical adequacy of a theory is not threatened by the existence of another that is empirically equivalent to it, fulfilling this aim has nothing to fear from the possibility of such empirical equivalents. In reply, many critics have suggested that van Fraassen gives no reasons for restricting belief to empirical adequacy that could not also be used to argue for suspending our belief in the future empirical adequacy of our best present theories. Of course there could be empirical equivalents to our best theories, but there could also be theories equally well-supported by all the evidence up to the present which diverge in their predictions about observables in future cases not yet tested. This challenge seems to miss the point of van Fraassen's epistemic voluntarism: his claim is that we should believe no more but also no less than we need to take full advantage of our scientific theories, and a commitment to the empirical adequacy of our theories, he suggests, is the least we can get away with in this regard. Of course it is true that we are running some epistemic risk in believing in even the full empirical adequacy of our present theories, but this is the minimum we need to take full advantage of the fruits of our scientific labors, and the risk is considerably less than what we assume in believing in their truth: as van Fraassen famously suggests, "it is not an epistemic principle that one might as well hang for a sheep as a lamb" (1980, 72). In an influential discussion, Larry Laudan and Jarrett Leplin (1991) argue that philosophers of science have invested even the bare possibility that our theories might have empirical equivalents with far too much epistemic significance. Notwithstanding the popularity of the presumption that there are empirically equivalent rivals to every theory, they argue, the conjunction of several familiar and relatively uncontroversial epistemological theses is sufficient to defeat it. Because the boundaries of what is observable change as we develop new experimental methods and instruments, because auxiliary assumptions are always needed to derive empirical consequences from a theory (cf. confirmational holism, above), and because these auxiliary assumptions are themselves subject to change over time, Laudan and Leplin conclude that there is no guarantee that any two theories judged to be empirically equivalent at a given time will remain so as the state of our knowledge advances. Accordingly, any judgment of empirical equivalence is both defeasible and relativized to a particular state of science. So even if two theories are empirically equivalent at a given time this is no guarantee that they will remain so, and thus there is no foundation for a general pessimism about our ability to distinguish theories that are empirically equivalent to each other on empirical grounds. Although they concede that we could have good reason to think that particular theories have empirically equivalent rivals, this must be established case-by-case rather than by any general argument or presumption. One fairly standard reply to this line of argument is to suggest that what Laudan and Leplin really show is that the notion of empirical equivalence must be applied to larger collections of beliefs than those traditionally identified as scientific theories--at least large enough to encompass the auxiliary assumptions needed to derive empirical predictions from them. At the extreme, perhaps this means that the notion of empirical equivalents (or at least timeless empirical equivalents) cannot be applied to anything less than "systems of the world" (i.e. total Quinean webs of belief), but even that is not fatal: what the champion of contrastive underdetermination asserts is that there are empirically equivalent systems of the world that incorporate different theories of the nature of light, or spacetime, or whatever (for useful discussion, see Okasha 2002). On the other hand, it might seem that quick examples like van Fraassen's variants of Newtonian cosmology do not serve to make this thesis as plausible as the more limited claim of empirical equivalence for individual theories. It seems equally natural, however, to respond to Laudan and Leplin simply by conceding the variability in empirical equivalence but insisting that this is not enough to undermine the problem. Empirical equivalents create a serious obstacle to belief in a theory so long as there is some empirical equivalent to that theory at any given time, but it need not be the same one at each time. On this line of thinking, cases like van Fraassen's Newtonian example illustrate how easy it is for theories to admit of empirical equivalents at any given time, and thus constitute a reason for thinking that there probably are or will be empirical equivalents to any given theory at any particular time, assuring that whenever the question of belief in a given theory arises, the challenge posed to it by contrastive underdetermination arises as well. Laudan and Leplin also suggest, however, that even if the universal existence of empirical equivalents were conceded, this would do much less to establish the significance of underdetermination than its champions have supposed, because "theories with exactly the same empirical consequences may admit of differing degrees of evidential support" (1991, 465). A theory may be better supported than an empirical equivalent, for instance, because the former but not the latter is derivable from a more general theory whose consequences include a third, well supported, hypothesis. More generally, the belief-worthiness of an hypothesis depends crucially on how it is connected or related to other things we believe and the evidential support we have for those other beliefs.[8] Laudan and Leplin suggest that we have invited the specter of rampant underdetermination only by failing to keep this familiar home truth in mind and instead implausibly identifying the evidence bearing on a theory exclusively with the theory's own entailments or empirical consequences (but cf. Tulodziecki 2012). This impoverished view of evidential support, they argue, is in turn the legacy of a failed foundationalist and positivistic approach to the philosophy of science which mistakenly assimilates epistemic questions about how to decide whether or not to believe a theory to semantic questions about how to establish a theory's meaning or truth-conditions. John Earman (1993) has argued that this dismissive diagnosis does not do justice to the threat posed by underdetermination. He argues that worries about underdetermination are an aspect of the more general question of the reliability of our inductive methods for determining beliefs, and notes that we cannot decide how serious a problem underdetermination poses without specifying (as Laudan and Leplin do not) the inductive methods we are considering. Earman regards some version of Bayesianism as our most promising form of inductive methodology, and he proceeds to show that challenges to the long-run reliability of our Bayesian methods can be motivated by considerations of the empirical indistinguishability (in several different and precisely specified senses) of hypotheses stated in any language richer than that of the evidence itself that do not amount simply to general skepticism about those inductive methods. In other words, he shows that there are more reasons to worry about underdetermination concerning inferences to hypotheses about unobservables than to, say, inferences about unobserved observables. He also goes on to argue that at least two genuine cosmological theories have serious, nonskeptical, and nonparasitic empirical equivalents: the first essentially replaces the gravitational field in Newtonian mechanics with curvature in spacetime itself, [9] while the second recognizes that Einstein's General Theory of Relativity permits cosmological models exhibiting different global topological features which cannot be distinguished by any evidence inside the light cones of even idealized observers who live forever.[10] And he suggests that "the production of a few concrete examples is enough to generate the worry that only a lack of imagination on our part prevents us from seeing comparable examples of underdetermination all over the map" (1993, 31) even as he concedes that his case leaves open just how far the threat of underdetermination extends (1993, 36). Most philosophers of science, however, have not embraced the idea that it is only lack of imagination which prevents us from finding empirical equivalents to our scientific theories generally. They note that the convincing examples of empirical equivalents we do have are all drawn from a single domain of highly mathematized scientific theorizing in which the background constraints on serious theoretical alternatives are far from clear, and suggest that it is therefore reasonable to ask whether even a small handful of such examples should make us believe that there are probably empirical equivalents to most of our scientific theories most of the time. They concede that it is always possible that there are empirical equivalents to even our best scientific theories concerning any domain of nature, but insist that we should not be willing to suspend belief in any particular theory until some convincing alternative to it can actually be produced: as Philip Kitcher puts it, "give us a rival explanation, and we'll consider whether it is sufficiently serious to threaten our confidence" (1993, 154; see also Leplin 1997, Achinstein 2002). That is, these thinkers insist that until we are able to actually construct an empirically equivalent alternative to a given theory, the bare possibility that such equivalents exist is insufficient to justify suspending belief in the best theories we do have. For this same reason most philosophers of science are unwilling to follow van Fraassen into what they regard as constructive empiricism's unwarranted epistemic modesty. Even if van Fraassen is right about the most minimal beliefs we must hold in order to take full advantage of our scientific theories, most thinkers do not see why we should believe the least we can get away with rather than believing the most we are entitled to by the evidence we have. Champions of contrastive underdetermination have most frequently responded by trying to establish that all theories have empirical equivalents, typically by proposing something like an algorithmic procedure for generating such equivalents from any theory whatsoever. Stanford (2001, 2006) suggests that these efforts to prove that all our theories must have empirical equivalents fall roughly but reliably into global and local varieties, and that neither makes a convincing case for a distinctive scientific problem of contrastive underdetermination. Global algorithms are well-represented by Andre Kukla's (1996) suggestion that from any theory T we can immediately generate such empirical equivalents as T' (the claim that T's observable consequences are true, but T itself is false), T'' (the claim that the world behaves according to T when observed, but some specific incompatible alternative otherwise), and the hypothesis that our experience is being manipulated by powerful beings in such a way as to make it appear that T is true. But such possibilities, Stanford argues, amount to nothing more than the sort of Evil Deceiver to which Descartes appealed in order to doubt any of his beliefs that could possibly be doubted (see Section 1, above). Such radically skeptical scenarios pose an equally powerful (or powerless) challenge to any knowledge claim whatsoever, no matter how it is arrived at or justified, and thus pose no special problem or challenge for beliefs offered to us by theoretical science. If global algorithms like Kukla's are the only reasons we can give for taking underdetermination seriously in a scientific context, then there is no distinctive problem of the underdetermination of scientific theories by data, only a salient reminder of the irrefutability of classically Cartesian or radical skepticism.[11] In contrast to such global strategies for generating empirical equivalents, local algorithmic strategies instead begin with some particular scientific theory and proceed to generate alternative versions that will be equally well supported by all possible evidence. This is what van Fraassen does with the example of Newtonian cosmology, showing that an infinite variety of supposed empirical equivalents can be produced by ascribing different constant absolute velocities to the universe as a whole. But Stanford suggests that empirical equivalents generated in this way are also insufficient to show that there is a distinctive and genuinely troubling form of underdetermination afflicting scientific theories, because they rely on simply saddling particular scientific theories with further claims for which those theories themselves (together with whatever background beliefs we actually hold) imply that we cannot have any evidence. Such empirical equivalents invite the natural response that they simply tack on to our theories further commitments that are or should be no part of those theories themselves. Such claims, it seems, should simply be excised from our theories, leaving over just the claims that sensible defenders would have held were all we were entitled to believe by the evidence in any case. In van Fraassen's Newtonian example, for instance, this could be done simply by undertaking no commitment concerning the absolute velocity and direction (or lack thereof) of the universe as a whole. Note also that if we believe a given scientific theory when one of the empirical equivalents we could generate from it by the local algorithmic strategy is correct instead, most of what we originally believed will nonetheless turn out to be straightforwardly true. Philosophers of science have responded in a variety of ways to the suggestion that a few or even a small handful of serious examples of empirical equivalents does not suffice to establish that there are probably such equivalents to most scientific theories in most domains of inquiry. One such reaction has been to invite more careful attention to the details of particular examples of putative underdetermination: considerable work has been devoted to assessing the threat of underdetermination in the case of particular scientific theories (for recent examples see Pietsch 2012; Tulodziecki 2013; Werndl 2013; Belot 2014; Butterfield 2014; Miyake 2015; Kovaka 2019; Fletcher 2021, and others). Other thinkers have sought to investigate whether certain types of scientific theories or theories in particular scientific fields differ in the extent to which they are genuinely threatened by underdetermination. Some have argued that underdetermination poses a less serious (Cleland 2002; Carman 2005) or a more serious (Turner 2005, 2007) challenge for 'historical' sciences like geology, paleontology, and archaeology than it does for 'experimental' sciences like particle physics. Others have resisted such generalizations but nonetheless argued that underdetermination predicaments arise in distinctive or characteristic ways in historical sciences and that scientists in these fields have different resources and strategies for addressing them (Currie 2018; Forber and Griffith 2011; Stanford 2010). Finally, some thinkers have sought to defend particular forms of explanation frequently deployed in historical sciences, such as narrative explanation (Sterelny and Currie 2017), from the charge that they suffer from rampant underdetermination. ### 3.3 Unconceived Alternatives and A New Induction Stanford (2001, 2006) concludes that no convincing general case has been made for the presumption that there are empirically equivalent rivals to all or most scientific theories, or to any theories besides those for which such equivalents can actually be constructed. But he goes on to insist that empirical equivalents are no essential part of the case for a significant problem of contrastive underdetermination. Our efforts to confirm scientific theories, he suggests, are no less threatened by what Larry Sklar (1975, 1981) has called "transient" underdetermination, that is, theories which are not empirically equivalent but are equally (or at least reasonably) well confirmed by all the evidence we happen to have in hand at the moment, so long as this transient predicament is also "recurrent", that is, so long as we think that there is (probably) at least one such (fundamentally distinct) alternative available--and thus the transient predicament re-arises--whenever we are faced with a decision about whether to believe a given theory at a given time. Stanford argues that a convincing case for contrastive underdetermination of this recurrent, transient variety can indeed be made, and that the evidence for it is available in the historical record of scientific inquiry itself. Stanford concedes that present theories are not transiently underdetermined by the theoretical alternatives we have actually developed and considered to date: we think that our own scientific theories are considerably better confirmed by the evidence than any rivals we have actually produced. The central question, he argues, is whether we should believe that there are well confirmed alternatives to our best scientific theories that are presently unconceived by us. And the primary reason we should believe that there are, he claims, is the long history of repeated transient underdetermination by previously unconceived alternatives across the course of scientific inquiry. In the progression from Aristotelian to Cartesian to Newtonian to contemporary mechanical theories, for instance, the evidence available at the time each earlier theory dominated the practice of its day also offered compelling support for each of the later alternatives (unconceived at the time) that would ultimately come to displace it. Stanford's "New Induction" over the history of science claims that this situation is typical; that is, that "we have, throughout the history of scientific inquiry and in virtually every scientific field, repeatedly occupied an epistemic position in which we could conceive of only one or a few theories that were well confirmed by the available evidence, while subsequent inquiry would routinely (if not invariably) reveal further, radically distinct alternatives as well confirmed by the previously available evidence as those we were inclined to accept on the strength of that evidence" (2006, 19). In other words, Stanford claims that in the past we have repeatedly failed to exhaust the space of fundamentally distinct theoretical possibilities that were well confirmed by the existing evidence, and that we have every reason to believe that we are probably also failing to exhaust the space of such alternatives that are well confirmed by the evidence we have at present. Much of the rest of his case is taken up with discussing historical examples illustrating that earlier scientists did not simply ignore or dismiss, but instead genuinely failed to conceive of the serious, fundamentally distinct theoretical possibilities that would ultimately come to displace the theories they defended, only to be displaced in turn by others that were similarly unconceived at the time. He concludes that "the history of scientific inquiry itself offers a straightforward rationale for thinking that there typically are alternatives to our best theories equally well confirmed by the evidence, even when we are unable to conceive of them at the time" (2006, 20; for reservations and criticisms concerning this line of argument, see Magnus 2006, 2010; Godfrey-Smith 2008; Chakravartty 2008; Devitt 2011; Ruhmkorff 2011; Lyons 2013). Stanford concedes, however, that the historical record can offer only fallible evidence of a distinctive, general problem of contrastive scientific underdetermination, rather than the kind of deductive proof that champions of the case from empirical equivalents have typically sought. Thus, claims and arguments about the various forms that underdetermination may take, their causes and consequences, and the further significance they hold for the scientific enterprise as a whole continue to evolve in the light of ongoing controversy, and the underdetermination of scientific theory by evidence remains very much a live and unresolved issue in the philosophy of science.
understanding	## 1. Contexts The concept of understanding has been sometimes prominent, sometimes neglected, and sometimes viewed with suspicion, across a number of different areas of philosophy (for a partial overview, see Zagzebski 2001). This section traces some of that background, beginning with the place of understanding in Ancient Greek philosophy. It then considers how the topic of understanding was lost and then "recovered" in contemporary discussions in epistemology and the philosophy of science. ### 1.1 Ancient Philosophy The ancient Greek word episteme is at the root of our contemporary word "epistemology", and among philosophers it has been common to translate episteme simply as "knowledge" (see, e.g., Parry 2003 [2020]). For the last several decades, however, a case has been made that "understanding" is a better translation of episteme. (For early influential arguments see, e.g., Moravscik 1979; Burnyeat 1980, 1981; Annas 1981.) To appreciate why, note that knowledge, as now commonly conceived, can apparently be quite easy to get. Thus it seems I can know a proposition such as that it is raining outside just by opening my eyes. It also seems that items of knowledge can be in principle isolated or atomistic. I can therefore apparently know that it is raining while knowing very little about other things, such as why it is raining, or what constitutes rain, or when it will stop. Understanding seems to be different than knowledge in both respects. For one thing, understanding typically seems harder to acquire, and more of an epistemic accomplishment, than knowledge (Pritchard 2010). For another, the objects of understanding seem more structured and interconnected (Zagzebski 2019). Thus the subject matters we try to understand are often highly complex (quantum mechanics, the U.S. Civil War), and even when we try to understand isolated events (such as the spilling of my coffee cup), we typically do so by drawing connections with other events (such as the jostling of the table by my knee). With contrasts such as these in mind, it has seemed to several scholars of Ancient philosophy that episteme has more in common with what we would now call "understanding" than what we would now call "knowledge". Thus Julia Annas notes that, for Plato, the person with episteme has a systematic understanding of things, and is not a mere possessor of various truths (Annas 1981: ch. 10). Jonathan Lear similarly claims that for Aristotle, > > > To have episteme one must not only know a thing, one must > also grasp its cause or explanation. This is to understand it: to know > in a deep sense what it is and how it has come to be. (Lear 1988: > 6) > > > Granted, what Greek philosophers had in mind by episteme often does not fully align with our contemporary ideas about understanding, because in the hands of philosophers such as Plato and Aristotle episteme is an exceptionally high-grade epistemic accomplishment. For Plato, full episteme seems to require a grasp of the basic elements of reality--in its most complete form, a grasp that traces back to the Form of the Good itself (Schwab 2016, 2020; Moss 2020). For Aristotle, it seems to require an appreciation of the deductive relationships that allegedly hold between natures or first principles and observable phenomena (Burnyeat 1981). In our contemporary use, by contrast, we often happily ascribe understanding to quite low-grade cases, where forms or first principles do not seem to be grasped or even relevant--as when we take ourselves to understand why the coffee cup spilled. Still, with its stress on systematicity and interconnectedness, episteme plausibly has more in common with our contemporary concept understanding than our contemporary concept knowledge. If an understanding-like state was of fundamental epistemic importance to the Greeks, it is interesting to ask why the focus of epistemology shifted over time, and why an interest in knowledge, especially propositional knowledge, came to predominate--including and maybe especially quite isolated bits of propositional knowledge, such as that I am sitting in front of a fire, or that I have two hands. Perhaps the shift occurred in response to the rise of scepticism in Hellenistic philosophy (Burnyeat 1980: 188; cf. Zagzebski 2001: 236). Or perhaps the modern focus on propositional knowledge was a response to the wars of religion in sixteenth and seventeenth century Europe, where it became increasingly important to distinguish good knowledge claims from bad, even with respect to quite isolated claims. (For more on the modern rise of interest in propositional knowledge, see Pasnau 2010; 2017.) Regardless of why the shift occurred, a desire to grasp "how things hang together" undoubtedly remained a part of the human condition. It is therefore unsurprising that a move to revive understanding as a topic of philosophical inquiry eventually emerged. ### 1.2 Epistemology & Philosophy of Science Although understanding as an epistemic good was largely neglected by modern epistemologists in favor of theorizing about knowledge (or related epistemic properties, such as justification and rationality), it reappeared as a central object of concern at the end of the twentieth century, for a few different reasons. Catherine Elgin, for one, influentially argued that we cannot make sense of some of our greatest intellectual accomplishments, especially the accomplishments we associate with science and art, without appreciating the way they are often oriented not towards knowledge, but rather understanding (see especially Elgin 1991, 1996). From the perspective of virtue epistemology, Linda Zagzebski claimed that if we think of an intellectual virtue as an "excellence of the mind", attuned to a variety of epistemic goods, then there is something one-sided about focusing attention only on the good of knowledge, while neglecting other highly prized goods such as understanding and wisdom (see especially Zagzebski 1996: 43-50; 2001). Finally, Jonathan Kvanvig forcefully argued that while understanding is distinctively valuable from an epistemic point of view--i.e., more valuable than any of its proper parts, such as truth, or justification, or a combination of the two--knowledge is not (Kvanvig 2003). For all these thinkers, the spotlight of concern within epistemology needed to be broadened so that goods such as understanding could be given their proper due, and their claims resonated with other epistemologists (for overviews, see Gordon 2017; Hannon forthcoming). While the notion of understanding was often simply neglected in epistemology, in the philosophy of science it was for many years actively downplayed. A primary figure in this dynamic was Carl Hempel (see especially Hempel 1965). Although Hempel helped bring the notion of explanation back into respectability in the philosophy of science, he had significant reservations about tying explanation too closely to the notion of understanding. Part of this seemed to stem from the fact that the idea of understanding that prevailed in his day was highly subjective and psychological--it emphasized more a subjective "sense" of understanding, often tied to a felt sense of familiarity. As Hempel notes, however, poor explanations might excel along this dimension because they might > > > give the questioner a sense of having attained some understanding; > they may resolve his perplexity and in this sense "answer" > his question. > > > "But", he continues, > > > however satisfactory these answers may be psychologically, they are > not adequate for the purposes of science, which, after all, is > concerned to develop a conception of the world that has a clear, > logical bearing on our experience and is thus capable of objective > test. (Hempel 1966: 47-48) > > > The goodness of an explanation therefore seems to have little obvious connection to whether it manages to generate understanding in a particular audience. A good explanation might do that. But then again, it might not. Patently poor explanations are also able to generate a rich "sense" of understanding in some audiences (think of conspiracy theorists), despite their shortcomings. Philosophers such as Michael Friedman responded to Hempel's concerns by noting that simply because there seems to be a psychological element to understanding it does not follow that understanding is merely subjective or up for grabs (Friedman 1974). After all, knowledge has a psychological element, in light of the belief condition, but few hold that knowledge is merely subjective or up for grabs. Others, such as Jaegwon Kim, argued that leaving considerations about understanding out of accounts of explanation was deeply mistaken (Kim 1994 [2010]). After all, Kim claimed, we desire to explain things because we want to understand them. Despite the efforts of Friedman, Kim, and others, significant reservations about the notion of understanding continued to linger in the philosophy of science (see, for example, Trout 2002). A notable shift occurred with Henk de Regt's distinction between the "feeling" or phenomenology of understanding and genuine understanding (de Regt 2004, 2009: ch. 1; cf. de Regt and Dieks 2005). In particular, he argued that the feeling is neither necessary nor sufficient for genuine understanding. De Regt's important distinction helped pave the way for a new surge of work on the topic over the last two decades, and helped philosophers move beyond thinking of understanding mainly in terms of felt "aha!" or "eureka!" experiences. ## 2. Theoretical Frameworks As we look to particular accounts of understanding, it will help to consider in turn: 1. understanding's distinctive object (or objects), 2. its distinctive psychology, and 3. the distinctive sort of normative relationship that needs to hold between the psychology of the person who understands and the object of his or her understanding. By way of comparison, consider the traditional "justified true belief" analysis of knowledge. On this view, knowledge involves a distinctive object (roughly, the truth, or a true proposition), a distinctive psychology (the psychological act of belief or assent), and a distinctive normative relationship that needs to hold between the psychology of the believer and the thing believed (namely, that one's belief in the true proposition needs to be justified, in some sense). What can be said, in a parallel way, about the elements of understanding? ### 2.1 Objects of Understanding At least at first glance, the objects of understanding appear to be so varied that it is not obvious where one might find a common thread. Thus, we can understand fields of study, particular states of affairs, institutions, other people, and on and on(cf. Elgin 1996: 123). In line with the discussion in Ancient philosophy, and setting aside for the moment special issues related to understanding other people (see Section 5), let us start with the generic view that the objects of understanding are something like "connections" or "relations". Following a distinction from Kim (1994 [2010]), we can contrast two ways of thinking about these connections and relations--i.e., these plausible objects of understanding. According to explanatory internalism, the connections or explanatory relations one grasps are "logico-linguistic" relations that hold among a person's beliefs or attitudes, or more exactly the contents of those beliefs or attitudes (Kim 1994 [2010]). What we grasp or see, when we understand some phenomenon, are how these various contents are logically or semantically related to one another. Kim argues that Hempel is a paradigm example of an explanatory internalist (Kim 1994 [2010]; cf. 1999 [2010]). For instance, suppose that one wants to explain and hence understand why a particular bar of metal began to rust (McCain 2016: ch. 9). A good Hempelian explanation would be one in which a sentence describing the rusting follows inferentially from (a) a statement of the initial conditions (the moisture in the air, the constitution of the bar) and (b) a further law-like statement connecting the moisture, the constitution of the bar, and the onset of rust. This Hempelian framework seems internalist because what you see or grasp, when you understand the phenomenon, are connections among the propositions you accept--more exactly, you see or grasp different inferential or probabilistic connections among the contents of your beliefs that bear on the rusting. As Kim puts it: > > > the basic relation that generates an explanatory relation is a > logico-linguistic one that connects descriptions of events, and the > job of formulating an explanation consists, it seems, in merely > re-arranging appropriate items in the body of propositions that > constitute our total knowledge at a time. In explaining something, > then, all action takes place within the epistemic system, on > the subjective side of the divide between knowledge and the reality > known, or between representation and the world represented. (Kim 1994 > [2010: 171-172]) > > > An explanatory externalist by comparison holds that the basic connections or relationship one grasps, when one understands, are not logico-linguistic but metaphysical. What you grasp, when you understand why the metal began to rust, are not primarily relationships among your beliefs or their contents. Rather, you primarily grasp real, mind-independent relationships that obtain in the world. Theorists who hold that the objects of understanding are internal--i.e., logico-linguistic relations, the grasp of which yields understanding--vary somewhat about which relations count. Although it seems clear in Hempel that the objects are inferential relationships of deductive and inductive support, epistemologists often appeal in a general way to relations of coherence. Thus Carter and Gordon write: > > > We think it is clear that objectual understanding--for > example, as one attains when one grasps the relevant coherence-making > relations between propositions comprising some subject matter--is > a particularly valuable epistemic good... Understanding wider > subject matters will tend to be more cognitively demanding than > understanding narrow subject matters because more > propositions must be believed and their relations grasped. (Carter > & Gordon 2014: 7-8) > > > Kvanvig likewise points to the importance of coherence: > > > Central to the notion of understanding are various coherence-like > elements: to have understanding is to grasp explanatory and conceptual > connections between various pieces of information involved in the > subject matter in question. Such language involves a subjective > element (the grasping or seeing of the connections in question) and a > more objective, epistemic element. The more objective, epistemic > element is precisely the kind of element identified by coherentists as > central to the notion of epistemic justification or rationality, as > clarified, in particular, by Lehrer (1974), BonJour (1985) and Lycan > (1988). (Kvanvig 2018: 699) > > > Other epistemologists similarly point to coherence relations as the objects of understanding (e.g., Riggs 2003: 192). Since "coherence" is presumably a relation that holds among the contents of beliefs, and not among items out there in the world, these views would qualify as internalist accounts of the objects of understanding, according to the Kimean framework. (For further examples and criticism, see Khalifa 2017a.) For externalists about the object of understanding, especially concerning phenomena "out there in the world", the proposed objects vary. For instance, there is some support for the idea that the objects are nomic relations, or relationships according to which individual events and other phenomena are explained by laws, and upper-level laws are explained by lower-level or more fundamental laws (Railton 1978). Another view is that the objects are causal relations (Salmon 1984; Lipton 1991 [2004]), or more generally dependence relations (Kim 1974, 1994; Grimm 2014, 2017; Greco 2014; Dellsen 2020), where causation is usually taken to be just one species of dependence. A variation on the "dependence relationships" idea is that the objects of understanding are "possibility spaces" (cf. le Bihan 2017). Indeed, on the plausible assumption that dependence is an essentially modal notion--that is, fundamentally tied to ideas of possibility and necessity--dependence relations ineliminably give rise to or generate possibility spaces. This is in keeping with a suggestion by Robert Nozick that > > > explanation locates something in actuality, showing its actual > connections with other things, while understanding locates it > in a network of possibility, showing the connections it would have to > other nonactual things or processes. (Nozick 1981: 12; cf. Grimm > 2008) > > > Note that the distinction between internal and external explanation relations is related to, but cross-cuts, another influential distinction by Wesley Salmon, between ontic and epistemic accounts of explanation (Salmon 1989; for discussion see Bechtel & Abrahamsen 2005 and Craver 2014). Salmon's distinction has come in for criticism (see, e.g., Illari 2013; Bokulich 2016, 2018), because among other things it is not clear why epistemic accounts should not be ontic or world-involving; knowledge is an epistemic concept, after all, but it is ontic or world-involving in virtue of the truth condition. The distinction between explanatory internalism and externalism plausibly avoids this concern. Suppose in any case that the objects of understanding--the connections or relations we grasp when we understand--are out there in the world, and are not simply logico-linguistic items or elements of our psychology. According to a further important distinction from John Greco (2014), it does not follow that these logico-linguistic/psychological items--or more generally these mental representations--do not play a crucial role in our coming to understand, because the representations typically play the role of vehicles of understanding, even if they are not themselves understanding's object. More generally, Greco notes, it is important to distinguish between the object of understanding vs. the vehicle of understanding, i.e., "between the thing understood and its representation" (Greco 2014: 293). Thus consider that a good map--say, of Midtown Manhattan--is typically a vehicle of understanding rather than an object of understanding. When it is accurate, it allows the mind to grasp how the streets and landmarks of Midtown are laid out, and to appreciate the relationships they bear to one another. Or again, typically when you take a look at your car's gas gauge and form the belief that the tank is half empty, the gauge is the vehicle by which you form your belief about the gas, but it is not the object of your belief (see Dretske 1981). Assuming that the gauge is functioning properly, the object of your belief is the gas itself, while the gauge is simply the vehicle that represents the state of the gas to you. (Of course, either the map or the gauge could become an object of understanding--one could think about how the streets and landmarks are represented, e.g., the different colors or shapes or fonts the map uses to represent Midtown, or the length of the gauge--but this would be to "involve a representation of a representation" (Greco 2014: 293). And while the mind is capable of such an act, it represents an element of abstraction, and plausibly a departure from our usual way of engaging with the world.) This distinction between objects and vehicles of understanding leaves open the possibility that mental representations might not be the only vehicles of understanding, or the only way of latching on to real relations in the world. Perhaps a person might come to understand the world by first manually manipulating it, thus noting the possibilities that it affords and how its different elements are connected (cf. Lipton 2009; Kuorikoski and Ylikoski 2015). This would be a way to "directly" grasp causal structure, a structure that could then--cognitively downstream, as it were--be represented by the mind, perhaps in the form of mental maps, or "dependence maps" (see Section 2.2). Appreciating the contrast between vehicle of understanding and object of understanding helps to reveal that in addition to pure internalist views about the object of understanding (where the objects of understanding are logico-linguistic relations), and pure externalist views (where the objects are worldly), there are also possible hybrid views. For instance, on the sort of view we find in Michael Strevens's Depth (2008), what we grasp in the first instance are relations that hold among the contents of our beliefs: relations of deductive entailment or coherence or probabilistic support. This view is not simply internalist, however, because grasping these logico-linguistic relationships, and in particular the relationships of deductive entailment, "mirror" the real relationships that obtain in the world, and thus provide a vehicle for apprehending those relationships (Strevens 2008: 72). Thus Strevens holds that his theory is able to: > > > show how a deductive argument, or similar formal structure, can > represent causal influence.... When a deductive argument is of > the right sort to represent a causal process that has as a consequence > the state of affairs asserted to hold by the argument's > conclusion, I say that the argument causally entails the > conclusion. A derivation that causally entails its conclusion, then, > is one that can be used to represent (whether truly or not) a causal > process that has the concluding state of affairs as its consequence. > (Strevens 2012: 449) > > > On this view, by grasping or appreciating the necessitating force of an entailment relation one thereby grasps or appreciates the necessitating force of a causal relation--or, at least, something importantly like it. According to another hybrid approach, the internal logical or probabilistic relationships do not mirror the relationships out there in the world, but rather provide evidence for the existence of relationships out there in the world. Thus some hold that when things go well an appreciation or grasp of the probabilistic connections among the various things we believe allows us to infer the existence of real causal connections in the world (seeSpirtes, Glymour, & Scheines 1993; Pearl 2000 [2009]). As Wesley Salmon characterizes an earlier version of this idea, > > > The explanatory significance of statistical relations is indirect. > Their fundamental import lies in the fact... that they constitute > evidence for causal relations. (Salmon 1984: 192) > > > Bearing in mind earlier distinctions, these would be accounts on which one's appreciation of statistical relations is the vehicle through which one grasps real (external) causal relations in the world. ### 2.2 Psychology of Understanding With respect to the psychology of understanding, recall that the psychological element of propositional knowledge is typically construed in terms of belief, where belief is taken to be a kind of assent or saying "Yes" to the content of the proposition. Thus to believe that the sky is blue is to assent to the proposition that the sky is blue; it is "taking it to be true" that the sky is blue. When we turn to understanding, by contrast, some have claimed that a new suite of cognitive abilities comes onto the scene, abilities that we did not find in ordinary cases of propositional knowledge. In particular, some philosophers claim that the kind of mental action verbs that naturally come to the fore when we think about understanding--"grasping" and "seeing", for example--evoke mental abilities "beyond belief", i.e., beyond simple assent or taking-to-be-true (for an overview, see Baumberger, Beisbart, & Brun 2017). For instance, Elgin, de Regt, and Wilkenfeld argue that those who grasp how things are connected are able to cognitively "do" things that others cannot--they can apply their understanding to new cases, for example, and draw new inferences. (See, for example, Elgin 1996: 122-24; de Regt 2004, 2017; Wilkenfeld 2013). Thus car mechanics who understand how engines work are able to make sense of engines that they have not encountered before, and to anticipate how changes in one part of the engine will typically lead (or fail to lead) to changes in other parts. Grimm claims that the distinctive nature of these abilities flows from the distinctive nature of the objects of understanding (Grimm 2017). Suppose that the connections or relations one grasps are complex enough to constitute what we might call "structures"--structures with parts that depend upon one another in various ways. Arguably, for the mind to take up these structures in the right way, it is not enough simply to assent to their existence. Rather, one needs to appreciate how the structure "works", or how changes in its various parts will lead, or fail to lead, to changes in other parts. For instance, suppose the structures to be grasped are represented by "causal maps" or "dependence maps"--maps with nodes representing variables that can take on different values and thus represent directions of dependence (Gopnik & Glymour 2002; Gopnik et al. 2004). In the schematic image below, from Gopnik et al. (2004), the map represents a system in which changes to the value of Z bring about changes to the value of S, changes to the value of S bring about changes to the values of X and R, and so on. ![6 nodes: Z, S, R, X, W, and Y. Z is above S, R is below S, X is to the right of S, W is to the right of X, Y is below X. Arrows go from Z to S, S to R, S to X, X to W, Y to X and Y to S](figure1.svg) According to Grimm (2017), a striking feature of these maps is that they are, as it were, "mobile" maps. That is, they are cognitive representations that by their very nature can adapt and change as the variables represented by the map take on different values. Put another way, they are "unsaturated" maps, in the sense that they are characterized in terms of unsaturated variables that can become saturated by taking on different values. What this means in terms of cognitive uptake is important, Grimm argues, because if the maps are mobile or unsaturated in this way, then the mind that aptly takes them up must itself be mobile. In particular, the mind that takes up causal maps in a way that yields understanding must be able to anticipate how varying or adjusting the value of one of the variables will lead (or fail to lead) to changes in the values of the other variables (cf. Woodward 2003). Relatedly, Alison Hills characterizes the distinctive psychological abilities that undergird understanding in terms of "cognitive control" (Hills 2016). For example, Hills claims that in order to understand why it is right to give money to charity, it is not enough to simply believe the proposition that it is right to give money to charity because we owe assistance to the very needy (Hills 2016: 669), because I could accept a "because" claim along these lines on the basis of testimony while having only a very dim sense of how the two ingredient claims (that it is right to give money and we owe assistance to the very needy) are related. What is needed in addition is an appreciation of the relationship between these two claims. And what this involves, according to the cognitive control view, is an ability to "manipulate" the things standing in relationship: for example, to be able to make a variety of inferences in light of the relationship (Hills 2016: 663). Although the views we have considered so far agree in thinking that there is "something special" from a psychological point of view about understanding, other accounts claim that there is nothing particularly special about understanding's psychological profile. More exactly, it is said that understanding does not appeal to any special abilities or capacities that we do not find in ordinary cases of knowledge. Emily Sullivan, for instance, claims that even if we grant that understanding involves abilities, it does not follow that there is something special about understanding from a psychological point of view, because ordinary propositional knowledge itself requires abilities (Sullivan 2018). Thus I can apparently only know that the traffic light is red, in the actual world, if I would have gotten the color right in close possible worlds as well. More generally, knowing seems to require the ability to track the truth about the world, so that when the world changes, my cognitive attitude about the world changes with it. But this ability to be responsive to changing conditions is not obviously anything exotic, and it does seem to entail that the object of the mind is not a proposition (that the traffic light is red seems like a paradigm example of a proposition, after all). Along the same lines, Kareem Khalifa claims that abilities are involved in understanding, but holds that they are "especially unspecial" (Khalifa 2017b: 56). For Khalifa, the modal character of abilities is evident in normal scientific practice, because as scientists evaluate explanations on the basis of testing they naturally rule out inadequate explanations and gravitate towards well-supported ones. But evaluating and testing hypotheses in this way does not appear to require anything exotic or special from a psychological point of view, or to entail that the object of understanding is not propositional. Finally, psychologists themselves have increasingly focused on characterizing the cognitive profile of understanding. Thus Tania Lombrozo and colleagues have explored the question of why we seek understanding, and how activities such as offering explanations aid in the acquisition and retention of understanding (Lombrozo, 2012; Williams & Lombrozo, 2010, 2013; Lombrozo & Wilkenfeld 2019). Psychologists have also explored the empirical question of how good (or bad) we are at identifying real relationships and dependencies in the world. According to Frank Keil, for example, we are not good at this at all, and we frequently fall prey to illusions of understanding (Keil 2006; cf. Ylikoski 2009). For instance, we often think we understand how a helicopter achieves lift, or how moving the pedals on a bicycle help to propel the bike forward. But in both cases, we are often well wide of the mark (Keil 2006; cf. Grimm 2009; Sloman & Fernbach 2017). ### 2.3 Normativity of Understanding Suppose you accurately grasp that your house burned down because of faulty wiring. You do not think it burned down because of a lightning strike, or a stray cigarette. Faulty wiring was the cause, and you accurately grasp it as the cause (see Pritchard 2009, 2010 for more details on a case like this). Is that all there is to understanding why your house burned down? Or does it also matter, for example, how one comes to grasp this relationship? These questions push us to ask about the normative dimension of understanding, and in particular to ask: Are there better and worse ways to come by one's grasp, and does this matter for the acquisition of understanding? For example, are careful experiments or good evidence needed? Or can one come by one's grasp more or less by luck, and yet still in a way that generates understanding? A parallel question with respect to knowledge would ask: Supposing you have a true belief, is that all there is to knowledge? And here most would say "no". In addition to the true belief, how you came by that true belief is also important. Is it based on good evidence, or a reliable process, or a well-functioning design plan? When it comes to knowledge, these further normative questions clearly matter. It is therefore not surprising that the same sorts of questions have been asked with respect to understanding. While some argue that the normative profile of understanding is essentially the same as the normative profile of knowledge (Grimm 2006; Khalifa 2013, 2017b; Greco 2014), others claim that they differ in important ways, and especially with respect to the way in which understanding, but not knowledge, seems compatible with luck. Among theorists who see a difference between knowledge and understanding here, we can distinguish those who claim that understanding can be fully externally lucky from those who think it can only be partly externally lucky. A leading advocate of the fully externally lucky view is Jonathan Kvanvig, who argues that how one comes by one's accurate grasp matters--what we might call the "etiology" of the accurate grasp--but it does not matter in all the same ways that we find in cases of knowledge (see especially Kvanvig 2003). In particular, it matters that the grasp was acquired in a way that was internally appropriate (especially, in accord with the evidence in one's possession), but it does not matter that the grasp was externally appropriate--for example, that one came by one's evidence in a reliable way. Thus Kvanvig argues: > > > What is distinctive about understanding has to do with the way in > which an individual combines pieces of information into a unified > body. This point is not meant to imply that truth is not important for > understanding, for we have noted already the factive character of both > knowledge and understanding. But once we move past its facticity, the > grasping of relations between items of information is central to the > nature of understanding. By contrast, when we move past the facticity > of knowledge, the central features involve non-accidental connections > between mind and world. (Kvanvig 2003: 197) > > > For instance, suppose you read a history book full of inaccurate facts about the Comanche dominance of the Southern Plains in the nineteenth century, but your dyslexia miraculously transforms them all into truths (Kvanvig 2009). For Kvanvig, despite the one-in-a-billion luckiness of your grasp's origins, you could nonetheless come to an understanding of this topic. So long as the accuracy and internal appropriateness conditions are satisfied, how you came by your accurate grasp is not important. Other philosophers have objected to Kvanvig's account. On their view, it is implausible to think you can come to understand the world through sheer external luck, or (perhaps worse) through being the victim of massive deception (Grimm 2006; Pritchard 2010; Khalifa 2017b: ch. 7; Kelp 2021). Thus Pritchard claims that one needs to acquire one's accurate grasp "in the right fashion" (Pritchard 2010: 108) or "in the right kind of way" (Pritchard 2010: 110)--in other words, by means of a reliable source or method. One's grasp therefore cannot be the result of "Gettier luck"--where, for instance, by sheer chance a story intended to deceive you happens to be right (Pritchard 2010). At the same time, philosophers such as Pritchard and Alison Hills hold that understanding does tolerate certain kinds of luck, and in a way that propositional knowledge does not (Pritchard 2010; Hills 2016). They therefore hold that understanding can be partly externally lucky. Thus imagine your environment is filled with misleading information about some event--the fire that destroyed your house, for example. Suppose that only Source X will offer you the truth about the fire--that faulty wiring was the cause--while all of the other sources will offer plausible but mistaken accounts. If you luckily rely on Source X, that source can enable you to understand why the event occurred, even though it cannot generate knowledge about why it occurred, due to the presence of what Pritchard calls "environmental luck". The upshot, on this view, is that while understanding is compatible with environmental luck, it is not compatible with Gettier luck. They conclude from this that since knowledge is compatible with neither Gettier luck nor environmental luck, understanding is not a species of knowledge. (For recent empirical studies regarding the compatibility of judgments of understanding with luck, and finding mixed support for this compatibility, see Wilkenfeld et al. 2018; Carter et al. 2019.) Regarding the normative profile of understanding and its relation to knowledge, others have argued that understanding is not vulnerable to defeat, especially in the face of known counterevidence, in the way that knowledge is (Hills 2016; Dellsen 2017). Dellsen for instance argues that if I come to believe or grasp that a car engine works a certain way, and it really does work that way, then my understanding of how the engine works is secure even if I have (misleading) evidence that the person telling me about the engine is unreliable (Dellsen 2017). To these philosophers, this provides still another reason for thinking that understanding is not a species of knowledge. ## 3. Special Issues in Epistemology ### 3.1 The Epistemic Value of Understanding As part of what Wayne Riggs has called the "value turn in epistemology", epistemologists have increasingly attempted to identify the fundamental bearers of epistemic value (Riggs 2008). From a purely epistemic point of view, is knowledge of just any sort valuable? Or is the fundamentally valuable thing instead the ability to provide reasons, or to possess beliefs that are in some way "Gettier proof"? Into this mix some have argued that the really valuable things from an epistemic point of view are "higher grade" cognitive accomplishments, such as understanding or wisdom (Riggs 2003; cf. Baehr 2014). Thus by nature we do not seem to have any desire to acquire trivial pieces of information, such as the name of the 589th person in the 1971 Dallas phone book. But any instance of understanding might seem worth pursuing or in some way worthwhile; thus Riggs writes, > > > it seems to me that any understanding, even of some subject matter we > may consider trivial or mundane, contributes to the epistemic value of > one's life. (2003: 217) > > > Understanding might therefore be a better candidate for a fundamentally valuable epistemic state than knowledge or truth. Just why understanding might have this property is up for debate. Some proposals focus on the "internal" benefits that understanding is alleged to provide. According to Zagzebski (2001), understanding has a kind of first-person transparency, tied to an ability to articulate reasons, that we do not always find in cases of knowledge. Thus to say a chicken-sexer knows the sex of a particular chicken without being able to cite his or her grounds is one thing, but to say that someone understands some subject matter--say, the U.S. Civil War--without being able to explain it or to describe how the subject's different elements depend upon and relate to one another seems like another, more implausible step (cf. Pritchard 2010). Understanding therefore arguably provides "a more natural home" (cf. Kvanvig 2003: 1993) for internalist intuitions in epistemology than knowledge, because it seems to more naturally appeal to notions such as transparency, articulacy, and reason-giving than knowledge. (For the view that understanding too can be inarticulate, see Grimm 2017.) Another view, from Pritchard (2009, 2010), is that any instance of understanding is valuable because it counts as an achievement, and achievements are the kinds of things that are distinctively and finally valuable. Instances of understanding count as achievements (and indeed what Pritchard calls strong achievements) either because they involve overcoming an obstacle or because they bring to bear "significant cognitive ability". For Pritchard, however, the same cannot be said for any instance of knowledge. For example, in many cases I might come to believe the truth on the basis of testimony, but the credit for my true belief seems to belong more to the testifier than to me (cf. Lackey 2007). I do not reach the truth on this question primarily because of my ability or skill, but because of the ability or skill of the testifier; such items of knowledge therefore do not count as achievements, and hence lack final value. Against this, some have objected that there are many cases of "easy understanding" where no significant obstacle seems to be involved, and no significant cognitive ability at play (e.g., Lawler 2019). For instance, it seems I could come to understand why my tumble-dryer isn't working--because it is unplugged--quite easily and without bringing to bear any particularly significant cognitive ability (Carter & Gordon 2014: 5). It has therefore been claimed that not just any item of understanding is valuable, and particularly not just any item of "understanding why". Rather, only so-called objectual understanding is distinctively valuable, where objectual understanding is said to concern "wide" subject matters that involve "a large network of propositions and relations between those propositions" (Carter & Gordon 2014: 8; cf. Khalifa 2017b: ch. 8). A final proposal is that understanding is a better candidate for the goal of inquiry than knowledge (Pritchard 2016; cf. Kvanvig 2003: 202 and Kelp 2021). Suppose you learn from a reliable source that the rising and falling of the tides are due to the moon's gravitational pull. You will then apparently have knowledge of why the tides rise and fall, but according to Pritchard this knowledge will not properly close your inquiry or satisfy your curiosity. > > > Indeed, one would expect our subject to continue asking questions of > her informant until she gains a proper explanatory grip on how cause > and effect are related; mere knowledge of the cause will not suffice. > (Pritchard 2016: 34) > > > On this view, an explanation of how cause and effect are related is essential to understanding why, and one's inquiry will not reach its natural end until one resolves this question. ### 3.2 Testimony A key issue in social epistemology is whether understanding, like ordinary propositional knowledge, can be transmitted via testimony. Thus it seems that I can transmit my knowledge to you that the next train is arriving at 4:15, just by telling you. Transmitting understanding, however, does not seem to work so easily, if it is possible at all. According to Myles Burnyeat, > > > Understanding is not transmissible in the same sense as knowledge is. > It is not the case that in normal contexts of communication the > expression of understanding imparts understanding to one's > hearer as the expression of knowledge can and often does impact > understanding. (Burnyeat 1980: 186) > > > Thus a teacher might try to convey her understanding of some subject matter--say, of how Type II Diabetes works--but there is no guarantee that the understanding will in fact be transmitted, or that her students will come to see or grasp what she herself sees and grasps. Supposing that understanding cannot be transmitted by testimony, there are a few explanations for why this might be the case. For one, Zagzebski argues that the kind of "seeing" or "grasping" that seems integral to understanding is something a person can only do "first hand"--it cannot be inherited from anyone else (Zagzebski 2008: ch. 6). For another, and according to Hills, if we agree that understanding is a skill or ability, then understanding will be difficult or impossible to transmit because skills and abilities in general are difficult or impossible to transmit (Hills 2009, 2020). It is therefore a specific case of a more general phenomena. Others have argued that understanding is not, in fact, so difficult to transmit. Suppose I ask why you are late for our meeting, and you tell me "traffic". It then seems like you have transmitted your understanding to me quite directly--apparently just as directly as when you communicate your knowledge to me that the next train is coming at 4:15 (Grimm 2020). What seems required for successful transmission of understanding is thus that the right conceptual scaffolding is in place, on the part of the recipient. But supposing the scaffolding is in place, understanding can plausibly be transmitted in much the way knowledge can (Boyd 2017; cf. Malfatti 2019, 2020). Others argue that since something like propositional knowledge of causes entails at least a low degree of understanding, a low degree of understanding can be transmitted via testimony in the same way as knowledge can (Hu 2019). It has also been pointed out that even if we agree that seeing or grasping needs to be first-hand, and that others cannot do this for me, the same "first-handedness" apparently holds for belief. That is, no one else can believe for me, because believing is a first-personal act (Hazlett forthcoming: sec. 2.3). But just as this fact about belief does not lead us to think that propositional knowledge cannot be transmitted, so too it should not lead us to think that understanding cannot be transmitted (Boyd 2017). In relation to moral testimony in particular, understanding has been invoked to explain why it often seems odd or "fishy" to defer to others about moral issues, such as whether eating meat is morally wrong (for an overview, see Callahan 2020; cf. Riaz 2015). One explanation of the fishiness is that the epistemic good we really want with respect to moral questions is not mere knowledge but rather understanding. It is then said, for reasons tied to those just mentioned, that understanding cannot be easily (if at all) transmitted by testimony--either because it is an ability, and abilities cannot easily be transmitted (Hills 2009, 2013, 2016) or because genuine understanding in moral matters involves a suite of emotional and affective responses that cannot easily be transmitted by testimony (Callahan 2018). Deferring to moral testimony is therefore fishy, it is argued, because it does not get us the epistemic good we really want--moral understanding. ## 4. Special Issues in the Philosophy of Science ### 4.1 Explanation and Understanding We seek explanations for an epistemic benefit, but how should we think about that benefit? We noted above (Section 1.2) that philosophers such as Hempel have significant reservations about thinking about the benefit in terms of understanding. As they point out, just as a mistaken explanation can generate a "feeling" of understanding in some people, so too an accurate explanation might leave people cold. In response to this concern, others note that the feeling of understanding (or the phenomenology of understanding) should not be confused with the epistemic state itself, any more than the feeling of knowing should be confused with the epistemic state of knowledge (de Regt 2004, 2017). But even if this move is granted, and we think of understanding as a full-bodied epistemic state and not a mere feeling, the relationship between explanation and understanding remains controversial. According to what we might call an "understanding-first" approach to the relationship, thinking about the epistemology of understanding is indispensable for thinking about what makes for a good explanation. More exactly, because we seem to assess the goodness or badness of an explanation in terms of its ability to generate understanding, understanding is in some sense conceptually prior to, or more basic than, the notion of explanation. Thus intuitions about understanding are often taken to be diagnostic of the goodness (the explanatoriness) of an explanation (Wilkenfeld 2013; cf. Wilkenfeld & Lombrozo 2020), and if we can think of a case where all of the conditions for the theory of explanation are met, but understanding does not result, that is a reason to reject the sufficiency of the conditions (see, e.g., Woodward 2003: 195, in diagnosing Hempel's view). Similarly, if we have understanding in ways not sanctioned by the theory, that is reason to think the conditions are not necessary. More generally, an advocate of the "understanding-first" approach would likely concur with Paul Humphreys's claim that: > > > Scientific understanding provides a far richer terrain than does > scientific explanation and the latter is best viewed as a vehicle to > understanding, rather than an end in itself. (Humphreys 2000: 267; cf. > Potochnik 2017: ch. 5) > > > Against understanding-first approaches, there are "debunking" approaches according to which understanding is of little if any help to accounts of explanation. (More positively, these might simply be considered "explanation-first" approaches.) Thus Khalifa argues that attempts to revive understanding as a central notion in the philosophy of science have amounted to little more than a repackaging of existing models of explanation (Khalifa 2012, 2017b: ch. 3), and that strictly speaking all one needs for a plausible account of understanding is a plausible account of what counts as a good or correct explanation, combined with a plausible account of knowledge. Understanding therefore amounts to knowing a correct explanation. But then nothing new or special is needed to theorize about understanding; our accounts of explanation and our theories of knowledge do all the important theoretical work (see especially Khalifa 2017b; cf. Kelp 2015). Similarly, consider the following attempt, from Bradford Skow, to characterize the relationship between explanation and understanding: > > > Something E* is an explanation of why Q only if > someone who possesses E understands why Q. (Skow > 2018: 214) > > > This condition on explanation cannot help us as theorists, Skow argues, because it is not even true. Thus suppose we construe "possessing" an explanation as knowing* an explanation. Plugged into the formula, the result is that one cannot know an explanation of why Q unless that knowledge constitutes an understanding of why Q. But, plausibly, we can know an explanation of why Q without understanding why Q. For example, someone can know why the litmus paper turned red--because it was dipped in acid--without understanding why it turned red (Skow 2018: 215). This suggests it takes more to understand why p than simply to know why p. In the case of the litmus paper, Skow claims one also needs to appreciate how the acid turns the paper red, or why it turns it red. Opinions differ about the force of such examples. According to some, to require that understanding why p involves knowledge of mechanisms or deeper processes in this way would annihilate most of our everyday understanding (Grimm 2019a). Thus I can apparently understand why my eyes are watering--because I am chopping onions--without appreciating anything about the mechanism or connection that underlies the watering. Others argue that there is indeed understanding in these cases, although perhaps not a lot (Sliwa 2015). ### 4.2 Understanding and Idealization A longstanding puzzle in the philosophy of science concerns how idealized models and representations ("idealizations", for short) allow cognitive access to the world. The puzzle is that although idealizations seem to provide real epistemic benefits, the benefits cannot apparently be identified with truth, because to the extent that they falsify or mispresent the world, idealizations often fail to reflect the truth. For example, idealizations often appeal to entities that do not and perhaps cannot exist--fully rational agents, frictionless planes, etc. Or they subtract important worldly elements from their accounts--e.g., long range inter-molecular forces. Yet if idealizations provide epistemic benefits, and we cannot readily think of the benefits in terms of truth, then how exactly should we think about them? According to some philosophers, we should think not in terms of truth but rather in terms of understanding. Understanding is the epistemic benefit we receive from idealizations, and understanding and truth can come apart. On this view, understanding (unlike knowledge) can therefore be "non-factive" (Elgin 2004, 2017; Potochnik 2017; cf. Sullivan & Khalifa 2019). For instance, Elgin asks us to consider a paint sample we might see in a hardware store--say, of the shade jonquil yellow (Elgin 2017: 187). While the sample might in fact be subtly different than the actual shade, and hence in that sense misrepresent it, it nonetheless seems to provide epistemic access to the shade. Idealizations such as the Ideal Gas Law or Snell's law work are then held to work in a similar way. They represent not by mirroring but rather by exemplifying certain aspects of the target system, and via the mechanism of exemplification enable understanding of the system (Elgin 2017). Others agree that idealizations enable scientists to understand the world even though, strictly speaking, the idealizations misrepresent their target systems. According to Angela Potochnik (2017), this is because the systems studied by scientists are enormously complex networks of causal patterns and interactions. Particular interests and cognitive limitations will lead scientists to focus on some of these causal patterns and neglect others, and they might get those particular patterns right. In the process, however, they will misrepresent the actual messy complexity of the target system, and the understanding of phenomena that results will thereby be non-factive. For Strevens, idealizations enable understanding because they draw attention to the difference makers that bring about the thing to be explained (Strevens 2017). More exactly, idealizations help us to appreciate the factors that make a difference to bringing about the phenomena we want to explain, and to identifying the factors that do not make a difference. The Ideal Gas Law helps us to understand the Boylean behavior of a gas, for example, even though the law imagines that there are zero long-range forces at work, because the existence of those long-rage forces does not make a difference to the derivation of the Boylean behavior: they are insignificant enough that their values can be set to zero without loss. One difference between theorists such as Strevens and Potochnik may lie in the nature of the explanandum. For Potochnik, it seems clear that the thing to be explained is a concrete, coarse-grained phenomenon: the behavior of a gas in a container, for example. For Strevens, it is a more abstract, fine-grained, and high-level event: the expansion of the gas. Thus for Strevens the thing to be explained is something along the lines of: that the gas expanded, not how it expanded. If one were to hold the explanandum consistent, there might be notable agreement between these views (though see Potochnik 2016 for further discussion). ## 5. Humanistic Understanding To this point we have mainly focused on what it takes to understand phenomena in the natural world, such as rusting metal bars, or the rising and falling of the tides, or the behavior of gases. A longstanding question, however, especially in the Continental tradition of philosophy and especially in the philosophy of the social sciences, is whether understanding human beings and their artifacts requires something different and distinctive from an epistemic point of view--perhaps, for example, a distinctive set of abilities or methodologies, tied to the distinctive objects (or, importantly, subjects!) we are trying to understand. Moreover, a long string of influential thinkers--starting roughly with Giambattista Vico and continuing through figures such Wilhelm Dilthey and R. G. Collingwood--answer this question affirmatively. Distinctive abilities and perhaps methodologies do come on the scene when we try to understand human beings, at least when we try to understand them in a particular way. In broad strokes, the idea is that there is a kind of understanding of other human beings that we can only acquire by reconstructing their perspectives--in some sense, "from within" those perspectives and according to their own terms (cf. Ismael 2018). Sometimes this approach has been referred to as "the verstehen tradition"--in honor of its special roots in the German tradition, and leaving untranslated the German word for understanding (Martin 2000; Feest 2010). Alternative labels for this approach include "the interpretative tradition", "the historicist tradition", "the hermeneutic tradition", and "the humanistic tradition" (for overviews, see Hiley et al. 1992; Stueber 2012). For convenience, we will refer to this as "the humanistic tradition". It is not an essential part of this tradition that human beings can only be understood by reconstructing their perspectives, but the tradition does characteristically claim that there is special value in trying to do so. It also typically claims that taking up perspectives "from within" requires distinctive cognitive resources and perhaps distinctive methods--resources and methods that are not clearly needed when we attempt to understand natural phenomena. Giambattista Vico (1668-1744) was among the first to try to articulate how exactly understanding human beings differs from understanding the natural world (see especially Vico 1725 [2002].) According to Vico, just as we have a special scienza (knowledge or understanding) of things we have made or produced ourselves, so too we can have special insight into things that other human beings have made or produced--where the things made or produced included not just physical artifacts but also human actions. Vico further postulated a special ability--fantasia, or reconstructive imagination--by which we can enter into the minds of others and see the world through their eyes and in terms of their categories of thought (Miller 1993: ch. 5). Further north in Europe, Johann Gottfried Herder (1744-1803) held that because human nature is not static, or fixed irrespective of time and place, understanding alien cultures requires a process of Einfuhlung--a "feeling one's way into"--the culture in question, in all its specificity and perhaps with an eye to its particular genius. In Herder's hands, however, Einfuhlung was not a quasi-mystical or irrational attempt to leap into the minds of others. Instead, it was often a slow, methodical process that needed to be aided by careful historical-philological inquiry (Forster 2002: xvii). (For central texts, see Herder 2002.) In the late nineteenth and early twentieth century, Wilhelm Dilthey (1833-1911) became the most celebrated advocate of the idea that the human sciences--in German, the Geisteswissenschaften, such as history--had a different focus than the natural sciences, and centered on the idea of inner or lived experience. (See Dilthey 1883 [1989] for a characteristic work.) Someone's lived experience, moreover, was constituted not only by their beliefs or attitudes, but also by their emotions and volitions. For Dilthey, as Frederick Beiser puts the idea, > > > We understand an episode in a person's life only when we see how > it plays a role in realizing his basic values, his conception of what > makes life worth living. (Beiser 2011: 334) > > > In England, the humanistic approach was championed especially by R. G. Collingwood (1889-1943). According to Collingwood, what distinguishes human history from, say, geological history, is that human history is suffused with thought--i.e., it is a record of human aspirations, goals, beliefs, frustrations, and so on. We can, moreover, have two possible stances towards these thoughts. We can try to map them from a third person point of view, and identify the various ways the thoughts depend upon and relate to one another. But we can also try to "re-enact" or "rethink" them (D'Oro 2000, 2004). Thus Collingwood held that, > > > The events of history are never mere phenomena, never mere spectacles > for contemplation, but things which the historian looks, not at, but > through, to discern the thought within them. (Collingwood 1946 [1993: > 214]) > > > ### 5.1 Perspective-taking and its Critics A central tenet of the humanistic tradition is that when an agent has a first-person perspective on the world, there is value in trying to "take up" or "assume" that perspective, rather than simply trying to map that agent's perspective from a detached, third-person perspective. Abraham Lincoln for instance, could presumably be understood from a third-person, detached perspective by a psychologist adept at mapping the various ways in which Lincoln's beliefs and desires, hopes and fears, seemed to depend upon and relate to one another. What the humanistic tradition says is that even if the psychologist were to do this exquisitely, so that all of these relationships were accurately plotted, there would still be an important sort of understanding missing--an understanding of what it was like to be Lincoln from the inside, or to regard the world as he regarded it, at least in part. Philosophers differ about the epistemic value of this project of perspective-taking. Thus some claim that there are certain facts that only show up from a first-person point of view (Nagel 1974), and hence can only be apprehended from that point of view. For instance, if I want to apprehend facts about "what it is like" to experience the world from your perspective, I arguably need to try to place myself, somehow, in that perspective. Others claim that certain concepts only show up for us from a first-personal, engaged perspective. For instance, perhaps the concept of chronic pain can only be acquired by someone who has experienced it, or the concept of romantic love, or racial bigotry. A related view is that while these concepts can perhaps be acquired without the relevant first-person experience, they cannot be adequately grasped or mastered without that experience. As Stephen Turner writes: When a mother tells her 13 year old daughter that she does not know what "love" is, she is not making a comment about semantics; she is pointing to the nonlinguistic experiential conditions that are bound up with the understanding of the term that the daughter does not share. (Turner 2019: 254) The thought here seems to be that while the daughter might have the concept of love, her grasp of the concept will be extremely poor without the benefit of the first-person experience, and her attempts to apply it will be unreliable. In fields such as anthropology, ethnography, and sociology, and in the philosophical areas that reflect on these disciplines (primarily, the philosophy of social science), questions related to the epistemic value of perspective-taking gave rise to debates about whether it is necessary to immerse oneself in a society in order to understand it adequately. According to some, the answer is "yes": if I want to understand a society or culture, I need to understand it from the participants' perspective, with special sensitivity to the concepts and rules that guide the participants' perspective, even if only implicitly (see especially Winch 1958, 1964). It is then sometimes claimed that one can only really grasp or master these concepts by participating in the relevant form of life. As the ethnographer James Spradley writes: > > > Immersion is the time-honored strategy used by most ethnographers. By > cutting oneself off from other interests and concerns, by listening to > informants hours on end, by participating in the cultural scene, and > by allowing one's mental life to be taken over by the new > culture, themes [the implicit beliefs of a culture] often emerge > ... This type of immersion will often reveal new relationships > among domains and bring to light cultural themes you cannot discover > any other way. (Spradley 1980: 145; cf. Kampa 2019) > > > Another view is that immersion, while perhaps helpful, is not strictly necessary. What is necessary is recognizing the priority of the participants' perspective, and using that perspective to characterize the activities that need to be understood in the first place (Taylor 1971; McCarthy 1973). If we want to make sense of what looks like (say) voting or praying in another culture, we need to adopt that culture's way of identifying what counts as voting or praying (which perhaps will turn out to be a distinct but closely related activity, such as voting\* or praying\). Carving up the behavior according to our own* categories will only misrepresent what is happening and lead to misunderstanding. Peter Winch, among others, further argues that it is hard or impossible to try to adopt another culture's way of carving up the world without somehow taking that culture on board in a holistic way, because it is hard or impossible to characterize particular bits of behavior in isolation (Winch 1958). To make sense of what a medieval knight means by praying (or praying\), for example, I might need to connect this with other concepts (perhaps such as honor, or duty, or salvation) that again might be quite different than my own. The result might then be that one gradually acquires what Alasdair MacIntyre calls a "second first language", as one's grasp of the relationships between different concepts and their applications increases (MacIntyre 1988: ch. 19). Against the push for perspective-taking as a source of understanding, especially in the social sciences, at least three strains of criticism arose: Positivist, Critical, and Gadamerian. The Positivist critique--associated with thinkers such as Theodore Abel (1948), Ernest Nagel (1953), and Richard Rudner (1966)--had a number of prongs. For one thing, there were grave doubts about the reliability of the processes associated with perspective taking. After all, it seems all too easy to project one's own cares and concerns onto the minds of others. It was also argued that reliving experiences, to the extent that this is possible, does not actually amount to an explanation of why the experiences happened. I could, after all, have lived through a series of events myself, but not been able to explain or understand them--so why should imaginatively living through someone else's experiences, to the extent this is possible, automatically grant me understanding? (For an overview of these objections, see Martin 2000; Fay 2016; Beiser 2019.) Critical Theorists have argued that the first-person perspective is not the most theoretically or politically important perspective, because it often masks deeper and more significant sources of behavior (see Warnke 2019). The deeper sources might include an agent's own, unacknowledged motivations, or they might include the presuppositions, power dynamics, and economic conditions of the society that shaped the possibility spaces in which the agent moves. It is thus what happens "behind the back" of the subject that is often crucial for truly understanding behavior--the hidden impulses and power dynamics and systems of oppression that are often obscured from the perspective of the agent, sometimes in an unacknowledged way by the agent herself. (Here we also find influential Marxist and Feminist analyses and critiques. See Alcoff 2005; Warnke 2014.) A hybrid view, from Habermas, is that we need to be able to shift back and forth between the subject's perspective and the structural perspective, if we are to properly understand the reality of social life (Habermas 1981 [1984/1987]; for an overview, see Baynes 2016). The Gadamerian critique, finally, is that we can never fully jump outside of our own cares and concerns in order to adopt the cares and concerns of others. Reliving or re-experiencing-- in the sense of transposing ourselves out* of our framework and into the framework of other agents or cultures--is thus an impossible ideal. Instead, we always take our frameworks with us (Gadamer 1960 [1989]). For Gadamer, what understanding others--and especially their written or spoken words--requires is not transposing ourselves into their worldviews, but trying to fuse our worldviews in some way (Horizontverschmelzung). Perhaps, for example, my conception of voting or prayer will become enlarged or modified by engaging with the conceptions I find in other cultures, and perhaps especially through a dialogue with the texts of those cultures. Whether this fusion essentially amounts to a project of translation (from one worldview to another), or something else is a matter for debate. (See Vessey 2009 for an overview.) ### 5.2 Understanding as an Ontological Category An additional question, especially in the Continental tradition, is whether we should think of understanding primarily in epistemic terms. Perhaps it is better to think of understanding as an ontological category--a way of being in the world--rather than an epistemic one (for an overview, see Feher 2016). According to Heidegger's influential account, for instance, understanding is not a cognitive act that we might or might not perform. Rather, it is our fundamental way of living in the world, and we are "always already" engaged in understanding. Thus he writes that: > > > Understanding is not an acquaintance derived from knowledge, but a > primordially existential kind of being, which, more than anything > else, makes such knowledge and acquaintance possible (123-4; > trans. by Wrathall 2013). > > > Just as Descartes spoke of himself as a res cogitans--a thinking thing--so for Heidegger we are, fundamentally, understanding things. As Gadamer elaborates the Heideggerian idea: > > > Understanding is... the original form of the realization of > Dasein, which is being-in-the-world. Before any differentiation of > understanding into the various directions of pragmatic or theoretical > interest, understanding is Dasein's mode of being. (Gadamer 1960 > [1989: 259]) > > > One central element of this view is that we are always projecting possibilities onto the world around us. Or rather, it is not as if the world first presents itself as a bare thing empty of possibilities, and then we construe it as possessing these possibilities; instead, our experience of the world is from the first suffused with this sense of possibility. To take an example that Samantha Matherne (2019) uses to illustrate Heidegger's view: when I first apprehend the martini in front of me, I take it as offering a variety of possibilities--to be sipped, to be thrown, to be shaken, to be stirred. If I then take the martini as to be sipped, I am seizing on one of these possibilities and interpreting the martini in light of this specific possibility. But my experience of the drink was never devoid of an apprehension of possibilities altogether. A further question is why, on the Heideggerian framework, the state of projecting possibilities does not count as epistemic or truth-evaluable, even if we allow that it is also in some sense ontological. After all, the possibilities that we project onto the world might not be genuine or grounded in reality. As Grimm (2008) notes, someone might think that apparently solid things (like baseballs) could not possibly pass through other apparently solid things (like tables), but learning about quantum tunneling might call this into question, and transform one's sense of possibility. There thus seem to be facts about the possibilities that the world affords, and it seems like a mind could either track these facts accurately, or not. But this appears to be a topic of interest not just to ontologists or metaphysicians, but to epistemologists as well (cf. Westphal 1999).
selection-units	## 1. Introduction When we think of evolutionary theory and natural selection, we usually think of organisms, say, a herd of deer, in which some deer are faster than others at escaping their predators. These swifter deer will, all things being equal, leave more offspring, and these offspring will have a tendency to be swifter than other deer. Thus, we get a change in the average swiftness of deer over evolutionary time. In a case like this, the unit of selection, sometimes called the "target" of selection, is the single organism, the individual deer, and the property being selected, swiftness, also lies at the organismic level, in that it is exhibited by the intact and whole deer, and not by either parts of deer, such as cells, or groups of deer, such as herds. But there are other levels of biological organization that have been proposed to be units or targets of selection--levels at which selection may act to increase a given property at that level, and at which units increase or decrease as a result of selection at that specific level of biological organization. But for over thirty years, some participants in the "units of selection" debates have argued that more than one issue is at stake. The notions of "replicator" and "vehicle" were introduced, to stand for different roles in the evolutionary process (Dawkins 1978, 1982a,b). In this case, the individual deer would be called the "vehicles" and their genes that make them tend to be swifter would be called the "replicators." The genic selection argument proceeded to assert that the units of selection debates should not be about vehicles, as they formerly had been, but about replicators. It was then asserted that the "replicator" actually subsumes two distinct functional roles, which can be broken up into "replicator" and "interactor": > > > Dawkins...has replicators interacting with their environment in > two ways--to produce copies of themselves and to influence their > own survival and the survival of their copies. (Hull 1980: 318) > > > The new view would call the individual deer the "interactors." It was then argued that the force of this distinction between replicator and interactor had been underappreciated, and if the units of selection controversies were analyzed further, that the question about interactors should more accurately be called the "levels of selection" debate to distinguish it from the dispute about replicators, which should be allowed to keep the "units of selection debate" title (Brandon 1982; Mitchell 1987). The purpose of this article is to delineate further the various questions pursued under the rubric of "units and levels of selection."[1] Four quite distinct questions will be isolated that have, in fact, been asked in the context of considering, what is a unit of selection? In section 2, these distinct questions are described. Section 3 returns to the sites of several very confusing, occasionally heated debates about "the" unit of selection. Several leading positions on the issues are analyzed utilizing the taxonomy of distinct questions. This analysis is not meant to resolve any of the conflicts about which research questions are most worth pursuing; moreover, there is no attempt to decide which of the questions or combinations of questions discussed ought to be considered "the" units of selection question. ## 2. Four Basic Questions Four basic questions can be delineated as distinct and separable. As will be demonstrated in section 3, these questions are often used in combination to represent the units of selection problem. But let us begin by clarifying terms (see Lloyd 1992, 2001). (See the entry on the biological notion of individual for more on this topic.) The term replicator, originally introduced in the 1970s but since modified by philosophers in the 1980s, is used to refer to any entity of which copies are made (Dawkins 1976, 1982a,b; Hull 1980; Brandon 1982). Replicators were originally described using two orthogonal distinctions. A "germ-line" replicator, as distinct from a "dead-end" replicator, is "the potential ancestor of an indefinitely long line of descendant replicators" (Dawkins 1982a: 46). For instance, DNA in a chicken's egg is a germ-line replicator, whereas that in a chicken's wing is a dead-end replicator. Note that DNA are, but chickens are not, replicators, since the latter do not replicate themselves as wholes. An "active" replicator is "a replicator that has some causal influence on its own probability of being propagated," whereas a "passive" replicator is never transcribed and has no phenotypic expression whatsoever (Dawkins 1982a: 47). There is special interest in active germ-line replicators, "since adaptations 'for' their preservation are expected to fill the world and to characterize living organisms" (Dawkins 1982a: 47). The original terminology of "replicator" was introduced along with the term "vehicle", which is defined as > > > any relatively discrete entity...which houses replicators, and > which can be regarded as a machine programmed to preserve and > propagate the replicators that ride inside it. (Dawkins 1982b: 295) > > > > On this view, most replicators' phenotypic effects are represented in vehicles, which are themselves the proximate targets of natural selection (Dawkins 1982a: 62). In the introduction of the term "interactor", it was observed that the previous theory has replicators interacting with their environments in two distinct ways: they produce copies of themselves, and they influence their own survival and the survival of their copies through the production of secondary products that ultimately have phenotypic expression (Hull 1980). The term "interactor" was suggested for the entities that function in this second process. An interactor denotes that entity which interacts, as a cohesive whole, directly with its environment in such a way that replication is differential--in other words, an entity on which selection acts directly (Hull 1980: 318). The process of evolution by natural selection is > > > a process in which the differential extinction and proliferation of > interactors cause the differential perpetuation of the replicators > that produced them. (Hull 1980: 318; see Brandon 1982: 317-318) > > > > One challenge to the term, "interactor," was that "interacting is not conspicuous during the process of elimination that results in natural selection" (Mayr 1997: 2093). It's difficult to imagine why anyone would say this, given the original description of the interactor as "an entity that directly interacts ... in such a way that replication is differential". Perhaps more interestingly, the "target of selection" language is rejected because selection is seen as more of an elimination process; thus, it would be misleading to call the "leftovers" of the elimination process the "targets" of selection. The term "selecton", was proposed, which is defined as > > > a discrete entity and a cohesive whole, an individual or a social > group, the survival and successful reproduction of which is favored by > selection owing to its possession of certain properties. (Mayr 1997: > 2093) > > > This seems remarkably similar to an interactor, with the difference that no differential reproduction, and thus no evolution, is mentioned. At the birth of the "interactor" concept, the concept of "evolvers" was also introduced, which are the entities that evolve as a result of selection on interactors: these are usually called lineages (Hull 1980). So far, no one has directly claimed that evolvers are units of selection. They can be seen, however, to be playing a role in considering the question of who owns an adaptation and who benefits from evolution by selection, which we will consider in sections 2.3 and 2.4. ### 2.1 The Interactor Question In its traditional guise, the interactor question is, what units are being actively selected in a process of natural selection? As such, this question is involved in the oldest forms of the units of selection debates (Darwin 1859 [1964], Haldane 1932, Wright 1945). In an early review on "units of selection", the purpose of the article was claimed as: "to contrast the levels of selection, especially as regards their efficiency as causers of evolutionary change" (Lewontin 1970: 7). Similarly, others assumed that a unit of selection is something that "responds to selective forces as a unit--whether or not this corresponds to a spatially localized deme, family, or population" (Slobodkin & Rapoport 1974: 184). Questions about interactors focus on the description of the selection process itself, that is, on the interaction between an entity, that entity's traits and environment, and on how this interaction produces evolution; they do not focus on the outcome of this process (see Wade 1977; Vrba & Gould 1986). The interaction between some interactor at a certain level and its environment is assumed to be mediated by "traits" that affect the interactor's expected survival and reproductive success. Here, the interactor is possibly at any level of biological organization, including a group, a kin-group, an organism, a gamete, a chromosome, or a gene. Some portion of the expected fitness of the interactor is directly correlated with the value of the trait in question. The expected fitness of the interactor is commonly expressed in terms of genotypic fitness parameters, that is, in terms of the fitness of combinations of replicators. Hence, interactor success is most often reflected in and counted through, replicator success, either through simple summation of the fitnesses of their traits, or some more complicated relation. Several methods are available for expressing the correlation between interactor trait and (genotypic or genic) fitness, including partial regression, variances, and covariances.[2] In fact, much of the interactor debate has been played out through the construction of mathematical genetic models--with the exception of work on group selection and on female-biased sex ratios (Wade 1980, 1985, 2016; D.S. Wilson & Colwell 1981; see especially Griesemer & Wade 1988). The point of building such models is to determine what kinds of selection, operating at which levels, may be effective in producing evolutionary change. It has been widely held, for instance, that the conditions under which group selection can effect evolutionary change are quite stringent and rare. Typically, group selection was seen to require small group size, low migration rate, and extinction of entire demes.[3] Some modelers, however, disagree that these stringent conditions are necessary, and show that in the evolution of altruism by group selection, very small groups may not be necessary (Matessi & Jayakar 1976: 384; contra Maynard Smith 1964). Others also argue that small effective deme size is not a necessary prerequisite to the operation of group selection (Wade & McCauley 1980: 811). Similarly, another shows that strong extinction pressure on demes is not necessary (Boorman 1978: 1909). And finally, there was an early group selection model that violates all three of the "necessary" conditions usually cited (Uyenoyama 1979; see Wade 2016). That different researchers reach such disparate conclusions about the efficacy of group selection is partly because they are using different models with different parameter values. Several assumptions, routinely used in group selection models, were highlighted, that biased the results of these models against the efficacy of group selection (Wade 1978). For example, many group selection models use a specific mechanism of migration; it is assumed that the migrating individuals mix completely, forming a "migrant pool" from which migrants are assigned to populations randomly. All populations are assumed to contribute migrants to a common pool from which colonists are drawn at random. Under this approach, which is used in all models of group selection prior to 1978, small sample size is needed to get a large genetic variance between populations (Wade 1978: 110; see discussion in Okasha 2003, 2006). If, in contrast, migration occurs by means of large populations, higher heritability of traits and a more representative sampling of the parent population will result. Each propagate is made up of individuals derived from a single population, and there is no mixing of colonists from the different populations during propagule formation. On the basis of further analysis, much more between-population genetic variance can be maintained with the propagule model (Slatkin and Wade 1978: 3531). Thus, by using propagule pools as the assumption about colonization, one can greatly expand the set of parameter values for which group selection can be effective (Slatkin and Wade 1978, cf. Craig 1982). Another aspect of this debate that has received a great deal of consideration concerns the mathematical tools necessary for identifying when a particular level of biological organization meets the criteria for being an interactor.[4] Overall, while many of the suggested techniques have had strengths, no one approach to this aspect of the interactor question has been generally accepted and indeed it remains the subject of debate in biological circles (Okasha 2004b,c). Detailed work on two of the major techniques, the Price equation and contextual analysis, has indicated that neither approach is universally applicable, on the grounds that neither provides a proper causal decomposition in all varieties of selection (Okasha 2006). Specifically, it appears that while contextual analysis may be superior in most cases of multi-level selection, the Price equation may be more useful in certain cases of genic selection (Okasha 2006). It is important to note that, even in the midst of deciding among the various methods for representing selection processes, these choices have consequences for the empirical adequacy of the selection models. This is true even if the models are denied to have a causal interpretation, as is done by those promoting a "statisticalist" interpretation of selection theory (Walsh, Lewens, & Ariew 2002). On this view, evolution is seen as a purely statistical phenomenon, and population genetics studies statistical relations estimated by census, and not causal relationships. The claim is that the "deeply uninteresting" units of selection problem has been dissolved, whereas in fact, it has simply been restricted to the interactor question (while ignoring the other three "units" questions entirely); the problem of how to deliver an empirically adequate selection model is not directly addressed (2002: 470-471). Instead, an unspecified method is assumed that "identifies classes [that] ... adequately predict and explain changes in the structure of the population" (Walsh, Lewens, & Ariew 2002: 471), with no acknowledgment that this involves making a commitment to one or another of the above methods of determining an interactor, whether under a causal interpretation or not. Thus, the interactor problem has not been escaped, whether or not it is interpreted causally (see Otsuka 2016). Note that the "interactor question" does not involve attributing adaptations or benefits to the interactors, or indeed, to any candidate unit of selection. Interaction at a particular level involves only the presence of a trait at that level with a special relation to genic or genotypic expected success that is not reducible to interactions at a lower level. A claim about interaction indicates only that there is an evolutionarily significant event occurring at the level in question; it says nothing about the existence of adaptations at that level. As we shall see, the most common error made in interpreting many of the interactor-based approaches is that the presence of an interactor at a level is taken to imply that the interactor is also a manifestor of an adaptation at that level. ### 2.2 The Replicator Question The focus of discussions about replicators concerns just which organic entities actually meet the definition of replicator. Answering this question obviously turns on what one takes the definition of replicator to be. In this connection the revision of the original meaning of "replicator" turned out to be central. The revised meaning refined and restricted the meaning of "replicator," which was defined as "an entity that passes on its structure directly in replication" (Hull 1980: 318). The terms replicator and interactor will be used in this latter sense in the rest of this entry. The revised definition of replicator corresponds more closely than the original one to a long-standing debate in genetics about how large or small a fragment of a genome ought to count as a replicating unit--something that is copied, and which can be treated separately in evolutionary theory (see especially Lewontin 1970; Hull 1980). This debate revolves critically around the issue of linkage disequilibrium and led some biologists to advocate the usage of parameters referring to the entire genome rather than to allele and genotypic frequencies in genetical models (Lewontin 1974).[5] The basic point is that with much linkage disequilibrium, individual genes cannot be considered as replicators because they do not behave as separate units during reproduction. Although this debate remains pertinent to the choice of state space of genetical model, it has been eclipsed by concerns about interactors in evolutionary genetics. This is not to suggest that the replicator question has been solved. Work on the replicator question is part of a rich and continuing research program; it is simply no longer a large part of the units debates. That this parting of ways took place is largely due to the fact that evolutionists working on the units problems tacitly adopted the original suggestion that the replicator, whatever it turned out to be, be called the "gene" (Dawkins 1982b, pp. 84-86; see section 3.3). This move neatly removes the replicator question from consideration. Exactly why this move should have met with near universal acceptance is to some extent historical, however the fact that the intellectual tools (largely mathematical models) of the participants in the units debates were better suited to dealing with aspects of that debate other than the replicator question which requires mainly bio-chemical investigation, surely contributed to this outcome. There is a very important class of exceptions to this general abandonment of the replicator question. Developmental Systems Theory was formulated as a radical alternative to the interactor/replicator dichotomy (Oyama 1985; Griffiths & Gray 1994, 1997; Oyama, Griffiths, & Gray 2001). Here the evolving unit is understood to be the developing system as a whole, privileging neither the replicator nor the interactor. There has also been a profound reconception of the evolution by selection process, which has rejected the role of replicator as misconceived. In its place the role of "reproducer" is proposed, which focuses on the material transference of genetic and other matter from generation to generation (Griesemer 2000a,b; see Forsdyke 2010; see section 3.5). On this approach, thinking in terms of reproducers incorporates development into heredity and the evolutionary process. It also allows for both epigenetic and genetic inheritance to be dealt with within the same framework. The reproducer plays a central role, along with a hierarchy of interactors, in work on the units of evolutionary transition (see Evolutionary Transition; Griesemer 2000c). This topic concerns the major transitions of life from one level of complexity to the next, for example, the transition from unicellularity to multicellularity. More recently, another notion was introduced of a "reproducer" that is more broadly inclusive, in that it relaxes the material overlap requirement and focuses on an understanding of "who came from whom, and roughly where one begins and another ends" (Godfrey-Smith 2009: 86). These two definitions of "reproducer" disagree about retroviral reproduction, and what counts as a salient material bond between generations. On one side is the claim that there is no material overlap in the case of retroviral reproduction, and that the key is formal or informational relations (Godfrey-Smith 2009). On the other side, is a view that sees material overlap due to RNA strand hybridization guiding and channeling flows of information (Griesemer 2014, 2016). There is also the introduction of a notion of reproducer that involves only the copying of a property, with no substance overlapping involved (Nanay 2011). Like the second view of reproducer, it appeals to the case of retroviruses having no material overlap (cf. Griesemer 2014, 2016). ### 2.3 The Beneficiary Question Who benefits from a process of evolution by selection? There are two predominant interpretations of this question: Who benefits ultimately in the long term, from the evolution by selection process? And who gets the benefit of possessing adaptations as a result of a selection process? Take the first of these, the issue of the ultimate beneficiary. There are two obvious answers to this question--two different ways of characterizing the long-term survivors and beneficiaries of the evolution by selection process. One might say that the species or lineages (the previous "evolvers") are the ultimate beneficiaries of the evolutionary process. Alternatively, one might say that the lineages characterized on the genic level, that is, the surviving alleles, are the relevant long-term beneficiaries. I have not located any authors holding the first view, but, for Richard Dawkins, the latter interpretation is the primary fact about evolution. To arrive at this conclusion, he adds the requirement of agency to the notion of beneficiary (see Hampe & Morgan 1988). A beneficiary, by definition, does not simply passively accrue credit in the long term; it must function as the initiator of a causal pathway (Dawkins 1982a,b). Under this definition, the replicator is causally responsible for all of the various effects that arise further down the biochemical or phenotypic pathway, irrespective of which entities might reap the long-term rewards (Sapienza 2010). A second and quite distinct version of the beneficiary question involves the notion of adaptation. The evolution by selection process may be said to "benefit" a particular level of entity under selection, through producing adaptations at that level (Williams 1966, Maynard Smith 1976, Eldredge 1985, Vrba 1984). On this approach, the level of entity actively selected (the interactor) benefits from evolution by selection at that level through its acquisition of adaptations. It is crucial to distinguish the question concerning the level at which adaptations evolve from the question about the identity of the ultimate beneficiaries of that selection process. One can think that organisms have adaptations without thinking that organisms are the "ultimate beneficiaries" of the selection process.[6] This sense of "beneficiary" that concerns adaptations will be treated as a separate issue, discussed in the next section. ### 2.4 The Manifestor-of-Adaptation Question At what level do adaptations occur? Or, "When a population evolves by natural selection, what, if anything, is the entity that does the adapting?" (Sober 1984: 204). As mentioned previously, the presence of adaptations at a given level of entity is sometimes taken to be a requirement for something to be a unit of selection.[7] Significantly, group selection for "group advantage" should be distinguished from group selection per se (Wright 1980). In fact, the combination of the interactor question with the question of what entity had adaptations had created a great deal of confusion in the units of selection debates in general. Some, if not most, of this confusion is a result of a very important but neglected duality in the meaning of "adaptation" (in spite of useful discussions in Brandon 1978, Burian 1983, Krimbas 1984, Sober 1984). Sometimes "adaptation" is taken to signify any trait at all that is a direct result of a selection process at that level. In this view, any trait that arises directly from a selection process is claimed to be, by definition, an adaptation (e.g., Sober 1984; Brandon 1985, 1990; Arnold & Fristrup 1982). Sometimes, on the other hand, the term "adaptation" is reserved for traits that are "good for" their owners, that is, those that provide a "better fit" with the environment, and that intuitively satisfy some notion of "good engineering."[8] These two meanings of adaptation, the selection-product and engineering definitions respectively, are distinct, and in some cases, incompatible. Consider the peppered moth case: natural selection is acting on the color of the moths over time, and the population evolves, but no "engineering" adaptation emerges. Rather, the proportion of dark moths simply increases over time, relative to the industrial environmental conditions, a clear case of evolution by natural selection, on which a good fit to the environment is reinforced. Note that the dark moths lie within the range of variation of the ancestral population; they are simply more frequent now, due to their superior fit with the environment. The dark moths are a "selection-product" adaptation. Contrast the moth case to the case of Darwin's finches, in which different species evolved distinct beak shapes specially adapted to their diet of particular seeds and foods (Grant & Grant 1989; Grant 1999). Natural selection here occurred against constantly changing genetic and phenotypic backgrounds in which accumulated selection processes had changed the shapes of the beaks, thus producing "engineering" adaptations when natural selection occurred. The finches now possess evolved traits that especially "fit" them to their environmental demands; their newly shaped beaks are new mechanisms beyond the original range of variation in the ancestral population (Lloyd 2015). Some evolutionary biologists have strongly advocated an engineering definition of adaptation (e.g., Williams 1966). The basic idea is that it is possible to have evolutionary change result from direct selection favoring a trait without having to consider that changed trait as an adaptation. Consider, for example, Waddington's (1956) genetic assimilation experiments. How should we interpret the results of Waddington's experiments in which latent genetic variability was made to express itself phenotypically because of an environmental pressure (Williams 1966: 70-81; see the lucid discussion in Sober 1984: 199-201)? The question is whether the bithorax condition (resulting from direct artificial selection on that trait) should be seen as an adaptive trait, and the engineering adaptationist's answer is that it should not. Instead, the bithorax condition is seen as "a disruption...of development," a failure of the organism to respond (Williams 1966: 75-78). Hence, this analysis drives a wedge between the notion of a trait that is a direct product of a selection process and a trait that fits the stronger engineering definition of an adaptation (see Gould & Lewontin 1979; Sober 1984: 201; cf. Dobzhansky 1956).[9] In sum, when asking whether a given level of entity possesses adaptations, it is necessary to state not only the level of selection in question but also which notion of adaptation--either selection-product or engineering--is being used. This distinction between the two meanings of adaptation also turns out to be pivotal in the debates about the efficacy of higher levels of selection, as we will see in sections 3.1 and 3.2. ### 2.5 Summary In this section, four distinct questions have been described that appear under the rubric of "the units of selection" problem, What is the interactor? What is the replicator? What is the beneficiary? And what entity manifests any adaptations resulting from evolution by selection? There is a serious ambiguity in the meaning of "adaptation"; which meaning is in play has had deep consequences for both the group selection debates and the species selection debates (Lloyd 2001). Commenting on this analysis, John Maynard Smith wrote in Evolution: > > > [Lloyd 2001] argues, correctly I believe, that much of the confusion > has arisen because the same terms have been used with different > meanings by different authors ... [but] I fear that the > confusions she mentions will not easily be ended. (Maynard Smith 2001: > 1497) > > > In section 3, this taxonomy of questions is used to sort out some of the most influential positions in five debates: group selection (3.1), species selection (3.2), genic selection (3.3), genic pluralism (3.4), as well as units of evolutionary transition (3.5). ## 3. An Anatomy of the Debates ### 3.1 Group Selection The near-deathblow in the nineteen sixties to group panselectionism was, oddly enough, about benefit (Williams 1966). The interest was in cases in which there was selection among groups and the groups as a whole benefited from organism-level traits (including behaviors) that seemed disadvantageous to the organism (Wynne Edwards 1962; Williams 1966; Maynard Smith 1964). The argument was that the presence of a benefit to the group was not sufficient to establish the presence of group selection. This was demonstrated by showing that a group benefit was not necessarily a group adaptation (Williams 1966). Hence, here the term "benefit" was being used to signify the manifestation of an adaptation at the group level. The assumption was that a genuine group selection process results in the evolution of a group-level trait--a real adaptation--that serves a design purpose for the group. The mere existence however, of traits that benefit the group is not enough to show that they are adaptations; in order to be an adaptation, under this view, the trait must be an engineering adaptation that evolved by natural selection. The argument was that group benefits do not, in general, exist because they benefit the group; that is, they do not have the appropriate causal history (Williams 1966; see Brandon 1981, 1985: 81; Sober 1984: 262 ff.; Sober & Wilson 1998). Implicit in this discussion is the assumption that being a unit of selection at the group level requires two things: (1) having the group as an interactor, and (2) having a group-level engineering-type adaptation. That is, the approach taken combines two different questions, the interactor question and the manifestor-of-adaptation question, and calls this combined set the unit of selection question. These requirements for "group selection" make perfect sense given that the prime target was a view of group selection that incorporated this same two-pronged definition of a unit of selection (see Borrello 2010 for a philosophically-oriented history of Wynne-Edwards and his views on group selection). This combined requirement of engineering group-level adaptation in addition to the existence of an interactor at the group level is a very popular version of the necessary conditions for being a unit of selection within the group selection debates. For example, it was claimed that the group selection issue hinges on "whether entities more inclusive than organisms exhibit adaptations" (Hull 1980: 325). Another view states that the unit of selection is determined by "Who or what is best understood as the possessor and beneficiary of the trait" (Cassidy 1978: 582). Similarly, paleontological approaches required adaptations for an entity to count as a unit of selection (Eldredge 1985: 108; Vrba 1983, 1984). The engineering notion of adaptation was also tied into the version of the units of selection question in other contexts (Maynard Smith 1976). In an argument separating group and kin selection, it was concluded that group selection is favored by small group size, low migration rates, and rapid extinction of groups infected with a selfish allele and that > > > the ultimate test of the group selection hypothesis will be whether > populations having these characteristics tend to show > "self-sacrificing" or "prudent" behavior more > commonly than those which do not. (Maynard Smith 1976: 282) > > > This means that the presence of group selection or the effectiveness of group selection is to be measured by the existence of nonadaptive behavior on the part of individual organisms along with the presence of a corresponding group-level adaptation. Therefore, this approach to kin and group selection does require a group-level adaptation from groups to count as units of selection. As with the previous view, it is significant that the engineering notion of adaptation is assumed rather than the weaker selection-product notion; > > > [A]n explanation in terms of group advantage should always be > explicit, and always calls for some justification in terms of the > frequency of group extinction. (Maynard Smith 1976: 278; cf. Wade > 1978; Wright 1980) > > > More recently, geneticists have attempted to make precise the notion of a group adaptation though the "Formal Darwinism Project" (Grafen 2008), in which the general concept of adaptation can be applied to groups (Gardner & Grafen 2009). However, it is unclear how the notion of adaptation developed within the formal Darwinism project relates to the previously discussed engineering notion of adaptation. Philosophers have offered a different analysis of group adaptation, one based on an earlier analysis of selection and adaptation (Okasha & Paternotte 2012; Grafen 2008). The key distinction, in the original view, is between a trait that is a trait that merely benefits the group, i.e., "fortuitous group benefit," and one that is a genuine group adaptation, a feature evolved because it benefited the group, i.e., "for the right reason" (Okasha & Paternotte 2012: 1137). Contextual analysis, as well as the Price equation, can provide a formal definition of group adaptation, but both need to be supplemented by causal reasoning (Okasha & Paternoster 2012). In contrast to the preceding approach, we can separate the interactor and manifestor-of-adaptation questions in our group selection models (Wright 1980; see Lewontin 1978; Gould & Lewontin 1979). This is done by distinguishing between what is called "intergroup selection," that is, interdemic selection in the shifting balance process, and "group selection for group advantage" (Wright 1980: 840; see Wright 1929, 1931). The term "altruist" originally denoted, in genetics, a phenotype "that contributes to group advantage at the expense of disadvantage to itself" (1980: 840; Haldane 1932). This earlier debate is connected to the main group selection debate in the 1960s, in which the group selectionists asserted the evolutionary importance of "group selection for group advantage" (Wright 1980). The argument is that the primary kin selection model is "very different" from "group selection for the uniform advantage of a group"(1980: 841; like Arnold & Fristrup 1982; Damuth & Heisler 1988; Heisler & Damuth 1987). There are excellent summaries of the empirical and theoretical discoveries enabled by "intergroup" selection models (Goodnight & Stevens 1997; Wade 2016). Those supporting a genic selection view in the 1970s were taken to task for mistakenly thinking that because they have successfully criticized group selection for group advantage, they can conclude that "natural selection is practically wholly genic"(Wright 1980: 841). > > > [N]one of them discussed group selection for organismic advantage to > individuals, the dynamic factor in the shifting balance process, > although this process, based on irreversible local peak-shifts is not > fragile at all, in contrast with the fairly obvious fragility of group > selection for group advantage, which they considered worthy of > extensive discussion before rejection. (Wright 1980: 841) > > > This is a fair criticism of the genic selectionist view. The problem is that these authors failed to distinguish between two questions: the interactor question and the manifestor-of-adaptation question. The form of group selection that involves interdemic group selection models involves groups only as interactors, not as manifestors of group-level adaptations. More recently, modelers following Sewall Wright's interest in structured populations have created a new set of genetical models that are also called "group selection" models and in which the questions of group adaptations and group benefit play little or no role.[10] For a period spanning two decades, however, the genic selectionists did not acknowledge that the position they attacked, namely group selection as engineering adaptation, is significantly different from other available approaches to group selection, such as those that primarily treat groups as interactors. Ultimately, however, genic selectionists did recognize the significance of the distinction between the interactor question and the manifestor-of-adaptation question. In 1985, for example, we have progress towards mutual understanding: > > > If some populations of species are doing better than others at > persistence and reproduction, and if such differences are caused in > part by genetic differences, this selection at the population level > must play a role in the evolution of the species.... [But] > selection at any level above the family (group selection in a broad > sense) is unimportant for the origin and maintenance of adaptation. > (Williams 1985: 7-8) > > > And in 1987, we have an extraordinary concession: > > > There has been some semantic confusion about the phrase "group > selection," for which I may be partly responsible. For me, the > debate about levels of selection was initiated by Wynne-Edwards' > book. He argued that there are group-level adaptations...which > inform individuals of the size of the population so that they can > adjust their breeding for the good of the population. He was clear > that such adaptations could evolve only if populations were > units of selection.... Perhaps unfortunately, he referred to the > process as "group selection." As a consequence, for me and > for many others who engaged in this debate, the phrase cane to imply > that groups were sufficiently isolated from one another reproductively > to act as units of evolution, and not merely that selection acted on > groups. > > > > The importance of this debate lay in the fact that group-adaptationist > thinking was at that time widespread among biologists. It was > therefore important to establish that there is no reason to expect > groups to evolve traits ensuring their own survival unless they are > sufficiently isolated for like to beget like.... When Wilson > (1975) introduced his trait-group model, I was for a long time > bewildered by his wish to treat it as a case of group selection and > doubly so by the fact that his original model...had interesting > results only when the members of the group were genetically related, a > process I had been calling kin selection for ten years. I think that > these semantic difficulties are now largely over. (Maynard Smith 1987: > 123). > > > Even the originator of the replicator/vehicle distinction also seems to have rediscovered the evolutionary efficacy of higher-level selection processes in an article on artificial life. In this article, the primary concern is with modeling the course of selection processes, and a species-level selection interpretation is offered for an aggregate species-level trait (Dawkins 1989a). Still, Dawkins seems not to have recognized the connection between this evolutionary dynamic and the controversies surrounding group selection because in his second edition of The Selfish Gene (Dawkins 1989b) he had yet to accept the distinction made so clearly by group selectionists in 1980 (Wright 1980). This was in spite of the fact that by 1987, the importance of distinguishing between evolution by selection processes and any engineering adaptations produced by these processes had been acknowledged by the workers he claimed to be following most closely (Williams 1985, 1992; Maynard Smith 1987). More recently, a related debate has fired up between genic selection and group selection in the journals, about the definitions of group and kin selection (E.O. Wilson 2008; see below). But this debate is bound for nowhere without tight enough definitions of these kinds of selection (Shavit & Millstein 2008). The adoption of Wade's strict definitions would help, following the prescriptions of early group selectionists (Shavit & Millstein 2008). There has been an even more recent challenge to the received understanding of kin selection, favoring a group selection interpretation, which has been rebutted by those defending a strict separation between kin and group selection (Nowak, Tarnita, & Wilson 2010; Holldobler & Wilson 2009; rebutted by Gardner, West. & Wild 2011; Abbot et al. 2011). This view, has in turn, been rebutted by others (van Veelen et al. 2012; Allen, Nowak, & Wilson 2013; E.O. Wilson & Nowak 2014). Philosophers have offered a very useful analysis of these debates about kin and group selection (Birch & Okasha 2015). The basic prediction of kin selection theory is that social behavior, especially social behavior that benefits others, should correlate with genetic relatedness. This is commonly expressed through Hamilton's rule, rb > c, where r is relatedness, b is the benefit that behavior offers the conspecific, and c is the cost to the actor (see Hamilton 1975 for an expansion to multilevel selection). The critics claiming that kin selection is a form of group selection, assert that Hamilton's rule "almost never holds" (Nowak et al. 2010: 1059). That is, it almost never states the true conditions under which a social behavior will evolve by selection. Their opponents claim the opposite: that it is incorrect to claim that Hamilton's rule requires restrictive assumptions, or that it almost never holds. On the contrary, they claim, it holds a great deal of the time (Gardner, West, & Wild 2011). On a philosophical analysis, there are really three distinct versions of Hamilton's rule, and thus three distinct versions of kin selection theory under discussion. One involves many substantial assumptions, including > > > weak selection, additive gene action, (i.e., no dominance or > epistasis), and the additivity of fitness payoffs (i.e., a relatively > simple payoff structure). (Birch & Okasha 2015: 23) > > > When these assumptions are weakened, we get more variants of Hamilton's rule. Particularly important are the payoff parameters c and b. Sometimes these denote the values of a particular model, called HRS (Hamilton's Rule, special), but other times, they denote averaging effects or partial regression coefficients, in the case of HRG (Hamilton's Rule, general). A third approach, HRA (Hamilton's Rule, approximate), which uses first-order approximates of regression coefficients, is the approach most commonly used in contemporary kin selection theory. The special version is very restrictive, while the general version allows a wide variety of cases. According to the philosophical analysis of the cases, Nowak et al. are using the special version of Hamilton's rule when they say it "almost never holds," whereas Gardner, West, and Wild are using the regression-based, general version of the rule, which allows a great deal of leeway in application. In other words, they are talking past each other (Birch & Okasha 2015). Significantly, > > > Neither [Nowak et al. nor Gardner, West, & Wild ] is referring to > HRA, even though this approximate version of the rule is the version > most commonly used by kin selection theorists. (Birch & Okasha > 2015: 24) > > > On the same philosophical analysis, it is also argued that kin selection and multilevel selection represented using the Price equation are formally equivalent, and that preferences for kin selection models may not be justified as they are usually (Birch & Okasha 2015; cf. West et al. 2008). Some biologists, for example, have argued that kin selection is more easily applicable than group selection, and that kin selection can be applied whenever there is group selection (West, Griffin, & Gardner 2007, 2008). Moreover, they deny that kin selection is a form of group selection, despite formal similarities and derivations (West, Griffin, & Gardner 2007: 424). Problems about using multilevel selection models may stem from the group beneficiary problem arising from the early group selection context wherein group selection was assumed to involve both group benefit and group engineering adaptation (Birch & Okasha 2015). > > > [A]lthough kin and multilevel selection are equivalent as statistical > decompositions of evolutionary change, there are situations in which > one approach provides a more accurate representation of the causal > structure of social interaction. (Birch & Okasha 2015: 30; see > also Okasha & Paternotte 2012 vs. Gardner & Grafen 2009 on > group adaptations) > > > Geneticists have offered an effective critique of the "Formal Darwinism Project" to units of selection and adaptation, arguing that the latter's preferences for the level of individual organism is arbitrary, as is their bias against multilevel selection (Shelton & Michod 2014a). In an analysis of the contextual analysis and Price equation methods of representing hierarchical selection models, it was argued that contextual analysis is superior overall, except in cases of meiotic drive (Okasha 2006). However, it was recently argued that contextual analysis can even handle cases of meiotic drive, thus making it the superior approach to multilevel selection (Earnshaw 2015). In a separate and helpful analysis, the relationship between kin and multilevel models was spelled out using causal graphs (Okasha 2015). Just because the two models can produce the same changes in gene frequencies, it does not follow that they represent the same causal structure, which is illustrated using causal graph theory and examples from biology, such as group adaptation and meiotic drive (Okasha 2015; see Genic Selection: The Pluralists). This goes very much against the claims of equivalence of the two model types, kin and group selection (West, Griffin, & Gardner 2007, 2008; West & Gardner 2013), in which these technical equivalences are taken to signify total equivalence of the evolutionary systems (see also Frank 2013; Queller 1992; Dugatkin & Reeve 1994; Sober & Wilson 1998). Take, for instance, the claim: "There is no theoretical or empirical example of group selection that cannot be explained with kin selection" (West, Griffin, & Gardner 2008: 375), It is good to keep in mind that there are dissenters from this claim of the equivalence between group and kin selection models (e.g., Lloyd, Lewontin, & Feldman 2008; van Veelen 2009; van Veelen et al. 2012; Holldobler & Wilson 2009; Traulsen 2010; Nowak, Tarnita, & Wilson 2010, 2011; E.O. Wilson 2012). Those who claim full equivalence discuss the evolution of cooperation and altruism, arguing that only kin selection allows an easy solution to these evolutionary problems (West, Griffin, & Gardner 2008). From the more robust group selectionist point of view, the so-called free rider problems with kin and group selection--such as those that are contemplated by evolutionists puzzling over the evolution of altruism--are pseudo-problems based on misconceptions (Wade 2016; see also Bowles & Gintis 2011; Planer 2015; Sterelny 2012). One approach notes that kin selection (either in its inclusive fitness form or the direct fitness approach) and multilevel selection "differ primarily in the types of questions being addressed" (Goodnight 2013: 1546). Whereas kin selection aims at identifying character states that maximize fitness, multilevel selection methods have the goal of looking at the effects of selection on trait changes. While the two have formal similarities, the kin selection models arose out of game theory and evolutionary stable strategies (ESS) and are used to identify the optimal solution, but they cannot be used to examine the process by which the population will achieve that optimum or equilibrium. In contrast, the multilevel selection methods, such as contextual analysis, which arise out of the quantitative genetics traditions, are used to describe the processes acting on the population in its current state.[11] Thus, the two methods are not the same, nor are they competing paradigms. > > > Rather they should be considered complementary approaches that when > used together give a clearer picture of social evolution than either > one can when used in isolation. (Goodnight 2013: 1547; cf. Maynard Smith > 1976; Dawkins 1982b; West, Griffin, & Garnder 2007, 2008; Gardner > & Grafen 2009) > > > In the laboratory, the hierarchical genetic approach of multilevel selection has been used to demonstrate that populations respond rapidly to experimentally imposed group selection, and that indirect genetic effects are primarily responsible for the surprising strength and effectiveness of group selection experiments, contra the full equivalence claims (Goodnight 2013; Goodnight & Stevens 1997; cf. West, Griffin, & Gardner 2007, 2008). Field studies using contextual analysis have shown that multilevel selection is far more common in nature than previously expected (Goodnight 2013; e.g., Stevens, Goodnight, & Kalisz 1995;Tsuji 1995; Aspi et al. 2003; Weinig et al. 2007; Eldakar et al. 2010; Wade 2016). There is much less emphasis on the evolution of altruism within the hierarchical genetic approach, as selection is observed as it is occurring, and this includes group selection going in the same direction as organismic selection, not just in opposition to it, contra early genic selectionist recommendations (Maynard Smith 1976). This sort of group selection of interactors is not based on group level engineering adaptations, although some still persist in confusing group selection itself with the combination of the two features, group selection and group engineering adaptation (e.g., Ramsey & Brandon 2011). Most recently, a new topic has arisen in the context of multilevel selection, involving the evolution of "holobionts," i.e., the combination of a eukaryotic organism with its microbiotic load (Zilber-Rosenberg & Rosenberg 2008). It has become clear that each "individual" human being is actually a community of organisms co-evolved for mutual benefit (Gilbert, Sapp, & Tauber 2012). Our microbiota (the collection of bacteria, viruses, and fungi living in our gut, mouth, and skin) is necessary for our survival and development, and our species is also needed in turn for their survival. Bacterial symbionts help induce and sustain the human immune system, T-cells, and B-cells (Lee & Mazamanian 2010; Round, O'Connell, & Mazamanian 2010), as well as providing essential vitamins to the host human being. Gut bacteria are necessary for cognitive development (Sampson & Mazamanian 2015), as well as for the development of blood vessels in the gut; lipid metabolism, detoxification of dangerous bacteria, viruses, and fungi; and the regulation of colonic pH and intestinal permeability (Nicholson et al. 2012). The holobiont--the combination of the host and its microbiota--functions as a unique biological entity anatomically, metabolically, immunologically, and developmentally (Gilbert, Rosenberg, & Zilber-Rosenberg forthcoming; Gilbert 2011). Similarly, a holobiont is seen as an "integrated community of species, [which] becomes a unit of natural selection" (Gilbert, Sapp, & Tauber 2012: 334). That is, in essence, theorists claim that the holobiont can function as an interactor since it has features that bind it together as a functional whole in such a way that it can interact in a natural selection process. So what ties the different species together to produce an interactor? According to pioneering philosophical thought on holobionts and symbionts, it is the community's common evolutionary fate, its being a "functioning whole," that characterizes it as an evolutionary interactor, "objects between which natural selection selects" (Dupre 2012: 160; see also Dupre & O'Malley 2013; Zilber-Rosenberg & Rosenberg 2008). This community can also be described as a "team" of consortia undergoing selection (Gilbert et al. forthcoming). Others describe them as "collaborators" or "polygenomic consortia", which has the advantage of encompassing both competition and cooperation within the holobiont (Dupre & O'Malley 2013: 314; Lloyd forthcoming; see also Huttegger & Smead 2011 on stag hunt game-theoretic results regarding the range of collaboration). Holobionts can also be reproducers, where the host usually reproduces vertically and the microbiota reproduce either vertically, horizontally, or both. This situation has provoked discussion among philosophers (Godfrey-Smith 2009, 2011; Sterelny 2011; Griesemer 2014, 2016; Booth 2014; Lloyd forthcoming). Holobionts' microbiota can reproduce outside the context of the original host organism, so some holobionts, e.g., the Hawaiian bobtail squid and its luminescent bacteria, are not "Darwinian populations" (Godfrey-Smith 2009, 2011), and therefore not units of selection (see Booth 2014). This approach contrasts with that of the original reproducer approach which would include the squid-bacteria system and also retroviruses excluded under the "Darwinian populations" account (Griesemer2000a, 2016). > > > As in [my book, Darwinian Populations and Natural Selection], > I hold that it is a mistake to see things that do not reproduce as > units of selection. (Godfrey-Smith 2011: 509; Booth 2014) > > > This exclusion rests on the merging of the interactor with the reproducer requirements, and as such will not hold sway over those who do not buy such a confounding of roles (e.g., Dupre & O'Malley 2013). This is yet another case wherein distinguishing the interactor question from the replicator/reproducer question can be "more illuminating" (Sterelny 2011: 496; see Dupre & O'Malley 2013; Gilbert, Sapp, & Tauber 2012; Lloyd forthcoming). Finally, holobionts can also be manifestors of adaptations, as in the case of the evolution of placental mammals in the acquisition by horizontal gene transfer from a retrovirus of a crucial gene coding for the protein syncytin (Lloyd forthcoming; Dupressoir, Lavialle, & Heidemann 2012). Syncytin allows fetuses to fuse to their mother's placenta, a role crucial to the evolution of placental mammals. Moreover, it seems that several retrovirally derived enhancers played critical roles in the formation of a key cell in the uterine wall, also crucial for maintaining pregnancy, enhancing the holobiont's role as a manifestor of adaptation (Wagner et al. 2014). There are several other significant entries into the group selection discussions, including the book, Unto Others: The Evolution and Psychology of Unselfish Behavior (Sober & Wilson 1998). Here, a case for group selection is developed based on the need to account for the existence of biological altruism. Biological altruism is any behaviour that benefits another organism at some cost to the actor. Such behavior must always reduce the actor's fitness but it may (following the work on interdemic selection), increase the fitness of certain groups within a structured population. There was a big benefit to bringing the attention of the larger philosophical community to group selection models, and explaining them in an accessible fashion. It has thus brought this aspect of the units of selection controversy out onto the main stage of philosophical thought. The "Darwinian populations" view previously mentioned provides a considerably different view of the necessary conditions for group selection, one which rejects many of the currently accepted cases of the phenomenon. For a given selection story to be descriptively valid, a "Darwinian population" must exist at the level of selection being described, which requires the presence of both an interactor and a reproducer at that level, thus putting together what others have pulled apart (Godfrey-Smith 2009: 112). A Darwinian population is conceived as, at minimum, > > > a collection of causally connected individual things in which there is > variation in character, which leads to differences in reproductive > output (differences in how much or how quickly individuals reproduce), > and which is inherited to some extent. (2009: 39) > > > There are further differentiations between paradigmatic, minimal, and marginal Darwinian populations based on a variety of criteria, such as the fidelity of heritability, continuity (the degree to which small shifts in phenotype correlate to small changes in fitness), and the dependence of reproductive differences on intrinsic features of individuals (Godfrey-Smith 2009: Chapter 3). For example, under this view, the case of the evolution of altruism, which is commonly attributed to group selection, should not be considered as such, because of the lack of a true reproducer at the group level; the group level description depicts at best a marginal Darwinian population (Godfrey-Smith 2009: 119). Rather, the argument is that a neighborhood selection model, in which individuals are affected by the phenotypes of their neighbors but cannot be seen as "collectives competing at a higher level," is fully capable of capturing the selective process involved, and represents a Darwinian population, in which the individuals are seen both as the interactors and the reproducers (2009: 118). This would seem to entail that many group selection accounts (e.g., Sober & Wilson 1998), as well as any models classified as Multi-level selection 1 (MLS1) models (Heisler & Damuth 1987), cannot be properly considered as such. This view grants that there are empirical examples in which a group-level reproducer clearly exists (for example, Wade & Griesemer 1998; Griesemer & Wade 2000; there is otherwise no discussion of Wrightian approaches to group selection). The approach using Darwinian populations and reproducers is claimed to present an advantage over other available analyses of units of selection because it can account for previously neglected examples such as epigenetic inheritance systems (Godfrey-Smith 2009). The question remains as to whether gaining an account to deal with these is worth rejecting an entire class of accepted group selection models, and whether such a loss is truly necessary to deal with epigenesis, given that we have an epigenetic account with reproducers that allows for group selection (see Griesemer 2000c). ### 3.2 Species Selection Ambiguities about the definition of a unit of selection have also snarled the debate about selection processes at the species level. One response to the notion of species selection comes with a classic confusion: "It is individual selection discriminating against the individuals of the losing species that causes the extinction" (Mayr 1997: 2093). The individual death of species members is confused with extinction: "the actual selection takes place at the level of competing individuals of the two species" (Mayr 1997: 2093). Once we overcome such difficulties, and succeed in conceiving of species as unified interactors, we are still faced with two questions. The combining of the interactor question and the manifestor-of-adaptation question (in the engineering sense) led to the rejection of research aimed at considering the role of species as interactors, simpliciter, in evolution. Once it is understood that species-level interactors may or may not possess design-type adaptations, it becomes possible to distinguish two research questions: Do species function as interactors, playing an active and significant role in evolution by selection? And does the evolution of species-level interactors produce species-level engineering adaptations and, if so, how often? For the early history of the species selection debate, these questions were lumped together; asking whether species could be units of selection meant asking whether they fulfilled both the interactor and manifestor-of-adaptation roles. For example, early species selection advocates used a genic selectionist treatment of the evolution of altruism as a touchstone in the definition of species selection (e.g., Vrba 1984). The relevant argument is that kin selection could cause the spread of altruistic genes but that it should not be called group selection (Maynard Smith 1976). Again, this was because the groups were not considered to possess design-type adaptations themselves. Some species selectionists agreed that the spread of altruism should not be considered a case of group selection because "there is no group adaptation involved; altruism is not emergent at the group level" (Vrba 1984: 319; Maynard Smith gives different reasons for his rejection). This amounts to assuming that there must be group benefit in the sense of a design-type group-level adaptation in order to say that group selection can occur. This species selection view was that evolution by selection is not happening at a given level unless there is a benefit or engineering adaptation at that level. The early species selection position explicitly equates units of selection with the existence of an interactor plus adaptation at that level (Vrba 1983: 388); furthermore, it seems that the stronger, engineering definition of adaptation had been adopted. It was generally accepted among early species selectionists that species selection does not happen unless there are species-level adaptations (Eldredge 1985: 196, 134). Certain cases are rejected as higher-level selection processes overall because > > > frequencies of the properties of lower-level individuals which are > part of a high-level individual simply do not make convincing > higher-level adaptations. (Eldredge 1985: 133) > > > Most of those defending species selection early on defined a unit of selection as requiring an emergent, adaptive property (Vrba 1983, 1984; Vrba and Eldredge 1984; Vrba and Gould 1986). This amounts to asking a combination of the interactor and manifestor-of-adaptation questions. But the relevant question is not "whether some particle-level causal processes or other bear the causal responsibility," but rather "whether particle-level selection bears the causal responsibility" (Okasha 2006: 107). An emergent character requirement conflates these two questions. Such a character may be the result of a selection process at the group/species level, but it should not be treated as a pre-condition of such a process. But consider the lineage-wide trait of variability. Treating species as interactors has a long tradition (Dobzhansky 1956, Thoday 1953, Lewontin 1958). If species are conceived as interactors (and not necessarily manifestors of adaptations), then the notion of species selection is not vulnerable to the original anti group-selection objections from the early genic selectionists (Williams 1966).[12] The old idea was that lineages with certain properties of being able to respond to environmental stresses would be selected for, and thus that the trait of variability itself would be selected for and would spread in the population of populations. In other words, lineages were treated as interactors. The earlier researchers spoke loosely of adaptations where adaptations were treated in the weak sense as equivalent simply to the outcome of selection processes (at any level). They were explicitly not concerned with the effect of species selection on organismic level traits but with the effect on species-level characters such as speciation rates, lineage-level survival, and extinction rates of species. Some argued, including the present author, that this sort of case represents a perfectly good form of species selection, using so-called "emergent fitnesses," even though some balk at the thought that variability would then be considered, under a weak definition, a species-level adaptation (Lloyd & Gould 1993; Lloyd 1988 [1994]). Paleontologists used this approach to species selection in their research on fossil gastropods (Jablonski 2008, 1987; Jablonski & Hunt 2006), and the approach has also been used in the leading text on speciation (Coyne & Orr 2004). Early species selectionists also eventually recognized the advantages of keeping the interactor question separate from a requirement for an engineering-type adaptation, dropping the former requirement that, in order for species to be units of selection, they must possess species-level adaptations (Vrba 1989). Ultimately, the current widely-accepted definition of species selection is in conformity with a simple interactor interpretation of a unit of selection (Vrba 1989; see Damuth & Heisler 1988; Lloyd 1988 [1994]; Jablonski 2008). It is easy to see how the two-pronged definition of a unit of selection--as interactor and manifestor of adaptation--held sway for so long in the species selection debates. After all, it had dominated much of the group selection debates for so long. Some of the confusion and conflict over higher-level units of selection arose because of an historical contingency--the early group selectionist's implicit definition of a unit of selection and the responses it provoked (Wynne Edwards 1962; Borrello 2010). ### 3.3 Genic Selection: The Originators One may understandably think that the early genic selectionists were interested in the replicator question because of the claims that the unit of selection ought to be the replicator. This would be a mistake. Rather, the primary interest is in a specific ontological issue about benefit (Dawkins 1976, 1982a,b). This amounts to asking a special version of the beneficiary question, and the answer to that question dictates the answers to the other three questions flying under the rubric of the "units of selection". Briefly, the argument is that because replicators are the only entities that "survive" the evolutionary process, they must be the beneficiaries (Dawkins 1982a,b). What happens in the process of evolution by natural selection happens for their sake, for their benefit. Hence, interactors interact for the replicators' benefit, and adaptations belong to the replicators. Replicators are the only entities with real agency as initiators of causal chains that lead to the phenotypes; hence, they accrue the credit and are the real units of selection. This version of the units of selection question amounts to a combination of the beneficiary question plus the manifestor-of-adaptation question. There is little evidence that they are answering the predominant interactor question; rather, the argument is that people who focus on interactors are laboring under a misunderstanding of evolutionary theory (Dawkins 1976, 1982a,b). One reason for thinking this might be that the opponents are taken to be those who hold a combination of the interactor plus manifestor-of-adaptations definition of a unit of selection (e.g., Wynne-Edwards). Unfortunately, leading genic selectionists ignore those who are pursuing the interactor question alone; these researchers are not vulnerable to the criticisms posed against the combined interactor-adaptation view (Dawkins 1982a,b; Williams 1966). Some insist that the early genic selectionists have misunderstood evolutionary selection, an argument that is based upon interpreting the units of selection controversy as a debate about interactors (Gould 1977; Istvan 2013); however, because the early genic selectionists say that the debate concerns the units of the ultimate beneficiary, they are arguing past one another (Istvan 2013). Section 3.4, Genic Selection: The Pluralists, addresses those who interpret themselves as arguing against the interactor question itself. In the next few paragraphs, two aspects of Dawkins' specific version of the units of selection problem shall be characterized. I will attempt to clarify the key issues of interest to Dawkins and to relate these to the issues of interest to others. There are two mistakes that Dawkins is not making. First, he does not deny that interactors are involved in the evolutionary process. He emphasizes that it is not necessary, under his view, to believe that replicators are directly "visible" to selection forces (1982b: 176). Dawkins has recognized from the beginning that his question is completely distinct from the interactor question. He remarks, in fact, that the debate about group versus organismic selection is "a factual dispute about the level at which selection is most effective in nature," whereas his own point is "about what we ought to mean when we talk about a unit of selection" (1982a: 46). He also states that genes or other replicators do not "literally face the cutting edge of natural selection. It is their phenotypic effects that are the proximal subjects of selection" (1982a: 47). We shall return to this issue in section 3.4, Genic Selection: The Pluralists. Second, Dawkins does not specify how large a chunk of the genome he will allow as a replicator; there is no commitment to the notion that single exons are the only possible replicators. He argues that if Lewontin, Franklin, Slatkin and others are right, his view will not be affected (see Replicators). If linkage disequilibrium is very strong, then the "effective replicator will be a very large chunk of DNA" (Dawkins 1982b: 89; Sapienza 2010). We can conclude from this that Dawkins is not interested in the replicator question at all; his claim here is that his framework can accommodate any of its possible answers. On what basis, then, does Dawkins reject the question about interactors? I think the answer lies in the particular question in which he is most interested, namely, What is "the nature of the entity for whose benefit adaptations may be said to exist?"[13] On the face of it, it is certainly conceivable that one might identify the beneficiary of the adaptations as--in some cases, anyway--the individual organism or group that exhibits the phenotypic trait taken to be the adaptation. In fact, some writers seem to have done just that in the discussion of group selection (see Williams 1966).[14] But the original genic selectionist rejects this move, introducing an additional qualification to be fulfilled by a unit of selection; it must be "the unit that actually survives or fails to survive" (Dawkins 1982a: 60). Because organisms, groups, and even genomes do not actually survive the evolution-by-selection process, the answer to the survival question must be the replicator. (Strictly speaking, this is false; it is copies of the replicators that survive. Replicators must therefore be taken in some sense as information and not as biological entities (see Hampe & Morgan 1988; cf. Griesemer 2005). But there is still a problem. Although the conclusion is that "there should be no controversy over replicators versus vehicles. Replicator survival and vehicle selection are two aspects of the same process" (1982a: 60), the genic selectionist does not just leave the vehicle selection debate alone. Instead, the argument is that we do not need the concept of discrete vehicles at all. This is what we shall investigate in section 3.4 Genic Selection: The Pluralists. The important point for now is that, on Dawkins' analysis, the fact that replicators are the only survivors of the evolution-by-selection process automatically answers also the question of who owns the adaptations. Adaptations must be seen as being designed for the good of the active-germ-line replicator for the simple reason that replicators are the only entities around long enough to enjoy them over the course of natural selection. The genic selectionist acknowledges that the phenotype is "the all important instrument of replicator preservation," and that genes' phenotypic effects are organized into organisms (that thereby might benefit from them in their lifetimes) (1982b: 114). But because only the active germ-line replicators survive, they are the true locus of adaptations (1982b: 113; emphasis added). The other things that benefit over the short term (e.g., organisms with adaptive traits) are merely the tools of the real survivors, the real owners. Hence, Dawkins rejects the vehicle approach partly because he identifies it with the manifestor-of-adaptation approach, which he has answered by definition, in terms of the long-term beneficiary. The second key aspect of genic selectionists' views on interactors is the desire to do away with them entirely. Dawkins is aware that the vehicle concept is "fundamental to the predominant orthodox approach to natural selection" (1982b: 116). Nevertheless, he rejects this approach in The Extended Phenotype, claiming, "the main purpose of this book is to draw attention to the weaknesses of the whole vehicle concept" (1982b: 115). But this "vehicle" approach is not equivalent to "the interactor question"; it encompasses a much more restricted approach. In particular, when arguing against "the vehicle concept," Dawkins is only arguing against the desirability of seeing the individual organism as the one and only possible vehicle. His target is explicitly those who hold what he calls the "Central Theorem," which says that individual organisms should be seen as maximizing their own inclusive fitness (1982b: 5, 55). These arguments are indeed damaging to the Central theorem, but they are ineffective against other approaches that define units of selection as interactors. One way to interpret the Central Theorem is that it implies that the individual organism is always the beneficiary of any selection process. The genic selectionists seem to mean by "beneficiary" both the manifestor of adaptation and that which survives to reap the rewards of the evolutionary process. Dawkins argues, rightly and persuasively, I think, that it does not make sense always to consider the individual organism to be the beneficiary of a selection process. But it is crucial to see that Dawkins is not arguing against the importance of the interactor question in general, but rather against a particular definition of a unit of selection. The view being criticized assumes that the individual organism is the interactor, and the beneficiary, and the manifestor of adaptation. Consider the main argument against the utility of considering vehicles: the primary reason to abandon thinking about vehicles is that it confuses people (1982b: 189). But look at the examples; their point is that it is inappropriate always to ask how an organism's behavior benefits that organism's inclusive fitness. We should ask instead, "whose inclusive fitness the behavior is benefiting" (1982b: 80). Dawkins states that his purpose in the book is to show that "theoretical dangers attend the assumption that adaptations are for the good of...the individual organism" (1982b: 91). So, Dawkins is quite clear about what he means by the "vehicle selection approach"; it always assumes that the organism is the beneficiary of its accrued inclusive fitness. Dawkins advances powerful arguments against the assumption that the organism is always the interactor cum beneficiary cum manifestor of adaptations. This approach is clearly not equivalent to the approach to units of selection characterized as the interactor approach. Unfortunately, genic selectionists extend Dawkins' conclusions to these other approaches, which he has, in fact, not addressed. The genic selectionists' lack of consideration of the interactor definition of a unit of selection leads to two grave problems with this view. One problem is the tendency to interpret all group selectionist claims as being about beneficiaries and manifestors of adaptations as well as interactors. This is a serious misreading of authors who are pursuing the interactor question alone. Consider, for example, this argument that groups should not be considered units of selection: > > > To the extent that active germ-line replicators benefit from the > survival of the group of individuals in which they sit, over and above > the [effects of individual traits and altruism], we may expect to see > adaptations for the preservation of the group. But all these > adaptations will exist, fundamentally, through differential replicator > survival. The basic beneficiary of any adaptation is the active > germ-line replicator (Dawkins 1982b: 85). > > > Notice that this argument begins by admitting that groups can function as interactors, and even that group selection may effectively produce group-level adaptations. The argument that groups should not be considered real units of selection amounts to the claim that the groups are not the ultimate beneficiaries. To counteract the intuition that the groups do, of course, benefit, in some sense, from the adaptations, the terms "fundamentally" and "basic" are used, thus signaling what the author considers the most important level. Even if a group-level trait is affecting a change in gene frequencies, "it is still genes that are regarded as the replicators which actually survive (or fail to survive) as a consequence of the (vehicle) selection process" (Dawkins 1982b: 115). Thus, the replicator is the unit of selection because it is the beneficiary, and the real owner, of all adaptations that exist. Saying all this does not, however, address the fact that other researchers investigating group selection are asking the interactor question and sometimes also the manifestor-of-adaptation question, rather than Dawkins' special version of the (ultimate) beneficiary question. He gives no additional reason to reject these other questions as legitimate; he simply reasserts the superiority of his own preferred unit of selection. In sum, Dawkins has identified three criteria as necessary for something to be a unit of selection: it must be a replicator; it must be the most basic beneficiary of the selection process; and it is automatically the ultimate manifestor of adaptation through being the beneficiary. Finally, further work in the philosophy of biology brings the level of the unit of selection down even further than the original genic selectionists do (Rosenberg 2006). Higher level selection is reducible to more fundamental levels.[15] Taking a reductionist stance, which is taken to be necessary to avoid an "untenable dualism" in biology between physicalism and antireductionism, the argument is that the principle of natural selection (PNS) should be properly viewed as a basic law of physical science (specifically chemistry), which can operate at the level of atoms and molecules (Rosenberg 2006: 189-191). Different molecular environments would favor different chemical types, and those that "more closely approximate an environmentally optimal combination of stability and replication," are thus the "fittest" and would predominate (2006: 190). This could then be applied at each step of the way from simple molecules to compounds, organelles, cells, tissues, and so on, such that > > > the result at each level of chemical aggregation is the instantiation > of another PNS, grounded in, or at least in principle derivable from, > the molecular interactions that follow the PNS in the environment > operating at one or more lower levels of aggregation. (Rosenberg 2006: > 192) > > > This approach addresses antireductionist arguments regarding group level properties. The claim is that this new envisioning of the PNS as a purely physical law allows us to better understand the lower level origins of apparently higher level causes, thus revealing that "the appearance of 'downward causation' is just that: mere appearance." (Rosenberg 2006: 197) For example, the claim is that group level selection explanations, such as are commonly given for altruism, do not require an antireductionist stance, since physical laws, such as that second law of thermodynamics, can allow for local unfavorable changes (in this case, local decreases in entropy) as long as compensation is made elsewhere. With regard to the physical PNS, > > > groups of biological individuals may experience fitness increases at > the expense of fitness decreases among their individual members for > periods of time that will depend on the size and composition of the > group and the fitness effects of their traits. What the PNS will not > permit is long-term fitness changes at the level of groups without > long-term fitness changes in the same direction among some or all of > the individuals composing them. (Rosenberg 2006: 198) > > > In other words, this is supposed to show that there is no need to think in terms of irreducible group level interactors. Again, note that this analysis merges characteristics of interactors and replicators. In the next section, we will consider some relatively more recent work in which genic selectionism is defended through a pluralist approach to modeling. What matters in the final analysis, though, is exactly what matters to the original genic selectionists, and that is the search for the ultimate beneficiary of the evolution by selection process. ### 3.4 Genic Selection: The Pluralists As we saw in the previous section, the original genic selectionists had particular problems with their treatment of the interactor. While they admitted that the "vehicle" was necessary for the selection process, they did not want to accord it any weight in the units of selection debate because it was not the beneficiary, but rather an agent of the beneficiary. Soon, however, there emerged a new angle available to genic selectionists (Waters 1986).[16] The new "genic pluralism" appears to let one bypass the interactor question, by, in effect, turning genes into interactors (Sterelny & Kitcher 1988). The proposal is that there are two "images" of natural selection, one in which selection accounts are given in terms of a hierarchy of entities and their traits' environments, the other of which is given in terms of genes having properties that affect their abilities to leave copies of themselves (Sterelny & Kitcher 1988; see Kitcher, Sterelny, & Waters 1990, Sterelny 1996a,b; Waters 1986, 1991).[17] Something significant follows from the fact that hierarchical models or selection processes can be reformulated in terms of the genic level. These claims have been resisted on a variety of grounds (see objections in R.A. Wilson 2003, Stanford 2001, Van der Steen & van den Berg 1999, Gannett 1999, Shanahan 1997, Glennan 2002, Sober 1990, Sober & Wilson 1998, Brandon & Nijhout 2006, Sarkar 2008). The big payoff of the genic point of view is: > > > Once the possibility of many, equally adequate, representations of > evolutionary processes has been recognized, philosophers and > biologists can turn their attention to more serious projects than that > of quibbling about the real unit of selection. (Kitcher, Sterelny, > & Waters 1990: 161) > > > By "quibbling about the real unit of selection," here, the authors seem to be referring to the large range of articles in which evolutionists have tried to give concrete evidence and requirements for something to serve as an interactor in a selection process. As an aside, it is important to note that none of the philosophers are advocating genic selectionism to the exclusion of other views. What interests them is a proposed equivalence between being able to tell the selection story one way, in terms of interactors and replicators, and to tell the same story another way, purely in terms of "genic agency". Thus, they are pluralists, in that they are not ultimately arguing in favor of the genic view; they are, however, expanding the genic selectionist view beyond its previous limits. The pluralists attack the view that "for any selection process, there is a uniquely correct identification of the operative selective forces and the level at which each impinges" (Waters 1991: 553). Rather, they claim, "We believe that asking about the real unit of selection is an exercise in muddled metaphysics" (Kitcher, Sterelny, & Waters 1990: 159). The basic view is that "the causes of one and the same selection process can be correctly described at different levels" (including the genic one) (Waters 1991: 555). Moreover, these descriptions are on equal ontological footing. Equal, that is, except for when Sterelny and Kitcher slip over into a genuinely reductionist genic view, when they state that it is an error to claim > > > that selection processes must be described in a particular way, and > their error involves them in positing entities, "targets of > selection," that do not exist. (1988: 359) > > > Here they seem to be denying the existence of interactors altogether. If interactors don't exist, then clearly a genic level account of the phenomena would be preferable to, not merely equivalent to, a hierarchical view. The pluralists do seem to be arguing against the utility of the notion of the interactor in studying the selection process. Echoing the original genic selectionists, their idea is that the whole causal story can be told at the level of genes, and that no higher level entities need be proposed or considered in order to have an accurate and complete explanation of the selection process. But, arguably, the genic level story cannot be told without taking the functional role of interactors into account, and thus the pluralists cannot avoid quibbling about interactors, as they claim (see Lloyd 2005). Nor is the genic account adequate to all selection cases; the genic account fails when drift is factored in (Brandon & Nijhout 2006). Let us recall what the interactor question in the units of selection debate amounts to: What levels of entities interact with their environments through their traits in such a way that it makes a difference to replicator success? As mentioned before, there has been much discussion in the literature about how to delineate and locate interactors among multilayered processes of selection. Each of these suggestions leads to slightly different results and different problems and limitations, but each also takes the notion of the interactor seriously as a necessary component to understanding a selection process. The genic pluralists state that "All selective episodes (or, perhaps, almost all) can be interpreted in terms of genic selection. That is an important fact about natural selection" (Kitcher, Sterelny, & Waters 1990: 160). Thus, the functional claim of the pluralists is that anything that a hierarchical selection model can do, a genic selection model can do just as well. Much attention is paid to showing that the two types of models can represent certain patterns of selection equally well, even those that are conventionally considered hierarchical selection. This is argued for using both specific examples and schema for translating hierarchical models into genic ones. Let us consider one challenging case here. Take the classic account of the efficacy of interdemic or group selection, the case that even G.C. Williams acknowledged was hierarchical selection. Lewontin and Dunn (Lewontin & Dunn 1960 and Lewontin 1962), in investigating the house mouse, found first, that there was segregation distortion, in that over 80% of the sperm from mice heterozygous for the t-allele also carried the t-allele, whereas the expected rate would be 50%. Second, they also found that male homozygotes (those with two t-alleles) tended to be sterile. (In several populations the t-alleles were homozygous lethal, but in the populations in question, homozygous males were sterile.) Third, they also found a substantial effect of group extinction based on the fact that female mice would often find themselves in groups in which all males were sterile, and the group itself would therefore go extinct. This, then, is how a genuine and empirically robust hierarchical model was developed. What the pluralists want to note about this case is very narrow, that is > > > whether there are real examples of processes that can be modeled as > group selection can be asked and answered entirely within the > genic point of view. (Kitcher, Sterelny, & Waters 1990: 160) > > > > Just as a warning to the unwary, the key to understanding the genic reinterpretation of this case is to grasp that the pluralists use a concept of genetic environment that their critics ignore. The pluralists tell how to "construct" a genic model of the causes responsible for the frequency of the t-allele. We must first distinguish > > > genetic environments that are contained within female mice that are > trapped in small populations with only sterile males from genetic > environments that are not contained within such females. In effect, > the interactions at the group level would be built in as a part of one > kind of genetic environment. (Waters 1991: 563) > > > In other words, various very detailed environments would have to be specified for various different t-alleles and wild-type alleles. In order to determine the invariant fitness parameter of a specific allele, let's call it "A" for example, we would need to know what kind of environment it is in at the allelic level, e.g., whether it is paired with a t-allele. Then we would need to know a further detailed layer of the environment of "A", such as what the sex is of the "environment" it is in. If it is in a t-allele arrangement, and it is also in a male environment, the allelic fitness of "A" would be changed. Finally, we need to know the type of subpopulation or deme the "A" allele is in. Is it in a small deme with many t-alleles? Then it is more likely to become extinct. So, as we can see, various aspects of the allele's environment are built up from the gene out, depending on what would make a difference to the gene's fitness in that very particular kind of environment. If you want to know the overall fitness of the "A" allele, you add up the fitnesses in each set of specialized, detailed environments and weight them according to the frequency of that environment. The idea is: > > > What appears as a multiple level selection process (e.g., selection of > the t-allele) to those who draw the conceptual divide [between > environments] at the traditional level, appears to genic selectionists > of Williams's style as several selection processes being carried > out at the same level within different genetic environments. (Waters > 1991: 571) > > > The "same level" here means the "genic level," while the genetic environments include everything from the other allele at the locus, to whether the genotype is present in a male or female mouse, to the size and composition of the deme the mouse is in. This completes the sketch of the genic pluralist position. We now turn to its reception. Genic pluralism's impact has been largely philosophical rather than biological (but see Shanahan 1997 and Van der Steen & Van den Berg 1999). Within philosophy, the view has been widely disseminated and taught, and a steady stream of critical responses to the genic pluralist position has been forthcoming. These responses fall into two main categories: pragmatic and causal. The pragmatic response to genic pluralism simply notes that in any given selective scenario the genic perspective provides no information that is not also available from the hierarchical point of view. This state of affairs is taken by critics of this type as sufficient reason to prefer whichever perspective is most useful for solving the problems facing a particular researcher (Glymour 1999; Van der Steen & Van den Berg 1999; and Shanahan 1997). The weakness of this approach as a critique of genic pluralism is that it does not so much criticize genic pluralism as simply ignore it. The other major form of critique of genic pluralism is based on arguments concerning the causal structure of selective episodes. The idea here is that while genic pluralism gets the "genetic book-keeping" (i.e., the input/output relations) correct, it does not accurately reflect the causal processes that bring about the result in question (Wimsatt 1980a,b). Some examples of this approach used against the genic pluralists (including Sober 1990; Sober & Wilson 1994, 1998) also appeal to aspects of the manifestor-of-adaptations and beneficiary questions to establish the failure of genic pluralism to represent certain selective events correctly. Causal concerns are also raised in some other work (Shanahan 1997, Van der Steen and Van den Berg 1999, Stanford 2001, and Glennan 2002), though without the focus on other units questions. The weakness of this line of criticism is its inability to isolate a notion of cause that is both plausible and plausibly true of hierarchical but not genic level models. This feature--that the genic and hierarchical models are so similar as to be indistinguishable--which appears as an insurmountable problem in the context of debates about differing causal structure, turns out to be the locus of critical response to genic pluralism, which denies that the genic selectionists have any distinct and coherent genic level causes at all (Lloyd 2005; Lloyd et al. 2005). Genic pluralism presents alleles as independent causal entities, with the claim that the availability of such models makes hierarchical selection models--and the ensuing debates about how to identify interactors in selection processes--moot. Or, in a less contentious version of the argument, the hierarchical and genic models are fully developed causal alternatives (Waters 1991). However, in each case of the causal allelic models, these models are directly and completely derived from precisely the hierarchical models the authors reject. Moreover, causal claims made on behalf of alleles are utterly dependent on hierarchically identified and established interactors as causes, thus undermining their claims that the units of selection (interactor) debates are mere "quibbles" and are irrelevant to the representation of selection processes. Moreover, and contrary to the claims of pluralists, cases of frequency-dependence, such as in heterosis and in game-theoretic models of selection, necessitate selection at higher than genic levels because the relevant properties of the entities at the genic level are only definable relative to higher levels of organization. Thus, they cannot be properly described as properties of alleles nor are they "even definable at the allelic level." (Sarkar 2008: 219) In addition, when drift is taken into account, the genic accounts fail to be empirically adequate (Brandon & Nijhout 2006). We can say that the allelic level models are completely derivative from higher level models of selection processes using the following guidelines (Lloyd 2005). Two models that are mathematically equivalent may be semantically different, that is, they have different interpretations. Such models can be independent from one another or one may be derivative of the other. In the genic selection case, the pluralists appear to be claiming that the genic level models are independent from the hierarchical models. The claim is: although the genic models are mathematically equivalent, they have different parameters, and a different interpretation, and they are completely independent from hierarchical models. But, despite the pluralists' repeated claims, we can see from their own calculations and examples that theirs are derivative models, and thus, that their "genic" level causes are derivative from and dependent on higher level causes. Their genic level models depend for their empirical, causal, and explanatory adequacy on entire mathematical structures taken from the hierarchical models and refashioned. As reviewed above, one example from their own writing comes from the treatment of the t-allele case, a universally recognized case of three levels of selection operating simultaneously on a single allele. Right before the t-allele case, a suggestion is offered that a Williams's type analysis could be based on an application of Lloyd's additivity criterion for identifying interactors, which is strictly hierarchical (Waters 1991: 563; Lloyd [1988] 1994: Ch. 5). Thus, the pluralist suggestion is to borrow a method for identifying potential higher-level interactors in order to determine the genic environments and thus to have more adequate genic level models. Similarly, other pluralists resort to a traditional approach to identifying interactors in order to make their genic models work. It had earlier been proposed that the statistical idea of screening off be used to identify which levels of entities are causally effective in the selection process; i.e., it is a method used to isolate interactors (Brandon 1982). But some pluralists propose using screening off to identify layers of allelic environments, and also show how Sober's probabilistic causal account could be used for a genic account (Sterelny & Kitcher 1988: 354). Hence, the pluralists all use the same methods for isolating relevant genic-level environments as others do for the traditional isolating of interactors. What, we may ask, is the real difference? Both can be seen as attempting to get the causal influences on selection right, because they are using the same methods. What is different is that the genic selectionists want to tell the causal story in terms of genes and not in terms of interactors and genes. Moreover, they propose doing away with interactors altogether, by renaming them the genic-level environments. Are we to think that renaming changes the metaphysics of the situation? It seems that levels of interaction important to the outcome of the selection process are being discovered in the usual ways, i.e., by using approaches to interactors and their environments, and that that exact same information is being translated into talk of the differentiated and layered environments of the genes. The issue concerning renaming model structures is especially confusing in the genic pluralists presentations, because they repeatedly rely on an assumption or intuition that, given an allelic state space, we are dealing with allelic causes. This last assumption is easily traced back to the original genic selectionist views that alleles are the ultimate beneficiaries of any long term selection process (Williams 1966; Dawkins 1982a,b); thus, the genic pluralist argument rests substantially on a view regarding the superior importance of the beneficiary question, which has been clearly delineated from the interactor question, above. Let us summarize the consequences of derivativeness in terms of the science and metaphysics of the processes discussed. First, the genic pluralists end up offering not, as they claim, a variety of genuinely diverse causal versions of the selection process at different levels. This is because the causes of the hierarchical models, however determined, are simply transformed and renamed in the lower level models, but remain fully intact as relevant causes at the full range of higher and lower levels. More importantly, no new allelic causes are introduced. Second, while genic models may be derived from hierarchical models, they fail to sustain the necessary supporting methodology. Third, the lack of genuine alternative causal accounts destroys the claims of pluralism or, at least, of any interesting philosophical variety, since there are no genuine alternatives being presented, unless you count renaming model structures as metaphysically significant (see also Okasha 2011).[18] Thus, the picture of proposing an alternative "interactor" at the genic level is not fulfilled (vs. Sterelny & Kitcher 1988). Perhaps the best way to save the pluralist vision is to appeal to the work on neighbor selection (Godfrey-Smith 2008), which can be cast within a pluralist program. This effort is to revive and discuss an alternative fashion of modeling altruism or group benefit, within the terms of a lower, individual level (see the discussion in section 3.1, Genic Selection: The Originators). There is a further complication with respect to the nature of the genic selection models put forward by genic pluralists. These models function under the presupposition that they are at least mathematically equivalent to hierarchical models. This claim has largely depended on the work of Dugatkin and Reeve in establishing this equivalence (Dugatkin & Reeve 1994, Sterelny 1996b, Sober & Wilson 1998, Sterelny & Griffiths 1999, Kerr & Godfrey-Smith 2002a, Waters 2005). However, foundational work has indicated that this equivalence does not in fact hold. In Dugatkin and Reeve and the rest of this literature, comparison of population genetic models was largely based on predictions of allele frequency changes; in other words, if two models made the same predictions as to the changes of allelic frequencies in a given situation, then the models are equivalent. However, this is an overly simplistic method for testing model equivalence which pays little mind to the details of the models themselves. When the notion of representational adequacy of the models is taken into account, specifically through the inclusion of parametric and dynamical sufficiency as important points of comparison, this equivalence between genic and hierarchical models disappears (Lloyd, Lewontin, & Feldman 2008; Lewontin 1974; see also group selection for more on formal equivalence). Parametric sufficiency concerns what state space and variables are sufficient to capture the relevant properties of a given system, while dynamical sufficiency > > > concerns what state space and variables are sufficient to describe the > evolution of a system given the parameters being used in the specific > system. (Lloyd, Lewontin, & Feldman 2008: 146; Lewontin 1974) > > > Utilizing these concepts allows for a more detailed and meaningful evaluation of a given mathematical model. And under such an analysis, the claims regarding the equivalency of genic and hierarchical models cannot be sustained. Since allelic parameters and the changes in allelic frequencies depend on genotypic fitnesses, the genic models claimed to be equivalent to the hierarchical models are neither parametrically nor dynamically sufficient.[19] ### 3.5 Units of Evolutionary Transition In our preceding discussions of units of selection, we have restricted ourselves to situations in which the various units were pre-established entities. Our approach has been synchronic, one in which the relevant units, be they genes, organisms, or populations, are the same both before and after a given evolutionary process. However, not all evolutionary processes may be able to be captured under such a perspective. In particular, recent discussions regarding so-called "evolutionary transitions" present a unique complication to the debates over units and levels of selection. Evolutionary transition is "the process that creates new levels of biological organization" (Griesemer 2000c: 69), such as the origins of chromosomes, multicellularity, eukaryotes, and social groups (Maynard-Smith and Szathmary 1995: 6-7). These transitions all share a common feature, namely that "entities that were capable of independent replication before the transition can replicate only as part of a larger whole after it" (Maynard-Smith and Szathmary 1995: 6). Evolutionary transitions create new potential levels and units of selection by creating new kinds of entities that can have variances in fitness. Thus, it is the "project of a theory of evolutionary transition to explain the evolutionary origin of entities with such capacities" (Griesemer 2000c: 70). However, since such cases involve the evolutionary origin of a given level of selection, traditional synchronic approaches to units and levels of selection, which assume the pre-existence of a "hierarchy of entities that are potential candidates for units of selection", may be insufficient, since it is the evolution of those very properties that allow entities to serve as, for example, interactors or replicators that is being addressed (Griesemer 2000c: 70). Such a task requires a diachronic perspective, one under which the properties of our currently extant units of selection cannot be presupposed. > > > ...[A]s long as evolutionary theory concerns the function of > contemporary units at fixed levels of the biological > hierarchy..., the functionalist approach may be adequate to its > intended task. However, if a philosophy of units is to address > problems going beyond this scope--for example to problems of > evolutionary transition,... then a different approach is > needed. (Griesemer 2003: 174) > > > The "reproducer" concept (discussed in section 2.2), which incorporates the notion of development into the treatment of units and levels of selection, is a step toward meeting the goal of addressing such evolutionary transitions, and > > > the dependency of formerly independent replicators on the > "replication" of the wholes--the basis for the > definition of evolutionary transition ... is a > developmental dependency that should be incorporated into the > analysis of units. (2000c: 75) > > > Those adopting the reproducer concept argue that thinking in broader terms of reproducers avoids the presupposition of evolved coding mechanisms implicit to the concept of replicators. In the case of evolutionary transitions, this allows us to separate the basic development involved in the origin of a new biological level from the later evolution of sophisticated developmental mechanisms for the "stabilization and maintenance of a new level of reproduction" (Griesemer 2000c: 77). Explaining evolutionary transitions in Darwinian terms poses a particular challenge: "Why was it advantageous for the lower-level units to sacrifice their individuality and form themselves into a corporate body?" (Okasha 2006: 218). On one analysis, three stages of such a transition, each defined in terms of the connection between fitness at the level of the collective and the individual fitness of its component particles, are identified (Okasha 2006: 238). Initially, collective fitness is simply defined as average particle fitness. As fitness at the two levels begins to be decoupled, collective fitness remains proportional to average particle fitness, but is not defined by it; at such a stage, "the emerging collective lacks 'individuality', and has no collective-level functions of its own" (Okasha 2006: 237). Finally, collective fitness "starts to depend on the functionality of the collective itself" (Okasha 2006: 237-8; see Okasha 2015 for a representation of this in terms of causal graphs). On this analysis, the different stages of an evolutionary transition involve different conceptions of multi-level selection (Okasha 2006, 2015). Using the distinction defended by Lorraine Heisler and John Damuth (Heisler & Damuth 1987; Damuth & Heisler 1988) in their "contextual analysis" of units of selection, this analysis claims that early on in the process of an evolutionary transition, multi-level selection 1 (MLS1), in which the particles themselves are the "'focal' units" upon which selection directly acts, applies. However, by the end of the transition, both the collectives and the particles are focal units of selection processes, with independent fitnesses, a case of Damuth and Heisler's multi-level selection 2 (MLS2) (Okasha 2006: 4). An easy way to capture this distinction is that, under MLS1, the lower level particles are the interactors as well as the replicators, while in MLS2, both the upper level collectives as well as the particles are interactors. Thus, the issues surrounding evolutionary transitions involve both the interactor question and the replicator question. Understanding evolutionary transitions hence provides additional significance to Damuth and Heisler's distinction: > > > Rather than simply describing selection processes of different sorts, > which should be kept separate in the interests of conceptual clarity, > MLS1 and MLS2 represent different temporal stages of an > evolutionary transition. (Okasha 2006: 239) > > > On a different approach, evolutionary transitions are seen as the appearance of a "new kind of Darwinian population", of "new entities that can enter into Darwinian processes in their own right" (Godfrey-Smith 2009: 122). These transitions involve a "de-Darwinizing" of the lower-level entities such that > > > an initial collective has come to engage in definite high-level > reproduction, and this has involved the curtailing of independent > evolution at the lower level. (Godfrey-Smith 2009: 124) > > > This can be accomplished in a variety of ways, such as through the bottleneck caused by the production of new collectives from single individuals, coupled with germ-line segregation (as in the transitions to multicellularity), or by a single member of the collective preventing all other members from reproducing (for example, among eusocial insects), or by single member having primary but not total control over the other constituents (as in the evolution of eukaryotes) (Godfrey-Smith 2009: 123-124). These processes all involve restrictions on the ability of the lower-level entities to function as interactors and replicators, and the emergence of upper-level collectives as both interactors and replicators. The degree to which lower-level entities are thus restricted can vary. For example, somatic cells are still capable of bearing individual fitness, of outcompeting neighboring cells, and of producing more progeny. Thus, they are not yet "post-populational"; they "retain crucial Darwinian features in their own right" (Godfrey-Smith 2009: 126). However, they are dependent on the germ-line cells for the propagation of new collectives, and thus their ability to act as replicators is necessarily curtailed. Thus, in order to prevent subversion and encourage cooperation, such a transition requires both the "generation of benefit" and the "alignment of reproductive interests" (Godfrey-Smith 2009: 124, with terminology from Calcott 2008; see Booth's 2014 analysis of heterokaryotic Fungi using Godfrey-Smith's approach). For example, in the case of multicellularity, the latter can be accomplished by "close kinship within the collective" (Godfrey-Smith 2009: 124). In a useful analysis of the volvocine algae, other hierarchical selectionists use optimality modeling at the group level to search for a group level adaptation, in aid of modeling evolutionary transitions (Shelton & Michod 2014b). They look for selection and adaptation at the higher level in their model of transition, which contrasts other views that look only for selection at the higher level, but not for engineering adaptations. The emphasis is on the distinction between fortuitous group benefit and real group adaptation. In places, however, they seem to embrace the product-of-selection definition of group adaptation, even though they are committed to denying its applicability (2014b: 454). Their point is to decompose levels of selection and adaptation using a model organism to get evolutionary emergence of levels, i.e., evolutionary transition. Thus, there are a variety of philosophical approaches to analyzing evolutionary transition on offer, whether in terms of reproducers, multilevel selection, or Darwinian populations. The essential diachronic nature of the problem poses a unique challenge, and involves not just the interactor and replicator (or reproducer) questions, but also the questions of who is the beneficiary of the selection process, and how that new level emerges. ## 4. Conclusion It makes no sense to treat different answers as competitors if they are answering different questions. We have reviewed a framework of four questions with which the debates appearing under the rubric of "units of selection" can be classified and clarified. The original discussants of the units of selection problem separated the classic question about the level of selection or interaction (the interactor question) from the issue of how large a chunk of the genome functions as a replicating unit (the replicator question). The interactor question should also be separated from the question of which entity should be seen as acquiring adaptations as a result of the selection process (the manifestor-of-adaptation question). In addition, there is a crucial ambiguity in the meaning of adaptation that is routinely ignored in these debates: adaptation as a selection product and adaptation as an engineering design. Finally, we can distinguish the issue of the entity that ultimately benefits from the selection process (the beneficiary question) from the other three questions. This set of distinctions has been used to analyze leading points of view about the units of selection and to clarify precisely the question or combination of questions with which each of the protagonists is concerned. There are many points in the debates in which misunderstandings may be avoided by a precise characterization of which of the units of selection questions is being addressed.
scientific-unity	## 1. Historical development in philosophy and science from Greek philosophy to Logical Empiricism in America ### 1.1 From Greek thought to Western science Unity has a history as well as a logic. Different formulations and debates express intellectual and other resources and interests in different contexts. Questions about unity belong partly in a tradition of thought that can be traced back to pre-Socratic Greek cosmology, in particular to the preoccupation with the question of the One and the Many. In what senses is the world and, as a result, our knowledge of it one? A number of representations of the world in terms of a few simple constituents that were considered fundamental emerged: Parmenides' static substance, Heraclitus' flux of becoming, Empedocles' four elements, Democritus' atoms, Pythagoras' numbers, Plato's forms, and Aristotle's categories. The underlying question of the unity of our types of knowledge was explicitly addressed by Plato in the Sophist as follows: "Knowledge also is surely one, but each part of it that commands a certain field is marked off and given a special name proper to itself. Hence language recognizes many arts and many forms of knowledge" (Sophist, 257c). Aristotle asserted in On the Heavens that knowledge concerns what is primary, and different "sciences" know different kinds of causes; it is metaphysics that comes to provide knowledge of the underlying kind. With the advent and expansion of Christian monotheism, the organization of knowledge reflected the idea of a world governed by the laws dictated by God, its creator and legislator. From this tradition emerged encyclopedic efforts such as the Etymologies, compiled in the sixth century by the Andalusian Isidore, Bishop of Seville, the works of the Catalan Ramon Llull in the Middle Ages and those of the Frenchman Petrus Ramus in the Renaissance. Llull introduced iconic tree-diagrams and forest-encyclopedias representing the organization of different disciplines including law, medicine, theology and logic. He also introduced more abstract diagrams--not unlike some found in Cabbalistic and esoteric traditions--in an attempt to combinatorially encode the knowledge of God's creation in a universal language of basic symbols. Their combination would be expected to generate knowledge of the secrets of creation and help articulate knowledge of universal order (mathesis universalis), which would, in turn, facilitate communication with different cultures and their conversion to Christianity. Ramus introduced diagrams representing dichotomies and gave prominence to the view that the starting point of all philosophy is the classification of the arts and sciences. The encyclopedia organization of knowledge served the project of its preservation and communication. The emergence of a distinctive tradition of scientific thought addressed the question of unity through the designation of a privileged method, which involved a privileged language and set of concepts. Formally, at least, it was modeled after the Euclidean ideal of a system of geometry. In the late-sixteenth century, Francis Bacon held that one unity of the sciences was the result of our organization of records of discovered material facts in the form of a pyramid with different levels of generalities. These could be classified in turn according to disciplines linked to human faculties. Concomitantly, the controlled interaction with phenomena of study characterized so-called experimental philosophy. In accordance with at least three traditions--the Pythagorean tradition, the Bible's dictum in the Book of Wisdom and the Italian commercial tradition of bookkeeping--Galileo proclaimed at the turn of the seventeenth century that the Book of Nature had been written by God in the language of mathematical symbols and geometrical truths, and in it, the story of Nature's laws was told in terms of a reduced set of objective, quantitative primary qualities: extension, quantity of matter and motion. A persisting rhetorical role for some form of theological unity of creation should not be neglected when considering pre-twentieth-century attempts to account for the possibility and desirability of some form of scientific knowledge. Throughout the seventeenth century, mechanical philosophy and Descartes' and Newton's systematization from basic concepts and first laws of mechanics became the most promising framework for the unification of natural philosophy. After the demise of Laplacian molecular physics in the first half of the nineteenth century, this role was taken over by ether mechanics and, unifying forces and matter, energy physics. ### 1.2 Rationalism and Enlightenment Descartes and Leibniz gave this tradition a rationalist twist that was centered on the powers of human reason and the ideal of system of knowledge founded on rational principles. It became the project of a universal framework of exact categories and ideas, a mathesis universalis (Garber 1992; Gaukroger 2002). Adapting the scholastic image of knowledge, Descartes proposed an image of a tree in which metaphysics is depicted by the roots, physics by the trunk, and the branches depict mechanics, medicine and morals. Leibniz proposed a general science in the form of a demonstrative encyclopedia. This would be based on a "catalogue of simple thoughts" and an algebraic language of symbols, characteristica universalis, which would render all knowledge demonstrative and allow disputes to be resolved by precise calculation. Both defended the program of founding much of physics on metaphysics and ideas from life science (Smith 2011) (Leibniz's unifying ambitions with symbolic language and physics extended beyond science, to settle religious and political fractures in Europe). By contrast, while sharing a model of a geometric, axiomatic structure of knowledge, Newton's project of natural philosophy was meant to be autonomous from a system of philosophy and, in the new context, still endorsed, for its model of organization and its empirical reasoning, values of formal synthesis and ontological simplicity (see the entry on Newton; also Janiak 2008). Belief in the unity of science or knowledge, along with the universality of rationality, was at its strongest during the European Enlightenment. The most important expression of the encyclopedic tradition came in the mid-eighteenth century from Diderot and D'Alembert, editors of the Encyclopedie, ou dictionnaire raisonne des sciences, des arts et des metiers (1751-1772). Following earlier classifications by Nichols and Bacon, their diagram presenting the classification of intellectual disciplines was organized in terms of a classification of human faculties. Diderot stressed in his own entry, "Encyclopaedia", that the word signifies the unification of the sciences. The function of the encyclopedia was to exhibit the unity of human knowledge. Diderot and D'Alembert, in contrast to Leibniz, made classification by subject the primary focus, and introduced cross-references instead of logical connections. The Enlightenment tradition in Germany culminated in Kant's critical philosophy. ### 1.3 German tradition since Kant For Kant, one of the functions of philosophy was to determine the precise unifying scope and value of each science. For him, the unity of science is not the reflection of a unity found in nature, or, even less, assumed in a real world behind the apparent phenomena. Rather, it has its foundations in the unifying a priori character or function of concepts, principles and of Reason itself. Nature is precisely our experience of the world under the universal laws that include such concepts. And science, as a system of knowledge, is "a whole of cognition ordered according to principles", and the principles on which proper science is grounded are a priori (Preface to Metaphysical Foundations of Natural Science). A devoted but not exclusive follower of Newton's achievements and insights, he maintained through most of his life that mathematization and a priori universal laws given by the understanding were preconditions for genuine scientific character (like Galileo and Descartes earlier, and Carnap later, Kant believed that mathematical exactness constituted the main condition for the possibility of objectivity). Here Kant emphasized the role of mathematics coordinating a priori cognition and its determined objects of experience. Thus, he contrasted the methods employed by the chemist, a "systematic art" organized by empirical regularities, with those employed by the mathematician or physicist, which were organized by a priori laws, and he held that biology is not reducible to mechanics--as the former involves explanations in terms of final causes (see Critique of Pure Reason, Critique of Judgment and Metaphysical Foundations of Natural Science). With regards to biology--insufficiently grounded in the fundamental forces of matter--its inclusion requires the introduction of the idea of purposiveness (McLaughlin 1991). More generally, for Kant, unity was a regulative principle of reason, namely, an ideal guiding the process of inquiry toward a complete empirical science with its empirical concepts and principles grounded in the so-called concepts and principles of the understanding that constitute and objectify empirical phenomena. (On systematicity as a distinctive aspect of this ideal and on its origin in reason, see Kitcher 1986 and Hoyningen-Huene 2013). Kant's ideas set the frame of reference for discussions of the unification of the sciences in German thought throughout the nineteenth century (Wood and Hahn 2011). He gave philosophical currency to the notion of worldview (Weltanschauung) and, indirectly, world-picture (Weltbild), establishing among philosophers and scientists the notion of the unity of science as an intellectual ideal. From Kant, German-speaking Philosophers of Nature adopted the image of Nature in terms of interacting forces or powers and developed it in different ways; this image found its way to British natural philosophy. In Great Britain this idealist, unifying spirit (and other notions of an idealist and romantic turn) was articulated in William Whewell's philosophy of science. Two unifying dimensions are these: his notion of mind-constructed fundamental ideas, which form the basis for organizing axioms and phenomena and classifying sciences, and the argument for the reality of explanatory causes in the form of consilience of induction, wherein a single cause is independently arrived at as the hypothesis explaining different kinds of phenomena. In face of expanding research, the unifying emphasis on organization, classification and foundation led to exploring differences and rationalizing boundaries. The German intellectual current culminated in the late-nineteenth century in the debates among philosophers such as Windelband, Rickert and Dilthey. In their views and those of similar thinkers, a worldview often included elements of evaluation and life meaning. Kant had established the basis for the famous distinction between the natural sciences (Naturwissenschaften) and the cultural, or social, sciences (Geisteswissesnschaften) popularized in theory of science by Wilhelm Dilthey and Wilhelm Windelband. Dilthey, Windelband, his student Heinrich Rickert, and Max Weber (although the first two preferred Kulturwissenschaften, which excluded psychology) debated over how differences in subject matter between the two kinds of sciences forced a distinctive difference between their respective methods. Their preoccupation with the historical dimension of the human phenomena, along with the Kantian emphasis on the conceptual basis of knowledge, led to the suggestion that the natural sciences aimed at generalizations about abstract types and properties, whereas the human sciences studied concrete individuals and complexes. The human case suggested a different approach based on valuation and personal understanding (Weber's verstehen). For Rickert, individualized concept formation secured knowledge of historical individuals by establishing connections to recognized values (rather than personal valuations). In biology, Ernst Haeckel defended a monistic worldview (Richards 2008). The Weltbild tradition influenced the physicists Max Planck and Ernst Mach, who engaged in a heated debate about the precise character of the unified scientific world-picture. Mach's more influential view was both phenomenological and Darwinian: the unification of knowledge took the form of an analysis of ideas into biologically embodied elementary sensations (neutral monism) and was ultimately a matter of adaptive economy of thought. Planck adopted a realist view that took science to gradually approach complete truth about the world, and he fundamentally adopted the thermodynamical principles of energy and entropy (on the Mach-Planck debate see Toulmin 1970). These world-pictures constituted some of the alternatives to a long-standing mechanistic view that, since the rise of mechanistic philosophy with Descartes and Newton, had informed biology as well as most branches of physics. In the background was the perceived conflict between the so-called mechanical and electromagnetic worldviews, which resulted throughout the first two decades of the twentieth century in the work of Albert Einstein (Holton 1998). In the same German tradition, and amidst the proliferation of work on energy physics and books on unity of science, the German energeticist Wilhelm Ostwald declared the twentieth century the "Monistic century". During the 1904 World's Fair in St. Louis, the German psychologist and Harvard professor Hugo Munsterberg organized a congress under the title "Unity of Knowledge"; invited speakers were Ostwald, Ludwig Boltzmann, Ernest Rutherford, Edward Leamington Nichols, Paul Langevin and Henri Poincare. In 1911, the International Committee of Monism held its first meeting in Hamburg, with Ostwald presiding.[1] Two years later it published Ostwald's monograph, Monism as the Goal of Civilization. In 1912, Mach, Felix Klein, David Hilbert, Einstein and others signed a manifesto aiming at the development of a comprehensive worldview. Unification remained a driving scientific ideal. In the same spirit, Mathieu Leclerc du Sablon published his L'Unite de la Science (1919), exploring metaphysical foundations, and Johan Hjorst published The Unity of Science (1921), sketching out a history of philosophical systems and unifying scientific hypotheses. ### 1.4 Positivism and logical empiricism The German tradition stood in opposition to the prevailing empiricist views that, since the time of Hume, Comte and Mill, held that the moral or social sciences (even philosophy) relied on conceptual and methodological analogies with geometry and the natural sciences, not just astronomy and mechanics, as well as with biology. In the Baconian tradition, Comte emphasized a pyramidal hierarchy of disciplines in his "encyclopedic law" or order, from the most general sciences about the simplest phenomena to the most specific sciences about the most complex phenomena, each depending on knowledge from its more general antecedent: from inorganic physical sciences (arithmetic, geometry, mechanics, astronomy, physics and chemistry) to the organic physical ones, such as biology and the new "social physics", soon to be renamed sociology (Comte 1830-1842). Mill, instead, pointed to the diversity of methodologies for generating, organizing and justifying associated knowledge with different sciences, natural and human, and the challenges to impose a single standard (Mill 1843, Book VI). He came to view political economy eventually as an art, a tool for reform more than a system of knowledge (Snyder 2006). Different yet connected currents of nineteenth-century positivism, first in European philosophy in the first half of the century--Auguste Comte, J.S. Mill and Herbert Spencer--and subsequently in North American philosophy--John Fiske, Chauncey Wright and William James--arose out of intellectual tensions between metaphysics and the sciences and identified positivism as synthetic and scientific philosophy; accordingly, they were concerned with the ordering and unification of knowledge through the organization of the sciences. The synthesis was either of methods alone--Mill and Wright--or else also of doctrines--Comte, Spencer and Fiske. Some used the term "system" especially in relation to a common logic or method (Pearce 2015). In the twentieth century the unity of science became a distinctive theme of the scientific philosophy of logical empiricism (Cat 2021). The question of unity engaged science and philosophy alike. In their manifesto, logical empiricists--known controversially also as logical positivists--and most notably the founding members of the Vienna Circle, adopted the Machian banner of "unity of science without metaphysics". This was a normative criterion of unity with a role in social reform based on the demarcation between science and metaphysics: a unity of method and language that included all the sciences, natural and social. A common method did not necessarily imply a more substantive unity of content involving theories and their concepts. A stronger reductive model within the Vienna Circle was recommended by Rudolf Carnap in his The Logical Construction of the World (1928). While embracing the Kantian connotation of the term "constitutive system", it was inspired by recent formal standards: Hilbert's axiomatic approach to formulating theories in the exact sciences and Frege's and Russell's logical constructions in mathematics. It was also predicated on the formal values of simplicity, rationality, (philosophical) neutrality and objectivity associated with scientific knowledge. In particular, Carnap tried to explicate such notions in terms of a rational reconstruction of science in terms of a method and a structure based on logical constructions out of (1) basic concepts in axiomatic structures and (2) rigorous, reductive logical connections between concepts at different levels. Different constitutive systems or logical constructions would serve different (normative) purposes: a theory of science and a theory of knowledge. Both foundations raised the issue of the nature and universality of a physicalist language. One such system of unified science is the theory of science, in which the construction connects concepts and laws of the different sciences at different levels, with physics and its genuine laws as fundamental, lying at the base of the hierarchy. Because of the emphasis on the formal and structural properties of our representations, objectivity, rationality and unity go hand in hand. Carnap's formal emphasis developed further in Logical Syntax of Language (1934). Alternatively, all scientific concepts could be constituted or constructed in a different system in the protocol language out of classes of elementary complexes of experiences, scientifically understood, representing experiential concepts. Carnap subsequently defended the epistemological and methodological universality of physicalist language and physicalist statements. The unity of science in this context was an epistemological project (for a survey of the epistemological debates, see Uebel 2007; on different strands of the anti-metaphysical normative project of unity see Frost-Arnold 2005). Whereas Carnap aimed at rational reconstructions, another member of the Vienna Circle, Otto Neurath, favored a more naturalistic and pragmatic approach, with a less idealized and reductive model of unity. His evolving standards of unity were generally motivated by the complexity of empirical reality and the application of empirical knowledge to practical goals. He spoke of an "encyclopedia-model", opposed to the classic ideal of a pyramidal, reductive "system-model". The encyclopedia-model took into account the presence within science of ineliminable and imprecise terms from ordinary language and the social sciences and emphasized a unity of language and the local exchanges of scientific tools. Specifically, Neurath stressed the material-thing language called "physicalism", not to be confounded with the emphasis on the vocabulary of physics. Its motivation was partly epistemological, and Neurath endorsed anti-foundationalism; no unified science, like a boat at sea, would rest on firm foundations. The scientific spirit abhorred dogmatism. This weaker model of unity emphasized empiricism and the normative unity of the natural and the human sciences. Like Carnap's unified reconstructions, Neurath's had pragmatic motivations. Unity without reductionism provided a tool for cooperation and it was motivated by the need for successful treatment--prediction and control--of complex phenomena in the real world that involved properties studied by different theories or sciences (from real forest fires to social policy): unity of science at the point of action. It is an argument from holism, the counterpart of Duhem's claim that only clusters of hypotheses are confronted with experience. Neurath spoke of a "boat", a "mosaic", an "orchestration", and a "universal jargon". Following institutions such as the International Committee on Monism and the International Council of Scientific Unions, Neurath spearheaded a movement for Unity of Science in 1934 that encouraged international cooperation among scientists and launched the project of an International Encyclopedia of Unity of Science. It expressed the internationalism of his socialist convictions and the international crisis that would lead to the Second World War (Kamminga and Somsen 2016). At the end of the Eighth International Congress of Philosophy, held in Prague in September of 1934, Neurath proposed a series of International Congresses for the Unity of Science. These took place in Paris, 1935; Copenhagen, 1936; Paris, 1937; Cambridge, England, 1938; Cambridge, Massachusetts, 1939; and Chicago, 1941. For the organization of the congresses and related activities, Neurath founded the Unity of Science Institute in 1936 (renamed in 1937 as the International Institute for the Unity of Science) alongside the International Foundation for Visual Education, founded in 1933. The Institute's executive committee was composed of Neurath, Philip Frank and Charles Morris. After the Second World War, a discussion of unity engaged philosophers and scientists in the Inter-Scientific Discussion Group, first as the Science of Science Discussion Group, in Cambridge, Massachusetts, founded in October 1940 primarily by Philip Frank and Carnap (themselves founders of the Vienna Circle), Quine, Feigl, Bridgman, and the psychologists E. Boring and S.S. Stevens. This would later become the Unity of Science Institute. The group was joined by scientists from different disciplines, from quantum mechanics (Kemble and Van Vleck) and cybernetics (Wiener) to economics (Morgenstern), as part of what was both a self-conscious extension of the Vienna Circle and a reflection of local concerns within a technological culture increasingly dominated by the interest in computers and nuclear power. The characteristic feature of the new view of unity was the ideas of consensus and, subsequently, especially within the USI, cross-fertilization. These ideas were instantiated in the emphasis on scientific operations (operationalism) and the creation of war-boosted cross-disciplines such as cybernetics, computation, electro-acoustics, psycho-acoustics, neutronics, game theory and biophysics (Galison 1998; Hardcastle 2003). In the late 1960s, Michael Polanyi and Marjorie Grene organized a series of conferences funded by the Ford Foundation on unity of science themes (Grene 1969a, 1969b, 1971). Their general character was interdisciplinary and anti-reductionist. The group was originally called "Study Group on Foundations of Cultural Unity", but this was later changed to "Study Group on the Unity of Knowledge". By then, a number of American and international institutions were already promoting interdisciplinary projects in academic areas (Klein 1990). For both Neurath and Polanyi, the organization of knowledge and science, the Republic of Science, was inseparable from ideals of political organization. Over the last four decades much historical and philosophical scholarship has challenged the ideals of monism and reductionism and pursued a growing critical interest in pluralism and interdisciplinary collaboration (see below). Along the way, the distinction between the historical human sciences and ahistorical natural sciences has received much critical and productive attention. One outcome has been the application and development of concepts and accounts in philosophy of history in understanding different human and natural sciences as historical, with a special focus on the epistemic role of unifying narratives and standards and problems in disciplines such as archeology, geology, cosmology and paleontology (see, for instance, Morgan and Wise 2017; Currie 2019). Another outcome has been a renewed critique of the autonomy of the social sciences. One concern is, for instance, the raising social and economic value of the natural sciences and their influence on the models and methods of the social sciences; another is the influence on both of certain particular political and economic views. Other related concerns are the role of the social sciences in political life and the assumption that humans are unique and superior to animals and emerging technologies (see, for instance, van Bouwel 2009b; Kincaid and van Bouwel 2023). ## 2. Varieties of Unity The historical introductory sections have aimed to show the intellectual centrality, varying formulations, and significance of the concept of unity. The rest of the entry presents a variety of modern themes and views. It will be helpful to introduce a number of broad categories and distinctions that can sort out different kinds of accounts and track some relations between them, as well as additional significant philosophical issues. (The categories are not mutually exclusive, and they sometimes partly overlap. Therefore, while they help label and characterize different positions, they cannot provide a simple, easy and neatly ordered conceptual map.) Connective unity is a weaker and more general notion than the specific ideal of reductive unity; this requires asymmetric relations of reduction (see below), which typically rely on assumptions about hierarchies of levels of description and the primacy--conceptual, ontological, epistemological and so on--of a fundamental representation. The category of connective unity helps accommodate and bring attention to the diversity of non-reductive accounts. Another useful distinction is between synchronic and diachronic unity. Synchronic accounts are ahistorical, assuming no meaningful temporal relations. Diachronic accounts, by contrast, introduce genealogical hypotheses involving asymmetric temporal and causal relations between entities or states of the systems described. Evolutionary models are of this kind; they may be reductive to the extent that the posited original entities are simpler and on a lower level of organization and size. Others simply emphasize connection without overall directionality. In general, it is useful to distinguish between ontological unity and epistemological unity, even if many accounts bear both characteristics and fall under both rubrics. In some cases, one kind supports the other salient kind in the model. Ontological unity is here broadly understood as involving relations between descriptive conceptual elements; in some cases, the concepts will describe entities, facts, properties or relations, and descriptive models will focus on metaphysical aspects of the unifying connections such as holism, emergence, or downwards causation. Epistemological unity applies to epistemic relations or goals such as explanation. Methodological connections and formal (logical, mathematical, etc.) models may belong in this kind. This article does not draw any strict or explicit distinction between epistemological and methodological dimensions or modes of unity. Additional possible categories and distinctions include the following: vertical unity or inter-level unity is unity of elements attached to levels of analysis, composition or organization on a hierarchy, whether for a single science or more, whereas horizontal unity or intra-level unity applies to one single level and to its corresponding kind of system (Wimsatt 2007). Global unity is unity of any other variety with a universal quantifier of all kinds of elements, aspects or descriptions associated with individual sciences as a kind of monism, for instance, taxonomical monism about natural kinds, while local unity applies to a subset. (Cartwright has distinguished this same-level global form of reduction, or "imperialism", in Cartwright 1999; see also Mitchell 2003). Obviously, vertical and horizontal accounts of unity can be either global or local. Finally, the rejection of global unity has been associated with both isolationism--keeping independent competing alternative representations of the same phenomena or systems--as well as with local integration--the local connective unity of the alternative perspectives. A distinction of methodological nature contrasts internal and external perspectives, according to whether the accounts are based naturalistically on the local contingent practices of certain scientific communities at a given time or based on universal metaphysical assumptions broadly motivated (Ruphy 2017). (Ruphy has criticized Cartwright and Dupre for having adopted external metaphysical positions and defended the internal perspective, also present in the program of the so-called Minnesota School, i.e., Kellert et al. 2006.) ## 3. Epistemological unities ### 3.1 Reduction The project of unity, as mentioned above, has long guided models of scientific understanding by privileging descriptions of more fundamental entities and phenomena such as the powers and behaviors of atoms, molecules and machines. Since at least the 1920s and until the 1960s, unity was understood in terms of formal approaches to theories, of semantic relations between vocabularies and of logical relations between linguistic statements in those vocabularies. The ideal of unity was formulated, accordingly, in terms of reduction relations to more fundamental terms and statements. Different accounts have since stood and fallen with any such commitments. Also, note that identifying relations of reduction must be distinguished from the ideal of reductionism. Reductionism is the adoption of reduction relations as the global ideal or standard of a proper unified structure of scientific knowledge; and distance from that ideal was considered a measure of its unifying progress. In general, reduction reconciles ontological and epistemological considerations of identity and difference. Claims such as "mental states are reduceable to neurochemical states", "chemical differences reduce to differences between atomic structures" and "optics can be reduced to electromagnetism" imply the truth of identity statements--a strong ontological reduction. However, the relations of reduction between entities or properties in such claims also get additional epistemic value from the expressed semantic diversity of conceptual content and the asymmetry of the relation described in such claims between the reducing and the reduced claims--concepts, laws, theories, etc. (Riel 2014). Elimination and eliminativism are the extreme forms of reduction and reductionism that commit to aligning the epistemic content with the ontological content fixed by the identity statements: since only x exists and claims about x best meet scientific goals, only talk of x merits scientific consideration and use. Two formulations of unification in the logical positivist tradition of the ideal logical structure of science placed the question of unity at the core of philosophy of science: Carl Hempel's deductive-nomological model of explanation and Ernst Nagel's model of reduction. Both are fundamentally epistemological models, and both are specifically explanatory, at least in the sense that explanation serves unification. The emphasis on language and logical structure makes explanatory reduction a form of unity of the synchronic kind. Still, Nagel's model of reduction is a model of scientific structure and explanation as well as of scientific progress. It is based on the problem of relating different theories as different sets of theoretical predicates (Nagel 1961). Reduction requires two conditions: connectability and derivability. Connectability of laws of different theories requires meaning invariance in the form of extensional equivalence between descriptions, with bridge principles between coextensive but distinct terms in different theories. Nagel's account distinguishes two kinds of reductions: homogenous and heterogeneous. When both sets of terms overlap, the reduction is homogeneous. When the related terms are different, the reduction is heterogeneous. Derivability requires a deductive relation between the laws involved. In the quantitative sciences, the derivation often involves taking a limit. In this sense the reduced science is considered an approximation to the reducing new one. Neo-Nagelian accounts have attempted to solve Nagel's problem of reduction between putatively incompatible theories. The following paragraphs list a few. Nagel's two-term relation account has been modified by weaker conditions of analogy and a role for conventions, requiring it to be satisfied not necessarily by the two original theories, $T\_1$ and $T\_2$, which are respectively new and old and more and less general, but by the modified theories $T'\_1$ and $T'\_2$. Explanatory reduction is strictly a four-term relation in which $T'\_1$ is "strongly analogous" to $T\_1$ and corrects, with the insight that the more fundamental theory can offer, the older theory, $T\_2$, changing it to $T'\_2$. Nagel's account also requires that bridge laws be synthetic identities, in the sense that they be factual, empirically discoverable and testable; in weaker accounts, admissible bridge laws may include elements of convention (Schaffner 1967; Sarkar 1998). The difficulty lay especially with the task of specifying or giving a non-contextual, transitive account of the relations between $T$ and $T'$ (Wimsatt 1976). An alternative set of semantic and syntactic conditions of reduction bears counterfactual interpretations. For instance, syntactic conditions may take the form of limit relations (e.g., classical mechanics reduces to relativistic mechanics for the value of the speed of light c[?] 0 and to quantum mechanics for the value of the Planck constant h[?] 0) and ceteris paribus assumptions (e.g., when optical laws can be identified with results of the theory of electromagnetism); then, each condition helps explain why the reduced theory works where it does and fails where it does not (Glymour 1969). A different approach to reductionism acknowledges a commitment to providing explanation but rejects the value of a focus on the role of laws. This approach typically draws a distinction between hard sciences such as physics and chemistry and special sciences such as biology and the social sciences. It claims that laws that are in a sense operative in the hard sciences are not available in the special ones, or play a more limited and weaker role, and this on account of historical character, complexity or reduced scope. The rejection of empirical laws in biology, for instance, has been argued on grounds of historical dependence on contingent initial conditions (Beatty 1995), and as matter of supervenience (see the entry on supervenience) of spatio-temporally restricted functional claims on lower-level molecular ones, and the multiple realization (see the entry on multiple realizability) of the former by the latter (Rosenberg 1994; Rosenberg's argument from supervenience to reduction without laws must be contrasted with Fodor's physicalism about the special sciences about laws without reduction (see below and the entry on physicalism); for a criticism of these views see Sober 1996). This non-Nagelian approach assumes further that explanation rests on identities between predicates and deductive derivations (reduction and explanation might be said to be justified by derivations, but not constituted by them; see Spector 1978). Explanation is provided by lower-level mechanisms; their explanatory role is to replace final why-necessarily questions (functional) with proximate how-possibly questions (molecular). One suggestion to make sense of the possibility of the supervening functional explanations without Nagelian reduction may rely on an ontological type of relation of reduction such as the composition of powers in explanatory mechanisms (Gillette 2010, 2016). The reductive commitment to the lower level is based on relations of composition, at play in epistemological analysis and metaphysical synthesis, but is merely formal and derivational. We may be able to infer what composes the higher level, but we cannot simply get all the relevant knowledge of the higher level from our knowledge of the lower level (see also Auyang 1998). This kind of proposal points to epistemic anti-reductionism. Note that significant proposals of reductive unity rely on ontological assumptions about some hierarchy of levels organizing reality and scientific theories of it (more below). A more general characterization views reductionism as a research strategy. On this methodological view reductionism can be characterized by a set of so-called heuristics (non-algorithmic, efficient, error-based, purpose-oriented, problem-solving tasks) (Wimsatt 2006): heuristics of conceptualization (e.g., descriptive localization of properties, system-environment interface determinism, level and entity-dependence), heuristics of model-building and theory construction (e.g., model intra-systemic localization with emphasis of structural properties over functional ones, contextual simplification and external generalization) and heuristics of observation and experimental design (e.g., focused observation, environmental control, local scope of testing, abstract shared properties, behavioral regularity and context-independence of results). ### 3.2 Antireductionism Since the 1930s, the focus was on a syntactic approach, with physics as the paradigm of science, deductive logical relations as the form of cognitive or epistemic goals such as explanation and prediction, and theory and empirical laws as paradigmatic units of scientific knowledge (Suppe 1977; Grunbaum and Salmon 1988). The historicist turn in the 1960s, the semantic turn in philosophy of science in the 1970s and a renewed interest in special sciences has changed this focus. The very structure of hierarchy of levels has lost its credibility, even for those who believe in it as a model of autonomy of levels rather than as an image of fundamentalism. The rejection of such models and their emendations have occupied the last four decades of philosophical discussion about unity in and of the sciences (especially in connection to psychology and biology, and more recently chemistry). A valuable consequence has been the strengthening of philosophical projects and communities devoting more sustained and sophisticated attention to special sciences, different from physics. The first target of antireductionist attacks has been Nagel's demand of extensional equivalence. It has been dismissed as an inadequate demand of "meaning invariance" and approximation, and with it the possibility of deductive connections. Mocking the positivist legacy of progress through unity, empiricism and anti-dogmatism, these constraints have been decried as intellectually dogmatic, conceptually weak and methodologically overly restrictive (Feyerabend 1962). The emphasis is placed, instead, on the merits of the new theses of incommensurability and methodological pluralism. A similar criticism of reduction involves a different move: that the deductive connection be guaranteed provided that the old, reduced theory was "corrected" beforehand (Shaffner 1967). The evolution and the structure of scientific knowledge could be neatly captured, using Schaffner's expression, by "layer-cake reduction". The terms "length" and "mass"--or the symbols $l$ and $m$--for instance, may be the same in Newtonian and relativistic mechanics, or the term "electron" the same in classical physics and quantum mechanics, or the term "atom" the same in quantum mechanics and in chemistry, or "gene" in Mendelian genetics and molecular genetics (see, for instance, Kitcher 1984). But the corresponding concepts, it is argued, are not. Concepts or words are to be understood as getting their content or meaning within a holistic or organic structure, even if the organized wholes are the theories that include them. From this point of view, different wholes, whether theories or Kuhnian paradigms, manifest degrees of conceptual incommensurability. As a result, the derived, reducing theories typically are not the allegedly reduced, older ones; and their derivation sheds no relevant insight into the relation between the original, older one and the new (Feyerabend 1962; Sklar 1967). From a historical standpoint, the positivist model collapsed the distinction between synchronic and diachronic reduction, that is, between reductive models of the structure and the evolution, or succession, of scientific theories. By contrast, historicism, as embraced by Kuhn and Feyerabend, drove a wedge between the two dimensions and rejected the linear model of scientific change in terms of accumulation and replacement. For Kuhn, replacement becomes partly continuous, partly non-cumulative change in which one world--or, less literally, one world-picture, one paradigm--replaces another (after a revolutionary episode of crisis and proliferation of alternative contenders) (Kuhn 1962). This image constitutes a form of pluralism, and, like the reductionism it is meant to replace, it can be either synchronic or diachronic. Here is where Kuhn and Feyerabend parted ways. For Kuhn, synchronic pluralism only describes the situation of crisis and revolution between paradigms. For Feyerabend, history is less monistic, and pluralism is and should remain a synchronic and diachronic feature of science and culture (Feyerabend, here, thought science and society inseparable, and followed Mill's philosophy of liberal individualism and democracy). A different kind of antireductionism addresses a more conceptual dimension, the problem of categorial reduction: Meta-theoretical categories of description and interpretation for mathematical formalisms, e.g., criteria of causality, may block full reduction. Basic interpretative concepts that are not just variables in a theory or model are not reducible to counterparts in fundamental descriptions (Cat 2000; the case of individuality in quantum physics has been discussed in Healey 1991; Redhead and Teller 1991; Auyang 1995; and, in psychology, in Block 2003). ### 3.3 Epistemic roles: From demarcation to explanation and evidence. Varieties of connective unity. Aesthetic value Unity has been considered an epistemic virtue and goal, with different modes of unification associated with roles such as demarcation, explanation and evidence. It can also be traced to synthetic cognitive tasks of categorization and reasoning--also applied in the sciences--especially through relations of similarity and difference (Cat 2022b). Demarcation. Certain models of unity, which we may call container models, attempt to demarcate science from non-science. The criteria adopted are typically methodological and normative, not descriptive. Unlike connective models, they serve a dual function of drawing up and policing a boundary that (1) encloses and endorses the sciences and (2) excludes other practices. As noted above, some demarcation projects have aimed to distinguish between natural and special sciences. The more notorious ones, however, have aimed to exclude practices and doctrines dismissed under the labels of metaphysics, pseudo-science or popular knowledge. Empirical or not, the applications of standards of epistemic purity are not merely identification or labeling exercises for the sake of carving out scientific inquiry as a natural kind or mapping out intellectual landscapes. The purpose is to establish authority and the stakes involve educational, legal and financial interests. Recent controversies include not just the teaching of creation science, but also polemics over the scientific status of, for instance, homeopathy, vaccination and models of plant neurology and climate change. The most influential demarcation criterion has been Popper's original anti-metaphysics barrier: the condition of empirical falsifiability of scientific statements. It required the logically possible relation to basic statements, linked to experience, that can prove general hypotheses to be false with certainty. For this purpose, he defended the application of a particular deductive argument, the modus tollens (Popper 1935/1951). Another demarcation criterion is explanatory unity, empirically grounded. Hempel's deductive-nomological model characterizes the scientific explanation of events as a logical argument that expresses their expectability in terms of their subsumption under an empirically testable generalization. Explanations in the historical sciences too must fit the model if they are to count as scientific. They could then be brought into the fold as bona fide scientific explanations even if they could qualify only as explanation sketches. Since their introduction, Hempel's model and its weaker versions have been challenged as neither generally applicable nor appropriate. The demarcation criterion of unity is undermined by criteria of demarcation between natural and historical sciences. For instance, historical explanations have a genealogical or narrative form, or else they require the historian's engaging problems or issuing a conceptual judgment that brings together meaningfully a set of historical facts (recent versions of such decades-old arguments are in Cleland 2002, Koster 2009, Wise 2011). According to more radical views, natural sciences such as geology and biology are historical in their contextual, causal and narrative forms; as well, Hempel's model, especially the requirement of empirically testable strict universal laws, is satisfied by neither the physical sciences nor the historical sciences, including archeology and biology (Ereshefsky 1992). A number of legal decisions have appealed to Popper's and Hempel's criteria, adding the epistemic role of peer review, publication and consensus around the sound application of methodological standards. A more recent criterion has sought a different kind of demarcation: it is comparative rather than absolute; it aims to compare science and popular science; it adopts a broader notion of the German tradition of Wissenschaften, that is, roughly of scholarly fields of research that include formal sciences, natural sciences, human sciences and the humanities; and it emphasizes the role of systematicity, with an emphasis on different forms of epistemic connectedness as weak forms of coherence and order (Hoyningen-Huene 2013). Explanation. Unity has been defended in the wake of authors such as Kant and Whewell as an epistemic criterion of explanation or at least fulfilling an explanatory role. In other words, rather than modeling unification in terms of explanation, explanation is modeled in terms of unification. A number of proposals introduce an explanatory measure in terms of the number of independent explanatory laws or phenomena conjoined in a theoretical structure. On this representation, unity contributes understanding and confirmation from the fewest basic kinds of phenomena, regardless of explanatory power in terms of derivation or argument patterns (Friedman 1974; Kitcher 1981; Kitcher 1989; Wayne 1996; within a probabilistic framework, Myrvold 2003; Sober 2003; Roche and Sober 2017; see below). A weaker position argues that unification is not explanation on the grounds that unification is simply systematization of old beliefs and operates as a criterion of theory-choice (Halonen and Hintikka 1999). The unification account of explanation has been defended within a more detailed cognitive and pragmatist approach. The key is to think of explanations as question-answer episodes involving four elements: the explanation-seeking question about $P, P$?, the cognitive state $C$ of the questioner/agent for whom $P$ calls for explanation, the answer $A$, and the cognitive state $C+A$ in which the need for explanation of $P$ has disappeared. A related account models unity in the cognitive state in terms of the comparative increase of coherence and elimination of spurious unity--such as circularity or redundancy (Schurz 1999). Unification is also based on information-theoretic transfer or inference relations. Unification of hypotheses is only a virtue if it unifies data. The last two conditions imply that unification yields also empirical confirmation. Explanations are global increases in unification in the cognitive state of the cognitive agent (Schurz 1999; Schurz and Lambert 1994). The unification-explanation link can be defended on the grounds that laws make unifying similarity expectable (hence Hempel-explanatory), and this similarity becomes the content of a new belief (Weber and Van Dyck 2002 contra Halonen and Hintikka 1999). Unification is not the mere systematization of old beliefs. Contra Schurz, they argue that scientific explanation is provided by novel understanding of facts and the satisfaction of our curiosity (Weber and Van Dyck 2002 contra Schurz 1999). In this sense, causal explanations, for instance, are genuinely explanatory and do not require an increase of unification. A contextualist and pluralist account argues that understanding is a legitimate aim of science that is pragmatic and not necessarily formal, or a subjective psychological by-product of explanation (De Regt and Dieks 2005). In this view explanatory understanding is variable and can have diverse forms, such as causal-mechanical and unification, without conflict (De Regt and Dieks 2005). In the same spirit, Salmon linked unification to the epistemic virtue or goal of explanation and distinguished between unification and causal-mechanical explanation as forms of scientific explanatory understanding (Salmon 1998). Explanation may also provide unification of different fields of research through explanatory dependence so that one uses the other's results to explain one's own (Kincaid 1997). The views on scientific explanation have evolved away from the formal and cognitive accounts of the epistemic categories. Accordingly, the source of understanding provided by scientific explanations has been misidentified according to some (Barnes 1992). The genuine source for important, but not all, cases lies in causal explanation or causal mechanism (Cartwright 1983; Cartwright 1989; see also Glennan 1996; Craver 2007). Mechanistic models of explanation have become entrenched in philosophical accounts of the life sciences (Darden 2006; Craven 2007). As an epistemic virtue, the role of unification has been traced to the causal form of the explanation, for instance, in statistical regularities (Schurz 2015). The challenge extends to the alleged extensional link between explanation on the one hand, and truth and universality on the other (Cartwright 1983; Dupre 1993; Woodward 2003). In this sense, explanatory unity, which rests on metaphysical assumptions about components and their properties, also involves a form of ontological or metaphysical unity. (For a methodological criticism of external, metaphysical perspectives, see Ruphy 2016.) Similar criticisms extend to the traditionally formalist arguments in physics about fundamental levels; there, unification fails to yield explanation in the formal scheme based on laws and their symmetries. Unification and explanation conflict on the grounds that in biology and physics only causal mechanical explanations answering why-questions yield understanding of the connections that contribute to "true unification" (Morrison 2000;[2] Morrison's choice of standard for evaluating the epistemic accounts of unity and explanation and her focus on systematic theoretical connections without reduction has not been without critics, e.g., Wayne 2002; Plutynski 2005; Karaca 2012).[3] Methodology. Unity has long been understood as a methodological principle, primarily, but not exclusively, in reductionist versions (e.g., for the case of biology, see Wimsatt 1976, 2006). This is different from the case of unity through methodological prescriptions. One methodological criterion appeals to the epistemic virtues of simplicity or parsimony, whether epistemological or ontological (Sober 2003). As a formal probabilistic principle of curve-fitting or average predictive accuracy, the relevance of unity is objective. Unity plays the role of an empirical background theory. The methodological role of unification may track scientific progress. Unlike in the case of the explanatory role of unification presented above, this account of progress and interdisciplinarity relies on a unifying role of explanation: as in evo-devo in biology, unification is a process of advancement in which two fields of research are in the process of unification through mutual explanatory relevance, that is, when results in one field are required to pose and address questions in the other, raising explananda and explanations (Nathan 2017). Heuristic dependence involves one influencing, accommodating, contributing to and addressing the other's research questions (for examples relating chemistry and physics see Cat and Best 2023). Evidence. As in the relation of unification to explanation, unification is considered an epistemic criterion of evidence in support of the unified account (for a non-probabilistic account of the relation between unification and confirmation, see Schurz 1999). The resulting evidence and demonstration may be called synthetic evidence and demonstration. Synthetic evidence may be the outcome of synthetic modes of reasoning that rely on assumptions of similarity and difference, for instance in cases of robustness, cross-checking and meta-analysis (Cat 2022b). Like probabilistic models of explanation, recent formal discussions of unity and coherence within the framework of Bayesianism place unity in evidentiary reasoning (Forster and Sober 1994, sect. 7; Schurz and Lambert 2005 is also a formal model, with an algebraic approach). More generally, the probabilistic framework articulates formal characterizations of unity and introduces its role in evaluations of evidence. A criterion of unity defended for its epistemic virtue in relation to evidence is simplicity or parsimony (Sober 2013, 2016). Comparatively speaking, simpler hypotheses, models or theories present a higher likelihood of truth, empirical support and accurate prediction. From a methodological standpoint, however, appeals to parsimony might not be sufficient. Moreover, the connection between unity as parsimony and likelihood is not interest-relative, at least in the way that the connection between unity and explanation is (Sober 2003; Forster and Sober 1994; Sober 2013, 2016). On the Bayesian approach, the rational comparison and acceptance of probabilistic beliefs in the light of empirical data is constrained by Bayes' Theorem for conditional probabilities (where $h$ and $d$ are the hypothesis and the data respectively): \[ \P(h \mid d) = \frac{\P(d \mid h) \cdot \P(h)}{P(d)} \] One explicit Bayesian account of unification as an epistemic, methodological virtue, has introduced the following measure of unity: a hypothesis $h$ unifies phenomena $p$ and $q$ to the degree that given $h, p$ is statistically/probabilistically relevant to (or correlated with) $q$ (Myrvold 2003; for a probabilistically equivalent measure of unity in Bayesian terms see McGrew 2003; on the equivalence, Schupbach 2005). This measure of unity has been criticized as neither necessary nor sufficient (Lange 2004; Lange's criticism assumes the unification-explanation link; in a rebuttal, Schupbach 2005 rejects this and other assumptions behind Lange's criticism). In a recent development, Myrvold argues for mutual information unification, i.e., that hypotheses are said to be supported by their ability to increase the amount of what he calls the mutual information of the set of evidence statements (see Myrvold 2017). The explanatory unification contributed by hypotheses about common causes is an instance of the information condition. Evidentiary unification may contribute to the unification of different fields of research in the form of evidentiary dependence: Evidentiary dependence involves the appeal to the other's results in the evaluation of one's own (Kincaid 1997). Aesthetic value. Finally, epistemic values of unity may rely on subsidiary considerations of aesthetic value. Nevertheless, consideration of beauty, elegance or harmony may also provide autonomous grounds for adopting or pursuing varieties of unification in terms of simplicity and patterns of order (regularity of specific relations) (McAllister 1996; Glynn 2010; Orrell 2012). Whether aesthetic judgments have any epistemic import depends on metaphysical, cognitive or pragmatic assumptions. Unification without reduction. Reduction is not the sole standard of unity, and models of unification without reduction have proliferated. In addition, such models introduce new units of analysis. An early influential account centers around the notion of interfield theories (Darden and Maull 1977; Darden 2006). The orthodox central place of theories as the unit of scientific knowledge is replaced by that of fields. Examples of such fields are genetics, biochemistry and cytology. Different levels of organization correspond in this view to different fields: Fields are individuated intellectually by a focal problem, a domain of facts related to the problem, explanatory goals, methods and a vocabulary. Fields import and transform terms and concepts from others. The model is based on the idea that theories and disciplines do not match neat levels of organization within a hierarchy; rather, many of them in their scope and development cut across different such levels. Reduction is a relation between theories within a field, not across fields. Interdependence and hybridity. In general, the higher-level theories (for instance, cell physiology) and the lower-level theories (for instance, biochemistry) are ontologically and epistemologically interdependent on matters of informational content and evidential relevance; one cannot be developed without the other (Kincaid 1996; Kincaid 1997; Wimsatt 1976; Spector 1977). The interaction between fields (through researchers' judgments and borrowings) may provide enabling conditions for subsequent interactions. For instance, Maxwell's adoption of statistical techniques in color research enabled the introduction of similar ideas from social statistics in his research on reductive molecular theories of gases. The reduction, in turn, enabled experimental evidence from chemistry and acoustics; similarly, different chemical and spectroscopic bases for colors provided chemical evidence in color research (Cat 2014). The emergence and development of hybrid disciplines and theories is another instance of non-reductive cooperation or interaction between sciences. Noted above is the post-war emergence of interdisciplinary areas of research: the so-called hyphenated sciences such as neuro-acoustics, radioastronomy, biophysics, etc. (Klein 1990, Galison 1997) On a smaller scale, in the domain of, for instance, physics, one can find semiclassical models in quantum physics or models developed around phenomena where the limiting reduction relations are singular or catastrophic, such as caustic optics and quantum chaos (Cat 1998; Batterman 2002; Belot 2005). Such semiclassical explanatory models have not found successful quantum substitutes and have placed structural explanations at the heart of the relation between classical and quantum physics (Bokulich 2008). The general form of pervasive cases of emergence has been characterized with the notion of contextual emergence (Bishop and Atmanspacher 2006) where properties, behaviors and their laws on a restricted, lower-level, single-scale domain are necessary but not sufficient for the properties and behaviors of another, e.g., higher-level one, not even of itself. The latter are also determined by contingent contexts (contingent features of the state space of the relevant system). The interstitial formation of more or less stable small-scale syntheses and cross-boundary "alliances" has been common in most sciences since the early twentieth century. Indeed, it is crucial to development in model building and growing empirical relevance in fields ranging anywhere from biochemistry to cell ecology, or from econophysics to thermodynamical cosmology. Similar cases can be found in chemistry and the biomedical sciences. Conceptual unity. The conceptual dimension of cross-cutting has been developed in connection with the possibility of cross-cutting natural kinds that challenges taxonomical monism. Categories of taxonomy and domains of description are interest-relative, as are rationality and objectivity (Khalidi 1998; his view shares positions and attitudes with Longino 1989; Elgin 1996, 1997). Cross-cutting taxonomic systems, then, are not conceptually inconsistent or inapplicable. Both the interest-relativity and hybridity feature prominently in the context of ontological pluralism (see below). Another, more general, unifying element of this kind is Holton's notion of themata. Themata are conceptual values that are a priori yet contingent (both individual and social). They are forming and organizing presuppositions that factor centrally in the evolution of the science and include continuity/discontinuity, harmony, quantification, symmetry, conservation, mechanicism and hierarchy (Holton 1973). Unity of some kind is itself a thematic element. A more complex and comprehensive unit of organized scientific practice is the notion of the various styles of reasoning, such as statistical, analogical modeling, taxonomical, genetic/genealogical or laboratory styles; each is a cluster of epistemic standards, questions, tools, ontology, and self-authenticating or stabilizing protocols (Hacking 1996; see below for the relevance of this account of a priori elements to claims of global disunity--the account shares distinctive features of Kuhn's notion of paradigm). Another model of non-reductive unification is historical and diachronic: it emphasizes the genealogical and historical identity of disciplines, which has become complex through interaction. The interaction extends to relations between specific sciences, philosophy and philosophy of science (Hull 1988). Hull has endorsed an image of science as a process, modeling historical unity after a Darwinian-style pattern of evolution (developing an earlier suggestion by Popper). Part of the account is the idea of disciplines as evolutionary historical individuals, which can be revised with the help of more recent ideas of biological individuality: hybrid unity as an external model of unity as integration or coordination of individual disciplines and disciplinary projects, e.g., characterized by a form of occurrence, evolution or development whose tracking and identification involves a conjunction with other disciplines, projects and domains of resources, from within science or outside science. This diachronic perspective can accommodate models of discovery, in which genealogical unity integrates a variety of resources that can be both theoretical and applied, or scientific and non-scientific (an example, from physics, the discovery of superconductivity, can be found in Holton, Chang and Jurkowitz 1996). Some models of unity below provide further examples. A generalization of the notion of interfield theories is the idea that unity is interconnection: Fields are unified theoretically and practically (Grantham 2004). This is an extension of the original modes of unity or identity that single out individual disciplines. Theoretical unification involves conceptual, ontological and explanatory relations. Practical unification involves heuristic dependence, confirmational dependence and methodological integration. The social dimension of the epistemology of scientific disciplines relies on institutional unity. With regard to disciplines as professions, this kind of unity has rested on institutional arrangements such as professional organizations for self-identification and self-regulation, university mechanisms of growth and reproduction through certification, funding and training, and communication and record through journals. Many examples of unity without reduction are local rather than global. These are not merely a phase in a global and linear project or tradition of unification (or integration), and they are typically focused on science as a human activity. From that standpoint, unification is typically understood or advocated as a piecemeal description and strategy of collaboration (see Klein 1990 on the distinction between global integration and local interdisciplinarity). Cases are restricted to specific models, phenomena or situations. Material unity. A more recent approach to the connection between different research areas has focused on a material level of scientific practice, with attention to the use of instruments and other material objects (Galison 1997; Bowker and Star 1999). For instance, the material unity of natural philosophy in the sixteenth and seventeenth centuries relied on the circulation, transformation and application of objects in their concrete and abstract representations (Bertoloni-Meli 2006). The latter correspond to the imaginary systems and their representations, which we call models. The evolution of objects and images across different theories and experiments and their developments in nineteenth-century natural philosophy provide a historical model of scientific development, but the approach is not meant to illustrate reductive materialism, since the same objects and models work and are perceived as vehicles for abstract ideas, institutions, cultures, etc., or are prompted by them. On one view, objects are regarded as elements in so-called trading zones (see below) with shifting meanings in the evolution of twentieth-century physics, such as with the cloud chamber which was first relevant to meteorology and next to particle physics (Galison 1997). Alternatively, material objects have been given the status of boundary objects, which provide the opportunity for experts from different fields to collaborate through their respective understandings of the system in question and their respective goals (Bowker and Star 1999). Graphic unity. At the concrete perceptual level, recent accounts emphasize the role of visual representations in the sciences and suggest what may be called graphic unification of the sciences. Their cognitive roles, methodological and rhetorical, include establishing and disseminating facts and their so-called virtual witnessing, revealing empirical relations, testing their fit with available patterns of more abstract theoretical relations (theoretical integration), suggesting new ones, aiding in computations, serving as aesthetic devices etc. But these uses are not homogeneous across different sciences and make visible disciplinary differences. We may equally speak of graphic pluralism. The rates in the use of diagrams in research publications appear to vary along the hard-soft axis of pyramidal hierarchy, from physics, chemistry, biology, psychology, economics and sociology, and political science (Smith et al. 2000). The highest use can be found in physics, intuitively identified by the highest degree of hardness understood as consensus, codification, theoretical integration and factual stability, to the highest interpretive and instability of results. Similarly, the same variation occurs among sub-disciplines within each discipline. The kinds of images and their contents also vary across disciplines and within disciplines, ranging from hand-made images of particular specimens to hand-made or mechanically generated images of particulars standing in for types, to schematic images of geometric patterns in space or time, or to abstract diagrams representing quantitative relations. Importantly, graphic tools circulate like other cognitive tools between areas of research that they in turn connect (Galison 1997; Daston and Galison 2007; Lopes 2009; see also Lynch and Woolgar 1990; Baigrie 1996; Jones and Galison 1998; Galison 1997; Cat 2014; Kaiser 2005). Disciplinary unity and collaboration. The relation between disciplines or fields of research has often been tracked by relations between their respective theories or epistemic products. But disciplines constitute broader and richer units of analysis of connections in the sciences characterized, for instance, by their domain of inquiry, cognitive tools and social structure (Bechtel 1987). Unification of disciplines, in that sense, has been variously categorized, for instance, as interdisciplinary, multidisciplinary, crossdisciplinary or transdisciplinary (Klein 1990; Graff 2005; Kellert 2008; Repko 2012). It might involve a researcher borrowing from different disciplines or the collaboration of different researchers. Neither modality of connection amounts to a straightforward generalization of, or reduction to, any single discipline, theory, etc. In either case, the strategic development is typically defended for its heuristic problem-solving or innovative powers, as it is prompted by a problem that is considered complex in that it does not arise or cannot be fully treated within the purview of one specific discipline unified or individuated around some potentially non-unique set of elements such as scope of empirical phenomena, rules, standards, techniques, conceptual and material tools, aims, social institutions, etc. Indicators of disciplinary unity may vary (Kuhn 1962; Klein 1990; Kellert 2008). Interdisciplinary research or collaboration creates a new discipline or project, such as interfield research, often leaving the existence of the original ones intact. Multidisciplinary work involves the juxtaposition of the treatments and aims of the different disciplines involved in addressing a common problem. Crossdisciplinary work involves borrowing resources from one discipline to serve the aims of a project in another. Transdisciplinary work is a synthetic creation that encompasses work from different disciplines (Klein 1990; Kellert 2008; Brigandt 2010; Hoffmann, Schmidt and Nersessian 2012; Osbeck et al. 2011; Repko 2012). These different modes of synthesis or connection are not mutually exclusive. Models of interdisciplinary cooperation and their corresponding outcomes are often described using metaphors of different kinds: cartographic (domains, boundaries, trading zone, etc.), linguistic (pidgin language, communication, translation, etc.), architectural (building blocks, tiles, etc.), socio-political (imperialism, hierarchy, republic, orchestration, negotiation, coordination, cooperation, etc.) or embodied (cross-training). Each selectively highlights and neglects different aspects of scientific practice and properties of scientific products. Cartographic and architectural images, for instance, focus on spatial and static synchronic relations and simply connected, compatible elements. Socio-political and embodied images emphasize activity and non-propositional elements (Kellert 2008 defends the image of cross-training). In this context, methodological unity often takes the form of borrowing standards and techniques for the application of formal and empirical methods. They range from calculational techniques and tools for theoretical modeling and simulation of phenomena to techniques for modeling of data, using instruments and conducting experiments (e.g., the culture of field experiments and, more recently, randomized control trials across natural and social sciences). A key element of scientific practice, often ignored by philosophical analysis, is expertise. As part of different forms of methodological unity, it is key to the acceptance and successful appropriation of techniques. Recent accounts of multidisciplinary collaboration as a human activity have focused on the dynamics of integrating different kinds of expertise around common systems or goals of research (Collins and Evans 2007; Gorman 2002). The same perspective can accommodate the recent interest in so-called mixed methods, e.g., different forms of integration of quantitative and qualitative methods and approaches in the social sciences (but mixed-method approaches do not typically involve mixed disciplines). As teamwork and collaboration within and between research units has steadily increased, so has the degree of specialization (Wutchy et al. 2007). Between both trends, new reconfigurations keep forming with different purposes and types of institutional expression and support; for example, STEM, Earth sciences, physical sciences, mind-brain sciences, etc., in addition to hybrid fields mentioned above. Yet, in educational research and funding, disciplinarity has acquired new normative purchase in opposite directions: while funding agencies and universities promote interdisciplinarity, university administrations encourage both disciplinary competition and more versatile adisciplinarity (Griffiths 2022). In the face of defenses of different forms of interdisciplinarity (Graff 2015), there is also a renewed attention to the critical value of autonomy of disciplines, or disciplinarity, as the key resource in division of labor and collaboration alike (Jacobs 2013). The social epistemology of interdisciplinarity is complex. It develops both top-down from managers and bottom-up from practitioners (Maki 2016; Maki and MacLeod 2016), relying on a variety of kinds of interactions with heuristic and normative dimensions (Boyer-Kassem et al. 2018). More generally, collaborative work arguably provides epistemic, ethical and instrumental advantages, e.g., more comprehensive relevant understanding of complex problems, expanded justice to interests of direct knowledge users and increased resources (Laursen et al. 2021). Yet, collaboration relies on a plurality of values or norms that may lead to conflicts such as a paralyzing practical incoherence of guiding values and moral incoherence and problematic forms of oppression (Laursen et al. 2021; see more on the limits of pluralism below). Empirical work in sociology and cognitive psychology on scientific collaboration has led to a broader perspective, including a number of dimensions of interdisciplinary cooperation involving identification of conflicts and the setting of sufficient so-called common ground integrators. These include shared (pre-existing, revised and newly developed) concepts, terminology, standards, techniques, aims, information, tools, expertise, skills (abstract, dialectical, creative and holistic thinking), cognitive and social ethos (curiosity, tolerance, flexibility, humility, receptivity, reflexivity, honesty, team-play) social interaction, institutional structures and geography (Cummings and Kiesler 2005; Klein 1990; Kockelmans 1979; Repko 2012). Sociological studies of scientific collaboration can in principle place the connective models of unity within the more general scope of social epistemology, for instance, in relation to distributive cognition (beyond the focus on strategies of consensus within communities). The broad and dynamical approach to processes of interdisciplinary integration may effectively be understood to describe the production of different sorts and degrees of epistemic emergence. The integrated accounts require shared (old or new) assumptions and may involve a case of ontological integration, for instance in causal models. Suggested kinds of interdisciplinary causal-model integration are the following: sequential causal order in a process or mechanism cutting across disciplinary divides; horizontal parallel integration of different causal models of different elements of a complex phenomenon; horizontal joint causal model of the same effect; and vertical or cross-level causal integration (see emergent or top-down causality, below) (Repko 2012; Kockelmans 1979). The study of the social epistemology of interdisciplinary and collaborative research has been carried out and proposed from different perspectives that illuminate different dimensions and implications. These include historical typology (Klein 1990), formal modeling (Boyer-Kassem et al. 2018), ethnography (Nersessian 2022), ethical, scientometrics and multidimensional philosophical perspectives (Maki 2016). Other approaches aim to design and offer tools that facilitate collaboration such as the Toolbox Dialogue Initiative (Hubbs et al. 2020). The different forms of plurality and connection also ultimately inform the organization of science in the social and political terms of diversity and democracy (Longino 1998, 2001). In terms of cooperation and coordination, unity, in this sense, cannot be reduced to consensus (Rescher 1993; van Bouwel 2009a; Repko 2012; Hoffmann, Schmidt and Nersessian 2012.) A general model of local interconnection, which has acquired widespread attention and application in different sciences, is the anthropological model of trading zone, where hybrid languages and meanings are developed that allow for interaction without straightforward extension of any party's original language or framework (Galison 1997). Galison has applied this kind of anthropological analysis to the subcultures of experimentation. This strategy aims to explain the strength, coherence and continuity of science in terms of local coordinations of intercalated levels of symbolic procedures and meanings, instruments and arguments. At the experimental level, instruments, as found objects, acquire new meanings, developments and uses as they bridge over the transitions between theories, observations or theory-laden observations. Instruments and experimental projects in the case of Big Science also bring together, synchronically and interactively, the skills, standards and other resources from different communities, and they change each in turn (on interdisciplinary experimentation see also Osbeck et al. 2011). Patterns of laboratory research are shared by the different sciences, including not just instruments but the general strategies of reconfiguration of human researchers and the natural entities researched (Knorr-Cetina 1992). This includes statistical standards (e.g., statistical significance) and ideals of replication. At the same time, attention has been paid to the different ways in which experimental approaches differ among the sciences (Knorr-Cetina 1992; Guala 2005; Weber 2005) as well as to how they have been transferred (e.g., field experiments and randomized control trials) or integrated (e.g., mixed methods combining quantitative and qualitative techniques). Interdisciplinary research has been claimed to revolve shared boundary objects (Gorman 2002) and to yield evidentiary and heuristic integration through cooperation (Kincaid 1997). Successful interdisciplinary research, however, does not seem to require integration, e.g., evolutionary game theory in economics and biology (Grune-Yanoff 2016). Heuristic cooperation has also led to new stable disciplines through stronger integrative forms of problem solving. In the life sciences, for instance, computational, mathematical and engineering techniques for modeling and data collection in molecular biology have led to integrative systems biology (MacLeod and Nersessian 2016; Nersessian 2022). Similarly, constraints and resources for problem-solving in physics, chemistry and biology led to the development of quantum chemistry (Gavroglu and Simoes 2012; see also the case of nuclear chemistry in Cat and Best 2023). ## 4. Ontological unities ### 4.1 Ontological unities and reduction Since Nagel's influential model of reduction by derivation, most discussions of the unity of science have been cast in terms of reductions between concepts and the entities they describe, and between theories incorporating the descriptive concepts. Ontological unity is expressed by a preferred set of such ontological units. In this regard, it should be noted that the selection of units typically relies on assumptions about classification or natural order such as commitments to natural kinds or a hierarchy of levels. In terms of concepts featured in preferred descriptions, explanatory or not, reduction endorses taxonomical monism: a privileged set of fundamental kinds of things. These privileged kinds are often known as so-called natural kinds; although, as it has been argued, monism does not entail realism (Slater 2005) and the notion admits of multiple interpretations, ranging from the more conventionalist to the more essentialist (Tahko 2021; see a critique in Cat 2022a). Natural kindness has further been debated in terms, for instance, of the contrast between so-called cluster and causal theories. Connectedly, pluralism does not entail anti-realism (Dupre 1993), nor does realism entail monism or essentialism (Khalidi 2021). Without additional metaphysical assumptions, the fundamental units are ambiguous with respect to their status as either entity or property. Reduction may determine the fundamental kinds or level through the analysis of entities. A distinctive ontological model is as follows. The hierarchy of levels of reduction is fixed by part-whole relations. The levels of aggregation of entities run all the way down to atomic particles and field parts, rendering microphysics the fundamental science (Gillett 2016). The focus of recent accounts has been placed on the relation between the causal powers at different levels and how lower-level entities and powers determine higher-level ones. Discussions have centered on whether the relation is of identity between types or tokens of things (Thalos 2013; Gillette 2016; Wilson 2021; see more below) or even if causal powers are the relevant unit of analysis (Thalos 2013). A classic reference to this compositional type of account is Oppenheim and Putnam's "The Unity of Science as a Working Hypothesis" (Oppenheim and Putnam 1958; Oppenheim and Hempel had worked in the 1930s on taxonomy and typology, a question of broad intellectual, social and political relevance in Germany at the time). Oppenheim and Putnam intended to articulate an idea of science as a reductive unity of concepts and laws reduced to those of the most elementary elements. They also defended it as an empirical hypothesis--not an a priori ideal, project or precondition--about science. Moreover, they claimed that its evolution manifested a trend in that unified direction out of the smallest entities and lowest levels of aggregation. In an important sense, the evolution of science recapitulates, in the reverse, the evolution of matter, from aggregates of elementary particles to the formation of complex organisms and species (we find a similar assumption in Weinberg's downward arrow of explanation). Unity, then, is manifested not just in mereological form, but also diachronically, genealogically or historically. A weaker form of ontological reduction advocated for the biomedical sciences with the causal notion of partial reductions: explanations of localized scope (focused on parts of higher-level systems only) laying out a causal mechanism connecting different levels in the hierarchy of composition and organization (Schaffner 1993; Schaffner 2006; Scerri has similarly discussed degrees of reduction in Scerri 1994). An extensional, domain-relative approach introduces the distinction between "domain preserving" and "domain combining" reductions. Domain-preserving reductions are intra-level reductions and occur between $T\_1$ and its predecessor $T\_2$. In this parlance, however, $T\_2$ "reduces" to $T\_1$. This notion of "reduction" does not refer to any relation of explanation (Nickles 1973). The claim that reduction, as a relation of explanation, needs to be a relation between theories or even involve any theory has also been challenged. One such challenge focuses on "inter-level" explanations in the form of compositional redescription and causal mechanisms (Wimsatt 1976). The role of biconditionals or even Schaffner-type identities, as factual relations, is heuristic (Wimsatt 1976). The heuristic value extends to the preservation of the higher-level, reduced concepts, especially for cognitive and pragmatic reasons, including reasons of empirical evidence. This amounts to rejecting the structural, formal approach to unity and reductionism favored by the logical-positivist tradition. Reductionism is another example of the functional, purposive nature of scientific practice. The metaphysical view that follows is a pragmatic and non-eliminative realism (Wimsatt 2006). As a heuristic, this kind of non-eliminative pragmatic reductionism is a complex stance. It is, across levels, integrative and intransitive, compositional, mechanistic and functionally localized, approximative and abstractive. It is bound to adopting false idealizations, focusing on regularities and stable common behavior, circumstances and properties. It is also constrained in its rational calculations and methods, tool-binding and problem-relative. The heuristic value of eliminative, inter-level reductions has been defended as well (Poirier 2006). The appeal to formal laws and deductive relations is dropped for sets of concepts or vocabularies in the replacement analysis (Spector 1978). This approach allows for talk of entity reduction or branch reduction, and even direct theory replacement, without the operation of laws, and it circumvents vexing difficulties raised by bridge principles and the deductive derivability condition (self-reduction, infinite regress, etc.). Formal relations only guarantee, but do not define, the reduction relation. Replacement functions are meta-linguistic statements. As Sellars argued in the case of explanation, this account distinguishes between reduction and the testing for reduction, and it highlights the role of derivations in both. Finally, replacement can be in practice or in theory. Replacement in practice does not advocate elimination of the reduced or replaced entities or concepts (Spector 1978). As indicated above, reductive models and associated organization of scientific theories and disciplines assume an epistemic hierarchy grounded on an ontological hierarchy of levels of organization of entities, properties or phenomena, from societies down to cells, molecules and subatomic elements. Patterns of behavior of entities at lower, more fundamental levels are considered more general, stable and explanatory. Levels forming a hierarchy are discrete and stratified so that entities at level n strongly depend on entities at level $n-1.$ The different levels may be distinguished in different ways: by differences in scale, by a relation of realization and, especially, by a relation of composition such that entities at level n are considered composed of, or decomposable into, entities at level $n-1.$ This assumption has been the target of criticism (Thalos 2013; Potochnik 2017). Neither scales, realization nor decomposition can fix an absolute hierarchy of levels. Critics have noted, for instance, that entities may participate in causal interactions at multiple levels and across levels in physical models (Thalos 2013) and across biological and neurobiological models (Haug 2010; Eronen 2013). In addition, the compartmentalization of theories and their concepts or vocabulary into levels neglects the existence of empirically meaningful and causally explanatory relations between entities or properties at different levels. If they are neglected as theoretical knowledge and left as only bridge principles, the possibility of completeness of knowledge is jeopardized. Maximizing completeness of knowledge requires a descriptive unity of all phenomena at all levels and anything between these levels. Any bounded region or body of knowledge neglecting such cross-boundary interactions is radically incomplete, and not just confirmationally or evidentially so; we may refer to this problem as the problem of cross-boundary incompleteness at either intra-level or horizontal incompleteness and, on a hierarchy, the problem of inter-level or vertical incompleteness (Kincaid 1997; Cat 1998). If levels cannot track, for instance, same-level causal relations, either their causal explanatory relevance is derivative and contextual (Craver 2007), their role is cognitive and pragmatic as in the role as conceptual coordinate systems or, more radically, altogether dispensable (Thalos 2013; Potochnik 2017). As a result, they fail to fix a corresponding hierarchy of fields and subfields of scientific research defended by reductionist accounts. The most radical form of reduction as replacement is often called eliminativism. The position has made a considerable impact in philosophy of psychology and philosophy of mind (Churchland 1981; Churchland 1986). On this view the vocabulary of the reducing theories (neurobiology) eliminates and replaces that of the reduced ones (psychology), leaving no substantive relation between them (which is only a replacement rule) (see also eliminative materialism). From a semantic standpoint, one may distinguish different kinds of reduction in terms of four criteria, two epistemological and two ontological: fundamentalism, approximation, abstract hierarchy and spatial hierarchy. Fundamentalism implies that the features of a system can be explained in terms only of factors and rules from another realm. Abstract hierarchy is the assumption that the representation of a system involves a hierarchy of levels of organization with the explanatory factors being located at the lower levels. Spatial hierarchy is a special case of abstract hierarchy in which the criterion of hierarchical relation is a spatial part-whole or containment relation. Strong reduction satisfies the three "substantive" criteria, whereas weak reduction only satisfies fundamentalism. Approximate reductions--strong and hierarchical--are those which satisfy the criterion of fundamentalism only approximately (Sarkar 1998; the merit of Sarkar's proposal resides in its systematic attention to hierarchical conditions and, more originally, to different conditions of approximation; see also Ramsey 1995; Lange 1995). The semantic turn extends to more recent notion of models that do not fall under the strict semantic or model-theoretic notion of mathematical structures (Giere 1999; Morgan and Morrison 1999). This is a more flexible framework about relevant formal relations and the scope of relevant empirical situations. It is implicitly or explicitly adopted by most accounts of unity without reduction. One may add the primacy of temporal representation and temporal parts, temporal hierarchy or temporal compositionality, first emphasized by Oppenheim and Putnam as a model of genealogical or diachronic unity. This framework applies to processes both of evolution and development (a more recent version is in McGivern 2008 and in Love and Hutteman 2011). The shift in the accounts of scientific theory from syntactic to semantic approaches has changed conceptual perspectives and, accordingly, formulations and evaluations of reductive relations and reductionism. However, examples of the semantic approach focusing on mathematical structures and satisfaction of set-theoretic relations have focused on syntactic features--including the axiomatic form of a theory--in the discussion of reduction (Sarkar 1998; da Costa and French 2003). In this sense, the structuralist approach can be construed as a neo-Nagelian account, while an alternative line of research has championed the more traditional structuralist semantic approach (Balzer and Moulines 1996; Moulines 2006; Ruttkamp 2000; Ruttkamp and Heidema 2005). ### 4.2 Ontological unities and antireductionism From the opposite direction, arguments concerning new concepts such as multiple realizability and supervenience, introduced by Putnam, Kim, Fodor and others, have led to talk of higher-level functionalism, a distinction between type-type and token-token reductions and the examination of its implications. The concepts of emergence, supervenience and downward causation are related metaphysical tools for generating and evaluating proposals about unity and reduction in the sciences. This literature has enjoyed its chief sources and developments in general metaphysics and in philosophy of mind and psychology (Davidson 1969; Putnam 1975; Fodor 1975; Kim 1993). Supervenience, first introduced by Davidson in discussions of mental properties, is the notion that a system with properties on one level is composed of entities on a lower level and that its properties are determined by the properties of the lower-level entities or states. The relation of determination is that no changes at the higher-level occur without changes at the lower level. Like token-reductionism, supervenience has been adopted by many as the poor man's reductionism (see the entry on supervenience). A different case for the autonomy of the macrolevel is based on the notion of multiple supervenience (Kincaid 1997; Meyering 2000). The autonomy of the special sciences from physics has been defended in terms of a distinction between type-physicalism and token-physicalism (Fodor 1974; Fodor countered Oppenheim and Putnam's hypothesis under the rubric "the disunity of science"; see the entry on physicalism). The key logical assumption is the type-token distinction, whereby types are realized by more specific tokens, e.g., the type "animal" is instantiated by different species, the type "tiger" or "electron" can be instantiated by multiple individual token tigers and electrons. Type-physicalism is characterized by a type-type identity between the predicates/properties in the laws of the special sciences and those of physics. By contrast, token-physicalism is based on the token-token identity between the predicates/properties of the special sciences and those of physics; every event under a special law falls under a law of physics and bridge laws express contingent token-identities between events. Token-physicalism operates as a demarcation criterion for materialism. Fodor argued that the predicates of the special sciences correspond to infinite or open-ended disjunctions of physical predicates, and these disjunctions do not constitute natural kinds identified by an associated law. Token-physicalism is the only alternative. All special kinds of events are physical, but the special sciences are not physics (for criticisms based on the presuppositions in Fodor's argument, see Sober 1999). The denial of remedial, weaker forms of reductionism is the basis for the concept of emergence (Humphreys 1997; Bedau and Humphreys 2008; Wilson 2021). Different accounts have attempted to articulate the idea of a whole being different from or more than the mere sum of its parts (see the entry on emergent properties). Emergence has been described beyond logical relations, synchronically as an ontological property and diachronically as a material process of fusion, in which the powers of the separate constituents lose their separate existence and effects (Humphreys 1997). This concept has been widely applied in discussions of complexity. Unlike the earliest antireductionist models of complexity in terms of holism and cybernetic properties, more recent approaches track the role of constituent parts (Simon 1996). Weak emergence has been opposed to nominal and strong forms of emergence. The nominal kind simply represents that some macro-properties cannot be properties of micro-constituents. The strong form is based on supervenience and irreducibility, with a role for the occurrence of autonomous downwards causation upon any constituents (see below). Weak emergence is linked to processes stemming from the states and powers of constituents, with a reductive notion of downwards causation of the system as a resultant of constituents' effects (Wilson 2021); however, the connection is not a matter of Nagelian formal derivation, but of so-called universality classes (Batterman 2002; Thalos 2013), self-organization (Mitchell 2012) and implementation through computational aggregation, compression and iteration. Weak emergence, then, can be defined in terms of simulation: a macro-property, state or fact is weakly emergent if and only if it can be derived from its macro-constituents only by simulation (Bedau and Humphreys 2008) The denial of remedial, weaker forms of reductionism is the basis for the concept of emergence (Humphreys 1997; Bedau 2008); see the entry on simulations in science). Computational models of emergence or complexity straddle the boundary between formal epistemology and ontology. They are based on simulations of chaotic dynamical processes such as cellular automata (Wolfram 1984, 2002). Their supposed superiority to combinatorial models based on aggregative functions of parts of wholes does not lack defenders (Crutchfield 1994; Crutchfield and Hanson 1997; Humphreys 2004, 2007, 2008; Humphreys and Huneman 2008; Huneman 2008a, 2008b, 2010). Connected to the concept of emergence is top-down or downward causation, which captures the autonomous and genuine causal power of higher-level entities or states, especially upon lower-level ones. The most extreme and most controversial version includes a violation of laws that regulate the lower level (Meehl and Sellars 1956; Campbell 1974). Weaker forms require compatibility with the microlaws (for a brief survey and discussion see Robinson 2005; on downward causation without top-down causes, see Craver and Bechtel 2007; Bishop 2012). The very concept has become the subject of some interdisciplinary interest in the sciences (Ellis, Noble and O'Connor 2012). Another general argument for the autonomy of the macrolevel in the form of non-reductive materialism has been a cognitive type of functionalism, namely, cognitive pragmatism (Van Gulick 1992). This account links ontology to epistemology. It discusses four pragmatic dimensions of representations: the nature of the causal interaction between theory-user and the theory, the nature of the goals to the realization of which the theory can contribute, the role of indexical elements in fixing representational content, and differences in the individuating principles applied by the theory to its types (Wimsatt and Spector's arguments above are of this kind). A more ontologically substantive account of functional reduction is Ramsey's bottom-up construction by reduction: transformation reductions streamline formulations of theories in such a way that they extend basic theories upwards by engineering their application to specific context or phenomena. As a consequence, they reveal, by construction, new relations and systems that are antecedently absent from a scientist's understanding of the theory--independently of a top or reduced theory (Ramsey 1995). A weaker framework of ontological unification is categorial unity, wherein abstract categories such as causality, information, etc., are attached to the interpretation of the specific variables and properties in models of phenomena. ## 5. Disunity and pluralism A more radical departure from logical-positivist standards of unity is the recent criticism of the methodological values of reductionism and unification in the sciences and also its position in culture and society. From the descriptive standpoint, many views under the rubric of disunity are versions of positions mentioned above. The difference is mainly normative and a matter of emphasis, scope and perspective. Such views reject global or universal standards of unity--including unity of method--by emphasizing disunity and endorsing different forms of epistemological and ontological pluralism. ### 5.1 The Stanford School An influential picture of disunity comes from related works by members of the so-called Stanford School such as John Dupre, Ian Hacking, Peter Galison, Patrick Suppes and Nancy Cartwright. Disunity is, in general terms, a rejection of universalism and uniformity, both methodological and metaphysical. Through their work, the rubric of disunity has acquired a visibility parallel to the one once acquired by unity, as an inspiring philosophical rallying cry. From a methodological point of view, members of the school have defended, from analysis of actual scientific practice, a model of local unity such as the so-called trading-zone, (Galison 1998), a plurality of scientific methods (Suppes 1978), a plurality of scientific styles with the function of establishing spaces of epistemic possibility and a plurality of kinds of unities (Hacking 1996; Hacking follows the historian A.A. Crombie; for a criticism of Hacking's historical epistemology see Kusch 2010). From a metaphysical point of view, the disunity of science can be given adequate metaphysical foundations that make pluralism compatible with realism (Dupre 1993; Cartwright 1983, 1999). Dupre opposes a mechanistic paradigm of unity characterized by determinism, reductionism and essentialism. The paradigm spreads the values and methods of physics to other sciences that he thinks are scientifically and socially deleterious. Disunity appears characterized by three pluralistic theses: against essentialism--there is always a plurality of classifications of reality into kinds; against reductionism--there exists equal reality and causal efficacy of systems at different levels of description (that is, the microlevel is not causally complete, leaving room for downward causation); and against epistemological monism--there is no single methodology that supports a single criterion of scientificity, nor a universal domain of its applicability, leaving only a plurality of epistemic and non-epistemic virtues. The unitary concept of science should be understood, following the later Wittgenstein, as a family-resemblance concept. (For a criticism of Dupre's ideas, see Mitchell 2003; Sklar 2003.) Against the universalism of explanatory laws, Cartwright has argued that laws cannot be both universal and exactly true, as Hempel required in his influential account of explanation and demarcation; there exist only patchworks of laws and local cooperation. Like Dupre, Cartwright adopts a kind of scientific realism but denies that there is a universal order, whether represented by a theory of everything or a corresponding a priori metaphysical principle (Cartwright 1983). Theories apply only locally, where and to the extent that their interpretive models fit the phenomena studied, ceteris paribus (Cartwright 1999). Cartwright's pluralism is not just opposed to vertical reductionism but also horizontal imperialism, or universalism and globalism. She explains their more or less general domain of application in terms of causal capacities and arrangements she calls nomological machines (Cartwright 1989, 1999). The regularities they bring about depend on a shielded environment. As a matter of empiricism, this is the reason that it is in the controlled environment of laboratories and experiments, where causal interference is shielded off, that factual regularities are manifested. The controlled, stable, regular world is an engineered world. Representation rests on intervention (cf. Hacking 1983; for criticisms see Winsberg et al. 2000; Hoefer 2003; Sklar 2003; Howhy 2003; Teller 2004; McArthur 2006; Ruphy 2016). Disunity and autonomy of levels have been associated, conversely, with antirealism, meaning instrumentalist or empiricist heuristics. This includes, for Fodor and Rosenberg, higher-level sciences such as biology and sociology (Fodor 1974; Rosenberg 1994; Huneman 2010). It is against this picture that Dupre's and Cartwright's attacks on uniformly global unity and reductionism, above, might seem surprising by including an endorsement, in causal terms, of realism.[4] Rohrlich has defended a similar realist position about weaker, conceptual (cognitive) antireductionism, although on the grounds of the mathematical success of derivational explanatory reductions (Rohrlich 2001). Ruphy, however, has argued that antireductionism merely amounts to a general methodological prescription and is too weak to yield uncontroversial metaphysical lessons; these are in fact based on general metaphysical commitments external to scientific practice (Ruphy 2005, 2016). ### 5.2 Pluralism Unlike more descriptive accounts of plurality, pluralism is a normative endorsement of plurality. The question of the metaphysical significance of disunity and anti-reductionism takes one straight to the larger issue of the epistemology and metaphysics (and aesthetics, social culture and politics) of pluralism. And here one encounters the familiar issues and notions such as conceptual schemes, frameworks and worldviews, incommensurability, relativism, contextualism and perspectivalism about goals and standards of concepts and methods (for a general discussion see Lynch 1998; on perspectivalism about scientific models see Giere 1999, 2006; Rueger 2005; Massimi and McCoy 2020). In connection with relativism and instrumentalism, pluralism has typically been associated with antirealism about taxonomical practices. But it has been defended from the standpoint of realism (for instance, Dupre 1993; Chakravartty 2011). Pluralism about knowledge of mind-independent facts can be formulated in terms of different ways to distribute properties (sociability-based pluralism), with more specific commitments about the ontological status of the related elements and their plural contextual manifestations of powers or dispositions (Chakravartty 2011; Cartwright 2007). From a more epistemological standpoint, pluralism applies widely to concepts, explanations, virtues, goals, methods, models and kinds of representations (see above for graphic pluralism), etc. In this sense, pluralism has been defended as a general framework that rejects the ideal of consensus in cognitive, evaluative and practical matters, against pure skepticism (nothing goes) or indifferentism (anything goes), including a defense of preferential and contextual rationality that notes the role of contextual rational commitments, by analogy with political forms of engagement (Rescher 1993; van Bouwel 2009a; Cat 2012). Consider at least four distinctions--they are formulated about concepts, facts, and descriptions, and they apply also to values, virtues, methods, etc.: * Vertical vs. horizontal pluralism. Vertical pluralism is inter-level pluralism, the view that there is more than one level of factual description or kind of fact and that each is irreducible, equally fundamental, or ontologically/conceptually autonomous. Horizontal pluralism is intra-level pluralism, the view that there may be incompatible descriptions or facts on the same level of discourse (Lynch 1998). For instance, the plurality of explanatory causes to be chosen from or integrated in biology or physics has been defended as a lesson in pluralism (Sober 1999). * Global vs. local pluralism. Global pluralism is pluralism about every type of fact or description. Global horizontal pluralism is the view that there may be incompatible descriptions of the same type of fact. Global vertical pluralism is the view that no type of fact or description reduces to any other. Local horizontal and vertical pluralism is about one type of fact or description (Lynch 1998). It may also concern situated standpoints informed by, among others, social differences (Wylie 2015). * Difference vs. integrative pluralism. Difference pluralism has been defended in terms of division of labor in the face of complexity and cognitive limitations (Giere 2006), epistemic humility (Feyerabend 1962 and his final writings in the 1990s; Chang 2002), scientific freedom, empirical testability and theory choice (Popper 1935; Feyerabend 1962) and underdetermination. Underdetermination arguments concern choices from a disjunction of equivalent types of descriptions (Mitchell 2003, 2009) or of incompatible partial representations or models of phenomena in the same intended scope (Longino 2002, 2013). The representational incompatibility may be traced to competing values or aims, or assumptions in ceteris paribus laws. Critiques of difference pluralism point to different consequences such as failure to address complex and boundary phenomena and problems in, for instance, the life and social sciences (Gorman 2002; Mitchell 2003; 2008); detrimental effects of taxonomic instability (Sullivan 2017) and so-called intraconcept variability (Cunningham 2021) in the mind-brain sciences. When the variability characterizes what are taken to be the same concepts and terminology, several contexts of detrimental effect have been identified: (1) science education, (2) collaborative research, intra- and cross-disciplinary (e.g., the practical and moral problems of value conflicts mentioned above), (3) clinical practice, and (4) metascientific research (Cunningham 2021). Integrative, or connective, types of pluralism are the conjunctive or holistic requirement of different types of descriptions, methods or perspectives (Mitchell 2003, 2009; contrast with the more isolationist position in Longino 2002 and her essay in Kellert, Longino and Waters 2006; Longino 2013). To a merely syncretistic or tolerant, non-interactive pluralism, a recent body of literature has opposed a dynamic, coordinated, interactive disciplinary kind of pluralism (Chang 2012; Wylie 2015; Sullivan 2017). The former requires only respectful division of labor; the latter may involve either a limited cross-fertilization through communication and assimilation--borrowing or co-optation--or a more robust, integrative epistemic engagement. The latter may involve, for instance, communicative expertise--also known as interactional expertise--without having contributory expertise in the other practice, exchange and collaboration on the same project. Borrowing and cross-fertilization across divides and over distances concern more than theories and change the respective practices, processes and products. They extend to data, concepts, models, methods, evidence, heuristic techniques, technology and other resources. To mention one distinction, while theoretical integration concerns concepts, ontology, explanations, models, etc., practical integration concerns methods, heuristics and testing (Grantham 2004). Thus, accounts including different versions of taxonomical pluralism range from the more conventional and contingent (from Elgin 1997 to astronomical kinds in Ruphy 2016) and the more grounded in contexts of practices (categorization work in Bowker and Star 1999; life sciences and chemistry in Kendig 2016) and the interactive (Hacking's interactive kinds in the human sciences) to the more metaphysically substantive. Some methodological prescriptions of pluralism rely on pluralism in metascientific research including history (Chang 2012). Connective varieties of pluralism have been endorsed on grounds of their epistemic value. Considerations of empirical adequacy and predictive power can be traced back to Neurath, the explanatory and methodological value of cross-fertilization, the epistemic benefits of a stance of openness to new kinds of facts and ways of inquiry and learning (Wylie 2015) and evidential value. From the standpoint of evidence, we find second-order varieties of mixed evidence patterns--triangulation, security and integration--that connect different kinds of evidence to provide enhanced evidential value. The relevant plurality or difference requires independence, a condition to be explicated in different ways (Kuorikoski and Marchionni 2022). Regarding triangulation, enhanced evidence results from multiple theory-laden but theoretically independent lines of evidence in, for instance, microscopy (Hacking 1983), archeology (Wylie 1999) and interdisciplinary research in the social sciences, for example, neuroeconomics in support of existence claims about phenomena and descriptions of phenomena but not to more general theories about them (Kuorikoski and Marchionni 2016). Connective forms of pluralism have been modeled in terms of relations between disciplines (see above) and defended at the level of social epistemology by analogy with political models of liberal democracy and as a model of social governance between the extremes of so-called consensual mainstreaming and antagonistic exclusivism (van Bouwel 2009a). Through dialogue, a plurality of represented perspectives enables the kind of critical scrutiny of questions and knowledge claims that exposes error, bias, unexamined norms of justification and acceptance and the complexity of issues and relevant implications. As a result, it has been argued, pluralism supports a corresponding standard of critical objectivity (Longino 2002; Wylie 2015). * Internal vs. external pluralism. From a methodological standpoint, an internal perspective is naturalistic in its reliance on the contingent plurality of scientific practice by any of its standards. This has been defended by members of the so-called Minnesota School (Kellert, Longino and Waters 2006) and Ruphy (2016). The alternative, which Ruphy has attributed to Dupre and Cartwright, is the adoption of a metaphysical commitment external to actual scientific practice. As a matter of actual practice, pluralism has been identified as part of a plurality of projects and perspectives in, for instance, cognitive and mind-brain sciences, where attitudes towards a plurality of ways of researching and understanding cognition vary (Milkowski and Hohol 2021). These attitudes include (1) lamenting the variability of meaning and systems of classification (Sullivan 2007; Cunningham 2021); (2) embracing complementarity between hierarchical mechanistic and computational approaches--in the face of, for instance, models with propositional declarative and lawlike statements and computational models with software codes; (3) seeking integration with mutual consistency and evidential support; (4) seeking reduction privileging neural models; and (5) seeking grand unification that values simplicity and generality over testability. ### 5.3 Metapluralism The preference for one kind of pluralism over another is typically motivated by epistemic virtues or constraints. Meta-pluralism, or pluralism about pluralism, is obviously conceivable in similar terms, as it can be found in the formulation of the so-called pluralist stance (Kellert, Longino and Waters 2006). The pluralist stance replaces metaphysical principles with scientific or empirical methodological rules and aims that have been "tested". Like Dupre's and Cartwright's metaphysical positions, its metascientific position must be empirically tested. Metascientific conclusions and assumptions cannot be considered universal or necessary, but are local, contingent and relative to scientific interests and purposes. Thus, on this view, complexity does not always require interdisciplinarity (Kellert 2008), and in some situations the pluralist stance will defend reductions or specialization over interdisciplinary integration (Kellert, Longino and Waters 2006; Cat 2012; Rescher 1993). ## 6. Conclusion: Why unity? And what difference does it really make? From Greek philosophy to current debates, justifications for adopting positions on matters of unification have varied from the metaphysical and theological to the epistemic, social and pragmatic. Whether as a matter of truth (Thalos 2013) or consequence, views on matters of unity and unification make a difference in both science and philosophy, and, by application, in society as well. In science they provide strong heuristic or methodological guidance and even justification for hypotheses, projects, and specific goals. In this sense, different rallying cries and idioms such as simplicity, unity, disunity, emergence or interdisciplinarity, have been endowed with a normative value. Their evaluative role extends broadly. They are used to provide legitimacy, even if rhetorically, in social contexts, especially in situations involving sources of funding and profit. They set a standard of what carries the authority and legitimacy of what it is to be scientific. As a result, they make a difference in scientific evaluation, management and application, especially in public domains such as healthcare and economic decision-making. For instance, pointing to the complexity of causal structures challenges traditional deterministic or simple causal strategies of policy decision-making with known risks and unknown effects of known properties (Mitchell 2009). Last but not least is the influence that implicit assumptions about what unification can do have on science education (Klein 1990). Philosophically, assumptions about unification help choose what sort of philosophical questions to pursue and what target areas to explore. For instance, fundamentalist assumptions typically lead one to address epistemological and metaphysical issues in terms of only results and interpretations of fundamental levels of disciplines. Assumptions of this sort help define what counts as scientific and shape scientistic or naturalized philosophical projects. In this sense, they determine, or at least strongly suggest, what relevant science carries authority in philosophical debate. At the end of the day, one should not lose sight of the larger context that sustains problems and projects in most disciplines and practices. We are as free to pursue them as Kant's dove is free to fly, that is, not without the surrounding air resistance to flap its wings upon and against. Philosophy was once thought to stand for the systematic unity of the sciences. The foundational character of unity became the distinctive project of philosophy, in which conceptual unity played the role of the standard of intelligibility. In addition, the ideal of unity, frequently under the guise of harmony, has long been a standard of aesthetic virtue, although this image has been eloquently challenged by, for instance, John Bailey and Iris Murdoch (Bailey 1976; Murdoch 1992). Unities and unifications help us meet cognitive and practical demands upon our life as well as cultural demands upon our self-images that are both cosmic and earthly. It is not surprising that talk of the many meanings and levels of unity--the fundamental level, unification, system, organization, universality, simplicity, atomism, reduction, harmony, complexity or totality--can place an urgent grip on our intellectual imagination.
binarium	## 1. The "most famous pair" Some historians of medieval philosophy describe what they see as an Augustinian "doctrinal complex" that emerged in the late twelfth and early thirteenth centuries and that was the "common teaching" among scholastics after the 1220s.[1] Weisheipl [1980], pp. 242-43, lists five ingredients of this "Augustinian" synthesis:[2] 1. voluntarism (an emphasis on the role of the will as distinct from the intellect) 2. universal hylomorphism 3. plurality of forms 4. divine illumination, interpreted through the influence of Avicenna 5. the real identity of the soul with its powers or faculties. The second and third items on this list together make up what is sometimes called the "binarium famosissimum," the "most famous pair." ### 1.1 Universal hylomorphism Paul Woodruff has defined hylomorphism as "the doctrine, first taught by Aristotle, that concrete substance consists of forms in matter (hyle)" (Audi [1999], p. 408). One might therefore expect universal hylomorphism to be the doctrine that all substances consist of forms in matter. But in fact the medieval theory of universal hylomorphism maintained something slightly weaker than that; it held that all substances except God were composed of matter and form, whereas God is entirely immaterial. This view seems to be the result of two more basic theses: * the explicit claim that only God is metaphysically simple in all respects, so that all creatures are metaphysically composite; * the view, not always explicitly spelled out, that all composition is in some way a composition of matter and form, an indeterminate factor and a determining element. The reasoning behind the second thesis is murky. On the other hand, it is easy to find medieval authors who argue in detail for the first claim.[3] Still, it is surprisingly hard to find any medieval author who gives a good motivation for it. That is, why should it be important to maintain that God and only God is metaphysically simple? What rests on the claim? It it tempting, and plausible, to suppose that the implicit reasoning goes something like this: "Composite" (com + positus = "put together") things don't just happen to be composite; something put them together. In short, composition requires an efficient cause. It follows therefore that God, as first cause, cannot be composite. Conversely, anything that is caused is in some sense composite. Hence, since everything besides God is created and therefore caused, everything other than God is composite. In short, the unique simplicity of God is important to maintain because it is required by the doctrine of creation. Nevertheless, while this line of reasoning is plausible, it is not found clearly stated in any medieval author I know of.[4] The notion that all creatures are composites of matter and form requires that something be said about what we might otherwise call "immaterial" substances--angels, Aristotelian "separated substances," the human soul after death. For universal hylomorphism, such entities cannot be truly immaterial, and yet they are obviously quite unlike familiar physical objects. As a result, universal hylomorphists distinguished between "corporeal" matter, i.e., the matter of physical, sensible objects, and another kind of matter sometimes called "spiritual matter."[5] ### 1.2 Plurality of forms The theory known as "plurality of forms" is not just the theory that there are typically many forms in a material substance. That would have been an innocuous claim; everyone agreed that material substances routinely have many accidental forms. The theory of plurality of forms is instead the theory that there is a plurality of substantial forms in a given material substance. Details of the theory varied widely from author to author.[6] There was some disagreement over how many substantial forms were involved. Most people who held a version of this theory agreed that at least a "form of corporeity" was required in all physical substances, but they disagreed over how many additional substantial forms were required for a given kind of body. Particular attention was given to the case of the human body. The arguments in support of this theory were quite diverse, and come from a variety of directions. William of Ockham, for instance, held that if the form of corporeity were not distinct from the intellective soul and were not essentially present in the human being both during his life and after the intellective soul departs at death, then once a saint had died it would be false to say that the body that remains is the body that saint ever had. Hence the cult of venerating the bodies of the saints would make no sense. (Quodlibet II, q. 11.[7]) Again, Thomas Aquinas, although he rejects plurality or forms, nevertheless records several arguments given on its behalf. Among them[8] > > > > Furthermore, before the coming of the rational soul the body in the > womb of the mother has some form. Now when the rational soul comes, it > cannot be said that this form disappears, because it does not lapse > into nothingness, nor would it be possible to specify anything into > which it might return. Therefore, some form exists in the matter > previous to the rational soul ... > > > > > Furthermore, in VII Metaphysica [11, 1036a 26] it is said > that every definition has parts, and that the parts of a definition are > forms. In anything that is defined, therefore, there must be several > forms. Since, therefore, man is a kind of defined thing, it is > necessary to posit in him several forms; and so some form exists before > the rational soul. > > > ## 2. Proponents and opponents The theories of universal hylomorphism and plurality of forms are found together in many authors from the twelfth and thirteenth century. They are both held, for instance, by the author known to the scholastics as "Avicebron" (or Avicebrol, Avencebrol, etc.), who is to be identified with the Spanish Jewish philosopher and poet Solomon Ibn Gabirol (c. 1022-c. 1052/c. 1070), and whose Mekor Hayyim (= Fons vitae, "Fountain of Life") was translated into Latin in the late-twelfth century.[9] Indeed, medieval as well as modern scholars have sometimes looked to Ibn Gabirol as the main source for the two doctrines in the thirteenth century.[10] Other authors who held both theories included the translator of the Fons vitae, one Dominic Gundisalvi (Gundissalinus)[11] as well as people as diverse as Thomas of York,[12] St. Bonaventure,[13] the anonymous thirteenth-century Summa philosophiae once ascribed to Robert Grosseteste,[14] John Pecham, Richard of Mediavilla,[15] and many others.[16] But other authors rejected these two views. The best known is no doubt Thomas Aquinas. Aquinas held that the uniqueness of God's absolute simplicity does not require positing a kind of matter in all creatures. Rather all creatures, including incorporeal substances such as angels or human souls, are composite insofar as they have a composition of essence and esse ("to be," the act of existing), whether or not they have an additional composition or matter and form.[17] Again, he argues that if any substance has a plurality of forms, only the first form that comes to it can be a substantial form; all the others must be accidental forms.[18] Godfrey of Fontaines likewise rejected both theories. John Duns Scotus accepted plurality of forms, but denied universal hylomorphism.[19] ## 3. The expression "binarium famosissimum" The expression "binarium famosissimum" is not a medieval label for this pairing of doctrines; its use in this sense appears to be purely a twentieth-century development. Oddly enough, only a single occurrence of the expression has been found in a medieval text, the anonymous Summa philosophiae cited above.[20] But it clear that this author uses the expression to refer not to a pair of doctrines, whether to universal hylomorphism and plurality of forms or to any other pair, but to the division of substance into corporeal and incorporeal[21]: > > Now the first contrariety in the categorial line of substance, from > the nature of genus, partly by reason of the matter related to both > sides [i.e., to corporeal and incorporeal], partly from the nature of > the most common form tending to particularity by degrees according to > the proportion of the matter's receptivity, contains the binarium > famosissimum, that is, corporeal and incorporeal. > On the other hand, as early as 1943 Daniel Callus writes of a certain John Blund, an early-thirteenth century Englishman who rejected both universal hylomorphism and plurality of forms. Callus says [1943], p. 252: > > More than Gundissalinus, we see delineated in Blund the great > questions which in the second half of that century were to divide the > different schools into two opposing armies, the outlining of the > conflict between philosophers and theologians, Aristotelians and the > so-called Augustinians. In Blund we meet with the earliest, clear, and > unmistakable account of the binarium famosissimum of the > Augustinians, Plurality of Forms and hylomorphic composition of > spiritual substances, the angels and the human soul. > Again, in a paper delivered in 1946, Callus [1955], p. 4, mentions the "binarium famosissimum, the twofold pillar on which the whole structure of the Augustinian school was supposed to stand." Although he does not there say what this "twofold pillar" is, a few pages later (p. 9) he cites "the famosissimum binarium Augustinianum, namely, the hylomorphic composition of all created beings, not only corporeal but also spiritual substances, the angels and the human soul; and plurality of forms in one and the same individual." A few years later Etienne Gilson [1955], p. 377, in the discussion of Thomas Aquinas in his monumental History of Christian Philosophy in the Middle Ages, says: > > > The radical elimination of the binarium famosissimum, i.e., > hylomorphism and the plurality of forms, was not due to a more correct > understanding of the metaphysics of Aristotle but to the introduction, > by Thomas Aquinas, of a new metaphysical notion of being. > > > Again, Weisheipl [1980], p. 250, writing about Albert the Great, says (the emphasis is Weisheipl's): > > Thus Albert is quick to point out that Avicebron in the Fons > vitae is "the only [philosopher] who says that from one simple > principle two [things] must immediately proceed in the order of > nature, since the number 'two' follows upon unity." And > Saint Thomas notes: "Some say that the soul and absolutely every > substance besides God is composed of matter and form; indeed the > first author to hold this position is Avicebron, the author > of Liber fons vitae." This is the origin of the later > binarium famosissimum: after One must come Two. > In this passage, Weisheipl uses the phrase "binarium famosissimum" in a sense perhaps loosely related to that used by the author of the Summa philosophiae. But he also links it, via the quotation from Aquinas, to the doctrine of universal hylomorphism.[22] A few years later, Weisheipl ([1984], p. 451), speaking about Robert Grosseteste, remarks that he: > > saw no problem whatever in accepting universal hylomorphism or a plurality of forms in a single material composite. Not only were these two tenets, known as the binarium famosissimum, common in the 1220s and 1230s, they were accepted as 'traditional' teaching by Franciscans at Oxford and Paris throughout the thirteenth century. John Peckham, Matthew of Aquasparta, Richard of Mediavilla, Roger Bacon, the pseudo-Grosseteste and even John Duns Scotus took the binarium famosissimum as orthodox Augustinian doctrine against the 'novelties' of Albert the Great and Thomas Aquinas.[23] > Again, E. A. Synan [1993], p. 236, refers to Aquinas's denial that there can be a plurality of substantial forms in any given substance, and calls it a "rejection of one half of the binarium famosissimum." ## 4. Conceptual issues Thus, although Gilson and some other twentieth-century scholars have paired the theories of universal hylomorphism and plurality of forms under the title "binarium famosissimum," and although it is certainly true that many medieval authors held both theories, there is no evidence that in medieval times they were ever thought of as a "pair" in this way. Why then have some recent scholars linked the two theories so closely? They certainly have done so. Weisheipl ([1980], p. 243), for example, describes the plurality of forms as "simply a logical consequence of" universal hylomorphism. And Zavalloni ([1951], p. 437, n. 61) likewise claims that universal hylomorphism necessarily implies plurality of forms, although he says the latter does not imply the former.[24] What may be going on is this. The theory of universal hylomorphism closely fits the view that the structure of what we truly say about things mirrors the structure of the things themselves. Thus, if I truly say 'The cat is black', then there is a cat that corresponds to the subject term, and that cat is qualified by the quality blackness corresponding to the predicate term. Without the blackness, the cat is to that extent indeterminate. It is the addition of blackness that determines the cat to the particular color it has. In general then, the relation of subject to predicate in a true affirmative judgment is the relation of what is at least relatively indeterminate to what at least partially determines it. Now the relation of something indeterminate to what determines it is a relation of "matter" to "form."[25] Hence everything we can truly say about a subject reflects the fact that the thing is composed of an indeterminate side and a determining element--of matter and form. In short, hylomorphic composition is involved in anything we can make true affirmations about--thus, universal hylomorphism.[26] Of course, we can truly say many things about a given subject. (E.g., 'The cat is black', 'The cat is fat', 'The cat is asleep', etc.) The predicates of all these true statements correspond to forms really inhering in the relatively indeterminate subject. Hence we can speak of a "plurality of forms." But the "plurality of forms," in the sense in which our authors speak of it, refers to something more restricted, to the fact that we can predicate predicates of a given subject in a certain "nested" order. We can, for example, while taking about the very same cat, say "This is a body" (i.e., it is corporeal), "This body is alive" (i.e., it is an organism), "This organism is sensate" (i.e., it has sensation, it is an animal), "This animal is a cat," "This cat is black," etc. Each such predication attributes a form to the underlying, "material," indeterminate subject, and each such subject is in turn a composite of a form and a deeper, underlying material subject. The picture we get then is the picture of some kind of primordial matter, corresponding to the bare 'this' of the first predication ("This is a body"), to which is added a series of forms one on top of the other in a certain order, each one limiting and narrowing down the preceding ones. The structure that results is a kind of laminated structure, a metaphysical "onion" with several layers. On this picture, of course, substantial and accidental forms are both "layers of the onion" in exactly the same sense. The distinction between essential and accidental features of a thing would therefore have to be drawn in some other way. If this reconstruction is more or less correct, then it is clear why universal hylomorphism and plurality of forms can be viewed as conceptually linked. Both fit nicely with the view that the structure of reality is accurately mirrored in true predication. Ibn Gabirol, who held both theories, seems to have been thinking along approximately these lines. But some of the arguments cited above in favor of plurality of forms show that at least that half of the "pair" was sometimes held for entirely different reasons. By contrast with the above reasoning, there is another view of predication, an "Aristotelian" (and later, Thomistic) view that rejects this picture. On this view, true predication in language is not a reliable guide to the metaphysical structure of what makes it true. One can truly describe a given subject in narrower and narrower terms without there being any corresponding distinction of real metaphysical components in what we are describing. We can, for example, describe our cat as a body, as an organism, as an animal, and as in particular a cat, all the while referring to a single metaphysical configuration, a particular combination (in this case) of matter and a feline form. Calling it a body, an organism, an animal, before calling it a cat in no way reflects any sequence of metaphysical distinctions in the entity itself. It is only when we call it "black" that we introduce a new entity into the structure, an accident. In short, on this alternative view, the structure of reality is not accurately mirrored in true predication, at least not in any straightforward way. This view is not committed to any "plurality" of substantial forms, and is not committed to universal hylomorphism either.[27] ## 5. Conclusion The issues here are complex and the historical facts are not yet well sorted out. But it appears that twentieth-century scholars who saw a close conceptual link between universal hylomorphism and the plurality of forms were perhaps thinking of the two theories as motivated by a common commitment to the first theory of predication described in SS 4 above. In some cases (for example the Fons vitae), they probably were so motivated. Still, the fact that the two theories are regarded as a "pair" is a recent phenomenon, not a medieval one. It seems to have arisen first in the writings of Daniel Callus.[28]
properties	## 1. Properties: Basic Ideas There are some crucial terminological and conceptual distinctions that are typically made in talking of properties. There are also various sorts of reasons that have been adduced for the existence of properties and different traditional views about whether and in what sense properties should be acknowledged. We shall focus on such matters in the following subsections. ### 1.1 How We Speak of Properties Properties are expressed, as meanings, by predicates. In the past "predicate" was often used as synonym of "property", but nowadays predicates are linguistic entities, typically contrasted with singular terms, i.e., simple or complex noun phrases such as "Daniel", "this horse" or "the President of France", which can occupy subject positions in sentences and purport to denote, or refer to, a single thing. Following Frege, predicates are verbal phrases such as "is French" or "drinks". Alternatively, predicates are general terms such as "French", with the copula "is" (or verbal inflection) taken to convey an exemplification link (P. Strawson 1959; Bergmann 1960). We shall conveniently use "predicate" in both ways. Predicates are predicated of singular terms, thereby generating sentences such as "Daniel is French". In the familiar formal language of first-order logic, this would be rendered, say, as "$F(d)$", thus representing the predicate with a capital letter. The set or class of objects to which a predicate veridically applies is often called the extension of the predicate, or of the corresponding property. This property, in contrast, is called the intension of the predicate, i.e., its meaning. This terminology traces back to the Middle Age and in the last century has led to the habit of calling sets and properties extensional and intensional entities, respectively. Extensions and intensions can hardly be identified; this is immediately suggested by paradigmatic examples of co-extensional predicates that appear to differ in meaning, such as "has a heart", and "has kidneys" (see SS3.1). Properties can also be referred to by singular terms, or so it seems. First of all, there are singular terms, e.g., "being honest" or "honesty", that result from the nominalization of predicates, such as "is honest" or "honest" (some think that "being $F$" and "$F$-ness" stand for different kinds of property (Levinson 1991). Further, there are definite descriptions, such as "Mary's favorite property". Finally, though more controversially, there are demonstratives, such as "that shade of red", deployed while pointing to a red object (Heal 1997). Frege (1892) and Russell (1903) had different opinions regarding the ontological import of nominalization. According to the former, nominalized predicates stand for a "correlate" of the "unsaturated" entity that the predicate stands for (in Frege's terminology they are a "concept correlate" and a "concept", respectively). According to the latter, who speaks of "inextricable difficulties" in Frege's view (Russell 1903: SS49), they stand for exactly the same entity. Mutatis mutandis, they similarly disagreed about other singular terms that seemingly refer to properties. The ontological distinction put forward by Frege is mainly motivated by the fact that grammar indeed forbids the use of predicates in subject position. But this hardly suffices for the distinction and it is dubious that other motivations can be marshalled (Parsons 1986). We shall thus take for granted Russell's line here, although many philosophers support Frege's view or at least take it very seriously (Castaneda 1976; Cocchiarella 1986a; Landini 2008). ### 1.2 Arguments for Properties Properties are typically invoked to explain phenomena of philosophical interest. The most traditional task for which properties have been appealed to is to provide a solution to the so-called "one over many problem" via a corresponding "one over many argument". This traces back at least to Socrates and Plato (e.g., Phaedo, 100 c-d) and keeps being rehearsed (Russell 1912: ch. 9; Butchvarov 1966; Armstrong 1978a: ch. 7; Loux 1998: ch.1). The problem is that certain things are many, they are numerically different, and yet they are somehow one: they appear to be similar, in a way that suggests a uniform classification, their being grouped together into a single class. For example, some objects have the same shape, certain others have the same color, and still others the same weight. Hence, the argument goes, something is needed to explain this phenomenon and properties fill the bill: the objects in the first group, say, all have the property spherical, those in the second red, and those in the third weighing 200 grams. Relatedly, properties have been called for to explain our use of general terms. How is it, e.g., that we apply "spherical" to those balls over there and refuse to do it for the nearby bench? It does not seem to be due to an arbitrary decision concerning where, or where not, to stick a certain label. It seems rather the case that the recognition of a certain property in some objects but not in others elicits the need for a label, "spherical", which is then used for objects having that property and not for others. Properties are thus invoked as meanings of general terms and predicates (Plato, Phaedo, 78e; Russell 1912: ch. 9). In contrast with this, Quine (1948; 1953 [1980: 11, 21, 131] ) influentially argued that the use of general terms and predicates in itself does not involve an ontological commitment to entities corresponding to them, since it is by deploying singular terms that we purport to refer to something (see also Sellars 1960). However, as noted, predicates can be nominalized and thus occur as singular terms. Hence, even if one agrees with Quine, nominalized predicates still suggest the existence of properties as their referents, at least to the extent that the use of nominalized predicates cannot be paraphrased away (Loux 2006: 25-34; entry on platonism in metaphysics, SS4). After Quine (1948), quantificational idiom is the landmark of ontological commitment. We can thus press the point even more by noting that we make claims and construct arguments that appear to involve quantification over properties, with quantifiers reaching (i) over predicate positions, or even (ii) over both subject and predicate positions (Castaneda 1976).[2] As regards (i), consider: this apple is red; this tomato is red; hence, there is something that this apple is and this tomato also is. As for (ii), consider: wisdom is more important than beauty; Mary is wise and Elisabeth is beautiful; hence, there is something that Mary is which is more important than something that Elisabeth is. Quantification over properties seems ubiquitous not just in ordinary discourse but in science as well. For example, the inverse square law for dynamics and the reduction of temperature to mean molecular energy can be taken to involve quantification over properties such as masses, distances and temperatures: the former tells us that the attraction force between any two bodies depends on such bodies' having certain masses and being at a certain distance, and the latter informs us that the fact that a sample of gas has a given temperature depends on its having such and such mean kinetic energy. Swoyer (1999: SS3.2) considers these points within a long list of arguments that have been, or can be, put forward to motivate an ontological commitment to properties. He touches on topics such as a priori knowledge, change, causation, measurement, laws of nature, intensional logic, natural language semantics, numbers (we shall cover some of this territory in SS5 and SS6). Despite all this, whether, and in what sense, properties should be admitted in one's ontology appears to be a perennial issue, traditionally shaped as a controversy about the existence of universals. ### 1.3 Traditional Views about the Existence of Universals Do universals really exist? There are three long-standing answers to this question: realism, nominalism, and conceptualism. According to realists, universals exist as mind-independent entities. In transcendent realism, put forward by Plato, they exist even if uninstantiated and are thus "transcendent" or "ante res" ("before the things"). In immanent realism, defended by Aristotle in opposition to his master, they are "immanent" or "in rebus" ("in things"), as they exist only if instantiated by objects. Contemporary notable supporters are Russell (1912) for the former and Armstrong (1978a) for the latter (for recent takes on this old dispute see the essays by Loux, Van Inwagen, Lowe and Galluzzo in Galluzzo & Loux 2015). Transcendentism is of course a less economical position and elicits epistemological worries regarding our capacity to grasp ante res universals. Nevertheless, such worries may be countered in various ways (cf. entry on platonism in metaphysics, SS5; Bealer 1982: 19-20; 1998: SS2; Linsky & Zalta 1995), and uninstantiated properties may well have work to do, particularly in capturing the intuitive idea that there are unrealized possibilities and in dealing with cognitive content (see SS3). A good summary of pro-transcendentist arguments and immanentist rejoinders is offered by Allen (2016: SS2.3). See also Costa forthcoming for a new criticism of immanentism, based on the notion of grounding. Nominalists eschew mind-independent universals. They may either resort to tropes in their stead, or accept predicate nominalism, which tries to make it without mind-independent properties at all, by taking the predicates themselves to do the classifying job that properties are supposed to do. This is especially indigestible to a realist, since it seems to put the cart before the horse by making language and mind responsible for the similarities we find in the rich varieties of things surrounding us. Some even say that this involves an idealist rejection of a mind-independent world (Hochberg 2013). Conceptualists also deny that there are mind-independent universals, and because of this they are often assimilated to nominalists. Still, they can be distinguished insofar as they replace such universals with concepts, understood as non-linguistic mind-dependent entities, typically functioning as meanings of predicates. The mind-dependence of concepts however makes conceptualism liable to the same kind of cart/horse worry just voiced above in relation to predicate nominalism.[3] The arguments considered in SS1.2 constitute the typical motivation for realism, which is the stance that we take for granted here. They may be configured as an abductive inference to the best explanation (Swoyer 1999). Thus, of course, they are not foolproof, and in fact nominalism is still a popular view, which is discussed in detail in the entry on nominalism in metaphysics, as well as in the entry on tropes. Conceptualism appears to be less common nowadays, although it still has supporters (cf. Cocchiarella 1986a: ch. 3; 2007), and it is worth noting that empirical research on concepts is flourishing. ### 1.4 Properties in Propositions and States of Affairs We have talked above in a way that might give the impression that predication is an activity that we perform, e.g., when we say or think that a certain apple is red. Although some philosophers might think of it in this way, predication, or attribution, may also be viewed as a special link that connects a property to a thing in a way that gives rise to a proposition, understood as a complex featuring the property and the thing (or concepts of them) as constituents with different roles: the latter occurs in the proposition as logical subject or argument, as is often said, and the former as attributed to such an argument. If the proposition is true (the predication is veridical), the argument exemplifies the property, viz. the former is an instance of the latter. The idea that properties yield propositions when appropriately connected to an argument motivates Russell's (1903) introduction of the term "propositional function" to speak of properties. We take for granted here that predication is univocal. However, according to some neo-Meinongian philosophers, there are two modes of predication, sometimes characterized as "external" and "internal" (Castaneda 1972; Rapaport 1978; Zalta 1983; see entry on nonexistent objects). Zalta (1983) traces back the distinction to Mally and uses "exemplification" to characterize the former and "encoding" to characterize the latter. Roughly, the idea is that non-existent objects may encode properties that existent objects exemplify. For instance, winged is exemplified by that bird over there and is encoded by the winged horse. It is often assumed nowadays that, when an object exemplifies a property, there is a further, complex, entity, a fact or state of affairs (Bergmann 1960; Armstrong 1997, entries on facts and states of affairs), having the property (qua attributed) and the object (qua argument) as constituents (this compositional conception is not always accepted; see, e.g., Bynoe 2011 for a dissenting voice). Facts are typically taken to fulfill the theoretical roles of truthmakers (the entities that make true propositions true, see entry on truthmakers) and causal relata (the entities connected by causal relations; see entry on the metaphysics of causation, SS1). Not all philosophers, however, distinguish between propositions and states of affairs; Russell (1903) acknowledges only propositions and, for a more recent example, so does Gaskin (2008). It appears that properties can have a double role that no other entities can have: they can occur in propositions and facts both as arguments and as attributed (Russell 1903: SS48). For example, in truly saying that this apple is red and that red is a color, we express a proposition wherein red occurs as attributed, and another proposition wherein red occurs as argument. Correspondingly, there are two facts with red in both roles, respectively. This duplicity grounds the common distinction between different orders or types of properties: first-order ones are properties of things that are not themselves predicables; second-order ones are properties of first-order properties; and so on. Even though the formal and ontological issues behind this terminology are controversial, it is widely used and is often connected to the subdivision between first-order and higher-order logics (see, e.g., Thomason 1974; Oliver 1996; Williamson 2013; entry on type theory). It originates from Frege's and Russell's logical theories, especially Russell's type theory, wherein distinctions of types and orders are rigidly regimented in order to circumvent the logical paradoxes (see SS6). ### 1.5 Relations A relation is typically attributed to a plurality of objects. These jointly instantiate the relation in question, if the attribution is veridical. In this case, the relata (as arguments) and the relation (as attributed) are constituents of a state of affairs. Depending on the number of objects that it can relate, a relation is usually taken to have a number of "places" or a "degree" ("adicity", "arity"), and is thus called "dyadic" ("two-place"), "triadic" ("three-place"), etc. For example, before and between are dyadic (of degree 2) and triadic (of degree 3), respectively. In line with this, properties and propositions are "monadic" and "zero-adic" predicables, as they are predicated of one, and of no, object, respectively, and may then be seen as limiting cases of relations (Bealer 1982, where properties, relations and propositions are suggestively grouped under the acronym "PRP;" Dixon 2018; Menzel 1986; 1993; Swoyer 1998; Orilia 1999; Van Inwagen 2004; 2015; Zalta 1983). This terminology is also applied to predicates and sentences; for example, the predicate "between" is triadic, and the sentence "Peter is between Tom and May" is zero-adic. Accordingly, standard first-order logic employs predicates with a fixed degree, typically indicated by a superscript, e.g., $P^1$, $Q^2$, $R^3$, etc. In natural language, however, many predicates appear to be multigrade or variably polyadic; i.e., they can be used with different numbers of arguments, as they can be true of various numbers of things. For example, we say "John is lifting a table", with "lifting" used as dyadic, as well as "John and Mary are lifting a table", with "lifting" used as triadic. Moreover, there is a kind of inference, called "argument deletion", which also suggests that many predicates that prima facie could be assigned a certain fixed degree are in fact multigrade. For example, "John is eating a cake" suggests that "is eating" is dyadic, but since, by argument deletion, it entails "John is eating", one could conclude that it is also monadic and thus multigrade. Often one can resist the conclusion that there are multigrade predicates. For example, it could be said that "John is eating" is simply short for "John is eating something". But it seems hard to find a systematic and convincing strategy that allows us to maintain that natural language predicates have a fixed degree. This has motivated the construction of logical languages that feature multigrade predicates in order to provide a more appropriate formal account of natural language (Grandy 1976; Graves 1993; Orilia 2000a). Since natural language predicates appear to be multigrade, one may be tempted to take the properties and relations that they express to also be multigrade, and the metaphysics of science may lend support to this conclusion (Mundy 1989). Seemingly, relations are not jointly instantiated simpliciter; how the instantiation occurs also plays a role. This comes to the fore in particular with non-symmetric relations such as loving. For example, if John loves Mary, then loving is jointly instantiated by John and Mary in a certain way, whereas if it is Mary who loves John, then loving is instantiated by John and Mary in another way. Accordingly, relations pose a special problem: explicating the difference between facts, such as Abelard loves Eloise and Eloise loves Abelard, that at least prima facie involve exactly the same constituents, namely a non-symmetric relation and two other items (loving, Abelard, Eloise). Such facts are often said to differ in "relational order" or in the "differential application" of the non-symmetric relation in question, and the problem then is that of characterizing what this relational order or differential application amounts to. Russell (1903: SS218) attributed an enormous importance to this issue and attacked it repeatedly. Despite this, it has been pretty much neglected until the end of last century, with only few others confronting it systematically (e.g., Bergmann 1992; Hochberg 1987). However, Fine (2000) has forcefully brought it again on the ontological agenda and proposed a novel approach that has received some attention. Fine identifies standard and positionalist views (analogous to two approaches defended by Russell at different times (1903; 1984); cf. Orilia 2008). According to the former, relations are intrinsically endowed with a "direction", which allows us to distinguish, e.g., loving and being loved: Abelard loves Eloise and Eloise loves Abelard differ, because they involve two relations that differ in direction (e.g., the former involves loving and the latter being loved). According to the latter, relations have different "positions" that can somehow host relata: Abelard loves Eloise and Eloise loves Abelard differ, because the two positions of the very same loving relation are differently occupied (by Abelard and Eloise in one case and by Eloise and Abelard in the other case). Fine goes on to propose and endorse an alternative, "anti-positionalist" standpoint, according to which relations have neither direction nor positions. The literature on this topic keeps growing and there are now various proposals on the market, including new versions of positionalism (Orilia 2014; Donnelly 2016; Dixon 2018), and primitivism, according to which differential application cannot be analyzed (MacBride 2014). Russell (1903: ch. 26) also had a key role in leading philosophers to acknowledge that at least some relations, in particular spatio-temporal ones, are external, i.e., cannot be reduced to monadic properties, or the mere existence, of the relata, in contrast to internal relations that can be so reduced. This was a breakthrough after a long tradition tracing back at least to Aristotle and the Scholastics wherein there seems to be hardly any place for external relations (see entry on medieval theories of relations).[4] ### 1.6 Universals versus Tropes According to some philosophers, universals and tropes may coexist in one ontological framework (see, e.g., Lowe 2006 for a well-known general system of this kind, and Orilia 2006a, for a proposal based on empirical data from quantum mechanics). However, nowadays they are typically seen as alternatives, with the typical supporter of universals ("universalist") trying to do without tropes (e.g., Armstrong 1997) and the typical supporter of tropes ("tropist") trying to dispense with universals (e.g., Maurin 2002).[5] In order to clarify how differently they see matters, we may take advantage of states of affairs. Both parties may agree, say, that there are two red apples, $a$ and $b$. They will immediately disagree, however, for the universalist will add that 1. there are two distinct states of affairs, that a is red and that b is red, 2. such states are similar in having the universal red as constituent, and 3. they differ insofar as the former has $a$ as constituent, whereas the latter has $b$. The tropist will reject these states of affairs with universals as constituents and rather urge that there are two distinct tropes, the redness of $a$ and the redness of $b$, which play a theoretical role analogous to the one that the universalist would invoke for such states of affairs. Hence, tropists claim that tropes can be causal relata (D. Williams 1953) and truthmakers (Mulligan, Simons, & Smith 1984). Tropes are typically taken to be simple, i.e., without any subconstituent (see Section 2.2 of the entry on tropes). Their playing the role of states of affairs with universals as constituents depends on this: universals combine two functions, only one of which is fulfilled by tropes. On the one hand, universals are characterizers, inasmuch as they characterize concrete objects. On the other hand, they are also unifiers, to the extent that different concrete objects may be characterized by the very same universal, which is thus somehow shared by all of them; when this is the case, there is, according to the universalist, an objective similarity among the different objects (see SS1.2). In contrast, tropes are only characterizers, for, at least as typically understood, they cannot be shared by distinct concrete objects. Given its dependency on one specific object, say, the apple $a$, a trope can do the work of a state of affairs with $a$ as constituent. But for tropes to play this role, the tropist will have to pay a price and introduce additional theoretical machinery to account for objective similarities among concrete objects. To this end, she will typically resort to the idea that there are objective resemblances among tropes, which can then be grouped together in resemblance classes. These resemblance classes play the role of unifiers for the tropist. Hence, from the tropist's point of view "property" is ambiguous, since it may stand for the characterizers (tropes) or for the unifiers (resemblance classes) (cf. entry on mental causation, SS6.5). Similarly, "exemplification" and related words may be regarded as ambiguous insofar as they can be used either to indicate that an object exemplifies a certain trope or to indicate that the object relates to a certain resemblance class by virtue of exemplifying a trope in that class.[6] ### 1.7 Kinds of Properties Many important and often controversial distinctions among different kinds of properties have been made throughout the whole history of philosophy until now, often playing key roles in all sorts of disputes and arguments, especially in metaphysics. Here we shall briefly review some of these distinctions and others will surface in the following sections. More details can be found in other more specialized entries, to which we shall refer. Locke influentially distinguished between primary and secondary qualities; the former are objective features of things, such as shapes, sizes and weights, whereas the latter are mind-dependent, e.g., colors, tastes, sounds, and smells. This contrast was already emphasized by the Greek atomists and was revived in modern times by Galileo, Descartes, and Boyle. At least since Aristotle, the essential properties of an object have been contrasted with its accidental properties; the object could not exist without the former, whereas it could fail to have the latter (see entry on essential vs. accidental properties). Among essential properties, some acknowledge individual essences (also called "haecceities" or "thisnesses"), which univocally characterize a certain individual. Adams (1979) conceives of such properties as involving, via the identity relation, the very individual in question, e.g., Socrates: being identical to Socrates, which cannot exist if Socrates does not exist. In contrast, Plantinga (1974) views them as capable of existing without the individuals of which they are essences, e.g., Socratizing, which could have existed even if Socrates had not existed. See SS5.2 on the issue of the essences of properties themselves. Sortal properties are typically expressed by count nouns like "desk" and "cat" and are taken to encode principles of individuation and persistence that allow us to objectively count objects. For example, there is a fact of the matter regarding how many things in this room instantiate being a desk and being a cat. On the other hand, non-sortal properties such as red or water do not allow us to count in a similarly obvious way. This distinction is often appealed to in contemporary metaphysics (P. Strawson 1959: ch. 5, SS2; Armstrong 1978a: ch. 11, SS4), where, in contrast, the traditional one between genus and species plays a relatively small role. The latter figured conspicuously in Aristotle and in much subsequent philosophy inspired by him. We can view a genus as a property more general than a corresponding species property, in a hierarchically relative manner. For example, being a mammal is a genus relative to the species being a human, but it is a species relative to the genus being an animal. The possession of a property called differentia is appealed to in order to distinguish between different species falling under a common genus; e.g., as the tradition has it, the differentia for being human is being rational (Aristotle, Categories, 3a). Similar hierarchies of properties, however without anything like differentiae, come with the distinction of determinables and determinates, which appears to be more prominent in current metaphysics. Color properties provide typical examples of such hierarchies, e.g., with red and scarlet as determinable and determinate, respectively. ## 2. Exemplification We saw right at the outset that objects exemplify, or instantiate, properties. More generally, items of all sorts, including properties themselves, exemplify properties, or, in different terminology, bear, have or possess properties. Reversing order, we can also say that properties characterize, or inhere in, the items that exemplify them. There is then a very general phenomenon of exemplification to investigate, which has been labeled in various ways, as the variety of terms of art just displayed testifies. All such terms have often been given special technical senses in the rich array of different explorations of this territory since Ancient and Medieval times up to the present age (see, e.g., Lowe 2006: 77). These explorations can hardly be disentangled from the task of providing a general ontological picture with its own categorial distinctions. In line with what most philosophers do nowadays, we choose "exemplification", or, equivalently, "instantiation" (and their cognates), to discuss this phenomenon in general and to approach some different accounts that have been given of it in recent times. This sweeping use of these terms is to be kept distinct from the more specialized uses of them that will surface below (and to some extent have already surfaced above) in describing specific approaches by different philosophers with their own terminologies. ### 2.1 Monist vs. Pluralist Accounts We have taken for granted that there is just one kind of exemplification, applying indifferently to different categories of entities. This monist option may indeed be considered the default one. A typical recent case of a philosopher who endorses it is Armstrong (1997). He distinguishes three basic categories, particulars, properties or relations, and states of affairs, and takes exemplification as cutting across them: properties and relations are exemplified not only by particulars, but by properties or relations and states of affairs as well. But some philosophers are pluralist: they distinguish different kinds of exemplification, in relation to categorial distinctions in their ontology. One may perhaps attribute different kinds of exemplification to the above-considered Meinongians in view of the different sorts of predication that they admit (see, e.g., Monaghan's (2011) discussion of Zalta's theory). A more typical example of the pluralist alternative is however provided by Lowe (2006), who distinguishes "instantiation", "characterization" and "exemplification" in his account of four fundamental categories: objects, and three different sorts of properties, namely kinds (substantial universals), attributes and modes (tropes).[7] To illustrate, Fido is a dog insofar as it instantiates the kind dog, $D$, which in turn is characterized by the attribute of barking, $B$. Hence, when Fido is barking, it exemplifies $B$ occurrently by virtue of being characterized by a barking mode, $b$, that instantiates $B$; and, when Fido is silent, it exemplifies $B$ dispositionally, since $D$, which Fido instantiates, is characterized by $B$ (see Gorman 2014 for a critical discussion of this sort of view). ### 2.2 Compresence and Partial Identity Most philosophers, whether tacitly or overtly, appear to take exemplification as primitive and unanalyzable. However, on certain views of particulars, it might seem that exemplification is reduced to something more fundamental. A well-known such approach is the bundle theory, which takes particulars to be nothing more than "bundles" of universals connected by a special relation, commonly called compresence, after Russell (1948: Pt. IV, ch. 8)[8]. Despite well-known problems (Van Cleve 1985), this view, or approaches in its vicinity, keep having supporters (Casullo 1988; Curtis 2014; Dasgupta 2009; Paul 2002; Shiver 2014; J. Russell 2018; see Sider 2020, ch. 3, for a recent critical analysis). From this perspective, that a particular exemplifies a property amounts to the property's being compresent with the properties that constitute the bundle with which the particular in question is identified. It thus looks as if exemplification is reduced to compresence. Nevertheless, compresence itself is presumably jointly exemplified by the properties that constitute a given bundle, and thus at most there is a reduction, to compresence, of exemplification by a particular (understood as a bundle), and not an elimination of exemplification in general. Another, more recent, approach is based on partial identity. Baxter (2001) and, inspired by him, Armstrong (2004), have proposed related assays of exemplification, which seem to analyze it in terms of such partial identity. These views have captured some interest and triggered discussions (see, e.g., Mumford 2007; Baxter's (2013) reply to critics and Baxter's (2018) rejoinder to Brown 2017). Baxter (2001) relies on the notion of aspect and on the relativization of numerical identity to counts. In his view, both particulars and properties have aspects, which can be similar to distinct aspects of other particulars or properties. The numerical identity of aspects is relative to standards for counting, counts, which group items in count collections: aspects of particulars in the particular collection, and aspects of universals, in the universal collection. There can then be a cross-count identity, which holds between an aspect in the particular collection, and an aspect in the universal collection, e.g., the aspects Hume as human and humanity as had by Hume. In this case, the universal and the particular in question (humanity and Hume, in our example) are partially identical. Instantiation, e.g., that Hume instantiates humanity, then amounts to this partial identity of a universal and a particular. One may have the feeling, as Baxter himself worries (2001: 449), that in this approach instantiation has been traded for something definitely more obscure, such as aspects and an idiosyncratic view of identity. It can also be suspected that particulars' and properties' having aspects is presupposed in this analysis, where this having is a relation rather close to exemplification itself. Armstrong (2004) tries to do without aspects. At first glance it seems as if he analyzes exemplification, for he takes the exemplification of a property (a universal) by a particular to be a partial identity of the property and the particular; as he puts it (2004: 47), "[i]t is not a mere mereological overlap, as when two streets intersect, but it is a partial identity". However, when we see more closely what this partial identity amounts to, the suspicion arises that it presupposes exemplification. For Armstrong appears to identify a particular via the properties that it instantiates and similarly a property via the particulars that instantiate it. So that we may identify a particular, $x$, via a collection of properties qua instantiated by $x$, say $\{F\_x,$ $G\_x,$ $H\_x,$ ..., $P\_x,$ $Q\_x,$ $\ldots\};$ and a property, $P$, via a collection of particulars qua instantiating $P$, say $\{P\_a, P\_b , \ldots ,P\_x, P\_y , \ldots \}$. By putting things in this way, we can then say that a particular is partially identical to a property when the collection that identifies the particular has an element in common with the collection that identifies the property. To illustrate, the $x$ and the $P$ of our example are partially identical because they have the element $P\_x$ in common. Now, the elements of these collections are neither properties tout court nor particulars tout court, which led us to talk of properties qua instantiated and particulars qua instantiating.[9] But this of course presupposes instantiation. Moreover, there is the unwelcome consequence that the world becomes dramatically less contingent than we would have thought at first sight, for neither a concrete particular nor a property can exist without it being the case that the former has the properties it happens to have, and that the latter is instantiated by the same particulars that actually instantiate it; we get, as Mumford puts it (2007: 185), "a major new kind of necessity in the world". ### 2.3 Bradley's Regress One important motivation, possibly the main one, behind attempts at analysis such as the ones we have just seen is the worry to avoid the so-called Bradley's regress regarding exemplification (Baxter 2001: 449; Mumford 2007: 185), which goes as follows. Suppose that the individual $a$ has the property $F$. For $a$ to instantiate $F$ it must be linked to $F$ by a (dyadic) relation of instantiation, $I\_1$. But this requires a further (triadic) relation of instantiation, $I\_2$, that connects $I\_1, F$ and $a$, and so on without end. At each stage a further connecting relation is required, and thus it seems that nothing ever gets connected to anything else (it is not clear to what extent Bradley had this version in mind; for references to analogous regresses prior to Bradley's, see Gaskin 2008: ch. 5, SS70). This regress has traditionally been regarded as vicious (see, e.g., Bergmann 1960), although philosophers such as Russell (1903: SS55) and Armstrong (1997: 18-19) have argued that it is not. In doing so, however, they seem to take for granted the fact that $a$ has the property $F$ (pretty much as in the brute fact approach; see below) and go on to see $a$'s and $F$'s instantiating $I\_1$ as a further fact that is merely entailed by the former, which in turn entails $a$'s, $F$'s and $I\_1$'s instantiating $I\_2$, and so on. This way of looking at the matter tends to be regarded as a standard response to the regress. But those who see the regress as vicious assume that the various exemplification relations are introduced in an effort to explain the very existence of the fact that $a$ has the property $F$. Hence, from their explanatory standpoint, taking the fact in question as an unquestioned ground for a chain of entailments is beside the point (cf. Loux 1998: 31-36; Vallicella 2002). It should be noted, however, that this perspective suggests a distinction between an "internalist" and an "externalist" version of the regress (in the terminology of Orilia 2006a). In the former, at each stage we postulate a new constituent of the fact, or state of affairs, $s$, that exists insofar as $a$ has the property $F$, and there is viciousness because $s$ can never be appropriately characterized.[10] In the latter, at each stage we postulate a new, distinct, state of affairs, whose existence is required by the existence of the state of affairs of the previous stage. This amounts to admitting infinite explanatory and metaphysical dependence chains. However, according to Orilia (2006b: SS7), since no decisive arguments against such chains exist, the externalist regress should not be viewed as vicious (for criticisms, see Maurin 2015 and Allen 2016: SS2.4.1; for a similar view about predication, see Gaskin 2008). A typical line for those convinced that the regress is vicious has consisted in proposing that instantiation is not a relation, or at least not a normal one. Some philosophers hold that it is a sui generis linkage that hooks things up without intermediaries. Peter Strawson (1959) calls it a non-relational tie and Bergmann (1960) calls it a nexus. Broad (1933: 85) likened instantiation to glue, which just sticks two sheets of paper together, without needing anything additional; similarly, instantiation just relates. An alternative line has been to reject instantiation altogether. According to Frege, it is not needed, because properties have "gaps" that can be filled, and according to a reading of Wittgenstein's Tractatus, because objects and properties can be connected like links in a chain. However, both strategies are problematic, as argued by Vallicella (2002). His basic point is that, if $a$ has property $F$, we need an ontological explanation of why $F$ and $a$ happen to be connected in such a way that $a$ has $F$ as one of its properties (unless $F$ is a property that $a$ has necessarily). But none of these strategies can provide this explanation. For example, the appeal to gaps is pointless: $F$ has a gap whether or not it is filled by $a$ (for example, it could be filled in by another object), and thus the gap cannot explain the fact that $a$ has $F$ as one of its properties. Before turning to exemplification as partial identity, Armstrong (1997: 118) has claimed that Bradley's regress can be avoided by taking a state of affairs, say $x$'s being $P$, as capable by itself of holding together its constituents, i.e., the object $x$ and the property $P$ (see also Perovic 2016). Thus, there is no need to invoke a relation of exemplification linking $x$ and $P$ in order to explain how $x$ and $P$ succeed in giving rise to a unitary item, namely the state of affairs in question. There seems to be a circularity here for it appears that we want to explain how an object and a property come to be united in a state of affairs by appealing to the result of this unification, namely the state of affairs itself. But perhaps this view can be interpreted as simply the idea that states of affairs should be taken for granted in a primitivist fashion without seeking an explanation of their unity by appealing to exemplification or otherwise; this is the brute fact approach, as we may call it (for supporters, see Van Inwagen (1993: 37) and Oliver (1996: 33); for criticisms and a possible defense, see Vallicella 2002, and Orilia 2016, respectively). Lowe (2006) has tried to tackle Bradley's regress within his pluralist approach to exemplification. In his view, characterization, instantiation and exemplification are "formal" and thus quite different from garden-variety relations such giving or loving. This guarantees that these three relations escape Bradley's regress (Lowe 2006: 30, 80, 90).[11] Let us illustrate how, by turning back to the Fido example of SS2.1. What a mode instantiates and what it characterizes belong to its essence. In other words, a mode cannot exist without instantiating the attribute it instantiates and characterizing the object it characterizes. Hence, mode $b$, by simply existing, instantiates attribute $B$ and characterizes Fido. Moreover, since exemplification (the occurrent one, in this case) results from "composing" characterization and instantiation, $b$'s existence also guarantees that Fido exemplifies $B$. According to Lowe, we thus have some truths, that $b$ characterizes Fido, that $b$ instantiates $B$, and that Fido exemplifies $B$ (i.e., is barking), all of which are made true by $b$. Hence, there is no need to postulate as truthmakers states of affairs with constituents, Fido and $b$, related by characterization, or $b$ and $B$, related by exemplification, or Fido and $B$, related by exemplification. This, in Lowe's opinion, eschews Bradley's regress, since this arises precisely because we appeal to states of affairs with constituents in need of a glue that contingently keep them together. Nevertheless, there is no loss of contingency in Lowe's world picture, for an object need not be characterized by the modes that happen to characterize it. Thus, for example, mode $b$ might have failed to exist and there could have been a Fido silence mode in its stead, in which case the proposition that Fido is barking would have been false and the proposition that Fido is silent would have been true. One may wonder however what makes it the case that a certain mode is a mode of just a certain object and not of another one, say another barking dog. Even granting that it is essential for $b$ to be a mode of Fido, rather than of another dog, it remains true that it is of Fido, rather than of the other dog, and one may still think that this being of is also glue of some sort, perhaps with a contingency inherited from the contingency of $b$ (which might have failed to exist). The suspicion then is that the problem of accounting for the relation between a mode and an object has replaced the Armstrongian one of what makes it the case that a universal $P$ and an object $x$ make up the state of affairs $x$'s being $P$. But the former problem, one may urge, is no less thorny than the latter, and some universalists like Armstrong may consider uneconomical Lowe's acceptance of tropes in addition to universals (for accounts of Bradley's regress analogous to Lowe's, but within a thoroughly tropist ontology, see Section 3.2 of the entry on tropes.) As it should be clear from this far from exhaustive survey, Bradley's regress deeply worries ontologists and the attempts to tame it keep flowing.[12] ### 2.4 Self-exemplification Presumably, properties exemplify properties. For example, if properties are abstract objects, as is usually thought, then seemingly every property exemplifies abstractness. But then we should also grant that there is self-exemplification, i.e., a property exemplifying itself. For example, abstractness is itself abstract and thus exemplifies itself. Self-exemplification however has raised severe perplexities at least since Plato. Plato appears to hold that all properties exemplify themselves, when he claims that forms participate in themselves. This claim is crucially involved in his so-called third man argument, which led him to worry that his theory of forms is incoherent (Parmenides, 132 ff.). As we see matters now, it is not clear why we should hold that all properties exemplify themselves (Armstrong 1978a: 71); for instance, people are honest, but honesty itself is not honest (see, however, the entry on Plato's Parmenides, and Marmodoro forthcoming). Nowadays, a more serious worry related to self-exemplification is Russell's famous paradox, constructed as regarding the property of non-self-exemplification, which appears to exemplify itself iff it does not, thus defying the laws of logic, at least classical logic. The discovery of his paradox (and then the awareness of related puzzles) led Russell to introduce a theory of types, which institutes a total ban on self-predication by a rigid segregation of properties into a hierarchy of types (more on this in SS6.1). The account became more complex and rigid, as Russell moved from simple to ramified type theory, which involves a distinction of orders within types (see entries on type theory and Russell's paradox, and, for a detailed reconstruction of how Russell reacted to the paradox, Landini 1998). In type theory all properties are, we may say, typed. This approach has never gained unanimous consensus and its many problematic aspects are well-known (see, e.g., Fitch 1952: Appendix C; Bealer 1989). Just to mention a few, the type-theoretical hierarchy imposed on properties appears to be highly artificial and multiplies properties ad infinitum (e.g., since presumably properties are abstract, for any property $P$ of type $n$, there is an abstractness of type $n+1$ that $P$ exemplifies). Moreover, many cases of self-exemplification are innocuous and common. For example, the property of being a property is itself a property, so it exemplifies itself. Accordingly, many recent proposals are type-free (see SS6.1) and thus view properties as untyped, capable of being self-predicated, sometimes veridically. An additional motivation to move in this direction is a new paradox proposed by Orilia and Landini (2019), which affects simple type theory. It is "contingent" in that it is derived from a contingent assumption, namely that someone, say John, is thinking just about the property of being a property $P$ such that John is thinking of something that is not exemplified by $P$. ## 3. Existence and Identity Conditions for Properties Quine (1957 [1969: 23]) famously claimed that there should be no entity without identity. His paradigmatic case concerns sets: two of them are identical iff they have exactly the same members. Since then it has been customary in ontology to search for identity conditions for given categories of entities and to rule out categories for want of identity conditions (against this, see Lowe 1989). Quine started this trend precisely by arguing against properties and this has strictly intertwined the issues of which properties there are and of their identity conditions.[13] ### 3.1 From Extensionality to Hyperintensionality In an effort to provide identity conditions for properties, one could mimic those for sets, or equate the former with the latter (as in class nominalism; see note 3), and provide the following extensionalist identity conditions: two properties are identical iff they are co-extensional. This criterion can hardly work, however, since there are seemingly distinct properties with the same extension, such as having a heart and having kidneys, and even wildly different properties such as spherical and weighing 2 kilos could by accident be co-extensive. One could then try the following intensional identity conditions: two properties are identical iff they are co-intensional, i.e., necessarily co-extensional, where the necessity in question is logical necessity. This guarantees that spherical and weighing 2 kilos are different even if they happen to be co-extensional. Following this line, one may take properties to be intensions, understood as, roughly, functions that assign extensions (sets of objects) to predicates at logically possible worlds. Thus, for instance, the predicates "has a heart" and "has kidneys" stand for different intensions, for even if they have the same extension in the actual world they have different extensions at worlds where there are creatures with heart and no kidneys or vice versa. This approach is followed by Montague (1974) in his pioneering work in natural language semantics, and in a similar way by Lewis (1986b), who reduces properties to sets of possible objects in his modal realism, explicitly committed to possible worlds and mere possibilia inhabiting them. Most philosophers find this commitment unappealing. Moreover, one may wonder how properties can do their causal work if they are conceived of in this way (for further criticisms see Bealer 1982: 13-19, and 1998: SS4; Egan 2004). However, the criterion of co-intensionality may be accepted without also buying the reduction of properties to sets of possibilia (Bealer views this as the identity condition for his conception 1 properties; see below). Still, co-intensionality must face two challenges coming from opposite fronts. On the one hand, from the perspective of empirical science, co-intensionality may appear too strong as a criterion of identity. For the identity statements of scientific reductions, such as that of temperature to mean kinetic energy, could suggest that some properties are identical even if not co-intensional. For example, one may accept that having absolute temperature of 300K is having mean molecular kinetic energy of $6.21 \times 10^{-21}$ (Achinstein 1974: 259; Putnam (1970: SS1) and Causey (1972) speak of "synthetic identity" and "contingent identity", respectively). Rather than logical necessity, it is nomological necessity, necessity on the basis of the causal laws of nature, that becomes central in this line of thought. Following it, some have focused on the causal and nomological roles of properties, i.e., roughly, the causes and effects of their being instantiated, and their involvement in laws of nature, respectively. They have thus advanced causal or nomic criteria. According to them, two properties are identical iff they have the same causal (Achinstein 1974: SSXI; Shoemaker 1980) or nomological role (Swoyer 1982; Kistler 2002). This line has been influential, as it connects to the "pure dispositionalism" discussed in SS5.2. There is however a suspicion of circularity here, since causal and nomological roles may be viewed as higher-order properties (see entry on dispositions, SS3). On the other hand, once matters of meaning and mental content are taken into account, co-intensionality might seem too weak, for it makes a property $P$ identical to any logically equivalent property, e.g., assuming classical logic, $P$ and ($Q$ or not $Q$). And with a sufficiently broad notion of logical necessity, even, for example, being triangular and being trilateral are identical. However, one could insist that "trilateral" and "triangular" appear to have different meanings, which somehow involve the different geometrical properties having a side, and having an angle, respectively. And if being triangular were really identical to being trilateral, from the fact that John believes that a certain object has the former property, one should be able to infer that John also believes that such an object has the latter property. Yet, John's ignorance may make this conclusion unwarranted. In the light of this, borrowing a term from Cresswell (1975), one may move from intensional to hyperintensional identity conditions, according to which two properties, such as trilaterality and triangularity, may be different even if they are co-intensional. In order to implement this idea, Bealer (1982) take two properties to be identical iff they have the same analysis, i.e., roughly, they result from the same ultimate primitive properties and the same logical operations applied to them (see also Menzel 1986; 1993). Zalta (1983; 1988) has developed an alternative available to those who admit two modes of predication (see SS1.4): two properties are identical iff they are (necessarily) encoded by the same objects. Hyperintensional conditions make of course for finer distinctions among entities than the other criteria we considered. Accordingly, the former are often called "fine-grained" and the latter "coarse-grained", and the same denominations are correspondingly reserved for the entities that obey the conditions in question. Coarse- or fine-grainedness is a relative matter. The extensional criterion is more coarse-grained than the intensional or causal/nomological ones. And hyperintensional conditions themselves may be more or less fine-grained: properties could be individuated almost as finely as the predicates expressing them, to the point that, e.g., even being $P$ and $Q$ and being $Q$ and $P$ are kept distinct, but one may also envisage less stringent conditions that make for the identification of properties of that sort (Bealer 1982: 54). It is conceivable, however, that one could be logically obtuse to the point of believing that something has a certain property without believing that it has a trivially equivalent property. Thus, to properly account for mental content, maximal hyperintensionality appears to be required, and it is in fact preferred by Bealer. Even so, the paradox of analysis raises a serious issue. One could say, for example, that being a circle is being a locus of points equidistant from a point, since the latter provides the analysis of the former. In reply, Bealer distinguishes between "being a circle" as designating a simple "undefined" property, which is an analysandum distinct from the analysans, being a locus of points equidistant from a point, and "being a circle" as designating a complex "defined" property, which is indeed identical to that analysans. Orilia (1999: SS5.5) similarly distinguishes between the simple analysans and the complex analysandum, without admitting that expressions for properties such as "being a circle" could be ambiguous in the way Bealer suggests. Orilia rather argues that the "is" of analysis does not express identity but a weaker relation, which is asymmetrical in that analysans and analysandum play different roles in it. The matter keeps being discussed. Rosen (2015) appeals to grounding to characterize the different role played by the analysans. Dorr (2016) provides a formal account of the "is" used in identity statements involving properties, according to which it stands for a symmetric "identification" relation. ### 3.2 The Sparse and the Abundant Conceptions Bealer (1982) distinguishes between conception 1 properties, or qualities, and conception 2 properties, or concepts (understood as mind-independent). With a different and now widespread terminology, Lewis (1983, 1986b) followed suit, speaking of a sparse and an abundant conception of properties. According to the former, there are relatively few properties, namely just those responsible for the objective resemblances and causal powers of things; they cut nature at its joints, and science is supposed to individuate them a posteriori. According to the latter, there are immensely many properties, corresponding to all meaningful predicates we could possibly imagine and to all sets of objects, and they can be assumed a priori. (It should be clear that "few" and "many" are used in a comparative sense, for the number of sparse properties may be very high, possibly infinite). To illustrate, the sparse conception admits properties currently accepted by empirical science such as having negative charge or having spin up and rejects those that are no longer supported such as having a certain amount of caloric, which featured in eighteenth century chemistry; in contrast, the abundant conception may acknowledge the latter property and all sorts of other properties, strange as they may be, e.g., negatively charged or disliked by Socrates, round and square or Goodman's (1983) notorious grue and bleen. Lewis (1986b: 60) tries to further characterize the distinction by taking the sparse properties to be intrinsic, rather than extrinsic (e.g., being 6 feet tall, rather than being taller than Tom), and natural, where naturalness is something that admits of degrees (for example, he says, masses and charges are perfectly natural, colors are less natural and grue and bleen are paradigms of unnaturalness). Much work on such issues has been done since then (see entries on intrinsic vs. extrinsic properties and natural properties). Depending on how sparse or abundant properties are, we can have two extreme positions and other more moderate views in between. At one end of the spectrum, there is the most extreme version of the sparse conception, minimalism, which accepts all of these principles: 1. there are only coarse-grained properties, 2. they exist only if instantiated and thus are contingent beings, 3. they are all instantiated by things in space-time (setting aside those instantiated by other properties), 4. they are fundamental and thus their existence must be sanctioned by microphysics. This approach is typically motivated by physicalism and epistemological qualms regarding transcendent universals. The best-known contemporary supporter of minimalism is Armstrong (1978a,b, 1984). Another minimalist is Swoyer (1996). By dropping or mitigating some of the above principles, we get less minimalist versions of the sparse conception. For example, some have urged uninstantiated properties to account for features of measurement (Mundy 1987), vectors (Bigelow & Pargetter 1991: 77), or natural laws (Tooley 1987), and some even that there are all the properties that can be possibly exemplified, where the possibility in question is causal or nomic (Cocchiarella 2007: ch. 12). In line with positions traditionally found in emergentism (see SS5.1), Schaffer (2004) proposes that there are sparse properties as fundamental, and in addition, as grounded on them, the properties that need be postulated at all levels of scientific explanations, e.g., chemical, biological and psychological ones. Even Armstrong goes beyond minimalist strictures, when in his later work (1997) distinguishes between "first-class properties" (universals), identified from a minimalist perspective, and "second-class properties" (supervening on universals). All these positions appear to be primarily concerned with issues in the metaphysics of science (see SS5) and typically display too short a supply of properties to deal with meaning and mental content, and thus with natural language semantics and the foundations of mathematics (see SS6). However, minimalists might want to resort to concepts, understood as mind-dependent entities, in dealing with such issues (e.g., along the lines proposed in Cocchiarella 2007). Matters of meaning and mental content are instead what typically motivate the views at the opposite end of the spectrum. To begin with, maximalism, i.e., the abundant conception in all its glory (Bealer 1982; 1993; Carmichael 2010; Castaneda 1976; Jubien 1989; Lewis 1986b; Orilia 1999; Zalta 1988; Van Inwagen 2004): properties are fine-grained necessary entities that exist even if uninstantiated, or even impossible to be instantiated. In its most extreme version, maximalism adopts identity conditions that differentiate properties as much as possible, but more moderate versions can be obtained by slightly relaxing such conditions. Views of this sort are hardly concerned with physicalist constraints or the like and rather focus on the explanatory advantages of hyperintensionality. These may well go beyond the typical motivations considered above: Nolan (2014) argues that hyperintensionality is increasingly important in metaphysics in dealing with issues such as counter-possible conditionals, explanation, essences, grounding, causation, confirmation and chance. Rather than choosing between the sparse and abundant conceptions, the very promoters of this distinction have opted in different ways for a dualism of properties, according to which there are properties of both kinds. Lewis endorses abundant properties as reduced to sets of possibilia, and sparse properties either viewed as universals, and corresponding to some of the abundant properties (1983), or as themselves sets of possibilia (1986b: 60). Bealer (1982) proposes a systematic account wherein qualities are the coarse-grained properties that "provide the world with its causal and phenomenal order" (1982: 183) and concepts are the fine-grained properties that can function as meanings and as constituents of mental contents. He admits a stock of simple properties, which are both qualities and concepts (1982: 186), wherefrom complex qualities and complex concepts are differently constructed: on the one hand, by thought-building operations, which give rise to fine-grained qualities; on the other hand, by condition-building operations, which give rise to coarse-grained qualities. Orilia (1999) has followed Bealer's lead in also endorsing both coarse-grained qualities and fine-grained concepts, without however identifying simple concepts and simple qualities. In Orilia's account concepts are never identical to qualities, but may correspond to them; in particular, the identity statements of intertheoretic reductions should be taken to express the fact that two different concepts correspond to the same quality. Despite its advantages in dealing with a disparate range of phenomena, dualism has not gained any explicit consensus. Its implicit presence may however be widespread. For example, Putnam's (1970: SS1) distinction between predicates and physical properties, and Armstrong's above-mentioned recognition of second-class properties may be seen as forms of dualism. ## 4. Complex Properties It is customary to distinguish between simple and complex properties, even though some philosophers take all properties to be simple (Grossmann 1983: SSSS58-61). The former are not characterizable in terms of other properties, are primitive and unanalyzable and thus have no internal structure, whereas the latter somehow have a structure, wherein other properties, or more generally entities, are parts or constituents. It is not obvious that there are simple properties, since one may imagine that all properties are analyzable into constituents ad infinitum (Armstrong 1978b: 67). Even setting this aside, to provide examples is not easy. Traditionally, determinate colors are cited, but nowadays many would rather appeal to fundamental physical properties such as having a certain electric charge. It is easier to provide putative examples of complex properties, once some other properties are taken for granted, e.g., blue and spherical or blue or non-spherical. These logically compound properties, which involve logical operations, will be considered in SS4.1. Next, in SS4.2 we shall discuss other kinds of complex properties, called structural (after Armstrong 1978b), which are eliciting a growing interest, as a recent survey testifies (Fisher 2018). Their complexity has to do with the subdivisions of their instances into subcomponents. Typical examples come from chemistry, e.g., H2O and methane understood as properties of molecules. It should be noted that it is not generally taken for granted that complex properties literally have parts or constituents. Some philosophers take this line (Armstrong 1978a: 36-39, 67ff.; Bigelow & Pargetter 1989; Orilia 1999), whereas others demur, and rather think that talk in terms of structures with constituents is metaphorical and dependent on our reliance on structured terms such as "blue and spherical" (Bealer 1982; Cocchiarella 1986a; Swoyer 1998: SS1.2). ### 4.1 Logically Compound Properties At least prima facie, our use of complex predicates suggests that there are corresponding complex properties involving all sorts of logical operations. Thus, one can envisage conjunctive properties such as blue and spherical, negative properties such as non-spherical, disjunctive properties such as blue or non-spherical, existentially or universally quantified properties such as loving someone or loved by everyone, reflexive properties such as loving oneself, etc. Moreover, one could add to the list properties with individuals as constituents, which are then denied the status of a purely qualitative property, e.g., taller than Obama. It is easy to construct complex predicates. But whether there really are corresponding properties of this kind is a much more difficult and controversial issue, tightly bound to the sparse/abundant distinction. In the sparse conception the tendency is to dispense with them. This is understandable since in this camp one typically postulates properties in rebus for empirical explanatory reasons. But then, if we explain some phenomena by attributing properties $F$ and $G$ to an object, while denying that it does not exemplify $H$, it seems of no value to add that it also has, say, F and G, F or H, and non-H. Nevertheless, Armstrong, the leading supporter of sparseness, has defended over the years a mixed stance without disjunctive and negative properties, but with conjunctive ones. Armstrong's line has of course its opponents (see, e.g., Paolini Paoletti 2014; 2017a; 2020a; Zangwill 2011 argues that there are negative properties, but they are less real than positive ones). On the other hand, in the abundant conception, all sorts of logically compound properties are acknowledged. Since the focus now is on meaning and mental content, it appears natural to postulate such properties to account for our understanding of complex predicates and of how they differ from simple ones. Even here, however, there are disagreements, when it comes to problematic predicates such the "does not exemplify itself" of Russell's paradox. Some suggest that in this case one should deny there is a corresponding property, in order to avoid the paradox (Van Inwagen 2006). However, we understand this predicate and thus it seems ad hoc that the general strategy of postulating a property fails. It seems better to confront the paradox with other means (see SS6). ### 4.2 Structural Properties Following Armstrong (1978b: 68-71; see also Fisher 2018: SS2), a structural property $F$ is typically viewed as a universal, and can be characterized thus: 1. an object exemplifying $F$, say $x$, must have "relevant" proper parts that do not exemplify $F$; 2. there must be some other "relevant" properties somehow involved in $F$ that are not exemplified by the object in question; 3. the relevant proper parts must rather exemplify one or the other of the relevant properties involved in $F$. For instance, a molecule $w$ exemplifying H2O has as parts two atoms $h\_1$ and $h\_2$ and an atom $o$ that do not exemplify H2O; this property involves hydrogen and oxygen, which are not exemplified by $w$; $h\_1$ and $h\_2$ exemplify hydrogen, and $o$ exemplifies oxygen. For relational structural properties, there is the further condition that 4. a certain "relevant" relation that links the relevant proper parts must also be involved. H2O is a case in point: the relevant (chemical) relation is bonding, which links the three atoms in question. Another, often considered, example is methane as a structural property of methane-molecules (CH4), which involves a bonding relation between atoms and the joint instantiation of hydrogen (by four atoms) and carbon (by one atom). Non-relational structural properties do not require condition (iv). For instance, mass 1 kg involves many relevant properties of the kind mass n kg, for $n \lt 1$, which are instantiated by relevant proper parts of any object that is 1 kg in mass, and these proper parts are not linked by any relevant relation (Armstrong 1978b: 70). In most approaches, including Armstrong's, the relevant properties and the relevant relation, if any, are conceived of as parts, or constituents, of the structural property in question, and thus their being involved in the latter is a parthood relation. For instance, hydrogen, oxygen and bonding are constituents of H2O. Moreover, the composition of structural properties is isomorphic to that of the complexes exemplifying them. These two theses characterize what Lewis (1986a) calls the "pictorial conception". It is worth noting that structural properties differ from conjunctive properties such as human and musician, in that the constituents of the latter (i.e., human and musician) are instantiated by whatever entity exemplifies the conjunctive property (i.e., a human being who is also a musician), whereas the constituents of a structural property (e.g., hydrogen, oxygen and bonding) are not instantiated by the entities that instantiate it (e.g., H2O molecules). Structural properties have been invoked for many reasons. Armstrong (1978b: ch. 22) appeals to them to explain the resemblance of universals belonging to a common class, e.g., lengths and colors. Armstrong (1988; 1989) also treats physical quantities and numbers through structural properties. The former involve, as in the above mass 1 kg example, smaller quantities such as mass 0.1 kg and mass 0.2 kg (this view is criticized by Eddon 2007). The latter are internal proportion relations between structural properties (e.g., being a nineteen-electrons aggregate) and unit-properties (e.g., being an electron). Moreover, structural properties have been appealed to in treatments of laws of nature (Armstrong 1989; Lewis 1986a), some natural kinds (Armstrong 1978b, 1997; Hawley & Bird 2011), possible worlds (Forrest 1986), ersatz times (Parsons 2005), emergent properties (O'Connor & Wong 2005), linguistic types (Davis 2014), ontic structural realism (Psillos 2012). However, structural universals have also been questioned on various grounds, as we shall now see. Lewis (1986a) raises two problems for the pictorial conception. First, it is not clear how one and the same universal (e.g., hydrogen) can recur more than once in a structural universal. Let us dub this "multiple recurrence problem". Secondly, structural universals violate the Principle of Uniqueness of Composition of classical mereology. According to this principle, given a certain collection of parts, there is only one whole that they compose. Consider isomers, namely, molecules that have the same number and types of atoms but different structures, e.g., butane and isobutane (C4H10). Here, butane and isobutane are different structural universals. Yet, they arise from the same universals recurring the same number of times. Let us dub this "isomer problem." Two further problems may be pointed out. First, even molecules with the same number and types of atoms and the same structures may vary in virtue of their spatial orientation (Kalhat 2008). This phenomenon is known as chirality. How can structural properties account for it? Secondly, the composition of structural properties is restricted: not every collection of properties gives rise to a structural property. This violates another principle of classical mereology: the Principle of Unrestricted Composition. Lewis dismisses alternatives to the pictorial conception, such as the "linguistic conception" (in which structural universals are set-theoretic constructions out of simple universals) and the "magic conception" (in which structural universals are not complex and are primitively connected to the relevant properties). Moreover, he also rejects the possibility of there being amphibians, i.e., particularized universals lying between particulars and full-fledged universals. Amphibians would solve the multiple recurrence problem, as they would be identical with multiple recurrences of single universals. For example, in methane, there would be four amphibians of hydrogen. To reply to Lewis' challenges, two main strategies have been adopted on behalf of the pictorial conception: a non-mereological composition strategy, according to which structural properties do not obey the principles of classical mereology, and a mereology-friendly strategy. Let us start with the former. Armstrong (1986) admits that the composition of structural universals is non-mereological. In his 1997, he emphasizes that states of affairs do not have a mereological kind of composition, and views universals, including structural ones, as states of affairs-types which, as such, are not mereological in their composition. Structural universals themselves turn out to be state of affairs-types of a conjunctive sort (1997: 34 ff.). To illustrate, let us go back to the above example with H2O exemplified by the water molecule $w$. It involves the conjunction of these states of affairs: 1. atom $h\_1$ being hydrogen, 2. atom $h\_2$ being hydrogen, 3. atom $o$ being oxygen, 4. $h\_1$, $h\_2$, and $o$ being bonded. This conjunction of states of affairs provides an example of a state of affairs-type identifiable with the structural universal H2O. In pursuit of the non-mereological composition strategy, Forrest (1986; 2006; 2016) relies on logically compound properties, in particular (in his 2016) conjunctive, existentially quantified and reflexive ones. Bennett (2013) argues that entities are part of further entities by occupying specific slots within the latter. Parthood slots are distinct from each other. Therefore, which entity occupies which slot matters to the composition of the resultant complex entity. Hawley (2010) defends the possibility of there being multiple composition relations. In a more general way, McDaniel (2009) points out that the relation of structure-making does not obey some of the principles of classical mereology. Mormann (2010) argues that distinct categories (to be understood as in category theory) come together with distinct parthood and composition relations. As regards the mereology-friendly strategy, it has been suggested that structural properties actually include extra components accounting for structures, so that the Principle of Uniqueness of Composition is safe. Considering methane, Pages (2002) claims that such a structural property is composed of the properties carbon and hydrogen, a bonding relation and a peculiar first-order, formal relation between the atoms. According to Kalhat (2008), the extra component is a second-order arrangement relation between the first-order properties. Ordering formal relations--together with causally individuated properties--are also invoked by McFarland (2018). At the intersection of these strategies, Bigelow and Pargetter (1989, 1991) claim that structural universals are internally relational properties that supervene on first-order properties and relations and on second-order proportion relations. Second-order proportion relations are relations such as having four times as. They relate first-order properties and relations. For example, in methane, the relation having four times as relates two conjunctive properties: (i) hydrogen and being part of this molecule; (ii) carbon and being part of this molecule. Campbell (1990) solves the multiple recurrence problem and the isomer problem by appealing to tropes. In methane, there are multiple hydrogen tropes. In butane and isobutane, distinct tropes entertain distinct structural relations. Despite Lewis' dismissal, a number of theories appeal to amphibians (i.e., particularized universals lying between particulars and full-fledged universals) or, more precisely, to entities similar to amphibians. Fine (2017) suggests that structural properties should be treated by invoking arbitrary objects (Fine 1985). Davis' (2014) account of linguistic types as structural properties confronts the problem of the multiple occurrence of one type, e.g., "dog", in another (e.g., "every dog likes another dog") (Wetzel 2009). In general, the idea that types are properties of tokens is a natural one, although there are also alternative views on the matter (see entry on types and tokens). Be this as it may, Davis proposes that types cannot occur within further types as tokens: they can only occur as subtypes. Therefore, subtypes lie, like amphibians, between types and tokens. Subtypes are individuated by their positions in asymmetric wholes, whereas they are primitively and non-qualitatively distinct in symmetric ones. The approach could be extended to properties such as methane, which would be taken to have four distinct hydrogen subtypes. ## 5. Properties in the Metaphysics of Science When it comes to the metaphysical underpinnings of scientific theories, properties play a prominent role: it appears that science can hardly be done without appealing to them. This adds up to the case for realism about properties and to our understanding of them. We shall illustrate this here first by dwelling on some miscellaneous topics and then by focusing on a debate regarding the very nature of the properties invoked in science, namely whether or not they are essentially dispositional. Roughly, an object exemplifies a dispositional property, such as soluble or fragile, by having a power or disposition to act or being acted upon in a certain way in certain conditions. For example, a glass' disposition of being fragile consists in its possibly shattering in certain conditions (e.g., if struck with a certain force). In contrast, something exemplifies a categorical property, e.g., made of salt or spherical, by merely being in a certain way (see entry on dispositions). ### 5.1 Miscellaneous Topics Many predicates in scientific theories (e.g., "being a gene" and "being a belief") are functionally defined. Namely, their meaning is fixed by appealing to some function or set of functions (e.g., encoding and transmitting genetic information). To make sense of functions, properties are needed. First, functions can be thought of as webs of causal relationships between properties or property-instances. Secondly, predicates such as "being a gene" may be taken to refer to higher-level properties, or at least to something that has such properties. In this case, the property of being a gene would be the higher-level property of possessing some further properties (e.g., biochemical ones) that play the relevant functions (i.e., encoding and transmitting genetic information). Alternatively, being a gene would be a property that satisfies the former higher-level property. Thirdly, the ensuing realization relationships between higher-level and function-playing properties can only be defined by appealing to properties (see entry on functionalism). More generally, many ontological dependence and reduction relationships primarily concern properties: type-identity (Place 1956; Smart 1959; Lewis 1966) and token-identity (Davidson 1980) (see entry on the mind/brain identity theory), supervenience (Horgan 1982; Kim 1993; entry on supervenience); realization (Putnam 1975; Wilson 1999; Shoemaker 2007), scientific reduction (Nagel 1961). Moreover, a non-reductive relation such as ontological emergence (Bedau 1997; Paolini Paoletti 2017b) is typically characterized as a relation between emergent properties and more fundamental properties. Even mechanistic explanations often appeal to properties and relations in characterizing the organization, the components, the powers and the phenomena displayed by mechanisms (Glennan & Illari 2018; entry on mechanisms in science). Entities in nature are typified by natural kinds. These are mostly thought of as properties or property-like entities that carve nature at its joints (Campbell, O'Rourke, & Slater 2011) at distinct layers of the universe: microphysical (e.g., being a neutron), chemical (e.g., being gold), biological (e.g., being a horse). Physical quantities such as mass or length are typically treated as properties, specifiable in terms of a magnitude, a certain number, and a unit measure, e.g., kg or meter, which can itself be specified in terms of properties (Mundy 1987; Swoyer 1987). Following Eddon's (2013) overview, it is possible to distinguish two main strands here: relational and monadic properties theories (see also Dasgupta 2013). In relational theories (Bigelow & Pargetter 1988; 1991), quantities arise from proportion relations, which may in turn be related by higher-order proportion relations. Consider a certain quantity, e.g., mass 3 kg. Here the thrice as massive as proportion relation is relevant, which holds of certain pairs of objects $a$ and $b$ such that $a$ is thrice as massive as $b$. By virtue of such relational facts, the first members of such pairs have a mass of 3 kg. Another relational approach is by Mundy (1988), who claims that mass 3 kg is a relation between (ordered pairs of) objects and numbers. For example, mass 3 kg is a relation holding between the ordered pair $\langle a, b\rangle$ (assuming that $a$ is thrice as massive as $b$, and not the other way round) and the number 3; or--if relations have built-in relational order--between $a, b$ and 3. Knowles (2015) holds a similar view, wherein physical quantities are relations that objects bear to numbers. According to the monadic properties approach, quantities are intrinsic properties of objects (Swoyer 1987; Armstrong 1988; 1989). As we saw, Armstrong develops such a view by taking quantities to be structural properties. Related metaphysical inquiries concern the dimensions of quantities (Skow 2017) and the status of forces (Massin 2009). Lastly, properties play a prominent role in two well-known accounts of the laws of nature: the nomological necessitation account and law dispositionalism. The former has been expounded by Tooley (1977; 1987), Dretske (1977) and Armstrong (1983). Roughly, following Armstrong, a law of nature consists in a second-order and external nomological necessitation relation $\rN$, contingently holding between first-order universals $P$ and $Q$: $\rN(P, Q)$. Such a higher-order fact necessitates certain lower-order regularities, i.e., all the objects that have $P$ also have $Q$ (by nomological necessity, in virtue of $\rN(P, Q))$. Law dispositionalism interprets laws of nature by appealing to dispositional properties. According to it, laws of nature "flow" from the essence of such properties. This implies that laws of nature hold with metaphysical necessity: whenever the dispositions are in place, the relevant laws must be in place too (Cartwright 1983; Ellis 2001; Bird 2007; Chakravartty 2007; Fischer 2018; see also Schrenk 2017 and Dumsday 2019). (For criticisms, see, e.g., van Fraassen 1989 as regards the former approach, and McKitrick 2018 and Paolini Paoletti 2020b, as regards the latter). ### 5.2. Essentially Categorical vs. Essentially Dispositional Properties There is a domestic dispute among supporters of properties in the metaphysics of science, regarding the very nature of such properties (or at least the fundamental ones). We may distinguish two extreme views, pure dispositionalism and pure categoricalism. According to the former, all properties are essentially dispositional ("dispositional", for brevity's sake, from now on), since they are nothing more than causal powers; their causative roles, i.e., which effects their instantiations can cause, exhaust their essences. According to pure categoricalism, all properties are essentially categorical (in brief, "categorical", in the following), because their causative roles are not essential to them. If anything is essential to a property, it is rather a non-dispositional and intrinsic aspect called "quiddity" (after Armstrong 1989), which need not be seen as an additional entity over and above the property itself (Locke 2012, in response to Hawthorne 2001). Between such views there lie a number of intermediate positions. Pure dispositionalism has been widely supported in the last few decades (Mellor 1974; 2000; Shoemaker 1984: ch. 10 and 11 [in terms of causal roles]; Mumford 1998; 2004; Bird 2005; 2007; Chakravartty 2007; Whittle 2008; see Tugby 2013 for an attempt to argue from this sort of view to a Platonist conception of properties). There are three main arguments in favor of it. First, pure dispositionalism easily accounts for the natural necessity of the laws of nature, insofar as such a necessity just "flows" from the essence of the dispositional properties involved. Secondly, dispositional properties can be easily known as they really are, because it is part of their essence that they affect us in certain ways. Thirdly, at the (presumably fundamental) micro-physical level, properties are only described dispositionally, which is best explained by their being dispositional (Ellis & Lierse 1994; N. Williams 2011). Nevertheless, pure dispositionalism is affected by several problems. First, some authors believe that it is not easy to provide a clear-cut distinction between dispositional and would-be non-dispositional properties (Cross 2005). Secondly, it seems that the essence of certain properties does not include, or is not exhausted by, their causative roles: qualia, inert and epiphenomenal properties,[14] structural and geometrical properties, spatio-temporal properties (Prior 1982; Armstrong 1999; for some responses, see Mellor 1974 and Bird 2007; 2009). Thirdly, there can be symmetrical causative roles. Consider three distinct properties $A$, $B$ and $C$ such that $A$ can cause $B$, $B$ can cause $A$, $A$ and $B$ can together cause $C$ and nothing else characterizes $A$ and $B$. $A$ and $B$ have the same causative role. Therefore, in pure dispositionalism, they turn out to be identical, against the hypothesis (Hawthorne 2001; see also Contessa 2019). Fourthly, according to some, pure dispositionalism falls prey to (at least) three distinct regresses (for a fourth regress, see Psillos 2006). Such regresses arise from the fact that the essential causative role of a dispositional property $P$ "points towards" further properties $S$, $T$, etc. qua possible effects. The essential causative roles of the latter "point towards" still other properties, and so on. The first regress concerns the identity of $P$, which is never fixed, as it depends on the identity of further properties, which depend for their identity on still other properties, and so on (Lowe 2006; Barker 2009; 2013). The second regress concerns the knowability of $P$ (Swinburne 1980). $P$ is only knowable through its possible effects (i.e., the instantiation of $S$, $T$, etc.), included in its causative role. Yet, such possible effects are only knowable through their possible effects, and so on. The third regress concerns the actuality of $P$ (Armstrong 1997). $P$'s actuality is never reached, since $P$ is nothing but the power to give rise to $S$, $T$, etc., which are nothing but the powers to give rise to further properties, and so on. For responses to these regresses, see Molnar 2003; Marmodoro 2009; Bauer 2012; McKitrick 2013; see also Ingthorsson 2013. Pure categoricalism seems to imply that causative roles are only contingently associated to a property. Therefore, on pure categoricalism, a property can possibly have distinct causative roles, which allows it to explain--among other things--the apparent contingency of causative roles and the possibility of recombining a property with distinct causative roles. Its supporters include Lewis (1986b, 2009), Armstrong (1999), Schaffer (2005), and more recently Livanios (2017), who provides further arguments based on the metaphysics of science. Kelly (2009) and Smith (2016) may be added to the list, although they take roles to be non-essential and necessary (for further options, see also Kimpton-Nye 2018, Yates 2018a, Coates forthcoming and Tugby forthcoming). However, pure categoricalism falls prey to two sorts of difficulties. First, the contingency of causative roles has some unpalatable consequences: unbeknownst to us, two distinct properties can swap their roles; they can play the same role at the same time; the same role can be played by one property at one time and by another property at a later time; an "alien" property can replace a familiar one by playing its role (Black 2000). Secondly, and more generally, we are never able to know which properties play which roles, nor are we able to know the intrinsic nature of such properties. This consequence should be accepted with "Ramseyan humility" (Lewis 2009; see also Langton 1998 for a related, Kantian, sort of humility) or it should be countered in the same way as we decide to counter any broader version of skepticism (Schaffer 2005). On this issue, see also Whittle (2006); Locke (2009); Kelly (2013); Yates (2018b). Let us now turn to some intermediate positions. According to dualism (Ellis 2001; Molnar 2003), there are both dispositional and categorical properties. Dualism is meant to combine the virtues of pure dispositionalism and pure categoricalism. It then faces the charge of adopting a less parsimonious ontology, since it accepts two classes of properties rather than one, i.e., dispositional and categorical ones. According to the identity theory (Heil 2003; 2012; Jaworski 2016; Martin 2008; G. Strawson 2008; see also N. Williams 2019), every property is both dispositional and categorical (or qualitative). The main problem here is how to characterize the distinction and relation of these two "sides". Martin and Heil suggest that they are two distinct ways of partially considering one and the same property, whereas Mumford (1998) explores the possibility of seeing them as two distinct ways of conceptualizing the property in question. Heil claims that the qualitative and the dispositional sides need to be identified with one another and with the whole property. Jacobs (2011) holds that the qualitative side consists in the possession of some qualitative nature by the property, whereas the dispositional side consists in that property being (part of) a sufficient truthmaker for certain counterfactuals. Dispositional and qualitative sides may also be seen as essential, higher-order properties of properties, as supervenient and ontologically innocent aspects of properties (Giannotti 2019), or as constituents of the essence of properties (Taylor 2018). In general, the identity theory is between Scylla and Charybdis. If it reifies the dispositional and qualitative sides, it runs the risk of implying some sort of dualism. If it insists on the identity between them, it runs the opposite risk of turning into a pure dispositionalist theory (Taylor 2018). ## 6. Formal Property Theories and their applications Formal property theories are logical systems that aim at formulating "general noncontingent laws that deal with properties" (Bealer & Monnich 1989: 133). In the next subsection we shall outline how they work. In the two subsequent subsections we shall briefly consider their deployment in natural language semantics and the foundations of mathematics, which can be taken to provide further reasons for the acknowledgment of properties in one's ontology, or at least of certain kinds of properties. ### 6.1 Logical Systems for Properties These systems allow for terms corresponding to properties, in particular variables that are meant to range over properties and that can be quantified over. This can be achieved in two ways. Either (option 1; Cocchiarella 1986a) the terms standing for properties are predicates or (option 2; Bealer 1982) such terms are subject terms that can be linked to other subject terms by a special predicate that is meant to express a predication relation (let us use "pred") pretty much as in standard set theory a special predicate, "$\in$", is used to express the membership relation. To illustrate, given the former option, an assertion such as "there is a property that both John and Mary have" can be rendered as \[\notag \exists P(P(j) \amp P(m)).\] Given the second option, it can be rendered as \[\notag \exists x(\textrm{pred}(x, j) \amp \textrm{pred}(x, m)).\] (The two options can be combined as in Menzel 1986; see Menzel 1993 for further discussion). Whatever option one follows, in spelling out such theories one typically postulates a rich realm of properties. Traditionally, this is done by a so-called comprehension principle which, intuitively, asserts that, for any well-formed formula ("wff") $A$, with $n$ free variables, $x\_1 , \ldots ,x\_n$, there is a corresponding $n$-adic property. Following option 1, it goes as follows: \[\tag{CP} \exists R^n \forall x\_1 \ldots \forall x\_n(R^n(x\_1 , \ldots ,x\_n) \leftrightarrow A). \] Alternatively, one can use a variable-binding operator, $\lambda$, that, given an open wff, generates a term (called a "lambda abstract") that is meant to stand for a property. This way to proceed is more flexible and is followed in the most recent versions of property theory. We shall thus stick to it in the following. To illustrate, we can apply "$\lambda$" to the open formula, "$(R(x) \amp S(x))$" to form the one-place complex predicate "[$\lambda x (R(x) \amp S(x))$]"; if "$R$" denotes being red and "$S$" denotes being square, then this complex predicate denotes the compound, conjunctive property being red and square. Similarly, we can apply the operator to the open formula "$\exists y(L(x, y))$" to form the one-place predicate "[$\lambda x \exists y(L(x, y))$]"; if "$L$" stands for loves, this complex predicate denotes the compound property loving someone (whereas "[$\lambda y \exists x(L(x, y))$]" would denote being loved by someone). To ensure that lambda abstracts designate the intended property, one should assume a "principle of lambda conversion". Given option 1, it can be stated thus: \[\tag{$\lambda$-conv} [\lambda x\_1\ldots x\_n A](t\_1 , \ldots ,t\_n) \leftrightarrow A(x\_1 /t\_1 , \ldots ,x\_n /t\_n).\] $A(x\_1 /t\_1 , \ldots ,x\_n /t\_n)$ is the wff resulting from simultaneously replacing each $x\_i$ in $A$ with $t\_i$ (for $1 \le i \le n)$, provided $t\_i$ is free for $x\_i$ in $A$). For example, given this principle, [$\lambda x (R(x) \amp S(x))](j)$ is the case iff $(R(j) \amp S(j))$ is also the case. Standard second-order logic allows for predicate variables bound by quantifiers. Hence, to the extent that these variables are taken to range over properties, this system could be seen as a formal theory of properties. Its expressive power is however limited, since it does not allow for subject terms that stand for properties. Thus, for example, one cannot even say of a property $F$ that $F = F$. This is a serious limitation if one wants a formal tool for a realm of properties whose laws one is trying to explore. Standard higher order logics beyond the second order obviate this limitation by allowing for predicates in subject position, provided that the predicates that are predicated of them belong to a higher type. This presupposes a grammar in which predicates are assigned types of increasing levels, which can be taken to mean that the properties themselves, for which the predicates stand for, are arranged into a hierarchy of types. Thus, such logics appropriate one version or another of the type theory concocted by Russell to tame his own paradox and related conundrums. If a predicate can be predicated of another predicate only if the former is of a type higher than the latter, then self-predication is banished and Russell's paradox cannot even be formulated. Following this line, we can construct a type-theoretical formal property theory. The simple theory of types, as presented, e.g., in Copi (1971), can be seen as a prototypical version of such a property theory (if we neglect the principle of extensionality assumed by Copi). The type-theoretical approach keeps having supporters. For example, it is followed in the property theories embedded in Zalta's (1983) theory of abstract objects and more recently in the metaphysical systems proposed by Williamson (2013) and Hale (2013). However, for reasons sketched in SS2.4, type theory is hardly satisfactory. Accordingly, many type-free versions of property theory have been developed over the years and no consensus on what the right strategy is appears to be in sight. Of course, without type-theoretical constraints, given $(\lambda$-conv) and classical logic (CL), paradoxes such as Russell's immediately follow (to see this, consider this instance of $(\lambda\textrm{-conv}): [\lambda x {\sim}x(x)]([\lambda x {\sim}x(x)]) \leftrightarrow{\sim}[\lambda x {\sim}x(x)]([\lambda x {\sim}x(x)])$). In formal systems where abstract singular terms or predicates may (but need not) denote properties (Swoyer 1998), formal counterparts of (complex) predicates like "being a property that does not exemplify itself" (formally, "[$\lambda x {\sim}x(x)$]") could exist in the object language without denoting properties; from this perspective, Russell's paradox merely shows that such predicates do not stand for properties (similarly, according to Schnieder 2017, it shows that it is not possible that a certain property exists). But we would like to have general criteria to decide when a predicate stands for a property and when it does not. Moreover, one may wonder what gives these predicates any significance at all if they do not stand for properties. There are then motivations for building type-free property theories in which all predicates stand for properties. We can distinguish two main strands of them: those that weaken CL and those that circumscribe $(\lambda$-conv) (some of the proposals to be mentioned below are formulated in relation to set theory, but can be easily translated into proposals for property theory). An early example of the former approach was offered in a 1938 paper by the Russian logician D. A. Bochvar (Bochvar 1938 [1981]), where the principle of excluded middle is sacrificed as a consequence of the adoption of what is now known as Kleene's weak three-valued scheme. An interesting recent attempt based on giving up excluded middle is Field 2004. A rather radical alternative proposal is to embrace a paraconsistent logic and give up the principle of non-contradiction (Priest 1987). A different way of giving up CL is by questioning its structural rules and turn to a substructural logic, as in Mares and Paoli (2014). The problem with all these approaches is whether their underlying logic is strong enough for all the intended applications of property theory, in particular to natural language semantics and the foundations of mathematics. As for the second strand (based on circumscribing $(\lambda$-conv)), it has been proposed to read the axioms of a standard set theory such as ZFC, minus extensionality, as if they were about properties rather than sets (Schock 1969; Bealer 1982; Jubien 1989). The problem with this is that these axioms, understood as talking about sets, can be motivated by the iterative conception of sets, but they seem rather ad hoc when understood as talking about properties (Cocchiarella 1985). An alternative can be found in Cocchiarella 1986a, where $(\lambda$-conv) is circumscribed by adapting to properties the notion of stratification used by Quine for sets. This approach is however subject to a version of Russell's paradox derivable from contingent but intuitively possible facts (Orilia 1996) and to a paradox of hyperintensionality (Bozon 2004) (see Landini 2009 and Cocchiarella 2009 for a discussion of both). Orilia (2000; Orilia & Landini 2019) has proposed another strategy for circumscribing $(\lambda$-conv), based on applying to exemplification Gupta's and Belnap's theory of circular definitions. Independently of the paradoxes (Bealer & Monnich 1989: 198 ff.), there is the issue of providing identity conditions for properties, specifying when it is the case that two properties are identical. If one thinks of properties as meanings of natural language predicates and tries to account for intensional contexts, one will be inclined to assume rather fine-grained identity conditions, possibly even allowing that [$\lambda x(R(x) \amp S(x))$] and [$\lambda x(S(x) \amp R(x))$] are distinct (see Fox & Lappin 2015 for an approach based on operational distinctions among computable functions). On the other hand, if one thinks of properties as causally operative entities in the physical world, one will want to provide rather coarse-grained identity conditions. For instance, one might require that [$\lambda x A$] and [$\lambda x B$] are the same property iff it is physically necessary that $\forall x(A \leftrightarrow B)$. Bealer (1982) tries to combine the two approaches (see also Bealer & Monnich 1989)[15]. ### 6.2 Semantics and Logical Form The formal study of natural language semantics started with Montague and gave rise to a flourishing field of inquiry (see entry on Montague semantics). The basic idea in this field is to associate to natural language sentences wffs of a formal language, in order to represent sentence meanings in a logically perspicuous manner. The association reflects the compositionality of meanings: different syntactic subcomponents of sentences correspond systematically to syntactic subcomponents of the wffs; subcomponents of wffs thus represent the meanings of the subcomponents of the sentences. The formal language eschews ambiguities and has its own formal semantics, which grants that formulas have logical properties and relations, such as logical truth and entailments, so that in particular certain sequences of formulas count as logically valid arguments. The ambiguities we normally find in natural language sentences and the entailment relations that link them are captured by associating ambiguous sentences to different unambiguous wffs, in such a way that when a natural language argument is felt to be valid there is a corresponding sequence of wffs that count as a logically valid argument. In order to achieve all this, Montague appealed to a higher-order logic. To see why this was necessary, one may focus on this valid argument: (1) every Greek is mortal; (2) the president of Greece is Greek; therefore, (3) the president of Greece is mortal. To grant compositionality in a way that respects the syntactic similarity of the three sentences (they all have the same subject-predicate form), and the validity of the argument, Montague associates (1)-(3) to formulas such as these: \[ \tag{1a} [\lambda F \forall x(\mathrm{G}(x) \rightarrow F(x)](\mathrm{M}); \] \[ \tag{2a} [\lambda F \exists x(\forall y(P(x) \rightarrow x = y) \amp F(x))](G); \] \[ \tag{3a} [\lambda F \exists x(\forall y(P(x) \rightarrow x = y) \amp F(x))](\mathrm{M}). \] The three lambda abstracts in (1a)-(3a) represent, respectively, the meanings of the three noun phrases in (1)-(3). These lambda-abstracts occur in predicate position, as predicates of predicates, so that (1a)-(3a) can be read, respectively, as: every Greek is instantiated by being mortal; the president of Greece is instantiated by being Greek; the president of Greece is instantiated by being mortal. Given lambda-conversion plus quantifier and propositional logic, the argument is valid, as desired. It should be noted that lambda abstracts such as these can be taken to stand for peculiar properties of properties, classifiable as denoting concepts (after Russell 1903; see Cocchiarella 1989). One may then say then this approach to semantics makes a case for the postulation of denoting concepts, in addition to the more obvious and general fact that it grants properties as meanings of natural language predicates (represented by symbols of the formal language). This in itself says nothing about the nature of such properties. As we saw in SS3.1, Montague took them to be intensions as set-theoretically characterized in terms of possible worlds. Moreover, he took them to be typed, since, to avoid logical paradoxes, he relied on type theory. After Montague, these two assumptions have been typically taken for granted in natural language semantics, though with attempts to recover hyperintensionality somehow (Cresswell 1985) in order to capture the semantics of propositional attitudes verbs such as "believe", affected by the phenomena regarding mental content hinted at in SS3.1. However, the development of type-free property theories has suggested the radically different road of relying on them to provide logical forms for natural language semantics (Chierchia 1985; Chierchia & Turner 1988; Orilia 2000b; Orilia & Landini 2019). This allows one to capture straightforwardly natural language inferences that appear to presuppose type-freedom, as they feature quantifiers that bind simultaneously subject and predicate positions (recall the example of SS1.2). Moreover, by endowing the selected type-free property theory with fine-grained identity conditions, one also accounts for propositional attitude verbs (Bealer 1989). Thus, we may say that this line makes a case for properties understood as untyped and highly fine-grained. ### 6.3 Foundations of Mathematics Since the systematization in the first half of last century, which gave rise to paradox-free axiomatizations of set theory such as ZFC, sets are typically taken for granted in the foundations of mathematics and it is well known that they can do all the works that numbers can do. This has led to the proposal of identifying numbers with sets. Russell's type theory was an alternative that rather relied on properties (viewed as propositional functions), in backing up a logicist reduction of mathematics to logic. In essence, the idea was that properties can do all the work that sets are supposed to do, thus making the latter dispensable. Hence, Russell spoke of his approach as a "no-classes" theory of classes (see Landini 2011: 115, and SS2.4 of the entry on Russell's logical atomism; cf. Bealer 1982: 111-119, and Jubien 1989 for followers of this line). Following this line, numbers are then seen as properties rather than sets. The Russellian approach did not encounter among mathematicians a success comparable to that of set theory. Nevertheless, from an ontological point of view, it appears to be more economical in its relying on properties, to the extent that the latter are needed any way for all sorts of explanatory jobs, reviewed above, which sets, qua extensional entities, can hardly perform. As we have seen, type theory is problematic. However, type-free property theories can come to the rescue by replacing typed properties with untyped ones in the foundations of mathematics. And in fact the advocates of such theories have often proposed to recover the logicist program, in particular by identifying natural numbers with untyped properties of properties (Bealer 1982; Cocchiarella 1986b; Orilia 2000b). (See also entry on logicism and neologicism).
universals-medieval	## 1. Introduction The inherent problems with Plato's original theory were recognized already by Plato himself. In his Parmenides Plato famously raised several difficulties, for which he apparently did not provide satisfactory answers. Aristotle (384-322 B.C.E.), with all due reverence to his teacher, consistently rejected Plato's theory, and heavily criticized it throughout his own work. (Hence the famous saying, amicus Plato sed magis amica veritas).[1] Nevertheless, despite this explicit doctrinal conflict, Neo-Platonic philosophers, pagans (such as Plotinus ca. 204-270, and Porphyry, ca. 234-305) and Christians (such as Augustine, 354-430, and Boethius, ca. 480-524) alike, observed a basic concordance between Plato's and Aristotle's approach, crediting Aristotle with an explanation of how the human mind acquires its universal concepts of particular things from experience, and Plato with providing an explanation of how the universal features of particular things are established by being modeled after their universal archetypes.[2] In any case, it was this general attitude toward the problem in late antiquity that set the stage for the ever more sophisticated medieval discussions.[3] In these discussions, the concepts of the human mind, therefore, were regarded as posterior to the particular things represented by these concepts, and hence they were referred to as universalia post rem ('universals after the thing'). The universal features of singular things, inherent in these things themselves, were referred to as universalia in re ('universals in the thing'), answering the universal exemplars in the divine mind, the universalia ante rem ('universals before the thing').[4] All these, universal concepts, universal features of singular things, and their exemplars, are expressed and signified by means of some obviously universal signs, the universal (or common) terms of human languages. For example, the term 'man', in English is a universal term, because it is truly predicable of all men in one and the same sense, as opposed to the singular term 'Socrates', which in the same sense, i.e., when not used equivocally, is only predicable of one man (hence the need to add an ordinal number to the names of kings and popes of the same name). Depending on which of these items (universal features of singular things, their universal concepts, or their universal names) they regarded as the primary, really existing universals, it is customary to classify medieval authors as being realists, conceptualists, or nominalists, respectively. The realists are supposed to be those who assert the existence of real universals in and/or before particular things, the conceptualists those who allow universals only, or primarily, as concepts of the mind, whereas nominalists would be those who would acknowledge only, or primarily, universal words. But this rather crude classification does not adequately reflect the genuine, much more subtle differences of opinion between medieval thinkers. (No wonder one often finds in the secondary literature distinctions between, "moderate" and "extreme" versions of these crudely defined positions.) In the first place, nearly all medieval thinkers agreed on the existence of universals before things in the form of divine ideas existing in the divine mind,[5] but all of them denied their existence in the form of mind-independent, real, eternal entities originally posited by Plato. Furthermore, medieval thinkers also agreed that particular things have certain features which the human mind is able to comprehend in a universal fashion, and signify by means of universal terms. As we shall see, their disagreements rather concerned the types of the relationships that hold between the particular things, their individual, yet universally comprehensible features, the universal concepts of the mind, and the universal terms of our languages, as well as the ontological status of, and distinctions between, the individualized features of the things and the universal concepts of the mind. Nevertheless, the distinction between "realism" and "nominalism", especially, when it is used to refer to the distinction between the radically different ways of doing philosophy and theology in late-medieval times, is quite justifiable, provided we clarify what really separated these ways, as I hope to do in the later sections of this article. In this brief summary account, I will survey the problem both from a systematic and from a historical point of view. In the next section I will first motivate the problem by showing how naturally the questions concerning universals emerge if we consider how we come to know a universal claim, i.e., one that concerns a potentially infinite number of particulars of a given kind, in a simple geometrical demonstration. I will also briefly indicate why a naive Platonic answer to these questions in terms of the theory of perfect Forms, however plausible it may seem at first, is inadequate. In the third section, I will briefly discuss how the specific medieval questions concerning universals emerged, especially in the context of answering Porphyry's famous questions in his introduction to Aristotle's Categories, which will naturally lead us to a discussion of Boethius' Aristotelian answers to these questions in his second commentary on Porphyry in the fourth section. However, Boethius' Aristotelian answers anticipated only one side of the medieval discussions: the mundane, philosophical theory of universals, in terms of Aristotelian abstractionism. But the other important, Neo-Platonic, theological side of the issue provided by Boethius, and, most importantly, by St. Augustine, was for medieval thinkers the theory of ontologically primary universals as the creative archetypes of the divine mind, the Divine Ideas. Therefore, the fifth section is going to deal with the main ontological and epistemological problems generated by this theory, namely, the apparent conflict between divine simplicity and the multiplicity of divine ideas, on the one hand, and the tension between the Augustinian theory of divine illumination and Aristotelian abstractionism, on the other. Some details of the early medieval Boethian-Aristotelian approach to the problem and its combination with the Neo-Platonic Augustinian tradition before the influx of the newly recovered logical, metaphysical, and physical writings of Aristotle and their Arabic commentaries in the second half of the 12th century will be taken up in the sixth section, in connection with Abelard's (1079-1142) discussion of Porphyry's questions. The seventh section will discuss some details of the characteristic metaphysical approach to the problem in the 13th century, especially as it was shaped by the influence of Avicenna's (980-1037) doctrine of common nature. The eighth section outlines the most general features of the logical conceptual framework that served as the common background for the metaphysical disagreements among the authors of this period. I will argue that it is precisely this common logical-semantical framework that allows the grouping together of authors who endorse sometimes radically different metaphysics and epistemologies (not only in this period, but also much later, well into the early modern period) as belonging to what in later medieval philosophy came to be known as the "realist" via antiqua, the "old way" of doing philosophy and theology. By contrast, it was precisely the radically different logical-semantical approach initiated by William Ockham (ca. 1280-1350), and articulated and systematized most powerfully by Jean Buridan (ca. 1300-1358), that distinguished the "nominalist" via moderna, the "modern way" of doing philosophy and theology from the second half of the 14th century. The general, distinctive characteristics of this "modern way" will be the discussed in the ninth section. Finally, the concluding tenth section will briefly indicate how the separation of the two viae, in addition to a number of extrinsic social factors, contributed to the disintegration of scholastic discourse, and thereby to the disappearance of the characteristically medieval problem of universals, as well as to the re-mergence of recognizably the same problem in different guises in early modern philosophy. ## 2. The Emergence of the Problem It is easy to see how the problem of universals emerges, if we consider a geometrical demonstration, for example, the demonstration of Thales' theorem. According to the theorem, any triangle inscribed in a semicircle is a right triangle, as is shown in the following diagram: ![Points A, B, and D are on a circle with center C. Line AB is a horizontal diameter through C. Lines AD, CD, and BD are also drawn forming 3 triangles: ABD, ACD, and DCB](image1.gif) Figure 1. Thales' theorem Looking at this diagram, we can see that all we need to prove is that the angle at vertex D of triangle ABD is a right angle. The proof is easy once we realize that since lines AC, DC, and BC are the radii of a circle, the triangles ACD and DCB are isosceles triangles, whence their base angles are equal. For then, if we denote the angles of ABD by the names of their vertices, this fact entails that D=A + B; and so, since A + B + D=180o, it follows that 2A + 2B=180o; therefore, A + B=90o, that is, D=90o, q. e. d. Of course, from our point of view, the important thing about this demonstration is not so much the truth of its conclusion as the way it proves this conclusion. For the conclusion is a universal theorem, which has to concern all possible triangles inscribed in any possible semicircle whatsoever, not just the one inscribed in the semicircle in the figure above. Yet, apparently, in the demonstration above we were talking only about that triangle. So, how can we claim that whatever we managed to prove concerning that particular triangle will hold for all possible triangles? If we take a closer look at the diagram, we can easily see the appeal of the Platonic answer to this question. For upon a closer look, it is clear that, despite appearances to the contrary, this demonstration cannot be about the triangle in this diagram. Indeed, in the demonstration we assumed that the lines AC, DC, and BC were all perfectly equal, straight lines. However, if we zoom in on the figure, we can clearly see that these lines are far from being equal; in fact, they are not even straight lines: ![Figure 1 magnified. The straight and circular lines of Figure 1 are no longer smooth; under increased magnification, they now appear as 'jagged' lines](image2.gif) Figure 2. The result of zooming in on Figure 1. The demonstration was certainly not about the collection of jagged black surfaces that we can see here. Rather, the demonstration concerned something we did not see with our bodily eyes, but what we had in mind all along, understanding it to be a triangle, with perfectly straight edges, touching a perfect circle in three unextended points, which are all perfectly equidistant from the center of the circle. The figure we could see was only a convenient "reminder" of what we are supposed to have in mind when we want to prove that a certain property, namely, that it is a right triangle, has to belong to the object in our mind in virtue of what it is, namely, a triangle inscribed in a semicircle. Obviously, the conclusion applies perfectly only to the perfect triangle we had in mind, whereas it holds for the visible figure only insofar as, and to the extent that, this figure resembles the object we had in mind. But this figure fails to have this property precisely insofar as, and to the extent that, it falls short of the object in our mind. However, on the basis of this point it should also be clear that the conclusion does apply to this figure, and every other visible triangle inscribed in a semicircle as well, insofar as, and to the extent that, it manages to imitate the properties of the perfect object in our mind. Therefore, the Platonic answer to the question of what this demonstration was about, namely, that it was about a perfect, ideal triangle, which is invisible to the eyes, but is graspable by our understanding, at once provides us with an explanation of the possibility of universal, necessary knowledge. By knowing the properties of the Form or Idea, we know all its particulars, i.e., all the things that imitate it, insofar as they imitate or participate in it. So, the Form itself is a universal entity, a universal model of all its particulars; and since it is the knowledge of this universal entity that can enable us to know at once all its particulars, it is absolutely vital for us to know what it is, what it is like, and exactly how it is related to its particulars. However, obviously, all these questions presuppose that it is at all, namely, that such a universal entity exists. But the existence of such an entity seems to be rather precarious. Consider, for instance, the perfect triangle we were supposed to have in mind during the demonstration of Thales' theorem. If it is a perfect triangle, it obviously has to have three sides, since a perfect triangle has to be a triangle, and nothing can be a triangle unless it has three sides. But of those three sides either at least two are equal or none, that is to say, the triangle in question has to be either isosceles or scalene (taking 'isosceles' broadly, including even equilateral triangles, for the sake of simplicity). However, since it is supposed to be the universal model of all triangles, and not only of isosceles triangles, this perfect triangle cannot be an isosceles, and for the same reason it cannot be a scalene triangle either. Therefore, such a universal triangle would have to have inconsistent properties, namely, both that it is either isosceles or scalene and that it is neither isosceles nor scalene. However, obviously nothing can have these properties at the same time, so nothing can be a universal triangle any more than a round square. So, apparently, no universal triangle can exist. But then, what was our demonstration about? Just a little while ago, we concluded that it could not be directly about any particular triangle (for it was not about the triangle in the figure, and it was even less about any other particular triangle not in the figure), and now we had to conclude that it could not be about a universal triangle either. But are there any further alternatives? It seems obvious that through this demonstration we do gain universal knowledge concerning all particulars. Yet it is also clear that we do not, indeed, we cannot gain this knowledge by examining all particulars, both because they are potentially infinite and because none of them perfectly satisfies the conditions stated in the demonstration. So, there must have been something wrong in our characterization of the universal, which compelled us to conclude that, in accordance with that characterization, universals could not exist. Therefore, we are left with a whole bundle of questions concerning the nature and characteristics of universals, questions that cannot be left unanswered if we want to know how universal, necessary knowledge is possible, if at all. ## 3. The Origin of the Specifically Medieval Problem of Universals What we may justifiably call the first formulation of "the medieval problem of universals" (distinguishing it from the both logically and historically related ancient problems of Plato's Theory of Forms) was precisely such a bundle of questions famously raised by Porphyry in his Isagoge, that is, his Introduction to Aristotle's Categories. As he wrote: > > (1) Since, Chrysaorius, to teach about Aristotle's > Categories it is necessary to know what genus and difference > are, as well as species, property, and accident, and since reflection > on these things is useful for giving definitions, and in general for > matters pertaining to division and demonstration, therefore I shall > give you a brief account and shall try in a few words, as in the > manner of an introduction, to go over what our elders said about these > things. I shall abstain from deeper enquiries and aim, as appropriate, > at the simpler ones. > > > (2) For example, I shall beg off saying anything about (a) whether > genera and species are real or are situated in bare thoughts alone, > (b) whether as real they are bodies or incorporeals, and (c) whether > they are separated or in sensibles and have their reality in > connection with them. Such business is profound, and requires another, > greater investigation. Instead I shall now try to show how the > ancients, the Peripatetics among them most of all, interpreted genus > and species and the other matters before us in a more logical fashion. > [Porphyry, Isagoge, in Spade 1994 (henceforth, Five > Texts), p. 1.] > > > Even though in this way, by relegating them to a "greater investigation", Porphyry left these questions unanswered, they certainly proved to be irresistible for his medieval Latin commentators, beginning with Boethius, who produced not just one, but two commentaries on Porphyry's text; the first based on Marius Victorinus's (fl. 4th c.) translation, and the second on his own.[6] In the course of his argument, Boethius makes it quite clear what sort of entity a universal would have to be. > > A universal must be common to several particulars > > 1. in its entirety, and not only in part > 2. simultaneously, and not in a temporal succession, and > 3. it should constitute the substance of its > particulars.[7] > > > However, as Boethius argues, nothing in real existence can satisfy these conditions. The main points of his argument can be reconstructed as follows. Anything that is common to many things in the required manner has to be simultaneously, and as a whole, in the substance of these many things. But these many things are several beings precisely because they are distinct from one another in their being, that is to say, the act of being of one is distinct from the act of being of the other. However, if the universal constitutes the substance of a particular, then it has to have the same act of being as the particular, because constituting the substance of something means precisely this, namely, sharing the act of being of the thing in question, as the thing's substantial part. But the universal is supposed to constitute the substance of all of its distinct particulars, as a whole, at the same time. Therefore, the one act of being of the universal entity would have to be identical with all the distinct acts of being of its several particulars at the same time, which is impossible.[8] This argument, therefore, establishes that no one thing can be a universal in its being, that is to say, nothing can be both one being and common to many beings in such a manner that it shares its act of being with those many beings, constituting their substance. This can easily be visualized in the following diagram, where the tiny lightning bolts indicate the acts of being of the entities involved, namely, a woman, a man, and their universal humanity (the larger dotted figure). ![Consists of three 'matchstick human' characters (as on public bathroom doors), a bigger one on the top, drawn in dashed lines, representing the universal humanity, a female on the lower left and a male character on the lower right, representing individual humans. Each has a lightning bolt symbol next to it, representing their acts of being, showing that the acts of being of the individuals are not identical, while the act of being of the universal would have to be identical with these distinct acts of being, which is impossible.](image3.gif) Figure 3. Illustration of the first part of Boethius' argument But then, Boethius goes on, we should perhaps say that the universal is not one being, but rather many beings, that is, [the collection of][9] those constituents of the individual essences of its particulars on account of which they all fall under the same universal predicable. For example, on this conception, the genus 'animal' would not be some one entity, a universal animality over and above the individual animals, yet somehow sharing its being with them all (since, as we have just seen, that is impossible), but rather [the collection of] the individual animalities of all animals. Boethius rejects this suggestion on the ground that whenever there are several generically similar entities, they have to have a genus; therefore, just as the individual animals had to have a genus, so too, their individual animalities would have to have another one. However, since the genus of animalities cannot be one entity, some 'super-animality' (for the same reason that the genus of animals could not be one entity, on the basis of the previous argument), it seems that the genus of animalities would have to be a number of further 'super-animalities'. But then again, the same line of reasoning should apply to these 'super-animalities', giving rise to a number of 'super-super-animalities', and so on to infinity, which is absurd. Therefore, we cannot regard the genus as some real being even in the form of [a collection of] several distinct entities. Since similar reasonings would apply to the other Porphyrian predicables as well, no universal can exist in this way. Now, a universal either exists in reality independently of a mind conceiving of it, or it only exists in the mind. If it exists in reality, then it either has to be one being or several beings. But since it cannot exist in reality in either of these two ways, Boethius concludes that it can only exist in the mind.[10] However, to complicate matters, it appears that a universal cannot exist in the mind either. For, as Boethius says, the universal existing in the mind is some universal understanding of some thing outside the mind. But then this universal understanding is either disposed in the same way as the thing is, or differently. If it is disposed in the same way, then the thing also must be universal, and then we end up with the previous problem of a really existing universal. On the other hand, if it is disposed differently, then it is false, for "what is understood otherwise than the thing is is false" (Five Texts, Spade 1994, p. 23 (21)). But then, all universals in the understanding would have to be false representations of their objects; therefore, no universal knowledge would be possible, whereas our considerations started out precisely from the existence of such knowledge, as seems to be clear, e.g., in the case of geometrical knowledge. ## 4. Boethius' Aristotelian Solution Boethius' solution of the problem stated in this form consists in the rejection of this last argument, by pointing out the ambiguity of the principle that "what is understood otherwise than the thing is is false". For in one sense this principle states the obvious, namely, that an act of understanding that represents a thing to be otherwise than the thing is is false. This is precisely the reading of this principle that renders it plausible. However, in another sense this principle would state that an act of understanding which represents the thing in a manner which is different from the manner in which the thing exists is false. In this sense, then, the principle would state that if the mode of representation of the act of understanding is different from the mode of being of the thing, then the act of understanding is false. But this is far from plausible. In general, it is simply not true that a representation can be true or faithful only if the mode of representation matches the mode of being of the thing represented. For example, a written sentence is a true and faithful representation of a spoken sentence, although the written sentence is a visible, spatial sequence of characters, whereas the spoken sentence is an audible, temporal pattern of articulated sounds. So, what exists as an audible pattern of sounds is represented visually, that is, the mode of existence of the thing represented is radically different from the mode of its representation. In the same way, when particular things are represented by a universal act of thought, the things exist in a particular manner, while they are represented in a universal manner, still, this need not imply that the representation is false. But this is precisely the sense of the principle that the objection exploited. Therefore, since in this sense the principle can be rejected, the objection is not conclusive.[11] However, it still needs to be shown that in the particular case of universal representation the mismatch between the mode of its representation and the mode of being of the thing represented does in fact not entail the falsity of the representation. This can easily be seen if we consider the fact that the falsity of an act of understanding consists in representing something to be in a way it is not. That is to say, properly speaking, it is only an act of judgment that can be false, by which we think something to be somehow. But a simple act of understanding, by which we simply understand something without thinking it to be somehow, that is, without attributing anything to it, cannot be false. For example, I can be mistaken if I form in my mind the judgment that a man is running, whereby I conceive a man to be somehow, but if I simply think of a man without attributing either running or not running to him, I certainly cannot make a mistake as to how he is.[12] In the same way, I would be mistaken if I were to think that a triangle is neither isosceles nor scalene, but I am certainly not in error if I simply think of a triangle without thinking either that it is isosceles or that it is scalene. Indeed, it is precisely this possibility that allows me to form the universal mental representation, that is, the universal concept of all particular triangles, regardless of whether they are isosceles or scalene. For when I think of a triangle in general, then I certainly do not think of something that is a triangle and is neither isosceles nor scalene, for that is impossible, but I simply think of a triangle, not thinking that it is an isosceles and not thinking that it is a scalene triangle. This is how the mind is able to separate in thought what are inseparable in real existence. Being either isosceles or scalene is inseparable from a triangle in real existence. For it is impossible for something to be a triangle, and yet not to be an isosceles and not to be a scalene triangle either. Still, it is not impossible for something to be thought to be a triangle and not to be thought to be an isosceles and not to be thought to be a scalene triangle either (although of course, it still has to be thought to be either-isosceles-or-scalene). This separation in thought of those things that cannot be separated in reality is the process of abstraction.[13] In general, by means of the process of abstraction, our mind (in particular, the faculty of our mind Aristotle calls active intellect (nous poietikos, in Greek, intellectus agens, in Latin) is able to form universal representations of particular objects by disregarding what distinguishes them, and conceiving of them only in terms of those of their features in respect of which they do not differ from one another. In this way, therefore, if universals are regarded as universal mental representations existing in the mind, then the contradictions emerging from the Platonic conception no longer pose a threat. On this Aristotelian conception, universals need not be thought of as somehow sharing their being with all their distinct particulars, for their being simply consists in their being thought of, or rather, the particulars' being thought of in a universal manner. This is what Boethius expresses by saying in his final replies to Porphyry's questions the following: > > > ... genera and species subsist in one way, but are understood in > an another. They are incorporeal, but subsist in sensibles, joined to > sensibles. They are understood, however, as subsisting by themselves, > and as not having their being in others. [Five Texts, Spade > 1994, p. 25] > > > But then, if in this way, by positing universals in the mind, the most obvious inconsistencies of Plato's doctrine can be avoided, no wonder that Plato's "original" universals, the universal models which particulars try to imitate by their features, found their place, in accordance with the long-standing Neo-Platonic tradition, in the divine mind.[14] It is this tradition that explains Boethius' cautious formulation of his conclusion concerning Aristotelianism pure and simple, as not providing us with the whole story. As he writes: > > > ... Plato thinks that genera and species and the rest are not > only understood as universals, but also exist and subsist apart from > bodies. Aristotle, however, thinks that they are understood as > incorporeal and universal, but subsist in sensibles. > > > > I did not regard it as appropriate to decide between their views. For > that belongs to a higher philosophy. But we have carefully followed > out Aristotle's view here, not because we would recommend it the > most, but because this book, [the Isagoge], is written about > the Categories, of which Aristotle is the author. [Five > Texts, Spade 1994, p. 25] > > > ## 5. Platonic Forms as Divine Ideas Besides Boethius, the most important mediator between the Neo-Platonic philosophical tradition and the Christianity of the Medieval Latin West, pointing out also its theological implications, was St. Augustine. In a passage often quoted by medieval authors in their discussions of divine ideas, he writes as follows: > > > ... in Latin we can call the Ideas "forms" or > "species", in order to appear to translate word for word. > But if we call them "reasons", we depart to be sure from a > proper translation -- for reasons are called "logoi" > in Greek, not Ideas -- but nevertheless, whoever wants to use > this word will not be in conflict with the fact. For Ideas are certain > principal, stable and immutable forms or reasons of things. They are > not themselves formed, and hence they are eternal and always stand in > the same relations, and they are contained in the divine > understanding. [Spade 1985, Other Internet Resources, p. > 383][15] > > > As we could see from Boethius' solution, in this way, if Platonic Forms are not universal beings existing in a universal manner, but their universality is due to a universal manner of understanding, we can avoid the contradictions arising from the "naive" Platonic conception. Nevertheless, placing universal ideas in the divine mind as the archetypes of creation, this conception can still do justice to the Platonic intuition that what accounts for the necessary, universal features of the ephemeral particulars of the visible world is the presence of some universal exemplars in the source of their being. It is precisely in virtue of having some insight into these exemplars themselves that we can have the basis of universal knowledge Plato was looking for. As St. Augustine continues: > > > And although they neither arise nor perish, nevertheless everything > that is able to arise and perish, and everything that does arise and > perish, is said to be formed in accordance with them. Now it is denied > that the soul can look upon them, unless it is a rational one, [and > even then it can do so] only by that part of itself by which it > surpasses [other things] -- that is, by its mind and reason, as > if by a certain "face", or by an inner and intelligible > "eye". To be sure, not each and every rational soul in > itself, but [only] the one that is holy and pure, that [is the one > that] is claimed to be fit for such a vision, that is, the one that > keeps that very eye, by which these things are seen, healthy and pure > and fair and like the things it means to see. What devout man imbued > with true religion, even though he is not yet able to see these > things, nevertheless dares to deny, or for that matter fails to > profess, that all things that exist, that is, whatever things are > contained in their own genus with a certain nature of their own, so > that that they might exist, are begotten by God their author, and that > by that same author everything that lives is alive, and that the > entire safe preservation and the very order of things, by which > changing things repeat their temporal courses according to a fixed > regimen, are held together and governed by the laws of a supreme God? > If this is established and granted, who dares to say that God has set > up all things in an irrational manner? Now if it is not correct to say > or believe this, it remains that all things are set up by reason, and > a man not by the same reason as a horse -- for that is absurd to > suppose. Therefore, single things are created with their own reasons. > But where are we to think these reasons exist, if not in the mind of > the creator? For he did not look outside himself, to anything placed > [there], in order to set up what he set up. To think that is > sacrilege. But if these reasons of all things to be created and > [already] created are contained in the divine mind, and [if] there > cannot be anything in the divine mind that is not eternal and > unchangeable, and [if] Plato calls these principal reasons of things > "Ideas", [then] not only are there Ideas but they are > true, because they are eternal and [always] stay the same way, and > [are] unchangeable. And whatever exists comes to exist, however it > exists, by participation in them. But among the things set up by God, > the rational soul surpasses all [others], and is closest to God when > it is pure. And to the extent that it clings to God in charity, to > that extent, drenched in a certain way and lit up by that intelligible > light, it discerns these reasons, not by bodily eyes but by that > principal [part] of it by which it surpasses [everything else], that > is, by its intelligence. By this vision it becomes most blessed. These > reasons, as was said, whether it is right to call them Ideas or forms > or species or reasons, many are permitted to call [them] whatever they > want, but [only] to a very few [is it permitted] to see what is true. > [Spade 1985, Other Internet Resources, pp. 383-384] > > > Augustine's conception, then, saves Plato's original intuitions, yet without their inconsistencies, while it also combines his philosophical insights with Christianity. But, as a rule, a really intriguing solution of a philosophical problem usually gives rise to a number of further problems. This solution of the original problem with Plato's Forms is no exception. ### 5.1 Divine Ideas and Divine Simplicity First of all, it generates a particular ontological/theological problem concerning the relationship between God and His Ideas. For according to the traditional philosophical conception of divine perfection, God's perfection demands that He is absolutely simple, without any composition of any sort of parts.[16] So, God and the divine mind are not related to one another as a man and his mind, namely as a substance to one of its several powers, but whatever powers God has He is. Furthermore, the Divine Ideas themselves cannot be regarded as being somehow the eternal products of the divine mind distinct from the divine mind, and thus from God Himself, for the only eternal being is God, and everything else is His creature. Now, since the Ideas are not creatures, but the archetypes of creatures in God's mind, they cannot be distinct from God. However, as is clear from the passage above, there are several Ideas, and there is only one God. So how can these several Ideas possibly be one and the same God? Augustine never explicitly raised the problem, but for example Aquinas, who (among others) did, provided the following rather intuitive solution for it (ST1, q. 15, a. 2). The Divine Ideas are in the Divine Mind as its objects, i.e., as the things understood. But the diversity of the objects of an act of understanding need not diversify the act itself (as when understanding the Pythagorean theorem, we understand both squares and triangles). Therefore, it is possible for the self-thinking divine essence to understand itself in a single act of understanding so perfectly that this act of understanding not only understands the divine essence as it is in itself, but also in respect of all possible ways in which it can be imperfectly participated by any finite creature. The cognition of the diversity of these diverse ways of participation accounts for the plurality of divine ideas. But since all these diverse ways are understood in a single eternal act of understanding, which is nothing but the act of divine being, and which in turn is again the divine essence itself, the multiplicity of ideas does not entail any corresponding multiplicity of the divine essence. To be sure, this solution may still give rise to the further questions as to what these diverse ways are, exactly how they are related to the divine essence, and how their diversity is compatible with the unity and simplicity of the ultimate object of divine thought, namely, divine essence itself. In fact, these are questions that were raised and discussed in detail by authors such as Henry of Ghent (c. 1217-1293), Thomas of Sutton (ca. 1250-1315), Duns Scotus (c. 1266-1308) and others.[17] ### 5.2 Illuminationism vs. Abstractionism Another major issue connected to the doctrine of divine ideas, as should also be clear from the previously quoted passage, was the bundle of epistemological questions involved in Augustine's doctrine of divine illumination. The doctrine -- according to which the human soul, especially "one that is holy and pure", obtains a specific supernatural aid in its acts of understanding, by gaining a direct insight into the Divine Ideas themselves -- received philosophical support in terms of a typically Platonic argument in Augustine's De Libero Arbitrio.[18] The argument can be reconstructed as follows. > > The Augustinian Argument for Illumination. > > 1. I can come to know from experience only something that can be > found in experience [self-evident] > 2. Absolute unity cannot be found in experience [assumed] > 3. Therefore, I cannot come to know absolute unity from experience. > [1,2] > 4. Whatever I know, but I cannot come to know from experience, I came > to know from a source that is not in this world of experiences. > [self-evident] > 5. I know absolute unity. [assumed] > 6. Therefore, I came to know absolute unity from a source that is not > in this world of experiences. [3,4,5] > > > > Proof of 2. Whatever can be found in experience is some > material being, extended in space, and so it has to have a multitude > of spatially distinct parts. Therefore, it is many in respect of those > parts. But what is many in some respect is not one in that respect, > and what is not one in some respect is not absolutely one. Therefore, > nothing can be found in experience that is absolutely one, that is, > nothing in experience is an absolute unity. > > > > Proof of 5. I know that whatever is given in experience has > many parts (even if I may not be able to discern those parts by my > senses), and so I know that it is not an absolute unity. But I can > have this knowledge only if I know absolute unity, namely, something > that is not many in any respect, not even in respect of its parts, > for, in general, I can know that something is F in a certain respect, > and not an F in some other respect, only if I know what it is for > something to be an F without any qualification. (For example, I know > that the two halves of a body, taken together, are not absolutely two, > for taken one by one, they are not absolutely one, since they are also > divisible into two halves, etc. But I can know this only because I > know that for obtaining absolutely two things [and not just two > multitudes of further things], I would have to have two things that in > themselves are absolutely one.) Therefore, I know absolute unity. > > > It is important to notice here that this argument (crucially) assumes that the intellect is passive in acquiring its concepts. According to this assumption, the intellect merely receives the cognition of its objects as it finds them. By contrast, on the Aristotelian conception, the human mind actively processes the information it receives from experience through the senses. So by means of its faculty appropriately called the active or agent intellect, it is able to produce from a limited number of experiences a universal concept equally representing all possible particulars falling under that concept. In his commentary on Aristotle's De Anima Aquinas insightfully remarks: > > > The reason why Aristotle came to postulate an active intellect was his > rejection of Plato's theory that the essences of sensible things > existed apart from matter, in a state of actual intelligibility. For > Plato there was clearly no need to posit an active intellect. But > Aristotle, who regarded the essences of sensible things as existing in > matter with only a potential intelligibility, had to invoke some > abstractive principle in the mind itself to render these essences > actually intelligible. [In De Anima, bk. 3, lc. 10] > > > On the basis of these and similar considerations, therefore, one may construct a rather plausible Aristotelian counterargument, which is designed to show that we need not necessarily gain our concept of absolute unity from a supernatural source, for it is possible for us to obtain it from experience by means of the active intellect. Of course, similar considerations should apply to other concepts as well. > > An Aristotelian-Thomistic counterargument from > abstraction. > > 1. I know from experience everything whose concept my active > intellect is able to abstract from experience. [self-evident] > 2. But my active intellect is able to abstract from experience the > concept of unity, since we all experience each singular thing as being > one, distinct from another. [self-evident, common > experience][19] > 3. Therefore, I know unity from experience by abstraction. [1,2] > 4. Whenever I know something from experience by abstraction, I know > both the thing whose concept is abstracted and its limiting conditions > from which its concept is abstracted. [self-evident] > 5. Therefore, I know both unity and its limiting conditions from > which its concept is abstracted. [3,4] > 6. But whenever I know something and its limiting conditions, and I > can conceive of it without its limiting conditions (and this is > precisely what happens in abstraction), I can conceive of its > absolute, unlimited realization. [self-evident] > 7. Therefore, I can conceive of the absolute, unlimited realization > of unity, based on the concept of unity I acquired from experience by > abstraction. [5,6] > 8. Therefore, it is not necessary for me to have a preliminary > knowledge of absolute unity before all experience, from a source other > than this world of experiences. [7] > > > To be sure, we should notice here that this argument does not falsify the doctrine of illumination. Provided it works, it only invalidates the Augustinian-Platonic argument for illumination. Furthermore, this is obviously not a sweeping, knock-down refutation of the idea that at least some of our concepts perhaps could not so simply be derived from experience by abstraction; in fact, in the particular case of unity, and in general, in connection with our transcendental notions (i.e., notions that apply in each Aristotelian category, so they transcend the limits of each one of them, such as the notions of being, unity, goodness, truth, etc.), even the otherwise consistently Aristotelian Aquinas would have a more complicated story to tell (see Klima 2000b). Nevertheless, although Aquinas would still leave some room for illumination in his epistemology, he would provide for illumination an entirely naturalistic interpretation, as far as the acquisition of our intellectual concepts of material things is concerned, by simply identifying it with the "intellectual light in us", that is, the active intellect, which enables us to acquire these concepts from experience by abstraction.[20] Duns Scotus, who opposed Aquinas on so many other points, takes basically the same stance on this issue. Other medieval theologians, especially such prominent "Augustinians" as Bonaventure, Matthew of Aquasparta, or Henry of Ghent, would provide greater room for illumination in the form of a direct, specific, supernatural influence needed for human intellectual cognition in this life besides the general divine cooperation needed for the workings of our natural powers, in particular, the abstractive function of the active intellect.[21] But they would not regard illumination as supplanting, but rather as supplementing intellectual abstraction. As we could see, Augustine makes recognition of truth dependent on divine illumination, a sort of irradiation of the intelligible light of divine ideas, which is accessible only to the few who are "holy and pure". But this seems to go against at least > > > 1. the experience that there are knowledgeable non-believers or > pagans > > > > 2. the Aristotelian insight that we can have infallible comprehension > of the first principles of scientific demonstrations for which we only > need the intellectual concepts that we can acquire naturally, from > experience by > abstraction,[22] > > > and > > > 3. the philosophical-theological consideration that if human reason, > man's natural faculty for acquiring truth were not sufficient > for performing its natural function, then human nature would be > naturally defective in its noblest part, precisely in which it was > created after the image of God. > > > In fact, these are only some of the problems explicitly raised and considered by medieval Augustinians, which prompted their ever more refined accounts of the role of illumination in human cognition. For example, Matthew of Aquasparta, recapitulating St. Bonaventure, writes as follows: > > > Plato and his followers stated that the entire essence of cognition > comes forth from the archetypal or intelligible world, and from the > ideal reasons; and they stated that the eternal light contributes to > certain cognition in its evidentness as the entire and sole reason for > cognition, as Augustine in many places recites, in particular in bk. > viii. c. 7 of The City of God: 'The light of minds for > the cognition of everything is God himself, who created > everything'. > > > > But this position is entirely mistaken. For although it appears to > secure the way of wisdom, it destroys the way of knowledge. > Furthermore, if that light were the entire and sole reason for > cognition, then the cognition of things in the Word would not differ > from their cognition in their proper kind, neither would the cognition > of reason differ from the cognition of revelation, nor philosophical > cognition from prophetic cognition, nor cognition by nature from > cognition by grace. > > > > The other position was apparently that of Aristotle, who claimed that > the entire essence of cognition is caused and comes from below, > through the senses, memory, and experience, [working together] with > the natural light of our active intellect, which abstracts the species > from phantasms and makes them actually understood. And for this reason > he did not claim that the eternal is light necessary for cognition, > indeed, he never spoke about it. And this opinion of his is obvious in > bk. 2 of the Posterior Analytics. [...] > > > > But this position seems to be very deficient. For although it builds > the way of knowledge, it totally destroys the way of wisdom. > [...] > > > > Therefore, I take it that one should maintain an intermediate position > without prejudice, by stating that our cognition is caused both from > below and from above, from external things as well as the ideal > reasons. > > > > [...] God has provided our mind with some intellectual light, by > means of which it would abstract the species of objects from the > sensibles, by purifying them and extracting their quiddities, which > are the per se objects of the intellect. [...] But this light is > not sufficient, for it is defective, and is mixed with obscurity, > unless it is joined and connected to the eternal light, which is the > perfect and sufficient reason for cognition, and the intellect attains > and somehow touches it by its upper part. > > > > However the intellect attains that light or those eternal reasons as > the reason for cognition not as sole reason, for then, as has been > said, cognition in the Word would not differ from cognition in proper > kind, nor the cognition of wisdom would differ from the cognition of > knowledge. Nor does it attain them as the entire reason, for then it > would not need the species and similitudes of things; but this is > false, for the Philosopher says, and experience teaches, that if > someone loses a sense, then he loses that knowledge of things which > comes from that sense. [DHCR, pp. 94-96] > > > In this way, taking the intermediate position between Platonism and Aristotelianism pure and simple, Matthew interprets Augustine's Platonism as being compatible with the Aristotelian view, crediting the Aristotelian position with accounting for the specific empirical content of our intellectual concepts, while crediting the Platonic view with accounting for their certainty in grasping the natures of things. Still, it may not appear quite clear exactly what the contribution of the eternal light is, indeed, whether it is necessary at all. After all, if by abstraction we manage to gain those intellectual concepts that represent the natures of things, what else is needed to have a grasp of those natures? Henry of Ghent, in his detailed account of the issue, provides an interesting answer to this question. Henry first distinguishes cognition of a true thing from the cognition of the truth of the thing. Since any really existing thing is truly what it is (even if it may on occasion appear something else), any cognition of any really existing thing is the cognition of a true thing. But cognition of a true thing may occur without the cognition of its truth, since the latter is the cognition that the thing adequately corresponds to its exemplar in the human or divine mind. For example, if I draw a circle, when a cat sees it, then it sees the real true thing as it is presented to it. Yet the cat is simply unable to judge whether it is a true circle in the sense that it really is what it is supposed to be, namely, a locus of points equidistant from a given point. By contrast, a human being is able to judge the truth of this thing, insofar as he or she would be able to tell that my drawing is not really and truly a circle, but is at best a good approximation of what a true circle would be. Now, in intellectual cognition, just as in the sensory cognition of things, when the intellect simply apprehends a true thing, then it still does not have to judge the truth of the thing, even though it may have a true apprehension, adequately representing the thing. But the cognition of the truth of the thing only occurs in a judgment, when the intellect judges the adequacy of the thing to its exemplar. But since a thing can be compared to two sorts of exemplar, namely, to the exemplar in the human mind, and to the exemplar in the divine mind, the cognition of the truth of a thing is twofold, relative to these two exemplars. The exemplar of the human mind, according to Henry, is nothing but the Aristotelian abstract concept of the thing, whereby the thing is simply apprehended in a universal manner, and hence its truth is judged relative to this concept, when the intellect judges that the thing in question falls under this concept or not. As Henry writes: > > > [...] attending to the exemplar gained from the thing as the > reason for its cognition in the cognizer, the truth of the thing can > indeed be recognized, by forming a concept of the thing that conforms > to that exemplar; and it is in this way that Aristotle asserted that > man gains knowledge and cognition of the truth from purely natural > sources about changeable natural things, and that this exemplar is > acquired from things by means of the senses, as from the first > principle of art and science. [...] So, by means of the universal > notion in us that we have acquired from the several species of animals > we are able to realize concerning any thing that comes our way whether > it is an animal or not, and by means of the specific notion of donkey > we realize concerning any thing that comes our way whether it is a > donkey or not. [HQO, a. 1, q. 2, fol. 5 E-F] > > > But this sort of cognition of the truth of a thing, although it is intellectual, universal cognition, is far from being the infallible knowledge we are seeking. As Henry argues further: > > > But by this sort of acquired exemplar in us we do not have the > entirely certain and infallible cognition of truth. Indeed, this is > entirely impossible for three reasons, the first of which is taken > from the thing from which this exemplar is abstracted, the second from > the soul, in which this exemplar is received, and the third from the > exemplar itself that is received in the soul about the thing. > > > > The first reason is that this exemplar, since it is abstracted from > changeable things, has to share in the nature of changeability. > Therefore, since physical things are more changeable than mathematical > objects, this is why the Philosopher claimed that we have a greater > certainty of knowledge about mathematical objects than about physical > things by means of their universal species. And this is why Augustine, > discussing this cause of the uncertainty of the knowledge of natural > things in q. 9 of his Eighty-Three Different Questions, says > that from the bodily senses one should not expect the pure truth > [syncera veritas] > > > > ... The second reason is that the human soul, since it is > changeable and susceptible to error, cannot be rectified to save it > from swerving into error by anything that is just as changeable as > itself, or even more; therefore, any exemplar that it receives from > natural things is necessarily just as changeable as itself, or even > more, since it is of an inferior nature, whence it cannot rectify the > soul so that it would persist in the infallible truth. > > > > ... The third reason is that this sort of exemplar, since it is > the intention and species of the sensible thing abstracted from the > phantasm, is similar to the false as well as to the true [thing], so > that on its account these cannot be distinguished. For it is by means > of the same images of sensible things that in dreams and madness we > judge these images to be the things, and in sane awareness we judge > the things themselves. But the pure truth can only be perceived by > discerning it from falsehood. Therefore, by means of such an exemplar > it is impossible to have certain knowledge, and certain cognition of > the truth. And so if we are to have certain knowledge of the truth, > then we have to turn our mind away from the senses and sensible > things, and from every intention, no matter how universal and > abstracted from sensible things, to the unchangeable truth existing > above the mind [...]. [ibid., fol. 5. F] > > > So, Henry first distinguished between the cognition of a true thing and the intellectual cognition of the truth of a thing, and then, concerning the cognition of the truth of the thing, he distinguished between the cognition of truth by means of a concept abstracted from the thing and "the pure truth" [veritas syncera vel liquida], which he says cannot be obtained by means of such abstracted concepts. But then the question naturally arises: what is this "pure truth", and how can it be obtained, if at all? Since cognition of the pure truth involves comparison of objects not to their acquired exemplar in the human mind, but to their eternal exemplar in the divine mind, in the ideal case it would consist in some sort of direct insight into the divine ideas, enabling the person who has this access to see everything in its true form, as "God meant it to be", and also see how it fails to live up to its idea due to its defects. So, it would be like the direct intuition of two objects, one sensible, another intelligible, on the basis of which one could also immediately judge how closely the former approaches the latter. But this sort of direct intuition of the divine ideas is only the share of angels and the souls of the blessed in beatific vision; it is generally not granted in this life, except in rare, miraculous cases, in rapture, or prophetic vision. Therefore, if there is to be any non-miraculous recognition of this pure truth in this life, then it has to occur differently. Henry argues that even if we do not have a direct intuition of divine ideas as the objects cognized (whereby their particulars are recognized as more or less approximating them), we do have the cognition of the quiddities of things as the objects cognized by reason of some indirect cognition of their ideas. The reason for this, Henry says, is the following: > > > ...for our concept to be true by the pure truth, the soul, > insofar as it is informed by it, has to be similar to the truth of the > thing outside, since truth is a certain adequacy of the thing and the > intellect. And so, as Augustine says in bk. 2 of On Free Choice of > the Will, since the soul by itself is liable to slip from truth > into falsity, whence by itself it is not informed by the truth of any > thing, although it can be informed by it, but nothing can inform > itself, for nothing can give what it does not have; therefore, it is > necessary that it be informed of the pure truth of a thing by > something else. But this cannot be done by the exemplar received from > the thing itself, as has been shown earlier [in the previously quoted > passage -- GK]. It is necessary, therefore, that it be informed > by the exemplar of the unchangeable truth, as Augustine intends in the > same place. And this is why he says in On True Religion that > just as by its truth are true those that are true, so too by its > similitude are similar those that are similar. It is necessary, > therefore, that the unchangeable truth impress itself into our > concept, and that it transform our concept to its own character, and > that in this way it inform our mind with the expressed truth of the > thing by the same similitude that the thing itself has in the first > truth. [HQO a. 1, q. 2, fol. 7, I] > > > So, when we have the cognition of the pure truth of a thing, then we cannot have it in terms of the concept acquired from the thing, yet, since we cannot have it from a direct intuition of the divine exemplar either, the way we can have it is that the acquired concept primarily impressed on our mind will be further clarified, but no longer by a similarity of the thing, but by the similarity of the divine exemplar itself. Henry's point seems to be that given that the external thing itself is already just a (more or less defective) copy of the exemplar, the (more or less defective) copy of this copy can only be improved by means of the original exemplar, just as a copy of a poor repro of some original picture can only be improved by retouching the copy not on the basis of the poor repro, but on the basis of the original. But since the external thing is fashioned after its divine idea, the "retouching" of the concept in terms of the original idea does yield a better representation of the thing; indeed, so much better that on the basis of this "retouched" concept we are even able to judge just how well the thing realizes its kind. For example, when I simply have the initial simple concept of circle abstracted from circular objects I have seen, that concept is good enough for me to tell circular objects apart from non-circular ones. But with this simple, unanalyzed concept in mind, I may still not be able to say what a true circle is supposed to be, and accordingly, exactly how and to what extent the more or less circular objects I see fail or meet this standard. However, when I come to understand that a circle is a locus of points equidistant from a given point, I will realize by means of a clear and distinct concept what it was that I originally conceived in a vague and confused manner in my original concept of circle.[23] To be sure, I do not come to this definition of circle by looking up to the heaven of Ideas; in fact, I may just be instructed about it by my geometry teacher. But what is not given to me by my geometry teacher is the understanding of the fact that what is expressed by the definition is indeed what I originally rather vaguely conceived by my concept abstracted from visible circles. This "flash" of understanding, when I realize that it is necessary for anything that truly matches the concept of a circle to be such as described in the definition, would be an instance of receiving illumination without any particular, miraculous revelation.[24] However, even if in this light Henry's distinctions between the two kinds of truths and the corresponding differences of concepts make good sense, and even if we accept that the concepts primarily accepted from sensible objects need to be further worked on in order to provide us with true, clear understanding of the natures of things, it is not clear that this further work cannot be done by the natural faculties of our mind, assuming only the general influence of God in sustaining its natural operations, but without performing any direct and specific "retouching" of our concepts "from above". Using our previous analogy of the acquired concept as the copy of a poor repro of an original, we may say that if we have a number of different poor, fuzzy repros that are defective in a number of different ways, then in a long and complex process of collating them, we might still be able discern the underlying pattern of the original, and thus produce a copy that is actually closer to the original than any of the direct repros, without ever being allowed a glimpse of the original. In fact, this was precisely the way Aristotelian theologians, such as Aquinas, interpreted Augustine' conception of illumination, reducing God's role to providing us with the intelligible light not by directly operating on any of our concepts in particular, but providing the mind with "a certain likeness of the uncreated light, obtained through participation" (ST1, q. 84, a. 5c), namely, the agent intellect. Matthew of Aquasparta quite faithfully describes this view, associating it with the Aristotelian position he rejects: > > > Some people engaged in "philosophizing" [quidam > philosophantes] follow this position, although not entirely, when > they assert that that light is the general cause of certain cognition, > but is not attained, and its special influence is not necessary in > natural cognition; but the light of the agent intellect is sufficient > together with the species and similitudes of things abstracted and > received from the things; for otherwise the operation of [our] nature > would be rendered vacuous, our intellect would understand only by > coincidence, and our cognition would not be natural, but supernatural. > And what Augustine says, namely, that everything is seen in and > through that light, is not to be understood as if the intellect would > somehow attain that light, nor as if that light would have some > specific influence on it, but in such a way that the eternal God > naturally endowed us with intellectual light, in which we naturally > cognize and see all cognizable things that are within the scope of > reason. [DHCR, p. 95] > > > Although Matthew vehemently rejects this position as going against Augustine's original intention ("which is unacceptable, since he is a prominent teacher, whom catholic teachers and especially theologians ought to follow" -- as Matthew says), this view, in ever more refined versions, gained more and more ground toward the end of the 13th century, adopted not only by Aquinas and his followers, but also by his major opponents, namely, Scotus and his followers.[25] Still, illuminationism and abstractionism were never treated by medieval thinkers as mutually exclusive alternatives. They rather served as the two poles of a balancing act in judging the respective roles of nature and direct divine intervention in human intellectual cognition.[26] Although Platonism definitely survived throughout the Middle Ages (and beyond), in the guise of the interconnected doctrines of divine ideas, participation, and illumination, there was a quite general Aristotelian consensus,[27] especially after Abelard's time, that the mundane universals of the species and genera of material beings exist as such in the human mind, as a result of the mind's abstracting from their individuating conditions. But consensus concerning this much by no means entailed a unanimous agreement on exactly what the universals thus abstracted are, what it is for them to exist in the mind, how they are related to their particulars, what their real foundation in those particulars is, what their role is in the constitution of our universal knowledge, and how they contribute to the encoding and communication of this knowledge in the various human languages. For although the general Aristotelian stance towards universals successfully handles the inconsistencies quite obviously generated by a naive Platonist ontology, it gives rise precisely to these further problems of its own. ## 6. Universals According to Abelard's Aristotelian Conception It was Abelard who first dealt with the problem of universals explicitly in this form. Having relatively easily disposed of putative universal forms as real entities corresponding to Boethius' definition, in his Logica Ingredientibus he concludes that given Aristotle's definition of universals in his On Interpretation as those things that can be predicated of several things, it is only universal words that can be regarded as really existing universals. However, since according to Aristotle's account in the same work, words are meaningful in virtue of signifying concepts in the mind, Abelard soon arrives at the following questions: 1. What is the common cause in accordance with which a common name is imposed? 2. What is the understanding's common conception of the likeness of things? 3. Is a word called "common" on account of the common cause things agree in, or on account of the common conception, or on account of both together? [Five Texts, Spade 1994, p. 41 (88)] These questions open up a new chapter in the history of the problem of universals. For these questions add a new aspect to the bundle of the originally primarily ontological, epistemological, and theological questions constituting the problem, namely, they add a semantic aspect. On the Aristotelian conception of universals as universal predicables, there obviously are universals, namely, our universal words. But the universality of our words is clearly not dependent on the physical qualities of our articulate sounds, or of the various written marks indicating them, but on their representative function. So, to give an account of the universality of our universal words, we have to be able to tell in virtue of what they have this universal representative function, that is to say, we have to be able to assign a common cause by the recognition of which in terms of a common concept we can give a common name to a potential infinity of individuals belonging to the same kind. But this common cause certainly cannot be a common thing in the way Boethius described universal things, for, as we have seen, the assumption of the existence of such a common thing leads to contradictions. To be sure, Abelard also provides a number of further arguments, dealing with several refinements of Boethius' characterization of universals proposed by his contemporaries, such as William of Champeaux, Bernard of Chartres, Clarembald of Arras, Jocelin of Soissons, and Walter of Mortagne - but I cannot go into those details here.[28] The point is that he refutes and rejects all these suggestions to save real universals either as common things, having their own real unity, or as collections of several things, having a merely collective unity. The gist of his arguments against the former view is that the universal thing on that view would have to have its own numerical unity, and therefore, since it constitutes the substance of all its singulars, all these singulars would have to be substantially one and the same thing which would have to have all their contrary properties at the same time, which is impossible. The main thrust of his arguments against the collection-theory is that collections are arbitrary integral wholes of the individuals that make them up, so they simply do not fill the bill of the Porphyrian characterizations of the essential predicables such as genera and species.[29] So, the common cause of the imposition of universal words cannot be any one thing, or a multitude of things; yet, being a common cause, it cannot be nothing. Therefore, this common cause, which Abelard calls the status[30] of those things to which it is common, is a cause, but it is a cause which is a non-thing. However strange this may sound, Abelard observes that sometimes we do assign causes which are not things. For example, when we say "The ship was wrecked because the pilot was absent", the cause that we assign, namely, that the pilot was absent is not some thing, it is rather how things were, i.e., the way things were, which in this case we signify by the whole proposition "The pilot was absent".[31] From the point of view of understanding what Abelard's status are, it is significant that he assimilates the causal role of status as the common cause of imposition to causes that are signified by whole propositions. These significata of whole propositions, which in English we may refer to by using the corresponding "that-clauses" (as I did above, referring to the cause of the ship's wreck by the phrase "that the pilot was absent"), and in Latin by an accusative-with-infinitive construction, are what Abelard calls the dicta of propositions. These dicta, not being identifiable with any single thing, yet, not being nothing, constitute an ontological realm that is completely different from that of ordinary things. But it is also in this realm that Abelard's common causes of imposition may find their place. Abelard says that the common cause of imposition of a universal name has to be something in which things falling under that name agree. For example, the name 'man' (in the sense of 'human being', and not in the sense of 'male human being') is imposed on all humans on account of something in which all humans, as such, agree. But that in which all humans as such agree is that each one of them is a man, that is, each one agrees with all others in their being a man. So, it is their being human [esse hominem] that is the common cause Abelard was looking for, and this is what he calls the status of man. The status of man is not a thing; it is not any singular man, for obviously no singular man is common to all men, and it is not a universal man, for there is no such a thing. But being a man is common in the required manner (i.e., it is something in which all humans agree), yet it is clearly not a thing. For let us consider the singular propositions 'Socrates is a man' [Socrates est homo], 'Plato is a man' [Plato est homo], etc. These signify their dicta, namely, Socrates's being a man [Socratem esse hominem], and Plato's being a man [Platonem esse hominem], etc. But then it is clear that if we abstract from the singular subjects and retain what is common to them all, we can get precisely the status in which all these subjects agree, namely, being a man [esse hominem]. So, the status, just like the dicta from which they can be obtained, constitute an ontological realm that is entirely different from that of ordinary things. Still, despite the fact that it clearly has to do something with abstraction, an activity of the mind, Abelard insists that a status is not a concept of our mind. The reason for his insistence is that the status, being the common cause of imposition of a common name, must be something real, the existence of which is not dependent on the activity of our minds. A status is there in the nature of things, regardless of whether we form a mental act whereby we recognize it or not. In fact, for Abelard, a status is an object of the divine mind, whereby God preconceives the state of his creation from eternity.[32] A concept, or mental image of our mind, however, exists as the object of our mind only insofar as our mind performs the mental act whereby it forms this object. But this object, again, is not a thing, indeed, not any more than any other fictitious object of our minds. However, what distinguishes the universal concept from a merely fictitious object of our mind is that the former corresponds to a status of really existing singular things, whereas the latter does not have anything corresponding to it. To be sure, there are a number of points left in obscurity by Abelard's discussion concerning the relationships of the items distinguished here. For example, Abelard says that we cannot conceive of the status. However, it seems that we can only signify by our words whatever we can conceive. Yet, Abelard insists that besides our concepts, our words must signify the status themselves.[33] A solution to the problem is only hinted at in Abelard's remark that the names can signify status, because "their inventor meant to impose them in accordance with certain natures or characteristics of things, even if he did not know how to think out the nature or characteristic of the thing" (Five Texts, Spade 1994, p. 46 (116)). So, we may assume that although the inventor of the name does not know the status, his vague, "senses-bound" conception, from which he takes his word's signification, is directed at the status, as to that which he intends to signify.[34] However, Abelard does not work out this suggestion in any further detail. Again, it is unclear how the status is related to the individualized natures of the things that agree in the status. If the status is what the divine mind conceives of the singulars in abstraction from them, why couldn't the nature itself be conceived in the same way? - after all, the abstract nature would not have to be a thing any more than a status is, for its existence would not be real being, but merely its being conceived. Furthermore, it seems quite plausible that Abelard's status could be derived by abstraction from singular dicta with the same predicate, as suggested above. But dicta are the quite ordinary significata of our propositions, which Abelard never treats as epistemologically problematic, so why would the status, which we could apparently abstract from them, be accessible only to the divine mind? I'm not suggesting that Abelard could not provide acceptable and coherent answers to these and similar questions and problems.[35] But perhaps these problems also contributed to the fact that by the 13th century his doctrine of status was no longer in currency. Another historical factor that may have contributed to the waning of Abelard's theory was probably the influence of the newly translated Aristotelian writings along with the Arabic commentaries that flooded the Latin West in the second half of the 12th century. ## 7. Universal Natures in Singular Beings and in Singular Minds The most important influence in this period from our point of view came from Avicenna's doctrine distinguishing the absolute consideration of a universal nature from what applies to the same nature in the subject in which it exists. The distinction is neatly summarized in the following passage. > > Horsehood, to be sure, has a definition that does not demand > universality. Rather it is that to which universality happens. Hence > horsehood itself is nothing but horsehood only. For in itself it is > neither many nor one, neither is it existent in these sensibles nor in > the soul, neither is it any of these things potentially or actually in > such a way that this is contained under the definition of horsehood. > Rather [in itself it consists] of what is horsehood > only.[36] > In his little treatise On Being and Essence, Aquinas explains the distinction in greater detail in the following words: > > > A nature, however, or essence ...can be considered in two ways. > First, we can consider it according to its proper notion, and this is > its absolute consideration; and in this way nothing is true of it > except what pertains to it as such; whence if anything else is > attributed to it, that will yield a false attribution. ...In the > other way [an essence] is considered as it exists in this or that > [individual]; and in this way something is predicated of it per > accidens [non-essentially or coincidentally], on account of that > in which it exists, as when we say that a man is white because > Socrates is white, although this does not pertain to man as such. > > > > A nature considered in this way, however, has two sorts of existence. > It exists in singulars on the one hand, and in the soul on the other, > and from each of these [sorts of existence] it acquires accidents. In > the singulars, furthermore, the essence has several [acts of] > existence according to the multiplicity of singulars. Nevertheless, if > we consider the essence in the first, or absolute, sense, none of > these pertain to it. For it is false to say that the essence of man, > considered absolutely, has existence in this singular, because if > existence in this singular pertained to man insofar as he is man, man > would never exist, except as this singular. Similarly, if it pertained > to man insofar as he is man not to exist in this singular, then the > essence would never exist in the singular. But it is true to say that > man, but not insofar as he is man, may be in this singular or in that > one, or else in the soul. Therefore, the nature of man considered > absolutely abstracts from every existence, though it does not exclude > any. And the nature thus considered is what is predicated of each > individual.[37] > > > So, a common nature or essence according to its absolute consideration abstracts from all existence, both in the singulars and in the mind. Yet, and this is the important point, it is the same nature that informs both the singulars that have this nature and the minds conceiving of them in terms of this nature. To be sure, this sameness is not numerical sameness, and thus it does not yield numerically one nature. On the contrary, it is the sameness of several, numerically distinct realizations of the same information-content, just like the sameness of a book in its several copies. Just as there is no such a thing as a universal book over and above the singular copies of the same book, so there is no such a thing as a universal nature existing over and above the singular things of the same nature; still, just as it is true to say that the singular copies are the copies of the same book, so it is true to say that these singulars are of the same nature. Indeed, this analogy also shows why this conception should be so appealing from the point of view of the original epistemological problem of the possibility of universal knowledge, without entailing the ontological problems of naive Platonism. For just as we do not need to read all copies of the same book in order to know what we can find on the same page in the next copy (provided it is not a corrupt copy),[38] so we can know what may apply to all singulars of the same nature without having to experience them all. Still, we need not assume that we can have this knowledge only if we can get somehow in a mysterious contact with the universal nature over and above the singulars; all we need is to learn how "to read" the singulars in our experience to discern the "common message", the universal nature, informing them all, uniformly, yet in their distinct singularity. (Note that "reading the singulars" is not a mere metaphor: this is precisely what geneticists are quite literally doing in the process of gene sequencing, for instance, in the human genome project.) Therefore, the same nature is not the same in the same way as the same individual having this nature is the same as long as it exists. For that same nature, insofar as it is regarded as the same, does not even exist at all; it is said to be the same only insofar as it is recognizable as the same, if we disregard everything that distinguishes its instances in several singulars. (Note here that whoever would want to deny such a recognizable sameness in and across several singulars would have to deny that he is able to recognize the same words or the same letters in various sentences; so such a person would not be able to read, write, or even to speak, or understand human speech. But then we shouldn't really worry about such a person in a philosophical debate.) However, at this point some further questions emerge. If this common nature is recognizably the same on account of disregarding its individuating conditions in the singulars, then isn't it the result of abstraction; and if so, isn't it in the abstractive mind as its object? But if it is, then how can Aquinas say that it abstracts both from being in the singulars and from being in the mind? Here we should carefully distinguish between what we can say about the same nature as such, and what we can say about the same nature on account of its conditions as it exists in this or that subject. Again, using our analogy, we can certainly consistently say that the same book in its first edition was 200 pages, whereas in the second only 100, because it was printed on larger pages, but the book itself, as such, is neither 200 nor 100 pages, although it can be either. In the same way, we can consistently say that the same nature as such is neither in the singulars nor in the mind, but of course it is only insofar as it is in the mind that it can be recognizably the same, on account of the mind's abstraction. Therefore, that it is abstract and is actually recognized as the same in its many instances is something that belongs to the same nature only on account of being conceived by the abstractive mind. This is the reason why the nature is called a universal concept, insofar as it is in the mind. Indeed, it is only under this aspect that it is properly called a universal. So, although that which is predicable of several singulars is nothing but the common nature as such, considered absolutely, still, that it is predicable pertains to the same nature only on account of being conceived by the abstractive intellect, insofar as it is a concept of the mind. At any rate, this is how Aquinas solves the paralogism that seems to arise from this account, according to which the true claims that Socrates is a man and man is a species would seem to entail the falsity that Socrates is a species. For if we say that in the proposition 'Socrates is a man' the predicate signifies human nature absolutely, but the same nature, on account of its abstract character, is a species, the false conclusion seems inevitable (Klima 1993a). However, since the common nature is not a species in its absolute consideration, but only insofar as it is in the mind, the conclusion does not follow. Indeed, this reasoning would be just as invalid as the one trying to prove that this book, pointing to the second edition which is actually 100 pages, is 200 pages, because the same book was 200 hundred pages in its first edition. For just as its being 200 pages belongs to the same book only in its first edition, so its being a species belongs to human nature only as it exists in the mind. So, to sum up, we have to distinguish here between the nature existing in this singular (such as the individualized human nature of Socrates, which is numerically one item, mind-independently existing in Socrates), the universal (such as the species of human nature existing only in the mind as its object considered in abstraction from the individuating conditions it has in the singular humans), and the nature according to its absolute consideration (such as human nature considered in abstraction both from its existence in the singulars as its subjects and in the mind as its object). What establishes the distinction of these items is the difference of what can be truly said of them on account of the different conditions they have in this or that. What establishes the unity of these items, however, is that they are somehow the same nature existing and considered under different conditions. For the human nature in Socrates is numerically one, it is numerically distinct from the human nature in Plato, and it has real, mind-independent existence, which is in fact nothing but the existence of Socrates, i.e., Socrates' life. However, although the human nature in Socrates is a numerically distinct item from the human nature in Plato, insofar as it is human nature, it is formally, in fact, specifically the same nature, for it is human nature, and not another, specifically different, say, feline or canine nature. It is precisely this formal, specific, mind-independent sameness of these items (for, of course, say, this cat and that cat do not differ insofar as they are feline, regardless of whether there is anyone to recognize this) that allows the abstractive human mind to recognize this sameness by abstracting from those individuating conditions on account of which this individualized nature in this individual numerically differs from that individualized nature in that individual. Thus, insofar as the formally same nature is actually considered by a human mind in abstraction from these individualizing conditions, it is a universal, a species, an abstract object of a mental act whereby a human mind conceives of any individualized human nature without its individuating conditions. But, as we could see earlier, nothing can be a human nature existing without its individuating conditions, although any individualized human nature can be thought of without thinking of its necessarily conjoined individuating conditions (just as triangular shape can be thought of without thinking its necessarily conjoined conditions of being isosceles or being scalene). So for this universal concept to be is nothing but to be thought of, to be an object of the abstractive human mind. Finally, human nature in its absolute consideration is the same nature abstracted even from this being, i.e., even from being an object of the mind. Thus, as opposed to both in its existence in individuals and in the mind, neither existence, nor non-existence, nor unity, nor disunity or multiplicity belongs to it, as it is considered without any of these; indeed, it is considered without considering its being considered, for it is considered only in terms of what belongs to it on account of itself, not considering anything that has to belong to it on account of something else in which it can only be (i.e., whether in the mind or in reality). So, the nature according to its absolute consideration does not have numerical unity or multiplicity, which it has as it exists in individuals, nor does it have the formal unity that it has in the consideration of the mind (insofar as it is one species among many), but it has that formal unity which precedes even the recognition of this unity by the abstractive mind.[39] Nevertheless, even if with these distinctions Aquinas' solution of the paralogism works and what he says about the existence and unity vs. multiplicity of a common nature can be given a consistent interpretation, the emergence of the paralogism itself and the complexities involved in explaining it away, as well as the problems involved in providing this consistent interpretation show the inherent difficulties of this account. The main difficulty is the trouble of keeping track of what we are talking about when it becomes crucial to know what pertains to what on account of what; in general, when the conditions of identity and distinction of the items we are talking about become variable and occasionally rather unclear. Indeed, we can appreciate just how acute these difficulties may become if we survey the items that needed to be distinguished in what may be described as the common conceptual framework of the "realist" via antiqua, the "old way" of doing philosophy and theology, before the emergence of the "modern way", the "nominalist" via moderna challenging some fundamental principles of the older framework, resulting mostly from the semantic innovations introduced by William Ockham. The survey of these items and the problems they generate will then allow us to see in greater detail the main motivation for Ockham's innovations. ## 8. Universals in the Via Antiqua In this framework, we have first of all the universal or common terms of spoken and written languages, which are common on account of being imposed upon universal concepts of the human mind. The concepts themselves are universal on account of being obtained by the activity of the abstractive human mind from experiences of singulars. But the process of concept formation also involves various stages. In the first place, the sensory information collected by the single senses is distinguished, synthesized, and collated by the higher sensory faculties of the common sense [sensus communis] and the so-called cogitative power [vis cogitativa], to be stored in sensory memory as phantasms, the sensory representations of singulars in their singularity. The active intellect [intellectus agens] uses this sensory information to extract its intelligible content and produce the intelligible species [species intelligibiles], the universal representations of several individuals in their various degrees of formal unity, disregarding their distinctive features and individuating conditions in the process of abstraction. The intelligible species are stored in the intellectual memory of the potential intellect [intellectus possibilis], which can then use them to form the corresponding concept in an act of thought, for example, in forming a judgment. The intelligible species and the concepts themselves, being formed by individual human minds, are individual in their being, insofar as they pertain to this or that human mind. However, since they are the result of abstraction, in their information content they are universal. Now insofar as this universal information content is common to all minds that form these concepts at all, and therefore it is a common intelligible content gained by these minds from their objects insofar as they are conceived by these minds in a universal manner, later scholastic thinkers refer to it as the objective concept [conceptus obiectivus], distinguishing it from the formal or subjective concepts [conceptus formales seu subiectivi], which are the individual acts of individual minds carrying this information (just as the individual copies of a book carry the information content of the book).[40] It is this objective concept that is identified as the universal of the human mind (distinguished from the universals of the divine mind), namely, a species, a genus, a difference, a property, or an accident. (Note that these are only the simple concepts. Complex concepts, such as those corresponding to complex terms and propositions are the products of the potential intellect using these concepts in its further operations.) These universals, then, as the objective concepts of the mind, would be classified as beings of reason [entia rationis], the being of which consists in their being conceived (cf. Klima 1993b and Schmidt 1966). To be sure, they are not merely fictitious objects, for they are grounded in the nature of things insofar as they carry the universal information content abstracted from the singulars. But then again, the universal information content of the objective concept itself, considered not insofar as it is in the mind as its object, but in itself, disregarding whatever may carry it, is distinguished from its carriers both in the mind and in the ultimate objects of the mind, the singular things, as the nature of these things in its absolute consideration. However, the common nature as such cannot exist on its own any more than a book could exist without any copies of it or any minds conceiving of it. So, this common nature has real existence only in the singulars, informing them, and giving them their recognizably common characteristics. However, these common characteristics can be recognized as such only by a mind capable of abstracting the common nature from experiencing it in its really existing singular instances. But it is on account of the real existence of these individualized instances in the singulars that the common nature can truly be predicated of the singulars, as long as they are actually informed by these individualized instances. The items thus distinguished and their interconnections can be represented by the following block-diagram. The dashed frames indicate that the items enclosed by them have a certain reduced ontological status, a "diminished" mode of being, while the boxes partly sharing a side indicate the (possible) partial identities of the items they enclose.[41] The arrows pointing from the common term to the singulars, their individualized natures and items in the mind on this diagram represent semantic relations, which I am going to explain later, in connection with Ockham's innovations. The rest of the arrows indicate the flow of information from experience of singulars through the sensory faculties to the abstractive mind, and to the application of the universal information abstracted by the mind to further singular experiences in acts of judgment. ![A box labeled (1) 'common term' has arrows pointing to boxes labeled (2) 'individual natures', (3) 'singulars', (4) 'absolute nature', (5) 'objective concept', and (6) 'subjective concept'. The arrow from (1) to (2) is labeled 'signification', the arrow from (1) to (3) is dashed and labeled 'supposition', the arrow from (1) to (6) is labeled 'subordination'. Boxes (2) and (3) share a side. Boxes (4) and (5) share a side and have dashed frames. Box (3) has an arrow to box (7) 'phantasms' which has an arrow to box (8) 'intelligible species' which has an arrow to box (9) 'subjective concept' which has an arrow to box (5). Boxes (7),(8), and (9) are inside a large box (10) labeled 'MIND'.](image4.gif) Figure 4. The via antiqua conception Obviously, this is a rather complicated picture. However, its complexity itself should not be regarded as problematic or even surprising, for that matter. After all, this diagram merely summarizes, and distinguishes the main stages of, how the human mind processes the intelligible, universal information received from a multitude of singular experiences, and then again, how it applies this information in classifying further experiences. This process may reasonably be expected to be complex, and should not be expected to involve fewer stages than, e.g., setting up, and retrieving information from, a computer database. What renders this picture more problematic is rather the difficulties involved in identifying and distinguishing these stages and the corresponding items. Further complications were also generated by the variations in terminology among several authors, and the various criteria of identity and distinctness applied by them in introducing various different notions of identity and distinctness. In fact, many of the great debates of the authors working within this framework can be characterized precisely as disputing the identity or distinctness of the items featured here, or the very criteria of identifying or distinguishing them. For example, already Abelard raised the question whether the concept or mental image, which we may identify in the diagram as the objective concept of later authors, should be identified with the act of thought, which we may identify as the subjective concept, or perhaps a further act of the mind, called formatio, namely, the potential intellect's act of forming the concept, using the intelligible species as the principle of its action. Such distinctions were later on severely criticized by authors such as John Peter Olivi and others, who argued for the elimination of intelligible species, and, in general, of any intermediaries between an act of the intellect and its ultimate objects, the singulars conceived in a universal manner.[42] Again, looking at the diagram on the side of the singulars, most 13th century authors agreed that what accounts for the specific unity of several individuals of the same species, namely, their specific nature, should be something other than what accounts for their numerical distinctness, namely, their principle of individuation. However, one singular entity in a species of several co-specific individuals has to contain both the principle of the specific unity of these individuals and its own principle of individuation. Therefore, this singular entity, being a composite at least of its specific nature and its principle of individuation, has to be distinct from its specific nature. At any rate, this is the situation with material substances, whose principle of individuation was held to be their matter. However, based on this reasoning, immaterial substances, such as angels, could not be regarded as numerically distinct on account of their matter, but only on account of their form. But since form is the principle of specific unity, difference in form causes specific diversity. Therefore, on this basis, any two angels had to be regarded as different in species. This conclusion was explicitly drawn by Aquinas and others, but it was rejected by Augustinian theologians, and it was condemned in Paris in 1277.[43] So, no wonder authors such as Henry of Ghent and Duns Scotus worked out alternative accounts of individuation, introducing not only different principles of individuation, such as the Scotists' famous (or infamous) haecceity, but also different criteria of distinctness and identity, such as those grounding Henry of Ghent's intentional distinction, or Scotus's formal distinction,[44] or even later Suarez' modal distinction.[45] But even further problems arose from considering the identity or distinctness of the individualized natures signified by several common terms in one and the same individual. The metaphysical debate over the real distinction of essence and existence from this point of view is nothing but the issue whether the individualized common nature signified by the definition of a thing is the same as the act of being signified by the verb 'is' in the same thing. In fact, the famous problem of the plurality vs. unity of substantial forms may also be regarded as a dispute over whether the common natures signified by the substantial predicates on the Porphyrian tree in the category of substance are distinct or the same in the same individual (cf. Callus 1967). Finally, and this appears to be the primary motivation for Ockham's innovations, there was the question whether one must regard all individualized common natures signified in the same individual by several predicates in the ten Aristotelian categories as distinct from one another. For the affirmative answer would involve commitment to a virtually limitless multiplication of entities. Indeed, according to Ockham, the via antiqua conception would entail that > > a column is to the right by to-the-rightness, God is creating by > creation, is good by goodness, just by justice, mighty by might, an > accident inheres by inherence, a subject is subjected by subjection, > the apt is apt by aptitude, a chimera is nothing by nothingness, > someone blind is blind by blindness, a body is mobile by mobility, and > so on for other, innumerable > cases.[46] > And this is nothing, but "multiplying beings according to the multiplicity of terms... which, however, is erroneous and leads far away from the truth".[47] ## 9. Universals in the Via Moderna To be sure, as the very debates within the via antiqua framework concerning the identity or non-identity of various items distinguished in that framework indicate, Ockham's charges are not quite justified.[48] After all, several via antiqua authors did allow the identification of the significata of terms belonging to various categories, so their "multiplication of beings" did not necessarily match the multiplicity of terms. Furthermore, since via antiqua authors also distinguished between various modes or senses of being, allowing various sorts of "diminished" kinds of being, such as beings of reason, their ontological commitments were certainly not as unambiguous as Ockham would have us believe in this passage. However, if we contrast the diagram of the via antiqua framework above with the following schematic representation of the via moderna framework introduced by Ockham, we can immediately appreciate the point of Ockham's innovations. ![shows the simplified version of Figure 4, what we get as a result of the nominalist reductions. It consists of three main boxes, labeled 'common term' on the top, 'singulars' on the lower left, and 'mind' on the lower right. The 'mind' box contains two sub-boxes, labeled 'phantasms' and 'common concepts', with an arrow pointing from the former to the latter, indicating the flow of information, as does the arrow pointing from the box 'singulars' to the box 'phantasms'. The rest of the arrows indicate semantic relations: the full arrow pointing from the box 'common term' to 'singulars' is labeled 'signification', the dashed arrow is labeled 'supposition'. The full arrow pointing from 'common term' to 'common concept' is labeled 'subordination'. Finally, the unlabeled full arrow pointing from 'common concept' to 'singulars' represents natural signification or signification, as is clear from the text.](image5.gif) Figure 5. The via moderna conception Without a doubt, it is the captivating simplicity of this picture, especially as compared with the complexity of the via antiqua picture, that was the major appeal of the Ockhamist approach. There are fewer items here, equally on the same ontological footing, distinguished from one another in terms of the same unambiguous distinction, the numerical distinction between individual real entities. To be sure, there still are universals in this picture. But these universals are neither common natures "contracted" to individuals by some really or merely formally distinct principle of individuation, nor some universal objects of the mind, which exist in a "diminished" manner, as beings of reason. Ockham's universals, at least in his mature theory,[49] are just our common terms and our common concepts. Our common terms, which are just singular utterances or inscriptions, are common in virtue of being subordinated to our common concepts. Our common concepts, on the other hand, are just singular acts of our singular minds. Their universality consists simply in the universality of their representative function. For example, the common term 'man' is a spoken or written universal term of English, because it is subordinated to that concept of our minds by which we conceive of each man indifferently. (See Klima, 2011) It is this indifference in its representative function that enables the singular act of my mind to conceive of each man in a universal manner, and the same goes for the singular act of your mind. Accordingly, there is no need to assume that there is anything in the individual humans, distinct from these humans themselves, a common yet individualized nature waiting to be abstracted by the mind. All we need to assume is that two humans are more similar to each other than either of them to a brute animal, and all animals are more similar to each other than any of them to a plant, etc., and that the mind, being able to recognize this similarity, is able to represent the humans by means of a common specific concept, the animals by means of a common generic concept, all living things by means of a more general generic concept, etc.[50] In this way, then, the common terms subordinated to these concepts need not signify some abstract common nature in the mind, and consequently its individualized instances in the singulars, for they directly signify the singulars themselves, just as they are directly conceived by the universally representative acts of the mind. So, what these common terms signify are just the singulars themselves, which are also the things referred to by these terms when they are used in propositions. Using the customary rendering of the medieval logical terminology, the things ultimately signified by a common term are its significata, while the things referred to by the same term when it is used in a proposition are their (personal) supposita.[51] Now if we compare the two diagrams representing the respective conceptions of the two viae, we can see just how radically Ockham's innovations changed the character of the semantic relations connecting terms, concepts and things. In both viae, common terms are subordinated to common concepts, and it is in virtue of this subordination that they ultimately signify what their concepts represent. In the via moderna, a concept is just an act of the mind representing singulars in a more or less indifferent manner, yielding a more or less universal signification for the term. In the via antiqua, however, the act of the mind is just one item in a whole series of intermediary representations, distinguished in terms of their different functions in processing universal information, and connected by their common content, ultimately representing the common, yet individualized natures of their singulars.[52] Accordingly, a common term, expressing this common content, is primarily subordinated to the objective concept of the mind. But of course, this objective concept is only the common content of the singular representative acts of singular minds, their subjective concepts, formed by means of the intelligible species, abstracted by their active intellects. On the other hand, the objective concept, abstracting from all individuating conditions, expresses only what is common to all singulars, namely, their nature considered absolutely. But this absolutely considered nature is only the common content of what informs each singular of the same nature in its actual real existence. So, the term's ultimate significata will have to be the individualized natures of the singulars. But these ultimate significata may still not be the singulars themselves, namely, when the things informed by these significata are not metaphysically simple. In the via moderna conception, therefore, the ultimate significata of a term are nothing but those singular things that can be the term's supposita in various propositions, as a matter of semantics. By contrast, in the via antiqua conception, a term's ultimate significata may or may not be the same things as the term's (personal) supposita, depending on the constitution of these supposita, as a matter of metaphysics. The singulars will be the supposita of the term when it is used as the subject term of a proposition in which something is predicated about the things informed by these ultimate significata (in the case of metaphysically simple entities, the term's significata and supposita coincide).[53] Nevertheless, despite the nominalists' charges to the contrary, the via antiqua framework, as far as its semantic considerations are concerned, was no more committed to the real distinction of the significata and supposita of its common terms than the via moderna framework was. For if the semantic theory in itself had precluded the identification of these semantic values, then the question of possible identity of these values could not have been meaningfully raised in the first place. Furthermore, in that case such identifications would have been precluded as meaningless even when talking about metaphysically simple entities, such as angels and God, whereas the metaphysical simplicity of these entities was expressed precisely in terms of such identifications. But also in the mundane cases of the significata and supposita of concrete and abstract universal terms in the nine accidental categories, several via antiqua authors argued for the identification of these semantic values both within and across categories. First of all there was Aristotle's authority for the claim that action and passion are the same motion,[54] so the significata of terms in these two categories could not be regarded as really distinct entities. But several authors also argued for the identification of relations with their foundations, that is to say, for the identity of the significata of relative terms with the significata of terms in the categories quantity and quality. (For example, on this conception, my equality in height to you would be just my height, provided you were of the same height, and not a distinct "equality-thing" somehow attached to my height, caused by our equal heights.)[55] By contrast, what makes the via moderna approach simpler is that it "automatically" achieves such identifications already on the basis of its semantic principles. Since in this approach the significata of concrete common terms are just the singulars directly represented by the corresponding concepts, the significata and (personal) supposita of terms are taken to be the same singulars from the beginning. So these common terms signify and supposit for the same things either absolutely, provided the term is absolute, or in relation to other singulars, provided the term is connotative. But even in the case of connotative terms, such as relative terms (in fact, all terms in the nine accidental categories, except for some abstract terms in the category quality, according to Ockham) we do not need to assume the existence of some mysterious relational entities informing singular substances. For example, the term 'father' need not be construed as signifying in me an inherent relation, my fatherhood, somehow connecting me to my son, and suppositing for me on that account in the context of a proposition; rather, it should merely be construed as signifying me in relation to my son, thereby suppositing for me in the context of a proposition, while connoting my son. ## 10. The Separation of the Viae, and the Breakdown of Scholastic Discourse in Late-Medieval Philosophy The appeal of the simplicity of the via moderna approach, especially as it was systematically articulated in the works of John Buridan and his students, had a tremendous impact on late-medieval philosophy and theology. To be sure, many late-medieval scholars, who were familiar with both ways, would have shared the sentiment expressed by the remark of Domingo Soto (1494-1560, describing himself as someone who was "born among nominalists and raised by realists")[56] to the effect that whereas the realist doctrine of the via antiqua was more difficult to understand, still, the nominalist doctrine of the via moderna was more difficult to believe.[57] Nevertheless, the overall simplicity and internal consistency of the nominalist approach were undeniable, gathering a strong following by the 15th century in all major universities of Europe, old and newly established alike.[58] The resulting separation and the ensuing struggle of the medieval viae did not end with the victory of the one over the other. Instead, due to the primarily semantic nature of the separation, getting the parties embroiled in increasingly complicated ways of talking past each other, thereby generating an ever growing dissatisfaction, even contempt, in a new, lay, humanist intelligentsia,[59] it ended with the demise of the characteristically medieval conceptual frameworks of both viae in the late-medieval and early modern period. These developments, therefore, also put an end to the specifically medieval problem of universals. However, the increasingly rarified late-medieval problem eventually vanished only to give way to several modern variants of recognizably the same problem, which keeps recurring in one form or another in contemporary philosophy as well. Indeed, one may safely assert that as long as there is interest in the questions of how a human language obviously abounding in universal terms can be meaningfully mapped onto a world of singulars, there is a problem of universals, regardless of the details of the particular conceptual framework in which the relevant questions are articulated. Clearly, in this sense, the problem of universals is itself a universal, the universal problem of accounting for the relationships between mind, language, and reality.
consequentialism	## 1. Classic Utilitarianism The paradigm case of consequentialism is utilitarianism, whose classic proponents were Jeremy Bentham (1789), John Stuart Mill (1861), and Henry Sidgwick (1907). (For predecessors, see Schneewind 1997, 2002.) Classic utilitarians held hedonistic act consequentialism. Act consequentialism is the claim that an act is morally right if and only if that act maximizes the good, that is, if and only if the total amount of good for all minus the total amount of bad for all is greater than this net amount for any incompatible act available to the agent on that occasion. (Cf. Moore 1912, chs. 1-2.) Hedonism then claims that pleasure is the only intrinsic good and that pain is the only intrinsic bad. These claims are often summarized in the slogan that an act is right if and only if it causes "the greatest happiness for the greatest number." This slogan is misleading, however. An act can increase happiness for most (the greatest number of) people but still fail to maximize the net good in the world if the smaller number of people whose happiness is not increased lose much more than the greater number gains. The principle of utility would not allow that kind of sacrifice of the smaller number to the greater number unless the net good overall is increased more than any alternative. Classic utilitarianism is consequentialist as opposed to deontological because of what it denies. It denies that moral rightness depends directly on anything other than consequences, such as whether the agent promised in the past to do the act now. Of course, the fact that the agent promised to do the act might indirectly affect the act's consequences if breaking the promise will make other people unhappy. Nonetheless, according to classic utilitarianism, what makes it morally wrong to break the promise is its future effects on those other people rather than the fact that the agent promised in the past (Sinnott-Armstrong 2009). Since classic utilitarianism reduces all morally relevant factors (Kagan 1998, 17-22) to consequences, it might appear simple. However, classic utilitarianism is actually a complex combination of many distinct claims, including the following claims about the moral rightness of acts: > > > Consequentialism = whether an act is morally right depends only on > consequences (as opposed to the circumstances or the > intrinsic nature of the act or anything that happens before the > act). > > > > Actual Consequentialism = whether an act is morally right depends only > on the actual consequences (as opposed to foreseen, > foreseeable, intended, or likely consequences). > > > > Direct Consequentialism = whether an act is morally right depends only > on the consequences of that act itself (as opposed to the > consequences of the agent's motive, of a rule or practice that > covers other acts of the same kind, and so on). > > > > Evaluative Consequentialism = moral rightness depends only on the > value of the consequences (as opposed to non-evaluative > features of the consequences). > > > > Hedonism = the value of the consequences depends only on the > pleasures and pains in the consequences (as opposed > to other supposed goods, such as freedom, knowledge, life, and so > on). > > > > Maximizing Consequentialism = moral rightness depends only on which > consequences are best (as opposed to merely satisfactory or > an improvement over the status quo). > > > > Aggregative Consequentialism = which consequences are best is some > function of the values of parts of those consequences (as > opposed to rankings of whole worlds or sets of consequences). > > > > Total Consequentialism = moral rightness depends only on the > total net good in the consequences (as opposed to the average > net good per person). > > > > Universal Consequentialism = moral rightness depends on the > consequences for all people or sentient beings (as opposed to > only the individual agent, members of the individual's society, > present people, or any other limited group). > > > > Equal Consideration = in determining moral rightness, benefits to one > person matter just as much as similar benefits to any other > person (as opposed to putting more weight on the worse or worst > off). > > > > Agent-neutrality = whether some consequences are better than others > does not depend on whether the consequences are evaluated from the > perspective of the agent (as opposed to an observer). > > > These claims could be clarified, supplemented, and subdivided further. What matters here is just that most pairs of these claims are logically independent, so a moral theorist could consistently accept some of them without accepting others. Yet classic utilitarians accepted them all. That fact makes classic utilitarianism a more complex theory than it might appear at first sight. It also makes classic utilitarianism subject to attack from many angles. Persistent opponents posed plenty of problems for classic utilitarianism. Each objection led some utilitarians to give up some of the original claims of classic utilitarianism. By dropping one or more of those claims, descendants of utilitarianism can construct a wide variety of moral theories. Advocates of these theories often call them consequentialism rather than utilitarianism so that their theories will not be subject to refutation by association with the classic utilitarian theory. ## 2. What is Consequentialism? This array of alternatives raises the question of which moral theories count as consequentialist (as opposed to deontological) and why. In actual usage, the term "consequentialism" seems to be used as a family resemblance term to refer to any descendant of classic utilitarianism that remains close enough to its ancestor in the important respects. Of course, different philosophers see different respects as the important ones (Portmore 2020). Hence, there is no agreement on which theories count as consequentialist under this definition. To resolve this vagueness, we need to determine which of the various claims of classic utilitarianism are essential to consequentialism. One claim seems clearly necessary. Any consequentialist theory must accept the claim that I labeled "consequentialism", namely, that certain normative properties depend only on consequences. If that claim is dropped, the theory ceases to be consequentialist. It is less clear whether that claim by itself is sufficient to make a theory consequentialist. Several philosophers assert that a moral theory should not be classified as consequentialist unless it is agent-neutral (McNaughton and Rawling 1991, Howard-Snyder 1994, Pettit 1997). This narrower definition is motivated by the fact that many self-styled critics of consequentialism argue against agent-neutrality. Other philosophers prefer a broader definition that does not require a moral theory to be agent-neutral in order to be consequentialist (Bennett 1989; Broome 1991, 5-6; and Skorupski 1995). Criticisms of agent-neutrality can then be understood as directed against one part of classic utilitarianism that need not be adopted by every moral theory that is consequentialist. Moreover, according to those who prefer a broader definition of consequentialism, the narrower definition conflates independent claims and obscures a crucial commonality between agent-neutral consequentialism and other moral theories that focus exclusively on consequences, such as moral egoism and recent self-styled consequentialists who allow agent-relativity into their theories of value (Sen 1982, Broome 1991, Portmore 2001, 2003, 2011). A definition solely in terms of consequences might seem too broad, because it includes absurd theories such as the theory that an act is morally right if it increases the number of goats in Texas. Of course, such theories are implausible. Still, it is not implausible to call them consequentialist, since they do look only at consequences. The implausibility of one version of consequentialism does not make consequentialism implausible in general, since other versions of consequentialism still might be plausible. Besides, anyone who wants to pick out a smaller set of moral theories that excludes this absurd theory may talk about evaluative consequentialism, which is the claim that moral rightness depends only on the value of the consequences. Then those who want to talk about the even smaller group of moral theories that accepts both evaluative consequentialism and agent-neutrality may describe them as agent-neutral evaluative consequentialism. If anyone still insists on calling these smaller groups of theories by the simple name, 'consequentialism', this narrower word usage will not affect any substantive issue. Still, if the definition of consequentialism becomes too broad, it might seem to lose force. Some philosophers have argued that any moral theory, or at least any plausible moral theory, could be represented as a version of consequentialism (Sosa 1993, Portmore 2009, Dreier 1993 and 2011; but see Brown 2011). If so, then it means little to label a theory as consequentialist. The real content comes only by contrasting theories that are not consequentialist. In the end, what matters is only that we get clear about which theories a particular commentator counts as consequentialist or not and which claims are supposed to make them consequentialist or not. Only then can we know which claims are at stake when this commentator supports or criticizes what they call "consequentialism". Then we can ask whether each objection really refutes that particular claim. ## 3. What is Good? Hedonistic vs. Pluralistic Consequentialisms Some moral theorists seek a single simple basic principle because they assume that simplicity is needed in order to decide what is right when less basic principles or reasons conflict. This assumption seems to make hedonism attractive. Unfortunately, however, hedonism is not as simple as they assume, because hedonists count both pleasures and pains. Pleasure is distinct from the absence of pain, and pain is distinct from the absence of pleasure, since sometimes people feel neither pleasure nor pain, and sometimes they feel both at once. Nonetheless, hedonism was adopted partly because it seemed simpler than competing views. The simplicity of hedonism was also a source of opposition. From the start, the hedonism in classic utilitarianism was treated with contempt. Some contemporaries of Bentham and Mill argued that hedonism lowers the value of human life to the level of animals, because it implies that, as Bentham said, an unsophisticated game (such as push-pin) is as good as highly intellectual poetry if the game creates as much pleasure (Bentham 1843). Quantitative hedonists sometimes respond that great poetry almost always creates more pleasure than trivial games (or sex and drugs and rock-and-roll), because the pleasures of poetry are more certain (or probable), durable (or lasting), fecund (likely to lead to other pleasures), pure (unlikely to lead to pains), and so on. Mill used a different strategy to avoid calling push-pin as good as poetry. He distinguished higher and lower qualities of pleasures according to the preferences of people who have experienced both kinds (Mill 1861, 56; compare Plato 1993 and Hutcheson 1755, 421-23). This qualitative hedonism has been subjected to much criticism, including charges that it is incoherent and does not count as hedonism (Moore 1903, 80-81; cf. Feldman 1997, 106-24). Even if qualitative hedonism is coherent and is a kind of hedonism, it still might not seem plausible. Some critics argue that not all pleasures are valuable, since, for example, there is no value in the pleasures that a sadist gets from whipping a victim or that an addict gets from drugs. Other opponents object that not only pleasures are intrinsically valuable, because other things are valuable independently of whether they lead to pleasure or avoid pain. For example, my love for my wife does not seem to become less valuable when I get less pleasure from her because she contracts some horrible disease. Similarly, freedom seems valuable even when it creates anxiety, and even when it is freedom to do something (such as leave one's country) that one does not want to do. Again, many people value knowledge of distant galaxies regardless of whether this knowledge will create pleasure or avoid pain. These points against hedonism are often supplemented with the story of the experience machine found in Nozick 1974 (42-45; cf. De Brigard 2010) and the movie, The Matrix. People on this machine believe they are spending time with their friends, winning Olympic gold medals and Nobel prizes, having sex with their favorite lovers, or doing whatever gives them the greatest balance of pleasure over pain. Although they have no real friends or lovers and actually accomplish nothing, people on the experience machine get just as much pleasure as if their beliefs were true. Moreover, they feel no (or little) pain. Assuming that the machine is reliable, it would seem irrational not to hook oneself up to this machine if pleasure and pain were all that mattered, as hedonists claim. Since it does not seem irrational to refuse to hook oneself up to this machine, hedonism seems inadequate. The reason is that hedonism overlooks the value of real friendship, knowledge, freedom, and achievements, all of which are lacking for deluded people on the experience machine. Some hedonists claim that this objection rests on a misinterpretation of hedonism. If hedonists see pleasure and pain as sensations, then a machine might be able to reproduce those sensations. However, we can also say that a mother is pleased that her daughter gets good grades. Such propositional pleasure occurs only when the state of affairs in which the person takes pleasure exists (that is, when the daughter actually gets good grades). But the relevant states of affairs would not really exist if one were hooked up to the experience machine. Hence, hedonists who value propositional pleasure rather than or in addition to sensational pleasure can deny that more pleasure is achieved by hooking oneself up to such an experience machine (Feldman 1997, 79-105; see also Tannsjo 1998 and Feldman 2004 for more on hedonism). A related position rests on the claim that what is good is desire satisfaction or the fulfillment of preferences; and what is bad is the frustration of desires or preferences. What is desired or preferred is usually not a sensation but is, rather, a state of affairs, such as having a friend or accomplishing a goal. If a person desires or prefers to have true friends and true accomplishments and not to be deluded, then hooking this person up to the experience machine need not maximize desire satisfaction. Utilitarians who adopt this theory of value can then claim that an agent morally ought to do an act if and only if that act maximizes desire satisfaction or preference fulfillment (that is, the degree to which the act achieves whatever is desired or preferred). What maximizes desire satisfaction or preference fulfillment need not maximize sensations of pleasure when what is desired or preferred is not a sensation of pleasure. This position is usually described as preference utilitarianism. One problem for preference utilitarianism concerns how to make interpersonal comparisons (though this problem also arises for several other theories of value). If we want to know what one person prefers, we can ask what that person would choose in conflicts. We cannot, however, use the same method to determine whether one person's preference is stronger or weaker than another person's preference, since these different people might choose differently in the decisive conflicts. We need to settle which preference (or pleasure) is stronger because we may know that Jones prefers A's being done to A's not being done (and Jones would receive more pleasure from A's being done than from A's not being done), whereas Smith prefers A's not being done (and Smith would receive more pleasure from A's not being done than from A's being done). To determine whether it is right to do A or not to do A, we must be able to compare the strengths of Jones's and Smith's preferences (or the amounts of pleasure each would receive in her preferred outcome) in order to determine whether doing A or not doing A would be better overall. Utilitarians and consequentialists have proposed many ways to solve this problem of interpersonal comparison, and each attempt has received criticisms. Debates about this problem still rage. (For a recent discussion with references, see Coakley 2015.) Preference utilitarianism is also often criticized on the grounds that some preferences are misinformed, crazy, horrendous, or trivial. I might prefer to drink the liquid in a glass because I think that it is beer, though it really is strong acid. Or I might prefer to die merely because I am clinically depressed. Or I might prefer to torture children. Or I might prefer to spend my life learning to write as small as possible. In all such cases, opponents of preference utilitarianism can deny that what I prefer is really good. Preference utilitarians can respond by limiting the preferences that make something good, such as by referring to informed desires that do not disappear after therapy (Brandt 1979). However, it is not clear that such qualifications can solve all of the problems for a preference theory of value without making the theory circular by depending on substantive assumptions about which preferences are for good things. Many consequentialists deny that all values can be reduced to any single ground, such as pleasure or desire satisfaction, so they instead adopt a pluralistic theory of value. Moore's ideal utilitarianism, for example, takes into account the values of beauty and truth (or knowledge) in addition to pleasure (Moore 1903, 83-85, 194; 1912). Other consequentialists add the intrinsic values of friendship or love, freedom or ability, justice or fairness, desert, life, virtue, and so on. If the recognized values all concern individual welfare, then the theory of value can be called welfarist (Sen 1979). When a welfarist theory of value is combined with the other elements of classic utilitarianism, the resulting theory can be called welfarist consequentialism. One non-welfarist theory of value is perfectionism, which claims that certain states make a person's life good without necessarily being good for the person in any way that increases that person's welfare (Hurka 1993, esp. 17). If this theory of value is combined with other elements of classic utilitarianism, the resulting theory can be called perfectionist consequentialism or, in deference to its Aristotelian roots, eudaemonistic consequentialism. Similarly, some consequentialists hold that an act is right if and only if it maximizes some function of both happiness and capabilities (Sen 1985, Nussbaum 2000). Disabilities are then seen as bad regardless of whether they are accompanied by pain or loss of pleasure. Or one could hold that an act is right if it maximizes fulfillment (or minimizes violation) of certain specified moral rights. Such theories are sometimes described as a utilitarianism of rights. This approach could be built into total consequentialism with rights weighed against happiness and other values or, alternatively, the disvalue of rights violations could be lexically ranked prior to any other kind of loss or harm (cf. Rawls 1971, 42). Such a lexical ranking within a consequentialist moral theory would yield the result that nobody is ever justified in violating rights for the sake of happiness or any value other than rights, although it would still allow some rights violations in order to avoid or prevent other rights violations. When consequentialists incorporate a variety of values, they need to rank or weigh each value against the others. This is often difficult. Some consequentialists even hold that certain values are incommensurable or incomparable in that no comparison of their values is possible (Griffin 1986 and Chang 1997). This position allows consequentialists to recognize the possibility of irresolvable moral dilemmas (Sinnott-Armstrong 1988, 81; Railton 2003, 249-91). Pluralism about values also enables consequentialists to handle many of the problems that plague hedonistic utilitarianism. For example, opponents often charge that classical utilitarians cannot explain our obligations to keep promises and not to lie when no pain is caused or pleasure is lost. Whether or not hedonists can meet this challenge, pluralists can hold that knowledge is intrinsically good and/or that false belief is intrinsically bad. Then, if deception causes false beliefs, deception is instrumentally bad, and agents ought not to lie without a good reason, even when lying causes no pain or loss of pleasure. Since lying is an attempt to deceive, to lie is to attempt to do what is morally wrong (in the absence of defeating factors). Similarly, if a promise to do an act is an attempt to make an audience believe that the promiser will do the act, then to break a promise is for a promiser to make false a belief that the promiser created or tried to create. Although there is more tale to tell, the disvalue of false belief can be part of a consequentialist story about why it is morally wrong to break promises. When such pluralist versions of consequentialism are not welfarist, some philosophers would not call them utilitarian. However, this usage is not uniform, since even non-welfarist views are sometimes called utilitarian. Whatever you call them, the important point is that consequentialism and the other elements of classical utilitarianism are compatible with many different theories about which things are good or valuable. Instead of turning pluralist, some consequentialists foreswear the aggregation of values. Classic utilitarianism added up the values within each part of the consequences to determine which total set of consequences has the most value in it. One could, instead, aggregate goods for each individual but not aggregate goods of separate individuals (Roberts 2002). Alternatively, one could give up all aggregation, including aggregation for individuals, and instead rank the complete worlds or sets of consequences caused by acts without adding up the values of the parts of those worlds or consequences. One motive for this move is Moore's principle of organic unity (Moore 1903, 27-36), which claims that the value of a combination or "organic unity" of two or more things cannot be calculated simply by adding the values of the things that are combined or unified. For example, even if punishment of a criminal causes pain, a consequentialist can hold that a world with both the crime and the punishment is better than a world with the crime but not the punishment, perhaps because the former contains more justice, without adding the value of this justice to the negative value of the pain of the punishment. Similarly, a world might seem better when people do not get pleasures that they do not deserve, even if this judgment is not reached by adding the values of these pleasures to other values to calculate any total. Cases like these lead some consequentialists to deny that moral rightness is any aggregative function of the values of particular effects of acts. Instead, they compare the whole world that results from an action with the whole world that results from not doing that action. If the former is better, then the action is morally right (J.J.C. Smart 1973, 32; Feldman 1997, 17-35). This approach can be called holistic consequentialism or world utilitarianism. Another way to incorporate relations among values is to consider distribution. Compare one outcome where most people are destitute but a few lucky people have extremely large amounts of goods with another outcome that contains slightly less total goods but where every person has nearly the same amount of goods. Egalitarian critics of classical utilitarianism argue that the latter outcome is better, so more than the total amount of good matters. Traditional hedonistic utilitarians who prefer the latter outcome often try to justify egalitarian distributions of goods by appealing to a principle of diminishing marginal utility. Other consequentialists, however, incorporate a more robust commitment to equality. Early on, Sidgwick (1907, 417) responded to such objections by allowing distribution to break ties between other values. More recently, some consequentialists have added some notion of fairness (Broome 1991, 192-200) or desert (Feldman 1997, 154-74) to their test of which outcome is best. (See also Kagan 1998, 48-59.) Others turn to prioritarianism, which puts more weight on people who are worse off (Adler and Norheim 2022, Arneson 2022). Such consequentialists do not simply add up values; they look at patterns. A related issue arises from population change. Imagine that a government considers whether to provide free contraceptives to curb a rise in population. Without free contraceptives, overcrowding will bring hunger, disease, and pain, so each person will be worse off. Still, each new person will have enough pleasure and other goods that the total net utility will increase with the population. Classic utilitarianism focuses on total utility, so it seems to imply that this government should not provide free contraceptives. That seems implausible to many utilitarians. To avoid this result, some utilitarians claim that an act is morally wrong if and only if its consequences contain more pain (or other disvalues) than an alternative, regardless of positive values (cf. R. N. Smart 1958). This negative utilitarianism implies that the government should provide contraceptives, since that program reduces pain (and other disvalues), even though it also decreases total net pleasure (or good). Unfortunately, negative utilitarianism also seems to imply that the government should painlessly kill everyone it can, since dead people feel no pain (and have no false beliefs, diseases, or disabilities - though killing them does cause loss of ability). A more popular response is average utilitarianism, which says that the best consequences are those with the highest average utility (cf. Rawls 1971, 161-75). The average utility would be higher with the contraceptive program than without it, so average utilitarianism yields the more plausible result--that the government should adopt the contraceptive program. Critics sometimes charge that the average utility could also be increased by killing the worst off, but this claim is not at all clear, because such killing would put everyone in danger (since, after the worst off are killed, another group becomes the worst off, and then they might be killed next). Still, average utilitarianism faces problems of its own (such as "the mere addition paradox" in Parfit 1984, chap. 19). In any case, all maximizing consequentialists, whether or not they are pluralists, must decide whether moral rightness depends on maximizing total good or average good. A final challenge to consequentialists' accounts of value derives from Geach 1956 and has been pressed by Thomson 2001. Thomson argues that "A is a good X" (such as a good poison) does not entail "A is good", so the term "good" is an attributive adjective and cannot legitimately be used without qualification. On this view, it is senseless to call something good unless this means that it is good for someone or in some respect or for some use or at some activity or as an instance of some kind. Consequentialists are supposed to violate this restriction when they say that the total or average consequences or the world as a whole is good without any such qualification. However, consequentialists can respond either that the term "good" has predicative uses in addition to its attributive uses or that when they call a world or total set of consequences good, they are calling it good for consequences or for a world (Sinnott-Armstrong 2003a). If so, the fact that "good" is often used attributively creates no problem for consequentialists. ## 4. Which Consequences? Actual vs. Expected Consequentialisms A second set of problems for classic utilitarianism is epistemological. Classic utilitarianism seems to require that agents calculate all consequences of each act for every person for all time. That's impossible. This objection rests on a misinterpretation. These critics assume that the principle of utility is supposed to be used as a decision procedure or guide, that is, as a method that agents consciously apply to acts in advance to help them make decisions. However, most classic and contemporary utilitarians and consequentialists do not propose their principles as decision procedures. (Bales 1971) Bentham wrote, "It is not to be expected that this process [his hedonic calculus] should be strictly pursued previously to every moral judgment." (1789, Chap. IV, Sec. VI) Mill agreed, "it is a misapprehension of the utilitarian mode of thought to conceive it as implying that people should fix their minds upon so wide a generality as the world, or society at large." (1861, Chap. II, Par. 19) Sidgwick added, "It is not necessary that the end which gives the criterion of rightness should always be the end at which we consciously aim." (1907, 413) Instead, most consequentialists claim that overall utility is the criterion or standard of what is morally right or morally ought to be done. Their theories are intended to spell out the necessary and sufficient conditions for an act to be morally right, regardless of whether the agent can tell in advance whether those conditions are met. Just as the laws of physics govern golf ball flight, but golfers need not calculate physical forces while planning shots; so overall utility can determine which decisions are morally right, even if agents need not calculate utilities while making decisions. If the principle of utility is used as a criterion of the right rather than as a decision procedure, then classical utilitarianism does not require that anyone know the total consequences of anything before making a decision. Furthermore, a utilitarian criterion of right implies that it would not be morally right to use the principle of utility as a decision procedure in cases where it would not maximize utility to try to calculate utilities before acting. Utilitarians regularly argue that most people in most circumstances ought not to try to calculate utilities, because they are too likely to make serious miscalculations that will lead them to perform actions that reduce utility. It is even possible to hold that most agents usually ought to follow their moral intuitions, because these intuitions evolved to lead us to perform acts that maximize utility, at least in likely circumstances (Hare 1981, 46-47). Some utilitarians (Sidgwick 1907, 489-90) suggest that a utilitarian decision procedure may be adopted as an esoteric morality by an elite group that is better at calculating utilities, but utilitarians can, instead, hold that nobody should use the principle of utility as a decision procedure. This move is supposed to make consequentialism self-refuting, according to some opponents. However, there is nothing incoherent about proposing a decision procedure that is separate from one's criterion of the right. Similar distinctions apply in other normative realms. The criterion of a good stock investment is its total return, but the best decision procedure still might be to reduce risk by buying an index fund or blue-chip stocks. Criteria can, thus, be self-effacing without being self-refuting (Parfit 1984, chs. 1 and 4). Others object that this move takes the force out of consequentialism, because it leads agents to ignore consequentialism when they make real decisions. However, a criterion of the right can be useful at a higher level by helping us choose among available decision procedures and refine our decision procedures as circumstances change and we gain more experience and knowledge. Hence, most consequentialists do not mind giving up consequentialism as a direct decision procedure as long as consequences remain the criterion of rightness (but see Chappell 2001). If overall utility is the criterion of moral rightness, then it might seem that nobody could know what is morally right. If so, classical utilitarianism leads to moral skepticism. However, utilitarians insist that we can have strong reasons to believe that certain acts reduce utility, even if we have not yet inspected or predicted every consequence of those acts. For example, in normal circumstances, if someone were to torture and kill his children, it is possible that this would maximize utility, but that is very unlikely. Maybe they would have grown up to be mass murders, but it is at least as likely that they would grow up to cure serious diseases or do other great things, and it is much more likely that they would have led normally happy (or at least not destructive) lives. So observers as well as agents have adequate reasons to believe that such acts are morally wrong, according to act utilitarianism. In many other cases, it will still be hard to tell whether an act will maximize utility, but that shows only that there are severe limits to our knowledge of what is morally right. That should be neither surprising nor problematic for utilitarians. If utilitarians want their theory to allow more moral knowledge, they can make a different kind of move by turning from actual consequences to expected or expectable consequences. Suppose that Alice finds a runaway teenager who asks for money to get home. Alice wants to help and reasonably believes that buying a bus ticket home for this runaway will help, so she buys a bus ticket and puts the runaway on the bus. Unfortunately, the bus is involved in a freak accident, and the runaway is killed. If actual consequences are what determine moral wrongness, then it was morally wrong for Alice to buy the bus ticket for this runaway. Opponents claim that this result is absurd enough to refute classic utilitarianism. Some utilitarians bite the bullet and say that Alice's act was morally wrong, but it was blameless wrongdoing, because her motives were good, and she was not responsible, given that she could not have foreseen that her act would cause harm. Since this theory makes actual consequences determine moral rightness, it can be called actual consequentialism. Other responses claim that moral rightness depends on foreseen, foreseeable, intended, or likely consequences, rather than actual ones. Imagine that Bob does not in fact foresee a bad consequence that would make his act wrong if he did foresee it, but that Bob could easily have foreseen this bad consequence if he had been paying attention. Maybe he does not notice the rot on the hamburger he feeds to his kids which makes them sick. If foreseen consequences are what matter, then Bob's act is not morally wrong. If foreseeable consequences are what matter, then Bob's act is morally wrong, because the bad consequences were foreseeable. Now consider Bob's wife, Carol, who notices that the meat is rotten but does not want to have to buy more, so she feeds it to her children anyway, hoping that it will not make them sick; but it does. Carol's act is morally wrong if foreseen or foreseeable consequences are what matter, but not if what matter are intended consequences, because she does not intend to make her children sick. Finally, consider Bob and Carol's son Don, who does not know enough about food to be able to know that eating rotten meat can make people sick. If Don feeds the rotten meat to his little sister, and it makes her sick, then the bad consequences are not intended, foreseen, or even foreseeable by Don, but those bad results are still objectively likely or probable, unlike the case of Alice. Some philosophers deny that probability can be fully objective, but at least the consequences here are foreseeable by others who are more informed than Don can be at the time. For Don to feed the rotten meat to his sister is, therefore, morally wrong if likely consequences are what matter, but not morally wrong if what matter are foreseen or foreseeable or intended consequences. Consequentialist moral theories that focus on actual or objectively probable consequences are often described as objective consequentialism (Railton 1984). In contrast, consequentialist moral theories that focus on intended or foreseen consequences are usually described as subjective consequentialism. Consequentialist moral theories that focus on reasonably foreseeable consequences are then not subjective insofar as they do not depend on anything inside the actual subject's mind, but they are subjective insofar as they do depend on which consequences this particular subject would foresee if he or she were better informed or more rational. One final solution to these epistemological problems deploys the legal notion of proximate cause. If consequentialists define consequences in terms of what is caused (unlike Sosa 1993), then which future events count as consequences is affected by which notion of causation is used to define consequences. Suppose I give a set of steak knives to a friend. Unforeseeably, when she opens my present, the decorative pattern on the knives somehow reminds her of something horrible that her husband did. This memory makes her so angry that she voluntarily stabs and kills him with one of the knives. She would not have killed her husband if I had given her spoons instead of knives. Did my decision or my act of giving her knives cause her husband's death? Most people (and the law) would say that the cause was her act, not mine. Why? One explanation is that her voluntary act intervened in the causal chain between my act and her husband's death. Moreover, even if she did not voluntarily kill him, but instead she slipped and fell on the knives, thereby killing herself, my gift would still not be a cause of her death, because the coincidence of her falling intervened between my act and her death. The point is that, when voluntary acts and coincidences intervene in certain causal chains, then the results are not seen as caused by the acts further back in the chain of necessary conditions (Hart and Honore 1985). Now, if we assume that an act must be such a proximate cause of a harm in order for that harm to be a consequence of that act, then consequentialists can claim that the moral rightness of that act is determined only by such proximate consequences. This position, which might be called proximate consequentialism, makes it much easier for agents and observers to justify moral judgments of acts because it obviates the need to predict non-proximate consequences in distant times and places. Hence, this move is worth considering, even though it has never been developed as far as I know and deviates far from traditional consequentialism, which counts not only proximate consequences but all upshots -- that is, everything for which the act is a causally necessary condition. ## 5. Consequences of What? Rights, Relativity, and Rules Another problem for utilitarianism is that it seems to overlook justice and rights. One common illustration is called Transplant. Imagine that each of five patients in a hospital will die without an organ transplant. The patient in Room 1 needs a heart, the patient in Room 2 needs a liver, the patient in Room 3 needs a kidney, and so on. The person in Room 6 is in the hospital for routine tests. Luckily (for them, not for him!), his tissue is compatible with the other five patients, and a specialist is available to transplant his organs into the other five. This operation would save all five of their lives, while killing the "donor". There is no other way to save any of the other five patients (Foot 1966, Thomson 1976; compare related cases in Carritt 1947 and McCloskey 1965). We need to add that the organ recipients will emerge healthy, the source of the organs will remain secret, the doctor won't be caught or punished for cutting up the "donor", and the doctor knows all of this to a high degree of probability (despite the fact that many others will help in the operation). Still, with the right details filled in (no matter how unrealistic), it looks as if cutting up the "donor" will maximize utility, since five lives have more utility than one life (assuming that the five lives do not contribute too much to overpopulation). If so, then classical utilitarianism implies that it would not be morally wrong for the doctor to perform the transplant and even that it would be morally wrong for the doctor not to perform the transplant. Most people find this result abominable. They take this example to show how bad it can be when utilitarians overlook individual rights, such as the unwilling donor's right to life. Utilitarians can bite the bullet, again. They can deny that it is morally wrong to cut up the "donor" in these circumstances. Of course, doctors still should not cut up their patients in anything close to normal circumstances, but this example is so abnormal and unrealistic that we should not expect our normal moral rules to apply, and we should not trust our moral intuitions, which evolved to fit normal situations (Sprigge 1965). Many utilitarians are happy to reject common moral intuitions in this case, like many others (cf. Singer 1974, Unger 1996, Norcross 1997). Most utilitarians lack such strong stomachs (or teeth), so they modify utilitarianism to bring it in line with common moral intuitions, including the intuition that doctors should not cut up innocent patients. One attempt claims that a killing is worse than a death. The doctor would have to kill the "donor" in order to prevent the deaths of the five patients, but nobody is killed if the five patients die. If one killing is worse than five deaths that do not involve killing, then the world that results from the doctor performing the transplant is worse than the world that results from the doctor not performing the transplant. With this new theory of value, consequentialists can agree with others that it is morally wrong for the doctor to cut up the "donor" in this example. A modified example still seems problematic. Just suppose that the five patients need a kidney, a lung, a heart, and so forth because they were all victims of murder attempts. Then the world will contain the five killings of them if they die, but not if they do not die. Thus, even if killings are worse than deaths that are not killings, the world will still be better overall (because it will contain fewer killings as well as fewer deaths) if the doctor cuts up the "donor" to save the five other patients. But most people still think it would be morally wrong for the doctor to kill the one to prevent the five killings. The reason is that it is not the doctor who kills the five, and the doctor's duty seems to be to reduce the amount of killing that she herself does. In this view, the doctor is not required to promote life or decrease death or even decrease killing by other people. The doctor is, instead, required to honor the value of life by not causing loss of life (cf. Pettit 1997). This kind of case leads some consequentialists to introduce agent-relativity into their theory of value (Sen 1982, Broome 1991, Portmore 2001, 2003, 2011). To apply a consequentialist moral theory, we need to compare the world with the transplant to the world without the transplant. If this comparative evaluation must be agent-neutral, then, if an observer judges that the world with the transplant is better, the agent must make the same judgment, or else one of them is mistaken. However, if such evaluations can be agent-relative, then it could be legitimate for an observer to judge that the world with the transplant is better (since it contains fewer killings by anyone), while it is also legitimate for the doctor as agent to judge that the world with the transplant is worse (because it includes a killing by him). In other cases, such as competitions, it might maximize the good from an agent's perspective to do an act, while maximizing the good from an observer's perspective to stop the agent from doing that very act. If such agent-relative value makes sense, then it can be built into consequentialism to produce the claim that an act is morally wrong if and only if the act's consequences include less overall value from the perspective of the agent. This agent-relative consequentialism, plus the claim that the world with the transplant is worse from the perspective of the doctor, could justify the doctor's judgment that it would be morally wrong for him to perform the transplant. A key move here is to adopt the agent's perspective in judging the agent's act. Agent-neutral consequentialists judge all acts from the observer's perspective, so they would judge the doctor's act to be wrong, since the world with the transplant is better from an observer's perspective. In contrast, an agent-relative approach requires observers to adopt the doctor's perspective in judging whether it would be morally wrong for the doctor to perform the transplant. This kind of agent-relative consequentialism is then supposed to capture commonsense moral intuitions in such cases (Portmore 2011). Agent-relativity is also supposed to solve other problems. W. D. Ross (1930, 34-35) argued that, if breaking a promise created only slightly more happiness overall than keeping the promise, then the agent morally ought to break the promise according to classic utilitarianism. This supposed counterexample cannot be avoided simply by claiming that keeping promises has agent-neutral value, since keeping one promise might prevent someone else from keeping another promise. Still, agent-relative consequentialists can respond that keeping a promise has great value from the perspective of the agent who made the promise and chooses whether or not to keep it, so the world where a promise is kept is better from the agent's perspective than another world where the promise is not kept, unless enough other values override the value of keeping the promise. In this way, agent-relative consequentialists can explain why agents morally ought not to break their promises in just the kind of case that Ross raised. Similarly, critics of utilitarianism often argue that utilitarians cannot be good friends, because a good friend places more weight on the welfare of his or her friends than on the welfare of strangers, but utilitarianism requires impartiality among all people. However, agent-relative consequentialists can assign more weight to the welfare of a friend of an agent when assessing the value of the consequences of that agent's acts. In this way, consequentialists try to capture common moral intuitions about the duties of friendship (see also Jackson 1991). One final variation still causes trouble. Imagine that the doctor herself wounded the five people who need organs. If the doctor does not save their lives, then she will have killed them herself. In this case, even if the doctor can disvalue killings by herself more than killings by other people, the world still seems better from her own perspective if she performs the transplant. Critics will object that it is, nonetheless, morally wrong for the doctor to perform the transplant. Many people will not find this intuition as clear as in the other cases, but consequentialists who do find it immoral for the doctor to perform the transplant even in this case will need to modify consequentialism in some other way in order to yield the desired judgment. This problem cannot be solved by building rights or fairness or desert into the theory of value. The five do not deserve to die, and they do deserve their lives, just as much as the one does. Each option violates someone's right not to be killed and is unfair to someone. So consequentialists need more than just new values if they want to avoid endorsing this transplant. One option is to go indirect. A direct consequentialist holds that the moral qualities of something depend only on the consequences of that very thing. Thus, a direct consequentialist about motives holds that the moral qualities of a motive depend on the consequences of that motive. A direct consequentialist about virtues holds that the moral qualities of a character trait (such as whether or not it is a moral virtue) depend on the consequences of that trait (Driver 2001a, Hurka 2001, Jamieson 2005, Bradley 2005). A direct consequentialist about acts holds that the moral qualities of an act depend on the consequences of that act. Someone who adopts direct consequentialism about everything is a global direct consequentialist (Pettit and Smith 2000, Driver 2012). In contrast, an indirect consequentialist holds that the moral qualities of something depend on the consequences of something else. One indirect version of consequentialism is motive consequentialism, which claims that the moral qualities of an act depend on the consequences of the motive of that act (compare Adams 1976 and Sverdlik 2011). Another indirect version is virtue consequentialism, which holds that whether an act is morally right depends on whether it stems from or expresses a state of character that maximizes good consequences and, hence, is a virtue. The most common indirect consequentialism is rule consequentialism, which makes the moral rightness of an act depend on the consequences of a rule (Singer 1961). Since a rule is an abstract entity, a rule by itself strictly has no consequences. Still, obedience rule consequentialists can ask what would happen if everybody obeyed a rule or what would happen if everybody violated a rule. They might argue, for example, that theft is morally wrong because it would be disastrous if everybody broke a rule against theft. Often, however, it does not seem morally wrong to break a rule even though it would cause disaster if everybody broke it. For example, if everybody broke the rule "Have some children", then our species would die out, but that hardly shows it is morally wrong not to have any children. Luckily, our species will not die out if everyone is permitted not to have children, since enough people want to have children. Thus, instead of asking, "What would happen if everybody did that?", rule consequentialists should ask, "What would happen if everybody were permitted to do that?" People are permitted to do what violates no accepted rule, so asking what would happen if everybody were permitted to do an act is just the flip side of asking what would happen if people accepted a rule that forbids that act. Such acceptance rule consequentialists then claim that an act is morally wrong if and only if it violates a rule whose acceptance has better consequences than the acceptance of any incompatible rule. In some accounts, a rule is accepted when it is built into individual consciences (Brandt 1992). Other rule utilitarians, however, require that moral rules be publicly known (Gert 2005; cf. Sinnott-Armstrong 2003b) or built into public institutions (Rawls 1955). Then they hold what can be called public acceptance rule consequentialism: an act is morally wrong if and only if it violates a rule whose public acceptance maximizes the good. The indirectness of such rule utilitarianism provides a way to remain consequentialist and yet capture the common moral intuition that it is immoral to perform the transplant in the above situation. Suppose people generally accepted a rule that allows a doctor to transplant organs from a healthy person without consent when the doctor believes that this transplant will maximize utility. Widely accepting this rule would lead to many transplants that do not maximize utility, since doctors (like most people) are prone to errors in predicting consequences and weighing utilities. Moreover, if the rule is publicly known, then patients will fear that they might be used as organ sources, so they would be less likely to go to a doctor when they need one. The medical profession depends on trust that this public rule would undermine. For such reasons, some rule utilitarians conclude that it would not maximize utility for people generally to accept a rule that allows doctors to transplant organs from unwilling donors. If this claim is correct, then rule utilitarianism implies that it is morally wrong for a particular doctor to use an unwilling donor, even for a particular transplant that would have better consequences than any alternative even from the doctor's own perspective. Common moral intuition is thereby preserved. Rule utilitarianism faces several potential counterexamples (such as whether public rules allowing slavery could sometimes maximize utility) and needs to be formulated more precisely (particularly in order to avoid collapsing into act-utilitarianism; cf. Lyons 1965). Such details are discussed in another entry in this encyclopedia (see Hooker on rule-consequentialism). Here I will just point out that direct consequentialists find it convoluted and implausible to judge a particular act by the consequences of something else (Smart 1956). Why should mistakes by other doctors in other cases make this doctor's act morally wrong, when this doctor knows for sure that he is not mistaken in this case? Rule consequentialists can respond that we should not claim special rights or permissions that we are not willing to grant to every other person, and that it is arrogant to think we are less prone to mistakes than other people are. However, this doctor can reply that he is willing to give everyone the right to violate the usual rules in the rare cases when they do know for sure that violating those rules really maximizes utility. Anyway, even if rule utilitarianism accords with some common substantive moral intuitions, it still seems counterintuitive in other ways. This makes it worthwhile to consider how direct consequentialists can bring their views in line with common moral intuitions, and whether they need to do so. ## 6. Consequences for Whom? Limiting the Demands of Morality Another popular charge is that classic utilitarianism demands too much, because it requires us to do acts that are or should be moral options (neither obligatory nor forbidden). (Scheffler 1982) For example, imagine that my old shoes are serviceable but dirty, so I want a new pair of shoes that costs $100. I could wear my old shoes and give the $100 to a charity that will use my money to save someone else's life. It would seem to maximize utility for me to give the $100 to the charity. If it is morally wrong to do anything other than what maximizes utility, then it is morally wrong for me to buy the shoes. But buying the shoes does not seem morally wrong. It might be morally better to give the money to charity, but such contributions seem supererogatory, that is, above and beyond the call of duty. Of course, there are many more cases like this. When I watch television, I always (or almost always) could do more good by helping others, but it does not seem morally wrong to watch television. When I choose to teach philosophy rather than working for CARE or the Peace Corps, my choice probably fails to maximize utility overall. If we were required to maximize utility, then we would have to make very different choices in many areas of our lives. The requirement to maximize utility, thus, strikes many people as too demanding because it interferes with the personal decisions that most of us feel should be left up to the individual. Some utilitarians respond by arguing that we really are morally required to change our lives so as to do a lot more to increase overall utility (see Kagan 1989, P. Singer 1993, and Unger 1996). Such hard-liners claim that most of what most people do is morally wrong, because most people rarely maximize utility. Some such wrongdoing might be blameless when agents act from innocent or even desirable motives, but it is still supposed to be moral wrongdoing. Opponents of utilitarianism find this claim implausible, but it is not obvious that their counter-utilitarian intuitions are reliable or well-grounded (Murphy 2000, chs. 1-4; cf. Mulgan 2001, Singer 2005, Greene 2013). Other utilitarians blunt the force of the demandingness objection by limiting direct utilitarianism to what people morally ought to do. Even if we morally ought to maximize utility, it need not be morally wrong to fail to maximize utility. John Stuart Mill, for example, argued that an act is morally wrong only when both it fails to maximize utility and its agent is liable to punishment for the failure (Mill 1861). It does not always maximize utility to punish people for failing to maximize utility. Thus, on this view, it is not always morally wrong to fail to do what one morally ought to do. If Mill is correct about this, then utilitarians can say that we ought to give much more to charity, but we are not required or obliged to do so, and failing to do so is not morally wrong (cf. Sinnott-Armstrong 2005). Many utilitarians still want to avoid the claim that we morally ought to give so much to charity. One way around this claim uses a rule-utilitarian theory of what we morally ought to do. If it costs too much to internalize rules implying that we ought to give so much to charity, then, according to such rule-utilitarianism, it is not true that we ought to give so much to charity (Hooker 2000, ch. 8). Another route follows an agent-relative theory of value. If there is more value in benefiting oneself or one's family and friends than there is disvalue in letting strangers die (without killing them), then spending resources on oneself or one's family and friends would maximize the good. A problem is that such consequentialism would seem to imply that we morally ought not to contribute those resources to charity, although such contributions seem at least permissible. More personal leeway could also be allowed by deploying the legal notion of proximate causation. When a starving stranger would stay alive if and only if one contributed to a charity, contributing to the charity still need not be the proximate cause of the stranger's life, and failing to contribute need not be the proximate cause of his or her death. Thus, if an act is morally right when it includes the most net good in its proximate consequences, then it might not be morally wrong either to contribute to the charity or to fail to do so. This potential position, as mentioned above, has not yet been developed, as far as I know. Yet another way to reach this conclusion is to give up maximization and to hold instead that we morally ought to do what creates enough utility. This position is often described as satisficing consequentialism (Slote 1984). According to satisficing consequentialism, it is not morally wrong to fail to contribute to a charity if one contributes enough to other charities and if the money or time that one could contribute does create enough good, so it is not just wasted. (For criticisms, see Bradley 2006.) A related position is progressive consequentialism, which holds that we morally ought to improve the world or make it better than it would be if we did nothing, but we don't have to improve it as much as we can (Elliot and Jamieson, 2009). Both satisficing and progressive consequentialism allow us to devote some of our time and money to personal projects that do not maximize overall good. A more radical set of proposals confines consequentialism to judgements about how good an act is on a scale (Norcross 2006, 2020) or to degrees of wrongness and rightness (Sinhababu 2018). This positions are usually described as scalar consequentialism. A scalar consequentialist can refuse to say whether it is absolutely right or wrong to give $1000 to charity, for example, but still say that giving $1000 to charity is better and more right than giving only $100 and simultaneously worse and more wrong than giving $10,000. A related contrastivist consequentialism could say that one ought to give $1000 in contrast with $100 but not in contrast with $10,000 (cf. Snedegar 2017). Such positions can also hold that less or less severe negative sanctions are justified when an agent's act is worse than a smaller set of alternatives compared to when the agent's act is worse than a larger set of alternatives. This approach then becomes less demanding, both because it sees less negative sanctions as justified when the agent fails to do the best act possible, and also because it avoids saying that everyday actions are simply wrong without comparison to any set of alternatives. Opponents still object that all such consequentialist theories are misdirected. When I decide to visit a friend instead of working for a charity, I can know that my act is not immoral even if I have not calculated that the visit will create enough overall good or that it will improve the world. These critics hold that friendship requires us to do certain favors for friends without weighing our friends' welfare impartially against the welfare of strangers. Similarly, if I need to choose between saving my drowning wife and saving a drowning stranger, it would be "one thought too many" (Williams 1981) for me to calculate the consequences of each act. I morally should save my wife straightaway without calculating utilities. In response, utilitarians can remind critics that the principle of utility is intended as only a criterion of right and not as a decision procedure, so utilitarianism does not imply that people ought to calculate utilities before acting (Railton 1984). Consequentialists can also allow the special perspective of a friend or spouse to be reflected in agent-relative value assessments (Sen 1982, Broome 1991, Portmore 2001, 2003) or probability assessments (Jackson 1991). It remains controversial, however, whether any form of consequentialism can adequately incorporate common moral intuitions about friendship. ## 7. Arguments for Consequentialism Even if consequentialists can accommodate or explain away common moral intuitions, that might seem only to answer objections without yet giving any positive reason to accept consequentialism. However, most people begin with the presumption that we morally ought to make the world better when we can. The question then is only whether any moral constraints or moral options need to be added to the basic consequentialist factor in moral reasoning. (Kagan 1989, 1998) If no objection reveals any need for anything beyond consequences, then consequences alone seem to determine what is morally right or wrong, just as consequentialists claim. This line of reasoning will not convince opponents who remain unsatisfied by consequentialist responses to objections. Moreover, even if consequentialists do respond adequately to every proposed objection, that would not show that consequentialism is correct or even defensible. It might face new problems that nobody has yet recognized. Even if every possible objection is refuted, we might have no reason to reject consequentialism but still no reason to accept it. In case a positive reason is needed, consequentialists present a wide variety of arguments. One common move attacks opponents. If the only plausible options in moral theory lie on a certain list (say, Kantianism, contractarianism, virtue theory, pluralistic intuitionism, and consequentialism), then consequentialists can argue for their own theory by criticizing the others. This disjunctive syllogism or process of elimination will be only as strong as the set of objections to the alternatives, and the argument fails if even one competitor survives. Moreover, the argument assumes that the original list is complete. It is hard to see how that assumption could be justified. Consequentialism also might be supported by an inference to the best explanation of our moral intuitions. This argument might surprise those who think of consequentialism as counterintuitive, but in fact consequentialists can explain many moral intuitions that trouble deontological theories. Moderate deontologists, for example, often judge that it is morally wrong to kill one person to save five but not morally wrong to kill one person to save a million. They never specify the line between what is morally wrong and what is not morally wrong, and it is hard to imagine any non-arbitrary way for deontologists to justify a cutoff point. In contrast, consequentialists can simply say that the line belongs wherever the benefits most outweigh the costs, including any bad side effects (cf. Sinnott-Armstrong 2007). Similarly, when two promises conflict, it often seems clear which one we should keep, and that intuition can often be explained by the amount of harm that would be caused by breaking each promise. In contrast, deontologists are hard pressed to explain which promise is overriding if the reason to keep each promise is simply that it was made (Sinnott-Armstrong 2009). If consequentialists can better explain more common moral intuitions, then consequentialism might have more explanatory coherence overall, despite being counterintuitive in some cases. (Compare Sidgwick 1907, Book IV, Chap. III; and Sverdlik 2011.) And even if act consequentialists cannot argue in this way, it still might work for rule consequentialists (such as Hooker 2000). Consequentialists also might be supported by deductive arguments from abstract moral intuitions. Sidgwick (1907, Book III, Chap. XIII) seemed to think that the principle of utility follows from certain very general self-evident principles, including universalizability (if an act ought to be done, then every other act that resembles it in all relevant respects also ought to be done), rationality (one ought to aim at the good generally rather than at any particular part of the good), and equality ("the good of any one individual is of no more importance, from the point of view ... of the Universe, than the good of any other"). Other consequentialists are more skeptical about moral intuitions, so they seek foundations outside morality, either in non-normative facts or in non-moral norms. Mill (1861) is infamous for his "proof" of the principle of utility from empirical observations about what we desire (cf. Sayre-McCord 2001). In contrast, Hare (1963, 1981) tries to derive his version of utilitarianism from substantively neutral accounts of morality, of moral language, and of rationality (cf. Sinnott-Armstrong 2001). Similarly, Gewirth (1978) tries to derive his variant of consequentialism from metaphysical truths about actions. Yet another argument for a kind of consequentialism is contractarian. Harsanyi (1977, 1978) argues that all informed, rational people whose impartiality is ensured because they do not know their place in society would favor a kind of consequentialism. Broome (1991) elaborates and extends Harsanyi's argument. Other forms of arguments have also been invoked on behalf of consequentialism (e.g. Cummiskey 1996, P. Singer 1993; Sinnott-Armstrong 1992). However, each of these arguments has also been subjected to criticisms. Even if none of these arguments proves consequentialism, there still might be no adequate reason to deny consequentialism. We might have no reason either to deny consequentialism or to assert it. Consequentialism could then remain a live option even if it is not proven.
utilitarianism-history	## 1. Precursors to the Classical Approach Though the first systematic account of utilitarianism was developed by Jeremy Bentham (1748-1832), the core insight motivating the theory occurred much earlier. That insight is that morally appropriate behavior will not harm others, but instead increase happiness or 'utility.' What is distinctive about utilitarianism is its approach in taking that insight and developing an account of moral evaluation and moral direction that expands on it. Early precursors to the Classical Utilitarians include the British Moralists, Cumberland, Shaftesbury, Hutcheson, Gay, and Hume. Of these, Francis Hutcheson (1694-1746) is explicitly utilitarian when it comes to action choice. Some of the earliest utilitarian thinkers were the 'theological' utilitarians such as Richard Cumberland (1631-1718) and John Gay (1699-1745). They believed that promoting human happiness was incumbent on us since it was approved by God. After enumerating the ways in which humans come under obligations (by perceiving the "natural consequences of things", the obligation to be virtuous, our civil obligations that arise from laws, and obligations arising from "the authority of God") John Gay writes: "...from the consideration of these four sorts of obligation...it is evident that a full and complete obligation which will extend to all cases, can only be that arising from the authority of God; because God only can in all cases make a man happy or miserable: and therefore, since we are always obliged to that conformity called virtue, it is evident that the immediate rule or criterion of it is the will of God" (R, 412). Gay held that since God wants the happiness of mankind, and since God's will gives us the criterion of virtue, "...the happiness of mankind may be said to be the criterion of virtue, but once removed" (R, 413). This view was combined with a view of human motivation with egoistic elements. A person's individual salvation, her eternal happiness, depended on conformity to God's will, as did virtue itself. Promoting human happiness and one's own coincided, but, given God's design, it was not an accidental coincidence. This approach to utilitarianism, however, is not theoretically clean in the sense that it isn't clear what essential work God does, at least in terms of normative ethics. God as the source of normativity is compatible with utilitarianism, but utilitarianism doesn't require this. Gay's influence on later writers, such as Hume, deserves note. It is in Gay's essay that some of the questions that concerned Hume on the nature of virtue are addressed. For example, Gay was curious about how to explain our practice of approbation and disapprobation of action and character. When we see an act that is vicious we disapprove of it. Further, we associate certain things with their effects, so that we form positive associations and negative associations that also underwrite our moral judgments. Of course, that we view happiness, including the happiness of others as a good, is due to God's design. This is a feature crucial to the theological approach, which would clearly be rejected by Hume in favor of a naturalistic view of human nature and a reliance on our sympathetic engagement with others, an approach anticipated by Shaftesbury (below). The theological approach to utilitarianism would be developed later by William Paley, for example, but the lack of any theoretical necessity in appealing to God would result in its diminishing appeal. Anthony Ashley Cooper, the 3rd Earl of Shaftesbury (1671-1713) is generally thought to have been the one of the earliest 'moral sense' theorists, holding that we possess a kind of "inner eye" that allows us to make moral discriminations. This seems to have been an innate sense of right and wrong, or moral beauty and deformity. Again, aspects of this doctrine would be picked up by Francis Hutcheson and David Hume (1711-1776). Hume, of course, would clearly reject any robust realist implications. If the moral sense is like the other perceptual senses and enables us to pick up on properties out there in the universe around us, properties that exist independent from our perception of them, that are objective, then Hume clearly was not a moral sense theorist in this regard. But perception picks up on features of our environment that one could regard as having a contingent quality. There is one famous passage where Hume likens moral discrimination to the perception of secondary qualities, such as color. In modern terminology, these are response-dependent properties, and lack objectivity in the sense that they do not exist independent of our responses. This is radical. If an act is vicious, its viciousness is a matter of the human response (given a corrected perspective) to the act (or its perceived effects) and thus has a kind of contingency that seems unsettling, certainly unsettling to those who opted for the theological option. So, the view that it is part of our very nature to make moral discriminations is very much in Hume. Further -- and what is relevant to the development of utilitarianism -- the view of Shaftesbury that the virtuous person contributes to the good of the whole -- would figure into Hume's writings, though modified. It is the virtue that contributes to the good of the whole system, in the case of Hume's artificial virtues. Shaftesbury held that in judging someone virtuous or good in a moral sense we need to perceive that person's impact on the systems of which he or she is a part. Here it sometimes becomes difficult to disentangle egoistic versus utilitarian lines of thought in Shaftesbury. He clearly states that whatever guiding force there is has made nature such that it is "...the private interest and good of every one, to work towards the general good, which if a creature ceases to promote, he is actually so far wanting to himself, and ceases to promote his own happiness and welfare..." (R, 188). It is hard, sometimes, to discern the direction of the 'because' -- if one should act to help others because it supports a system in which one's own happiness is more likely, then it looks really like a form of egoism. If one should help others because that's the right thing to do -- and, fortunately, it also ends up promoting one's own interests, then that's more like utilitarianism, since the promotion of self-interest is a welcome effect but not what, all by itself, justifies one's character or actions. Further, to be virtuous a person must have certain psychological capacities -- they must be able to reflect on character, for example, and represent to themselves the qualities in others that are either approved or disapproved of. > > ...in this case alone it is we call any creature worthy or > virtuous when it can have the notion of a public interest, and can > attain the speculation or science of what is morally good or ill, > admirable or blameable, right or wrong....we never say > of....any mere beast, idiot, or changeling, though ever so > good-natured, that he is worthy or virtuous. (Shaftesbury IVM; BKI, > PII, sec. iii) > Thus, animals are not objects of moral appraisal on the view, since they lack the necessary reflective capacities. Animals also lack the capacity for moral discrimination and would therefore seem to lack the moral sense. This raises some interesting questions. It would seem that the moral sense is a perception that something is the case. So it isn't merely a discriminatory sense that allows us to sort perceptions. It also has a propositional aspect, so that animals, which are not lacking in other senses are lacking in this one. The virtuous person is one whose affections, motives, dispositions are of the right sort, not one whose behavior is simply of the right sort and who is able to reflect on goodness, and her own goodness [see Gill]. Similarly, the vicious person is one who exemplifies the wrong sorts of mental states, affections, and so forth. A person who harms others through no fault of his own "...because he has convulsive fits which make him strike and wound such as approach him" is not vicious since he has no desire to harm anyone and his bodily movements in this case are beyond his control. Shaftesbury approached moral evaluation via the virtues and vices. His utilitarian leanings are distinct from his moral sense approach, and his overall sentimentalism. However, this approach highlights the move away from egoistic views of human nature -- a trend picked up by Hutcheson and Hume, and later adopted by Mill in criticism of Bentham's version of utilitarianism. For writers like Shaftesbury and Hutcheson the main contrast was with egoism rather than rationalism. Like Shaftesbury, Francis Hutcheson was very much interested in virtue evaluation. He also adopted the moral sense approach. However, in his writings we also see an emphasis on action choice and the importance of moral deliberation to action choice. Hutcheson, in An Inquiry Concerning Moral Good and Evil, fairly explicitly spelled out a utilitarian principle of action choice. (Joachim Hruschka (1991) notes, however, that it was Leibniz who first spelled out a utilitarian decision procedure.) > > > ....In comparing the moral qualities of actions...we are led > by our moral sense of virtue to judge thus; that in equal > degrees of happiness, expected to proceed from the action, the > virtue is in proportion to the number of persons to whom the > happiness shall extend (and here the dignity, or moral > importance of persons, may compensate numbers); and, in > equal numbers, the virtue is the quantity of the > happiness, or natural good; or that the virtue is in a compound ratio of the > quantity of good, and number of enjoyers....so > that that action is best, which procures > the greatest happiness for the greatest numbers; and > that worst, which, in like manner, occasions > misery. (R, 283-4) > Scarre notes that some hold the moral sense approach incompatible with this emphasis on the use of reason to determine what we ought to do; there is an opposition between just apprehending what's morally significant and a model in which we need to reason to figure out what morality demands of us. But Scarre notes these are not actually incompatible: > > The picture which emerges from Hutcheson's discussion is of a > division of labor, in which the moral sense causes us to look with > favor on actions which benefit others and disfavor those which harm > them, while consequentialist reasoning determines a more precise > ranking order of practical options in given situations. (Scarre, > 53-54) Scarre then uses the example of telling a lie to illustrate: lying is harmful to the person to whom one lies, and so this is viewed with disfavor, in general. However, in a specific case, if a lie is necessary to achieve some notable good, consequentialist reasoning will lead us to favor the lying. But this example seems to put all the emphasis on a consideration of consequences in moral approval and disapproval. Stephen Darwall notes (1995, 216 ff.) that the moral sense is concerned with motives -- we approve, for example, of the motive of benevolence, and the wider the scope the better. It is the motives rather than the consequences that are the objects of approval and disapproval. But inasmuch as the morally good person cares about what happens to others, and of course she will, she will rank order acts in terms of their effects on others, and reason is used in calculating effects. So there is no incompatibility at all. Hutcheson was committed to maximization, it seems. However, he insisted on a caveat -- that "the dignity or moral importance of persons may compensate numbers." He added a deontological constraint -- that we have a duty to others in virtue of their personhood to accord them fundamental dignity regardless of the numbers of others whose happiness is to be affected by the action in question. Hume was heavily influenced by Hutcheson, who was one of his teachers. His system also incorporates insights made by Shaftesbury, though he certainly lacks Shaftesbury's confidence that virtue is its own reward. In terms of his place in the history of utilitarianism we should note two distinct effects his system had. Firstly, his account of the social utility of the artificial virtues influenced Bentham's thought on utility. Secondly, his account of the role sentiment played in moral judgment and commitment to moral norms influenced Mill's thoughts about the internal sanctions of morality. Mill would diverge from Bentham in developing the 'altruistic' approach to Utilitarianism (which is actually a misnomer, but more on that later). Bentham, in contrast to Mill, represented the egoistic branch -- his theory of human nature reflected Hobbesian psychological egoism. ## 2. The Classical Approach The Classical Utilitarians, Bentham and Mill, were concerned with legal and social reform. If anything could be identified as the fundamental motivation behind the development of Classical Utilitarianism it would be the desire to see useless, corrupt laws and social practices changed. Accomplishing this goal required a normative ethical theory employed as a critical tool. What is the truth about what makes an action or a policy a morally good one, or morally right? But developing the theory itself was also influenced by strong views about what was wrong in their society. The conviction that, for example, some laws are bad resulted in analysis of why they were bad. And, for Jeremy Bentham, what made them bad was their lack of utility, their tendency to lead to unhappiness and misery without any compensating happiness. If a law or an action doesn't do any good, then it isn't any good. ### 2.1 Jeremy Bentham Jeremy Bentham (1748-1832) was influenced both by Hobbes' account of human nature and Hume's account of social utility. He famously held that humans were ruled by two sovereign masters -- pleasure and pain. We seek pleasure and the avoidance of pain, they "...govern us in all we do, in all we say, in all we think..." (Bentham PML, 1). Yet he also promulgated the principle of utility as the standard of right action on the part of governments and individuals. Actions are approved when they are such as to promote happiness, or pleasure, and disapproved of when they have a tendency to cause unhappiness, or pain (PML). Combine this criterion of rightness with a view that we should be actively trying to promote overall happiness, and one has a serious incompatibility with psychological egoism. Thus, his apparent endorsement of Hobbesian psychological egoism created problems in understanding his moral theory since psychological egoism rules out acting to promote the overall well-being when that it is incompatible with one's own. For the psychological egoist, that is not even a possibility. So, given 'ought implies can' it would follow that we are not obligated to act to promote overall well-being when that is incompatible with our own. This generates a serious tension in Bentham's thought, one that was drawn to his attention. He sometimes seemed to think that he could reconcile the two commitments empirically, that is, by noting that when people act to promote the good they are helping themselves, too. But this claim only serves to muddy the waters, since the standard understanding of psychological egoism -- and Bentham's own statement of his view -- identifies motives of action which are self-interested. Yet this seems, again, in conflict with his own specification of the method for making moral decisions which is not to focus on self-interest -- indeed, the addition of extent as a parameter along which to measure pleasure produced distinguishes this approach from ethical egoism. Aware of the difficulty, in later years he seemed to pull back from a full-fledged commitment to psychological egoism, admitting that people do sometimes act benevolently -- with the overall good of humanity in mind. Bentham also benefited from Hume's work, though in many ways their approaches to moral philosophy were completely different. Hume rejected the egoistic view of human nature. Hume also focused on character evaluation in his system. Actions are significant as evidence of character, but only have this derivative significance. In moral evaluation the main concern is that of character. Yet Bentham focused on act-evaluation. There was a tendency -- remarked on by J. B. Schneewind (1990), for example -- to move away from focus on character evaluation after Hume and towards act-evaluation. Recall that Bentham was enormously interested in social reform. Indeed, reflection on what was morally problematic about laws and policies influenced his thinking on utility as a standard. When one legislates, however, one is legislating in support of, or against, certain actions. Character -- that is, a person's true character -- is known, if known at all, only by that person. If one finds the opacity of the will thesis plausible then character, while theoretically very interesting, isn't a practical focus for legislation. Further, as Schneewind notes, there was an increasing sense that focus on character would actually be disruptive, socially, particularly if one's view was that a person who didn't agree with one on a moral issues was defective in terms of his or her character, as opposed to simply making a mistake reflected in action. But Bentham does take from Hume the view that utility is the measure of virtue -- that is, utility more broadly construed than Hume's actual usage of the term. This is because Hume made a distinction between pleasure that the perception of virtue generates in the observer, and social utility, which consisted in a trait's having tangible benefits for society, any instance of which may or may not generate pleasure in the observer. But Bentham is not simply reformulating a Humean position -- he's merely been influenced by Hume's arguments to see pleasure as a measure or standard of moral value. So, why not move from pleasurable responses to traits to pleasure as a kind of consequence which is good, and in relation to which, actions are morally right or wrong? Bentham, in making this move, avoids a problem for Hume. On Hume's view it seems that the response -- corrected, to be sure -- determines the trait's quality as a virtue or vice. But on Bentham's view the action (or trait) is morally good, right, virtuous in view of the consequences it generates, the pleasure or utility it produces, which could be completely independent of what our responses are to the trait. So, unless Hume endorses a kind of ideal observer test for virtue, it will be harder for him to account for how it is people make mistakes in evaluations of virtue and vice. Bentham, on the other hand, can say that people may not respond to the actions good qualities -- perhaps they don't perceive the good effects. But as long as there are these good effects which are, on balance, better than the effects of any alternative course of action, then the action is the right one. Rhetorically, anyway, one can see why this is an important move for Bentham to be able to make. He was a social reformer. He felt that people often had responses to certain actions -- of pleasure or disgust -- that did not reflect anything morally significant at all. Indeed, in his discussions of homosexuality, for example, he explicitly notes that 'antipathy' is not sufficient reason to legislate against a practice: > > > The circumstances from which this antipathy may have taken its rise > may be worth enquiring to.... One is the physical antipathy to > the offence.... The act is to the highest degree odious and > disgusting, that is, not to the man who does it, for he does it only > because it gives him pleasure, but to one who thinks [?] of it. Be it > so, but what is that to him? (Bentham OAO, v. 4, 94) Bentham then notes that people are prone to use their physical antipathy as a pretext to transition to moral antipathy, and the attending desire to punish the persons who offend their taste. This is illegitimate on his view for a variety of reasons, one of which is that to punish a person for violations of taste, or on the basis of prejudice, would result in runaway punishments, "...one should never know where to stop..." The prejudice in question can be dealt with by showing it "to be ill-grounded". This reduces the antipathy to the act in question. This demonstrates an optimism in Bentham. If a pain can be demonstrated to be based on false beliefs then he believes that it can be altered or at the very least 'assuaged and reduced'. This is distinct from the view that a pain or pleasure based on a false belief should be discounted. Bentham does not believe the latter. Thus Bentham's hedonism is a very straightforward hedonism. The one intrinsic good is pleasure, the bad is pain. We are to promote pleasure and act to reduce pain. When called upon to make a moral decision one measures an action's value with respect to pleasure and pain according to the following: intensity (how strong the pleasure or pain is), duration (how long it lasts), certainty (how likely the pleasure or pain is to be the result of the action), proximity (how close the sensation will be to performance of the action), fecundity (how likely it is to lead to further pleasures or pains), purity (how much intermixture there is with the other sensation). One also considers extent -- the number of people affected by the action. Keeping track of all of these parameters can be complicated and time consuming. Bentham does not recommend that they figure into every act of moral deliberation because of the efficiency costs which need to be considered. Experience can guide us. We know that the pleasure of kicking someone is generally outweighed by the pain inflicted on that person, so such calculations when confronted with a temptation to kick someone are unnecessary. It is reasonable to judge it wrong on the basis of past experience or consensus. One can use 'rules of thumb' to guide action, but these rules are overridable when abiding by them would conflict with the promotion of the good. Bentham's view was surprising to many at the time at least in part because he viewed the moral quality of an action to be determined instrumentally. It isn't so much that there is a particular kind of action that is intrinsically wrong; actions that are wrong are wrong simply in virtue of their effects, thus, instrumentally wrong. This cut against the view that there are some actions that by their very nature are just wrong, regardless of their effects. Some may be wrong because they are 'unnatural' -- and, again, Bentham would dismiss this as a legitimate criterion. Some may be wrong because they violate liberty, or autonomy. Again, Bentham would view liberty and autonomy as good -- but good instrumentally, not intrinsically. Thus, any action deemed wrong due to a violation of autonomy is derivatively wrong on instrumental grounds as well. This is interesting in moral philosophy -- as it is far removed from the Kantian approach to moral evaluation as well as from natural law approaches. It is also interesting in terms of political philosophy and social policy. On Bentham's view the law is not monolithic and immutable. Since effects of a given policy may change, the moral quality of the policy may change as well. Nancy Rosenblum noted that for Bentham one doesn't simply decide on good laws and leave it at that: "Lawmaking must be recognized as a continual process in response to diverse and changing desires that require adjustment" (Rosenblum 1978, 9). A law that is good at one point in time may be a bad law at some other point in time. Thus, lawmakers have to be sensitive to changing social circumstances. To be fair to Bentham's critics, of course, they are free to agree with him that this is the case in many situations, just not all -- and that there is still a subset of laws that reflect the fact that some actions just are intrinsically wrong regardless of consequences. Bentham is in the much more difficult position of arguing that effects are all there are to moral evaluation of action and policy. ### 2.2 John Stuart Mill John Stuart Mill (1806-1873) was a follower of Bentham, and, through most of his life, greatly admired Bentham's work even though he disagreed with some of Bentham's claims -- particularly on the nature of 'happiness.' Bentham, recall, had held that there were no qualitative differences between pleasures, only quantitative ones. This left him open to a variety of criticisms. First, Bentham's Hedonism was too egalitarian. Simple-minded pleasures, sensual pleasures, were just as good, at least intrinsically, than more sophisticated and complex pleasures. The pleasure of drinking a beer in front of the T.V. surely doesn't rate as highly as the pleasure one gets solving a complicated math problem, or reading a poem, or listening to Mozart. Second, Bentham's view that there were no qualitative differences in pleasures also left him open to the complaint that on his view human pleasures were of no more value than animal pleasures and, third, committed him to the corollary that the moral status of animals, tied to their sentience, was the same as that of humans. While harming a puppy and harming a person are both bad, however, most people had the view that harming the person was worse. Mill sought changes to the theory that could accommodate those sorts of intuitions. To this end, Mill's hedonism was influenced by perfectionist intuitions. There are some pleasures that are more fitting than others. Intellectual pleasures are of a higher, better, sort than the ones that are merely sensual, and that we share with animals. To some this seems to mean that Mill really wasn't a hedonistic utilitarian. His view of the good did radically depart from Bentham's view. However, like Bentham, the good still consists in pleasure, it is still a psychological state. There is certainly that similarity. Further, the basic structures of the theories are the same (for more on this see Donner 1991). While it is true that Mill is more comfortable with notions like 'rights' this does not mean that he, in actuality, rejected utilitarianism. The rationale for all the rights he recognizes is utilitarian. Mill's 'proof' of the claim that intellectual pleasures are better in kind than others, though, is highly suspect. He doesn't attempt a mere appeal to raw intuition. Instead, he argues that those persons who have experienced both view the higher as better than the lower. Who would rather be a happy oyster, living an enormously long life, than a person living a normal life? Or, to use his most famous example -- it is better to be Socrates 'dissatisfied' than a fool 'satisfied.' In this way Mill was able to solve a problem for utilitarianism. Mill also argued that the principle could be proven, using another rather notorious argument: > > The only proof capable of being given that an object is visible is > that people actually see it.... In like manner, I apprehend, the > sole evidence it is possible to produce that anything is desirable is > that people do actually desire it. If the end which the utilitarian > doctrine proposes to itself were not, in theory and in practiced, > acknowledged to be an end, nothing could ever convince any person that > it was so. (Mill, U, 81) Mill then continues to argue that people desire happiness -- the utilitarian end -- and that the general happiness is "a good to the aggregate of all persons." (81) G. E. Moore (1873-1958) criticized this as fallacious. He argued that it rested on an obvious ambiguity: > > > Mill has made as naive and artless a use of the naturalistic > fallacy as anybody could desire. "Good", he tells us, > means "desirable", and you can only find out what is > desirable by seeking to find out what is actually desired.... > The fact is that "desirable" does not mean "able to > be desired" as "visible" means "able to be > seen." The desirable means simply what ought to be > desired or deserves to be desired; just as the detestable means not > what can be but what ought to be detested... (Moore, PE, 66-7) > It should be noted, however, that Mill was offering this as an alternative to Bentham's view which had been itself criticized as a 'swine morality,' locating the good in pleasure in a kind of indiscriminate way. The distinctions he makes strike many as intuitively plausible ones. Bentham, however, can accommodate many of the same intuitions within his system. This is because he notes that there are a variety of parameters along which we quantitatively measure pleasure -- intensity and duration are just two of those. His complete list is the following: intensity, duration, certainty or uncertainty, propinquity or remoteness, fecundity, purity, and extent. Thus, what Mill calls the intellectual pleasures will score more highly than the sensual ones along several parameters, and this could give us reason to prefer those pleasures -- but it is a quantitative not a qualitative reason, on Bentham's view. When a student decides to study for an exam rather than go to a party, for example, she is making the best decision even though she is sacrificing short term pleasure. That's because studying for the exam, Bentham could argue, scores higher in terms of the long term pleasures doing well in school lead to, as well as the fecundity of the pleasure in leading to yet other pleasures. However, Bentham will have to concede that the very happy oyster that lives a very long time could, in principle, have a better life than a normal human. Mill's version of utilitarianism differed from Bentham's also in that he placed weight on the effectiveness of internal sanctions -- emotions like guilt and remorse which serve to regulate our actions. This is an off-shoot of the different view of human nature adopted by Mill. We are the sorts of beings that have social feelings, feelings for others, not just ourselves. We care about them, and when we perceive harms to them this causes painful experiences in us. When one perceives oneself to be the agent of that harm, the negative emotions are centered on the self. One feels guilt for what one has done, not for what one sees another doing. Like external forms of punishment, internal sanctions are instrumentally very important to appropriate action. Mill also held that natural features of human psychology, such as conscience and a sense of justice, underwrite motivation. The sense of justice, for example, results from very natural impulses. Part of this sense involves a desire to punish those who have harmed others, and this desire in turn "...is a spontaneous outgrowth from two sentiments, both in the highest degree natural...; the impulse of self-defense, and the feeling of sympathy." (Chapter 5, Utilitarianism) Of course, he goes on, the justification must be a separate issue. The feeling is there naturally, but it is our 'enlarged' sense, our capacity to include the welfare of others into our considerations, and make intelligent decisions, that gives it the right normative force. Like Bentham, Mill sought to use utilitarianism to inform law and social policy. The aim of increasing happiness underlies his arguments for women's suffrage and free speech. We can be said to have certain rights, then -- but those rights are underwritten by utility. If one can show that a purported right or duty is harmful, then one has shown that it is not genuine. One of Mills most famous arguments to this effect can be found in his writing on women's suffrage when he discusses the ideal marriage of partners, noting that the ideal exists between individuals of "cultivated faculties" who influence each other equally. Improving the social status of women was important because they were capable of these cultivated faculties, and denying them access to education and other opportunities for development is forgoing a significant source of happiness. Further, the men who would deny women the opportunity for education, self-improvement, and political expression do so out of base motives, and the resulting pleasures are not ones that are of the best sort. Bentham and Mill both attacked social traditions that were justified by appeals to natural order. The correct appeal is to utility itself. Traditions often turned out to be "relics" of "barbarous" times, and appeals to nature as a form of justification were just ways to try rationalize continued deference to those relics. In the latter part of the 20th century some writers criticized utilitarianism for its failure to accommodate virtue evaluation. However, though virtue is not the central normative concept in Mill's theory, it is an extremely important one. In Chapter 4 of Utilitarianism Mill noted > ... does the utilitarian doctrine deny that people > desire virtue, or maintain that virtue is not a thing to be desired? > The very reverse. It maintains not only that virtue is to be desired, > but also that it is to be desired disinterestedly, for > itself. Whatever may be the opinion of utilitarian moralists as to the > original conditions by which virtue is made virtue ... they not only > place virtue at the very head of things which are good as a means to > the ultimate end, but they also recognize as a psychological fact the > possibility of its being, to the individual, a good in itself, without > looking to any end beyond it; and hold, that the mind is not in a > right state, not in a state conformable to Utility, not in the state > most conducive to the general happiness, unless it does love virtue in > this manner ... In Utilitarianism Mill argues that virtue not only has instrumental value, but is constitutive of the good life. A person without virtue is morally lacking, is not as able to promote the good. However, this view of virtue is someone complicated by rather cryptic remarks Mill makes about virtue in his A System of Logic in the section in which he discusses the "Art of Life." There he seems to associate virtue with aesthetics, and morality is reserved for the sphere of 'right' or 'duty'. Wendy Donner notes that separating virtue from right allows Mill to solve another problem for the theory: the demandingness problem (Donner 2011). This is the problem that holds that if we ought to maximize utility, if that is the right thing to do, then doing right requires enormous sacrifices (under actual conditions), and that requiring such sacrifices is too demanding. With duties, on Mill's view, it is important that we get compliance, and that justifies coercion. In the case of virtue, however, virtuous actions are those which it is "...for the general interest that they remain free." ## 3. Henry Sidgwick Henry Sidgwick's (1838-1900) The Methods of Ethics (1874) is one of the most well known works in utilitarian moral philosophy, and deservedly so. It offers a defense of utilitarianism, though some writers (Schneewind 1977) have argued that it should not primarily be read as a defense of utilitarianism. In The Methods Sidgwick is concerned with developing an account of "...the different methods of Ethics that I find implicit in our common moral reasoning..." These methods are egoism, intuition based morality, and utilitarianism. On Sidgwick's view, utilitarianism is the more basic theory. A simple reliance on intuition, for example, cannot resolve fundamental conflicts between values, or rules, such as Truth and Justice that may conflict. In Sidgwick's words "...we require some higher principle to decide the issue..." That will be utilitarianism. Further, the rules which seem to be a fundamental part of common sense morality are often vague and underdescribed, and applying them will actually require appeal to something theoretically more basic -- again, utilitarianism. Yet further, absolute interpretations of rules seem highly counter-intuitive, and yet we need some justification for any exceptions -- provided, again, by utilitarianism. Sidgwick provides a compelling case for the theoretical primacy of utilitarianism. Sidgwick was also a British philosopher, and his views developed out of and in response to those of Bentham and Mill. His Methods offer an engagement with the theory as it had been presented before him, and was an exploration of it and the main alternatives as well as a defense. Sidgwick was also concerned with clarifying fundamental features of the theory, and in this respect his account has been enormously influential to later writers, not only to utilitarians and consequentialists, generally, but to intuitionists as well. Sidgwick's thorough and penetrating discussion of the theory raised many of the concerns that have been developed by recent moral philosophers. One extremely controversial feature of Sidgwick's views relates to his rejection of a publicity requirement for moral theory. He writes: > > > Thus, the Utilitarian conclusion, carefully stated, would seem to be > this; that the opinion that secrecy may render an action right which > would not otherwise be so should itself be kept comparatively secret; > and similarly it seems expedient that the doctrine that esoteric > morality is expedient should itself be kept esoteric. Or, if this > concealment be difficult to maintain, it may be desirable that Common > Sense should repudiate the doctrines which it is expedient to confine > to an enlightened few. And thus a Utilitarian may reasonably desire, > on Utilitarian principles, that some of his conclusions should be > rejected by mankind generally; or even that the vulgar should keep > aloof from his system as a whole, in so far as the inevitable > indefiniteness and complexity of its calculations render it likely to > lead to bad results in their hands. (490) This accepts that utilitarianism may be self-effacing; that is, that it may be best if people do not believe it, even though it is true. Further, it rendered the theory subject to Bernard Williams' (1995) criticism that the theory really simply reflected the colonial elitism of Sidgwick's time, that it was 'Government House Utilitarianism.' The elitism in his remarks may reflect a broader attitude, one in which the educated are considered better policy makers than the uneducated. One issue raised in the above remarks is relevant to practical deliberation in general. To what extent should proponents of a given theory, or a given rule, or a given policy -- or even proponents of a given one-off action -- consider what they think people will actually do, as opposed to what they think those same people ought to do (under full and reasonable reflection, for example)? This is an example of something that comes up in the Actualism/possibilism debate in accounts of practical deliberation. Extrapolating from the example used above, we have people who advocate telling the truth, or what they believe to be the truth, even if the effects are bad because the truth is somehow misused by others. On the other hand are those who recommend not telling the truth when it is predicted that the truth will be misused by others to achieve bad results. Of course it is the case that the truth ought not be misused, that its misuse can be avoided and is not inevitable, but the misuse is entirely predictable. Sidgwick seems to recommending that we follow the course that we predict will have the best outcome, given as part of our calculations the data that others may fail in some way -- either due to having bad desires, or simply not being able to reason effectively. The worry Williams points to really isn't a worry specifically with utilitarianism (Driver 2011). Sidgwick would point out that if it is bad to hide the truth, because 'Government House' types, for example, typically engage in self-deceptive rationalizations of their policies (which seems entirely plausible), then one shouldn't do it. And of course, that heavily influences our intuitions. Sidgwick raised issues that run much deeper to our basic understanding of utilitarianism. For example, the way earlier utilitarians characterized the principle of utility left open serious indeterminacies. The major one rests on the distinction between total and average utility. He raised the issue in the context of population growth and increasing utility levels by increasing numbers of people (or sentient beings): > > > Assuming, then, that the average happiness of human beings is a > positive quantity, it seems clear that, supposing the average > happiness enjoyed remains undiminished, Utilitarianism directs us to > make the number enjoying it as great as possible. But if we foresee as > possible that an increase in numbers will be accompanied by a decrease > in average happiness or vice versa, a point arises which has > not only never been formally noticed, but which seems to have been > substantially overlooked by many Utilitarians. For if we take > Utilitarianism to prescribe, as the ultimate end of action, happiness > on the whole, and not any individual's happiness, unless > considered as an element of the whole, it would follow that, if the > additional population enjoy on the whole positive happiness, we ought > to weigh the amount of happiness gained by the extra number against > the amount lost by the remainder. (415) For Sidgwick, the conclusion on this issue is not to simply strive to greater average utility, but to increase population to the point where we maximize the product of the number of persons who are currently alive and the amount of average happiness. So it seems to be a hybrid, total-average view. This discussion also raised the issue of policy with respect to population growth, and both would be pursued in more detail by later writers, most notably Derek Parfit (1986). ## 4. Ideal Utilitarianism G. E. Moore strongly disagreed with the hedonistic value theory adopted by the Classical Utilitarians. Moore agreed that we ought to promote the good, but believed that the good included far more than what could be reduced to pleasure. He was a pluralist, rather than a monist, regarding intrinsic value. For example, he believed that 'beauty' was an intrinsic good. A beautiful object had value independent of any pleasure it might generate in a viewer. Thus, Moore differed from Sidgwick who regarded the good as consisting in some consciousness. Some objective states in the world are intrinsically good, and on Moore's view, beauty is just such a state. He used one of his more notorious thought experiments to make this point: he asked the reader to compare two worlds, one was entirely beautiful, full of things which complemented each other; the other was a hideous, ugly world, filled with "everything that is most disgusting to us." Further, there are not human beings, one imagines, around to appreciate or be disgusted by the worlds. The question then is, which of these worlds is better, which one's existence would be better than the other's? Of course, Moore believed it was clear that the beautiful world was better, even though no one was around to appreciate its beauty. This emphasis on beauty was one facet of Moore's work that made him a darling of the Bloomsbury Group. If beauty was a part of the good independent of its effects on the psychological states of others -- independent of, really, how it affected others, then one needn't sacrifice morality on the altar of beauty anymore. Following beauty is not a mere indulgence, but may even be a moral obligation. Though Moore himself certainly never applied his view to such cases, it does provide the resources for dealing with what the contemporary literature has dubbed 'admirable immorality' cases, at least some of them. Gauguin may have abandoned his wife and children, but it was to a beautiful end. Moore's targets in arguing against hedonism were the earlier utilitarians who argued that the good was some state of consciousness such as pleasure. He actually waffled on this issue a bit, but always disagreed with Hedonism in that even when he held that beauty all by itself was not an intrinsic good, he also held that for the appreciation of beauty to be a good the beauty must actually be there, in the world, and not be the result of illusion. Moore further criticized the view that pleasure itself was an intrinsic good, since it failed a kind of isolation test that he proposed for intrinsic value. If one compared an empty universe with a universe of sadists, the empty universe would strike one as better. This is true even though there is a good deal of pleasure, and no pain, in the universe of sadists. This would seem to indicate that what is necessary for the good is at least the absence of bad intentionality. The pleasures of sadists, in virtue of their desires to harm others, get discounted -- they are not good, even though they are pleasures. Note this radical departure from Bentham who held that even malicious pleasure was intrinsically good, and that if nothing instrumentally bad attached to the pleasure, it was wholly good as well. One of Moore's important contributions was to put forward an 'organic unity' or 'organic whole' view of value. The principle of organic unity is vague, and there is some disagreement about what Moore actually meant in presenting it. Moore states that 'organic' is used "...to denote the fact that a whole has an intrinsic value different in amount from the sum of the values of its parts." (PE, 36) And, for Moore, that is all it is supposed to denote. So, for example, one cannot determine the value of a body by adding up the value of its parts. Some parts of the body may have value only in relation to the whole. An arm or a leg, for example, may have no value at all separated from the body, but have a great deal of value attached to the body, and increase the value of the body, even. In the section of Principia Ethica on the Ideal, the principle of organic unity comes into play in noting that when persons experience pleasure through perception of something beautiful (which involves a positive emotion in the face of a recognition of an appropriate object -- an emotive and cognitive set of elements), the experience of the beauty is better when the object of the experience, the beautiful object, actually exists. The idea was that experiencing beauty has a small positive value, and existence of beauty has a small positive value, but combining them has a great deal of value, more than the simple addition of the two small values (PE, 189 ff.). Moore noted: "A true belief in the reality of an object greatly increases the value of many valuable wholes..." (199). This principle in Moore -- particularly as applied to the significance of actual existence and value, or knowledge and value, provided utilitarians with tools to meet some significant challenges. For example, deluded happiness would be severely lacking on Moore's view, especially in comparison to happiness based on knowledge. ## 5. Conclusion Since the early 20th Century utilitarianism has undergone a variety of refinements. After the middle of the 20th Century it has become more common to identify as a 'Consequentialist' since very few philosophers agree entirely with the view proposed by the Classical Utilitarians, particularly with respect to the hedonistic value theory. But the influence of the Classical Utilitarians has been profound -- not only within moral philosophy, but within political philosophy and social policy. The question Bentham asked, "What use is it?," is a cornerstone of policy formation. It is a completely secular, forward-looking question. The articulation and systematic development of this approach to policy formation is owed to the Classical Utilitarians.
consequentialism-rule	## 1. Utilitarianism A moral theory is a form of consequentialism if and only if it assesses acts and/or character traits, practices, and institutions solely in terms of the goodness of the consequences. Historically, utilitarianism has been the best-known form of consequentialism. Utilitarianism assesses acts and/or character traits, practices, and institutions solely in terms of overall net benefit. Overall net benefit is often referred to as aggregate well-being or welfare. Aggregate welfare is calculated by counting a benefit or harm to any one individual the same as the same size benefit or harm to any other individual, and then adding all the benefits and harms together to reach an aggregate sum. There is considerable dispute among consequentialists about what the best account of welfare is. ## 2. Welfare Classical utitilitarians (i.e., Jeremy Bentham, J.S. Mill, and Henry Sidgwick) took benefit and harm to be purely a matter of pleasure and pain. The view that welfare is a matter of pleasure minus pain has generally been called hedonism. It has grown in sophistication (Parfit 1984: Appendix I; Sumner 1996; Crisp 2006; de Lazari-Radek and Singer 2014: ch. 9) but remains committed to the thesis that how well someone's life goes depends entirely on his or her pleasure minus pain, albeit with pleasure and pain being construed very broadly. Even if pleasures and pains are construed very broadly, hedonism encounters difficulties. The main one is that many (if not all) people care very strongly about things other than their own pleasures and pains. Of course these other things can be important as means to pleasures and to the avoidance of pain. But many people care very strongly about things over and beyond their hedonistic instrumental value. For example, many people want to know the truth about various matters even if this won't increase their (or anyone else's) pleasure. Another example is that many people care about achieving things over and beyond the pleasure such achievements might produce. Again, many people care about the welfare of their family and friends in a non-instrumental way. A rival account of these points, especially the last, is that people care about many things other than their own welfare. On any plausible view of welfare, the satisfaction people can feel when their desires are fulfilled constitutes an addition to their welfare. Likewise, on any plausible view, frustration felt as a result of unfulfilled desires constitutes a reduction in welfare. What is controversial is whether the fulfilment of someone's desire constitutes a benefit to that person apart from any effect that the fulfilment of the desire has on that person's felt satisfaction or frustration. Hedonism answers No, claiming that only effects on felt satisfaction or felt frustration matter. A different theory of welfare answers Yes. This theory holds that the fulfilment of any desire of the agent's constitutes a benefit to the agent, even if the agent never knows that desire has been fulfilled and even if the agent derives no pleasure from its fulfilment. This theory of human welfare is often referred to as the desire-fulfillment theory of welfare. Clearly, the desire-fulfillment theory of welfare is broader than hedonism, in that the desire-fulfillment theory accepts that what can constitute a benefit is wider than merely pleasure. But there are reasons for thinking that this broader theory is too broad. For one thing, people can have sensible desires that are simply too disconnected from their own lives to be relevant to their own welfare (Williams 1973a: 262; Overvold 1980, 1982; Parfit 1984: 494). I desire that the starving in far-away countries get food. But the fulfilment of this desire of mine does not benefit me. For another thing, people can have desires for absurd things for themselves. Suppose I desire to count all the blades of grass in the lawns on this road. If I get satisfaction out of doing this, the felt satisfaction constitutes a benefit to me. But the bare fulfilment of my desire to count all the blades of grass in the lawns on this road does not (Rawls 1971: 432; Parfit 1984: 500; Crisp 1997: 56). On careful reflection, we might think that the fulfilment of someone's desire constitutes an addition to that person's welfare if and only if that desire has one of a certain set of contents. We might think, for example, that the fulfilment of someone's desire for pleasure, friendship, knowledge, achievement, or autonomy for herself does constitute an addition to her welfare, and that the fulfilment of any desires she might have for things that do not fall into these categories do not directly benefit her (though, again, the pleasure she derives from their satisfaction does). If we think this, it seems we think there is a list of things that constitute anyone's welfare (Parfit 1984: Appendix I; Brink 1989: 221-36; Griffin 1996: ch. 2; Crisp 1997: ch. 3; Gert 1998: 92-4; Arneson 1999a). Insofar as the goods to be promoted are parts of welfare, the theory remains utilitarian. There is a lot to be said for utilitarianism. Obviously, how lives go is important. And there is something deeply attractive (if not downright irresistible) in the idea that morality is fundamentally impartial, i.e., the idea that, at the most fundamental level of morality, everyone is equally important -- women and men, strong and weak, rich and poor, Blacks, Whites, Hispanics, Asians, etc. And utilitarianism plausibly interprets this equal importance as dictating that in the calculation of overall welfare a benefit or harm to any one individual counts neither more nor less that the same size benefit or harm to any other individual. ## 3. Other Goods To Be Promoted The nonutilitarian members of the consequentialist family are theories that assess acts and/or character traits, practices, and institutions solely in terms of resulting good, where good is not restricted to welfare. "Nonutilitarian" here means "not purely utilitarian", rather than "completely unutilitarian". When writers describe themselves as consequentialists rather than as utilitarians, they are normally signalling that their fundamental evaluations will be in terms of not only welfare but also some other goods. What are these other goods? The most common answers have been justice, fairness, and equality. Justice, according to Plato, is "rendering to each his due" (Republic, Bk. 1). We might suppose that what people are due is a matter of what people are owed, either because they deserve it or because they have a moral right to it. Suppose we plug these ideas into consequentialism. Then we get the theory that things should be assessed in terms of not only how much welfare results but also the extent to which people get what they deserve and the extent to which moral rights are respected. For consequentialism to take this line, however, is for it to restrict its explanatory ambitions. What a theory simply presupposes, it does not explain. A consequentialist theory that presupposes both that justice is constituted by such-and-such and that justice is one of the things to be promoted does not explain why the components of justice are important. It does not explain what desert is. It does not explain the importance of moral rights, much less try to determine what the contents of these moral rights are. These are matters too important and contentious for a consequentialist theory to leave unexplained or open. If consequentialism is going to refer to justice, desert, and moral rights, it needs to analyze these concepts and justify the role it gives them. Similar things can be said about fairness. If a consequentialist theory presupposes an account of fairness, and simply stipulates that fairness is to be promoted, then this consequentialist theory is not explaining fairness. But fairness (like justice, desert, and moral rights) is a concept too important for consequentialism not to try to explain. One way for consequentialists to deal with justice and fairness is to contend that justice and fairness are constituted by conformity with a certain set of justified social practices, and that what justifies these practices is that they generally promote overall welfare and equality. Indeed, the contention might be that what people are due, what people have a moral right to, what justice and fairness require, is conformity to whatever practices promote overall welfare and equality. Whether equality needs to be included in the formula, however, is very controversial. Many think that a purely utilitarian formula has sufficiently egalitarian implications. They think that, even if the goal is promotion of welfare, not the promotion of welfare-plus-equality, there are some contingent but pervasive facts about human beings that push in the direction of equal distribution of material resources. According to the "law of diminishing marginal utility of material resources", the amount of benefit a person gets out of a certain unit of material resources is less the more units of that material good the person already has. Suppose I go from having no way of getting around except by foot to having a bicycle, or, though I live in a place where one can get very cold, I go from having no warm coat to having one. I will benefit more from getting that first bicycle or coat than I would if I go from having nine bicycles or coats to having ten. There are exceptions to the law of diminishing marginal utility. In most of these exceptions, an additional unit of material resource pushes someone over some important threshold. For example, consider the meal or pill or gulp of air that saves someone's life, or the car whose acquisition pushes the competitive collector into first place. In such cases, the unit that puts the person over the threshold might well be as beneficial to that person as any prior unit was. Still, as a general rule, material resources do have diminishing marginal utility. To the assumption that material resources have diminishing marginal utility, let us add the assumption that different people generally get roughly the same benefits from the same material resources. Again, there are exceptions. If you live in a freezing climate and I live in a hot climate, then you would benefit much more from a warm coat than I would. But suppose we live in the same place, which has freezing winters, good paths for riding bicycles, and no public transportation. And suppose you have ten bicycles and ten coats (though you are not vying for some bicycle- or coat-collector prize). Meanwhile, I am so poor that I have none. Then, redistributing one of your bicycles and one of your coats to me will probably harm you less than it will benefit me. This sort of phenomenon pervades societies where resources are unequally distributed. Wherever the phenomenon occurs, a fundamentally impartial morality is under pressure to redistribute resources from the richer to the poorer. However, there are also contingent but pervasive facts about human beings that pull in favor of practices that have the foreseen consequence of material inequality. First of all, higher levels of overall welfare can require higher levels of productivity (think of the welfare gains resulting from improvements in agricultural productivity). In many areas of the economy, the provision of material rewards for greater productivity seems the most efficient acceptable way of eliciting higher productivity. Some individuals and groups will be more productive than others (especially if there are incentive schemes). So the practice of providing material rewards for greater productivity will result in material inequality. Thus, on the one hand, the diminishing marginal utility of material resources exerts pressure in favor of more equal distributions of resources. On the other hand, the need to promote productivity exerts pressure in favor of incentive schemes that have the foreseen consequence of material inequality. Utilitarians and most other consequentialists find themselves balancing these opposed pressures. Note that those pressures concern the distribution of resources. There is a further question about how equally welfare itself should be distributed. Many recent writers have taken utilitarianism to be indifferent about the distribution of welfare. Imagine a choice between an outcome where overall welfare is large but distributed unequally and an outcome where overall welfare is smaller but distributed equally. Utilitarians are taken to favor outcomes with greater overall welfare even if it is also less equally distributed. To illustrate this, let us take an artificially simple population, divided into just two groups. > > > > \| \| \| \| > \| --- \| --- \| --- \| > \| \| Units of welfare \| Total welfare for both groups \| > \| Alternative 1 \| Per person \| Per group \| \| > \| 10,000 people in group A \| 1 \| 10,000 \| \| > \| 100,000 people in group B \| 10 \| 1,000,000 \| \| > \| \| \| \| Impartially calculated: 1,010,000 \| > > > > > > > \| \| \| \| > \| --- \| --- \| --- \| > \| \| Units of welfare \| Total welfare for both groups \| > \| Alternative 2 \| Per person \| Per group \| \| > \| 10,000 people in group A \| 8 \| 80,000 \| \| > \| 100,000 people in group B \| 9 \| 900,000 \| \| > \| \| \| \| Impartially calculated: 980,000 \| > > > Many people would think Alternative 2 above better than Alternative 1, and might think that the comparison between these alternatives shows that there is always pressure in favor of greater equality of welfare. As Derek Parfit (1997) in particular has argued, however, we must not be too hasty. Consider the following choice: > > > > \| \| \| \| > \| --- \| --- \| --- \| > \| \| Units of welfare \| Total welfare for both groups \| > \| Alternative 1 \| Per person \| Per group \| \| > \| 10,000 people in group A \| 1 \| 10,000 \| \| > \| 100,000 people in group B \| 10 \| 1,000,000 \| \| > \| \| \| \| Impartially calculated: 1,010,000 \| > > > > > > > \| \| \| \| > \| --- \| --- \| --- \| > \| \| Units of welfare \| Total welfare for both groups \| > \| Alternative 3 \| Per person \| Per group \| \| > \| 10,000 people in group A \| 1 \| 10,000 \| \| > \| 100,000 people in group B \| 1 \| 100,000 \| \| > \| \| \| \| Impartially calculated: 110,000 \| > > > Is equality of welfare so important that Alternative 3 is superior to Alternative 1? To take an example of Parfit's, suppose the only way to make everyone equal with respect to sight is to make everyone totally blind. Is such "levelling down" required by morality? Indeed, is it in any way at all morally desirable? If we think the answer is No, then we might think that equality of welfare as such is not really an ideal (cf. Temkin 1993). Losses to the better off are justified only where this benefits the worse off. What we had thought of as pressure in favor of equality of welfare was instead pressure in favor of levelling up. We might say that additions to welfare matter more the worse off the person is whose welfare is affected. This view has come to be called prioritarianism (Parfit 1997; Arneson 1999b). It has tremendous intuitive appeal. For a simplistic example of how prioritarianism might work, suppose the welfare of the worst off counts five times as much as the welfare of the better off. Then Alternative 1 from the tables above comes out at: > > $(1 \times 5 \times 10,000) + (10 \times 100,000)$, > which comes to 1,050,000 total units of welfare. Again with the welfare of the worst off counting five times as much, Alternative 2 comes out at: > > $(8 \times 5 \times 10,000) + (9 \times 100,000)$, > which comes to 1,300,000 total units of welfare. This accords with the common reaction that Alternative 2 is morally superior to Alternative 1. Of course in real examples there is never only one division in society. Rather there is a scale from the worst off, to the not quite so badly off, and so on up to the best off. Prioritarianism is committed to variable levels of importance of benefitting people at different places on this scale: the worse off a person is, the greater the importance attached to that person's gain. This raises two serious worries about prioritarianism. The first concerns prioritarianism's difficulty in nonarbitrarily determining how much more importance to give to the welfare of the worse off. For example, should a unit of benefit to the worst off count 10 times more than the same size benefit to the best off and 5 times more than the same size benefit to the averagely well off? Or should the multipliers be 20 and 10, or 4 and 2, or other amounts? The second worry about prioritarianism is whether attaching greater importance to increases in welfare for some than to the same size increases in welfare for others contradicts fundamental impartiality (Hooker 2000: 60-2). This is not the place to go further into debates between prioritarianism and its critics. So the rest of this article sets aside those debates. ## 4. Full Rule-consequentialism Consequentialists have distinguished three components of their theory: (1) their thesis about what makes acts morally wrong, (2) their thesis about the procedure agents should use to make their moral decisions, and (3) their thesis about the conditions under which moral sanctions such as blame, guilt, and praise are appropriate. What we might call full rule-consequentialism consists of rule-consequentialist criteria for all three. Thus, full rule-consequentialism claims that an act is morally wrong if and only if it is forbidden by rules justified by their consequences. It also claims that agents should do their moral decision-making in terms of rules justified by their consequences. And it claims that the conditions under which moral sanctions should be applied are determined by rules justified by their consequences. Full rule-consequentialists may think that there is really only one set of rules about these three different subject matters. Or they may think that there are different sets that in some sense correspond to or complement one another. Much more important than the distinction between different kinds of full rule-consequentialism is the distinction between full rule-consequentialism and partial rule-consequentialism. Partial rule-consequentialism might take many forms. Let us focus on the most common form. The most common form of partial rule-consequentialism claims that agents should make their moral decisions about what to do by reference to rules justified by their consequences, but does not claim that moral wrongness is determined by rules justified by their consequences. Partial rule-consequentialists typically subscribe to the theory that moral wrongness is determined directly in terms of the consequences of the act compared to the consequences of alternative possible acts. This theory of wrongness is called act-consequentialism. Distinguishing between full and partial rule-consequentialism clarifies the contrast between act-consequentialism and rule-consequentialism. Act-consequentialism is best conceived of as maintaining merely the following: > > Act-consequentialist criterion of wrongness: An act is wrong if and > only if it results in less good than would have resulted from some > available alternative act. > When confronted with that criterion of moral wrongness, many people naturally assume that the way to decide what to do is to apply the criterion, i.e., > > Act-consequentialist moral decision procedure: On each occasion, an > agent should decide what to do by calculating which act would produce > the most good. > However, consequentialists nearly never defend this act-consequentialist decision procedure as a general and typical way of making moral decisions (Mill 1861: ch 2; Sidgwick 1907: 405-6, 413, 489-90; Moore 1903: 162-4; Smart 1956: 346; 1973: 43, 71; Bales 1971: 257-65; Hare 1981; Parfit 1984: 24-9, 31-43; Railton 1984: 140-6, 152-3; Brink 1989: 216-7, 256-62, 274-6; Pettit and Brennan 1986; Pettit 1991, 1994, 1997: 156-61, 2017; de Lazari-Radek and Singer 2014: ch. 10). There are a number of compelling consequentialist reasons why the act-consequentialist decision procedure would be counter-productive. First, very often the agent does not have detailed information about what the consequences would be of various acts. Second, obtaining such information would often involve greater costs than are at stake in the decision to be made. Third, even if the agent had the information needed to make calculations, the agent might make mistakes in the calculations. (This is especially likely when the agent's natural biases intrude, or when the calculations are complex, or when they have to be made in a hurry.) Fourth, there are what we might call expectation effects. Imagine a society in which people know that others are naturally biased towards themselves and towards their loved ones but are trying to make their every moral decision by calculating overall good. In such a society, each person might well fear that others will go around breaking promises, stealing, lying, and even assaulting whenever they convinced themselves that such acts would produce the greatest overall good (Woodard 2019: 149). In such a society, people would not feel they could trust one another. This fourth consideration is more controversial than the first three. For example, Hodgson 1967, Hospers 1972, and Harsanyi 1982 argue that trust would break down. Singer 1972 and Lewis 1972 argue that it would not. Nevertheless, most philosophers accept that, for all four of the reasons above, using an act-consequentialist decision procedure would not maximize the good. Hence even philosophers who espouse the act-consequentialist criterion of moral wrongness reject the act-consequentialist moral decision procedure. In its place, they typically advocate the following: > > Rule-consequentialist decision procedure: At least normally, agents > should decide what to do by applying rules whose acceptance will > produce the best consequences, rules such as "Don't harm > innocent others", "Don't steal or vandalize > others' property", "Don't break your > promises", "Don't lie", "Pay special > attention to the needs of your family and friends", "Do > good for others generally". > Since act-consequentialists about the criterion of wrongness typically accept this decision procedure, act-consequentialists are in fact partial rule-consequentialists. Often, what writers refer to as indirect consequentialism is this combination of act-consequentialism about wrongness and rule-consequentialism about the appropriate decision procedure. Standardly, the decision procedure that full rule-consequentialism endorses is the one that it would be best for society to accept. The qualification "standardly" is needed because there are versions of rule-consequentialism that let the rules be relativised to small groups or even individuals (D. Miller 2010; Kahn 2012). And act-consequentialism insists upon the decision procedure it would be best for the individual to accept. So, according to act-consequentialism, since Jack's and Jill's capacities and situations may be very different, the best decision procedure for Jack to accept may be different from the best decision procedure for Jill to accept. However, in practice act-consequentialists typically ignore for the most part such differences and endorse the above rule-consequentialist decision procedure (Hare 1981, chs. 2, 3, 8, 9, 11; Levy 2000). When act-consequentialists endorse the above rule-consequentialist decision procedure, they acknowledge that following this decision procedure does not guarantee that we will do the act with the best consequences. Sometimes, for example, our following a decision procedure that rules out harming an innocent person will prevent us from doing that act that would produce the best consequences. Similarly, there will be some circumstances in which stealing, breaking our promises, etc., would produce the best consequences. Still, our following a decision procedure that generally rules out such acts will in the long run and on the whole probably produce far better consequences than our trying to run consequentialist calculations on an act-by-act basis. Because act-consequentialists typically agree with a rule-consequentialist decision procedure, whether to classify some philosopher as an act-consequentialist or as a rule-consequentialist can be problematic. For example, G.E. Moore (1903, 1912) is sometimes classified as an act-consequentialist and sometimes as a rule-consequentialist. Like so many others, including his teacher Henry Sidgwick, Moore combined an act-consequentialist criterion of moral wrongness with a rule-consequentialist procedure for deciding what to do. Moore simply went further than most in stressing the danger of departing from the rule-consequentialist decision procedure (see Shaw 2000). Hare (1981) and Pettit (1991, 1994, 1997) have also been especially influential act-consequentialists about what makes acts right or wrong while holding that everyday decision-making should be conducted in terms of familiar rules focused on things other than consequences. ## 5. Global Consequentialism Some writers propose that the purest and most consistent form of consequentialism is the view that absolutely everything should be assessed by its consequences, including not only acts but also rules, motives, the imposition of sanctions, etc. Let us follow Pettit and Smith (2000) in referring to this view as global consequentialism. Kagan (2000) pictures it as multi-dimensional direct consequentialism, in that each thing is assessed directly in terms of whether its own consequences are as good as the consequences of alternatives. How does this global consequentialism differ from what we have been calling partial rule-consequentialism? What we have been calling partial rule-consequentialism is nothing but the combination of the act-consequentialist criterion of moral wrongness with the rule-consequentialist decision procedure. So defined, partial rule-consequentialism leaves open the question of when moral sanctions are appropriate. Some partial rule-consequentialists say that agents should be blamed and feel guilty whenever they fail to choose an act that would result in the best consequences. A much more reasonable position for a partial rule-consequentialist to take is that agents should be blamed and feel guilty whenever they choose an act that is forbidden by the rule-consequentialist decision procedure, whether or not that individual act fails to result in the best consequences. Finally, partial rule-consequentialism, as we have defined it, is compatible with the claim that whether agents should be blamed or feel guilty depends not on the wrongness of what they did, nor on whether the recommended procedure for making moral decisions would have led them to choose the act they choose, but instead solely on whether this blame or guilt will do any good. This is precisely the view of sanctions that global consequentialism takes. One objection to global consequentialism is that simultaneously applying a consequentialist criterion to acts, decision procedures, and the imposition of sanctions leads to apparent paradoxes (Crisp 1992; Streumer 2003; Lang 2004). Suppose, on the whole and in the long run, the best decision procedure for you to accept is one that leads you to do act x now. But suppose also that in fact the act with the best consequences in this situation is not x but y. So global consequentialism tells you (a) to use the best possible decision procedure but also (b) not to do the act picked out by this decision procedure. That seems paradoxical. Things get worse when we consider blame and guilt. Suppose you follow the best possible decision procedure but fail to do the act with the best consequences. Are you to be blamed? Should you feel guilty? Global consequentialism claims that you should be blamed if and only if blaming you will produce the best consequences, and that you should feel guilty if and only if this will produce the best consequences. Suppose that for some reason the best consequences would result from blaming you for following the prescribed decision procedure (and thus doing x). But surely it is paradoxical for a moral theory to call for you to be blamed although you followed the moral decision procedure mandated by the theory. Or suppose that for some reason the best consequences would result from blaming you for intentionally choosing the act with the best consequences (y). Again, surely it is paradoxical for a moral theory to call for you to be blamed although you intentionally chose the very act required by the theory. So one problem with global consequentialism is that it creates potential gaps between what acts it claims to be required and what decision procedures it tells agents to use, and between each of these and blamelessness. (For explicit replies to this line of attack, see Driver 2014: 175 and de Lazari-Radek and Singer 2014: 315-16.) That is not the most familiar problem with global consequentialism. The most familiar problem with it is instead its maximising act-consequentialist criterion of wrongness. According to this maximising criterion, an act is wrong if and only if it fails to result in the greatest good. This criterion judges some acts to be not wrong which certainly seem to be wrong. It also judges some acts that seem not wrong to be wrong. For example, consider an act of murder that results in slightly more good than any other act would have produced. According to the most familiar, maximising act-consequentialist criterion of wrongness, this act of murder is not wrong. Many other kinds of act such as assaulting, stealing, promise breaking, and lying can be wrong even when doing them would produce slightly more good than not doing them would. Again, the familiar, maximising form of act-consequentialism denies this. Or consider someone who gives to her child, or keeps for herself, some resource of her own instead of contributing it to help some stranger who would have gained slightly more from that resource. Such an action hardly seems wrong. Yet the maximising act-consequentialist criterion judges it to be wrong. Indeed, imagine how much self-sacrifice an averagely well-off person would have to make before her further actions satisfied the maximising act-consequentialist criterion of wrongness. She would have to give to the point where further sacrifices from her in order to benefit others would harm her more than they would benefit the others. Thus, the maximising act-consequentialist criterion of wrongness is often accused of being unreasonably demanding. The objections just directed at maximising act-consequentialism could be side-stepped by a version of act-consequentialism that did not require maximising the good. This sort of act-consequentialism is now called satisficing consequentialism. See the entry on consequentialism/ for more on such a theory. ## 6. Formulating Full Rule-consequentialism There are a number of different ways of formulating rule-consequentialism. For example, it can be formulated in terms of the good that actually results from rules or in terms of the rationally expected good of the consequences of rules. It can be formulated in terms of the consequences of compliance with rules or in terms of the wider consequences of acceptance of rules. It can be formulated in terms of the consequences of absolutely everyone's accepting the rules or in terms of the rules' acceptance by something less than everyone. Rule-consequentialism can also be formulated in terms of the teaching of a code to everyone in the next generation, in the full realization that the degrees of resulting acceptance will vary (Mulgan 2006, 2017, 2020; D. Miller 2014, 2021; T. Miller 2016, 2021). Rule-consequentialism is more plausible if formulated in some ways than it is if formulated in other ways. This is explained in the following three subsections. Questions of formulation are also relevant in later sections on objections to rule-consequentialism. ### 6.1 Actual versus Expected Good As indicated, full rule-consequentialism consists in rule-consequentialist answers to three questions. The first is, what makes acts morally wrong? The second is, what procedure should agents use to make their moral decisions? The third is, what are the conditions under which moral sanctions such as blame, guilt, and praise are appropriate? As we have seen, the answer that full rule-consequentialists give to the question about decision procedure is much like the answer that other kinds of consequentialist give to that question. So let us focus on the points of contrast, i.e., the other two questions. These two questions -- about what makes acts wrong and about when sanctions are appropriate -- are more tightly connected than sometimes realized. Indeed, J.S. Mill, one of the fathers of consequentialism, affirmed their tight connection: > > We do not call anything wrong, unless we mean to imply that a person > ought to be punished in some way or other for doing it; if not by law, > by the opinion of his fellow creatures; if not by opinion, by the > reproaches of his own conscience. (1861: ch. 5, para. 14) > Let us assume that Mill took "ought to be punished, at least by one's own conscience if not by others" to be roughly the same as "blameworthy". With this assumption in hand, we can interpret Mill as tying wrongness tightly to blameworthiness. In a moment, we can consider what follows if Mill is mistaken that wrongness is tied tightly to blameworthiness. First, let us consider what follows if Mill is correct that wrongness is tied tightly to blameworthiness. Consider the following argument, whose first premise comes from Mill: > > If an act is wrong, it is blameworthy. > Surely, an agent cannot rightly be blamed for accepting and following rules that the agent could not foresee would have sub-optimal consequences. From this, we get our second premise: > > If an act is blameworthy, the sub-optimal consequences of rules > allowing that act must have been foreseeable. > From these two premises we get the conclusion: > > So if an act is wrong, the sub-optimal consequences of rules allowing > that act must have been foreseeable. > Of course, the actual consequences of accepting a set of rules may not be the same as the foreseeable consequences of accepting that set. Hence, if full rule-consequentialism claims that an act is wrong if and only if the foreseeable consequences of rules allowing that act are sub-optimal, rule-consequentialism cannot also hold that an act is wrong if and only if the actual consequences of rules allowing that act will be sub-optimal. Now suppose instead the relation between wrongness and blameworthiness is far looser than Mill suggested (cf. Sorensen 1996). That is, suppose that our criterion of wrongness can be quite different from our criterion of blameworthiness. In that case, we could hold: > > Actualist rule-consequentialist criterion of > wrongness: An act is wrong if and only if it is forbidden by > rules that would actually result in the greatest good. > and > > Expectablist rule-consequentialist criterion of > blameworthiness: An act is blameworthy if and only if it is > forbidden by the rules that would foreseeably result in the > greatest good. > Let us replace "foreseeably result in the greatest good" with "result in the greatest expected good". Here is how expected good of a set of rules is calculated. Suppose we can identify the value or disvalue of each possible outcome a set of rules might have. Multiply the value of each possible outcome by the probability of that outcome's occurring. Take all the products of these multiplications and add them together. The resulting number quantifies the expected good of that set of rules. Expected good is not to be calculated by employing whatever crazy estimates of probabilities people might assign to possible outcomes. Rather, expected good is calculated by multiplying the value or disvalue of possible outcomes by rational or justified probability estimates. Different agents have different evidence and thus have different rational and justified probability estimates. Such differences are sometimes exactly what explains disagreements about what changes to an extant moral code would be improvements. In some cases of disagreements, the cause of such disagreement is that at least one of the parties is not aware of, or has not fully assimilated, evidence that is available. Expectablist rule-consequentialist would likely want to tie rational or justified probability estimates to the evidence that is available at the time, even if some people are not aware of it or have not fully appreciated its implications. There might be considerable scepticism about how calculations of expected good are even possible (Lenman 2000). Even where such calculations are possible, they will often be quite impressionistic and imprecise. Nevertheless, we can reasonably hope to make at least some informed judgements about the likely consequences of alternative possible rules (Burch-Brown 2014). And we could be guided by such judgements, as legislators often say they are. In contrast, which rules would actually have the very best consequences will normally be inaccessible. Hence, the expectablist rule-consequentialist criterion of blameworthiness is appealing. Now return to the proposal that, while the criterion of blameworthiness is the expectablist rule-consequentialist one, the correct criterion of moral wrongness is the actualist rule-consequentialist one. This proposal rejects Mill's move of tying moral wrongness to blameworthiness. There is, however, a very strong objection to this proposal. What is the role and importance of moral wrongness if it is disassociated from blameworthiness? In order to retain an obvious role and importance for moral wrongness, those committed to the expectablist rule-consequentialist criterion of blameworthiness are likely to endorse: > > Simple expectablist rule-consequentialist criterion of moral > wrongness: An act is morally wrong if and only if it is forbidden by > the rules that would result in the greatest expected good. > Indeed, once we have before us the distinction between (a) the amount of value that actually results and (b) the rationally expected good, the full rule-consequentialist is likely to go for expectablist criteria of moral wrongness, blameworthiness, and decision procedures. What if, as far as we can tell, no one code has greater expected value than its rivals? We will need to amend our expectablist criteria in order to accommodate this possibility: > > Sophisticated expectablist rule-consequentialist criterion of > moral wrongness: An act is morally wrong if and only if it is > forbidden either by the rules that would result in the greatest > expected good, or, if two or more alternative codes of rules > are equally best in terms of expected good, by the one of these codes > closest to conventional morality. > The argument for using closeness to conventional morality to break ties between otherwise equally promising codes begins with the observation that social change regularly has unexpected consequences. And these unexpected consequences usually seem to be negative. Furthermore, the greater the difference between a new code and the one already conventionally accepted, the greater the scope for unexpected consequences. So, as between two codes we judge to have equally high expected value, we should choose the one closest to the one we already know. (For discussion of the situation where two codes have equally high expected value and seem equally close to conventional morality, see Hooker 2008: 83-4.) An implication is that people should make changes to the status quo where but only where these changes have greater expected value than sticking with the status quo. Rule-consequentialism manifestly has the capacity to recommend change. But it does not favor change for the sake of change. Rule-consequentialism most definitely does need to be formulated so as to deal with ties in expected value. However, for most of the rest of this article, I will ignore this complication. ### 6.2 Compliance and Acceptance There are other important issues of formulation that rule-consequentialists face. One is the issue of whether rule-consequentialism should be formulated in terms of compliance with rules, in terms of acceptance of rules, or in terms of the teaching of rules. Admittedly, the most important consequence of teaching and accepting rules is compliance with them. And early formulations of rule-consequentialism did indeed explicitly mention compliance. For example, they said an act is morally wrong if and only if it is forbidden by rules the compliance with which will maximize the good (or the expected good). (See Austin 1832; Brandt 1959; M. Singer 1955, 1961.) However, acceptance of a rule can have consequences other than compliance with the rule. As Kagan (2000: 139) writes, "once embedded, rules can have an impact on results that is independent of their impact on acts: it might be, say, that merely thinking about a set of rules reassures people, and so contributes to happiness." (For more on what we might call these 'beyond-compliance consequences' of rules, see Sidgwick 1907: 405-6, 413; Lyons 1965: 140; Williams 1973: 119-20, 122, 129-30; Adams 1976: 470; Scanlon 1998: 203-4; Kagan 1998: 227-34.) These consequences of acceptance of rules should most definitely be part of a cost-benefit analysis of rules. Formulating rule-consequentialism in terms of the consequences of acceptance allows them to be part of this analysis. Important consequences of communal acceptance of a set of rules include assurance, incentive, and deterrent effects. And consideration of assurance and incentive effects has played a large role in the development of rule-consequentialism (Harsanyi 1977, 1982: 56-61; 1993: 116-18; Brandt 1979: 271-77; 1988: 346ff [1992: 142ff.]; 1996: 126, 144; Johnson 1991, especially chs. 3, 4, 9). Hence, we should not be surprised that rule-consequentialism has gone from being formulated in terms of compliance with rules to being formulated in terms of acceptance of rules. However, just as we need to move from thinking about the consequences of compliance to thinking about the wider consequences of acceptance, we need to go further. Focusing purely on the consequences of acceptance of rules ignores the "transition" costs for the teachers of teaching those rules, and the costs to the teaching's recipients of internalizing these rules. And yet these costs can certainly be significant (Brandt 1963: section 4; 1967 [1992: 126]; 1983: 98; 1988: 346-47, 349-50 [1992: 140-143, 144-47]; 1996: 126-28, 145, 148, 152, 223). Suppose, for example, that, once a fairly simple and relatively undemanding code of rules Code A has been accepted, the expected value of Code A would be n. Suppose a more complicated and demanding alternative Code B would have an expected value of $n + 5$ once Code B has been accepted. So if we just consider the expected values of acceptance of the two alternative codes, Code B wins. But now let us factor into our cost/benefit analysis of rival codes the relative costs of teaching the two codes and of getting them internalized by new generations. Since Code A is fairly simple and relatively undemanding, the cost of getting it internalized is [?]1. Since Code B is more complicated and demanding, the cost of getting it internalized is [?]7. So if our comparison of the two codes considers the respective costs of getting them internalized, Code A's expected value is $n-1$, and Code B's is $n+5-7$. Once we include the respective costs of getting the codes internalized, Code A wins. As indicated, the costs of teaching a code successfully, so that the code is very widely internalized, are "transition costs". But of course such transitions are always to one arrangement from another. The arrangement we are imagining the transition being to is the acceptance of a certain proposed code. The arrangement we are imagining the transition being from is ... well, what? One answer is that the arrangement from which the transition is supposed to be starting is whatever moral code the society happens to accept already. That might seem like the natural answer. However, there is a strong objection to this answer, namely that rule-consequentialism should not let the cost/benefit analysis of a proposed code be influenced by the costs of getting people to give up whatever rules they may have already internalised. This is for two reasons. Most importantly, rule-consequentialist assessment of codes needs to avoid giving weight directly or indirectly to moral ideas that have their source in other moral theories but not in rule-consequentialism itself. Suppose people in a given society were brought up to believe that women should be subservient to men. Should rule-consequentialist evaluation of a proposed non-sexist code have to count the costs of getting people to give up the sexist rules they have already internalised so as to accept the new, non-sexist ones? Since the sexist rules are unjustifiable, that they were accepted should not be allowed to infect rule-consequentialist assessment. Another reason for rejecting the answer we are considering is that it threatens to underwrite an unattractive relativism. Different societies may differ considerably in their extant moral beliefs. Thus, a way of assessing proposed codes that considers the costs of getting people already committed to some other code will end up having to countenance different transition costs to get to the same code. For example, the transition costs to a non-racist code are much higher from an already accepted racist code than from an already accepted non-racist one. Formulating rule-consequentialism so that it endorses the same code for 1960s Michigan as for 1960s Mississippi is desirable. (For opposing arguments that rule-consequentialism should be formulated so as to coutenance social relativism, see Kahn 2012; D. Miller 2014, 2021; T. Miller 2021.) A way to avoid ending up with the social relativism just identified is to formulate rule-consequentialism in terms of acceptance by new generations of humans. The proposal might be that we compare the respective "teaching costs" of alternative codes, on the assumption that these codes will be taught to new generations of children, i.e., children who have not already been educated to accept a moral code. We are to imagine the children start off with natural (non-moral) inclinations to be very partial towards themselves and a few others. We should also assume that there is a cognitive cost associated with the learning of each rule. These are realistic assumptions, with big implications. One is that a cost/benefit analysis of alternative codes of rules would have reason to favor simpler codes over more complex ones. Of course there can also be benefits from having more, or more complicated, rules. Yet there is probably a limit on how complicated or complex a code can be and still have greater expected value than simpler codes, once teaching costs are included. Another implication concerns prospective rules about making sacrifices to help others. Since children start off focused on their own gratifications, getting them to internalize a kind of impartiality that requires them to make large sacrifices repeatedly for the sake of others would have extremely high costs. There would also, of course, be enormous benefits from the internalization of such a rule -- predominately, benefits to others. Would the benefits be greater than the costs? At least since Sidgwick (1907: 434), many utilitarians have taken for granted that human nature is such that the real possibilities are (1) that human beings care passionately about some and less about each of the rest of humanity or (2) that human beings care weakly but impartially about everyone. In other words, what is not a realistic possibility, according to this view of human nature, is human beings' caring strongly and impartially about everyone in the world. If this view of human nature is correct, then one enormous cost of successfully making people completely impartial is that doing so would leave them with only weak concerns. Even if that picture of human nature is not correct, that is, even if making people completely impartial could be achieved without draining them of enthusiasm and passion, the cost of successfully making people care as much about every other individual as they do about themselves would be prohibitive. At some point on the spectrum running from complete partiality to complete impartiality, the costs of pushing and inducing everyone further along the spectrum outweigh the benefits. A complication worth mentioning comes from the obvious point that moral education is developmental. How many rules are to be taught to children, how complicated these rules are, how demanding the rules are, and how to resolve conflicts among the rules depend on the developmental stage of the children. Hence, there can be conflict between the simpler and less demanding rules taught to the very young and the more elaborate, nuanced, and demanding rules taught at higher developmental stages. Of course, rule-consequentialism is best formulated in terms of the rules featured in the end of the development rather than the ones figuring in earlier stages. ### 6.3 Complete Acceptance versus Incomplete Acceptance While rule-consequentialist cost/benefit analyses of codes should count the cost of getting those codes internalised by new generations, such analyses should acknowledge that the internalisation will not be achieved in every last person (D. Miller 2021: 130). No matter how good the teaching is, the results will be imperfect. Some people will learn rules that differ to some degree from the ones that were taught, and some of these people will end up with very mistaken views about what is morally required, morally optional, or morally wrong. Other people will end up with insufficient moral motivation. Other people will end up with no moral motivation at all (psychopaths). Rule-consequentialism needs to have rules for coping with the inevitably imperfect results of moral teaching. Such rules will crucially include rules about punishment. From a rule-consequentialist point of view, one point of punishment is to deter certain kinds of act. Another point of punishment is to get undeterred, dangerous people off the streets. Perhaps rule-consequentialism can admit that another point of punishment is to appease the primitive lust for revenge on the part of victims of such acts and their family and friends. Finally, there is the expressive and reinforcing power of rules about punishment. Some ways of formulating rule-consequentialism make having rules about punishment difficult to explain. One such way of formulating rule-consequentialism is: > > An act is morally wrong if and only if it is prohibited by the code of > rules the full acceptance of which by absolutely everyone would > produce the greatest expected good. > Imagine a world in which absolutely every adult human fully accepts rules forbidding (for example) physical attacks on the innocent, stealing, promise breaking, and lying. Suppose these rules have been internalized so deeply by everyone in this world that in this world there is complete compliance with these rules. Also assume that, if everyone in this world always complies with these rules, this perfect compliance becomes common knowledge. In this world, there would be little or no need for rules about punishment and thus little or no benefit from having such rules. But there are teaching and internalization costs associated with each rule taught and internalized. So there are teaching and internalization costs associated with the inclusion of any rule about punishment. The combination of costs without benefits is repellant. Therefore, for a world of complete compliance, the version of rule-consequentialism immediately above would not endorse rules about punishment. We need a form of rule-consequentialism that includes rules for dealing with people who are not committed to the right rules, indeed even for people who are irredeemable. In other words, rule-consequentialism needs to be formulated so as to conceptualize society as containing some people insufficiently committed to the right rules, and even some people not committed to any moral rules. Here is a way of doing so: > > An act is wrong if and only if it is prohibited by a code of rules the > acceptance of which by the overwhelming majority of people in each new > generation would have the greatest expected value. > Note that rule-consequentialism neither endorses nor condones the non-acceptance of the code by those outside the overwhelming majority. On the contrary, rule-consequentialism claims those people are morally mistaken. Indeed, the whole point of formulating rule-consequentialism this way is to make room for rules about how to respond negatively to such people. Of course, the term "overwhelming majority" is very imprecise. Suppose we remove the imprecision by picking a precise percentage of society, say 90%. Picking any precise percentage as an obvious element of arbitrariness to it. For example, if we pick 90%, why not 89% or 91%? Perhaps, we can argue for a number in the range of 90% as a reasonable compromise between two pressures. On the one hand, the percentage we pick should be close enough to 100% to retain the idea that, ideally, moral rules would be accepted by everyone. On the other hand, the percentage needs to be far enough short of 100% to leave considerable scope for rules about punishment. It seems that 90% is in a defensible range, given the need to balance those considerations. (For dissent from this, see Ridge 2006; for a reply to Ridge, see Hooker and Fletcher 2008. The matter receives further discussion in H. Smith 2010; Tobia 2013, 2018; T. Miller 2014, 2021; Toppinen 2016; Portmore 2017; Yeo 2017; Podgorski 2018; Perl 2021.) Holly Smith (2010) pointed out that a cost/benefit analysis of the acceptance of any particular code by a positive percentage of the population less than 100% depends on what the rest of the population accepts and is disposed to do. Consider the following contrast. One imagined scenario is that 90% of the population accept one code, and the other 10% accept a very similar code, such that the two codes rarely diverge in practice. A second imagined scenario is that 90% of the population accept one code, and the other 10% accept various codes that frequently and dramatically conflict in practice with code accepted by 90%. Conflict resolution and enforcement are less important in the first imagined scenario than in the second. Hence, if rule-consequentialism is formulated in terms of the acceptance of a code by less than 100% of people, it matters what assumptions are made about whatever percentage of the population do not accept this code. Some theorists propose that formulating rule-consequentialism in terms of the code the teaching of which has the greatest expected value is superior to formulating the theory in terms of either a fixed or variable acceptance rate for codes (Mulgan 2006, 141, 147; 2017, 291; 2020, 12-21; T. Miller 2016; 2021; D. Miller 2021). A "teaching formulation" of rule-consequentialism holds: An act is morally prohibited, obligatory, or optional if and only if and because it is prohibited, obligatory, or permitted by the code of rules the teaching of which to everyone has at least as much expected value as the teaching of any other code. Two clarifications are needed immediately. First, "everyone" here needs to be qualified so as not to include people "with significant, cognitive, or conative deficits" (D. Miller 2021: 10). Second, teaching a code to everyone is not assumed to lead to everyone's internalization of this code. Hence, even if everyone is taught a certain code, foreseeably there will be some who internalize rules that are more or less different from the rules that were taught. There will also be some whose moral motivation is unreliable or even indiscernible. And there will be some who take themselves to be sceptics about moral rules and even about the rest of morality. There are some definite advantages of teaching formulations of rule-consequentialism. To be sure, we have to build into our cost/benefit analysis that enough costs are incurred to make the teaching at least partly successful. In other words, we must insist that the teaching be successful enough to get enough people to internalize rules such that a good degree of cooperation and security results. But we do not need to be precise about what percentage of people taught this code remain amoralists, what percentage of people taught this code end up internalizing codes somewhat different from the one being taught, or what percentage end up insufficiently motivated to comply with the moral code they have internalized unless effective enforcement is in place (D. Miller 2021; T. Miller 2021). Another advantage of teaching formulations is that the idea of teaching a code to everyone connects tightly to the idea that this code should be public knowledge. The idea that a moral code must be suitable for being public knowledge is very appealing (Baier 1958: 195-196; Rawls 1971: 133; Gert 1998: 10-13, 225, 239-240; Hill 2005; Cureton 2015; Pettit 2017: 39, 65, 102). And rule-consequentialists have championed the idea that moral rules are subject to this "publicity condition" (Brandt 1992: 136, 144; Hooker 2010; 2016, forthcoming; Parfit 2011, 2017a; D. Miller 2021: 131-32). ## 7. Three Ways of Arguing for Rule-consequentialism What rules will rule-consequentialism endorse? It will endorse rules prohibiting physically attacking innocent people, taking or harming the property of others, breaking one's promises, and lying. It will also endorse rules requiring one to pay special attention to the needs of one's family and friends, but more generally to be willing to help others with their (morally permissible) projects. A society where such rules are prominent in a public code would be likely to have more good in it than one lacking such rules. The fact that these rules are endorsed by rule-consequentialism makes rule-consequentialism attractive. For, intuitively, these rules seem right. However, other moral theories endorse these rules as well. Most obviously, a familiar kind of moral pluralism contends that these intuitively attractive rules constitute the most basic level of morality, i.e., that there is no deeper moral principle underlying and unifying these rules. Call this view Rossian pluralism (in honor of its champion W.D. Ross (1930, 1939)). Rule-consequentialism may agree with Rossian pluralism in endorsing rules against physically attacking the innocent, stealing, promise breaking, and rules requiring various kinds of loyalty and more generally doing good for others. But rule-consequentialism goes beyond Rossian pluralism by specifying an underlying unifying principle that provides impartial justification for such rules. Other moral theories try to do this too. Such theories include some forms of Kantianism (Audi 2001, 2004) and some forms of contractualism (Scanlon 1998; Parfit 2011; Levy 2013). In any case, the first way of arguing for rule-consequentialism is to argue that it specifies an underlying principle that provides impartial justification for intuitively plausible moral rules, and that no rival theory does this as well (Urmson 1953; Brandt 1967; Hospers 1972; Hooker 2000). (Attacks on this line of argument for rule-consequentialism include Stratton-Lake 1997; Thomas 2000; D. Miller 2000; Montague 2000; Arneson 2005; Moore 2007; Hills 2010; Levy 2014.) This first way of arguing for rule-consequentialism might be seen as drawing on the idea that a theory is better justified to us to the extent that it increases coherence within our beliefs (Rawls 1951, 1971: 19-21, 46-51; DePaul 1987; Ebertz 1993; Sayre-McCord 1986, 1996). [See the entry on coherentist theories of epistemic justification.] But the approach might also be seen as moderately foundationalist in that it begins with a set of beliefs (in various moral rules) to which it assigns independent credibility though not infallibility (Audi 1996, 2004; Crisp 2000). [See the entry on foundationalist theories of epistemic justification.] Admittedly, coherence with our moral beliefs does not make a moral theory true, since our moral beliefs might be mistaken. Nevertheless, if a moral theory fails significantly to cohere with our moral beliefs, this undermines the theory's ability to be justified to us. Wolf (2016) and Copp (2020) argue that the meta-ethical view that morality is a social practice with a particular function might lead us to rule-consequentialism. However, Hooker (forthcoming) contends that the meta-ethical view that morality is a social practice with a particular function stands in need of justification in terms of whether it coheres with our considered moral principles and more specific considered moral judgements. In other words, that meta-ethical view about the function of morality needs to be judged in terms of whether it helps us achieve a reflective equilibrium among our beliefs or not. The second way of arguing for rule-consequentialism is very different. It begins with a commitment to consequentialist assessment. With that commitment as a first premise, the point is then made that assessing acts indirectly, e.g., by focusing on the consequences of communal acceptance of rules, will in fact produce better consequences than assessing acts directly in terms of their own consequences (Austin 1832; Brandt 1963, 1979; Harsanyi 1982: 58-60; 1993; Riley 2000). After all, making decisions about what to do is the main point of moral assessment of acts. So if a way of morally assessing acts is likely to lead to bad decisions, or more generally lead to bad consequences, then, according to a consequentialist point of view, so much the worse for that way of assessing acts. Earlier we saw that all consequentialists now accept that assessing each act individually by its expected value is in general a terrible procedure for making moral decisions. Agents should decide how to act by instead appealing to certain rules such as "don't physically attack others", "don't steal", "don't break your promises", "pay special attention to the needs of your family and friends", and "be generally helpful to others". And these are the rules that rule-consequentialism endorses. Many consequentialists, however, think this hardly shows that full rule-consequentialism is the best form of consequentialism. Once a distinction is made between, on the one hand, the best procedure for making moral decisions about what to do and, on the other hand, the criteria of moral rightness and wrongness, all consequentialists can admit that we need rule-consequentialism's rules for our decision procedure. But consequentialists who are not rule-consequentialists contend that such rules play no role in the criterion of moral rightness. Hence these consequentialists reject what this article has called full rule-consequentialism. However, whether the objection we have just been considering to the second way of arguing for rule-consequentialism is a good objection depends on whether it is legitimate to distinguish between procedures appropriate for making moral decisions and the criteria of moral rightness or wrongness. That matter remains contentious (Hooker 2010; de Lazari-Radek and Singer 2014: ch. 10). Yet the second way of arguing for rule-consequentialism runs into another and quite different objection. This objection is that the first step in this argument for rule-consequentialism is a commitment to consequentialist assessment. This first step itself needs justification. Why assume that assessing things in a consequentialist way is uniquely justified? It might be said that consequentialist assessment is justified because promoting the impartial good has an obvious intuitive appeal. But that won't do, since there are alternatives to consequentialist assessment that also have obvious intuitive appeal, for example, "act on the code that no one could reasonably reject". What we need is a way of arguing for a moral theory that does not start by begging the question which kind of theory is most plausible. A third way of arguing for rule-consequentialism is contractualist (Harsanyi 1953, 1955, 1982, 1993; Brandt 1979, 1988, 1996; Scanlon 1982, 1998; Parfit 2011; Levy 2013). Suppose we can specify reasonable conditions under which everyone would choose, or at least would have sufficient reason to choose, the same code of rules. Intuitively, such an idealized agreement would legitimate that code of rules. Now if those rules are the ones whose internalisation would maximise the expected good, contractualism is leading us to rule-consequentialism's rules. There are different views about what would be reasonable conditions for choosing among alternative possible moral rules. One view is that everyone's impartiality would have to be insured by the imposition of a hypothetical "veil of ignorance" behind which no one knew any specific facts about himself or herself (Harsanyi 1953, 1955). Another view is that we should imagine that people would be choosing a moral code on the basis of (a) full empirical information about the different effects on everyone, (b) normal concerns (self-interested as well as altruistic), and (c) roughly equal bargaining power (Brandt 1979; cf. Gert 1998). Parfit (2011) proposes seeking rules that everyone has (personal or impartial) reason to choose or will that everyone accept. If impartial reasons are always sufficient even when opposed by personal ones, then everyone has sufficient reason to will that everyone accept the rules whose universal acceptance will have the best consequences impartially considered. Similarly, Levy (2013) supposes that no one could reasonably reject a code of rules that would impose on her burdens that add up to less than the aggregate of burdens that every other code would impose on others. Such arguments suggest the extensional equivalence of contractualism and rule-consequentialism. (For assessment of whether Parfit's contractualist arguments for rule-consequentialism succeed, see J. Ross 2009; Nebel 2012; Hooker 2014. For similarities between rule-consequentialism and Scanlon's contractualism, and the difficulty of deciding between these two theories, see Suikkanen 2022.) ## 8. Must Rule-consequentialism Be Guilty of Collapse, Incoherence, or Rule-worship? An oft-repeated line of objection to rule-consequentialism from the mid 1960s to the mid 1990s was that this theory is fatally impaled on one or the other horn of the following dilemma: Either rule-consequentialism collapses into practical equivalence with the simpler act-consequentialism, or rule-consequentialism is incoherent. Here is why some have thought rule-consequentialism collapses into practical equivalence with act-consequentialism. Consider a rule that rule-consequentialism purports to favor -- e.g., "don't steal". Now suppose an agent is in some situation where stealing would produce more good than not stealing. If rule-consequentialism selects rules on the basis of their expected good, rule-consequentialism seems driven to admit that compliance with the rule "don't steal except when ... or ... or ..." is better than compliance with the simpler rule "don't steal". This point generalizes. In other words, for every situation where compliance with some rule would not produce the greatest expected good, rule-consequentialism seems driven to favor instead compliance with some amended rule that does not miss out on producing the greatest expected good in the case at hand. But if rule-consequentialism operates this way, then in practice it will end up requiring the very same acts that act-consequentialism requires. If rule-consequentialism ends up requiring the very same acts that act-consequentialism requires, then rule-consequentialism is indeed in terrible trouble. Rule-consequentialism is the more complicated of the two theories. This leads to the following objection. What is the point of rule-consequentialism with its infinitely amended rules if we can get the same practical result much more efficiently with the simpler act-consequentialism? Rule-consequentialists in fact have an excellent reply to the objection that their theory collapses into practical equivalence with act-consequentialism. This reply relies on the point that the best kind of rule-consequentialism ranks systems of rules not in terms of the expected good of complying with them, but in terms of the expected good of their teaching and acceptance. Now if a rule forbidding stealing, for example, has exception clause after exception clause after exception clause tacked on to it, the rule with all these exception clauses will provide too much opportunity for temptation to convince agents that one of the exception clauses applies, when in fact stealing would be advantageous to the agent. And this point about temptation will also undermine other people's confidence that their property won't be stolen. The same is true of most other moral rules: incorporating too many exception clauses could undermine people's assurance that others will behave in certain ways (such as keeping promises and avoiding stealing). Furthermore, when comparing alternative rules, we must also consider the relative costs of getting them internalised by new generations. Clearly, the costs of getting new generations to learn either an enormous number of rules or hugely complicated rules would be prohibitive. So rule-consequentialism will favor a code of rules without too many rules, and without too much complication within the rules. There are also costs associated with getting new generations to internalise rules that require one to make enormous sacrifices for others with whom one has no particular connection. Of course, following such demanding rules will produce many benefits, mainly to others. But the costs associated with internalising such rules should be weighed against the benefits of following them. At some level of demandingness, the costs of getting such demanding rules internalised will outweigh the benefits that following them will produce. Hence, doing a careful cost/benefit analysis of internalising demanding rules will come out opposing rules' being too demanding. The code of rules that rule-consequentialism favours a code comprised of rules that are not too numerous, too complicated, or too demanding can sometimes lead people to do acts that do not have the greatest expected value. For example, following the simpler rule "Don't steal" will sometimes produce less good consequences than following a more complicated rule "Don't steal except when ... or ... or ... or ... or ...". Another example might be following a rule that allows people to give some degree of priority to their own projects, even when they could produce more good by sacrificing their own projects in order to help others. Still, rule-consequentialism's contention is that bringing about widespread acceptance of a simpler and less demanding code, even if acceptance of that code does sometimes lead people to do acts with sub-optimal consequences, has higher expected value in the long run than bringing about widespread acceptance of a maximally complicated and demanding code. Since rule-consequentialism can tell people to follow this simpler and less demanding code, even when following it will not to maximise expected good, rule-consequentialism escapes collapse into practical equivalence to act-consequentialism. To the extent that rule-consequentialism circumvents collapse, this theory is accused of incoherence. Rule-consequentialism is accused of incoherence for maintaining that an act can be morally permissible or even required though the act fails to maximise expected good. Behind this accusation must be the assumption that rule-consequentialism contains an overriding commitment to maximise the good. It is incoherent to have this overriding commitment and then to oppose an act required by the commitment. (For developments of this line of thought, see Arneson 2005; Card 2007; Wall 2009.) In order to evaluate the incoherence objection to rule-consequentialism, we need to be clearer about the supposed location of an overriding commitment to maximize the good. Is this commitment supposed to be part of the rule-consequentialist agent's moral psychology? Or is it supposed to be part of the theory rule-consequentialism? Well, rule-consequentialists need not have maximizing the good as their ultimate and overriding moral goal. Instead, they could have a moral psychology as follows: > > > Their fundamental moral motivation is to do what is impartially > defensible. > > > > They believe acting on impartially justified rules is impartially > defensible. > > > > They also believe that rule-consequentialism is on balance the best > account of impartially justified rules. > > > Agents with this moral psychology -- i.e., this combination of moral motivation and beliefs -- would be morally motivated to do as rule-consequentialism prescribes. This moral psychology is certainly possible. And, for agents who have it, there is nothing incoherent about following rules when doing so will not maximize the expected good. So, even if rule-consequentialist agents need not have an overriding commitment to maximize expected good, does their theory contain such a commitment? No, rule-consequentialism is essentially the conjunction of two claims: (1) that rules are to be selected solely in terms of their consequences and (2) that these rules determine which kinds of acts are morally wrong. This is really all there is to the theory -- in particular, there is not some third component consisting in or entailing an overriding commitment to maximize expected good. Without an overriding commitment to maximize the expected good, there is nothing incoherent in rule-consequentialism's forbidding some kinds of act, even when they maximize the expected good. Likewise, there is nothing incoherent about rule-consequentialism's requiring other kinds of act, even when they conflict with maximizing the expected good. The best known objection to rule-consequentialism dies once we realize that neither the rule-consequentialist agent nor the theory itself contains an overriding commitment to maximize the good. The viability of this defense of rule-consequentialism against the incoherence objection may depend in part on what the argument for rule-consequentialism is supposed to be. The defense seems less viable if the argument for rule-consequentialism starts from a commitment to consequentialist assessment. For starting with such a commitment seems very close to starting from an overriding commitment to maximize the expected good. The defence against the incoherence objection seems far more secure, however, if the argument for rule-consequentialism is that this theory does better than any other moral theory at specifying an impartial justification for intuitively plausible moral rules. (For more on this, see Hooker 2005, 2007; Rajczi 2016; Wolf 2016; Copp 2020.) Another old objection to rule-consequentialism is that rule-consequentialists must be "rule-worshipers" -- i.e., people who will stick to the rules even when doing so will obviously be disastrous. The answer to this objection is that rule-consequentialism endorses a rule requiring one to prevent disaster, even if doing so requires breaking other rules (Brandt 1992: 87-8, 150-1, 156-7). To be sure, there are many complexities about what counts as a disaster. Think about what counts as a disaster when the "prevent disaster" rule is in competition with a rule against lying. Now think about what counts as a disaster when the "prevent disaster" rule is in competition with a rule against stealing, or even more when in competition with a rule against physically harming the innocent. Rule-consequentialism may need to be clearer about such matters. But at least it cannot rightly be accused of potentially leading to disaster. An important confusion to avoid is to think that rule-consequentialism's including a "prevent disaster" rule means that rule-consequentialism collapses into practical equivalence with maximising act-consequentialism. Maximising act-consequentialism holds that we should lie, or steal, or harm the innocent whenever doing so would produce even a little more expected good than not doing so would. A rule requiring one to prevent disaster does not have this implication. Rather, the "prevent disaster" rule comes into play only when there is a very much larger difference in the amounts of expected value at stake. Woodard (2022) contends that the "prevent disaster" rule needs reconceptualizing. Many rules identify kinds of reasons. Examples are the reason to prevent harm, the reason not to steal, and the reason to keep your promises. But, Woodard holds, we should not think of the "prevent disaster" rule as being a rule identifying a kind of reason, which then is claimed to have overriding force. We should rather think that the "prevent disaster" rule distinguishes between cases in which the reason to prevent harm overrides opposing moral reasons and cases in which the reason to prevent harm is weaker than opposing reasons. Even if Woodard is correct about this, a "prevent disaster" rule can be accused of vagueness and indeterminacy. Indeed, the line between cases in which the moral reason to prevent harm overrides opposing moral reasons and cases in which the moral reason to prevent harm is weaker than opposing moral reasons might well be indeterminant. Woollard (2022) argues that admitting such vagueness and indeterminacy does not weaken rule-consequentialism. And she draws on her earlier work (2015) to explain how rule-consequentialism's "prevent disaster" rule need not impale the theory on the dilemma of either being overly demanding or placing a counterintuitive limit on a requirement to come to the aid of others. ## 9. Other Objections to Rule-consequentialism From the mid 1960s until the mid 1990s, most philosophers thought rule-consequentialism couldn't survive the objections discussed in the previous section. So, during those three decades, most philosophers didn't bother with other objections to the theory. However, if rule-consequentialism has convincing replies to all three of the objections just discussed, then a good question is whether or not there are other fatal objections to the theory. Some other objections try to show that, given the theory's criterion for selecting rules, there are conditions under which it selects intuitively unacceptable rules. For example, Tom Carson (1991) argued that rule-consequentialism turns out to be extremely demanding in the real world. Mulgan (2001, esp. ch. 3) agreed with Carson about that, and went on to argue that, even if rule-consequentialism's implications in the actual world are fine, the theory has counterintuitive implications in possible worlds. If Mulgan were right about that, this would cast doubt on rule-consequentialism's claim to explain why certain demands are appropriate in the actual world. Debate about such matters continues (Hooker 2003; Lawlor 2004; Parfit 2011, 2017a, 2017b; Woollard 2015: 181-205; Rajczi 2016; Portmore 2017; Podgorski 2018; D. Miller 2021; Perl 2021, 2022). And Mulgan has become a developer of rule-consequentialism rather than a critic (Mulgan 2006, 2009, 2015, 2017, 2020). A related objection to rule-consequentialism is that rule-consequentialism makes the justification of familiar rules contingent on various empirical facts, such as what human nature is like, and how many people there are in need or in positions to help. The objection to rule-consequentialism is that some familiar moral rules are necessarily, not merely contingently, justified (McNaughton and Rawling 1998; Gaut 1999, 2002; Montague 2000; Suikkanen 2008). A sibling of this objection is that rule-consequentialism makes the justification of rules depend on the wrong facts (Arneson 2005; Portmore 2009; cf. Woollard 2015: esp. pp. 185-86, 203-205; 2022). The mechanics of teaching new codes throws up serious questions for forms of rule-consequentialism that count the costs of getting rules internalised by new generations. As explained earlier, limiting the targets of the teaching to new generations is meant to avoid having to count the costs of getting rules internalised by existing generations of people who have already internalised some other moral rules and ideas. But can we come up with a coherent description of those who are supposed to do the teaching of these new generations? If the teachers are imagined to have already internalised the ideal code themselves, then how is that supposed to have happened? If these teachers are imagined not to have already internalised the ideal code, then there will be costs associated with the conflict between the ideal code and whatever they have already internalised. (This objection was formulated by John Andrews, Robert Ehman, and Andrew Moore. Cf. Levy 2000; D. Miller, 2021.) A related objection is that rule-consequentialism has not yet been formulated in a way that enables it to deal plausibly with conflicts among rules (Eggleston 2007). But see Copp 2020; D. Miller 2021; Woodard 2022.
vagueness	## 1. Inquiry Resistance If you cut one head off of a two-headed man, have you decapitated him? What is the maximum height of a short man? When does a fertilized egg develop into a person? These questions are impossible to answer because they involve absolute borderline cases. In the vast majority of cases, the unknowability of a borderline statement is only relative to a given means of settling the issue (Sorensen 2001, chapter 1). For instance, a boy may count as a borderline case of 'obese' because people cannot tell whether he is obese just by looking at him. His curious mother could try to settle the matter by calculating her son's body mass index. The formula is to divide his weight (in kilograms) by the square of his height (in meters). If the value exceeds 30, this test counts him as obese. The calculation will itself leave some borderline cases. The mother could then use a weight-for-height chart. These charts are not entirely decisive because they do not reflect the ratio of fat to muscle, whether the child has large bones, and so on. The boy will only count as an absolute borderline case of 'obese' if no possible method of inquiry could settle whether he is obese. When we reach this stage, we start to suspect that our uncertainty is due to the concept of obesity rather than to our limited means of testing for obesity. Absolute borderline cases are first officially targeted by Charles Sanders Peirce's entry for 'vague' in the 1902 Dictionary of Philosophy and Psychology: > > A proposition is vague when there are possible states of things > concerning which it is intrinsically uncertain whether, had > they been contemplated by the speaker, he would have regarded them as > excluded or allowed by the proposition. By intrinsically uncertain we > mean not uncertain in consequence of any ignorance of the interpreter, > but because the speaker's habits of language were indeterminate. > (Peirce 1902, 748) > Peirce connects intrinsic uncertainty with the sorites paradox: "vagueness is an indeterminacy in the applications of an idea, as to how many grains of sand are required to make a heap, and the like" (1892, 167). In the case of relative borderline cases, the question is determinate but our means for answering it are incomplete. For absolute borderline cases, there is incompleteness in the question itself. When a term is applied to one of its absolute borderline cases the result is a statement that resists all attempts to settle whether it is true or false (Hu 2021). For instance, no amount of conceptual analysis or empirical inquiry can establish the minimum number of grains needed for a heap. Equally futile is inquiry into the qualitative vagueness illustrated by the decapitation riddle. An inventive speaker could give the appearance of settling the matter by stipulating that 'decapitate' means 'remove a head' (as opposed to 'make headless' or 'remove the head'). But that semantic revision would change the topic to an issue that merely sounds the same as decapitation. The switch in meaning might not be detected if the inventive speaker was misperceived as merely applying antecedent knowledge of decapitation. We should expect absolute borderline cases to arise for unfamiliar cases. Where there is no perceived need for a decision, criteria are left undeveloped. Since philosophical thought experiments often involve unfamiliar cases, there is apt to be intrinsic uncertainty. W. V. O. Quine was especially suspicious of the far-out hypotheticals popular in the literature on personal identity. He globalizes his doubts about determinate answers in his thesis of radical translation: for any language, there are indefinitely many, equally adequate rival translation manuals. The indeterminacy of translation applies to one's own language. Introspection will not settle whether '$x$ is the same person as $y$' means '$x$ has the same body as $y$' or '$x$ has same memories as $y$'. You have never encountered a case in which these two possible rules of usage diverge. John Locke imagines a prince and cobbler who awake with each other's memories. Is the prince now in the cobbler's bed or is the prince in his royal bed with errant memories? Quine sees a deflating resemblance with the decapitation riddle. Either of the incompatible answers fits past usage. Yet Locke answers that the prince and cobbler have switched bodies. Materialists counter that the prince and cobbler have instead switched psychologies. Locke and his adversary should have instead shrugged their shoulders. The Cambridge University philosophers Bertrand Russell (1923) and Frank Ramsey were heavily influenced by American pragmatism (Misak 2016). Their reading of Peirce made semantic indeterminacy grow into an international pre-occupation among analytic philosophers. Although vagueness seems holistic, Peirce's definition is reductive. 'Tall' is vague by virtue of there being cases in which there is no fact of the matter. A man who is 1.8 meters in height is neither clearly tall nor clearly non-tall. No amount of conceptual analysis or empirical investigation can settle whether a 1.8 meter man is tall. Borderline cases are inquiry resistant. Inquiry resistance recurs. For in addition to the unclarity of the borderline case, there is normally unclarity as to where the unclarity begins. Twilight governs times that are borderline between day and night. But we are equally uncertain as to when twilight begins. This suggests there are borderline cases of borderline cases of 'day'. Consequently, 'borderline case' has borderline cases. This higher-order vagueness seems to show that 'vague' is vague (Hu 2017). We are slow to recognize higher-order vagueness because we are under a norm to assert only what we know. When we discuss cases of twilight, we focus on definite cases of this borderline between day and night, not the marginal cases between definite day and definite twilight. The vagueness of 'vague' would have two destructive consequences. First, Gottlob Frege could no longer coherently characterize vague predicates as incoherent. For this attribution of incoherence uses 'vague'. Frege's ideal of precision is itself vague because 'precise' is the complement of 'vague'. Second, the vagueness of 'vague' dooms efforts to avoid a sharp line between true and false with a buffer zone that is neither true nor false. If the line is not drawn between the true and the false, then it will be between the true and the intermediate state. Any finite number of intermediates just delays the inevitable. Several philosophers repudiate higher-order vagueness as an illusion (Wright 2021, chapter 5). They deny that there is an open-ended iteration of borderline status. They find it telling that speakers do not go around talking about borderline borderline cases and borderline borderline borderline cases and so forth (Raffman 2005, 23). Defenders of higher-order vagueness reply that ordinary speakers avoid iterating 'borderline' for the same reason they avoid iterating 'million' or 'know'. The iterations are confusing but perfectly meaningful. 'Borderline' behaves just like a vague predicate. For instance, 'borderline' can be embedded in a sorites argument. Defenders of higher-order vagueness have also tried to clinch the case with particular-specimens such as borderline hermaphrodites (reasoning that these individuals are borderline borderline males) (Sorensen 2010). Second thoughts about higher-order vagueness are difficult to sustain unless one also rethinks first-order vagueness. Perhaps Peirce overreacted to the threat of futile inquiry! Where does the tail of a snake begin? When posed as a rhetorical question, the speaker intimates that there is no definite answer. But the tail can be located by tracing down from the snake's rib cage. A false attribution of indeterminacy will lead to the premature abandonment of inquiry. Is heat the absence of cold or cold the absence of heat? Many physicists ceased to inquire after Ernst Mach ridiculed the ancient riddle as indeterminate. Israel Scheffler (1979, 77-78) draws a moral from such hasty desertions. Never give up! In an internal criticism of his colleague Quine, Scheffler condemns any attribution of semantic indeterminacy as a defeatist relic of the distinction between analytic and synthetic statements. Appeals to meaning never conclude inquiry or preclude inquiry. No one pursues the many lines of inquiry Scheffler refuses to close. Yet some agree that the attribution of borderline status is never mandatory. Crispin Wright (2021) concludes there are no definite borderline cases. Diana Raffman (2014, 153) offers some empirical support for borderline status always being optional. A minority of fluent speakers will draw a sharp line between yellow and orange. Mandatory status is defended by listing paradigm cases of 'borderline'. Just as a language teacher tests for competence by having students distinguish clear positives from clear negatives, she also tests by their recognition of borderline cases. Any pupil who uses '-ish' without recognizing borderline cases of 'noonish' has not yet mastered the suffix. 'Noonish' is not a synonym of 'within ten minutes of noon' or any other term with a precisely delimited interval. ## 2. Comparison with Ambiguity and Generality 'Tall' is relative. A 1.8 meter pygmy is tall for a pygmy but a 1.8 meter Masai is not tall for a Masai. Although relativization disambiguates, it does not eliminate borderline cases. There are shorter pygmies who are borderline tall for a pygmy and taller Masai who are borderline tall for a Masai. The direct bearers of vagueness are a word's full disambiguations such as 'tall for an eighteenth-century French man'. Words are only vague indirectly, by virtue of having a sense that is vague. In contrast, an ambiguous word bears its ambiguity directly--simply in virtue of having multiple meanings. This contrast between vagueness and ambiguity is obscured by the fact that most words are both vague and ambiguous. 'Child' is ambiguous between 'offspring' and 'immature offspring'. The latter reading of 'child' is vague because there are borderline cases of immature offspring. The contrast is further complicated by the fact that most words are also general. For instance, 'child' covers both boys and girls. Ambiguity and vagueness also contrast with respect to the speaker's discretion. If a word is ambiguous, the speaker can resolve the ambiguity without departing from literal usage. For instance, he can declare that he meant 'child' to express the concept of an immature offspring. If a word is vague, the speaker cannot resolve the borderline case. For instance, the speaker cannot make 'child' literally mean anyone under eighteen just by intending it. That concept is not, as it were, on the menu corresponding to 'child'. He would be understood as taking a special liberty with the term to suit a special purpose. Acknowledging departure from ordinary usage would relieve him of the obligation to defend the sharp cut-off. When the movie director Alfred Hitchcock mused 'All actors are children' he was taking liberties with clear negative cases of 'child' rather than its borderline cases. The aptness of his generalization is not judged by its literal truth-value (because it is obviously untrue). Likewise, we do not judge precisifications of borderline cases by their truth-values (because they are obviously not ascertainable as true or false). We instead judge precisifications by their simplicity, conservativeness, and fruitfulness. A precisification that draws the line across the borderline cases conserves more paradigm usage than one that draws the line across clear cases. But conservatism is just one desideratum among many. Sometimes the best balance is achieved at the cost of turning former positive cases into negative cases. Once we shift from literal to figurative usage, we gain fictive control over our entire vocabulary--not just vague words. When a travel agent says 'France is a hexagon', we do not infer that she has committed the geometrical error of classifying France as a six-sided polygon. We instead interpret the travel agent as speaking figuratively, as meaning that France is shaped like a hexagon. Similarly, when the travel agent says 'Reno is the biggest little city', we do not interpret her as overlooking the vagueness of 'little city'. Just as she uses the obvious falsehood of 'France is a hexagon' to signal a metaphor, she uses the obvious indeterminacy of 'Reno is the biggest little city' to signal hyperbole. Given that speakers lack any literal discretion over vague terms, we ought not to chide them for indecisiveness. Where there is no decision to be made, there is no scope for vice. Speakers would have literal discretion if statements applying a predicate to its borderline cases were just permissible variations in linguistic usage. For the sake of comparison, consider discretion between alternative spellings. Professor Letterman uses 'judgment' instead of 'judgement' because he wants to promote the principle that a silent E signals a long vowel. He still has fond memories of Tom Lehrer's 1971 children's song "Silent E": > > Who can turn a can into a cane? > > > Who can turn a pan into a pane? > > > It's not too hard to see, > > > It's Silent E. > > > > > Who can turn a cub into a cube? > > > Who can turn a tub into a tube? > > > It's elementary > > > For Silent E. > Professor Letterman disapproves of those who add the misleading E but concedes that 'judgement' is a permissible spelling; he does not penalize his students for misspelling when they make their hard-hearted choice of 'judgement'. Indeed, Professor Letterman scolds students if they fail to stick with the same spelling throughout the composition. Choose but stick to your choice! Professor Letterman's assertion 'The word for my favorite mental act is spelled j-u-d-g-m-e-n-t' is robust with respect to the news that it is also spelled j-u-d-g-e-m-e-n-t. He would continue to assert it. He can conjoin the original assertion with information about the alternative: 'The word for my favorite mental act is spelled j-u-d-g-m-e-n-t and is also spelled j-u-d-g-e-m-e-n-t'. In contrast, Professor Letterman's assertion that 'Martha is a woman' is not robust with respect to the news that Martha is a borderline case of 'woman' (say, Letterman learns Martha is younger than she looks). The new information would lead Letterman to retract his assertion in favor of a hedged remark such as 'Martha might be a woman and Martha might not be a woman'. Professor Letterman's loss of confidence is hard to explain if the information about her borderline status were simply news of a different but permissible way of describing her. Discoveries of notational variants do not warrant changes in former beliefs. News of borderline status has an evidential character. Loss of clarity brings loss of warrant. If you do not lower your confidence, you are open to the charge of dogmatism. To concede that your ferry is a borderline case of 'seaworthy' is to concede that you do not know that your ship is seaworthy. That is why debates can be dissolved by showing that the dispute is over an absolute borderline case. The debaters should both become agnostic. After all, they do not have a license to form beliefs beyond their evidence. If your ferry is borderline seaworthy, you are not free to believe it is seaworthy. According to W. K. Clifford, belief beyond the evidence is always morally forbidden, not just irrational. William James countered that belief beyond the evidence is permissible when there is a forced, momentous, live option. Can I survive the destruction of my body, simply by virtue of a strong psychological resemblance with a future person? The answer affects my prospects for immortality. James says that my belief that this is borderline survival does not forbid me from believing that it is survival. Whether I exercise this prerogative depends on my temperament. Since momentous borderline cases are so prevalent in speculative matters, much philosophical disagreement is a matter of personality differences. The Clifford drop into agnosticism is especially quick for those who believe that evidence supports a unique judgment. A detective who thinks the evidence shows Gray is a murderer cannot concede that the evidence equally permits the judgment that Gray is not a murderer. If the detective concedes that Gray's act is borderline murder, then he cannot shed the hedge and believe the act is murder. In the case of relative borderline cases, the hedge can be shed by the discovery of a novel species of evidence. Murder inquiries were sometimes inconclusive because the suspect had an identical twin. After detectives learned that identical twins have distinct fingerprints, some of these borderline cases became decidable. Presently, there are some borderline cases arising from the extreme similarity in DNA between identical twins. Soon, geneticists will be able to reliably discern the slight differences between twin DNA that arise from mutations. News of an alternative sense is like news of an alternative spelling; there is no evidential impact (except for meta-linguistic beliefs about the nature of words). Your assertion that 'All bachelors are men' is robust with respect to the news that 'bachelor' has an alternative sense in which it means a male seal. Assertions are not robust with respect to news of hidden generality. If a South African girl says 'No elephant can be domesticated' but learns there is another species of elephant indigenous to Asia, she will lose some confidence; maybe Asian elephants can be domesticated. News of hidden generality has evidential impact. When it comes to robustness, vagueness resembles generality more than vagueness resembles ambiguity. Mathematical terms such as 'prime number' show that a term can be general without being vague. A term can also be vague without being general. Borderline cases of analytically empty predicates illustrate this possibility. Generality is obviously useful. Often, lessons about a particular $F$ can be projected to other $F$s in virtue of their common $F$-ness. When a girl learns that her cat has a nictitating membrane that protects its eyes, she rightly expects that her neighbor's cat also has a nictitating membrane. Generality saves labor. When the girl says that she wants a toy rather than clothes, she narrows the range of acceptable gifts without going through the trouble of specifying a particular gift. The girl also balances values: a gift should be intrinsically desired and yet also be a surprise. If uncertain about which channel is the weather channel, she can hedge by describing the channel as 'forty-something'. There is an inverse relationship between the contentfulness of a proposition and its probability: the more specific a claim, the less likely it is to be true. By gauging generality, we can make sensible trade-offs between truth and detail. 'Vague' has a sense that is synonymous with abnormal generality. This precipitates many equivocal explanations of vagueness. For instance, many commentators say that vagueness exists because broad categories ease the task of classification. If I can describe your sweater as red, then I do not need to ascertain whether it is scarlet. This freedom to use wide intervals obviously helps us to learn, teach, communicate, and remember. But so what? The problem is to explain the existence of borderline cases. Are they present because vagueness serves a function? Or are borderline cases side-effects of ordinary conversation--like the echoes indoor listeners learn to ignore? Every natural language is both vague and ambiguous. However, both features seem eliminable. Indeed, both are eliminated in miniature languages such as checkers notation, computer programming languages, and mathematical descriptions. Moreover, it seems that both vagueness and ambiguity ought to be minimized. 'Vague' and 'ambiguous' are pejorative terms. And they deserve their bad reputations. Think of all the automotive misery that has been prefaced by > > Driver: Do I turn left? > > > Passenger: Right. > English can be lethal. Philosophers have long motivated appeals for an ideal language by pointing out how ambiguity creates the menace of equivocation: > > No child should work. > > > Every person is a child of someone. > > > Therefore, no one should work. > Happily, we know how to criticize and correct all equivocations. Indeed, every natural language is self-disambiguating in the sense that each has all the resources needed to uniquely specify any reading one desires. ## 3. The Philosophical Challenge Posed by Vagueness Vagueness, in contrast, precipitates a profound problem: the sorites paradox. For instance, * Base step: A one-day-old human being is a child. * Induction step: If an $n$-day-old human being is a child, then that human being is also a child when it is $n + 1$ days old. * Conclusion: Therefore, a 36,500-day-old human being is a child. The conclusion is false because a 100-year-old man is clearly a non-child. Since the base step of the argument is also plainly true and the argument is valid by mathematical induction, we seem to have no choice but to reject the second premise. George Boolos (1991) observes that we have an autonomous case against the induction step. In addition to implying plausible conditionals such as "If a 1-day-old human being is a child, then that human being is also a child when it is 2 days old", the induction step also implies ludicrous conditionals such as "If a 1-day-old human being is a child, then that human being is also a child when 36,500 days old". Boolos is puzzled why we overlook these clear counterexamples. One explanation is that we tend to treat the induction step as a generic generalization such as "People have ten toes" (Sorensen 2012) rather than a generalization with a quantifier such as 'all' or 'most'. Whereas the universal generalization "All people have ten toes" is refuted by a single person with eleven toes, the generic generalization tolerates exceptions. This hypothesis is plausible for newcomers to the sorites paradox. But it is less plausible for those being tutored by Professor Boolos. He guides logic students to the correct, universal reading of the induction step. When students drift to a generic reading, Boolos reminds them that the induction step is a generalization that tolerates no exceptions. Guided by Boolos' firm hand, logic students drive a second stake into the heart of the induction step (the entailment of preposterous conditionals). Yet the paradox seems far from dead. The hitch is that death for the induction step is life for its negation. The negation of the second premise classically implies a sharp threshold for childhood. For the falsehood of the induction step implies the existential generalization that there is a number $n$ such that an $n$-day-old human being is a child but is no longer a child one day later. Epistemicists accept this astonishing consequence. They stand by classical logic and conclude that vagueness is a form of ignorance. For any day in the borderline region of 'child', there is a small probability that it is the day one stopped being a child. Normally, the probability is negligible. But if we round down to zero, we fall into into inconsistency. This diagnosis is reminiscent of Carl Hempel's solution to the Raven Paradox. At first blush, we deny that a white handkerchief could confer any degree of confirmation on 'All ravens are black'. But this is incompatible with us regarding a white handkerchief as providing some confirmation of 'All non-black things are non-ravens'. Since this contrapositive is logically equivalent to 'All ravens are black', whatever confirms one hypothesis confirms the other. Hempel advises us to regain consistency by conceding that a white handkerchief slightly confirms 'All ravens are black'. Similarly, Hempel might advise us to slightly raise the probability of a borderline child being a child upon learning that she is a day younger than previously believed. Hempel's only reservation about indoor ornithology is inefficiency. Given how the actual world is arranged, more news about 'All ravens are black' can be gained outdoors. We do not bother to look indoors because the return on inquiry is too low. Bottom-up Bayesians envisage extraordinary scenarios in which the return on inquiry could be high. However, even their ample imagination fails to yield a possible world in which a single day has a more discernible opportunity to make a difference to whether someone is a child. All pressure to shift credence is exerted top down from probability theory. Timothy Williamson (1994) traces our ignorance of the threshold for childhood to "margin for error" principles. If one knows that an $n$-day-old human being is a child, then that human being must also be a child when $n + 1$ days old. Otherwise, one is right by luck. Given that there is a threshold, we would be ignorant of its location. Under a use theory of meaning, the threshold depends on many speakers. Williamson characterizes this threshold as sharing the unpredictability of long-term weather forecasts. According to the meteorologist Edward Lorenz, a tornado in Kansas could be precipitated by the flapping of a butterfly's wings in Tokyo. The use theory of meaning does not apply to asocial languages. For instance, psycholinguists postulate a language of thought that is not used for communication. This innate language is vague. Natural languages lexicalize only a small share of our concepts. The primary bearers of vagueness are concepts rather than words (Bacon 2018). Aphasia does not end vague thinking. Insofar as we attribute thinking to non-linguistic animals, we attribute vague thinking. Peirce's emphasis on a community of inquirers encourages a sociology of language. Ludwig Wittgenstein's attack on the possibility of a private language was intended to remove any rival to a community-based approach to language. Yet asocial vagueness remains viable. This motivates some epistemicists to explain our ignorance more metaphysically, say, as an effect of there being no truth-maker to control the truth-value of a borderline statement (Sorensen 2001, chapter 11). Debate over Williamson's margin for error principle draws us deep into epistemology and modality (Yli-Vakkuri 2016). Preferring to explore the shallows, some commentators survey attitudes weaker than knowledge. According to Nicholas Smith (2008, 182) we cannot even guess that the threshold for baldness is the 400th hair. Hartry Field (2010, 203) denies that a rational man can fear that he has just passed the threshold into being old. Hope, speculation, and wonder do not require evidence but they do require understanding. So it is revealing that these attitudes have trouble getting a purchase on the threshold of oldness (or any other vague predicate). A simple explanation is that bare linguistic competence gives us knowledge that there are no such thresholds. This accounts for the comical air of the epistemicist. Just as there is no conceptual room to worry that there is a natural number between sixty and sixty one, there is no conceptual room to worry that one has passed the threshold of oldness between one's sixtieth and sixty-first birthday. An old epistemicist might reply: My piecemeal confidence that a given number is not the threshold for oldness does not agglomerate into collective confidence that there is no such number. If I wager against each number being the threshold, then I must have placed a losing bet somewhere. For if I won each bet then there was no opportunity for me to make the transition to oldness. My bookie could have made a "Dutch book" against me. He would have been entitled to payment without having to identify which bet I lost. Since probabilities may be extracted from hypothetical betting behavior, I must actually assign some small (normally negligible) probability to hypotheses identifying particular thresholds. So must you. Stephen Schiffer (2003, 204) denies that classical probability calculations apply in vague contexts. Suppose Donald is borderline old and borderline bald. According to Schiffer we should be just as confident in the conjunction 'Donald is old and bald' as in either conjunct. Adding conjuncts does not reduce confidence because we have a "vague partial belief" rather than the standard belief assumed by mathematicians developing probability theory. Schiffer offers a calculus for this vagueness-related propositional attitude. He crafts the rules for vague partial belief to provide a psychological solution to the sorites paradox. The project is complicated by the fact that vague partial beliefs interact with precise beliefs (MacFarlane 2010). Consider a statement that has a mixture of vague and precise conjuncts: 'Donald is old and bald and has an even number of hairs'. Adding the extra precise conjunct should diminish confidence. Schiffer also needs to accommodate the fact that some speakers are undecided about whether the nature of the uncertainty involves vagueness. Even an idealized speaker may be unsure because there is vagueness about the borders between vagueness-related uncertainty and other sorts of uncertainty. Attitudes toward determinacy vary over time. Bernard Williams (2010, 162) says Thucydides (460-400 BC) invented historical time. Thucydides insists that any two events occurred simultaneously or one occurred before the other. Thucydides does not countenance a vague "olden days"--a period in which events fail to be chronologically ordered. Every event is situated on a single timeline. There can be ignorance of when an event took place but there is always a fact of the matter. With similar focus, Archimedes (287-212 BC) insists that every quantity corresponds to a specific number. No quantity, however large, is innumerable. In "The Sand Reckoner" Archimedes devises a special notation to express the total number of grains of sand. The reductionism of Enlightenment thinkers led them to further analysis of shades of gray into black and white dots of precision. In a holistic reaction, the Romantics portray the new boundaries as no more real than the grid lines of a microscope. The specimen under the slide lacks boundaries. Understanding does not require boundaries. Organizing items along a spectrum is a natural form of classification. We comprehend this analog grouping more readily and smoothly than we grasp the digital groupings instilled by Descartes' algebraic representation of space. If our metaphysics of classification is constricted by a logic designed for mathematics, the sorites paradox will make us see disorder where there is really order of a more fluid sort. Spontaneous arrangements will bring what order is permitted by the subject matter. Romantic biologists take to the field and refuse to pen species within the axes of the Cartesian coordinate system. Confronted with novel phenomena such as evolution, magnetism and electricity, Romantic scientists bypassed boundaries. A geologist views a pair of aerial photographs cross-eyed. His brain recognizes the two images as representations of the same landscape. The flat images fuse into a unifying stereoscopic perspective. The Romantics portray vague language as perceptively de-focussed. Vagueness is not the cloudy vision of an old man with cataracts. The epistemicist fits into the Thucydidean tradition of replacing indeterminacy with determinacy. Crispin Wright (2021, 393) compares the epistemicist's postulation of hidden boundaries to the mathematical realist's postulation of a hidden Platonic realm. Both the epistemicist and the mathematical realist add metaphysical infrastructure to ensure truth will transcend our means of proof. This infrastructure validates proofs of generalizations that are not constructed from examples. The impossibility of finding a witness for an existential generalization, such as 'There is a youngest old man', is compatible with there being a non-constructive proof. Despite rearguard defenses by the Romantics, Newtonian physics appeared to show that empirical reality fits classical logic. The hard sciences illustrated the crystalline growth of determinacy. But in a shocking reversal, quantum mechanics suggests the possibility of reversing from determinacy to indeterminacy. What had been formerly regarded as determinate, the position and velocity of an object, was regarded as indeterminate by Werner Heisenberg. Quantum logicians, such as Hilary Putnam (1976), abandoned the distributive laws of propositional logic. Putnam (1983) went on propose a solution that seems to apply L. E. J. Brouwer's intuitionism to the sorites paradox. (In response to a rebuttal, Putnam [1985] surprisingly denied that intuitionistic semantics was part of the proposal.) Brouwer had achieved fame with his fixed point theorem. But reflection on Kant's philosophy of mathematics led Brouwer to retract his lovely proof. Any non-constructive proof had to be replaced! Brouwer's recall of established theorems yielded more specific proofs that had higher explanatory power. Theorems that could not be retrofitted became objects of agnosticism. This was sad news for Brouwer's fixed point theorem. But Putnam welcomes the recall of the epistemicist's theorem that there is a smallest natural number. The classical logician relies on double negation to deduce a hidden boundary from the negation of the generalization "If $n$ is small, then $n + 1$ is small". Since the intuitionist does not accept this classical rule, his refusal to acquiesce to the induction step does not saddle him with hidden thresholds. Like the epistemicist, the intuitionist treats vagueness as a cognitive problem that does not require rejecting the law of bivalence. Unlike the epistemicist, the intuitionist must not claim to know the law of bivalence. To sustain this agnosticism about bivalence, the intuitionist must parry the sorites monger's reformulations (Wright 2021, chapter 14). For instance, the intuitionist's least number principle threatens to re-ignite the sorites. Many commentators concede to the epistemicist that it is logically possible for vague predicates to have thresholds. They just think it would be a miracle: > > It is logically possible that the words on this page will come to life > and sort my socks. But I know enough about words to dismiss this as a > serious possibility. So I am right to boggle at the possibility that > our rough and ready terms such as 'red' could so > sensitively classify objects. > Epistemicists counter that this bafflement rests on an over-estimate of stipulation's role in meaning. Epistemicists say much meaning is acquired passively by default rather than actively by decision. If some boundaries are more eligible for reference than others, then the environment does the work. If nothing steps in to make a proposition true, then it is false. Or so opines Timothy Williamson. Most doubt whether precise analytical tools fit vague arguments. The scientific Romantic H. G. Wells was amongst the first to suggest that we must moderate the application of logic: > > Every species is vague, every term goes cloudy at its edges, and so in > my way of thinking, relentless logic is only another name for > stupidity--for a sort of intellectual pigheadedness. If you push > a philosophical or metaphysical enquiry through a series of valid > syllogisms--never committing any generally recognised > fallacy--you nevertheless leave behind you at each step a certain > rubbing and marginal loss of objective truth and you get deflections > that are difficult to trace at each phase in the process. Every > species waggles about in its definition, every tool is a little loose > in its handle, every scale has its individual error. (1908, 18) > Many more believe that the problem is with logic itself rather than the manner in which it is applied. They favor solving the sorites paradox by replacing standard logic with an earthier deviant logic. There is a desperately wide range of opinions as to how the revision of logic should be executed. Every form of deviant logic has been applied in the hope of resolving the sorites paradox. ## 4. Many-valued Logic An early favorite was many-valued logic. On this approach, borderline statements are assigned truth-values that lie between full truth and full falsehood. Some logicians favor three truth-values, others prefer four or five. The most popular approach is to use an infinite number of truth-values represented by the real numbers between 0 (for full falsehood) and 1 (for full truth). This infinite spectrum of truth-values might be of service for a continuous sorites argument involving 'small real number' (Weber and Colyvan 2010). Classical logic does not enforce a sharp boundary for infinite domains. But it still enforces unlimited sensitivity for vague predicates. If two quantities $A$ and $B$ are growing but $A$ is given a headstart, then $A$ will cease to equal a small real number before $B$. Since this transitional difference holds regardless of the lead's magnitude, "$A$ equals a small real number" passes from true to false within an arbitrarily narrow interval. If the number of truth-values is instead as numerous as the real numbers, there is no longer any mismatch between the tiny size of the transition and the large difference in truth-value. There is a new mismatch, however. The number of truth-values now dwarfs the number of truth-value bearers. Since a natural language can express any proposition, the number of propositions equals the number of sentences. The number of sentences is $\aleph\_0$ while (assuming the continuum hypothesis) the number of real numbers is at least $\aleph\_1$. Consequently, there are infinitely many truth-values without bearers. This new mismatch seems more bearable than the old mismatch it is designed to avoid. Yet some find even a lower infinity of truth-values over-bearing. Instead of having just one artificially sharp line between the true and the false, the many-valued logician has infinitely many sharp lines such as that between statements with a truth of .323483925 and those with a higher truth-value. Mark Sainsbury grimaces, '... you do not improve a bad idea by iterating it' (1996, 255). A proponent of an infinite-valued logic might cheer up Sainsbury with an analogy. It is a bad idea to model a circle with a straight line. Using two lines is not much better, nor is there is much improvement using a three-sided polygon (a triangle). But as we add more straight lines to the polygon (square, pentagon, hexagon, and so on) we make progress--by iterating the bad idea of modeling a circle with straight lines. Indeed, it would be tempting to triumphantly conclude "The circle has been modeled as an infinite-sided polygon". But has the circle been revealed to be an infinite-sided polygon? Have curved lines been replaced by straight lines? Have curved lines (and hence circles) been proven to not exist? A model can succeed without being clear what has been achieved. But it is premature to dwell on the analogy "Precision is to vagueness as straightness is to curvature". The many-valued logician must first vindicate the comparison by providing details about how to calculate the truth-values of vague statements from the truth-values of their component statements. Proponents of many-valued logic approach this obligation with great industry. Precise new rules are introduced to calculate the truth-values of compound statements that contain statements with intermediate truth-values. For instance, the revised rule for conjunctions assigns the conjunction the same truth-value as the conjunct with the lowest truth-value. These rules are designed to yield all standard theorems when all the truth-values are 1 and 0. In this sense, classical logic is a limiting case of many-valued logic. Classical logic is agreed to work fine in the area for which it was designed--mathematics. Most theorems of standard logic break down when intermediate truth-values are involved. (An irregular minority, such as 'If $P$, then $P$', survive.) Even the classical contradiction 'Donald is bald and it is not the case that he is bald' receives a truth-value of .5 when 'Donald is bald' has a truth-value of .5. Many-valued logicians note that the error they are imputing to classical logic is often so small that classical logic can still be fruitfully applied. But they insist that the sorites paradox illustrates how tiny errors accumulate into a big error. Critics of the many-valued approach complain that it botches phenomena such as hedging. If I regard you as a borderline case of 'tall man', I cannot sincerely assert that you are tall and I cannot sincerely assert that you are of average height. But I can assert the hedged claim 'Either you are tall or of average height'. The many-valued rule for disjunction is to assign the whole statement the truth-value of its highest disjunct. Normally, the added disjunct in a hedged claim is not more plausible than the other disjuncts. Thus it cannot increase the degree of truth. Disappointingly, the proponent of many-valued logic cannot trace the increase of assertibility to an increase in the degree of truth. Epistemicists ascribe the increase in assertibility to the increasing probability of truth. Since the addition of disjuncts can raise probability indefinitely, the epistemicists correctly predict that we can hedge our way to full assertibility. However, epistemicists do not have a monopoly on this prediction. ## 5. Supervaluationism According to supervaluationists, borderline statements lack a truth-value. This neatly explains why it is universally impossible to know the truth-value of a borderline statement. Supervaluationists offer details about the nature of absolute borderline cases. Simple sentences about borderline cases lack a truth-value. Compounds of these statements can have a truth-value if they come out true regardless of how the statement is admissibly precisified. For instance, 'Either Mr. Stoop is tall or it is not the case that Mr. Stoop is tall' is true because it comes out true under all ways of sharpening 'tall'. Thus the method of supervaluations allows one to retain all the theorems of standard logic while admitting "truth-value gaps". One may wonder whether this striking result is a genuine convergence with standard logic. Is the supervaluationist characterizing vague statements as propositions? Or is he merely pointing out that certain non-propositions have a structure isomorphic to logical theorems? (Some electrical circuits are isomorphic to tautologies but this does not make the circuits tautologies.) Kit Fine (1975, 282), and especially David Lewis (1982), characterize vagueness as hyper-ambiguity. Instead of there being one vague concept, there are many precise concepts that closely resemble each other. 'Child' can mean a human being at most one day old or mean a human being at most two days old or mean a human being at most three days old .... Thus the logic of vagueness is a logic for equivocators. Lewis' idea is that ambiguous statements are true when they come out true under all disambiguations. But logicians normally require that a statement be disambiguated before logic is applied. The mere fact that an ambiguous statement comes out true under all its disambiguations does not show that the statement itself is true. Sentences which are actually disambiguated may have truth-values. But the best that can be said of those that merely could be disambiguated is that they would have had a truth-value had they been disambiguated (Tye 1989). Supervaluationism will converge with classical logic only if each word of the supervaluated sentence is uniformly interpreted. For instance, 'Either a carbon copy of Teddy Roosevelt's signature is an autograph or it is not the case that a carbon copy of Teddy Roosevelt's signature is an autograph' comes out true only if 'autograph' is interpreted the same way in both disjuncts. Vague sentences resist mixed interpretations. However, mixed interpretations are permissible for ambiguous sentences. As Lewis himself notes in a criticism of relevance logic, 'Scrooge walked along the bank on his way to the bank' can receive a mixed disambiguation. When exterminators offer 'non-toxic ant poison', we charitably switch relativizations within the noun phrase: the substance is safe for human beings but deadly for ants. Even if one agrees that supervaluationism converges with classical logic about theoremhood, they clearly differ in other respects. Supervaluationism requires rejection of inference rules such as contraposition, conditional proof and reductio ad absurdum (Williamson 1994, 151-152). In the eyes of the supervaluationist, a demonstration that a statement is not true does not guarantee that the statement is false. The supervaluationist is also under pressure to reject semantic principles which are intimately associated with the application of logical laws. According to Alfred Tarski's Convention T, a statement '$S$' is true if and only if $S$. In other words, truth is disquotational. Supervaluationists say that being supertrue (being true under all precisifications) suffices for being true. But given Convention T, supertruth would then be disquotational. Since the supervaluationists accept the principle of excluded middle, the combination of Convention T and 'P or ~P' being supertrue would force them to say '$P$' is supertrue or '$\neg P$' is supertrue (even if '$P$' applies a predicate to a borderline case). This would imply that either '$P$' is true or '$\neg P$' is true (Williamson 1994, 162-163). And that would be a fatal loss of truth-value gaps for supervaluationism. There is a final concern about the "ontological honesty" of the supervaluationist's existential quantifier. As part of his solution to the sorites paradox, the supervaluationist asserts, "There is a human being who, for some $n$, was a child when $n$ days old but not when $n + 1$ days old." For this statement comes out true under all admissible precisifications of 'child'. However, when pressed the supervaluationist adds a codicil: "Oh, of course I do not mean that there really is a sharp threshold for childhood." After the clarification, some wonder how supervaluationists differs from drastic metaphysical skeptics. In his nihilist days, Peter Unger (1979) admitted that it is useful to talk as if there are children. But he insisted that strictly speaking, vague terms such as 'child' cannot apply to anything. Unger was free to use supervaluationism as a theory to explain our ordinary discourse about children. (Unger instead used other resources to explain how we fruitfully apply empty predicates.) But once the dust had cleared and the precise rubble came into focus, Unger had to conclude that there are no children. Officially, the supervaluationist rejects the induction step of the sorites argument. Unofficially, he seems to instead reject the base step of the sorites argument. Supervaluationists encourage the view that all vagueness is a matter of linguistic indecision: the reason why there are borderline cases is that we have not bothered to make up our minds. The method of supervaluation allows us to assign truth-values prior to any decisions. Expressivists think this is a mistake akin to assigning truth-values to normative claims (MacFarlane 2016). They model vagueness as practical uncertainty as to whether to treat a borderline $F$ as an $F$. The deliberator may accept the tautologies of classical logic as constraints governing competing plans for drawing lines. I can accept "Either Donald is bald or not" without accepting either disjunct. An existentially quantified sentence can be accepted even when no instance is. A shrug of the shoulders signals readiness to go either way, not ignorance as to which possible world one inhabits. The expressivist is poised to explain how supervaluationism developed into the most respected theory of vagueness. Frege portrayed vagueness as negligent under-definition: One haphazardly introduces a necessary condition here, a sufficient condition there, but fails to supply a condition that is both necessary and sufficient. Supervaluationists counter that indecision can be both intentional and functional. Instead of committing ourselves prematurely, we fill in meanings as we go along in light of new information and interests. Psychologists may deny that we could really commit ourselves to the complete precisifications envisaged by the supervaluationist (which encompass an entire language). Many of these comprehensive alternatives are too complex for memory. For instance, any precisification admitting a long band of random verdicts will be algorithmically complex and so not compressible in a rule. Realistic alternatives are only modestly more precise than ordinary usage. Nevertheless, the supervaluationist's conjecture about gradual precisification is popular for the highly stipulative enterprise of promulgating and enforcing laws (Endicott 2000). Judges frequently seem to exercise and control discretion by means of vague language. Uncertainties about the scope of discretion may arise from higher-order vagueness (Schauer 2016). Discretion through gap-filling pleases those who regard adjudication as a creative process. It alarms those who think we should be judged by laws rather than men. The doctrine of discretion through indeterminacy has also been questioned on the grounds that the source of the discretion is the generality of the legal terms rather than their vagueness (Poscher 2012). By David Lanius's (2019) reckoning, the only function for vagueness in law is strategic. Drunk driving is discouraged by the vagueness of 'drunk' in the way it is discouraged by random testing of motorists. Scattershot enforcement becomes a welcome feature of the legal system rather than a bug in the legal programming. Motorists stay sober to avoid participation in a punishment lottery. Judicial variance is just what one expects if the judges are making a forced choice on borderline cases. Judges cannot confess ignorance and so are compelled to assert beyond their evidence. Little wonder that what comes out of their mouth is affected by irrelevancies such as what went in their mouths for lunch. This susceptibility to bias by irrelevant factors (weather, order effects, anchoring) could be eliminated by methodical use of a lottery. The lottery could be weighted to the fact that borderline crimes vary in how close they come to being clear crimes. Hrahn Asgeirsson (2020, chapter 3) admits that the descriptive question 'Is this drunk driving?' cannot be more reliably answered by a judge than anyone else when it is a borderline case. But he thinks the normative question, 'Should this be counted as drunk driving?' could be better answered by those with superior knowledge of legislative intent. ## 6. Subvaluationism Whereas the supervaluationist analyzes borderline cases in terms of truth-value gaps the dialetheist analyzes them in terms of truth-value gluts. A glut is a proposition that is both true and false. The rule for assigning gluts is the mirror image of the rule for assigning gaps: A statement is true exactly if it comes out true on at least one precisification. The statement is false just if it comes out false on at least one precisification. So if the statement comes out true under one precisification and false under another precisification, the statement is both true and false. To avoid triviality, the dialetheist must adopt a paraconsistent logic that stops two contradictory statements from jointly implying everything. The resulting "subvaluationism" is a dual of supervaluationism. The spiritual father of subvaluationism is Georg Hegel. For Hegel, the basic kind of vagueness is conflict vagueness. The shopper in between the wings of a revolving doorway is both inside the building and outside the building. Degree vagueness is just a special case of the conflict inherent in becoming. Any process requires a gradual manifestation of a contradiction inherent in the original state. At some stage, a metamorphosizing caterpillar is not a butterfly (by virtue of what it was) and a butterfly (by virtue of what it will be). Hegelians believed this dialectical conception of vagueness solved the sorites and demonstrated the inadequacy of classical logic (Weber 2010). The Russian Marxist Georgi Plekhanov (1908 [1937]) proposed a logic of contradiction to succeed classical logic (Hyde 2008, 93-5). One of his students, Henry Mehlberg (1958) went on to substitute gaps for gluts. The first version of supervaluationism is thus a synthesis, reconciling the thesis of classical logic with the anti-thesis posed by the logic of contradiction. From a formal point of view, there seems no more reason to prefer one departure from classical logic rather than the other (Cobreros and Tranchini 2019). Since Western philosophers abominate contradiction, parity with dialetheism would diminish the great popularity of supervaluationism. A Machiavellian epistemicist will welcome this battle between the gaps and the gluts. He roots for the weaker side. Although he does not want the subvaluationist to win, the Machiavellian epistemicist does want the subvaluationist to achieve mutual annihilation with his supervaluationist doppelganger. His political calculation is: Gaps + Gluts = Bivalence. Pablo Cobreros (2011) has argued that subvaluationism provides a better treatment of higher-order vagueness than supervaluationism. But for the most part, the subvaluationists (and their frenemies) have merely claimed subvaluationism to be at least as attractive as supervaluationism (Hyde and Colyvan 2008). This modest ambition seems prudent. After all, truth-value gaps have far more independent support from the history of philosophy (at least Western philosophy). Prior to the explosive growth of vagueness research after 1975, ordinary language philosophers amassed a panoramic battery of analyses suggesting that gaps are involved in presupposition, reference failure, fiction, future contingent propositions, performatives, and so on and so on. Supervaluationism rigorously consolidated these appeals to ordinary language. Dialetheists characterize intolerance for contradiction as a shallow phenomenon, restricted to a twentieth-century Western academic milieu (maybe even now being eclipsed by the rise of China). Experimental philosophers have challenged the old appeals to ordinary language with empirical results suggesting that glutty talk is as readily stimulated by borderline cases as gappy talk (Alxatib and Pelletier 2011, Ripley 2011). ## 7. Contextualism Just as contextualism in epistemology runs orthogonal to the familiar divisions amongst epistemologists (foundationalism, reliabilism, coherentism, etc.), there are contextualists of every persuasion amongst vagueness theorists. They develop an analogy between the sorites paradox and indexical sophistries such as: * Base step: The horizon is more than 1 meter away. * Induction step: If the horizon is more than $n$ meters away, then it is more than $n + 1$ meters away. * Conclusion: The horizon is more than a billion meters away. The horizon is where the earth meets the sky and is certainly less than a billion meters away. (The circumference of the earth is only forty million meters.) Yet when you travel toward the horizon to specify the $n$ at which the induction step fails, your trip is as futile as the pursuit of the rainbow. You cannot reach the horizon because it shifts with your location. All contextualists accuse the sorites monger of equivocating. In one sense, the meaning of 'child' is uniform; the context-invariant rule for using the term (its "character") is constant. However, the set of things to which the term applies (its "content") shifts with the context. In this respect, vague words resemble indexical terms such as: 'I', 'you', 'here', 'now', 'today', 'tomorrow'. When a debtor tells his creditor on Monday 'I will pay you back tomorrow' and then repeats the sentence on Tuesday, there is a sense in which he has said the same thing (the character is the same) and a sense in which he has said something different (the content has shifted because 'tomorrow' now picks out Wednesday). According to the contextualists, the rules governing the shifts prohibit us from interpreting any instance of the induction step as having a true antecedent and a false consequent. The very process of trying to refute the induction step changes the context so that the instance will not come out false. Indeed, contextualists typically emphasize that each instance is true. Assent is mandatory. Consequently, direct attacks on the induction step must fail. One is put in mind of Seneca's admonition to his student Nero: "However many you put to death, you will never kill your successor." Hans Kamp (1981), the founder of contextualism, maintained that the extension of vague words orbits the speaker's store of conversational commitments. He notes that the commitment pressures the speaker to adjust past judgments to current judgments. This backward smoothing undermines the search for a sharp transition. The diagnostic potential of this smoothing is first recognized by Diana Raffman (1994). Switching categories (e.g., from blue to green) between two neighboring, highly similar items in a sorites series, without installing a boundary, is enabled by a gestalt switch that occurs between the two judgments. Switching categories comes at a cost, however; so once the classifier has switched to a new category, she will stay in the new category as long as she can. In particular, if she now reverses direction and goes back down the series, she will continue to classify as green some items she previously classified as blue. This pattern of judgments, hysteresis, has the effect of smoothing out the category shift so that the seeming continuity of the series is preserved. Instead of merely inviting psychologists to verify her predictions about hysteresis effects in sorites series, Raffman (2014, 146-156) collaborates with two psychologists to run an experiment that confirms them. Stewart Shapiro integrates Kamp's ideas with Friedrich Waismann's concept of open texture. Shapiro thinks speakers have discretion over borderline cases because they are judgment dependent. They come out true in virtue of the speaker judging them to be true. Given that the audience does not resist, borderline cases of 'child' can be correctly described as children. The audience recognizes that other competent speakers could describe the borderline case differently. As Waismann lyricizes, "Every description stretches, as it were, into a horizon of open possibilities: However far I go, I shall always carry this horizon with me" (1945, 124). American pragmaticism colors Delia Graff Fara's contextualism. Consider dandelion farms. Why would someone grow weeds? The answer is that 'weed' is relative to interests. Dandelions are unwanted by lawn caretakers but are desired by farmers for food and wine. Fara thinks this interest relativity extends to all vague words. For instance, 'child' means a degree of immaturity that is significant to the speaker. Since the interests of the speaker shift, there is an opportunity for a shift in the extension of 'child'. Fara is reluctant to describe herself as a contextualist because the context only has an indirect effect on the extension via the changes it makes to the speaker's interest. How strictly are we to take the comparison between vague words and indexical terms? Scott Soames (2002, 445) answers that all vague words literally are indexical. This straightforward response is open to the objection that the sorites monger could stabilize reference. When the sorites monger relativizes 'horizon' to the northeast corner of the Empire State Building's observation deck, he seems to generate a genuine sorites paradox that exploits the vagueness of 'horizon' (not its indexicality). All natural languages have stabilizing pronouns, ellipsis, and other anaphoric devices. For instance, in 'Jack is tired now and Jill is too', the 'too' forces a uniform reading of 'tired'. Jason Stanley suggests that the sorites monger employ the premise: > > If that1 is a child then that2 is too, and if > that2 is too, then that3 is too, and if > that3 is too, then that4 is too, ... and > then that$\_i$ is too. > Each 'that$\_n$' refers to the $n$th element in a sequence of worsening examples of 'child'. The meaning of 'child' is not shifting because the first occurrence of the term governs all the subsequent clauses (thanks to 'too'). If vague terms were literally indexical, the sorites monger would have a strong reply. If vague terms only resemble indexicals, then the contextualist needs to develop the analogy in a way that circumvents Stanley's counsel to the sorites monger. The contextualist would also need to address a second technique for stabilizing the context. R. M. Sainsbury (2013) advises the sorites monger to present his premises in apparently random order. No pair of successive cases raises an alarm that similar cases are being treated differently. Unless the hearer has extraordinary memory, he will not feel pressured to adjust the context. The contextualist must find enough shiftiness to block every sorites argument. Since vagueness seeps into every syntactic category, critics complain that contextualism exceeds the level of ambiguity countenanced by linguists and psycholinguists. Another concern is that some sorites arguments involve predicates that do not give us an opportunity to equivocate. Consider a sorites with a base step that starts from a number too large for us to think about. Or consider an inductive predicate that is too complex for us to reason with. One example is obtained by iterating 'mother of' a thousand times (Sorensen 2001, 33). This predicate could be embedded in a mind-numbing sorites that would never generate context shifts. Other unthinkable sorites arguments use predicates that can only be grasped by individuals in other possible worlds or by creatures with different types of minds than ours. More fancifully, there could be a vague predicate, such as Saul Kripke's "killer yellow", that instantly kills anyone who wields it. The basic problem is that contextualism is a psychologistic theory of the sorites. Since arguments can exist without being propounded, they float free of attempts to moor them to arguers. ## 8. Global Indeterminacy without Local Indeterminacy? Contextualists base their holism about vagueness on top-down psychology. Holism can also be logical. We encountered this in the supervaluationist's rejection of truth-functionality. Instead of calculating the truth-values of all compound statements bottom-up from the truth-values of their component statements, Kit Fine (1975) assigned truth-values top-down by counting a statement as true if it comes out true under all admissible precisifications of the entire language. Fine's (2020) new holism parachutes in from R. M. Sainsbury's (1996) characterization of vagueness in terms of boundarylessness. A precise predicate such as 'thirty-something' organizes cases into pigeonholes. Positive cases commence at thirty and cease at thirty nine. But paradigms and foils of 'thirty-ish' operate as poles of a magnet. Candidates for 'thirty-ish' resemble iron filings. This force field prevents completion of a forced-march sorites argument (in which we are marched from a definite instance of a predicate to a definite non-instance). Judgment must be suspended somewhere in the series. But it does not follow that there is any specific case such that one must suspend judgment about it. Borderline status has the mobility of a bubble in a level. The precisificationist's mistake is to remove the bubble. Peirce's mistake is to freeze the bubble into intrinsic uncertainty. This is the dual of the error Peirce attributes to Rene Descartes' method of universal doubt. Doubting everything is impossible because doubt only gets leverage against a fulcrum of presuppositions. (Ludwig Wittgenstein's analogy is with the hinges of a door that hold fast in order that the door of doubt can open and close.) These presuppositions are indubitable in the sense that they cannot be challenged within the inquiry. When the skeptic claims to doubt everything, he winds up doubting nothing. The skeptic's sham doubts are belied by the absence of any interruption of his habits. A genuine doubt triggers inquiry. Inquiry requires presupposition. These background beliefs must be taken for granted. But the necessary existence of certainty does not entail any intrinsic certitudes. What must be presupposed for one inquiry need not be presupposed for another. By Fine's reckoning, Peirce should have issued a parallel rejection of intrinsic uncertitudes. Within any inquiry, there must be some suspension of judgment. But it does not follow that any proposition must resist all inquiry. Consider a literary historiographer who, weary of debate about the identity of the first book, demonstrates that 'book' is vague by assembling a slippery slope ranging from early non-books to paradigm books. According to Peirce, the demonstration succeeds only if the spectrum contains some intrinsically doubtful cases. They are what makes 'book' vague. Fine's negative thesis is that this reductionist explanation relies on impossible explanans. A booklet can no more be an intrinsically borderline book than it can be intrinsically misshelved. Fine's positive thesis is that slippery slopes suffice to demonstrate the vagueness of 'book'. There are borderline books only in the way there are heavy books. In contrast to 'mass', 'heavy' requires relativization to a gravitational field. 'Borderline' requires relativization to a range of cases. Many-valued logicians have long contended that local indeterminacy is incompatible with the law of excluded middle. Fine's (1975) supervaluationism was designed to reconcile the two. But now Fine concedes the incompatibility. Unlike the many-valued logician, Fine blames all the inconsistency on local indeterminacy. Indeed, Fine continues to think the law of excluded middle is undeniable (which is not to say it must be affirmed). What Fine does deny is conjunctive excluded middle: ($B \vee \neg B) \wedge (C \vee \neg C) \wedge (D \vee \neg D) \wedge \ldots .$ For this principle implies that there is a sharp cut-off in a forced-march sorites paradox. Since conjunctive excluded middle is a theorem of classical logic, Fine proposes a compatibility semantics that bears a family resemblance to intuitionism. Fine reconstructs a pedigree by modifying Saul Kripke's semantics for intuitionistic logic. Timothy Williamson (2020) objects that the resulting logic condemns even more innocent inferences than intuitionism. Others defend local indeterminacy by saying that there are paradigm cases of intrinsically uncertain propositions such as those opening section one. Paradigms such as the decapitation riddle concern qualitative vagueness rather than the quantitative vagueness that generates a forced-march sorites argument. Kit Fine is at work on a future book that may address the worry that his account is incomplete or disjunctive. ## 9. Is All Vagueness Linguistic? Supervaluationists emphasize the distinction between words and objects. Objects themselves do not seem to be the sort of thing that can be general, ambiguous, or vague (Eklund 2011). From this perspective, Georg Hegel commits a category mistake when he characterizes clouds as vague. Although we sometimes speak of clouds being ambiguous or even being general to a region, this does not entitle us to infer that there is metaphysical ambiguity or metaphysical generality. Supervaluationists are seconding a fallacy attribution dating back to Bertrand Russell's seminal article "Vagueness" (1923). This consensus was re-affirmed by Michael Dummett (1975) and ritualistically re-avowed by subsequent commentators. In 1978 Gareth Evans focused opposition to vague objects with a short proof modeled after Saul Kripke's attack on contingent identity. If there is a vague object, then some statement of the form '$a = b$' must be vague (where each of the flanking singular terms precisely designates that object). For the vagueness is allegedly due to the object rather than its representation. But any statement of form '$a = a$' is definitely true. Consequently, $a$ has the property of being definitely identical to $a$. Since $a = b$, then $b$ must also have the property of being definitely identical to $a$. Therefore '$a = b$' must be definitely true! Evans agrees that there are vague identity statements in which one of the flanking terms is vague (just as Kripke agrees that there are contingent identity statements when one of the flanking terms is a flaccid designator). But then the vagueness is due to language, not the world. Despite Evans' impressive assault, there was a renewal of interest in vague objects in the 1980s. As a precedent for this revival, Peter van Inwagen (1990, 283) recalls that in the 1960s, there was a consensus that all necessity is linguistic. Most philosophers now take the possibility of essential properties seriously. Some of the reasons are technical. Problems with Kripke's refutation of contingent identity have structural parallels that affect Evans' proof. Evans also relies on inferences that deviant logicians challenge (Parsons 2000). In the absence of a decisive reductio ad absurdum, many logicians feel their role to be the liberal one of articulating the logical space for vague objects. There should be 'vague objects for those who want them' (Cowles and White 1991). Logic should be ontologically neutral. Since epistemicists try to solve the sorites with little more than a resolute application of classical logic, they are methodologically committed to a partisan role for logic. Instead of looking for loopholes, we should accept the consequence (Williamson 2015). Some non-enemies of vague objects also have an ambition to consolidate various species of indeterminacy (Barnes and Williams 2011). Talk of indeterminacy is found in quantum mechanics, analyses of the open future, fictional incompleteness, and the continuum hypothesis. Perhaps vagueness is just one face of indeterminacy. This panoramic vision contrasts with the continuing resolution of many to tether vagueness to the sorites paradox (Eklund 2011). They fear that the clarity achieved by semantic ascent will be muddied by metaphysics. But maybe the mud is already on the mountain top. Trenton Merricks (2001) claims that standard characterizations of linguistic vagueness rely on metaphysical vagueness. If 'Donald is bald' lacks a truth-value because there is no fact to make the statement true, then the shortage appears to be ontological. The view that vagueness is always linguistic has been attacked from other directions. Consider the vagueness of maps (Varzi 2001). The vagueness is pictorial rather than discursive. So one cannot conclude that vagueness is linguistic merely from the premise that vagueness is representational. Or consider vague instrumental music such as Claude Debussy's "The Clouds". Music has syntax but too little semantics to qualify as language. There is a little diffuse reference through devices such as musical quotation, leitmotifs, and homages. These referential devices are not precise. Therefore, some music is vague (Sorensen 2010). The strength and significance of this argument depends on the relationship between music and language. Under the musilanguage hypothesis, language and music branched off from a common "musilanguage" with language specializing in semantics and music specializing in the expression of emotion. This scenario makes it plausible that purely instrumental music could have remnants of semantic meaning. Mental imagery also seems vague. When rising suddenly after a prolonged crouch, I 'see stars before my eyes'. I can tell there are more than ten of these hallucinated lights but I cannot tell how many. Is this indeterminacy in thought to be reduced to indeterminacy in language? Why not vice versa? Language is an outgrowth of human psychology. Thus it seems natural to view language as merely an accessible intermediate bearer of vagueness.
problem-of-many	## 1. Introduction In his (1980), Peter Unger introduced the "Problem of the Many". A similar problem appeared simultaneously in P. T. Geach (1980), but Unger's presentation has been the most influential over recent years. The problem initially looks like a special kind of puzzle about vague predicates, but that may be misleading. Some of the standard solutions to Sorites paradoxes do not obviously help here, so perhaps the Problem reveals some deeper truths involving the metaphysics of material constitution, or the logic of statements involving identity. The puzzle arises as soon as there is an object without clearly demarcated borders. Unger suggested that clouds are paradigms of this phenomenon, and recent authors such as David Lewis (1993) and Neil McKinnon (2002) have followed him here. Here is Lewis's presentation of the puzzle: > > Think of a cloud--just one cloud, and around it a clear blue sky. > Seen from the ground, the cloud may seem to have a sharp boundary. Not > so. The cloud is a swarm of water droplets. At the outskirts of the > cloud, the density of the droplets falls off. Eventually they are so > few and far between that we may hesitate to say that the outlying > droplets are still part of the cloud at all; perhaps we might better > say only that they are near the cloud. But the transition is gradual. > Many surfaces are equally good candidates to be the boundary of the > cloud. Therefore many aggregates of droplets, some more inclusive and > some less inclusive (and some inclusive in different ways than > others), are equally good candidates to be the cloud. Since they have > equal claim, how can we say that the cloud is one of these aggregates > rather than another? But if all of them count as clouds, then we have > many clouds rather than one. And if none of them count, each one being > ruled out because of the competition from the others, then we have no > cloud. How is it, then, that we have just one cloud? And yet we do. > (Lewis 1993: 164) > The paradox arises because in the story as told the following eight claims each seem to be true, but they are mutually inconsistent. 0. There are several distinct sets of water droplets sk such that for each such set, it is not clear whether the water droplets in sk form a cloud. 1. There is a cloud in the sky. 2. There is at most one cloud in the sky. 3. For each set sk, there is an object ok that the water droplets in sk compose. 4. If the water droplets in si compose oi, and the objects in sj compose oj, and the sets si and sk are not identical, then the objects oi and oj are not identical. 5. If oi is a cloud in the sky, and oj is a cloud in the sky, and oi is not identical with oj, then there are at least two clouds in the sky. 6. If any of these sets si is such that its members compose a cloud, then for any other set sj, if its members compose an object oj, then oj is a cloud. 7. Any cloud is composed of a set of water droplets. To see the inconsistency, note that by 1 and 7 there is a cloud composed of water droplets. Say this cloud is composed of the water droplets in si, and let sj be any other set whose members might, for all we can tell, form a cloud. (Premise 0 guarantees the existence of such a set.) By 3, the water droplets in sj compose an object oj. By 4, oj is not identical to our original cloud. By 6, oj is a cloud, and since it is transparently in the sky, it is a cloud in the sky. By 5, there are at least two clouds in the sky. But this is inconsistent with 2. A solution to the paradox must provide a reason for rejecting one of the premises, or a reason to reject the reasoning that led us to the contradiction, or the means to live with the contradiction. Since none of the motivations for believing in the existence of dialetheia apply here, let us ignore the last possibility. And since 0 follows directly from the way the story is told, let us ignore that option as well. That leaves open eight possibilities. (The classification of the solutions here is slightly different from that in Chapter One of Hud Hudson's "A Materialist Metaphysics of the Human Person." But it has a deep debt to Hudson's presentation of the range of solutions, which should be clear from the discussion that follows.) ## 2. Nihilism Unger's original solution was to reject 1. The concept of a cloud involved, he thought, inconsistent presuppositions. Since those presuppositions were not satisfied, there are no clouds. This is a rather radical move, since it applies not just to clouds, but to any kind of sortal for which a similar problem can be generated. And, Unger pointed out, this includes most sortals. As Lewis puts it, "Think of a rusty nail, and the gradual transition from steel ... to rust merely resting on the nail. Or think of a cathode, and its departing electrons. Or think of anything that undergoes evaporation or erosion or abrasion. Or think of yourself, or any organism, with parts that gradually come loose in metabolism or excretion or perspiration or shedding of dead skin" (Lewis 1993: 165). Despite Lewis's presentation, the Problem of the Many is not a problem about change. The salient feature of these examples is that, in practice, change is a slow process. Hence whenever a cathode, or a human, is changing, be it by shedding electrons, or shedding skin, there are some things that are not clearly part of the object, nor clearly not part of it. Hence there are distinct sets that each have a good claim to being the set of parts of the cathode, or of the human, and that is what is important. It would be profoundly counterintuitive if there were no clouds, or no cathodes, or no humans, and that is probably enough to reject the position, if any of the alternatives are not also equally counterintuitive. It also, as Unger noted, creates difficulties for many views about singular thought and talk. Intuitively, we can pick out one of the objects composed of water droplets by the phrase 'that cloud'. But if it is not a cloud, then possibly we cannot. For similar reasons, we may not be able to name any such object, if we use any kind of reference-fixing description involving 'cloud' to pick it out from other objects composed of water droplets. If the Problem of the Many applies to humans as well as clouds, then by similar reasoning we cannot name or demonstrate any human, or, if you think there are no humans, any human-like object. Unger was happy to take these results to be philosophical discoveries, but they are so counterintuitive that most theorists hold that they form a reductio of his theory. Bradley Rettler (2018) argues that the nihilist has even more problems than this. Nihilism solves some philosophical problems, such as explaining which of 0-7 is false. But, he argues, for any problem it solves, there is a parallel problem which it does not solve, but rival solutions do solve. For instance, if you think of the problem here as a version of a Sorites paradox, nihilism does not help with versions of the paradox which concern predicates applied to simples. It is interesting that some other theories of vagueness have adopted positions resembling Unger's in some respects, but without the extreme conclusions. Matti Eklund (2002) and Roy Sorensen (2001) have argued that all vague concepts involve inconsistent presuppositions. Sorensen spells this out by saying that there are some inconsistent propositions that anyone who possesses a vague concept should believe. In the case of a vague predicate F that is vulnerable to a Sorites paradox, the inconsistent propositions are that some things are Fs, some things are not Fs, any object that closely resembles (in a suitable respect) something that is F is itself F, and that there are chains of 'suitably resembling' objects between an F and a non-F. Here the inconsistent propositions are that a story like Lewis's is possible, and in it 0 through 7 are true. Neither Eklund nor Sorensen conclude from this that nothing satisfies the predicates in question; rather they conclude that some propositions that we find compelling merely in virtue of possessing the concepts from which they are constituted are false. So while they don't adopt Unger's nihilist conclusions, two contemporary theorists agree with him that vague concepts are in some sense incoherent. ## 3. Overpopulation A simple solution to the puzzle is to reject premise 2. Each of the relevant fusions of water droplets looks and acts like a cloud, so it is a cloud. As with the first option this leads to some very counterintuitive results. In any room with at least one person, there are many millions of people. But this is not as bad as saying that there are no people. And perhaps we don't even have to say the striking claim. In many circumstances, we quantify over a restricted domain. We can say, "There's no beer," even when there is beer in some non-salient locales. With respect to some restricted quantifier domains, it is true that there is exactly one person in a particular room. The surprising result is that with respect to other quantifier domains, there are many millions of people in that room. The defender of the overpopulation theory will hold that this shows how unusual it is to use unrestricted quantifiers, not that there really is only one person in the room. The overpopulation solution is not popular, but it is not without defenders. J. Robert G. Williams (2006) endorses it, largely because of a tension between the supervaluationist solution (that will be discussed in section 7) and what supervaluationism says about the Sorites paradox. James Openshaw (2021) and Alexander Sandgren (forthcoming) argue that the overpopulation solution is true, and each offer a theory of how singular thought about the cloud is possible given overpopulation. Sandgren also points out that there might be multiple sources of overpopulation. Even given a particular set of water droplets, some metaphysical theories will say that there are multiple objects those droplets compose, which differ in their temporal or modal properties. Hudson (2001: 39-44) draws out a surprising consequence of the overpopulation solution as applied to people. Assume that there are really millions of people just where we'd normally say there was exactly one. Call that person Charlie. When Charlie raises her arm, each of the millions must also raise their arms, for the millions differ only in whether or not they contain some borderline skin cells, not in whether their arm is raised or lowered. Normally, if two people are such that whenever one acts a certain way, then so must the other, we would say that at most one of them is acting freely. So it looks like at most one of the millions of people around Charlie are free. There are a few possible responses here, though whether a defender of the overpopulation view will view this consequence as being more counter-intuitive than other claims to which she is already committed, and hence whether it needs a special response, is not clear. There are some other striking, though not always philosophically relevant, features of this solution. To quote Hudson: > > Among the most troublesome are worries about naming and singular > reference ... how can any of us ever hope to successfully refer > to himself without referring to his brothers as well? Or how might we > have a little private time to tell just one of our sons of our > affection for him without sharing the moment with uncountably many of > his brothers? Or how might we follow through on our vow to practice > monogamy? (Hudson 2001: 39) > ## 4. Brutalism As Unger originally states it, the puzzle relies on a contentious principle of mereology. In particular, it assumes mereological Universalism, the view that for any objects, there is an object that has all of them as its fusion. (That is, it has each of those objects as parts, and has no parts that do not overlap at least one of the original objects.) Without this assumption, the Problem of the Many may have an easy solution. The cloud in the sky is the object up there that is a fusion of water droplets. There are many other sets of water droplets, other than the set of water droplets that compose the cloud, but since the members of those sets do not compose an object, they do not compose a cloud. There are two kinds of theories that imply that only one of the sets of water droplets is such that there exists a fusion of its atoms. First, there are principled restrictions on composition, theories that say that the xs compose an object y iff the xs are F, for some natural property F. Secondly, there are brutal theories, which say it's just a brute fact that in some cases the xs compose an object, and in others they do not. It would be quite hard to imagine a principled theory solving the Problem of the Many, since it is hard to see what the principle could be. (For a more detailed argument for this, set against a somewhat different backdrop, see McKinnon 2002.) But a brutal theory could work. And such a theory has been defended. Ned Markosian (1998) argues that not only does brutalism, the doctrine that there are brute facts about when the xs compose a y, solve the Problem of the Many, the account of composition it implies fits more naturally with our intuitions about composition. It seems objectionable, in some not easy-to-pin-down way, to rely on brute facts in just this way. Here is how Terrence Horgan puts the objection: > > In particular, a good metaphysical theory or scientific theory should > avoid positing a plethora of quite specific, disconnected, sui > generis, compositional facts. Such facts would be ontological > danglers; they would be metaphysically queer. Even though explanation > presumably must bottom out somewhere, it is just not credible--or > even intelligible--that it should bottom out with specific > compositional facts which themselves are utterly unexplainable and > which do not conform to any systematic general principles. (Horgan > 1993: 694-5) > On the other hand, this kind of view does provide a particularly straightforward solution to the Problem of the Many. As Markosian notes, if we have independent reason to view favourably the idea that facts about when some things compose an object are brute facts, which he thinks is provided by our intuitions about cases of composition and non-composition, the very simplicity of this solution to the Problem of the Many may count as an argument in favour of brutalism. ## 5. Relative Identity Assume that the brutalist is wrong, and that for every set of water droplets, there is an object those water droplets compose. Since that object looks for all the world like a cloud, we will say it is a cloud. The fourth solution accepts those claims, but denies that there are many clouds. It is true that there are many fusions of atoms, but these are all the same cloud. This view adopts a position most commonly associated with P. T. Geach (1980), that two things can be the same F but not the same G, even though they are both Gs. To see the motivation for that position, and a discussion of its strengths and weaknesses, see the article on relative identity. Here is one objection that many have felt is telling against the relative identity view: Let w be a water droplet that is in s1 but not s2. The relative identity solution says that the droplets in s1 compose an object o1, and the droplets in s2 compose an object o2, and though o1 and o2 are different fusions of water droplets, they are the same cloud. Call this cloud c. If o1 is the same cloud as o2, then presumably they have the same properties. But o1 has the property of having w as a part, while o2 does not. Defenders of the relative identity theory here deny the principle that if two objects are the same F, they have the same properties. Many theorists find this denial to amount to a reductio of the view. ## 6. Partial Identity Even if o1 and o2 exist, and are clouds, and are not the same cloud, it does not immediately follow that there are two clouds. If we analyze "There are two clouds" as "There is an x and a y such that x is a cloud and y is a cloud, and x is not the same cloud as y" then the conclusion will naturally follow. But perhaps that is not the correct analysis of "There are two clouds." Or, more cautiously, perhaps it is not the correct analysis in all contexts. Following some suggestions of D. M. Armstrong's (Armstrong 1978, vol. 2: 37-8), David Lewis suggests a solution along these lines. The objects o1 and o2 are not the same cloud, but they are almost the same cloud. And in everyday circumstances (AT) is a good-enough analysis of "There is one cloud" (AT) There is a cloud, and all clouds are almost identical with it. As Lewis puts it, we 'count by almost-identity' rather than by identity in everyday contexts. And when we do, we get the correct result that there is one cloud in the sky. Lewis notes that there are other contexts in which we count by some criteria other than identity. > > If an infirm man wishes to know how many roads he must cross to reach > his destination, I will count by identity-along-his-path rather than > by identity. By crossing the Chester A. Arthur Parkway and Route 137 > at the brief stretch where they have merged, he can cross both by > crossing only one road. (Lewis 1976: 27) > There are two major objections to this theory. First, as Hudson notes, even if we normally count by almost-identity, we know how to count by identity, and when we do it seems that there is one cloud in the sky, not many millions. A defender of Lewis's position may say that the only reason this seems intuitive is that it is normally intuitive to say that there is only one cloud in the sky. And that intuition is respected! More contentiously, it may be argued that it is a good thing to predict that when we count by identity we get the result that there are millions of clouds. After all, the only time we'd do this is when we're doing metaphysics, and we have noted that in the metaphysics classroom, there is some intuitive force to the argument that there are millions of clouds in the sky. It would be a brave philosopher to endorse this as a virtue of the theory, but it may offset some of the costs. Secondly, something like the Problem of the Many can arise even when the possible objects are not almost identical. Lewis notes this objection, and provides an illustrative example to back it up. A similar kind of example can be found in W. V. O. Quine's Word and Object (1960). Lewis's example is of a house with an attached garage. It is unclear whether the garage is part of the house or an external attachment to it. So it is unclear whether the phrase 'Fred's house' denotes the basic house, call it the home, or the fusion of the home and the garage. What is clear is that there is exactly one house here. However, the home might be quite different from the fusion of the home and the garage. It will probably be smaller and warmer, for example. So the home and the home-garage fusion are not even almost identical. Quine's example is of something that looks, at first, to be a mountain with two peaks. On closer inspection we find that the peaks are not quite as connected as first appeared, and perhaps they could be properly construed as two separate mountains. What we could not say is that there are three mountains here, the two peaks and their fusion, but since neither peak is almost identical to the other, or to the fusion, this is what Lewis's solution implies. But perhaps it is wrong to understand almost-identity in this way. Consider another example of Lewis's, one that Dan Lopez de Sa (2014) argues is central to a Lewisian solution to the problem. > > You draw two diagonals in a square; you ask me how many triangles; I > say there are four; you deride me for ignoring the four large > triangles and counting only the small ones. But the joke is on you. > For I was within my rights as a speaker of ordinary language, and you > couldn't see it because you insisted on counting by strict > identity. I meant that, for some w, x, y, > z, (1) w, x, y, and z are > triangles; (2) w and x are distinct, and ... and so > are y and z (six clauses); (3) for any triangle > t, either t and w are not distinct, or ... > or t and z are not distinct (four clauses). And by > 'distinct' I meant non-overlap rather than non-identity, > so what I said was true. (Lewis 1993, fn. 9) > One might think this is the general way to understand counting sentences in ordinary language. So There is exactly one F gets interpreted as There is an F, and no F is wholly distinct from it; and There are exactly two Fs gets interpreted as There are wholly distinct things x and y that are both F, and no F is wholly distinct from both of them, and so on. Lewis writes as if this is an explication of the almost-identity proposal, but this is at best misleading. A house with solar panels partially overlaps the city's electrical grid, but it would be very strange to call them almost-identical. It sounds like a similar, but distinct, proposed solution. However we understand the proposal, Lopez de Sa notes that it has a number of virtues. It seems to account for the puzzle involving the house, what he calls the Problem of the Two. If in general counting involves distinctness, then we have a good sense in which there is one cloud in the sky, and Fred owns one house. There still remain two challenges for this view. First, one could still follow Hudson and argue that even if we ordinarily understand counting sentences this way, we do still know how to count by identity. And when we do, it seems that there is just one cloud, not millions of them. Second, it isn't that clear that we always count by distinctness, in the way Lopez de Sa suggests. If I say there are three ways to get from my house to my office, I don't mean to say that these three are completely distinct. Indeed, they probably all start with going out my door, down my driveway etc., and end by walking up the stairs into my office. So the general claim about how to understand counting sentences seems false. C. S. Sutton (2015) argued that we can get around something like the second of these problems if we do two things. First, the rule isn't that we don't normally quantify over things that overlap, just that we don't normally quantify over things that substantially overlap. We can look at a row of townhouses and say that there are seven houses there even if the walls of the house overlap. Second, the notion of overlap here is not sensitive to the quantity of material that the objects have in common, but to their functional role. If two objects play very different functional roles, she argues that we will naturally count them as two, even if they have a lot of material in common. This could account for a version of the getting to work example where the three different ways of getting to work only differ on how to get through one small stretch in the middle. That is, if there are three (wholly distinct ways) to get from B to C, and the way to get from A to D is to go A-B-C-D, then there is a good sense in which there are three ways to get from A to D. Sutton's theory explains how this could be true even if the B-C leg is a short part of the trip. David Liebesman (2020) argued for a different way of implementing this kind of theory. He argues that the kinds of constraints that Lewis, Lopez de Sa and Sutton have suggested don't get incorporated into our theory of counting, but into the proper interpretation of the nouns involved in counting sentences. It helps to understand Liebesman's idea with an example. There is, we'll presumably all agree, just one colour in Yves Klein's Blue Monochrome. (It says so in the title.) But Blue Monochrome has blue in it, and it has ultramarine in it, and blue doesn't equal ultramarine. What's gone on? Well, says Liebesman, whenever a noun occurs under a determiner, it needs an interpretation. That interpretation will almost never be maximal. When we ask how many animals a person has in their house, we typically don't mean to count the insects. When we say every book is on the shelf, we don't mean every book in the universe. And typically, the relevant interpretation of the determiner phrase (like 'every book', or 'one cloud') will exclude overlapping objects. Typically but not, says Liebesman, always. We can say, for instance, that every shade of a colour is a colour, and in that sentence 'colour' includes both blue and ultramarine. This offers a new solution to the Problem of the Many. He argues that all of 0 to 7 are true, but they are true for different interpretations of phrases including the word 'cloud'. When we interpret it in a way that rules out overlap, then 6 is false. When we interpret it in the maximal way, like the way we interpret 'colour' in Every shade of a colour is a colour, then 2 is false. But to get the inconsistency, we have to equivocate. It's an easy equivocation to make, since each of the meanings is one that we frequently use. ## 7. Vagueness Step 6 in the initial setup of the problem says that if any of the oi is a cloud, then they all are. There are three important arguments for this premise, two of them presented explicitly by Unger, and the other by Geach. Two of the arguments seem to be faulty, and the third can be rejected if we adopt some familiar, though by no means universally endorsed, theories of vagueness. ### 7.1 Argument from Duplication The first argument, due essentially to Geach, runs as follows. Geach's presentation did not involve clouds, but the principles are clearly stated in his version of the argument. (The argument shows that if an ok is a cloud for arbitrary k, we can easily generalize to the claim that for every i, oi is a cloud.) D1. If all the water droplets not in sk did not exist, then ok would be a cloud. D2. Whether ok is a cloud does not depend on whether things distinct from it exist. C. ok is a cloud. D2 implies that being a cloud is an intrinsic property. The idea is that by changing the world outside the cloud, we do not change whether or not it is a cloud. There is, however, little reason to believe this is true. And given that it leads to a rather implausible conclusion, that there are millions of clouds where we think there is one, there is some reason to believe it is false. We can argue directly for the same conclusion. Assume many more water droplets coalesce around our original cloud. There is still one cloud in the sky, but it determinately includes more water droplets than the original cloud. The fusion of those water droplets exists, and we may assume that they did not change their intrinsic properties, but they are now a part of a cloud, rather than a cloud. Even if something looks like a cloud, smells like a cloud and rains like a cloud, it need not be a cloud, it may only be a part of a cloud. ### 7.2 Argument from Similarity Unger's primary argument takes a quite different tack. S1. For some j, oj is a typical cloud. S2. Anything that differs minutely from a typical cloud is a cloud. S3. ok differs minutely from oj. C. ok is a cloud. Since we only care about the conditional if oj is a cloud, so is ok, it is clearly acceptable to assume that oj is a cloud for the sake of the argument. And S3 is guaranteed to be true by the setup of the problem. The main issue then is whether S2 is true. As Hudson notes, there appear to be some clear counterexamples to it. The fusion of a cloud with one of the water droplets in my bathtub is clearly not a cloud, but by most standards it differs minutely from a cloud, since there is only one droplet of water difference between them. ### 7.3 Argument from Meaning The final argument is not set out as clearly, but it has perhaps the most persuasive force. Unger says that if exactly one of the oi is a cloud, then there must be a 'selection principle' that picks it over the others. But it is not clear just what kind of selection principle that could be. The underlying argument seems to be something like this: M1. For some j, oj is a cloud. M2. If oj is a cloud and the rest of the oi are not, then some principle selects oj to be the unique cloud. M3. There is no principle that selects oj to be the unique cloud. C. At least one of the other oi is a cloud. The idea behind M2 is that word meanings are not brute facts about reality. As Jerry Fodor put it, "if aboutness is real, it must be really something else" (Fodor 1987: 97). Something makes it the case that oj is the unique thing (around here) that satisfies our term 'heap'. Maybe that could be because oj has some unique properties that make it suitable to be in the denotation of ordinary terms. Or maybe it is something about our linguistic practices. Or maybe it is some combination of these things. But something must determine it, and whatever it is, we can (in theory) say what that is, by giving some kind of principled explanation of why oj is the unique cloud. It is at this point that theories of vagueness can play a role in the debate. Two of the leading theories of vagueness, epistemicism and supervaluationism, provide principled reasons to reject this argument. The epistemicist says that there are semantic facts that are beyond our possible knowledge. Arguably we can only know where a semantic boundary lies if that boundary was fixed by our use or by the fact that one particular property is a natural kind. But, say the epistemicists, there are many other boundaries that are not like this, such as the boundary between the heaps and the non-heaps. Here we have a similar kind of situation. It is vague just which of the oi is a cloud. What that means is that there is a fact about which of them is a cloud, but we cannot possibly know it. The epistemicist is naturally read as rejecting the very last step in the previous paragraph. Even if something (probably our linguistic practices) makes it the case that oj is the unique cloud, that need not be something we can know and state. The supervaluationist response is worth spending more time on here, both because it engages directly with the intuitions behind this argument and because two of its leading proponents (Vann McGee and Brian McLaughlin, in their 2001) have responded directly to this argument using the supervaluationist framework. Roughly (and for more detail see the section on supervaluations in the entry on vagueness) supervaluationists say that whenever some terms are vague, there are ways of making them more precise consistent with our intuitions on how the terms behave. So, to use a classic case, 'heap' is vague, which to the supervaluationist means that there are some piles of sand that are neither determinately heaps nor determinately non-heaps, and a sentence saying that that object is a heap is neither determinately true nor determinately false. However, there are many ways to extend the meaning of 'heap' so it becomes precise. Each of these ways of making it precise is called a precisification. A precisification is admissible iff every sentence that is determinately true (false) in English is true (false) in the precisification. So if a is determinately a heap, b is determinately not a heap and c is neither determinately a heap nor determinately not a heap, then every precisification must make 'a is a heap' true and 'b is a heap' false, but some make 'c is a heap' true and others make it false. To a first approximation, to be admissible a precisification must assign all the determinate heaps to the extension of 'heap' and assign none of the determinate non-heaps to its extension, but it is free to assign or not assign things in the 'penumbra' between these groups to the extension of 'heap'. But this is not quite right. If d is a little larger than c, but still not determinately a heap, then the sentence "If c is a heap so is d" is intuitively true. As it is often put, following Kit Fine (1975), a precisification must respect 'penumbral connections' between the borderline cases. If d has a better case for being a heap than c, then a precisification cannot make c a heap but not d. These penumbral connections play a crucial role in the supervaluationist solution to the Problem of the Many. Finally, a sentence is determinately true iff it is true on all admissible precisifications, determinately false iff it is false on all admissible precisifications. In the original example, described by Lewis, the sentence "There is one cloud in the sky" is determinately true. None of the sentences "o1 is a cloud", "o2 is a cloud" and so on are determinately true. So a precisification can make each of these either true or false. But, if it is to preserve the fact that "There is one cloud in the sky" is determinately true, it must make exactly one of those sentences true. McGee and McLaughlin suggest that this combination of constraints lets us preserve what is plausible about M2, without accepting that it is true. The term 'cloud' is vague; there is no fact of the matter as to whether its extension includes o1 or o2 or o3 or .... If there were such a fact, there would have to be something that made it the case that it included oj and not ok, and as M3 correctly points out, no such facts exist. But this is consistent with saying that its extension does contain exactly one of the oi. The beauty of the supervaluationist solution is that it lets us hold these seemingly contradictory positions simultaneously. We also get to capture some of the plausibility of S2--it is consistent with the supervaluationist position to say that anything similar to a cloud is not determinately not a cloud. Penumbral connections also let us explain some other puzzling situations. Imagine I point cloudwards and say, "That is a cloud." Intuitively, what I have said is true, even though 'cloud' is vague, and so is my demonstrative 'that'. (To see this, note that there's no determinate answer as to which of the oi it picks out.) On different precisifications, 'that' picks out different oi. But on every precisification it picks out the oi that is in the extension of 'cloud', so "That is a cloud" comes out true as desired. Similarly, if I named the cloud 'Edgar', then a similar trick lets it be true that "Edgar" is vague, while "Edgar is a cloud" is determinately true. So the supervaluationist solution lets us preserve many of the intuitions about the original case, including the intuitions that seemed to underwrite M2, without conceding that there are millions of clouds. But there are a few objections to this package. * Objection: There are many telling objections to supervaluationist theories of vagueness. + Reply: This may be true, but it would take us well beyond the scope of this entry to outline them all. See the entries on vagueness (the section on supervaluation) and the Sorites Paradox for more detail. * Objection: The supervaluationist solution makes some existentially quantified sentences, like "There is a cloud in the sky" determinately true even though no instance of them is determinately true. + Reply: As Lewis says, this is odd, but no odder than things that we learn to live with in other contexts. Lewis compares this to "I owe you a horse, but there is no particular horse that I owe you." * Objection: The penumbral connections here are not precisely specified. What is the rule that says how much overlap is required before two objects cannot both be clouds? + Reply: It is true that the connections are not precisely specified. It would be quite hard to carefully analyze 'cloud' to work out exactly what they are. But that we cannot say exactly what the rules are is no reason for saying no such rules exist, any more than our inability to say exactly what knowledge is provides a reason for saying that no one ever knows anything. Scepticism can't be proven that easily. * Objection: The penumbral connections appealed to here are left unexplained. At a crucial stage in the explanation, it seems to be just assumed that the problem can be solved, and that it is a determinate truth that there is one cloud in the sky. + Reply 1: We have to start somewhere in philosophy. This kind of reply can be spelled out in two ways. There is a 'Moorean' move that says that the premise that there is one cloud in the sky is more plausible than the premises that would have to be used in an argument against supervaluationism. Alternatively, it might be claimed that the main argument for supervaluationism is an inference to the best explanation. In that case, the intuition that there is exactly one cloud in the sky, but it is indeterminate just which object it is, is something to be explained, not something that has to be proven. This is the kind of position defended by Rosanna Keefe in her book Theories of Vagueness. Although Keefe does not apply this directly to the Problem of the Many, the way to apply her position to the Problem seems clear enough. + Reply 2: The penumbral connections we find for most words are generated by the inferential role provided by the meaning of the terms. It is because the inference from "This pile of sand is a heap", and "That pile of sand is slightly larger than this one, and arranged roughly the same way," to "That pile of sand is a heap" is generally acceptable that precisifications which make the premises true and the conclusions false are inadmissible. (We have to restrict this inferential rule to the case where 'this' and 'that' are ordinary demonstratives, and not used to pick out arbitrary fusions of grains, or else we get bizarre results for reasons that should be familiar by this point in the story.) And it isn't too hard to specify the inferential rule here. The inference from "oj is a cloud" and "oj and ok massively overlap" to "ok is a cloud" is just as acceptable as the above inference involving heaps. Indeed, it is part of the meaning of 'cloud' that this inference is acceptable. (Much of this reply is drawn from the discussion of 'maximal' predicates in Sider 2001 and 2003, though since Sider is no supervaluationist, he would not entirely endorse this way of putting things.) * Objection: The second reply cannot explain the existence of penumbral connections between 'cloud' and demonstratives like 'that' and names like 'Edgar'. It would only explain the existence of those penumbral connections if it was part of the meaning of names and demonstratives that they fill some kind of inferential role. But this is inconsistent with the widely held view that demonstratives and names are directly referential. (For more details on debates about the meanings of names, see the entries on propositional attitude reports and singular propositions.) + Reply: One response to this would be to deny the view that names and demonstratives are directly referential. Another would be to deny that inferential roles provide the only penumbral constraints on precisifications. Weatherson 2003b sketches a theory that does exactly this. The theory draws on David Lewis's response to some quite different work on semantic indeterminacy. As many authors (Quine 1960, Putnam 1981, Kripke 1982) showed, the dispositions of speakers to use terms are not fine-grained enough to make the language as precise as we ordinarily think it is. As far as our usage dispositions go, 'rabbit' could mean undetached rabbit part, 'vat' could mean vat image, and 'plus' could mean quus. (Quus is a function defined over pairs of numbers that yields the sum of the two numbers when they are both small, and 5 when one is sufficiently large.) But intuitively our language is not that indeterminate: 'plus' determinately does not mean quus. Lewis (1983, 1984) suggested the way out here is to posit a notion of 'naturalness'. Sometimes a term t denotes the concept C1 rather than C2 not because we are disposed to use t as if it meant C1 rather than C2, but simply because C1 is a more natural concept. 'Plus' means plus rather than quus simply because plus is more natural than quus. Something like the same story applies to names and demonstratives. Imagine I point in the direction of Tibbles the cat and say, "That is Edgar's favourite cat." There is a way of systematically (mis)interpreting all my utterances so 'that' denotes the Tibbles-shaped region of space-time exactly one metre behind Tibbles. (We have to reinterpret what 'cat' means to make this work, but the discussions of semantic indeterminacy in Quine and Kripke make it clear how to do this.) So there's nothing in my usage dispositions that makes 'that' mean Tibbles, rather than the region of space-time that 'follows' him around. But because Tibbles is more natural than that region of space-time, 'that' does pick out Tibbles. It is the very same naturalness that makes 'cat' denote a property that Tibbles (and not the trailing region of space-time) satisfies that makes 'that' denote Tibbles, a fact that will become important below. The same kind of story can be applied to the cloud. It is because the cloud is a more natural object than the region of space-time a mile above the cloud that our demonstrative 'that' denotes the cloud and not the region. However, none of the oi are more natural than any other, so there is still no fact of the matter as to whether 'that' picks out oj or ok. Lewis's theory does not eliminate all semantic indeterminacy; when there are equally natural candidates to be the denotation of a term, and each of them is consistent with our dispositions to use the term, then the denotation of the term is simply indeterminate between those candidates. Weatherson's theory is that the role of each precisification is to arbitrarily make one of the oi more natural than the rest. Typically, it is thought that the denotation of a term according to a precisification is determined directly. It is a fact about a precisification P that, according to it, 'cloud' denotes property c1. On Weatherson's theory this is not the case. What the precisification does is provide a new, and somewhat arbitrary, standard of naturalness, and the content of the terms according to the precisification is then determined by Lewis's theory of content. The denotations of 'cloud' and 'that' according to a precisification P are those concepts and objects that are the most natural-according-to-P of the concepts and objects that we could be denoting by those terms, for all one can tell from the way the terms are used. The coordination between the two terms, the fact that on every precisification 'that' denotes an object in the extension of 'cloud' is explained by the fact that the very same thing, naturalness-according-to-P, determines the denotation of 'cloud' and of 'that'. * Objection (From Stephen Schiffer 1998): The supervaluationist account cannot handle speech reports involving vague names. Imagine that Alex points cloudwards and says, "That is a cloud." Later Sam points towards the same cloud and says, "Alex said that that is a cloud." Intuitively, Sam's utterance is determinately true. But according to the supervaluationist, it is only determinately true if it is true on every precisification. So it must be true that Alex said that o1 is a cloud, that Alex said that o2 is a cloud, and so on, since these are all precisifications of "Alex said that that is a cloud." But Alex did not say all of those things, for if she did she would be committed to saying that there are millions of clouds in the sky, and of course she is not, as the supervaluationists have been arguing. + Reply: There is a little logical slip here. Let Pi be a precisification of Sam's word 'that' that makes it denote oi. All the supervaluationist who holds that Sam's utterance is determinately true is committed to is that for each i, according to Pi, Alex said that oi is a cloud. And this will be true if the denotation of Alex's word 'that' is also oi according to Pi. So as long as there is a penumbral connection between Sam's word 'that' and Alex's word 'that', the supervaluationist avoids the objection. Such a connection may seem mysterious at first, but note that Weatherson's theory predicts that just such a penumbral connection obtains. So if that theory is acceptable, then Schiffer's objection misfires. * Objection (From Neil Mackinnon 2002 and Thomas Sattig 2013): It is part of our notion of a mountain that facts about mountain-hood are not basic. If something is a mountain, there are facts in virtue of which it is a mountain, and nearby things are not. Yet on each precisification of 'mountain', this won't be true; it will be completely arbitrary which fusion of rocks is a mountain. + Reply: It is true that on any precisification there will be no principled reason why this fusion of rocks is a mountain, and another is not. And it is true that there should be such a principled reason; mountainhood facts are not basic. But that problem can be avoided by the theory that denies that "the supervaluationist rule [applies] to any statement whatever, never mind that the statement makes no sense that way" (Lewis 1993, 173). Lewis's idea, or at least the application of Lewis's idea to this puzzle, is that we know how to understand the idea that mountainhood facts are non-arbitrary: we understand it as a claim that there is some non-arbitrary explanation of which precisifications of 'mountain' are and are not admissible. If we must apply the supervaluationist rule to every statement, including the statement that it is not arbitrary which things are mountains, this understanding is ruled out. Lewis's response is to deny that the rule must always be applied. As long as there is some sensible way to understand the claim, we don't have to insist on applying the supervaluationist machinery to it. That said, it does seem like this is likely to be somewhat of a problem for everyone (even the theorist like Lewis who uses the supervaluationist machinery only when it is helpful). Sattig himself claims to avoid the problem by making the mountain be a maximal fusion of candidates. But for any plausible mountain, it will be vague and somewhat arbitrary what the boundary is between being a mountain-candidate and not being one. The lower boundaries of mountains are not, in practice, clearly marked. Similarly, there will be some arbitrariness in the boundaries between the admissible and inadmissible precisifications of 'mountain'. We may have to live with some arbitrariness. * Objection (From J. Robert G. Williams 2006): When we look at an ordinary mountain, there definitely is (at least) one mountain in front of us. That seems clear. The vagueness solution respects this fact. But in many contexts, people tend to systematically confuse "Definitely, there is an F" with "There is a definite F." Indeed, the standard explanation of why Sorites arguments seem, mistakenly, to be attractive is that this confusion gets made. Yet on the vagueness account, there is no definite mountain; all the candidates are borderline cases. So by parity of reasoning, we should expect intuition to deny that there is definitely a mountain. And intuition does not deny that, it loudly confirms it. At the very least, this shows a tension between the standard account of the Sorites paradox, and the vagueness solution to the Problem of the Many. + Reply: This is definitely a problem for the views that many philosophers have put forward. As Williams stresses, it isn't on its own a problem for the vagueness solution to the Problem of the Many, but it is a problem for the conjunction of that solution with a widely endorsed, and independently plausible, explanation of the Sorites paradox. In his dissertation, Nicholas K. Jones (2010) argues that the right response is to give up the idea that speakers typically confuse "Definitely, there is an F" with "There is a definite F", and instead use a different resolution of the Sorites. Space prevents a further discussion of all possible objections to the supervaluationist account, but interested readers are particularly encouraged to look at Neil McKinnon's objection to the account (see the Other Internet Resources section), which suggests that distinctive problems arise for the supervaluationist when there really are two or more clouds involved. Even if the supervaluationist solution to the Problem of the Many has responses to all of the objections that have been levelled against it, some of those objections rely on theories that are contentious and/or underdeveloped. So it is far from clear at this stage how well the supervaluationist solution, or indeed any solution based on vagueness, to the Problem of the Many will do in future years. ## 8. Rethinking Parthood Some theorists have argued that the underlying cause of the problem is that we have the wrong theory about the relation between parts and wholes. Peter van Inwagen (1990) argues that the problem is that we have assumed that the parthood relation is determinate. We have assumed that it is always determinately true or determinately false that one object is a part of another. According to van Inwagen, sometimes neither of these options applies. He thinks that we need to adopt some kind of fuzzy logic when we are discussing parts and wholes. It can be true to degree 0.7, for example, that one object is part of another. Given these resources, van Inwagen says, we are free to conclude that there is exactly one cloud in the sky, and that some of the 'outer' water droplets are part of it to a degree strictly between 0 and 1. This lets us keep the intuition that it is indeterminate whether these outlying water droplets are members of the cloud without accepting that there are millions of clouds. Note that this is not what van Inwagen would say about this version of the paradox, since he holds that some simples only constitute an object when that object is alive. For van Inwagen, as for Unger, there are no clouds, only cloud-like swarms of atoms. But van Inwagen recognises that a similar problem arises for cats, or for people, two kinds of things that he does believe exist, and he wields this vague constitution theory to solve the problems that arise there. Traditionally, many philosophers thought that such a solution was downright incoherent. A tradition stretching back to Bertrand Russell (1923) and Michael Dummett (1975) held that vagueness was always and everywhere a representational phenomenon. From this perspective, it didn't make sense to talk about it being vague or indeterminate whether a particular droplet was part of a particular cloud. But this traditional view has come under a lot of pressure in recent years; see Barnes (2010) for one of the best challenges, and Sorensen (2013, section 8) for a survey of more work. So let us assume here it is legitimate to talk about the possibility that parthood itself, and not just our representation of it, is vague. As Hudson (2001) notes though, it is far from clear just how the appeal to fuzzy logic is meant to help here. Originally it was clear for each of n water droplets whether they were members of the cloud to degree 1 or degree 0. So there were 2n candidate clouds, and the Problem of the Many is finding out how to preserve the intuition when faced with all these objects. It is unclear how increasing the range of possible relationships between each particle and the cloud from 2 to continuum-many should help here, for now it seems there are at least continuum-many cloud-like objects to choose between, one for each function from each of the n droplets to [0, 1], and we need a way of saying exactly one of them is a cloud. Assume that some droplet is part of the cloud to degree 0.7. Now consider the object (or perhaps possible object) that is just like the cloud, except this droplet is only part of it to degree 0.6. Does that object exist, and is it a cloud? Van Inwagen says, in a way reminiscent of Markosian's brutal composition solution, that such an 'object' does not even exist. A different kind of solution is offered by Mark Johnston (1992) and E. J. Lowe (1982, 1995). Both of them suggest that the key to solving the Problem is to distinguish cloud-constituters from clouds. They say it is a category mistake to identify clouds with any fusion of water droplets, because they have different identity conditions. The cloud could survive the transformation of half its droplets into puddles on the footpath (or whatever kind of land it happens to be raining over), it would just be a smaller cloud, the fusion could not. As Johnston says, "Hence Unger's insistent and ironic question 'But which of o1, o2, o3, ... is our paradigm cloud c?' has as its proper answer 'None'" (1992: 100, numbering slightly altered). Lewis (1993) listed several objections to this position, and Lowe (1995) responds to them. (Lewis and Lowe discuss a version of the problem using cats not clouds, and we will sometimes follow them below.) Lewis's first objection is that positing clouds as well as cloud-constituting fusions of atoms is metaphysically extravagant. As Lowe (and, for separate reasons, Johnston) point out, these extra objects are arguably needed to solve puzzles to do with persistence. Hence it is no objection to a solution to the Problem of the Many that it posits such objects. Resolving these debates would take us too far afield, so let us assume (as Lewis does) that we have reason to believe that these objects exist. Secondly, Lewis says that even with this move, we still have a Problem of the Many applied to cloud-constituters, rather than to clouds. Lowe responds that since 'cloud-constituter' is not a folk concept, we don't really have any philosophically salient intuitions here, so this cannot be a way in which the position is unintuitive. Finally, Lewis says that each of the constituters is so like the object it is meant to merely constitute (be it a cloud, or a cat, or whatever), it satisfies the same sortals as that object. So if we were originally worried that there were 1001 cats (or clouds) where we thought there was one, now we should be worried that there are 1002. But as Lowe points out, this argument seems to assume that being a cat, or being a cloud, is an intrinsic property. If we assume that it is extrinsic, if it turns on the history of the object, perhaps its future or its possible future, and on which object it is embedded in, then the fact that a cloud-constituter looks, when considered in isolation, to be a cloud is little reason to think it actually is a cloud. Johnston provides an argument that the distinction between clouds and cloud-constituting fusions of water droplets is crucial to solving the Problem. He thinks that the following principle is sound, and not threatened by examples like our cloud. > > (9') If y is a paradigm F, and x is > an entity that differs from y in any respect relevant to > being an F only very minutely, and x is of the right > category, i.e. is not a mere quantity or piece of matter, then > x is an F. (Johnston 1992: 100) > The theorist who thinks that clouds are just fusions of water droplets cannot accept this principle, or they will conclude that every oi is a cloud, since for them each oi is of the right category. On the other hand, Johnston himself cannot accept it either, unless he denies there can be another object c' which is in a similar position to c, and is of the same category as c, but differs with respect to which water droplets constitute it. It seems that what is doing the work in Johnston's solution is not just the distinction between constitution and identity, but a tacit restriction on when there is a 'higher-level' object constituted by certain 'lower-level' objects. To that extent, his theory also resembles Markosian's brutal composition theory, though since Johnston can accept that every set of atoms has a fusion his theory has different costs and benefits to Markosian's theory. A recent version of this kind of view comes from Nicholas K. Jones (2015), though he focusses on constitution, not composition. (Indeed, a distinctive aspect of his view is that he takes constitution to be metaphysically prior to composition.) Jones rejects the following principle, which is similar to 4 in the original inconsistent set. * If the water droplets in si constitute oi, and the objects in sj constitute oj, and the sets si and sk are not identical, then the objects oi and oj are not identical. He rejects this claim. He argues that some water droplets can constitute a cloud, and some other water droplets can constitute the very same cloud. On this view, the predicate constitute x behaves a bit like the predicate surround the building. It can be true that the Fs surround the building, and the Gs surround the building, without the Fs being the Gs. And on Jones's view, it can be true that the Fs constitute x, and the Gs constitute x, without the Fs being the Gs. This resembles Lewis's solution in terms of almost-identity, since both Jones and Lewis say that there is one cloud, yet both si and sk can be said to compose it. But for Lewis, this is possible because he rejects the inference from There is one cloud, to If a and b are clouds, they are identical. Jones accepts this inference, and rejects the inference from the premise that si and sk are distinct, and each compose a cloud, to the conclusion that they compose non-identical clouds. ## 9. Rethinking Parthood After concluding that all of these kinds of solutions face serious difficulties, Hudson (2001: Chapter 2) outlines a new solution, one which rejects so many of the presuppositions of the puzzle that it is best to count him as rejecting the reasoning, rather than rejecting any particular premise. (Hudson is somewhat tentative about endorsing this view, as opposed to merely endorsing the claim that it looks better than its many rivals, but for expository purposes let us refer to it here as his view.) To see the motivation behind Hudson's approach, consider a slightly different case, a variant of one discussed in Wiggins 1968. Tibbles is born at midnight Sunday, replete with a splendid tail, called Tail. An unfortunate accident involving a guillotine sees Tibbles lose his tail at midday Monday, though the tail is preserved for posterity. Then midnight Monday, Tibbles dies. Now consider the timeless question, "Is Tail part of Tibbles?" Intuitively, we want to say the question is underspecified. Outside of Monday, the question does not arise, for Tibbles does not exist. Before midday Monday, the answer is "Yes", and after midday the answer is "No". This suggests that there is really no proposition that Tail is part of Tibbles. There is a proposition that Tail is part of Tibbles on Monday morning (that's true) and that Tail is part of Tibbles on Monday afternoon (that's false), but no proposition involving just the parthood relation and two objects. Parthood is a three-place relation between two objects and a time, not a two-place relation between two objects. Hudson suggests that this line of reasoning is potentially on the right track, but that the conclusion is not quite right. Parthood is a three-place relation, but the third place is not filled by a time, but by a region of space-time. To a crude approximation, x is part of y at s is true if (as we'd normally say) x is a part of y and s is a region of space-time containing no region not occupied by y and all regions occupied by x. But this should be taken as a heuristic guide only, not as a reductive definition, since parthood is really a three-place relation, so the crude approximation does not even express a proposition according to Hudson. To see how this applies to the Problem of the Many, let's simplify the case a little bit so there are only two water droplets, w1 and w2, that are neither determinately part of the cloud nor determinately not a part of it. As well there is the core of the cloud, call it a. On an orthodox theory, there are four proto-clouds here, a, a + w1, a + w2 and a + w1 + w 2. On Hudson's theory the largest and the smallest proto-clouds still exist, but in the middle there is a quite different kind of object, which we'll call c. Let r1 be the region occupied by a and w1, and r2 the region occupied by a and w2. Then the following claims are all true according to Hudson: * c exactly occupies r1; * c exactly occupies r2; * c does not occupy the region consisting of the union of r1 and r2; * c has w1 as a part at r1, but not at r2; * c has w2 as a part at r2, but not at r1; * c has no parts at the region consisting of the union of r1 and r2. Hudson defines "x exactly occupies s" as follows: * x has a part at s, * there is no region of space-time, s\, such that s\ has s as a subregion, while x has a part at s\, and for every subregion of s, s', x has a part at s'. (Hudson 2001: 63) At first, it might look like not much has been accomplished here. All that we did was turn a Problem of 4 clouds into a Problem of 3 clouds, replacing the fusions a + w1 and a + w2 with the new, and oddly behaved, c. But that is to overlook a rather important feature of the remaining proto-clouds. The three remaining proto-clouds can be strictly ordered by the 'part of' relation. This was not previously possible, since neither a + w1 nor a + w2 were part of the other. If we adopt the principle that 'cloud' is a maximal predicate, so no cloud can be a proper part of another cloud, we now get the conclusion that exactly one of the proto-clouds is a cloud, as desired. This is a quite ingenious approach, and it deserves some attention in the future literature. It is hard to say what will emerge as the main costs and benefits of the view in advance of that literature, but the following two points seem worthy of attention. First, if we are allowed to appeal to the principle that no cloud is a proper part of another, why not appeal to the principle that no two clouds massively overlap, and get from 4 proto-clouds to one actual cloud that way? Secondly, why don't we have an object that is just like the old a + w1, that is, an object that has w1 as a part at r1, and does not have w2 (or anything else) as a part at r2? If we get it back, as well as a + w2, then all of Hudson's tinkering with mereology will just have converted a problem of 4 clouds into a problem of 5 clouds. Neither of these points should be taken to be conclusive refutations. As things stand now, Hudson's solution joins the ranks of the many and varied proposed solutions to the Problem of the Many. For such a young problem, the variety of these solutions is rather impressive. Whether the next few years will see these ranks whittled down by refutation, or swelled by imaginative theorising, remains to be seen.
vaihinger	## 1. Brief Biography Hans (born Johannes) Vaihinger's early life was in many ways that of a typical intellectual in nineteenth-century Germany. Born on 25 September 1852 near Tubingen, the son of the pastor Johann Georg Vaihinger, Hans was intended for the clergy. He began his studies in theology at the University of Tubingen in 1870, but, never especially devout, he soon turned his attentions to other subjects, primarily philosophy and natural science. Throughout the next four years, Vaihinger's course of study emphasized the classics of German philosophy--Kant, Fichte, Schelling, Hegel, and Schleiermacher--but he also devoted himself intensely to the independent study of Schopenhauer and Darwin. In 1874, Vaihinger completed his dissertation under the supervision of the logician Christoph von Sigwart with a prize-winning essay, entitled Recent Theories of Consciousness according to their Metaphysical Foundation and their Significance for Psychology. Later in the same year, he reported to Leipzig for compulsory military service, but was excused due to his poor eyesight. Free of his military duties, Vaihinger had the opportunity to attend lectures at the University from, among others, the founder of empirical psychology Wilhelm Wundt. It was during this time that Vaihinger first encountered the work of a figure who, next to Kant, would be his most important and lasting philosophical influence, the Neo-Kantian Friedrich Lange. Vaihinger describes the impact Lange's History of Materialism made on him as follows: "Now at last I had found the man whom I had sought in vain during those four years [at Tubingen]. I found a master, a guide, an ideal teacher... . All that I had striven for and aimed at stood before my eyes as a finished masterpiece. From this time onwards I called myself a disciple of F.A. Lange" (PAO, xxi-xxii). After a brief tenure in Leipzig, Vaihinger moved to Berlin, where he continued his studies with Hermann von Helmholtz and the Neo-Kantian Eduard Zeller. In 1876, Vaihinger published his first work, an exposition and defense of Lange's brand of Neo-Kantianism, entitled Hartmann, Duhring, and Lange: Towards a History of German Philosophy in the Nineteenth Century. It was here that the views which would form the basis of the PAO first began to take shape. In autumn of the same year, Vaihinger again relocated--this time to the University of Strassburg--and by early 1877 he had habilitated with a work entitled Logical Studies on Fictions. Part I: The Theory of Scientific Fictions. Though the PAO would not be published for more than 30 years, the Logical Studies is already, according to Vaihinger, "exactly the same" as Part I of the former work (PAO xxiii-xxiv). The extremely long gestation period of the PAO seems to have been the product of a variety of factors. First and foremost, monetary considerations compelled Vaihinger to find permanent academic employment. To further his candidacy, Vaihinger decided to bring out a Commentary on Kant's Critique of Pure Reason in honor of the centenary of the first Critique's publication. The first volume of the Commentary was released in 1881, and helped Vaihinger secure a permanent position at the University of Halle. The second volume followed in 1892. A sense of the scholarly depth of Vaihinger's work is given by the fact that the two-volume Commentary, which deals only with the KrV's Prefaces, Introductions, and Transcendental Aesthetic, reaches nearly 1,100 pages. Moreover, a number of personal and professional engagements occupied Vaihinger in the last decade of the century. In 1889 he married Elisabeth Schweigger, the daughter of a Berlin bookseller. In 1892 they had a son, Richard, and in 1895 a daughter, Erna. In 1896, he founded the journal Kant-Studien, and in 1901 the Kant Gesellschaft. Finally, in 1911, Vaihinger published the first edition of PAO. The work made him something like a philosophical celebrity virtually overnight. It proved so popular that it had gone through no fewer than 10 editions by the time of his death in 1933, attracting the attention of luminaries from a wide variety of academic fields (including Einstein, Ostwald, and Freud). Halle even became informally known as the "Vaihinger-Stadt" (Vaihinger city). The PAO's success enabled Vaihinger to found, in 1918, yet another journal, Annalen der Philosophie und philosophischen Kritik, which dealt specifically with the themes broached in the PAO, and included contributions from figures such as the mathematician Moritz Pasch, the embryologist Wilhelm Roux, and the positivist philosophers Rudolf Carnap and Hans Reichenbach. In 1930, Carnap and Reichenbach took over editorship of the Annalen and renamed it Erkenntnis. Though this period of Vaihinger's life was marked by tremendous professional success, it was also a period of personal struggle. In 1906, suffering badly from cataracts, Vaihinger was forced to discontinue his lecturing. Eventually, he was completely blinded. In 1913, there occurred a rather bizarre incident: an anti-Semitic publication, the Semi-Kurschner, "accused" Vaihinger of being a Jew. With anti-Semitism already on the rise in Germany, Vaihinger felt compelled to defend his reputation in court, and sued the publication for defamation. Though the Semi-Kurschner ultimately removed his name, the incident perhaps explains in part why his reputation would continue to suffer under the Nazis. Additionally, Vaihinger had invested significant portions of his assets in Russia; the October Revolution, together with the economic crisis after the First World War left him in dire financial straits, and he was forced to sell the bookstore he had inherited from Elisabeth's parents. Struggle turned to tragedy when, in 1918, Vaihinger's daughter Erna committed suicide. His son Richard also began suffering from neuropathy brought on by the war, and by 1929 he was completely incapacitated. Vaihinger died in 1933, on the eve of the rise of the Third Reich. A long-time liberal and pacifist, Vaihinger and his work were treated with hostile silence throughout the Nazi period. A final, posthumous tragedy would thus befall him during these years: once among Germany's most celebrated and influential philosophers, Vaihinger became a virtual nobody, and his works were quickly forgotten. Though contemporary philosophical literature reveals a renewed interest in his ideas, it is fair to say that his reputation has never fully recovered. For a more detailed treatment of Vaihinger's biography, see (Simon 2014). Vaihinger himself discusses his intellectual development at length in his autobiographical (1921b) (also included as the introduction to C.K. Ogden's 1925 English translation of PAO). ## 2. Early Views and Intellectual Context With the hegemony of absolute idealism broken, Germany was awash with competing philosophical systems by the time Vaihinger habilitated in 1877. Three important movements are worth emphasizing (for detailed studies of the late nineteenth-century context, see Beiser [2013; 2014; 2015; 2017]). First, under the influence of Lange and others, Neo-Kantianism had become a major force on the intellectual scene. Second, due partly to Schopenhauer's rising star, and partly to the continued influence of Hegel and Schelling, grand metaphysical theorizing after the idealist model was experiencing a resurgence in the work of Adolf Trendelenburg, Hermann Lotze, and Eduard von Hartmann. Finally, traditional materialism and empiricism, though very much on the defensive, had nevertheless gained back considerable ground since their nadir in early decades of the century. Vaihinger's first intervention in these foundational debates came in the form of his short tract Hartmann, Duhring, and Lange. Its title notwithstanding, the book is less a work of historical exegesis, and more a polemical defense of Lange's Kantian standpoint against the dual assault of naive realism (represented by the positivism of Eugen Duhring) and unchecked metaphysical speculation (exemplified by Hartmann's Neo-Schopenhauerian view, according to which fundamental reality is a monistic vital substance). Vaihinger's chief charge against Hartmann and Duhring is that they are dogmatists, that they systematically overstep the critical strictures Kant had placed on cognition, claiming knowledge of things in themselves (1876, 10, 17). Both commit "the original sin of post-Kantian philosophy"--Hegel's "fantastic petitio principii" of "the unity of thought and being" (1876, 67). Lange's Neo-Kantianism, by contrast, represents something like the best of both worlds, and so promises a middle path in a seemingly intractable dispute. With Duhring, Lange emphasizes the importance of grounding philosophy in the results of the empirical sciences. But with Hartmann, he avoids naively eliding the empirical objects studied by the sciences with things in themselves. It is not, however, only Lange's in-principle restriction of knowledge to appearances that Vaihinger wishes to champion. More importantly, he holds that Lange has demonstrated that the empirical sciences themselves conclusively support the transcendental idealist's position. In Chapter IV, Part 3 of the second volume of his magnum opus The History of Materialism and Critique of its Contemporary Significance, Lange draws on the work of Johannes Muller and Helmholtz to argue that "the physiology of the sense organs is the developed, or corrected Kantianism, and Kant's system can, as it were, be viewed as a program for the more recent discoveries in this field" (1873-75, 2:409). The basic idea was that such research had shown that our perceptual apparatus plays an active role in structuring our experience, and that the output of such a process does not qualitatively resemble the initial external stimulus, for which it is a mere "sign." Since what we perceive is always the product of such unconscious physiological processes, it supposedly follows that we have no knowledge of the objects that ultimately cause our perceptions (for more on these sorts of arguments, see [Hatfield 1990, 165-71, 195-208, and esp. 208-218; Beiser 2014, 199-205, 381-86], and the entries on Hermann von Helmholtz, and Friedrich Lange). Thus, if Duhring rejects transcendental idealism, he simply has not consistently followed his own prescription that philosophy should base itself on the empirical sciences. And if Hartmann claims to know the nature of things in themselves, he can do so only by flagrantly ignoring the results of those sciences. Though perhaps rhetorically effective, this argument leads to a basic tension in Vaihinger's early view. Vaihinger sums up Lange's conclusion as follows: "the result of the physiology of the senses is that we do not ... perceive external objects, but rather ourselves first produce the appearance of such things, specifically as a consequence of the affection by transcendent objects" (1876, 56). These appearances are the result of our "physiological organization" operating on sensory stimuli. But he goes on to concede that this "'organization' signifies only the unknown ... Y whose collaboration with the 'thing in itself', X, produces the world of inner and outer experience" (1876, 57), that "even our own body ... is only a product of our optical apparatus" (1876, 57), and that "not merely the external world, but also the organs with which we grasp it are mere images of what truly exists, which itself remains unknown to us" (1876, 58). So, according to the Lange-Vaihinger view, the objects of our experience are the joint product of affection by things in themselves and the operations of our physiological organization. This is supposed to have the conclusion that things in themselves are entirely unknowable, a fortiori not knowable through the methods of the empirical sciences. The problem becomes clear, then, when we ask what sort of thing 'physiological organization' is supposed to signify. If it refers to a thing in itself (the cognitive apparatus in itself), then the empirical sciences could establish nothing about it. In this case, the Langean argument is entirely impotent. If it refers to an appearance, then there is evidently a circle in the argument: the existence of appearances is supposed to be explained by the causal collaboration of things in themselves and the perceptual apparatus, but the perceptual apparatus is itself an appearance. Finally, if the physiological organization is both an appearance and a thing in itself, then the empirical sciences can in fact yield knowledge of at least some things in themselves. This, of course, is precisely what the argument is meant to forestall. Vaihinger does little to assuage these problems in (1876). Sometimes he suggests that they are rooted in the fact that the very notion of a thing in itself involves a contradiction, albeit one which is "grounded in the contradictory constitution of our cognitive faculty itself" (1876, 34). In other words, Vaihinger seeks to view the issue of things in themselves along the lines of a Kantian antinomy--a conflict which arises necessarily from the nature of our reason, but which can at least be recognized as such. This, however, is of little help. For one, it is questionable whether the basic epistemological standpoint of (1876) can even be coherently stated without commitment to things in themselves. For another, a crucial feature of Kant's antinomies is that they are resolved by his transcendental idealism, thus lending that doctrine an indirect proof (KrV, A 506/B 534). If the critical philosophy itself were to generate insoluble contradictions, that would be reason to reject, not accept, it. Vaihinger would continue to struggle with these issues in the PAO, which still aims to advance the broadly Langean project of naturalizing Kantianism. As we shall see in SS6, however, Vaihinger's conception of Kantianism undergoes a considerable shift, now taking the form of what he calls "idealistic positivism", "positivist idealism", or even (in what is evidently the first use of the phrase) "logical positivism." For more on Vaihinger's debt to Lange, see (Ceynowa 1993, 133-72; Heidelberger 2014). ## 3. Interpretation of Kant Before returning to Vaihinger's own positive philosophy, it is worth examining the task that occupied him intensely between the publication of (1876) and the PAO: the interpretation of Kant. The rise of Neo-Kantianism in the second half of the nineteenth century brought with it a renewed interest in the historical study of the critical philosophy. The majority of the works written during this period tended to approach Kant's texts polemically, either with the aim of defending and "updating" them, or with the intention of refuting them. In terms of its scholarly sobriety, depth of textual analysis, and sensitivity to Kant's immediate intellectual context, Vaihinger's Commentary was unique. He notes that his approach will be that of those who analyze ancient texts "with philological sobriety and strict rigor ... in order to determine their genuine meaning from an historical standpoint" (1881-92, 1:iii). The extent to which Vaihinger resists allowing his own philosophical commitments to influence his interpretation is indeed impressive. In fact, the very idea that understanding Kant's texts might require a distinctively historical kind of approach seems to have been something of a novelty and is among Vaihinger's lasting contributions to Kant scholarship. There are other important, and more substantive, respects in which the Commentary was at the vanguard of Kant scholarship. The dominant trend in the interpretation of Kant's theoretical philosophy throughout most of the nineteenth century had been the physio-psychological reading advanced by such figures as Schopenhauer, Jakob Fries, Friedrich Beneke, Jurgen Bona Meyer, Helmholtz, and of course Lange (Beiser 2018, 64-72, 81-84; 2014, 209-11, 460-63). According to this tradition, Kant's main goal in the Critique was the discovery of the causal mechanisms underlying perception. Kant's conception of the a priori was accordingly understood as referring to innate or ingrained properties of the psychological subject that played an active role in the construction of perception and acquisition of empirical knowledge. The first to break decisively with this tradition was probably Hermann Cohen. Cohen (1871) highlighted the epistemological side of Kant's project--his interest in justifying various claims to knowledge. Accordingly, Cohen interpreted the a priori as referring to the logical, rather than psychological, conditions of knowledge (1871, 208). In this respect, Cohen's influence on the Commentary was decisive. Vaihinger insists at the outset that Kant's "philosophy is, in the first instance, epistemology [Erkenntnistheorie]" (1881-92, 1:8). Vaihinger certainly does not, however, accept Cohen's interpretation tout court. Though Cohen too had emphasized the need for historical sensitivity in interpreting Kant, his aims were ultimately still polemical: he believed more careful attention to Kant's texts could exonerate his system from some of the perennial charges of inconsistency. For example, Kant's doctrine that things in themselves exist independently of us and causally affect our minds to produce our representations had long been thought to violate his doctrine that the categories have validity only for objects of possible experience. Another charge, very much alive at the time, was that Kant's key arguments for the non-spatiotemporality of things in themselves failed on the grounds that they showed, at best, only the subjectivity of space and time, not their mere subjectivity. According to this criticism, Kant overlooked the possibility that space and time might be both subjective forms of intuition and objective properties of things in themselves (the classic statement of this criticism is [Trendelenburg 1867]). Cohen's response to both charges was to argue that interpreters were wrong to read Kant's doctrine of things in themselves as positing a class of entities ontologically distinct from appearances. Instead, he maintained, the Kantian thing in itself is a mere "limiting concept" (Grenzbegriff), to whose actual existence Kant was never committed (1871, 252, 268). Vaihinger, by contrast, does not hesitate to admit that the above objections are genuine problems for Kant. Indeed, he believes they pinpoint serious inconsistencies in Kant's position. For example, against Cohen's deflationary treatment of the thing in itself, Vaihinger notes, "Kant speaks thousands of times of affecting things in themselves; the impossibility of the existence and causal efficacy of things in themselves is, by contrast, merely a consequence that Kant's readers must certainly draw from his theory of the categories, but which Kant himself hints at only seldom, and even then only timidly" (1881-92, 2:47) (the actual target of this remark is Fichte, but the point applies to Cohen's reading too). Vaihinger's goal in the Commentary is not to exonerate Kant from such inconsistencies, but to explain why he fell into them. In particular, Vaihinger argues that careful attention to both Kant's pre-critical writings and his unpublished notes reveals that the Critique is really a "patchwork" of inconsistent views and arguments from various periods of Kant's intellectual development. It consists of "geological strata which differ in both time of composition and content" (1902, 27). In the case of the Deduction, Vaihinger even claims to have been able to trace precisely each portion of the text to a particular series of notes from the decade between the Dissertation and the KrV (1902). In this respect, then, Kant's great book resembles more the texts of Homer than the work of a single author, writing at a single time, with a single set of views. More specifically, Vaihinger believes the development of Kant's thought proceeds through six stages. From 1750-1760 Kant is a dogmatic rationalist after the Leibnizian-Wolffian model. Then, under the influence of Locke and Hume, he endorses empiricism from 1760-1764. With the publication of Dreams of a Spirit Seer, he begins to adopt the standpoint of the critical philosophy, but by the time of the 1770 Inaugural Dissertation, the influence of Leibniz (particularly of the Nouveaux Essais) reasserts itself. This reversion to dogmatism lasts until 1772 when he becomes a skeptic, again under the influence of Hume. Finally, he arrives at the mature critical standpoint with the publication of the Critique of Pure Reason in 1781 (1881-92, 1:47-49). Unfortunately, on Vaihinger's reading, doctrines from the first five periods continue to figure in Kant's arguments for and even the core tenets of the critical philosophy. It is this which explains the numerous inconsistencies besetting Kant's system. A clear example of Vaihinger's basic approach is his treatment of Kant's famous argument for the subjectivity of space from incongruent counterparts. In the Prolegomena, Kant argues thus: > > > > There are no inner differences here [in the case of incongruent > counterparts] that any understanding could merely think; and yet the > differences are inner as far as the senses teach ... one > hand's glove cannot be used on the other... . These objects > are surely not representations of things as they are in themselves, > and > as the pure understanding would cognize them, rather, they are > sensory intuitions, i.e., appearances. (4:286) > > Roughly, Vaihinger reconstructs the argument as follows: (1) Some differences in the spatial properties of objects cannot be known through the pure understanding; (2) All differences between things in themselves can be known through the pure understanding; (C) Space is not a thing in itself. Premise (2), however, violates Kant's noumenal ignorance doctrine, according to which things in themselves cannot be known through the understanding's categories. Thus, Vaihinger concludes, "this assumption [that things in themselves are knowable through the pure understanding] is obviously anathema to Kantian criticism, and an archaic reversion to dogmatism" (1881-92, 2:528). More specifically, his claim is that the premise is residue from Kant's view in the 1770 Dissertation, according to which things in themselves are objects as the pure understanding knows them (1881-92, 2:354, 453, 522). Vaihinger's reading of the KrV, and especially of the Deduction, has become known as the "patchwork thesis." The patchwork reading has been influential in Anglophone scholarship, due largely to its adoption by Norman Kemp Smith (1918), but it remains controversial. Since the reading apparently requires us to assume that Kant failed to understand that combining views he knew were inconsistent with one another would produce a view that is itself inconsistent, it may be objected that, "if such a conclusion is true, we must say that Immanuel Kant was either abnormally stupid, or else that he took his thinking in a frivolous spirit" (Paton 1930, 157). However, the thesis that the Deduction, regardless of its precise method of composition, at least contains multiple inconsistent lines of argument, rather than a single unified argument, has found able defenders since Vaihinger (see e.g., Wolff 1963; Guyer 1987). For a recent challenge to this revised patchwork reading see (Allison 2015). Vaihinger's interpretive work during the 1880's and 90's did not extend to the other major part of the KrV, the Transcendental Dialectic. The PAO, however, includes a supplementary appendix in which Vaihinger lays out a novel interpretation of this chapter (PAO 201-35). The Dialectic was the part of the KrV that clearly had the most significant impact on Vaihinger's own thought, and his interpretation of it is radical and interesting. It will therefore be worth briefly examining before turning to the main argument of the PAO. In the Dialectic, Kant claims that certain concepts (e.g., those of an immaterial soul, freedom of the will, and God) have their root in pure reason. He argues that such concepts are mere "Ideas" (Ideen), in the sense that they are not constitutive principles of experience, and thus yield no "cognition" (Erkenntnis) of objects. Nevertheless, he suggests they are "not arbitrarily thought up" (A 327/B 384), but have a number of important "regulative" uses. In his works on moral philosophy, Kant goes on to argue that we can even have a certain kind of "practical cognition" of or "rational faith" in the Ideas. A natural interpretation of Kant's position regarding the Ideas holds that we lack theoretical justification for believing that they refer to objects, and this lack of justification is such that we cannot claim knowledge of the existence or non-existence of such objects. Practical reason, however, may provide justification for the belief in (say) the existence of God, even if not of a sort sufficient to yield knowledge of him. Vaihinger disputes this reading. Instead, he interprets the Ideas as self-conscious fictions, which are nevertheless licensed by their theoretical and practical utility. This utility justifies us in "making use" of such concepts--looking, e.g., at nature as though it were the product of an intelligent designer--but never in believing they refer to objects. The interpretation relies crucially on two factors: first, a suggestive passage in which Kant calls Ideas "heuristic fictions" (A 771/B 799), and contrasts them with hypotheses; and second, Kant's frequent use of 'als ob' and related locutions when describing the regulative employment of the Ideas. Both the distinction between fictions and hypotheses, and the idea of treating reality as if certain concepts appropriately applied to it are major themes that find extensive development in the PAO. Vaihinger's interpretation of the Ideas as fictions is highly controversial, and was already subjected to withering criticism in his own day by Erich Adickes (1927). Adickes accuses Vaihinger of ignoring countless passages that flatly contradict his fictionalist reading, and of misleadingly paraphrasing or eliding passages he does cite. In one way, at least, this criticism is unfair. Vaihinger explicitly concedes: "in Kant we also find in the same contexts many passages which permit or even demand a contrary interpretation... . Two tendencies are revealed, a critical and a dogmatic, a revolutionary and a conservative" (PAO 212). Vaihinger's interpretation is not, unlike his work in the Commentary, intended to be strictly historical; he is rather presenting one strand in Kant's thought that he finds especially promising. Adickes nevertheless makes a compelling case that even those passages on which Vaihinger does rely do not speak decisively in favor of his interpretation. Most contemporary Kant scholars follow Adickes in rejecting the fictionalist interpretation of the Ideas, though it continues to find some supporters (see e.g., Shaper 1966; Korsgaard 1996, 162-76; and esp. Rauscher 2015). ## 4. The Epistemology of the PAO I turn now to Vaihinger's chief systematic work, the PAO. That this work represents his attempt to continue Lange's project of naturalizing Kantian epistemology (Heidelberger 2014, 51-55) may appear surprising given Vaihinger's interpretation of Kant. Confusion here may be forestalled by attending to something Vaihinger himself says about the aim of the work: > > > > There were two possible ways of working out the Neo-Kantianism of > F.A. Lange. Either the Kantian standpoint could be developed, on the > basis of a closer and more accurate study of Kant's teaching, and > this is what Cohen has done, or one could bring Lange's > Neo-Kantianism into relation with empiricism and positivism. This has > been done in my philosophy of "As if," which also leads to > a more thorough study of Kant's "As if" theory. > (PAO xxii) > > So Vaihinger distinguishes between the views maintained by the historical Kant, and Lange's brand of Kantianism, which his own systematic work defends and develops. But, the PAO also builds on Lange's position in non-trivial ways. Vaihinger's autobiographical essay "The Origin of the Philosophy of 'As If'" points to two further significant influences, Arthur Schopenhauer and Charles Darwin. Vaihinger's interest in Schopenhauer was sparked by the doctrine that the intellect is "a mere tool in the service of the will" (Schopenhauer 1958, 2:205). This view is based in part on Schopenhauer's metaphysics, according to which the ultimate nature of reality is nothing other than a suprapersonal, aimless, quasi-volitional striving or "will." Schopenhauer holds that this "will" expresses itself empirically in the striving of organic beings to preserve their individual existence, and the intellect has accordingly developed as a mere tool to aid more complex organisms in this task. Vaihinger believed that Darwin's theory of natural selection had provided an empirical grounding for Schopenhauer's hypothesis (PAO xviii). This was crucial, since Vaihinger's own Kantianism prevented him from endorsing Schopenhauer's speculative vitalist metaphysics (PAO xvii). At the same time, Vaihinger comes to view Kant's doctrine of the essential limits of our cognition through this Schopenhauerian-Darwinian lens: the "limitations of human knowledge," he says, are "a necessary and natural result of the fact that thought and knowledge are originally only a means, to attain the Life-purpose, so that their actual independence signifies a breaking-away from their original purpose; indeed, by the fact of this breaking-loose, thought is confronted by impossible problems" (PAO xviii). Vaihinger will use this idea to motivate his own fictionalism, and to provide a radical reinterpretation of Kantianism. These various influences are drawn together in the introductory chapters of PAO. Here, Vaihinger presents a view of the "psyche" as a purely natural faculty, akin to a bodily organ, and bound by the same laws as the rest of the organic world. All psychic activities, including "scientific thought," are "to be considered from the point of view of an organic function" (PAO 1). Like bodily organs, the psyche is designed (or rather, selected) to perform a particular role within the total economy of the organism. Its purpose is "to change and elaborate the perceptual material into those ideas, associations of ideas, and conceptual constructs," which produce "a world [in which] objective happenings can be calculated and our behavior successfully carried out in relation to phenomena"(PAO 2). Its purpose is therefore not "to be a copy [Abbild] of reality," but rather "an instrument for finding our way about more easily in this world" (PAO 22, translation altered). Though broadly in the spirit of Schopenhauer and Darwin, this instrumentalist conception of the mind is, as Ceynowa has argued (1993, 35-132), also more specifically indebted to the nineteenth century psychologist Adolf Horwicz. Whatever its exact provenance, Vaihinger's basic thought is that what is selected for in the evolution of our cognitive capacities is the ability to aid us in carrying out activities that are useful for the purposes of survival. Specifically, Vaihinger seems to reason that any course of action--say, procuring food for oneself--involves at least some minimal degree of planning, and that in turn requires some basic ability to predict what the future will be like, and to know how it will be different if we choose to ph rather than to ps. Thus, he suggests that the purpose of thought is "to calculate those events that occur without our intervention and to realize our impulses appropriately" (PAO 3). For the bare purposes of carrying out those actions necessary for survival, Vaihinger is claiming, the intellect would not have to be a reliable guide to the way the world really is. On the contrary, fictions may have an indispensable role to play here. What are we to make of this sort of argument? One might suggest that the fact that our cognitive apparatus is a product of natural selection supports, rather than undermines, the idea that it is a reliable guide to the way the world is. For, being mistaken about how things stand in one's environment is surely an impediment, not an aid, to survival. Vaihinger would likely respond that this only shows that we have to get the right things about the world right. It does not mean that our total image of the world is, or even could be, a copy of reality, accurate in all its details. Vaihinger's view is that the picture of the world that is easiest to work with will depart from reality in certain crucial respects. One may now wonder, however, what is to be made of Vaihinger's near exclusive emphasis in PAO on fictions in the natural sciences and mathematics. After all, the theorizing in fundamental physics is a rather different matter than determining how to find a meal, and even if such theories turn out to be useful down the line, it seems that life would (and does) manage just fine without them. Here, Vaihinger relies on another Darwinian idea, which he calls the "Law of Preponderance of the Means over the End," according to which "means which serve a purpose often undergo a more complete development than is necessary for the attainment of their purpose" (PAO xxix). In the language of contemporary evolutionary biology, scientific thought is like a "spandrel." At the same time, Vaihinger is keen to emphasize the connection between scientific theorizing and more humdrum cognitive processes. He adopts the broadly Kantian view that what is given to us in experience is merely a "chaos of sensations" (PAO 117), which needs to be ordered and refined by the logical functions of thought, or categories. For example, the repeated co-occurrence of sensations of a branching shape and sensations of green leads the psyche to postulate a relation of substantial inherence between a thing or substance (the tree) and its attribute (green) (PAO 122-23). However, he argues in a manner evocative of Berkeley, the notion of a substance is incoherent since it implies the existence of a bare substrate with no properties whatsoever (PAO 125-26). The fiction is nevertheless justified, he suggests, because it facilitates the recognition of regularities in the occurrences of our sensations, and the communication of these regularities to others (PAO 130). Given that our most immediate relation to the world is thus mediated by the categories (which also include the relations of part-whole, cause-effect, inter alia), it is natural to suppose that these fictions will resurface in our more sophisticated theories, e.g., in the form of the atom (a simple material substance). Vaihinger is not suggesting that genuine knowledge that serves no particular practical aim is impossible; but he is suggesting that our total theoretic image of the world is ultimately limited by those fictions that have been preserved as most adaptive for our basic practical aims (PAO 130). Vaihinger's emphasis on the deep connection between cognition and action have led many interpreters to liken his views to those of the American pragmatists (e.g., Appiah 2017, 4; Koridze 2014; Fine 1993, 4; and esp. Ceynowa 1993). Others have disputed this connection (Bouriau 2016). Vaihinger himself tells us that fictionalism and pragmatism share important similarities, but nevertheless are distinguished by the fact that pragmatism accepts, whereas fictionalism rejects, the principle that "an idea which is found to be useful in practice proves thereby that it is also true in theory" (PAO viii). So, if one defines a pragmatist narrowly as someone who endorses this Jamesian definition of truth, Vaihinger is no pragmatist; his core contention is that "we may not argue from the utility of a psychical and logical construct that it is right" (PAO 51). Nevertheless, given Vaihinger's emphasis on the inseparability of practical and theoretical thought, his view that the use of a concept can be justified on purely practical grounds, and so forth, it does not seem unfair to liken him to the pragmatist tradition in a wider sense. ## 5. The Nature of Vaihinger's Fictionalism Central to Vaihinger's project in the PAO is a distinction between two sorts of fictions: (1) what he (somewhat inaptly) terms 'semi-fictions,' which are "methods and concepts based upon a simple deviation from reality" (PAO 59); and (2) 'real' or 'genuine fictions' which are self-contradictory concepts (ibid.). In other words, semi-fictions are statements or propositions that happen not to correspond to the world; genuine fictions are those that claim something impossible about the world. An important example the first kind is what Vaihinger calls the "abstractive" or "neglective" fiction (PAO, 14-17, 140-41). Such fictions are roughly what we have in mind today when we speak of 'idealized models' (contemporary philosophers of science sometimes distinguish further between "Aristotelian" and "Galilean" methods of idealization. See the article on Models in Science. Vaihinger does not clearly distinguish these). A standard example is the model of the planetary system given in classical mechanics: > > > > In physics we find such a fiction in the fact that masses of > undeniable extension, e.g., the sun and the earth, in connection with > the derivation of certain basic concepts of mechanics and the > calculation of their reciprocal attraction are reduced to points or > concentrated into points (gravitational points) in order, by means of > this fiction, to facilitate the presentation of the more composite > phenomena. Such a neglect of elements is especially resorted to where > a very small factor is assumed to be zero. (PAO > 16) > > The idea is familiar. If what we are interested in is determining the trajectory and velocity of the planets around the Sun, then it is useful to ignore or iron-out a wide variety of features of the system, e.g., the size of the planets, the presence of friction, the gravitational pull that the planets and their moons exert on one another. Here, then, we evidently have a clear-cut case in which "pretending" that certain things are true of the solar system can greatly aid our aim of prediction. As Vaihinger notes, we get the "right" results here because the properties of the system we are ignoring have only a negligible influence on the phenomena we are interested in studying. The solar system behaves nearly just as it would if the planets were mass points, etc. This suggests a counterfactual understanding of fictions: by reasoning from assumptions that are false--sometimes radically false--at the actual world, we may still arrive at conclusions that are true, or quite nearly true, at the actual world. If those results are arrived at more efficiently by entertaining the fiction, then the fiction is justified. If this is Vaihinger's view, however, it would appear to face a serious difficulty. For, his main interest is in the so-called "real fictions," which, as noted above, are not merely false, but contradictory. Since everything follows from a contradiction, any counterfactual whose antecedent is a real fiction will be trivially true, and this is to say they cannot possibly do the work Vaihinger envisions (Appiah 2017, 11; Cohen 1923, 485). One way to respond, here, is to note that the logic with which Vaihinger operates is still essentially the version of Aristotelian logic found in Kant and throughout much of the nineteenth century. Such pre-twentieth-century logics often implicitly reject the principle ex contradictione quodlibet. See, e.g., the discussion of conditionals in the influential Port-Royal Logic (Arnauld and Nicole 1996, 100). Relatedly, but from a more contemporary perspective, one might read Vaihinger as anticipating some recent views on counterpossible reasoning (Pollard 2010). Consider the following counterfactual conditionals: * (1) If Fermat's last theorem were false, then no argument for it would be sound. * (2) If Fermat's last theorem were false, then Andrew Wiles would deserve eternal fame. * (3) If intuitionistic logic were correct, then the Law of Excluded Middle would fail. * (4) If intuitionistic logic were correct, then the Law of Excluded Middle would hold. Intuitively, (1) and (3) are true while (2) and (4) are false. Moreover, someone who rejects intuitionism might well wish to argue against intuitionism by relying on something like (3), even though what the antecedent states is, by her lights, impossible. These facts have led some to suppose that (1) and (3) are not merely true, but non-trivially true (for more on these sorts of arguments, see the entry for Impossible Worlds). Vaihinger's fictionalism might thus be read as adding a heap of substantive examples to the list of alleged non-trivial counterpossible conditionals. This sort of reading seems supported by examples that Vaihinger himself gives, and the theory of "fictive judgments" he bases on them. One of Vaihinger's favorite examples is: * (5) If the circle were a polygon, then it would be subject to the laws of rectilinear figures. (PAO, 193) Though (5)'s antecedent and consequent both state something impossible, Vaihinger contends it forms the basis of Archimedes's proof of the area of a circle using the method of exhaustion (PAO 177). Here, then, is a case in which a proof no one would wish to call trivial allegedly proceeds from a contradictory assumption. Generalizing, Vaihinger claims, "in a hypothetical connection [sc. conditional proposition], not only real and possible but also unreal and impossible things can be introduced, because it is merely the connection between the two presuppositions and not their actual reality that is being expressed" (PAO 193-94). Anthony Appiah rejects this kind of interpretation, on the grounds that such counterpossibles are unintelligible (2017, 11-17; cf. Wilholt 2014, 112). Instead, he suggests Vaihinger's talk of useful contradictions is more fruitfully understood as referring to the use of multiple mutually incompatible models. In particular, the point is that we may find it useful to regard certain phenomena in one way for some purposes, but in another, incompatible way for other purposes. Even when it turns out to be useful to combine incompatible models for understanding one and the same phenomenon (or range of phenomena), Appiah suggests, Vaihinger need not be committed to a paraconsistent logic. Rather, drawing on (Cartwright 1999), he suggests that using a model does not involve drawing out each of its logical implications, but instead a kind of skill of "knowing which lines of inference one should and which one should not follow" (2017, 13). Moreover, because our main aim in using models is not ultimately to obtain the truth about the physical world, but to get around in it, there is no reason to avoid such inconsistency (2017, 14-17). Appiah's interpretation allows those who are wary of paraconsistent logics to salvage something philosophically promising from Vaihinger's talk of contradictions. It also perhaps has the advantage of doing greater justice to the multifarious character of Vaihinger's theory of fictions. For, while Vaihinger claims that the fictive judgment has a "hypothetical element," he does not simply reduce fictive judgments to conditionals (PAO 194). On the other hand, the interpretation stumbles against texts in which Vaihinger suggests that the contradictions of real fictions exist within theories or models, not between them. For example, the concept of a (classical) atom is supposed to be a real fiction because it is absurd to suppose something which occupies space, but nevertheless has no extension. This is to say that the atomistic fiction is itself contradictory, not merely that it conflicts with other ways of modeling matter. Another way of putting the worry is to note that the interpretation blurs the apparently sharp distinction between semi-fictions and real fictions: any model will be a real fiction as long as we use it together with another model incompatible with it. For yet another approach, which denies that Vaihinger has or needs a general account of how contradictory fictions can be fruitful, see (Fine 1993, 9-11). On this reading, Vaihinger is interested only in surveying actual scientific practice, and revealing that, as a matter of fact, it does make fruitful use of real fictions; the question of how this is possible is beside the point. There is something to this characterization of Vaihinger's procedure, which trades more in examples than in general pronouncements. However, it will also be difficult for proponents of this reading to explain passages like these: "This purpose [of thought] can only be that of facilitating conceptual activity, of effecting a safe and rapid connection of sensations. What we have to show, therefore, is how fictional methods and constructs render this possible" (PAO 52, emphasis added); "the philosophy of as if shows that and why these false and contradictory representations are still useful" (1921a, 532, emphasis added). Here, Vaihinger seems to indicate that his purpose is not just to enumerate instances of useful fictions, but to explain how and why they are useful. In addition to Vaihinger's specific analysis of fictions, one might wonder about his precise brand of fictionalism. Contemporary philosophers distinguish between "hermeneutic fictionalism" and "revolutionary fictionalism" (see the entry on Fictionalism). Hermeneutic fictionalism holds that certain domains of discourse are fictional, in the sense that their participants are not aiming at literal truth, but only seeming to do so. Some textual evidence speaks in favor of reading Vaihinger as a hermeneutic fictionalist. For example, he writes: "the axiom and the hypothesis ... endeavor to be expressions of reality. The fiction, on the other hand, is not such an expression nor does it claim to be one" (PAO 60). And Vaihinger's theory of the fictive judgment maintains that many judgements of the syntactic form 'A is B' are properly analyzed as, or merely elliptical for, 'A is to be regarded as if it were B'--"the 'is' is a very short abbreviation for an exceedingly complicated train of thought" (PAO 195). On the other hand, in many places Vaihinger clearly indicates that he thinks we often treat what would be better regarded as fictional discourse as aiming at truth. Many disputes, he suggests, can be resolved by reinterpreting such discourse fictionally. Thus, for example, proponents of classical atomism have been wrong to think that their theory is true, but their opponents have also been wrong to believe that the theory should therefore be given up (PAO 53). He makes an analogous point about the dispute between substantivalists and relationalists over the nature of space (PAO 170-71). So, Vaihinger appears to be a revolutionary fictionalist at least about some discourses in which he is interested. ## 6. The As-If Philosophy and Kantianism The most obvious aim of the PAO is to establish the centrality of fictions to our cognitive life. But there is also a broader, and in some ways more ambitious, aim to the book--to provide the kind of "corrected Kantianism" Lange had tried to establish in the History of Materialism. As we saw above, Vaihinger views Kant's system as riddled with inconsistencies, which arise from the alleged fact that Kant retains certain elements of his pre-critical dogmatism even after adopting the critical standpoint. The As-If philosophy can be seen as an attempt to retain what Vaihinger takes to be Kant's core philosophical insights, while removing the inconsistencies that beset the surrounding system in which those insights were originally couched. As was noted above, a major problem for Vaihinger's early view, as for many of the Neo-Kantians of his day, was the status of the thing in itself. Vaihinger now consistently rejects this notion, predictably labelling it a fiction. Given that things in themselves are supposed to be transcendent substances that causally affect us, this rejection follows immediately from his view that the concepts of substance and cause are themselves fictions (PAO 55-56). True to form, however, Vaihinger still holds that the concept of things in themselves has some utility: as Cohen had argued, it serves as a "limiting concept" (PAO 55), ensuring that we restrict our inquiries to what is empirically given to us. As we have seen, Vaihinger puts an instrumentalist spin on this Kantian idea; all knowledge is empirical in the sense that our guiding cognitive aim is the prediction and control of empirical phenomena, not correspondence to objective reality (though, once again, it bears emphasizing that Vaihinger does not seem to be claiming that such correspondence is impossible, only that it is incidental to our main epistemic aim). Vaihinger's rejection of the thing in itself is implicit in the general label he gives to his own philosophical view, "idealistic positivism." The 'positivism' here is in part an epistemological, and in part a metaphysical thesis. The epistemological thesis is that "what is given consists only of sensations" (PAO 124; cf. xxviii). The metaphysical thesis is that "what we usually term reality consists of our sensational contents [Empfindungsinhalte]" (PAO xxx, emphasis added); "Nothing exists except sensations" (PAO 44, emphasis added; cf. 1921a, 532-33). This extreme view (for which Vaihinger does not argue) has the immediate conclusion that all knowledge is of appearances, since nothing exists except appearances. In addition to this radical kind of empiricism, the PAO continues to uphold the broadly Kantian view that cognition is necessarily limited, or, as Vaihinger sometimes puts it, that there is no "identity between thought and being." However, it is no longer limited in the sense of being cut off from some transcendent realm of things in themselves, but in being essentially incomplete. The point is best appreciated by attending to another distinction central to the PAO--that between fictions and hypotheses. Hypotheses (not to be confused with so-called "hypothetical judgments") are judgments that are "problematic" in form, i.e., which have the form 'it is possible that A is B' (where the relevant sense of possibility is epistemic). Thus, if the judgment 'matter is composed of atoms' is a hypothesis, the person making it is maintaining that it may be true, and holds out hope that it may be confirmed or falsified in the future. By contrast, and as we have seen already, the fictive judgment has the form 'A is to be regarded as if it were B (when in fact A is not B).' Since Vaihinger holds that such judgments can be licensed by their practical utility, and because it is our practical aims that ultimately guide and constrain theoretical inquiry, Vaihinger infers that every theory will necessarily involve some fictions; not even the best theory will consist entirely of hypotheses, much less true hypotheses. The distinction between hypotheses and fictions is important for understanding Vaihinger's revised brand of Kantianism in another way as well. In the Dialectic, Kant had argued that dogmatic metaphysics necessarily leads reason into a series of conflicts, or "antinomies," with itself. It can produce, that is, equally valid arguments putatively establishing both that freedom exists and that freedom does not exist, that matter is composed of simples and that matter is not composed of simples, and so on. Kant claims that these conflicts are "natural" to reason, since reason has an innate drive to know the "unconditioned." He argues, however, that these conflicts can be avoided if (and only if) one recognizes the non-spatiotemporality of things in themselves. Transcendental idealism allows us to maintain that in some of the antinomial conflicts it is possible for both the thesis and antithesis to be false without contradiction, and that in others it is possible for both to be true without contradiction. The critical philosophy thus has the distinct advantage of being able to explain how certain apparently insoluble metaphysical debates arise, and of providing the only possible resolution of those debates. Vaihinger similarly posits a natural human tendency to confuse, not appearances and things in themselves, but fictions and hypotheses. Many statements that are introduced into our discourse as fictions end up being taken for hypotheses. And vice versa, many propositions that are originally intended as hypotheses are retained with mere fictional significance. Vaihinger calls this the "law of ideational shifts" (Gesetz der Ideenverschiebung) (PAO 92-99). Seemingly intractable debates in the sciences can be resolved, he maintains, by systematically recognizing this distinction. Thus, for instance, certain arguments against atomism are cogent if intended to establish that atomism is false (since it is contradictory); but they are erroneous if intended to imply that atomism should be rejected. Likewise, arguments for atomism are sure to fail if they are meant to establish the existence of atoms; but they are perfectly acceptable if they only aim to show that atomistic discourse is useful and indeed essential for scientific practice. For, the atomist's discourse is best understood as a conscious fiction. As Vaihinger suggests, the confusion arises from ignoring the essential practical dimension of thought (PAO xviii). How successful is Vaihinger in resolving the alleged tensions in Kant's critical philosophy? On one level, straightforwardly rejecting things in themselves and endorsing an ontology only of sensations clearly avoids the supposed problem of attributing causal efficacy to transcendent objects. On another, however, it would appear that Vaihinger cannot so easily reject things in themselves without causing problems for his basic epistemological view. For one, an obvious question arises as to what it is that has the sensational contents that serve as Vaihinger's ontological basics. For another, we have seen that Vaihinger's arguments in the Introduction hold that the mind or "psyche" operates on sensory stimuli, and that it has certain ends to which fictions are expedient means. If nothing exists except sensations, then it seems we must read Vaihinger's talk of 'mind' loosely, perhaps as referring to some particular kind of collection of or relation between sensations. But, one is tempted to say, it is simply incoherent to suggest that sensations themselves are the bearers of sensations, that they have aims, that they operate on and systematize other sensations, etc. At the very least, "operation" would seem to be a causal notion, so that Vaihinger's causal fictionalism threatens to undermine the arguments at the foundation of his system. Vaihinger sometimes attempts to respond to these sorts of worries. He suggests, for example, "From our point of view the sequence of sensations constitutes ultimate reality, and two poles are mentally added, subject and object" (PAO 56); "nothing exists except sensations which we analyze into two poles, subject and object ... . In other words, the ego and the 'thing in itself' are fictions" (PAO 44). The question of course is who the we is here that is doing the mental adding. Perhaps Vaihinger intends such talk to itself be fictional. But he cannot hold that without undermining the very motivation for his fictionalism. Nor does Vaihinger ever make clear and precise exactly what "poles" of sensations are supposed to be. One may also worry that, throughout the PAO, Vaihinger has reverted to a crudely psychologistic version of Kantianism that both he and Cohen had earlier rejected. On this score, Vaihinger is more easily defended. Though his fictionalism draws on certain biological and psychological theories, his main concern is still epistemological: "Apart from the general warning not to confuse fictions with reality, we may also call attention to the fact that every fiction must be able to justify itself, i.e., must justify itself by what it accomplishes for the progress of science" (PAO 79). There would arguably be a confusion if Vaihinger held that the fact that certain concepts (or dispositions to form them) are psychologically innate justifies us in believing that there exist objects exemplifying them. But that is precisely the view he wishes to reject. The appeal to the theory of natural selection is intended to suggest a pragmatic, rather than evidential, justification for the use of those concepts. In this sense, Vaihinger sees himself as drawing on Kant's method of justifying the concepts of God, freedom, and immortality on practical grounds, but extending it beyond the domain of moral philosophy.
logical-truth	## 1. The Nature of Logical Truth ### 1.1 Modality As we said above, it seems to be universally accepted that, if there are any logical truths at all, a logical truth ought to be such that it could not be false, or equivalently, it ought to be such that it must be true. But as we also said, there is virtually no agreement about the specific character of the pertinent modality. Except among those who reject the notion of logical truth altogether, or those who, while accepting it, reject the notion of logical form, there is wide agreement that at least part of the modal force of a logical truth ought to be due to its being a particular case of a true universal generalization over the possible values of the schematic letters in "formal" schemata like $(1')-(3')$. (These values may but need not be expressions.) On what is possibly the oldest way of understanding the logical modality, that modal force is entirely due to this property: thus, for example, on this view to say that (1) must be true can only mean that (1) is a particular case of the true universal generalization "For all suitable $P$, $Q$, $a$ and $b$, if $a$ is a $P$ only if $b$ is a $Q$, and $a$ is a $P$, then $b$ is a $Q$". On one traditional (but not uncontroversial) interpretation, Aristotle's claim that the conclusion of a syllogismos must be true if the premises are true ought to be understood in this way. In a famous passage of the Prior Analytics, he says: "A syllogismos is speech (logos) in which, certain things being supposed, something other than the things supposed results of necessity (ex anankes) because they are so" (24b18-20). Think of (2) as a syllogismos in which the "things supposed" are (2a) and (2b), and in which the thing that "results of necessity" is (2c): * (2a) No desire is voluntary. * (2b) Some beliefs are desires. * (2c) Some beliefs are not voluntary. On the interpretation we are describing, Aristotle's view is that to say that (2c) results of necessity from (2a) and (2b) is to say that (2) is a particular case of the true universal generalization "For all suitable $P$, $Q$ and $R$, if no $Q$ is $R$ and some $P$s are $Q$s, then some $P$s are not $R$". For this interpretation see e.g. Alexander of Aphrodisias, 208.16 (quoted by Lukasiewicz 1957, SS41), Bolzano (1837, SS155) and Lukasiewicz (1957, SS5). In many other ancient and medieval logicians, "must" claims are understood as universal generalizations about actual items (even if they are not always understood as universal generalizations on "formal" schemata). Especially prominent is Diodorus' view that a proposition is necessary just in case it is true at all times (see Mates 1961, III, SS3). Note that this makes sense of the idea that (2) must be true but, say, "People watch TV" could be false, for surely this sentence was not true in Diodorus' time. Diodorus' view appears to have been very common in the Middle Ages, when authors like William of Sherwood and Walter Burley seem to have understood the perceived necessity of conditionals like (2) as truth at all times (see Knuuttila 1982, pp. 348-9). An understanding of necessity as eternity is frequent also in later authors; see e.g., Kant, Critique of Pure Reason, B 184. In favor of the mentioned interpretation of Aristotle and of the Diodorean view it might be pointed out that we often use modal locutions to stress the consequents of conditionals that follow from mere universal generalizations about the actual world, as in "If gas prices go up, necessarily the economy slows down". Many authors have thought that views of this sort do not account for the full strength of the modal import of logical truths. A nowadays very common, but (apparently) late view in the history of philosophy, is that the necessity of a logical truth does not merely imply that some generalization about actual items holds, but also implies that the truth would have been true at a whole range of counterfactual circumstances. Leibniz assigned this property to necessary truths such as those of logic and geometry, and seems to have been one of the first to speak of the counterfactual circumstances as "possible universes" or worlds (see the letter to Bourguet, pp. 572-3, for a crisp statement of his views that contrasts them with the views in the preceding paragraph; Knuuttila 1982, pp. 353 ff. detects the earliest transparent talk of counterfactual circumstances and of necessity understood as at least implying truth in all of these in Duns Scotus and Buridan; see also the entry on medieval theories of modality). In contemporary writings the understanding of necessity as truth in all counterfactual circumstances, and the view that logical truths are necessary in this sense, are widespread--although many, perhaps most, authors adopt "reductivist" views of modality that see talk of counterfactual circumstances as no more than disguised talk about certain actualized (possibly abstract) items, such as linguistic descriptions. Even Leibniz seems to have thought of his "possible universes" as ideas in the mind of God. (See Lewis 1986 for an introduction to the contemporary polemics in this area.) However, even after Leibniz and up to the present, many logicians seem to have avoided a commitment to a strong notion of necessity as truth in all (actual and) counterfactual circumstances. Thus Bolzano, in line with his interpretation of Aristotle mentioned above, characterizes necessary propositions as those whose negation is incompatible with purely general truths (see Bolzano 1837, SS119). Frege says that "the apodictic judgment [i.e., roughly, the judgment whose content begins with a 'necessarily' governing the rest of the content] is distinguished from the assertory in that it suggests the existence of universal judgments from which the proposition can be inferred, while in the case of the assertory one such a suggestion is lacking" (Frege 1879, SS4). Tarski is even closer to the view traditionally attributed to Aristotle, for it is pretty clear that for him to say that e.g. (2c) "must" be true if (2a) and (2b) are true is to say that (2) is a particular case of the "formal" generalization "For all suitable $P$, $Q$ and $R$, if no $Q$ is $R$ and some $P$s are $Q$s, then some $P$s are not $R$" (see Tarski 1936a, pp. 411, 415, 417, or the corresponding passages in Tarski 1936b; see also Ray 1996). Quine is known for his explicit rejection of any modality that cannot be understood in terms of universal generalizations about the actual world (see especially Quine 1963). In some of these cases, this attitude is explained by a distrust of notions that are thought not to have reached a fully respectable scientific status, like the strong modal notions; it is frequently accompanied in such authors, who are often practicing logicians, by the proposal to characterize logical truth as a species of validity (in the sense of 2.3 below). (See Williamson 2013, ch. 3, and 2017, for a recent endorsement of the idea that logical truths are to be seen as instances of true universal generalizations over their logical forms, based on the scientific or "abductive" fruitfulness of the idea, though not on a disdain of modal notions.) On a recent view developed by Beall and Restall (2000, 2006), called by them "logical pluralism", the concept of logical truth carries a commitment to the idea that a logical truth is true in all of a range of items or "cases", and its necessity consists in the truth of such a general claim (see Beall and Restall 2006, p. 24). However, the concept of logical truth does not single out a unique range of "cases" as privileged in determining an extension for the concept, or, what amounts to the same thing, the universal generalization over a logical form can be interpreted via different equally acceptable ranges of quantification; instead, there are many such equally acceptable ranges and corresponding extensions for "logical truth", which may be chosen as a function of contextual interests. This means that, for the logical pluralist, many sets have a right to be called "the set of logical truths" (and "the set of logical necessities"), each in the appropriate context.[3] (See the entry on logical pluralism.) On another recent understanding of logical necessity as a species of generality, proposed by Rumfitt (2015), the necessity of a logical truth consists just in its being usable under all sets of subject-specific ways of drawing implications (provided these sets satisfy certain structural rules); or, more roughly, just in its being applicable no matter what sort of reasoning is at stake. On this view, a more substantive understanding of the modality at stake in logical truth is again not required. It may be noted that, although he postulates a variety of subject-specific implication relations, Rumfitt rejects pluralism about logical truth in the sense of Beall and Restall (see his 2015, p. 56, n. 23.), and in fact thinks that the set of logical truths in a standard quantificational language is characterized by the standard classical logic. Yet another sense in which it has been thought that truths like (1)-(3), and logical truths quite generally, "could" not be false or "must" be true is epistemic. It is an old observation, going at least as far back as Plato, that some truths count as intuitively known by us even in cases where we don't seem to have any empirical grounds for them. Truths that are knowable on non-empirical grounds are called a priori (an expression that begins to be used with this meaning around the time of Leibniz; see e.g. his "Primae Veritates", p. 518). The axioms and theorems of mathematics, the lexicographic and stipulative definitions, and also the paradigmatic logical truths, have been given as examples. If it is accepted that logical truths are a priori, it is natural to think that they must be true or could not be false at least partly in the strong sense that their negations are incompatible with what we are able to know non-empirically. Assuming that such a priori knowledge exists in some way or other, much recent philosophy has occupied itself with the issue of how it is possible. One traditional ("rationalist") view is that the mind is equipped with a special capacity to perceive truths through the examination of the relations between pure ideas or concepts, and that the truths reached through the correct operation of this capacity count as known a priori. (See, e.g., Leibniz's "Discours de Metaphysique", SSSS23 ff.; Russell 1912, p. 105; BonJour 1998 is a very recent example of a view of this sort.) An opposing traditional ("empiricist") view is that there is no reason to postulate that capacity, or even that there are reasons not to postulate it, such as that it is "mysterious". (See the entry on rationalism vs. empiricism.) Some philosophers, empiricists and otherwise, have attempted to explain a priori knowledge as arising from some sort of convention or tacit agreement to assent to certain sentences (such as (1)) and use certain rules. Hobbes in his objections to Descartes' Meditations ("Third Objections", IV, p. 608) proposes a wide-ranging conventionalist view. The later Wittgenstein (on one interpretation) and Carnap are distinguished proponents of "tacit agreement" and conventionalist views (see e.g. Wittgenstein 1978, I.9, I.142; Carnap 1939, SS12, and 1963, p. 916, for informal exposition of Carnap's views; see also Coffa 1991, chs. 14 and 17). Strictly speaking, Wittgenstein and Carnap think that logical truths do not express propositions at all, and are just vacuous sentences that for some reason or other we find useful to manipulate; thus it is only in a somewhat diminished sense that we can speak of (a priori) knowledge of them. However, in typical recent exponents of "tacit agreement" and conventionalist views, such as Boghossian (1997), the claim that logical truths do not express propositions is rejected, and it is accepted that the existence of the agreement provides full-blown a priori knowledge of those propositions. The "rational capacity" view and the "conventionalist" view agree that, in a broad sense, the epistemic ground of logical truths resides in our ability to analyze the meanings of their expressions, be these understood as conventions or as objective ideas. For this reason it can be said that they explain the apriority of logical truths in terms of their analyticity. (See the entry on the analytic/synthetic distinction.) Kant's explanation of the apriority of logical truths has seemed harder to extricate.[4] A long line of commentators of Kant has noted that, if Kant's view is that all logical truths are analytic, this would seem to be in tension with his characterizations of analytic truths. Kant characterizes analytic truths as those where the concept of the predicate is contained in or identical with the concept of the subject, and, more fundamentally, as those whose denial is contradictory. But it has appeared to those commentators that these characterizations, while applying to strict tautologies such as "Men are men" or "Bearded men are men", would seem to leave out much of what Kant himself counts as logically true, including syllogisms such as (2) (see e.g. Mill 1843, bk. II, ch. vi, SS5; Husserl 1901, vol. II, pt. 2, SS66; Kneale and Kneale 1962, pp. 357-8; Parsons 1967; Maddy 1999). This and the apparent lack of clear pronouncements of Kant on the issue has led at least Maddy (1999) and Hanna (2001) to consider (though not accept) the hypothesis that Kant viewed some logical truths as synthetic a priori. On an interpretation of this sort, the apriority of many logical truths would be explained by the fact that they would be required by the cognitive structure of the transcendental subject, and specifically by the forms of judgment.[5] But the standard interpretation is to attribute to Kant the view that all logical truths are analytic (see e.g. Capozzi and Roncaglia 2009). On an interpretation of this sort, Kant's forms of judgment may be identified with logical concepts susceptible of analysis (see e.g. Allison 1983, pp. 126ff.). An extended defense of the interpretation that Kant viewed all logical truths as analytic, including a vindication of Kant against the objections of the line of commentators mentioned above, can be found in Hanna (2001), SS3.1. A substantively Kantian contemporary theory of the epistemology of logic and its roots in cognition is developed in Hanna (2006); this theory does not seek to explain the apriority of logic in terms of its analyticity, and appeals instead to a specific kind of logical intuition and a specific cognitive logic faculty. (Compare also the anti-aprioristic and anti-analytic but broadly Kantian view of Maddy 2007, mentioned below.) The early Wittgenstein shares with Kant the idea that the logical expressions do not express meanings in the way that non-logical expressions do (see 1921, 4.0312). Consistently with this view, he claims that logical truths do not "say" anything (1921, 6.11). But he seems to reject conventionalist and "tacit agreement" views (1921, 6.124, 6.1223). It is not that logical truths do not say anything because they are mere instruments for some sort of extrinsically useful manipulation; rather, they "show" the "logical properties" that the world has independently of our decisions (1921, 6.12, 6.13). It is unclear how apriority is explainable in this framework. Wittgenstein calls the logical truths analytic (1921, 6.11), and says that "one can recognize in the symbol alone that they are true" (1921, 6.113). He seems to have in mind the fact that one can "see" that a logical truth of truth-functional logic must be valid by inspection of a suitable representation of its truth-functional content (1921, 6.1203, 6.122). But the extension of the idea to quantificational logic is problematic, despite Wittgenstein's efforts to reduce quantificational logic to truth-functional logic; as we now know, there is no algorithm for deciding if a quantificational sentence is valid. What is perhaps more important, Wittgenstein gives no discernible explanation of why in principle all the "logical properties" of the world should be susceptible of being reflected in an adequate notation. Against the "rational capacity", "conventionalist", Kantian and early Wittgensteinian views, other philosophers, especially radical empiricists and naturalists (not to speak of epistemological skeptics), have rejected the claim that a priori knowledge exists (hence by implication also the claim that analytic propositions exist), and they have proposed instead that there is only an illusion of apriority. Often this rejection has been accompanied by criticism of the other views. J.S. Mill thought that propositions like (2) seem a priori merely because they are particular cases of early and very familiar generalizations that we derive from experience, like "For all suitable $P$, $Q$ and $R$, if no $Q$ is $R$ and some $P$s are $Q$s, then some $P$s are not $R$" (see Mill 1843, bk. II, ch. viii). Bolzano held a similar view (see Bolzano 1837, SS315). Quine (1936, SSIII) famously criticized the Hobbesian view noting that since the logical truths are infinite in number, our ground for them must not lie just in a finite number of explicit conventions, for logical rules are presumably needed to derive an infinite number of logical truths from a finite number of conventions (an argument derived from Carroll 1895; see Soames 2018, ch. 10, for exposition and Gomez-Torrente 2019 for criticism of the argument). Later Quine (especially 1954) criticized Carnap's conventionalist view, largely on the grounds that there seems to be no non-vague distinction between conventional truths and truths that are tacitly left open for refutation, and that to the extent that some truths are the product of convention or "tacit agreement", such agreement is characteristic of many scientific hypotheses and other postulations that seem paradigmatically non-analytic. (See Grice and Strawson 1956 and Carnap 1963 for reactions to these criticisms.) Quine (especially 1951) also argued that accepted sentences in general, including paradigmatic logical truths, can be best seen as something like hypotheses that are used to deal with experience, any of which can be rejected if this helps make sense of the empirical world (see Putnam 1968 for a similar view and a purported example). On this view there cannot be strictly a priori grounds for any truth. Three recent subtle anti-aprioristic positions are Maddy's (2002, 2007), Azzouni's (2006, 2008), and Sher's (2013). For Maddy, logical truths are a posteriori, but they cannot be disconfirmed merely by observation and experiment, since they form part of very basic ways of thinking of ours, deeply embedded in our conceptual machinery (a conceptual machinery that is structurally similar to Kant's postulated transcendental organization of the understanding). Similarly, for Azzouni logical truths are equally a posteriori, though our sense that they must be true comes from their being psychologically deeply ingrained; unlike Maddy, however, Azzouni thinks that the logical rules by which we reason are opaque to introspection. Sher provides an attempt at combining a Quinean epistemology of logic with a commitment to a metaphysically realist view of the modal ground of logical truth. One way in which a priori knowledge of a logical truth such as (1) would be possible would be if a priori knowledge of the fact that (1) is a logical truth, or of the universal generalization "For all suitable $a$, $P$, $b$ and $Q$, if $a$ is $P$ only if $b$ is $Q$, and $a$ is $P$, then $b$ is $Q$" were possible. One especially noteworthy kind of skeptical consideration in the epistemology of logic is that the possibility of inferential a priori knowledge of these facts seems to face a problem of circularity or of infinite regress. If we are to obtain inferential a priori knowledge of those facts, then we will presumably follow logical rules at some point, including possibly the rule of modus ponens whose very correctness might well depend in part on the fact that (1) is a logical truth or on the truth of the universal generalization "For all suitable $a$, $P$, $b$ and $Q$, if $a$ is $P$ only if $b$ is $Q$, and $a$ is $P$, then $b$ is $Q$". In any case, it seems clear that not all claims of this latter kind, expressing that a certain truth is a logical truth or that a certain logical schema is truth-preserving, could be given an a priori inferential justification without the use of some of the same logical rules whose correctness they might be thought to codify. The point can again be reasonably derived from Carroll (1895). Some of the recent literature on this consideration, and on anti-skeptical rejoinders, includes Dummett (1973, 1991) and Boghossian (2000). ### 1.2 Formality On most views, even if it were true that logical truths are true in all counterfactual circumstances, a priori, and analytic, this would not give sufficient conditions for a truth to be a logical truth. On most views, a logical truth also has to be in some sense "formal", and this implies at least that all truths that are replacement instances of its form are logical truths too (and hence, on the assumption of the preceding sentence, true in all counterfactual circumstances, a priori, and analytic). To use a slight modification of an example of Albert of Saxony (quoted by Bochenski 1956, SS30.07), "If a widow runs, then a female runs" should be true in all counterfactual circumstances, a priori, and analytic if any truth is. However, "If a widow runs, then a log runs" is a replacement instance of its form, and in fact it even has the same form on any view of logical form (something like "If a $P$ $Q$s, then an $R$ $Q$s"), but it is not even true simpliciter. So on most views, "If a widow runs, then a female runs" is not a logical truth. For philosophers who accept the idea of formality, as we said above, the logical form of a sentence is a certain schema in which the expressions that are not schematic letters are widely applicable across different areas of discourse.[6] If the schema is the form of a logical truth, all of its replacement instances are logical truths. The idea that logic is especially concerned with (replacement instances of) schemata is of course evident beginning with Aristotle and the Stoics, in all of whom the word usually translated by "figure" is precisely schema. In Aristotle a figure is actually an even more abstract form of a group of what we would now call "schemata", such as (2'). Our schemata are closer to what in the Aristotelian syllogistic are the moods; but there seems to be no word for "mood" in Aristotle (except possibly ptoseon in 42b30 or tropon in 43a10; see Smith 1989, pp. 148-9), and thus no general reflection on the notion of formal schemata. There is explicit reflection on the contrast between the formal schemata or moods and the matter (hyle) of syllogismoi in Alexander of Aphrodisias (53.28ff., quoted by Bochenski 1956, SS24.06), and there has been ever since. The matter are the values of the schematic letters. The idea that the non-schematic expressions in logical forms, i.e. the logical expressions, are widely applicable across different areas of discourse is also present from the beginning of logic, and recurs in all the great logicians. It appears indirectly in many passages from Aristotle, such as the following: "All the sciences are related through the common things (I call common those which they use in order to demonstrate from them, but not those that are demonstrated in them or those about which something is demonstrated); and logic is related to them all, as it is a science that attempts to demonstrate universally the common things" (Posterior Analytics, 77a26-9); "we don't need to take hold of the things of all refutations, but only of those that are characteristic of logic; for these are common to every technique and ability" (Sophistical Refutations, 170a34-5). (In these texts "logic" is an appropriate translation of dialektike; see Kneale and Kneale 1962, I, SS3, who inform us that logike is used for the first time with its current meaning in Alexander of Aphrodisias.) Frege says that "the firmest proof is obviously the purely logical, which, prescinding from the particularity of things, is based solely on the laws on which all knowledge rests" (1879, p. 48; see also 1885, where the universal applicability of the arithmetical concepts is taken as a sign of their logicality). The same idea is conspicuous as well in Tarski (1941, ch. II, SS6). That logical expressions include paradigmatic cases like "if", "and", "some", "all", etc., and that they must be widely applicable across different areas of discourse is what we might call "the minimal thesis" about logical expressions. But beyond this there is little if any agreement about what generic feature makes an expression logical, and hence about what determines the logical form of a sentence. Most authors sympathetic to the idea that logic is formal have tried to go beyond the minimal thesis. It would be generally agreed that being widely applicable across different areas of discourse is only a necessary, not sufficient property of logical expressions; for example, presumably most prepositions are widely applicable, but they are not logical expressions on any implicit generic notion of a logical expression. Attempts to enrich the notion of a logical expression have typically sought to provide further properties that collectively amount to necessary and sufficient conditions for an expression to be logical. One idea that has been used in such characterizations, and that is also present in Aristotle, is that logical expressions do not, strictly speaking, signify anything; or, that they do not signify anything in the way that substantives, adjectives and verbs signify something. "Logic [dialektike] is not a science of determined things, or of any one genus" (Posterior Analytics, 77a32-3). We saw that the idea was still present in Kant and the early Wittgenstein. It reemerged in the Middle Ages. The main sense of the word "syncategorematic" as applied to expressions was roughly this semantic sense (see Kretzmann 1982, pp. 212 ff.). Buridan and other late medieval logicians proposed that categorematic expressions constitute the "matter" of sentences while the syncategorematic expressions constitute their "form" (see the text quoted by Bochenski 1956, SS26.11). (In a somewhat different, earlier, grammatical sense of the word, syncategorematic expressions were said to be those that cannot be used as subjects or predicates in categorical propositions; see Kretzmann 1982, pp. 211-2.) The idea of syncategorematicity is somewhat imprecise, but there are serious doubts that it can serve to characterize the idea of a logical expression, whatever this may be. Most prepositions and adverbs are presumably syncategorematic, but they are also presumably non-logical expressions. Conversely, predicates such as "are identical", "is identical with itself", "is both identical and not identical with itself", etc., which are resolutely treated as logical in recent logic, are presumably categorematic. (They are of course categorematic in the grammatical sense, in which prepositions and adverbs are equally clearly syncategorematic.) Most other proposals have tried to delineate in some other way the Aristotelian idea that the logical expressions have some kind of "insubstantial" meaning, so as to use it as a necessary and sufficient condition for logicality. One recent suggestion is that logical expressions are those that do not allow us to distinguish different individuals. One way in which this has been made precise is through the characterization of logical expressions as those whose extension or denotation over any particular domain of individuals is invariant under permutations of that domain. (See Tarski and Givant 1987, p. 57, and Tarski 1966; for related proposals see also McCarthy 1981, Sher 1991, ch. 3, McGee 1996, Feferman 1999, Bonnay 2008, Woods 2016 and Griffiths and Paseau 2022, among others.) A permutation of a domain is a one-to-one correspondence between the domain and itself. For example, if $D$ is the domain {Aristotle, Caesar, Napoleon, Kripke}, one permutation is the correspondence that assigns each man to himself; another is the correspondence $P$ that assigns Caesar to Aristotle (in mathematical notation, $P(\text{Aristotle})=\text{Caesar}$), Napoleon to Caesar, Kripke to Napoleon, and Aristotle to Kripke. That the extension of an expression over a domain is invariant under a permutation of that domain means that the induced image of that extension under the permutation is the extension itself (the "induced image" of an extension under a permutation $Q$ is what the extension becomes when in place of each object $o$ one puts the object $Q(o)$). The extension of "philosopher" over $D$ is not invariant under the permutation $P$ above, for that extension is $\{\text{Aristotle}, \text{Kripke}\}$, whose induced image under $P$ is $\{\text{Caesar}, \text{Aristotle}\}$. This is favorable to the proposal, for "philosopher" is certainly not widely applicable, and so non-logical on most views. On the other hand, the predicate "are identical" has as its extension over $D$ the set of pairs > > $\{ \langle \text{Aristotle, Aristotle} \rangle, \langle > \text{Caesar, Caesar} \rangle, \langle \text{Napoleon, Napoleon} > \rangle, \langle \text{Kripke, Kripke} \rangle\};$ > its induced image under $P$, and under any other permutation of $D$, is that very same set of pairs (as the reader may check); so again this is favorable to the proposal. (Other paradigmatic logical expressions receive more complicated extensions over domains, but the extensions they receive are invariant under permutations. For example, on one usual way of understanding the extension of "and" over a domain, this is the function that assigns, to each pair $\langle S\_1, S\_2 \rangle$, where $S\_1$ and $S\_2$ are sets of infinite sequences of objects drawn from $D$, the intersection of $S\_1$ and $S\_2$; and this function is permutation invariant.) One problem with the proposal is that many expressions that seem clearly non-logical, because they are not widely applicable, are nevertheless invariant under permutations, and thus unable to distinguish different individuals. The simplest examples are perhaps non-logical predicates that have an empty extension over any domain, and hence have empty induced images as well. "Male widow" is one example; versions of it can be used as counterexamples to the different versions of the idea of logicality as permutation invariance (see Gomez-Torrente 2002), and it's unclear that the proponent of the idea can avoid the problem in any non ad hoc way. Another popular recent way of delineating the Aristotelian intuition of the semantic "insubstantiality" of logical expressions appeals to the concept of "pure inferentiality". The idea is that logical expressions are those whose meaning, in some sense, is given by "purely inferential" rules. (See Kneale 1956, Hacking 1979, Peacocke 1987, Hodes 2004, among others.) A necessary property of purely inferential rules is that they regulate only inferential transitions between verbal items, not between extra-verbal assertibility conditions and verbal items, or between verbal items and actions licensed by those items. A certain inferential rule licenses you to say "It rains" when it rains, but it's not "purely inferential". A rule that licenses you to say "A is a female whose husband died before her" when someone says "A is a widow", however, is not immediately disqualified as purely inferential. Now, presumably in some sense the meaning of "widow" is given by this last rule together perhaps with the converse rule, that licenses you to say "A is a widow" when someone says "A is a female whose husband died before her". But "widow" is not a logical expression, since it's not widely applicable; so one needs to postulate more necessary properties that "purely inferential" rules ought to satisfy. A number of such conditions are postulated in the relevant literature (see e.g. Belnap 1962 (a reply to Prior 1960), Hacking 1979 and Hodes 2004). However, even when the notion of pure inferentiality is strengthened in these ways, problems remain. Most often the proposal is that an expression is logical just in case certain purely inferential rules give its whole meaning, including its sense, or the set of aspects of its use that need to be mastered in order to understand it (as in Kneale 1956, Peacocke 1987 and Hodes 2004). However, it seems clear that some paradigmatic logical expressions have extra sense attached to them that is not codifiable purely inferentially. For example, inductive reasoning involving "all" seems to be part of the sense of this expression, but it's hard to see how it could be codified by purely inferential rules (as noted by Sainsbury 1991, pp. 316-7; see also Dummett 1991, ch. 12). A different version of the proposal consists in saying that an expression is logical just in case certain purely inferential rules that are part of its sense suffice to determine its extension (as in Hacking 1979). But it seems clear that if the extension of, say, "are identical" is determined by a certain set of purely inferential rules that are part of its sense, then the extension of "are identical and are not male widows" is equally determined by the same rules, which arguably form part of its sense; yet "are identical and are not male widows" is not a logical expression (see Gomez-Torrente 2002). In view of problems of these and other sorts, some philosophers have proposed that the concept of a logical expression is not associated with necessary and sufficient conditions, but only with some necessary condition related to the condition of wide applicability, such as the condition of "being very relevant for the systematization of scientific reasoning" (see Warmbrod 1999 for a position of this type). Others (Gomez-Torrente 2021) have proposed that there may be a set of necessary and sufficient conditions, if these are not much related to the idea of semantic "insubstantiality" and are instead pragmatic and suitably vague; for example, many expressions are excluded directly by the condition of wide applicability; and prepositions are presumably excluded by some such implicit condition as "a logical expression must be one whose study is useful for the resolution of significant problems and fallacies in reasoning". To be sure, these proposals give up on the extended intuition of semantic "insubstantiality", and may be somewhat unsatisfactory for that reason. Some philosophers have reacted even more radically to the problems of usual characterizations, claiming that the distinction between logical and non-logical expressions must be vacuous, and thus rejecting the notion of logical form altogether. (See e.g. Orayen 1989, ch. 4, SS2.2; Etchemendy 1990, ch. 9; Read 1994; Priest 2001.) These philosophers typically think of logical truth as a notion roughly equivalent to that of analytic truth simpliciter. But they are even more liable to the charge of giving up on extended intuitions than the proposals of the previous paragraph. For more thorough treatments of the ideas of formality and of a logical expression see the entry logical constants, and MacFarlane 2000. ## 2. The Mathematical Characterization of Logical Truth ### 2.1 Formalization One important reason for the successes of modern logic is its use of what has been called "formalization". This term is usually employed to cover several distinct (though related) phenomena, all of them present in Frege (1879). One of these is the use of a completely specified set of artificial symbols to which the logician assigns unambiguous meanings, related to the meanings of corresponding natural language expressions, but much more clearly delimited and stripped from the notes that in those natural language expressions seem (to the logician) irrelevant to truth-conditional content; this is especially true of symbols meant to represent the logical expressions of natural language. Another phenomenon is the stipulation of a completely precise grammar for the formulae construed out of the artificial symbols, formulae that will be "stripped" versions of correlate sentences in natural language; this grammar amounts to an algorithm for producing formulae starting from the basic symbols. A third phenomenon is the postulation of a deductive calculus with a very clear specification of axioms and rules of inference for the artificial formulae (see the next section); such a calculus is intended to represent in some way deductive reasoning with the correlates of the formulae, but unlike ordinary deductions, derivations in the calculus contain no steps that are not definite applications of the specified rules of inference. Instead of attempting to characterize the logical truths of a natural language like English, the Fregean logician attempts to characterize the artificial formulae that are "stripped" correlates of those logical truths in a Fregean formalized language. In first-order Fregean formalized languages, among these formulae one finds artificial correlates of (1), (2) and (3), things like * $((\text{Bad}(\textit{death}) \rightarrow \text{Good}(\textit{life})) \ \& \ \text{Bad}(\textit{death})) \rightarrow \text{Good}(\textit{life}).$ * $(\forall x(\text{Desire}(x) \rightarrow \neg \text{Voluntary}(x)) \ \&\ \exists x(\text{Belief}(x) \ \&\ \text{Desire}(x)))$ $\rightarrow \exists x(\text{Belief}(x) \ \&\ \neg \text{Voluntary}(x)).$ * $(\text{Cat}(\textit{drasha}) \ \&\ \forall x(\text{Cat}(x) \rightarrow \text{Mysterious}(x)))$ $\rightarrow \text{Mysterious}(\textit{drasha}).$ (See the entry on logic, classical.) Fregean formalized languages include also classical higher-order languages. (See the entry on logic, second-order and higher-order.) The logical expressions in these languages are standardly taken to be the symbols for the truth-functions, the quantifiers, identity and other symbols definable in terms of those (but there are dissenting views on the status of the higher-order quantifiers; see 2.4.3 below). The restriction to artificial formulae raises a number of questions about the exact value of the Fregean enterprise for the demarcation of logical truths in natural language; much of this value depends on how many and how important are perceived to be the notes stripped from the natural language expressions that are correlates of the standard logical expressions of formalized languages. But whatever one's view of the exact value of formalization, there is little doubt that it has been very illuminating for logical purposes. One reason is that it's sometimes relatively clear that the stripped notes are irrelevant to truth-conditional content (this is especially true of the use of natural language logical expressions for doing mathematics). Another of the reasons is that the fact that the grammar and meaning of the artificial formulae is so well delimited has permitted the development of proposed characterizations of logical truth that use only concepts of standard mathematics. This in turn has allowed the study of the characterized notions by means of standard mathematical techniques. The next two sections describe the two main approaches to characterization in broad outline.[7] ### 2.2 Derivability We just noted that the Fregean logician's formalized grammar amounts to an algorithm for producing formulae from the basic artificial symbols. This is meant very literally. As was clear to mathematical logicians from very early on, the basic symbols can be seen as (or codified by) natural numbers, and the formation rules in the artificial grammar can be seen as (or codified by) simple computable arithmetical operations. The grammatical formulae can then be seen as (or codified by) the numbers obtainable from the basic numbers after some finite series of applications of the operations, and thus their set is characterizable in terms of concepts of arithmetic and set theory (in fact arithmetic suffices, with the help of some tricks). Exactly the same is true of the set of formulae that are derivable in a formalized deductive calculus. A formula $F$ is derivable in a Fregean calculus $C$ just in case $F$ is obtainable from the axioms of $C$ after some finite series of applications of the rules of inference of $C$. But the axioms are certain formulae built by the process of grammatical formation, so they can be seen as (or codified by) certain numbers; and the rules of inference can again be seen as (or codified by) certain computable arithmetical operations. So the derivable formulae can be seen as (or codified by) the numbers obtainable from the axiom numbers after some finite series of applications of the inference operations, and thus their set is again characterizable in terms of concepts of standard mathematics (again arithmetic suffices). In the time following Frege's revolution, there appears to have been a widespread belief that the set of logical truths of any Fregean language could be characterized as the set of formulae derivable in some suitably chosen calculus (hence, essentially, as the set of numbers obtainable by certain arithmetical operations). Frege himself says, speaking of the higher-order language in his "Begriffsschrift", that through formalization (in the third sense above) "we arrive at a small number of laws in which, if we add those contained in the rules, the content of all the laws is included, albeit in an undeveloped state" (Frege 1879, SS13). The idea follows straightforwardly from Russell's conception of mathematics and logic as identical (see Russell 1903, ch. I, SS10; Russell 1920, pp. 194-5) and his thesis that "by the help of ten principles of deduction and ten other premises of a general logical nature (...), all mathematics can be strictly and formally deduced" (Russell 1903, ch. I, SS4). See also Bernays (1930, p. 239): "[through formalization] it becomes evident that all logical inference (...) can be reduced to a limited number of logical elementary processes that can be exactly and completely enumerated". In the opening paragraphs of his paper on logical consequence, Tarski (1936a, 1936b) says that the belief was prevalent before the appearance of Godel's incompleteness theorems (see subsection 2.4.3 below for the bearing of these theorems on this issue). In recent times, apparently due to the influence of Tarskian arguments such as the one mentioned towards the end of subsection 2.4.3, the belief in the adequacy of derivability characterizations seems to have waned (see e.g. Prawitz 1985 for a similar appraisal). ### 2.3 Model-theoretic Validity Even on the most cautious way of understanding the modality present in logical truths, a sentence is a logical truth only if no sentence which is a replacement instance of its logical form is false. (This idea is only rejected by those who reject the notion of logical form.) It is a common observation that this property, even if it is necessary, is not clearly sufficient for a sentence to be a logical truth. Perhaps there is a sentence that has this property but is not really logically true, because one could assign some unexpressed meanings to the variables and the schematic letters in its logical form, and under those meanings the form would be a false sentence.[8] On the other hand, it is not clearly incorrect to think that a sentence is a logical truth if no collective assignment of meanings to the variables and the schematic letters in its logical form would turn this form into a false sentence. Say that a sentence is universally valid when it has this property. A standard approach to the mathematical characterization of logical truth, alternative to the derivability approach, uses always some version of the property of universal validity, proposing it in each case as both necessary and sufficient for logical truth. Note that if a sentence is universally valid then, even if it's not logically true, it will be true. So all universally valid sentences are correct at least in this sense. Apparently, the first to use a version of universal validity and explicitly propose it as both necessary and sufficient for logical truth was Bolzano (see Bolzano 1837, SS148; and Coffa 1991, pp. 33-4 for the claim of priority). The idea is also present in other mathematicians of the nineteenth century (see e.g. Jane 2006), and was common in Hilbert's school. Tarski (1936a, 1936b) was the first to indicate in a fully explicit way how the version of universal validity used by the mathematicians could itself be given a characterization in terms of concepts of standard mathematics, in the case of Fregean formalized languages with an algorithmic grammar. Essentially Tarski's characterization is widely used today in the form of what is known as the model-theoretic notion of validity, and it seems fair to say that it is usually accepted that this notion gives a reasonably good delineation of the set of logical truths for Fregean languages. The notion of model-theoretic validity mimics the notion of universal validity, but is defined just with the help of the set-theoretic apparatus developed by Tarski (1935) for the characterization of semantic concepts such as satisfaction, definability, and truth. (See the entry on Tarski's truth definitions.) Given a Fregean language, a structure for the language is a set-theoretical object composed of a set-domain taken together with an assignment of extensions drawn from that domain to its non-logical constants. A structure is meant by most logicians to represent an assignment of meanings: its domain gives the range or "meaning" of the first-order variables (and induces ranges of the higher-order variables), and the extensions that the structure assigns to the non-logical constants are "meanings" that these expressions could take. Using the Tarskian apparatus, one defines for the formulae of the Fregean language the notion of truth in (or satisfaction by) a set-theoretic structure (with respect to an infinite sequence assigning an object of the domain to each variable). And finally, one defines a formula to be model-theoretically valid just in case it is true in all structures for its language (with respect to all infinite sequences). Let's abbreviate "$F$ is true in all structures" as "MTValid$(F)$". The model-theoretic characterization makes it clear that "MTValid$(F)$" is definable purely in terms of concepts of set theory. (The notion of model-theoretic validity for Fregean languages is explained in thorough detail in the entries on classical logic and second-order and higher-order logic; see also the entry on model theory.)[9] (If $F$ is a formula of a first-order language without identity, then if no replacement instance of the form of $F$ is false, this is a sufficient condition for $F$'s being model-theoretically valid. As it turns out, if $F$ is not model-theoretically valid, then some replacement instance of its form whose variables range over the natural numbers and whose non-logical constants are arithmetical expressions will be false. This can be justified by means of a refinement of the Lowenheim-Skolem theorem. See the entry on logic, classical, and Quine 1970, ch. 4, for discussion and references. No similar results hold for higher-order languages.) The "MT" in "MTValid$(F)$" stresses the fact that model-theoretic validity is different from universal validity. The notion of a meaning assignment which appears in the description of universal validity is a very imprecise and intuitive notion, while the notion of a structure appearing in a characterization of model-theoretic validity is a fairly precise and technical one. It seems clear that the notion of a structure for Fregean formalized languages is minimally reasonable, in the sense that a structure models the power of one or several meaning assignments to make false (the logical form of) some sentence. As we will mention later, the converse property, that each meaning assignment's validity-refuting power is modeled by some structure, is also a natural but more demanding requirement on a notion of structure. ### 2.4 The Problem of Adequacy The fact that the notions of derivability and model-theoretic validity are definable in standard mathematics seems to have been a very attractive feature of them among practicing logicians. But this attractive feature of course does not justify by itself taking either notion as an adequate characterization of logical truth. On most views, with a mathematical characterization of logical truth we attempt to delineate a set of formulae possessing a number of non-mathematical properties. Which properties these are varies depending on our pretheoretic conception of, for example, the features of modality and formality. (By "pretheoretic" it's not meant "previous to any theoretical activity"; there could hardly be a "pretheoretic" conception of logical truth in this sense. In this context what's meant is "previous to the theoretical activity of mathematical characterization".) But on any such conception there will be external, non-mathematical criteria that can be applied to evaluate the question whether a mathematical characterization is adequate. In this last section we will outline some of the basic issues and results on the question whether derivability and model-theoretic validity are adequate in this sense. #### 2.4.1 Analysis and Modality One frequent objection to the adequacy of model-theoretic validity is that it does not provide a conceptual analysis of the notion of logical truth, even for sentences of Fregean formalized languages (see e.g. Pap 1958, p. 159; Kneale and Kneale 1962, p. 642; Field 1989, pp. 33-4; Etchemendy 1990, ch. 7). This complaint is especially common among authors who feel inclined to identify logical truth and analyticity simpliciter (see e.g. Kneale and Kneale, ibid., Etchemendy 1990, p. 126). If one thinks of the concept of logical truth simply as the concept of analytic truth, it is especially reasonable to accept that the concept of logical truth does not have much to do with the concept of model-theoretic validity, for presumably this concept does not have much to do with the concept of analyticity. To say that a formula is model-theoretically valid means that there are no set-theoretic structures in which it is false; hence, to say that a formula is not model-theoretically valid means that there are set-theoretic structures in which it is false. But to say that a sentence is or is not analytic presumably does not mean anything about the existence or non-existence of set-theoretic structures. Note that we could object to derivability on the same grounds, for to say that a sentence is or is not analytic presumably does not mean anything about its being or not being the product of a certain algorithm (compare Etchemendy 1990, p. 3). (One further peculiar, much debated claim in Etchemendy 1990 is that true claims of the form "$F$ is logically true" or "$F$ is not logically true" should themselves be logical truths (while the corresponding claims "MTValid$(F)$" and "Not MTValid$(F)$" are not logical truths). Etchemendy's claim is perhaps defensible under a conception of logical truth as analyticity simpliciter, but certainly doubtful on more traditional conceptions of logical truth, on which the predicate "is a logical truth" is not even a logical expression. See Gomez-Torrente 1998/9 and Soames 1999, ch. 4 for discussion.) Analogous "no conceptual analysis" objections can be made if we accept that the concept of logical truth has some other strong modal notes unrelated to analyticity; for example, if we accept that it is part of the concept of logical truth that logical truths are true in all counterfactual circumstances, or necessary in some other strong sense. Sher (1996) accepts something like the requirement that a good characterization of logical truth should be given in terms of a modally rich concept. However, she argues that the notion of model-theoretic validity is strongly modal, and so the "no conceptual analysis" objection is actually wrong: to say that a formula is or is not model-theoretically valid is to make a mathematical existence or non-existence claim, and according to Sher these claims are best read as claims about the possibility and necessity of structures. (Shalkowski 2004 argues that Sher's defense of model-theoretic validity is insufficient, on the basis of a certain metaphysical conception of logical necessity. Etchemendy 2008 relatedly argues that Sher's defense is based on inadequate restrictions on the modality relevant to logical truth. See also the critical discussion of Sher in Hanson 1997.) Garcia-Carpintero (1993) offers a view related to Sher's: model-theoretic validity provides a (correct) conceptual analysis of logical truth for Fregean languages, because the notion of a set-theoretical structure is in fact a subtle refinement of the modal notion of a possible meaning assignment. Azzouni (2006), ch. 9, also defends the view that model-theoretic validity provides a correct conceptual analysis of logical truth (though restricted to first-order languages), on the basis of a certain deflationist conception of the (strong) modality involved in logical truth. The standard view of set-theoretic claims, however, does not see them as strong modal claims--at best, some of them are modal in the minimal sense that they are universal generalizations or particular cases of these. But it is at any rate unclear that this is the basis for a powerful objection to model-theoretic validity or to derivability, for, even if we accept that the concept of logical truth is strongly modal, it is unclear that a good characterization of logical truth ought to be a conceptual analysis. An analogy might help. It is widely agreed that the characterizations of the notion of computability in standard mathematics, e.g. as recursiveness, are in some sense good characterizations. Note that the concept of computability is modal, in a moderately strong sense; it seems to be about what beings like us could do with certain symbols if they were free from certain limitations--not about, say, what existing beings have done or will do. However, to say that a certain function is recursive is not to make a modal claim about it, but a certain purely arithmetical claim. So recursiveness is widely agreed to provide a good characterization of computability, but it clearly does not provide a conceptual analysis. Perhaps it could be argued that the situation with model-theoretic validity, or derivability, or both, is the same. A number of philosophers explicitly reject the requirement that a good characterization of logical truth should provide a conceptual analysis, and (at least for the sake of argument) do not question the usual view of set-theoretic claims as non-modal, but have argued that the universe of set-theoretic structures somehow models the universe of possible structures (or at least the universe of possible set-theoretic structures; see McGee 1992, Shapiro 1998, Sagi 2014). In this indirect sense, the characterization in terms of model-theoretic validity would grasp part of the strong modal force that logical truths are often perceived to possess. McGee (1992) gives an elegant argument for this idea: it is reasonable to think that given any set-theoretic structure, even one construed out of non-mathematical individuals, actualized or not, there is a set-theoretic structure isomorphic to it but construed exclusively out of pure sets; but any such pure set-theoretic structure is, on the usual view, an actualized existent; so every possible set-theoretic structure is modeled by a set-theoretic structure, as desired. (The significance of this relies on the fact that in Fregean languages a formula is true in a structure if and only if it is true in all the structures isomorphic to it.) But model-theoretic validity (or derivability) might be theoretically adequate in some way even if some possible meaning-assignments are not modeled straightforwardly by (actual) set-theoretic structures. For model-theoretic validity to be theoretically adequate, it might be held, it is enough if we have other reasons to think that it is extensionally adequate, i.e. that it coincides in extension with our preferred pretheoretic notion of logical truth. In subsections 2.4.2 and 2.4.3 we will examine some existing arguments for and against the plain extensional adequacy of derivability and model-theoretic validity for Fregean languages. #### 2.4.2 Extensional Adequacy: A General Argument If one builds one's deductive calculus with care, one will be able to convince oneself that all the formulae derivable in the calculus are logical truths. The reason is that one can have used one's intuition very systematically to obtain that conviction: one can have included in one's calculus only axioms of which one is convinced that they are logical truths; and one can have included as rules of inference rules of which one is convinced that they produce logical truths when applied to logical truths. Using another terminology, this means that, if one builds one's calculus with care, one will be convinced that the derivability characterization of logical truth for formulae of the formalized language will be sound with respect to logical truth. It is equally obvious that if one has at hand a notion of model-theoretic validity for a formalized language which is based on a minimally reasonable notion of structure, then all logical truths (of that language) will be model-theoretically valid. The reason is simple: if a formula is not model-theoretically valid then there is a structure in which it is false; but this structure must then model a meaning assignment (or assignments) on which the formula (or its logical form) is false; so it will be possible to construct a formula with the same logical form, whose non-logical expressions have, by stipulation, the particular meanings drawn from that collective meaning assignment, and which is therefore false. But then the idea of formality and the weakest conception of the modal force of logical truths uncontroversially imply that the original formula is not logically true. Using another terminology, we can conclude that model-theoretic validity is complete with respect to logical truth. Let's abbreviate "$F$ is derivable in calculus $C$" by "DC$(F)$" and "$F$ is a logical truth (in our preferred pretheoretical sense)" by "LT$(F)$". Then, if $C$ is a calculus built to suit our pretheoretic conception of logical truth, the situation can be summarized thus: * (4) $\text{DC}(F) \Rightarrow \text{LT}(F) \Rightarrow \text{MTValid}(F).$ The first implication is the soundness of derivability; the second is the completeness of model-theoretic validity. In order to convince ourselves that the characterizations of logical truth in terms of DC$(F)$ and MTValid$(F)$ are extensionally adequate we should convince ourselves that the converse implications hold too: * (5) $\text{MTValid}(F) \Rightarrow \text{LT}(F) \Rightarrow \text{DC}(F).$ Obtaining this conviction, or the conviction that these implications don't in fact hold, turns out to be difficult in general. But a remark of Kreisel (1967) establishes that a conviction that they hold can be obtained sometimes. In some cases it is possible to give a mathematical proof that derivability (in some specified calculus $C$) is complete with respect to model-theoretic validity, i.e. a proof of * (6) $\text{MTValid}(F) \Rightarrow \text{DC}(F).$ Kreisel called attention to the fact that (6) together with (4) implies that model-theoretic validity is sound with respect to logical truth, i.e., that the first implication of (5) holds. (Strictly speaking, this is a strong generalization of Kreisel's remark, which in place of "$\text{LT}(F)$" had something like "$F$ is true in all class structures" (structures with a class, possibly proper, as domain of the individual variables).) This means that when (6) holds the notion of model-theoretic validity offers an extensionally correct characterization of logical truth. (See Etchemendy 1990, ch. 11, Hanson 1997, Gomez-Torrente 1998/9, and Field 2008, ch. 2, for versions of this observation, and Smith 2011 and Griffiths 2014 for objections.) Also, (6), together with (4), implies that the notion of derivability is complete with respect to logical truth (the second implication in (5)) and hence offers an extensionally correct characterization of this notion. Note that this reasoning is very general and independent of what our particular pretheoretic conception of logical truth is. An especially significant case in which this reasoning can be applied is the case of standard Fregean first-order quantificational languages, under a wide array of pretheoretic conceptions of logical truth. It is typical to accept that all formulae derivable in a typical first-order calculus are universally valid, true in all counterfactual circumstances, a priori and analytic if any formula is.[10] So (4) holds under a wide array of pretheoretic conceptions in this case. (6) holds too for the typical calculi in question, in virtue of Godel's completeness theorem, so (5) holds. This means that one can convince oneself that both derivability and model-theoretic validity are extensionally correct characterizations of our favorite pretheoretic notion of logical truth for first-order languages, if our pretheoretic conception is not too eccentric. The situation is not so clear in other languages of special importance for the Fregean tradition, the higher-order quantificational languages. #### 2.4.3 Extensional Adequacy: Higher-order Languages It follows from Godel's first incompleteness theorem that already for a second-order language there is no calculus $C$ where derivability is sound with respect to model-theoretic validity and which makes true (6) (for the notion of model-theoretic validity as usually defined for such a language). We may call this result the incompleteness of second-order calculi with respect to model-theoretic validity. Said another way: for every second-order calculus $C$ sound with respect to model-theoretic validity there will be a formula $F$ such that $\text{MTValid}(F)$ but it is not the case that $\text{DC}(F)$. In this situation it's not possible to apply Kreisel's argument for (5). In fact, the incompleteness of second-order calculi shows that, given any calculus $C$ satisfying (4), one of the implications of (5) is false (or both are): either derivability in $C$ is incomplete with respect to logical truth or model-theoretic validity is unsound with respect to logical truth. Different authors have extracted opposed lessons from incompleteness. A common reaction is to think that model-theoretic validity must be unsound with respect to logical truth. This is especially frequent in philosophers on whose conception logical truths must be a priori or analytic. One idea is that the results of a priori reasoning or of analytic thinking ought to be codifiable in a calculus. (See e.g. Wagner 1987, p. 8.) But even if we grant this idea, it's doubtful that the desired conclusion follows. Suppose that (i) every a priori or analytic reasoning must be reproducible in a calculus. We accept also, of course, that (ii) for every calculus $C$ sound with respect to model-theoretic validity there is a model-theoretically valid formula that is not derivable in $C$. From all this it doesn't follow that (iii) there is a model-theoretically valid formula $F$ such that for every calculus $C$ sound for model-theoretic validity $F$ is not derivable in C. From (iii) and (i) it follows of course that there are model-theoretically valid formulae that are not obtainable by a priori or analytic reasoning. But the step from (ii) to (iii) is a typical quantificational fallacy. From (i) and (ii) it doesn't follow that there is any model-theoretically valid formula which is not obtainable by a priori or analytic reasoning. The only thing that follows (from (ii) alone under the assumptions that model-theoretic validity is sound with respect to logical truth and that logical truths are a priori and analytic) is that no calculus sound with respect to model-theoretic validity can by itself model all the a priori or analytic reasonings that people are able to make. But it's not sufficiently clear that this should be intrinsically problematic. After all, a priori and analytic reasonings must start from basic axioms and rules, and for all we know a reflective mind may have an inexhaustible ability to find new truths and truth-preserving rules by a priori or analytic consideration of even a meager stock of concepts. The claim that all analytic truths ought to be derivable in one single calculus is perhaps plausible on the view that analyticity is to be explained by conventions or "tacit agreements", for these agreements are presumably finite in number, and their implications are presumably at most effectively enumerable. But this view is just one problematic idea about how apriority and analyticity should be explicated. (See also Etchemendy 1990, chs. 8, 9, for an argument for the unsoundness of higher-order model-theoretic validity based on the conception of logical truth as analyticity simpliciter, and Gomez-Torrente 1998/9, Soames 1999, ch. 4, Paseau 2014, Florio and Incurvati 2019, 2021, and Griffiths and Paseau 2022, ch. 9, for critical reactions.) Another type of unsoundness arguments attempt to show that there is some higher-order formula that is model-theoretically valid but is intuitively false in a structure whose domain is a proper class. (The "intended interpretation" of set theory, if it exists at all, might be one such structure, for it is certainly not a set; see the entry on set theory.) These arguments thus question the claim that each meaning assignment's validity-refuting power is modeled by some set-theoretic structure, a claim which is surely a corollary of the first implication in (5). (In McGee 1992 there is a good example; there is critical discussion in Gomez-Torrente 1998/9.) The most widespread view among set theorists seems to be that there are no formulae with that property in Fregean languages, but it's certainly not an absolutely firm belief of theirs. Note that these arguments offer a challenge only to the idea that universal validity (as defined in section 2.3) is adequately modeled by set-theoretic validity, not to the soundness of a characterization of logical truth in terms of universal validity itself, or in terms of a species of validity based on some notion of "meaning assignment" different from the usual notion of a set-theoretic structure. (The arguments we mentioned in the preceding paragraph and in 2.4.1 would have deeper implications if correct, for they easily imply challenges to all characterizations in terms of species of validity as well). In fact, worries of this kind have prompted the proposal of a different kind of notions of validity (for Fregean languages), in which set-theoretic structures are replaced with suitable values of higher-order variables in a higher-order language for set theory, e.g. with "plural interpretations" (see Boolos 1985, Rayo and Uzquiano 1999, Williamson 2003, Florio and Incurvati 2021; see also Florio and Linnebo 2021 and the entry on plural quantification). Both set-theoretic and proper class structures are modeled by such values, so these particular worries of unsoundness do not affect this kind of proposals. In general, there are no fully satisfactory philosophical arguments for the thesis that model-theoretic validity is unsound with respect to logical truth in higher-order languages. Are there then any good reasons to think that derivability (in any calculus sound for model-theoretic validity) must be incomplete with respect to logical truth? There don't seem to be any absolutely convincing reasons for this view either. The main argument (the first version of which was perhaps first made explicit in Tarski 1936a, 1936b) seems to be this. As noted above, Godel's first incompleteness theorem implies that for any calculus for a higher-order language there will be a model-theoretically valid formula that will not be derivable in the calculus. As it turns out, the formula obtained by the Godel construction is also always intuitively true in all domains (set-theoretical or not), and it's reasonable to think of it as universally valid. (It's certainly not a formula false in a proper class structure.) The argument concludes that for any calculus there are logically true formulae that are not derivable in it. From this it has been concluded that derivability (in any calculus) must be incomplete with respect to logical truth. But a fundamental problem is that this conclusion is based on two assumptions that will not necessarily be granted by the champion of derivability: first, the assumption that the expressions typically cataloged as logical in higher-order languages, and in particular the quantifiers in quantifications of the form $\forall X$ (where $X$ is a higher-order variable), are in fact logical expressions; and second, the assumption that being universally valid is a sufficient condition for logical truth. On these assumptions it is certainly very reasonable to think that derivability, in any calculus satisfying (4), must be incomplete with respect to logical truth. But in the absence of additional considerations, a critic may question the assumptions, and deny relevance to the argument. The second assumption would probably be questioned e.g. from the point of view that logical truths must be analytic, for there is no conclusive reason to think that universally valid formulae must be analytic. The first assumption actually underlies any conviction one may have that (4) holds for any one particular higher-order calculus. (Note that if we denied that the higher-order quantifiers are logical expressions we could equally deny that the arguments presented above against the soundness of model-theoretic validity with respect to logical truth are relevant at all.) That the higher-order quantifiers are logical has often been denied on the grounds that they are semantically too "substantive". It is often pointed out in this connection that higher-order quantifications can be used to define sophisticated set-theoretic properties that one cannot define just with the help of first-order quantifiers. (Defenders of the logical status of higher-order quantifications, on the other hand, point to the wide applicability of the higher-order quantifiers, to the fact that they are analogous to the first-order quantifiers, to the fact that they are typically needed to provide categorical axiomatizations of mathematical structures, etc. See Quine (1970), ch. 5, for the standard exponent of the restrictive view, and Boolos (1975) and Shapiro (1991) for standard exponents of the liberal view.)
lorenzo-valla	## 1. Life and Works Valla did not have an easy life. Equipped with a sharp and polemical mind, an even sharper pen and a sense of self-importance verging on the pathological, he made many enemies throughout his life. Born in Rome in (most likely) 1406 to a family with ties to the papal curia, Valla as a young man was already in close contact with some major humanists working as papal secretaries such as Leonardo Bruni (1370-1444) and Poggio Bracciolini (1380-1459). Another was his uncle Melchior Scrivani, whom Valla had hoped to succeed after his death; but opposition from Poggio and Antonio Loschi (1365/8-1441) must have led the pope to refuse to employ him. Valla had criticized an elegy by Loschi, and had also boldly favored Quintilian over Cicero in a treatise which has long considered to be lost; but a lengthy anonymous prefatory letter has recently been found and ascribed, convincingly, to the young Valla (Pagliaroli 2008; it turns out not to be a comparison between Cicero's theory of rhetoric and Quintilian's handbook, as scholars have always assumed, but a comparison between a declamation of Ps-Quintilian, Gladiator--the declamations were considered to be the work of Quintilian in Valla's time--and one of Cicero's orations, Pro Ligario.) Valla's Roman experience of humanist conversations found an outlet in his dialogue De voluptate (On Pleasure), in which the Christian concepts of charity and beatitude are identified with hedonist pleasure, while the "Stoic" concept of virtue is rejected (see below). Valla would later revise the dialogue and change the names of the interlocutors, but his Epicurean-Christian position remained the same. Meanwhile he had moved to Pavia in 1431, stimulated by his friend Panormita (Antonio Beccadelli, 1394-1471)--with whom he was soon to quarrel--and had begun to teach rhetoric. He had to flee Pavia, however, in 1433 after having aroused the anger of the jurists. In a letter to a humanist jurist friend of his, Catone Sacco (1394-1463), Valla had attacked the language of one of the lawyers' main authorities, Bartolus of Sassoferrato (1313-1357). After some travelling, in 1435 Valla found employment at the court of Alfonso of Aragon (1396-1458), who was trying to capture Naples. Though complaining about the lack of time, books and fellow humanists, Valla was immensely productive in this phase of his career. In 1439 he finished the first version of his critique of scholastic philosophy. Two years later he finished his Elegantiae linguae Latinae, a manual for the correct use of Latin syntax and vocabulary, which became a bestseller throughout Europe. As a humanist in the court of a king who was fighting against the pope, Valla demonstrated that the Donation of Constantine, which had served the papacy to claim worldly power, was a forgery. In the same years he composed a dialogue on free will and began working on his annotations to the standard Latin translation of the Bible, comparing it with the Greek text of the New Testament. He also wrote a dialogue On the Profession of the Religious (De professione religiosorum), in which he attacked the vow of obedience and asceticism taken by members of religious orders. Further, he worked on the text of Livy, wrote a history of the deeds of Alfonso's father, and began re-reading and annotating Quintilian's Institutio oratoria (Education of the Orator) in a manuscript which is still extant (Paris, Bibliotheque Nationale de France, lat. 7723). But Valla's philological approach and his penchant for quarreling made him enemies at the Aragonese court. After his patron Alfonso had made peace with the pope in 1443, Valla's role as an anti-papal polemicist may have shrunk. His enemies took advantage of the situation. The immediate occasion was Valla's denial of the apostolic origin of the Symbolum Apostolicum (Apostle's Creed). In a letter, now lost, to the Neapolitan lawyers he argued that a passage from Gratian's Decretum that formed the basis of the belief in the apostolic origin of the Creed was corrupt and needed emendation. Powerful men at the court staged an inquisitorial trial to determine whether Valla's works contained heretical and heterodox opinions. In preparation for the trial Valla wrote a self-defense (and later an "Apology"), but was rescued from this perilous situation by the intervention of the king. The whole affair must have fuelled Valla's wish to return to Rome. In 1447 he made peace with the pope and became an apostolic scriptor (scribe), and later, in 1455, a papal secretary. In these years he revised some of his earlier works such as the Repastinatio and his notes on the New Testament, and translated Thucydides and Herodotus into Latin; his work on Thucydides, in particular, was to have an important impact on the study of this difficult Greek author. Always an irascible man, he continued to engage in quarrels and exchanged a series of invectives with his arch-enemy Poggio. He died in 1457, still working on a second revision of his Dialectics (that is, the third version). He was buried in the Lateran, where his grave was removed in the sixteenth century, probably between the 1560s and 1580s at a time when his works were put on the Index; the present memorial is not Valla's tombstone. Valla's impact on the humanist movement was long-lasting and varied. His philological approach was developed by subsequent generations of humanists, and found, arguably, its first systematic expression in the work of Angelo Poliziano (1454-1494). His Elegantiae was printed many times, either in the original or in one of its many adaptations and abridgments made by later scholars. A copy of his annotations on the New Testament was found near Louvain by Erasmus (1467?-1534), who published it. Though many theologians complained about his ignorance of theology, Luther and Leibniz, each for his own reasons, found occasion to refer to Valla's dialogue on free will. Humanists found Valla's criticisms of Aristotelian-scholastic thought congenial, though not a few regarded his style as too aggressive and polemical. And though some humanists such as Juan Luis Vives (1492-1540) and Johann Eck (1486-1543) complained about Valla's lack of philosophical acumen, his critique of scholasticism has a place in the long transformation from medieval to modern thought--in which humanism played an important though by no means exclusive role. ## 2. Valla's Critique of Scholastic Philosophy In his Repastinatio dialectice et philosophie, which is extant in three versions with slightly different titles, Valla attacks what he sees as the foundations of scholastic-Aristotelian philosophy and sets out to transform Aristotelian dialectic. As the title of the first version clearly indicates, he wants to "re-plough" the ground traditionally covered by the Aristotelian scholastics. The term repastinatio means not only "re-ploughing" or "re-tilling" but also "cutting back" and "weeding out." Valla desires to weed out everything he regards as barren and infertile in scholastic thought and to re-cultivate the ground by sprinkling it with the fertile waters of rhetoric and grammar. His use of repastinatio indicates that he is setting out a program of reform rather than of destruction, in spite of his often aggressive and polemical tone. Book I is devoted mainly to metaphysics, but also contains chapters on natural philosophy and moral philosophy, as well a controversial chapter on the Trinity. In Books II and III Valla discusses propositions and forms of argumentation such as the syllogism. His main concern in the first book is to simplify the Aristotelian-scholastic apparatus. For Valla, the world consists of things, simply called res. Things have qualities and do or undergo things (which he refers to as "things"). Hence, there are three basic categories: substance, quality, and action. At the back of Valla's mind are the grammatical categories of noun, adjective, and verb; but in many places he points out that we cannot assume that, for instance, an adjective always refer to a quality or a verb to an action (Repastinatio, 1:134-156; 425-442; DD 1:240-80). These three categories are the only ones Valla admits; the other Aristotelian categories of accidents such as place, time, relation and quantity can all be reduced to quality or action. Here, too, grammar plays a leading role in Valla's thought. From a grammatical point of view, qualities such as being a father, being in the classroom, or being six-feet tall all tell us something about how a particular man is qualified; and there is, consequently, no need to preserve the other Aristotelian categories. In reducing the categories to his triad of substance, quality, and action, Valla does not seem to have in mind "realist" philosophers who accepted the independent existence of entities such as relations and quantities over and above individual things. Instead, his aim is to show that many terms traditionally placed in other categories, in fact, point to qualities or actions: linguistic usage (loquendi consuetudo) teaches us, for example, that quality is the overarching category. Thus, to the question "of what kind", we often give answers containing quantitative expressions. Take, for instance, the question: what sort of horse should I buy? It may be answered: erect, tall, with a broad chest, and so on. Valla's reduction of the categories often assumes the form of grammatical surveys of certain groups of words associated with a particular category: thus, in his discussion of time, he studies words like "day," "year," and a host of others; and his discussion of quantitative words treats mathematical terms such as line, point and circle. Of course, Valla does not deny that we can speak of quantity or time or place. But the rich array of Latin terms signify, in the final analysis, the qualities or actions of things, and nothing exists apart from concrete things. (Substance is a shady category for Valla, who says that he cannot give an example of it, much as Locke was later to maintain that "a substance is something I know not what"). It is tempting to connect this lean ontology to that of William of Ockham (c. 1287-1347). The interests, approach and arguments of the two thinkers, however, differ considerably. Unlike Valla, Ockham does not want to get rid of the system of categories. As long as one realizes, Ockham says, that categories do not describe things in the world but categorize terms by which we signify real substances or real inhering qualities in different ways, the categories can be maintained and the specific features of, for example, relational or quantitative terms can be explored. Thus, Ockham's rejection of a realist interpretation of the categories is accompanied by a wish to defend the categories as distinct groups of terms. Valla, on the other hand, sees the categories as summing up the real aspects of things: hence, there are only substances, qualities, and actions, and his reductive program consists in showing that we have a vast and rich vocabulary in Latin that we can use for referring to these things. His questions about words and classes of words are not unlike those of Priscian (fl. 500-530) and other grammarians. (Priscian, for instance, had stated that a noun signifies substance plus quality, and pronouns substance without quality.) Another good example of Valla's reduction of scholastic terminology, distinctions, and concepts is his critique of the transcendental terms (Repastinatio 1:11-21; DD 1:18-36). According to Valla, the traditional six terms--"being," "thing," "something," "one," "true," and "good"--should be reduced to "thing" (Latin res) since everything that exists, including a quality or an action of a thing (also called a "thing") is a thing. A good thing, for example, is a thing, and so, too, is a true thing. From a grammatical point of view, such terms as "good" and "true" are adjectives, and when used as substantives (the good, the one), they refer to qualities; hence, there is nothing "transcendental" about them. Valla's grammatical approach is evident in his rejection of the scholastic term ens. Just as "running" (currens) can be resolved into "he who runs" (is qui currit), so "being" (ens) can be resolved into "that which is" (id quod est). "That," however, is nothing other than "that thing" (ea res); so we get as a result the laborious formula: "that thing which is". (Repastinatio 1:14; DD 1:23-25.) We do not, however, need the phrase "that which is" (ea que est): "a stone is a being" (lapis est ens), or the equivalent phrase into which it can be resolved, "a stone is a thing which is" (lapis est res que est), are unclear, awkward, and absurd ways of saying simply that "a stone is a thing" (lapis est res). Valla also rejects other scholastic terms such as entitas ("entity"), hecceitas ("this-ness") and quidditas ("quidity") for grammatical reasons: these terms do not conform to the rules of word formation in Latin. While he is not against the introduction of new words for things unknown in antiquity (e.g., bombarda for "cannonball"), the terminology coined by the scholastics is a different matter entirely. Related to this analysis is Valla's repudiation of what he presents as the scholastic view of the distinction between abstract and concrete terms, that is, the view that abstract terms ("whiteness," "fatherhood") always refer solely to quality, while concrete terms ("white," "father") refer to both substance and quality (Repastinatio, 1:21-30; DD 1:36-54). In a careful discussion of this distinction, taking into account the grammatical categories of case, number, and gender, Valla rejects the ontological commitments which such a view seems to imply, and shows, on the basis of a host of examples drawn from classical Latin usage, that abstract terms often have the same meaning as their concrete counterparts (useful/utility, true/truth, honest/honesty). These terms refer to the concrete thing itself, that is, to the substance, its quality or action (or a combination of these three components into which a thing can be resolved). Again, Valla's main concern is to study the workings of language and how these relate to the world of everyday things, the world that we see and experience. In describing and analyzing this world of things Valla is not only guided by grammatical considerations. He also uses "common sense" and the limits of our imagination as yardsticks against which to measure scholastic notions and definitions. He thus thinks that it is ridiculous to imagine prime matter without any form or form without any matter, or to define a line as that which has no width and a point as an indivisible quantity that occupies no space. Valla's idea is that notions such as divisibility and quantity are properly at home only in the world of ordinary things. For him, there is only the world of bodies with actual shapes and dimensions; lines and points are parts of these things, but only, as he seems to suggest, in a derivative sense, in other words, as places or spaces that are filled by the body or parts of that body. If we want to measure or sketch a (part of a) body, we can select two spots on it and measure the length between them by drawing points and lines on paper or in our mind, a process through which these points and lines become visible and divisible parts of our world (Repastinatio, 1:142-147; 2:427-431; DD 254-64). But it would be wrong to abstract from this diagramming function and infer a world of points and lines with their own particular quantity. They are merely aids for measuring or outlining bodies. In modern parlance, Valla seems to be saying that ontological questions about these entities--do they exist? how do they exist?--amount to category mistakes, equivalent to asking the color of virtue. The appeal to common sense (or what Valla considers as such) informs his critique of Aristotelian natural philosophy. He insists on commonplace observations and experiences as criteria for testing ideas and hypotheses. On this basis, many of Aristotle's contentions, so Valla argues, are not true to the facts (Repastinatio, 1:98-112; DD 1:174-99). He rejects or qualifies a number of fundamental tenets of Aristotelian physics, for instance that movement is the cause of heat, that a movement is always caused by another movement, that elements can be transformed into one another, that each has its own proper qualities (heat and dryness for fire, heat and humidity for air, etc.), that there are pure elements, that the combination of heat and humidity is a sufficient condition for the generation of life, and so forth. Valla often uses reductio ad absurdum as an argumentative strategy: if Aristotle's theory is true, one would expect to observe phenomena quite different from the ones we do, in fact, observe. In arguing, for instance, for the existence of a fiery sphere below the moon, Aristotle had claimed that "leaden missiles shot out by force melt in the air" (De caelo II.7, 289a26-28). Valla rejects this claim by appealing to ordinary experience: we never see balls--whether leaden, iron, or stone, shot out of a sling or a cannon--heat up in the air; nor do the feathers of arrows catch fire. If movement is sufficient to produce heat, the spheres would set the air beneath in motion; but no one has ever observed this. Arguably, the importance of Valla's polemic here is not so much the quality of his arguments as the critical tendency they reveal, the awareness that Aristotle's conclusions often do not conform to daily experience. While he does not develop his critique in the direction of an alternative natural philosophy as later Renaissance philosophers such as Bernardino Telesio (1509-1588) and Francesco Patrizi of Cherso (1529-1597) would do, Valla contributed to undermining faith in the exclusive validity of the Aristotelian paradigm. Valla also criticizes Aristotle's natural philosophy because, according to him, it detracts from God's power. In strongly polemical terms, he attacks Aristotle for his "polytheistic" ideas and for what Valla sees as his equation of God with nature (Repastinatio, 1:54-59; DD 1:94-103). Valla wants to reinstate God as the sole creator of heaven and earth. To think of the cosmos in terms of a living animal or the heavens in terms of celestial orbs moved by intelligences is anathema to Valla (as it was to many medieval scholastics as well). The notion of God as First Mover is also rejected, since movement and rest are terms which should not be applied (except perhaps metaphorically) to spiritual beings such as God, angels, and souls. Religious considerations also led Valla to find fault with another fundamental tenet of Aristotelian scholastic thought: the Tree of Porphyry (Repastinatio, 1:46-50; 2:389-391; DD 1:82-88). Valla has several problems with the Tree. First of all, it puts substance, rather than thing, on top. For Valla, however, pure substance does not exist, since a thing is always already a qualified substance. He also thinks that there is no place for a human being in the Tree of Porphyry. Since the Tree divides substance into the corporeal and the spiritual, it is difficult to find a place for a human being, consisting of both soul and body. Moreover, the Tree covers both the divine and the created order, which leads to inappropriate descriptions of God and angels, to which the term "animal" should not be applied, since they do not have a body. Valla, therefore, divides Porphyry's Tree into three different trees: one for spiritual substance, one for corporeal substance, and one for what he calls "animal," that is, those creatures which consist of both body and soul (Repastinatio, 1:49-50; 2:422-424; DD 1:88). One might argue that what Valla gains over Porphyry by disentangling the supernatural from the natural order, he loses by having to admit that he cannot place Christ in any of his three trees, since he is not only human but also God. The soul as an incorporeal substance is treated by Valla in a separate chapter (Repastinatio, 1:59-73; 2:408-410; 418-419; DD 1:104-29). Rejecting the Aristotelian hylomorphic account, he returns to an Augustinian picture of the soul as a wholly spiritual and immaterial substance made in the image of God, and consisting of memory, intellect, and will. He rejects without much discussion the various functions of the soul (vegetative, sensitive, imaginative, intellectual), which would entail, he thinks, a plurality of souls. He briefly treats the five exterior senses but is not inclined to deal with the physiological aspects of sensation. The term "species" (whether sensible or intelligible) does not occur at all. The Aristotelian sensus communis--which the medieval commentary tradition on De anima had viewed as one of the internal senses, alongside imagination (sometimes distinguished from phantasia), memory, and the vis aestimativa (foresight and prudence)--is mentioned only to be rejected without further argument (Repastinatio, 1:73; DD 1:128). Imagination and the vis aestimativa are absent from Valla's account, while memory, as the soul's principal capacity, appears to have absorbed all the functions which scholastics had divided among separate faculties of the sensitive soul. What may have provoked Valla's anger about the traditional picture is the seemingly passive role allotted to the soul in perception and knowledge: it seems to come only at the very end of a long chain of transmission, which starts with outer objects and concludes with a merely receptive tabula rasa. In his view, the soul is far more noble than the hylomorphic account of Aristotle implies, at least as Valla understands that account. He therefore stresses on various occasions the soul's dignified nature, its immortality, unity, autonomy, and superiority to both the body and the animal soul, comparing it to the sun's central place in the cosmos. ## 3. Valla's "Reform" of Aristotelian Logic After Valla's attack on what he calls the fundamenta (foundations) of Aristotelian-scholastic metaphysics and natural philosophy, he turns to dialectic in Books II and III of his Repastinatio. For Valla, argumentation should be approached from an oratorical rather than a logical point of view. What counts is whether an argument works, which means whether it convinces one's adversary or public. The form of the argument is less important. Dialectic is a species of confirmation and refutation; and, as such, it is merely a component of invention, one of the five parts of rhetoric (Repastinatio, 1:175; 2:447; DD 2:3). Compared to rhetoric, dialectic is an easy subject and does not require much time to master, since it considers and uses only the syllogism "bare", as Valla puts it, that is, in isolation from its wider argumentative context; its sole aim is to teach. The rhetorician, on the other hand, uses not only the syllogism, but also the enthymeme (incomplete syllogism), epicheireme (a kind of extended reasoning) and example. The orator has to clothe everything in persuasive arguments, since his task is not only to teach but also to please and to move. This leads Valla to downplay the importance of the Aristotelian syllogism and to consider forms of argumentation that are not easily forced into its straightjacket. Among these are captious forms of reasoning such as the dilemma, paradox, and heap argument (sorites), and Valla offers an interesting analysis of these forms in the last book of the Repastinatio. Without rejecting the syllogism tout court, Valla is scathing about its usefulness. He regards it as an artificial type of reasoning, unfit to be employed by orators since it is does not reflect the natural way of speaking and arguing. What is the use, for example, of concluding that Socrates is an animal if one has already stated that every man is an animal and that Socrates is a man? It is a simple, puerile, and pedantic affair, hardly amounting to a real ars (art). Valla's treatment of the syllogism clearly shows his oratorical perspective. Following Quintilian, he stresses that the nature of syllogistic reasoning is to establish proof. One of the two premises contains what is to be proven (que probatur), and the other offers the proof (que probat), while the conclusion gives the result of the proof--into which the proof "goes down" (in quam probatio descendit). It is not always necessary, therefore, to have a fixed order (major, minor, conclusion). If it suits the occasion better, we can just as well begin with the minor, or even with the conclusion. The order is merely a matter of convention and custom (Repastinatio, 1:282-286; 2:531-534; DD 2:216-39). These complaints about the artificiality of the syllogism inform Valla's discussion of the three figures of the syllogism. Aristotle had proven the validity of the moods of Figure 2 and 3 by converting them to four moods of Figure 1; and this was taught, for example, by Peter of Spain (thirteenth century) in his widely read handbook on logic, the Summulae logicales, certainly consulted by Valla here. Valla regards this whole business of converting terms and transposing propositions in order to reduce a particular syllogism to one of these four moods of Figure 1 as useless and absurd. While he does not question the validity of these four moods, he believes that there are many deviant syllogisms that are also valid, for instance: God is in every place; Tartarus is a place; therefore, God is in Tartarus. Here the "every" or "all" sign is added to the predicate in the major proposition. He says, moreover, that an entirely singular syllogism can be valid: Homer is the greatest of poets; this man is the greatest of poets; therefore, this man is Homer. And he gives many other examples of such deviant schemes. Valla thus deliberately ignores the criteria employed by Aristotle and his commentators--that at least one premise must be universal, and at least one premise must be affirmative, and that if the conclusion is to be negative, one premise must be negative--or, at any rate, he thinks that they unnecessarily restrict the number of possible valid figures. In his discussion of the syllogism Valla does not refer to an important principle employed by Aristotle and his commentators: dici de omni et nullo (to be said/predicated about all and about none). To quote Peter of Spain's Summulae logicales: "To be said of every [dici de omni] is when there is nothing to to be taken under the subject [nichil est sumere sub subiecto] of which the predicate may not be said, like 'every man runs': here, running is said of every man, and under man there is nothing to to be taken of which running is not said. To be said of none [de nullo] is when there is nothing to be taken under the subject from which the predicate may not be eliminated, like 'no man runs': here, running is eliminated from any man at all [quolibet homine]" (ed. L. M. de Rijk 1972, 43; ed. Copenhaver et al., 2014, 170). This principle led Aristotle to conclude that only four moods of the first figure were immediately valid. That Valla does not make use of this fundamental condition is understandable from his oratorical point of view, since it would be an uninteresting or even irrelevant criterion of validity. It is therefore not surprising that he thinks that we might as well "reduce" the first figure to the second rather than vice versa. Likewise, from his oratorical point of view, he can only treat the third figure of the syllogism with contempt: it is a "completely foolish" form of reasoning; no one reasons as follows: every man is a substance; every man is an animal; therefore, some animal is a substance. In a similar vein, he rejects the use of letters in the study of syllogisms (Repastinatio, 1:297-300; 2:546-548; DD 2:264-71). Valla's insistence on examining and assessing arguments in terms of persuasion and usefulness leads him to criticize not only the syllogism but also other less formal modes of argumentation. These modes usually involve interrogation, resulting in an unexpected or unwanted conclusion or an aporetic situation. Some scholars have regarded Valla's interest in these less formal or non-formal arguments as an expression of a skeptical attitude towards the possibility of certainty in knowledge in general. Others have raised serious doubts about this interpretation. What we can say for sure, however, is that Valla was one of the first to study and analyze the heap argument, dilemma, and suchlike. The heap argument is supposed to induce doubts about the possibility of determining precise limits, especially to quantities. If I subtract one grain from a heap, is it still a heap? Of course. What if I subtract two grains? And so forth, until the heap consists of just one grain, which, to be sure, is an unacceptable conclusion. It seems impossible to determine the exact moment when the heap ceases to be a heap, and any attempt to determine this moment seems to involve an ad hoc decision, for a heap does not cease to be a heap due to the subtraction of just one single grain. Valla discusses a number of similar cases, and comments on their fallacious nature. Dilemma, too, receives extensive treatment from him. This type of argument had been widely studied in antiquity. The basic structure is a disjunction of propositions, usually in the form of a double question in an interrogation, which sets a trap for the respondent, since whichever horn of the dilemma he chooses, he seems to be caught up in a contradiction and will lose the debate ("If he is modest, why should you accuse someone who is a good man? If he is bad, why should you accuse someone who is unconcerned by such a charge?", Cicero, De inventione 1.45.83, cited by Valla Repastinatio, 1:321; DD 2:332). It was also recognized that the respondent could often counter the dilemma by duplicating the original argument and "turning it back" (convertere) on the interrogator, using it as a kind of boomerang ("if he is modest, you should accuse him because he will be concerned by such a charge; if he is bad, you should also accuse him because he is not a good man"). Alternatively, he could escape the dilemma by questioning the disjunction and showing that there is a third possibility. There were many variations of this simple scheme, and it was studied from a logical as well as a rhetorical point of view with considerable overlap between these two perspectives. In medieval times, dilemma does not seem to have attracted much theoretical reflection, though there was an extensive literature on related genres such as insolubilia and paradoxes, which were generally treated in a logical manner. It is, therefore, interesting to see Valla discussing a whole range of examples of dilemma. The rhetoric textbook by the Byzantine emigre George of Trebizond (1396-1486), composed about 1433, was probably an important source for him. This work might also have led Valla to explore the relevant places in Quintilian, Cicero, the Greek text of Aristotle's Rhetoric and perhaps other Greek sources. The example Valla discusses most extensively comes from Aulus Gellius and concerns a lawsuit between Protagoras and his pupil Euathlus (Repastinatio, 1:312-319; 2:562-568; DD 2:312-28. Aulus Gellius, Noctes Atticae, 5.10.5-16). The pupil has promised to pay the second installment of the fees as soon as he has won his first case. He refuses to undertake any cases, however, and Protagoras takes him to court, putting the case in the form of a dilemma. If Euathlus loses the case, he will have to pay the rest of the fee, on account of the verdict of the judges; but if Euathlus wins, he will also have to pay, this time on account of his agreement with Protagoras. Euathlus, however, cleverly converts the argument: in neither case will he have to pay, on account of the court's decision (if he wins), or on account of the agreement with Protagoras (if he loses). Aulus Gellius thinks that the judges should have refrained from passing judgment because any decision would be inconsistent with itself. But Valla rejects such a rebuttal (antistrephon or conversio) of the dilemma and thinks that a response may be formulated as long as one concentrates on the relevant aspects of the case. He carefully considers the perspectives of all parties, the words employed, the feelings of the judges, and all the circumstances of the case. In a highly rhetorical analysis of the case in which Valla makes speeches on behalf of Protagoras, Valla says in the end that Euathlus cannot have it both ways and must choose one or the other alternative, not both: he must comply either with his agreement with Protagoras (and pay the rest of the fees) or with the verdict passed by the judges. If he loses the case, a refusal to obey the sentence of the judges shows contempt; if he wins the case, he has to pay the rest of the fee to Protagoras. At any rate, there is no reason for the judges to despair; Aulus Gellius is, therefore, wrong in thinking that they should have refrained from passing a judgment. In all such cases, Valla argues, the conversion is not a rebuttal, but at best a correction of the initial argument (a correction, however, is not a refutation), and at worst a simple repetition or illegitimate shift of the initial position. An important way of seeing through deceptive arguments is to consider the weight of words carefully, and Valla gives some further examples of fallacies which can easily be refuted by examining the meaning and usage of words and the contexts in which they occur (Repastinatio, 1:320-334; 2:568-578; DD 2:328-69). He considers the fallacies "collected by Aristotle in his Sophistical Refutations as for the most part a puerile art," quoting the Rhetorica ad Herennium (2.11.16) to the effect that "knowledge of ambiguities as taught by dialecticians is of no help at all but rather a most serious hindrance." Valla is not interested in providing a comprehensive list of deceptive arguments and errors or in studying rules for resolving them. He does not mention, for example, the basic division between linguistic (in dictione) and extra-linguistic (extra dictionem) fallacies. In a letter to a friend, Valla lists some "dialecticians"--Albert of Saxony (c. 1316-1390), Albert the Great (c. 1220-1280), Ralph Strode (fl. 1350-1400), William of Ockham, and Paul of Venice (c. 1369-1429) --but there is no sign that he had done more than leaf through their works on sophismata and insolubilia (Valla 1984, 201). The few examples he gives seem to come from Peter of Spain's Summulae logicales. Nor was it necessary to have more than a superficial acquaintance with the works of these dialecticians in order to realize that their approach differed vastly from his. As he repeatedly states, what is required in order to disambiguate fallacies is not a deeper knowledge of the rules of logic but a recognition that arguments need to be evaluated within their wider linguistic and argumentative context. Such an examination of how words and arguments function will easily lay bare the artificial and sophistical nature of these forms of argumentation. This approach is also evident in Valla's analysis of the proposition of which a syllogism or argument consists. Propositions are traditionally divided according to quantity (universal of particular) and to quality (affirmative and negative). Quantity and quality are indicated by words that are called signa (markers, signs): "all," "any," "not," "no one," and so forth. In Book II of the Repastinatio, Valla considers a much wider range of words than the medieval logicians, who had mainly worked with "all," "some," "none," and "no one". To some extent, his aim is not unlike that of the dialecticians whom he so frequently attacks, that is, to study signs of quality and negation and how they determine the scope of a proposition. But for him there is only one proper method of carrying out such a study: examining carefully the multifarious ways in which these words are used in refined and grammatically correct Latin. Not surprisingly, Valla criticizes the square of contraries--the fourfold classification of statements in which the distinction between universal and particular and that between affirmative and negative are combined. A similar critique of the rather arbitrary restriction to a limited set of words is applied to the scholastic notion of modality (Valla Repastinatio, 1:237-244; 2:491-497; DD 2:126-43). Scholastics usually treat only the following six terms as modals: "possible," "impossible," "true," "false," "necessary," and "contingent." Latin, however, is much more resourceful in expressing modality. Using criteria such as refinement and utility, Valla considers terms such as "likely/unlikely," "difficult/easy," "certain/uncertain," "useful/useless," "becoming/unbecoming," and "honorable/dishonorable." This amounts to introducing a wholly new concept of modality, which comes close to an adverbial qualification of a given action. Valla's principal betes noires are Aristotle, Boethius, Porphyry, and Peter of Spain, but he also speaks in general terms of the entire natio peripatetica (nation of Peripatetics). He frequently refers to isti (those), a suitably vague label for the scholastic followers of Aristotle--including both dialecticians and theologians. It has often been claimed that Valla is attacking late medieval scholasticism; but it must be said that he does not quote any late medieval scholastic philosophers or theologians. He generally steers clear of their questions, arguments, and terminology. If we compare Valla's Repastinatio with, for example, Paul of Venice's Logica parva (The Small Logic), we quickly perceive the immense difference in attitude, argument, and approach. Moreover, as noted above, Valla's grammatical and oratorical approach is fundamentally different from Ockham's terminism. He probably thought that he did not need to engage with the technical details of scholastic works. It was sufficient for him to establish that there was a huge distance between his own approach and that of the scholastics. Once he had shown that the scholastic-Aristotelian edifice was built on shaky foundations, he did not care to attack the superstructure, so to speak. And Valla proved to his own satisfaction that these foundations were shaky by showing that the terminology and vocabulary of the scholastics rested on a misunderstanding of Latin and of the workings of language in general. But even though his criticisms are mainly addressed to Aristotle, Boethius, and Porphyry, his more general opinion of Aristotelianism was, of course, formed by what he saw in his own time. He loathed the philosophical establishment at the universities, their methods, genres, and, above all, their style and terminology. He thought that they were slavishly following Aristotle. In his view, however, a true philosopher does not hesitate to re-assess any opinion from whatever source; refusing to align himself with a sect or school, he presents himself as a critical, independent thinker (Repastinatio, 1:1-8; 2:359-363; DD 1:2-12). Whether scholasticism had, indeed, ossified by the time Valla came on the scene is a matter of debate; but there is no question that this is how he and the other humanists saw it. ## 4. Moral Philosophy The same critical spirit also infuses Valla's work on moral philosophy. In his dialogue, published as De voluptate in 1431, when he was still in his mid-twenties, and revised two years later under the title De vero bono (On the True Good), Valla presents a discussion between an "Epicurean," a "Stoic," and a "Christian" on an age-old question: what is the highest ethical good? The result of this confrontation between pagan and Christian moral thought is a combination of Pauline fideism and Epicurean hedonism, in which the Christian concepts of charity and beatitude are identified with hedonist pleasure, and the "Stoic" concept of virtue is rejected (Valla, De vero falsoque bono). Valla thus treats Epicureanism as a stepping-stone to the development of a Christian morality based on the concept of pleasure, and repudiates the traditional synthesis of Stoicism and Christianity, popular among scholastics and humanists alike. The substance of the dialogue is repeated in a long chapter in his Repastinatio (Repastinatio, 1:73-98; 2:411-418; DD 1:130-75). Valla's strategy is to reduce the traditional four virtues-- prudence, justice, fortitude, and propriety (or temperance)--to fortitude, and then to equate fortitude with charity and love. For Valla, fortitude is the essential virtue, since it shows that we do not allow ourselves to be conquered by the wrong emotions, but instead to act for the good. As a true virtue of action, it is closely connected to justice and is defined as "a certain resistance against both the harsh and the pleasant things which prudence has declared to be evils." It is the power to tolerate and suffer adversity and bad luck, but also to resist the blandishments of a fortune which can be all too good, thus weakening the spirit. Fortitude is the only true virtue, because virtue resides in the will, since our actions, to which we assign moral qualifications, proceed from the will. Valla's reductive strategy has a clear aim: to equate this essential virtue of action, fortitude, with the biblical concept of love and charity. This step requires some hermeneutic manipulation, but the Stoic overtones of Cicero's account in De officiis have prepared the way for it--ironically, perhaps, in view of Valla's professed hostility towards Stoicism--since enduring hardship with Stoic patience is easily linked to the Pauline message that we become strong by being tested (II Cor. 12:10, quoted by Valla). The labor, sweat, and trouble we must bear, though bad in themselves, "are called good because they lead to that victory," Valla writes, echoing St Paul (Repastinatio, 1:88-89; 2:415; DD 1:156). We do not, then, strive to attain virtue for its own sake, since it is full of toil and hardship, but rather because it leads us to our goal. This is one of Valla's major claims against the Stoics and the Peripatetics, who--at least in Valla's interpretation-- regarded virtue as the end of life, that is, the goal which is sought for its own sake. Because virtuous behavior is difficult, requiring us to put up with harsh and bitter afflictions, no one naturally and voluntarily seeks virtue as an end in itself. What we seek is pleasure or delectation, both in this life and--far more importantly --in the life to come. By equating pleasure with love, Valla can argue that it is love or pleasure that is our ultimate end. This entails the striking notion that God is not loved for his own sake, but for the sake of love: "For nothing is loved for its own sake or for the sake of something else as another end, but the love itself is the end" (Repastinatio, 2:417). This is a daring move. Traditionally, God was said to be loved for his own sake, not for his usefulness in gaining something else. Many thinkers agreed with Augustine that concupiscent love was to be distinguished from friendship, and, with respect to heavenly beatitude, use from fruition. We can love something as a means to an end (use), and we can love something for its own sake (fruition). But because Valla has maintained that pleasure is our highest good, God can only be loved as a means to that end. It is therefore a moot point whether Valla successfully integrated Epicurean hedonism with Christian morality. He seems to argue that the Epicurean position is valid only for the period before the coming of Christ. In our unredeemed state, we are rightly regarded as pleasure-seeking animals, governed by self-interest and utilitarian motives. After Christ's coming, however, we have a different picture: repudiating Epicurean pleasure, we should choose the harsh and difficult life of Christian honestas (virtue) as a step towards heavenly beatitude. Yet, the two views of human beings are not so readily combined. On the one hand, there is the positive evaluation of pleasure as the fundamental principle in human psychology--which is confirmed and underscored by the terminological equation of voluptas (pleasure), beatitudo (beatitude), fruitio (fruition), delectatio (delectation), and amor (love). On the other hand, Valla states apodictically that there are two pleasures: an earthly one, which is the mother of vice, and a heavenly one, which is the mother of virtue; that we should abstain from the former if we want to enjoy the latter; and that the natural, pre-Christian life is "empty and worthy of punishment" if not put in the wider perspective of human destiny. In other words, we are commanded to live the arduous and difficult life of Christian honestas, ruled by restraint, self-denial, and propriety (temperance), and, at the same time, to live a hedonist life, which consists of the joyful, free, and natural gratification of the senses. Another of his targets is the Aristotelian account of virtue as a mean between two extremes. According to Valla, each individual vice is instead the opposite of an individual virtue. He makes this point by distinguishing between two different senses of the same virtue, showing that they have different opposites. So, while Aristotle regards fortitude as the mean between the vices of rashness and cowardice, Valla argues that there are two aspects to fortitude: fighting bravely and being cautious (for instance, in yielding to the victorious enemy), with cowardice and rashness as their respective opposites. Likewise, generosity is not the mean between avarice and prodigality, but has two aspects: giving and not giving. Prodigality is the opposite of the first aspect, avarice of the second, for which we should use the term frugality or thrift rather than generosity. More generally, the terminology of vices as defects and excesses and virtue as a mean is misleading; virtues and vices should not be ranked "according to whether they are at the bottom, or halfway up, or at the top." Interestingly, a similar critique of Aristotle's notion of virtue as a mean between two extremes has been raised in modern scholarship (for example, by W. D. Ross). Valla regards the Aristotelian notion of virtue as too static and inflexible, so that it does not do justice to the impulsive nature of our moral behavior. For him, virtue is not a habit, as Aristotle believed, but rather an affect, an emotion or feeling that can be acquired and lost in a moment's time. Virtue is the domain solely of the will. The greatest virtue or the worse vice may arise out of one single act. And because virtue is painful and vice tempting, one may easily slide from the one into the other, unlike knowledge, which does not turn into ignorance all of a sudden. Valla therefore frequently removes the notions of knowledge, truth, and prudence from the sphere of moral action. Virtues as affects are located in the rear part of the soul, the will, while the domains of knowledge, truth, and opinion reside in the other two faculties, memory and reason (Repastinatio, 1:73-74; DD 1:130). This is not to say that the will is independent from the intellectual capacities. The affects need reason as their guide, and the lack of such guidance can result in vice. But Valla is not entirely clear as to what element we should assign the moral qualification "good" or "bad." He identifies virtues with affects, and says that only these merit praise and blame (Repastinatio, 1:74; DD 1:130-32); however, he also writes that the virtues, as affects, cannot be called good or bad in themselves, but that these judgments apply only to the will, that is, to the will's choice. This is underlined by his remark that virtue resides in the will rather than in an action (Repastinatio, 1:77; DD 1:136). In his discussion of the soul, however, he frequently calls reason the will's guide (e.g., Valla Repastinatio, 1:75; DD 1:132), and also says that the affects should follow reason, so that it, too, may be held responsible, in the final analysis, for moral behavior (even though he also explicitly denies that the will is determined by reason). Finally, pleasure, delectation, or beatitude are also called virtue by the equation of virtue with love and with charity (Repastinatio, 1:85; DD 1:150-52). Moreover, Valla's insistence on the will as the locus of moral behavior seems compromised by the predestinarianism advocated by the interlocutor "Lorenzo" in his dialogue De libero arbitrio (On Free Will). In this highly rhetorical work, Valla--if we can assume that Lorenzo represents Valla's own position--stresses that in his inscrutable wisdom God hardens the hearts of some, while saving those of others. We do not know why, and it is presumptuous and vain to inquire into the matter. Yet, somehow we do have free will, "Now, indeed, He brings no necessity, and His hardening one and showing another mercy does not deprive us of free will, since He does this most wisely and in full holiness" (Valla 1948, 177). So, we are free, after all, and God's foreknowledge does not necessitate the future; but how exactly Valla thinks his views settle these issues is not clear. His fideistic point is that we humans cannot settle the issue; 'the cause of the divine will which hardens one and shows mercy to another is known neither to man nor to angels'. Yet, in spite of his fideism and anti-intellectualism, his own discussion of God and free will, as well as his account of the Trinity in the Repastinatio, makes substantive claims about what God is. In conclusion, Valla's moral thought can be described--with some justice--as hedonistic, voluntarist, and perhaps also empirical (in the sense of taking account of how people actually behave). On closer inspection, however, his account seems to contain the seeds of several ideas that are not so easily reconciled with one another. This is doubtless due, in no small measure, to his eclecticism, his attempt to bring into one picture Aristotelian ethics, the Stoic virtues of Cicero, the biblical concepts of charity and beatitude, and the Epicurean notion of hedonist pleasure-- each with its own distinctive terminology, definitions, and philosophical context. ## 5. Evaluation Valla's contributions to historical, classical, and biblical scholarship are beyond doubt, and helped to pave the way for the critical textual philology of Poliziano, Erasmus, and later generations of humanists. Valla grasped the important insight--which was not unknown to medieval philosophers and theologians--that the meaning of a text can be understood only when it is seen as the product of its original historical and cultural context. Yet his attempt to reform or transform the scholastic study of language and argumentation--and, indeed, their entire mode of doing philosophy--is likely to be met with skepticism or even hostility by the historian of medieval philosophy who is dedicated to the argumentative rigor and conceptual analysis which are the hallmarks of scholastic thought. Nevertheless, while it may be true that Valla's individual arguments are sometimes weak, superficial, and unfair, his critique as a whole does have an important philosophical and historical significance. The following two points, in particular, should be mentioned. First, the humanist study of Stoicism, Epicureanism, skepticism, and Neoplatonism widened the philosophical horizon and eroded faith in the universal truth of Aristotelian philosophy--an essential preparatory stage for the rise of early modern thought. In Valla's day, Aristotle was still "the Philosopher," and scholastics put considerable effort into explaining his words. Valla attacks what he sees as the ipse dixit attitude of the scholastics. For him, a true philosopher does not follow a single master but instead says whatever he thinks. Referring to Pythagoras' modest claim that he was a not a wise man but a lover of wisdom (Repastinatio, 1:1; DD 1:2), Valla maintains that he does not belong to any sect (including that of the skeptics) and wants to retain his independence as critical thinker. What the scholastics forget, he thinks, is that there were many alternatives in antiquity to the supposedly great master, many sects, and many other types of philosopher. In criticizing Aristotle's natural philosophy, for instance, he gave vent to a sentiment which ultimately eroded faith in the Aristotelian system. This does not, however, mean that he was developing a non-Aristotelian natural philosophy; his rejection of Aristotle's account of nature was primarily motivated by religious and linguistic considerations. Indeed, Valla's insistence on common linguistic usage, combined with his appeal to common sense and his religious fervor, seems at times to foster a fideism that is at odds with an exploratory attitude towards the natural world. But with hindsight we can say that any undermining of the faith in the exclusiveness of the scholastic-Aristotelian worldview contributed to its demise and, ultimately, to its replacement by a different, mechanistic one. And although the humanist polemic was only one factor among many others, its role in this process was by no means negligible. Likewise, in attacking Peripatetic moral philosophy, Valla showed that there were alternatives to the Aristotelian paradigm, even though his use of Epicureanism and Stoicism was rhetorical rather than historical. The second point relates to the previous one. In placing himself in opposition to what he regarded as the Aristotelian paradigm, Valla often interprets certain doctrines--the syllogism, hypothetical syllogism, modal propositions, and the square of contraries--in ways they were not designed for. In such cases, we can see Valla starting, as it were, from the inside of the Aristotelian paradigm, from some basic assumptions and ideas of his opponents, in order to refute them by using a kind of reductio ad absurdum or submitting them to his own criteria, which are external to the paradigm. This moving inside and outside of the Aristotelian paradigm can explain (and perhaps excuse) Valla's inconsistency, for it is an inconsistency which is closely tied to his tactics and his agenda. He does not want to be consistent if this means merely obeying the rules of the scholastics, which in his view amounted to rigorously defining one's terms and pressing these into the straightjacket of a syllogistic argument, no matter what common sense and linguistic custom teach us. Behind this inconsistency, therefore, lies a consistent program of replacing philosophical speculation and theorizing with an approach based on common linguistic practice and common sense. But arguably it has also philosophical relevance; for throughout the history of philosophy a warning can be heard against abstraction, speculation, and formalization. One need not endorse this cautionary note in order to see that philosophy thrives on the creative tension between, on the one hand, a tendency to abstract, speculate, and formalize, and, on the other, a concern that the object of philosophical analysis should not be lost from sight, that philosophy should not become a game of its own--an abstract and theoretical affair that leaves the world it purports to analyze and explain far behind, using a language that can be understood only by its own practitioners.
value-incommensurable	## 1. Value Incommensurability Incommensurability between values must be distinguished from the kind of incommensurability associated with Paul Feyerabend (1978, 1981, 1993) and Thomas Kuhn (1977, 1983, 1996) in epistemology and the philosophy of science. Feyerabend and Kuhn were concerned with incommensurability between rival theories or paradigms -- that is, the inability to express or comprehend one conceptual scheme, such as Aristotelian physics, in terms of another, such as Newtonian physics. In contrast, contemporary inquiry into value incommensurability concerns comparisons among abstract values (such as liberty or equality) or particular bearers of value (such as a certain institution or its effects on liberty or equality). The term "bearer of value" is to be understood broadly. Bearers of value can be objects of potential choice (such as a career) or states of affairs that cannot be chosen (such as a beautiful sunset). Such bearers of value are valuable in virtue of the abstract value or values they instantiate or display (so, for example, an institution might be valuable in virtue of the liberty or equality that it engenders or embodies). ### 1.1 Measurement and Comparison The term "incommensurable" suggests the lack of a common measure. This idea has its historical roots in mathematics. For the ancient Greeks, who had not recognized irrational numbers, the dimensions of certain mathematical objects were found to lack a common unit of measurement. Consider the side and the diagonal of a unit square. These can be compared or ranked ordinally, since the diagonal is longer. However, without the use of irrational numbers, there is no way to specify with cardinal numbers exactly how much longer the diagonal is than the side of a unit square. The significance of this kind of incommensurability, especially for the Pythagoreans, is a matter of some debate (Burkert 1972, 455-465). Hippasus of Metapontum, who was thought by many to have demonstrated this kind of incommensurability, is held by legend to have been drowned by the gods for revealing his discovery (Heath 1921, 154; von Fritz 1970, 407). Given these historical roots, some authors reserve the term "incommensurable" for comparisons that can be made, but not cardinally (Stocker 1980, 176; Stocker 1997, 203; Chang 1997b, 2). Others interpret the idea of a common measure more broadly. On this broader interpretation, for there to be a common measure, all that is required is that ordinal comparisons or rankings are possible. Values or bearers of value are then incommensurable only when not even an ordinal comparison or ranking is possible (e.g., Raz 1986; Rabinowicz 2021a). On this interpretation, incommensurability is defined as the relation that holds between two items when neither is better than the other nor are they equally as good. Others have not given an exact definition of the term, but use it as an inclusive umbrella term for comparability problems in general. This inclusive interpretation has the advantage of encompassing a field of diverse philosophical discussions which engages in problems concerning value comparisons. This entry encourages the use of "incommensurable" in the etymologically correct way i.e., to refer to the lack of a cardinal scale. ### 1.2 Incommensurable or Incomparable? Because the idea of comparison is closely tied to the topic of value incommensurability, this has led to use of the term "incomparable" alongside "incommensurable" in the literature. Some authors use the terms interchangeably (e.g., Raz 1986). Others use them to refer to distinct concepts (e.g., Chang 1997b). This entry will use the term "incomparable" to refer to the possibility that no positive value relation holds between two value bearers (Chang 1997b). Positive value relations specify how two items compare (e.g., "better than") rather than how two items do not compare (e.g., "not better than"). Incomparability is often taken to be a three-place relation: A is incomparable with B with respect to V. "V" is here some specific consideration for which the items are being compared. One career may be incomparable to another with respect to the sense of purpose it will bring to your life, yet they may be comparable with respect to financial stability since one is clearly better than the other in this regard. Some argue that without specifying this "covering consideration" the comparative claim makes no sense (Chang 1997; Thomson 1997; Andersson 2016b). Often, however, the covering consideration is not explicitly expressed, but implicitly assumed by the context of utterance. Specifying the covering consideration also allows us to identify "noncomparability". If no single covering consideration is applicable to the things we are comparing, then the things are noncomparable. For example, the number four is noncomparable with the color blue with respect to their tastiness. This form of comparative failure is considered to be of no interest from the perspective of practical reasoning and is consequently not given much consideration in the literature. From the fact that two things are incommensurable i.e., lack a common unit of measurement, it does not follow that they are incomparable. It is possible to correctly judge that one thing is better than the other while it being impossible to measure how much better it is. However, when there is incomparability there is also incommensurability. If no positive value relation holds between two things, then they cannot be placed on the same cardinal scale. While both incommensurability and incomparability are important concepts, much of the recent research has focused on the more specific possibility of incomparable value bearers. Different competing accounts have been given for examples that seem to point to the possibility of things being incomparable. These examples often take the form similar to that of Joseph Raz's example in which a person faces the choice between two successful careers: one as a lawyer and one as a clarinetist. Neither career seems better than the other, and they also do not appear to be equally good. If they were of equal value, then a slightly improved version of the legal career would be better than the musical career, but this judgment appears incorrect (Raz 1986, 332). The focus on incomparability can be motived by its possible significance for rational choice. Comparing alternatives are central for rational choice, we want to know what alternative is the best, and thus examples of incomparability are taken to be a threat to rational decision-making. Incommensurability on the other hand is mostly of theoretical interest. The possibility of incommensurable values introduces restrictions for normative theories, in the sense that they should not assume that all values can be represented on the same cardinal scale. The focus on incomparability can thus be justified by referring to its relation to practical reason and rational choice. The remainder of section 1 considers proposed ways in which to conceive of value incommensurability. ### 1.3 Conceptions of Value Incommensurability This section outlines three conceptions of value incommensurability, each one capturing some sense in which incommensurability is due to the lack of a common measure. The first conception characterizes value incommensurability in terms of restrictions on how the further realization of one value outranks realization of another value. James Griffin has proposed forms of value incommensurability of this sort. One form involves what he calls "trumping." In a conflict between values A and B, A is said to trump B if "any amount of A, no matter how small, is more valuable than any amount of B, no matter how large" (Griffin 1986, 83). A weaker form of value incommensurability involves what Griffin calls "discontinuity." Two values, A and B, are incommensurable in this sense if "so long as we have enough of B any amount of A outranks any further amount of B; or that enough of A outranks any amount of B" (Griffin 1986, 85). If values are incommensurable in this first sense, there is no ambiguity whether the realization of one value outranks realization of the other. Ambiguity as to whether the realization of one value outranks realization of the other, however, is thought by many theorists to be a central feature of incommensurable values. The second and third conceptions of value incommensurability aim to capture this feature. According to the second conception, values are incommensurable if and only if there is no true general overall ranking of the realization of one value against the realization of the other value. David Wiggins, for example, puts this forward as one conception of value incommensurability. He writes that two values are incommensurable if "there is no general way in which A and B trade off in the whole range of situations of choice and comparison in which they figure" (1997, 59). This second conception of value incommensurability denies what Henry Richardson calls "strong commensurability" (1994, 104-105). Strong commensurability is the thesis that there is a true ranking of the realization of one value against the realization of the other value in terms of one common value across all conflicts of value. A denial of such a singular common value, however, does not rule out what Richardson calls "weak commensurability" (1994, 105). Weak commensurability is the thesis that in any given conflict of values, there is a true ranking of the realization of one value against the realization of the other value in terms of some value. This value may be one of the values in question or some independent value. This value also may differ across value conflicts. The denial of strong commensurability does not entail a denial of weak commensurability. Even if there is no systematic or general way to resolve any given conflict of values, there may be some value in virtue of which the realization of one value ranks against realization of the other. Donald Regan defends the thesis of strong commensurability (Regan 1997). The third conception of value incommensurability denies both strong and weak commensurability (Richardson 1994, Wiggins 1997, Williams 1981). This conception claims that in some conflicts of values, there is no true ranking of values. This third conception of value incommensurability is sometimes said to be necessary to explain why, in conflicts of value, a gain in one value does not always cancel the loss in another value. This view assumes that, whenever there is a true ranking between the realization of one value and the realization of another value, the gain in one of the values cancels the loss in the other. Many commentators question that assumption. This entry leaves open the possibility that when there is a true ranking between the realization of one value and the realization of another value, the gain in one of the two values need not cancel the loss in the other. If we accept this possibility, a number of questions arise. One question is what a ranking of realizations of values means if a gain in one value does not cancel the loss in the other. A second question is whether the first and second conceptions of value incommensurability each admit of two versions: one version in which the gain in one value cancels the loss of the other and one version in which it does not. A third question concerns the relation between value incommensurability and tragedy. It may be thought that what makes a choice tragic is that, no matter which alternative is chosen, the gain in value cannot cancel the loss in the other value. Not all authors, however, regard all value conflicts involving incommensurable values as tragic (Richardson 1994, 117). Wiggins, for example, reserves the second conception of value incommensurability for what he calls "common or garden variety incommensurable" choices and the third conception for what he calls "circumstantially cum tragically incommensurable" choices (1997, 64). ## 2. Incomparability Rather than focus on commensurability between abstract values, recent research focuses on the incomparability between concrete bearers of value, often in the context of choice (Broome 1997, 2000; Chang 1997, 2002; Griffin 1986; Raz 1986). Bearers of value sometimes appear incomparable in cases like Joseph Raz's example of the choice between careers (described above in subsection 1.2). The case for incomparability in such examples relies in part on what Ruth Chang calls the "Small Improvement Argument" (Chang 2002b, 667). As noted in the initial discussion of the example, if the legal and musical careers were of equal value, then a slightly improved version of the legal career would be better than the musical career, but this judgment appears incorrect. The Small Improvement Argument takes the following general form: "if (1) A is neither better nor worse than B (with respect to V), (2) A+ is better than A (with respect to V), (3) A+ is not better than B (with respect to V), then (4) A and B are not related by any of the standard trichotomy of relations (relativized to V)" where V represents the relevant set of considerations for purposes of the comparison (Chang 2002b, 667-668). In addition to Raz, Derek Parfit and Walter Sinnott-Armstrong are among those who have advanced the Small Improvement Argument (Parfit 1984; Sinnott-Armstrong 1985). A similar argument, but for preference relations has, however, a longer history. Leonard J. Savage hints towards this possibility already 1954, in 1958 R. Duncan Luce presents a version of the argument that he attributes to Howard Raiffa, and it is also discussed by Ronald de Sousa in 1974 (Savage 1954; Luce 1958; de Sousa 1974). Focusing on the incomparability of bearers of value has given rise to two lines of inquiry in the literature. The first concerns the relation between incomparability and vagueness; some argue that alleged examples of incomparability are better understood in terms of vagueness. The second concerns the range of comparative relations that can hold between two items; some argue that the Small Improvement Argument does not establish incomparability but rather that there are more value relations than previously believed. This section summarizes debate within each line of inquiry. ### 2.1 Vagueness Raz distinguishes incomparability from what he calls the "indeterminacy" of value. Recall that Raz defines two bearers of value as incomparable if and only if it is not true that "either one is better than the other or they are of equal value." The indeterminacy of value is a case of vagueness: it is neither true nor false of two items that "either one is better than the other or they are of equal value." Raz regards the indeterminacy of value to result from the "general indeterminacy of language" (1986, 324). In contrast, other philosophers argue for interpreting alleged examples of incomparability as vagueness (Griffin 1986, 96; Broome 1997, 2000; Andersson 2017; Elson 2017; Dos Santos 2019). Some have argued that the Small Improvement Argument fails to rule out the possibility that it is indeterminate how the value bearers relate; they may be related by one of the standard trichotomous relations, but it could be indeterminate which (Wasserman 2004; Klockseim 2010; Gustafsson 2013). There are also more positive arguments in favor of the vagueness interpretation. For example, Broome introduces what he calls a "standard configuration" (1997, 96; 2000, 23). Imagine the musical and legal careers from Raz's example. Fix the musical career as the "standard." Now imagine variations in the legal career arranged in a line such that in one direction, the variations are increasingly better than the standard and in the other direction, the standard is increasingly better than the variations. There is an intermediate zone of legal careers that are not better than the standard and such that the standard is not better than the legal careers. If this zone contains one item, Broome defines this legal career to be equally good with the standard. If this zone contains more than one item, the zone is either one of "hard indeterminacy" or one of "soft indeterminacy." A zone cannot be both one of hard indeterminacy and one of soft indeterminacy. In a zone of hard indeterminacy, it is false that the legal careers are better than the standard and false that the standard is better than the legal careers (1997, 73, 76). In a zone of soft indeterminacy, it is neither true nor false that the legal careers are better than the standard and neither true nor false that the standard is better than the legal careers (1997, 76). The latter is a zone of vagueness. Broome argues that indeterminate comparatives, including "better than" are softly indeterminate, thus there is no hard indeterminacy and no room for the possibility suggested by the Small Improvement Argument. By understanding incomparability to entail vagueness, Broome disagrees with Raz (Broome 2000, 30). Raz defines incomparability so that it is compatible with vagueness, but not so that it entails vagueness. Griffin also argues that incomparability entails vagueness (1986, 96). Where Broome disagrees with Griffin is with regard to the width and significance of the zone of soft indeterminacy. Broome takes Griffin to suggest that if there is a zone of soft indeterminacy, it is narrow and unimportant. Broome argues that vagueness need not imply either narrowness or lack of importance (2000, 30-31). Central to Broome's argument is his controversial "collapsing principle" for comparatives: For any x and y, if it is false that y is Fer than x and not false that x is Fer than y, then it is true that x is Fer than y (1997, 74). Many counterexamples have been presented to show the implausibility of the principle (Carlson 2004, 2013; Elson 2014b; Gustafsson 2018). While there are attempts to defend the collapsing principle, or versions of it, (Constantinescu 2012; Andersson & Herlitz 2018) the counterexamples make Broome's argument less convincing. Another line of argument has consequently been developed in defense of the vagueness interpretation. It is argued that the vagueness interpretation is theoretically parsimonious since it is natural to accept the existence of evaluative vagueness and if it can effortlessly account for alleged examples of incomparability then there is no reason to accept the more mysterious notion of incomparability or to introduce more value relations beyond the standard trichotomous relations (Andersson 2017; Elson 2017). ### 2.2 "Roughly Equal" and "On a Par" The second line of inquiry concerns the set of possible comparative relations that can obtain between two items. The small improvement argument for the incomparability of the musical career and the legal career in Raz's example assumes what Chang calls the "trichotomy thesis." The trichotomy thesis holds that if two items can be compared in terms of some value or set of values, then the two items are related by one of the standard trichotomy of comparative relations, "better than," "worse than," or "equally good" (2002b, 660). A number of authors have argued that these three comparative relations do not exhaust the space of comparative relations. If they are correct, the musical career and the legal career may, in fact, be comparable. James Griffin and Derek Parfit argue that items may in fact be "roughly equal" and hence comparable (Griffin 1986, 80-81, 96-98, and 104; 1997, 38-39; 2000, 285-289; Parfit 1987, 431). As an illustration, Parfit imagines comparing two poets and a novelist for a literary prize (1987, 431). Neither the First Poet nor the Novelist is worse than the other and the Second Poet is slightly better than the First Poet. If the First Poet and the Novelist were equally good, it would follow that the Second Poet is better than the Novelist. This judgment, according to Parfit, need not follow. Instead, the First Poet and the Novelist may be roughly equal. The intuition is that even though three items display the respects in virtue of which the comparisons are made, some comparisons are inherently rough so that even though two alternatives are not worse than one other, they are not equally good. In turn, the musical and legal careers in Raz's example may be roughly equal. Parfit later referred to this possibility as there being "evaluative imprecision" (Parfit 2016, 113). "Roughly equal," as used here, is to be distinguished from two other ways in which the term has been used: (1) to refer to a small difference in value between two items and (2) to refer to a choice of little significance (Raz 1986, 333). As used here, two items A and B are said to be roughly equal if neither is worse than the other and C's being better than B does not imply that C is better than A when the comparisons are all in virtue of the same set of respects. There is some debate as to whether "roughly equal" is in fact a fourth comparative relation to be considered in addition to the three standard relations, "better than," "worse than," and "equally good." One way to conceive of "roughly equal" is as a "roughed up" version of "equally good." On this interpretation, the trichotomy thesis basically holds; there are simply precise and rough versions (Chang 2002b, 661, fn. 5). Furthermore, "roughly equal" is a relation that can be defined only in virtue of three items, and as such, appears to be something distinct from a standard comparative relation. The standard comparative relations are binary; transitivity with respect to them is a separate condition (Hsieh 2005, 195). A separate proposal for a genuine fourth relationship is Ruth Chang's argument for "on a par" (Chang 1997; Chang 2002b). Two items are said to be on a par if neither is better than the other, their differences preclude their being equally good, and yet they are comparable. Whereas "roughly equal" is invoked to allow for comparability among alternatives that display the same respects (e.g., literary merit), "on a par" is invoked to allow for comparability between alternatives that are different in the respects that they display. Imagine comparing Mozart and Michelangelo in terms of creativity. According to Chang, neither Mozart nor Michelangelo is less creative than the other. Because the two artists display creativity in such different fields, however, it would be mistaken to judge them to be equally creative. Nevertheless, according to Chang, they are comparable with respect to creativity. Something can be said about their relative merits with respect to the same consideration. According to Chang, they are on a par. Chang's argument relies on invoking a continuum that resembles Broome's standard configuration. Chang asks us to imagine a sequence of sculptors who are successively worse than Michelangelo until we arrive at a sculptor who is clearly worse than Mozart in terms of creativity. Chang then brings to bear the intuition that "between two evaluatively very different items, a small unidimensional difference cannot trigger incomparability where before there was comparability" (2002b, 673). In the light of this intuition, because Mozart is comparable to this bad sculptor, Mozart is also comparable to each of the sculptors in the sequence, including Michelangelo. The soundness of this so-called Chaining Argument has, however, been questioned (Boot 2009; Elson 2014a; Andersson 2016a). Whether "roughly equal" or "on a par" imply comparability is a matter of some debate. "Roughly equal" and "on a par," for example, are intransitive, so to recognize them as distinct comparative relations would require us to reconsider the transitivity of comparative relations (Hsieh 2007). Some authors suggest that one of the three standard comparative relations obtains between all items that are claimed to be incomparable, roughly equal, or on a par (Regan 1997). In the case of "on a par," Joshua Gert and Erik Carlson have argued that it can be defined in terms of the three standard comparative relations (Gert 2004; 2015; Carlson 2010). Gert's suggestion is developed further by Wlodek Rabinowicz who provides a Fitting Attitudes account of value relations (Rabinowicz 2008; 2012). The Fitting Attitudes account typically analyzes goodness in terms of a normative component and an attitudinal component. By acknowledging that there can be two levels of normativity, requirement and permissibility, the account leaves room for both parity and standard value relations. X is better than Y if and only if it is rationally required to prefer X to Y, and X and Y are on a par if and only if it is rationally permissible to prefer X to Y and also rationally permissible to prefer Y to X. Interestingly, preferences and the lack of preferences together with the two levels of normativity can be combined in 15 different ways. This means that the fitting attitudes analysis provides the conceptual space for 15 possible value relations. Another view is that values can be "clumpy," meaning that the values sort items into clumps. According to this view, once we recognize the way in which the relation "equally good" functions in the context of clumpy values, items that appear to require "roughly equal" or "on a par" can be judged equally good (Hsieh 2005). ## 3. Arguments for Value Incommensurability and Incomparability While recent research has focused on value incomparability and how to interpret examples that have the structure described by the Small Improvement Argument, earlier research had less focus on the structure of value but focused more on substantial consideration that talks in favor of incomparability. In this section, such considerations will be discussed. First, a note on terminology. Most of the arguments discuss something that is similar to the phenomenon of incomparability but refer to the phenomenon in terms of "incommensurability". In this section, the terminology of the discussed authors will be respected, and thus "incommensurability" will sometimes be used, even though, the authors possibly refer to the phenomenon of incomparability. It should be noted that value pluralism seems central for the possibility of incomparability. After all, the most direct argument against the possibility that values are incommensurable or incomparable is based on value monism (e.g., Mill 1861 [1979] and Sidgwick 1874 [1981]). Donald Regan, for example, argues that "given any two items (objects, experiences, states of affairs, whatever) sufficiently well specified so that it is apposite to inquire into their (intrinsic) value in the Moorean sense, then either one is better than the other, or the two are precisely equal in value" (Regan 1997, 129). This argument has been questioned. As Ruth Chang points out, by referring to John Stuart Mill, that values may have both qualitative and quantitative aspects could in principle allow for value bearers to be incomparable due to a difference in their qualitative features. (Chang 1997b, 16-17). For purposes of this entry, it will not be assumed that value monism rules out incomparability. In any case, many philosophers argue for some form of value pluralism (Berlin 1969, Finnis 1981, Nagel 1979, Raz 1986, Stocker 1990, Taylor 1982, Williams 1981). Although this entry will not assume value pluralism, most philosophers who argue for incommensurability or incomparability are value pluralists, so it will be simpler to present their views by talking of separate values. ### 3.1 Constitutive Incommensurability It has been argued that value incomparability is constitutive of certain goods and values and consequently we should assume its existence. Two versions of this second argument will be discussed here. One version of this argument comes from Joseph Raz. Consider being offered a significant amount of money to leave one's spouse for a month. The indignation that is typically experienced in response to such an offer, according to Raz, is grounded in part in the symbolic significance of certain actions (1986, 349). In this case, "what has symbolic significance is the very judgment that companionship is incommensurable with money" (1986, 350). Although this form of value incommensurability looks like trumping, Raz does not see this as a case of trumping. He rejects the view that companionship is more valuable than money. If such a view were correct, then those who forgo companionship for money would be acting against reason (1986, 352). Instead, Raz takes the view that a "belief in incommensurability is itself a qualification for having certain relations" (1986, 351). Someone who does not regard companionship and money as incommensurable simply has chosen a kind of life that may be fulfilling in many ways, but being capable of having companionship is not one of them. In Raz's account, the symbolic significance of judging money to be incommensurable with companionship involves the existence of a social convention that determines participation in that convention (e.g., marriage) that requires a belief in value incommensurability. This conventional nature of belief in value incommensurability in Raz's account raises a question for some authors about its robustness as an account of value incommensurability. For example, Chang objects that incommensurability appears to become relative to one's participation in social conventions (2001, 48). It remains an open question how much of a problem this point raises. Raz's account appears to illustrate a basic sense in which the values of money and companionship can be incommensurable. Insofar as it is not against reason to choose money over companionship, there is no general way to resolve a conflict of values between money and companionship. In Raz's account, the resolution depends upon which social convention one has chosen to pursue. Elizabeth Anderson advances a second argument for constitutive incommensurability. Her account is grounded in a pragmatic account of value. Anderson reduces "'x is good' roughly to 'it is rational to value x,' where to value something is to adopt toward it a favorable attitude susceptible to rational reflection" (1997, 95). She argues that in virtue of these attitudes there may be no good reason to compare the overall values of two goods. Pragmatism holds that if such a comparison serves no practical function, then the comparative value judgment has no truth value, meaning that the goods are incommensurable (1997, 99). Because the favorable attitudes one adopts toward goods help to make them good, Anderson's account can be seen as an argument for constitutive incommensurability (Chang 2001, 49). Anderson advances three ways in which there may be no good reason to compare the overall values of goods. First, it may be boring or pointless to engage in comparison. To illustrate, "the project of comprehensively ranking all works of art in terms of their intrinsic aesthetic value is foolish, boring, and stultifying" (1997, 100). Second, Anderson points to instances in which "it makes sense to leave room for the free play of nonrational motivations like whims and moods" as in the choice of what to do on a leisurely Sunday afternoon (1997, 91). Third, Anderson argues that the roles that goods play in deliberation can be so different that "attempts to compare them head to head are incoherent" (1997, 91). Imagine that the only way to save one's dying mother is to give up a friendship. Rather than compare their overall values, argues Anderson, ordinary moral thinking focuses on what one owes to one's mother and one's friends (1997, 102). This focus on obligation recognizes mother and friend each to be intrinsically valuable and yet valuable in different ways (1997, 103). There is no good reason, according to Anderson, to compare their overall values with regard to some common measure. Chang argues against each of the three points raised by Anderson (Chang 2001). In response to the first point, Chang notes there are occasions in which comparisons do need to be made between goods for which Anderson argues there is no good reason to make comparisons. In response to the second point, Chang argues that the range of instances for which the second argument applies is small. In response to the third point, Chang contends that Anderson's argument assumes that if goods are comparable then they have some value or evaluative property in common. Chang points out that this need not be the case. ### 3.2 Moral Dilemmas Both value incommensurability and incomparability have been invoked to make sense of a central feature of supposed moral dilemmas--namely, that no matter which alternative the agent chooses, she fails to do something she ought to do. The apparent value conflicts involved in these choices have led some philosophers to relate moral dilemmas to the incomparability or incommensurability of values. Henry Richardson, for example, takes the situation confronting Sophie in the novel Sophie's Choice--that one of her two children will be spared death, but only if she chooses which one to save--to point to the incommensurability of values (1994, 115-117). It has been argued that the mere fact of a moral dilemma does not imply incomparability. James Griffin, for example, argues that the feature of "irreplaceability" in moral dilemmas often may be mistaken as evidence for incomparability (1997, 37). Irreplaceability is the feature that what is lost in choosing one alternative over another cannot be replaced by what is gained in choosing another alternative. Although a conflict of values displays this feature, not all instances of irreplaceability need involve plural values. Some moral dilemmas, for example, may involve not a conflict of values, but a conflict of obligations that arises from the same consideration. Consider forced choices in saving lives. If there is a dilemma, it need not involve conflicting values, but rather conflicting obligations that arise from the same consideration. Walter Sinnott-Armstrong calls such dilemmas "symmetrical" (1988, 54-58). The dilemma encountering Sophie, it may be said, does not point to the incommensurability of values. Richardson acknowledges that the moral considerations underlying Sophie's dilemma are not incommensurable. Nonetheless, he takes value incommensurability to be essential to understanding the tragedy of the dilemma that Sophie encounters. "It is a distinguishing feature of love, including parental love," he writes, "that it cherishes the particular and unique features of the beloved" (115). He concludes, "the fact that she cannot adequately represent each child's value on a single scale is what makes the choosing tragic" (116). By locating the incommensurability of values at the level of what is valuable about each of her children, Richardson argues that the tragedy of the dilemma points to incomparability. Another common approach to argue for value incommensurability is with reference to "non-symmetrical" dilemmas. As the name suggests, in non-symmetrical dilemmas, the alternatives are favored by different values (Sinnott-Armstrong 1988). If these values are incommensurable in the third sense as discussed in subsection 1.3, there is no systematic resolution of the value conflict. Consider Jean-Paul Sartre's well-known example of his pupil who faced the choice between going to England to join the Free French Forces and staying at home to help his mother live (Sartre 1975, 295-296). No matter which alternative he chooses, certain values will remain unrealized. An idea along these lines is considered, for example, by Walter Sinnott-Armstrong (1988, 69-71) and Bernard Williams (1981, 74-78) in their discussions of moral dilemmas. ### 3.3 Akrasia Value incommensurability also features in debates about akrasia (Nussbaum 2001, 113-117). David Wiggins, for example, invokes the idea of value incommensurability to suggest "the heterogeneity of the psychic sources of desire satisfaction and of evaluation" (1998, 266). This heterogeneity, according to Wiggins, allows for a coherent account of the agent's attraction to what is not best. It allows for a divergence between desire and value such that the akratic individual can be attracted to a value that should not be sought at that point in time. Wiggins invokes value incommensurability to capture the idea that a gain in the value that should not be sought does not reduce the loss in choosing what is not best. In contrast, Michael Stocker denies that value incommensurability is required for a coherent account of akrasia (1990, 214-240). For Stocker, coherent akrasia is possible with a single value in just the way that it is possible to be attracted to two objects that differ with respect to the same value (e.g., "a languorous lesser pleasure and a piquant better pleasure" (1990, 230)). Recall the discussion of quantitative and qualitative aspects of a single value at the outset of section 3. ## 4. Deliberation and Choice As suggested above much of the inquiry into value incomparability is motivated more generally by theories of practical reason and rational choice. Even if a conception of justified choice can accommodate value incomparability, questions remain about how to justify choice on the basis of incomparability and how to reason about incomparability. This section considers these issues as they have been discussed in the contemporary philosophical literature. It is worth noting the potential for drawing connections between the philosophical literature and the literature in psychology and the social sciences on decision-making. One area for such potential is the psychological literature on the difficulty of making decisions (Yates, Veinott, and Patalano 2003). Jane Beattie and Sema Barlas (2001), for example, advance the thesis that the observed variation in the difficulty of making trade-offs between alternatives can be explained in part by the category to which the alternatives belong. The authors consider three categories: commodities ("objects that are appropriately bought and sold in markets"), currencies ("objects that act as stand-ins for commodities"), and noncommodities ("objects that either cannot be transferred (e.g., pain) or that lose some of their value by being traded in markets (e.g., friendship)") (Beattie and Barlas 2001, 30). The authors' experimental findings are consistent with participants holding normative commitments about exchanging currencies and noncommodities similar to those considered in the discussion of constitutive incommensurability in subsection 3.2. For example, Beattie and Barlas report that participants "had a significant tendency to choose noncommodities over commodities and currencies" and that choices between noncommodities and currencies were the easiest for participants (2001, 50-51). The authors interpret these results to support the view that people choose noncommodities over currencies on the basis of a rule without engaging in a calculation of trade-offs (2001, 51-53). In the context of empirical research, it should also be noted that philosophers have contributed with empirical research on the characterization of hard choices (Messerli and Reuter 2017a, 2017b). ### 4.1 Optimization and Maximization At a minimum, for the choice of an alternative to qualify as justified, there must not be an overriding reason against choosing it. Beyond that, conceptions of justified choice differ in what is required for the choice of an alternative to qualify as justified. Ruth Chang defines "comparativism" as the view that "comparative facts are what make a choice objectively correct; they are that in virtue of which a choice is objectively rational or what one has most or sufficient normative reason to do. So, whether you are a consequentialist, deontologist, virtue theorist, perfectionist, contractualist, etc., about the grounds of rational choice, you should be, first and foremost, I suggest, a comparativist" (2016, 213). If alternatives are incomparable, then the lack of a comparative fact precludes the possibility of justified choice. One response has been to argue that apparently incomparable alternatives are, in fact, comparable. As discussed in subsection 2.2, judgments of incomparability frequently involve comparisons that are difficult and it may be that judgments of incomparability are mistaken (Regan 1997) or that, due to vagueness, it is indeterminate which value relation holds. In addition, as discussed, alternatives that appear incomparable by way of "better than," "worse than," or "equally good" may be comparable by way of some fourth comparative relation, such as "roughly equal" or "on a par." This means that there is a comparative fact that makes a choice objectively correct in cases of apparent incomparability. It does not, however, help us identify the objectively correct alternative. A common form of comparativism is optimization. According to optimization, the fact that an alternative is at least as good as the other alternatives is what justifies its choice. Consequently, if two alternatives are incomparable no justified choice can be made between them. Another line of response, one from within the economics literature, has been to distinguish between "optimization" and "maximization" as theories of justified choice (Sen 1997, 746; Sen 2000, 486). The theory of maximization as justified choice only requires the choice of an alternative that is not worse than other alternatives. Because incomparable alternatives are not worse than one another, the choice of either is justified according to the theory of maximization as justified choice. Maximization is thus a rejection of comparativism. However, Hsieh (2007) argues that proponents of comparativism have no reason to reject maximization as an account of justified choice and consequently incomparability need not pose a problem for the possibility of justified choice. ### 4.2 Cyclical Choice One objection voiced against accounts that permit justified choice between alternatives that are roughly equal or on a par or between incomparable alternatives is that such accounts may justify a series of choices that leaves a person worse off. Consider Raz's example of career choice. Suppose the person chooses the musical career over the legal career. At a future time, she has the opportunity to pursue a legal career that is slightly worse than the initial legal career. Suppose this slightly worse legal career and the musical career are judged roughly equal or on a par. If justified choice permits her to choose either of two alternatives when they are roughly equal or on a par, then she would be justified in choosing the slightly worse legal career. Similarly, if justified choice does not require the comparability of alternatives, she could be justified in choosing the slightly worse legal career. Through a series of such apparently justified choices, she would be left significantly worse off in a manner analogous to a "money pump" (Chang 1997, 11). One response is to question whether the problem posed by choices of this sort is serious. John Broome, for example, notes that after having chosen one kind of career, a person may change her mind and choose the kind of career she previously rejected. According to Broome, there would be a puzzle only if she did not repudiate her previous choice (2000, 34). Another line of response is that the considerations that make some alternatives worthy of choice count against the constant switching among alternatives envisioned in this objection. First, the constant switching among alternatives is akin to not choosing an alternative. If the alternatives are such that choosing either is better than choosing neither, then the considerations that make the alternatives worthy of choice count against constantly switching among them. Second, to switch constantly among careers appears to misunderstand what makes the alternatives worthy of choice. Not only is pursuing a career the kind of activity that depends upon continued engagement for its success, but it is also the kind of activity that is unlikely to be judged truly successful unless one demonstrates some commitment to it. Third, for a career to be considered successful, it may require the chooser to adopt a favorable attitude toward the considerations that favor it over other careers. In turn, when subsequently presented with the choice of a legal career, the considerations favoring it may no longer apply to her in the same way as they did before (Hsieh 2007). Ruth Chang advances a hybrid view on rational choice to meet the challenge of cyclical choice. She distinguishes "given reasons" from "will-based" reasons. Roughly put, given reasons could here be understood as those that are grounded in normative facts while will-based reasons are "reasons in virtue of some act of the will; they are a matter of our creation. They are voluntarist in their normative source. In short, we create will-based reasons and receive given ones." (Chang 2013, 177). Will-based reasons come into play when the given reasons no longer can guide our actions. When two alternatives are on a par, given reasons fail to guide us and at this stage, will-based reasons can provide guidance and guarantee that the agent commits to her choice. Anders Herlitz (2019) argues that Chang's approach and other "two-step models" violate an intuitively plausible constraint on rational choice called Basic Contraction Consistency. By violating this constraint two-step models also allow for money pumps. This could, however, possibly be avoided by introducing additional formal conditions for the second step in the model (Herlitz 2019, 299-307). ### 4.3 Rational Eligibility The idea expressed in the distinction between optimization and maximization in subsection 4.1 can be expressed more generally in terms of the idea of rational eligibility. To say that an alternative is rationally eligible is to mean that choosing it would not be an unjustified choice. Which alternatives are judged rationally eligible may vary with the theory of justified choice. According to maximization as a theory of justified choice, an alternative is rationally eligible if and only if there is no better alternative. Isaac Levi (1986; 2004) argues for "v-admissibility" as a criterion of justified choice. For v-admissibility, value incomparability does not pose a problem for the possibility of justified choice. An alternative is v-admissible if and only if it is optimal according to at least one of the relevant considerations at hand. In some conflicts of value, maximization and v-admissibility specify the same alternatives as rationally eligible. Suppose that alternative X is better than alternative Y with respect to consideration A and alternative Y is better than alternative X with respect to consideration B. According to maximization as a theory of justified choice, both X and Y are rationally eligible; neither alternative is better than the other with respect to A and B taken together. Both alternatives are also rationally eligible according to v-admissibility; X is optimal with respect to A and Y is optimal with respect to B. In some conflicts of value, maximization and v-admissibility specify different sets of alternatives as rationally eligible. V-admissibility is more restrictive than maximization (Levi 2004). Add to the above example alternative Z. Suppose that X is better than Z, which is better than Y, all with respect to A. Suppose also that Y is better than Z, which is better than X, all with respect to B. According to maximization as a theory of justified choice, X, Y, and Z are rationally eligible; no alternative is better than the other with respect to A and B taken together. However, Z is not rationally eligible according to v-admissibility. Z is not optimal with respect to A. Nor is it optimal with respect to B. Only X and Y are rationally eligible according to v-admissibility. According to Levi, v-admissibility captures what he takes to be a plausible judgment--namely, that it would be unjustified to choose the alternative that is second worse on all relevant respects (Levi 2004). The plausibility of this judgment might be questioned. Suppose that Z is only slightly worse than X with respect to A and Z is only slightly worse than Y with respect to B. Does the judgment still hold? For Joseph Raz, incomparability also does not pose a problem for the possibility of justified choice (1997). If incomparable options give us reasons to choose both alternatives, they are both rationally eligible from the perspective of justified choice. As such, the choice of either alternative is justified on the basis of reason. One question that arises is this. If an agent has reason to choose either alternative and they are not equally good, what makes her choice of one alternative over another intelligible to her? For Raz, what explains the choice of one alternative over the other is the exercise of the will. By the will, Raz has in mind "the ability to choose and perform intentional actions" and "the most typical exercise or manifestation of the will is in choosing among options that reason merely renders eligible" (1997, 111). John Finnis advances a similar view in response to the question of intelligibility. Finnis writes, "in free choice, one has reasons for each of the alternative options, but these reasons are not causally determinative.... No factor but the choosing itself settles which alternative is chosen" (1997, 220). In a choice between alternatives each favored by different, incomparable values, even though there are reasons to choose both alternatives, because reasons are not causally determinative, the lack of a reason to choose one alternative over another need not render the choice unintelligible. Donald Regan challenges this view. According to Regan, unless grounded in an adequate reason, "a decision to go one way rather than another will be something that happened to the agent rather than something she did" and hence be unintelligible to the agent herself (1997, 144). Suppose the agent has no more reason to choose one alternative over another and the choice, as suggested above, is settled by her wants. On Regan's view, if the agent's choice is to be intelligible to her, her wants must be grounded in reasons. Because she has no more reason to choose one alternative over another initially, the reasons grounding her wants must be available to her only after the initially relevant reasons are exhausted. This strikes Regan as implausible (1997, 150). Regan concludes that no choice between incomparable bearers of value is intelligible in the ways suggested by Raz or Finnis. ### 4.4 External Resources A number of authors have argued that practical reason has available to it the resources to accommodate incomparability in ways that would appear to address the concern raised by Regan. Charles Taylor outlines two such sets of resources. First, we are able to appeal to "constitutive goods that lie behind the life goods" our sense of which is "fleshed out, and passed on, in a whole range of media: stories, legends, portraits of exemplary figures and their actions and passions, as well as in artistic works, music, dance, ritual, modes of worship, and so on" (1997, 179). Second, we cannot escape the need to live an integrated life or to have this, at a minimum, as an aspiration (1997, 180). Given that a life is finite, leading a life involves an articulation of how different goods fit into it relative to one another. More generally, our lives take on a certain "shape" and this shape provides guidance in making choices in the face of incomparability (1997, 183). Michael Stocker similarly points out that alternatives are rarely considered in the abstract (1997). Instead, they are usually considered in concrete ways in which they are valuable--for example, as part of one's life. Once considered in such concrete contexts, there are considerations according to which alternatives can be evaluated for purposes of justified choice. In his analysis of incommensurability, Fred D'Agostino discusses the role of social institutions in resolving value conflicts (2003). Another resource that has been invoked is morality. Recall Elizabeth Anderson's discussion of the example of the choice between saving the life of one's mother and maintaining a close friendship. Anderson regards the attempt to invoke a comparison of values as incoherent from the perspective of practical reason. Instead, ordinary moral thinking focuses on the obligations one has to one's mother and one's friends. In turn, Anderson suggests that the obligations themselves will provide guidance (1997, 106). The availability of such resources to reason is independent of her theories of value and rationality. John Finnis also points to principles of morality as guiding reason which avoids comparability problems (1997). It remains an open question as to how wide a range of cases it is in which morality can provide such guidance. One worry that may arise in appealing to such external resources is that they address the problem of incomparability by simply denying it. The shape of one's life or the "constitutive goods that lie behind the life goods," for example, would appear to be sources of value. Insofar as they provide a value against which to resolve the initial value conflict, it might be said that there was no problem of incomparability in the first place. Two responses to this worry can be given. First, even if these external resources can ground comparisons, it does not follow that they provide a systematic resolution to value conflicts. Insofar as the shapes of people's lives differ, the way in which two individuals resolve the same value conflict may be different. The fact that it may be consistent with reason to resolve the same value conflict in different ways points to the possibility of incomparability. Second, in the case of moral considerations, there may not be any denial of incomparability. Moral considerations may provide guidance without comparing alternative courses of action (Finnis 1997, 225-226). Furthermore, some philosophers argue that the moral wrongness of certain actions is intelligible only in virtue of incomparability. For example, Alan Strudler argues that to deliberate about the permissibility of lying in terms of comparable options is to misunderstand the wrong in lying (1998, 1561-1564). ### 4.5 Non-Maximizing Choice It would appear that conceptions of justified choice that do not rely on comparisons avoid the problems posed by incomparability. Stocker aims to provide one such account of justified choice (1990; 1997). Two features help to distinguish his account. First, he argues for evaluating alternatives as "the best" in an absolute, rather than a relative, sense. Optimization and maximization rely on choosing "the best" in a relative sense: given the relevant comparison class, there is no better alternative. To be "the best" in an absolute sense, an alternative "must be, of its kind, excellent--satisfying, or coming close to, ideals and standards" (1997, 206). This is the sense, for example, in which a person can be the best of friends even if we may have even better friends. More generally, an alternative can be absolutely best even if there are better alternatives, and even if an alternative is relatively best, it may not be absolutely best. The second distinctive feature of Stocker's account is his appeal to a good life, for example, or part of a good life or project. This appeal does not require the alternative to be the best for that life or project. Instead, the alternative need only be good enough for that life or project to qualify as a justified choice. Stocker's account of justified choice differs from the concept of "satisficing" as used in the economics and rational choice literature. Introduced into the economics literature by Herbert Simon (1955), satisficing is the process of choosing in which it is rational to stop seeking alternatives when the agent finds one that is "good enough" (Byron 2004). Where satisficing differs from Stocker's account is that satisficing is instrumentally rational in virtue of a more general maximizing account of choice. In situations in which finding the best alternative is prohibitively costly or impossible, for example, satisficing becomes rational. In contrast, on Stocker's account, choosing the alternative that is "good enough" is non-instrumentally rational (Stocker 2004). Objections discussed in previous sections may be raised with respect to Stocker's account. Suppose two alternatives are both good in an absolute sense and incomparable. According to Stocker's account, insofar as both alternatives are good enough for an agent's life, the choice of either alternative appears justified. As discussed in subsection 4.2, Regan and others may object that the choice of either alternative is not intelligible to the agent. Also, the appeal to a good life may raise the worry discussed in subsection 4.4 that Stocker's account addresses the problem of incomparability by simply denying it. ### 4.6 Deliberation about Ends The topics of value incommensurability and incomparability also arise in accounts that concern deliberation about ends. This section discusses two accounts. The first account is by Henry Richardson (1994). Richardson argues for the possibility of rational deliberation about ends. According to Richardson, if comparability is a prerequisite for rational choice, choices involving conflicting values could be made rationally if each of the values were expressed in terms of their contribution to furthering some common end. This conception of choice treats this common end as the one final end relevant for the choice. According to Richardson, the idea that comparability is a prerequisite for rational choice seems to rule out rational deliberation about ends (1994, 15). Richardson defends an account of rational deliberation about ends that does not depend upon comparability. In his account, rational deliberation involves achieving coherence among one's ends. Briefly, coherence is "a matter of finding or constructing intelligible connections or links or mutual support among them and of removing relations of opposition or conflict" (Richardson 1994, 144). For Richardson, coherence among ends need not result in their being comparable (1994, 180). The second account is by Elijah Millgram (1997). Like Richardson, Millgram argues that comparability is not a prerequisite for deliberation. However, in contrast to Richardson, Millgram argues that practical deliberation results in the comparability of ends (1997, 151). By this he means that through deliberation, one develops a coherent conception of what matters and a "background against which one can judge not only that one consideration is more important than another, but how much more important" (1997, 163). For Millgram, comparing one's ends is a "central part of attaining unified agency" (1997, 162). Millgram proposes two ways in which deliberation results in the comparability of ends. The first proposal is that one can learn what is important and how it is important through experience (1997, 161). The second proposal is that deliberation about ends is "something like the construction by the agent of a conception of what matters ... out of raw materials such as desires, ends, preferences, and reflexes" (1997, 168). Millgram identifies an objection to each proposal and briefly responds to each. The objection to the first proposal is this. If the incomparability of ends can be resolved by experience, this suggests that the judgment of incomparability reflects incomplete knowledge about the values expressed in these ends. In other words, experience seems to help only if the values expressed in these ends are themselves comparable (1997, 168). Millgram responds as follows. The agent who uses her experience to develop a coherent conception of what matters is like the painter who uses her experience to paint a picture that is not a copy of an existing image. Even if the ends are comparable in her conception of what matters, the values they reflect need not be comparable just as the painting need not be an exact copy of an existing image. The objection to the second proposal is this. The second proposal suggests that deliberation is not driven by experience, which places it in tension with the first proposal. Millgram responds by continuing the analogy with the painter. Imagine a painter who paints a picture that is not a copy of an existing image. Just because the painting is not a copy of an existing image does not mean there is no source of correction and constraint on it. Similarly, deliberation may involve the construction by the agent of a conception of what matters, but this does not imply that it is not informed or constrained by experience (1997, 168-169). ### 4.7 Risky Actions Caspar Hare (2010) introduces and discusses a problem relating to how rational choice theory can encompass risky actions involving outcomes that are on a par. If we assume that two outcomes, A and B, are on a par, it follows that a small improvement, or mild sweetening as Hare phrases it, to one of the outcomes would not break the parity relation. We can imagine two actions, X and Y. If we opt for X then we will either end up with a sweetened A in state 1 or a sweetened B in state 2, and the outcomes are equiprobable. If we opt for Y, we will end up with B in state 1 or A in state 2, and the outcomes are equiprobable. On the one hand, it seems as if we ought to opt for X since we will end up with a sweetened alternative, A+ or B+. On the other hand, it is difficult to say why X is better than Y. The outcome in state 1 is not better if we do X rather than Y since the outcomes, A+ and B, are on a par, and the outcome in state 2 is not better if we do Y rather than X since the outcomes, B+ and A, are on a par. The problem can be construed as a lottery where one lottery is deemed comparable with another, yet their outcomes are incomparable (see Rabinowicz, 2021b, for this axiological formulation of Hare's problem). Hare argues that we should "take the sugar", i.e., we ought to opt for X (see also Bader 2019). Inspired by a Fitting-Attitudes Analysis of value Wlodek Rabinowicz argues that even though outcomes may be on a par, it is possible for an action to be better than another (2021b). Others such as Schoenfield (2014), Bales, Cohen & Handfield (2014), and Doody (2019) reach the opposite conclusion. ## 5. Social Choices and Institutions As discussed above, social practices and institutions play a role in the inquiry into value incommensurability and incomparability. They are claimed by some philosophers to ground the possibility of these phenomena as in the case of constitutive incommensurability (subsection 3.2), and to help resolve value conflicts, as in the case of providing external resources for practical reason (subsection 4.4). This section discusses additional areas in which considerations about social institutions intersect with the inquiry at hand. To begin, some philosophers have pointed to a structural similarity between a single individual trying to choose in the face of incomparability and the process of incorporating the varied interests and preferences of the members of society into a single decision. The preferences of different individuals, for example, may be taken to reflect differing evaluative judgments about alternatives so that combining these different preferences into a single decision becomes analogous to resolving value conflicts in the individual case. Given this analogy, Fred D'Agostino has applied methods of decision-making at the social level from social choice theory and political theory to consider the resolution of value conflicts at the individual level (2003). At the same time, Richard Pildes and Elizabeth Anderson caution against wholeheartedly adopting this analogy. The analogy, according to them, assumes that individuals already have rationally ordered preferences. Given incomparability, however, there is no reason to make such an assumption. In turn, Pildes and Anderson argue that "individuals need to participate actively in democratic institutions to enable them to achieve a rational ordering of their preferences for collective choices" (1990, 2177). Value incommensurability and incomparability also have been considered with respect to the law. Matthew Adler discusses the variety of ways in which legal scholars have engaged the topic of value incommensurability (1998). One question is whether the possibility of value incommensurability poses a problem for evaluating government policy options and laws, more generally. Some authors respond that it does not. Cass Sunstein, for example, argues that recognition of value incommensurability helps "to reveal what is at stake in many areas of the law" (1994, 780). According to Sunstein, important commitments of a well-functioning legal system are reflected in recognizing value incommensurability. More generally, a number of scholars have focused on the relation between incomparability and the structure of social and political institutions. John Finnis, for example, takes the open-endedness of social life to render it impossible to treat legal or policy choices as involving comparable alternatives (1997, 221-222). Michael Walzer's account of distributive justice also relates incomparability to the structure of social and political institutions. According to Walzer (1983), different social goods occupy different "spheres," each one governed by a distinct set of distributive norms. What is unjust is to convert the accumulation of goods in one sphere into the accumulation of goods in another sphere without regard for that second sphere's distributive norms. Underlying Walzer's account, it seems, is a commitment to a kind of constitutive incommensurability. Given its connection to the possibility of plural and incompatible ways of life, the concepts of value incommensurability and incomparability also play a role in many accounts of political liberalism, including Joseph Raz's account (1986) and Isaiah Berlin's account (1969). The latter's inquiry into the relation between incommensurable values and political institutions motivated much of the further inquiry into the topic.
value-intrinsic-extrinsic	## 1. What Has Intrinsic Value? The question "What is intrinsic value?" is more fundamental than the question "What has intrinsic value?," but historically these have been treated in reverse order. For a long time, philosophers appear to have thought that the notion of intrinsic value is itself sufficiently clear to allow them to go straight to the question of what should be said to have intrinsic value. Not even a potted history of what has been said on this matter can be attempted here, since the record is so rich. Rather, a few representative illustrations must suffice. In his dialogue Protagoras, Plato [428-347 B.C.E.] maintains (through the character of Socrates, modeled after the real Socrates [470-399 B.C.E.], who was Plato's teacher) that, when people condemn pleasure, they do so, not because they take pleasure to be bad as such, but because of the bad consequences they find pleasure often to have. For example, at one point Socrates says that the only reason why the pleasures of food and drink and sex seem to be evil is that they result in pain and deprive us of future pleasures (Plato, Protagoras, 353e). He concludes that pleasure is in fact good as such and pain bad, regardless of what their consequences may on occasion be. In the Timaeus, Plato seems quite pessimistic about these consequences, for he has Timaeus declare pleasure to be "the greatest incitement to evil" and pain to be something that "deters from good" (Plato, Timaeus, 69d). Plato does not think of pleasure as the "highest" good, however. In the Republic, Socrates states that there can be no "communion" between "extravagant" pleasure and virtue (Plato, Republic, 402e) and in the Philebus, where Philebus argues that pleasure is the highest good, Socrates argues against this, claiming that pleasure is better when accompanied by intelligence (Plato, Philebus, 60e). Many philosophers have followed Plato's lead in declaring pleasure intrinsically good and pain intrinsically bad. Aristotle [384-322 B.C.E.], for example, himself a student of Plato's, says at one point that all are agreed that pain is bad and to be avoided, either because it is bad "without qualification" or because it is in some way an "impediment" to us; he adds that pleasure, being the "contrary" of that which is to be avoided, is therefore necessarily a good (Aristotle, Nicomachean Ethics, 1153b). Over the course of the more than two thousand years since this was written, this view has been frequently endorsed. Like Plato, Aristotle does not take pleasure and pain to be the only things that are intrinsically good and bad, although some have maintained that this is indeed the case. This more restrictive view, often called hedonism, has had proponents since the time of Epicurus [341-271 B.C.E.].[1] Perhaps the most thorough renditions of it are to be found in the works of Jeremy Bentham [1748-1832] and Henry Sidgwick [1838-1900] (see Bentham 1789, Sidgwick 1907); perhaps its most famous proponent is John Stuart Mill [1806-1873] (see Mill 1863). Most philosophers who have written on the question of what has intrinsic value have not been hedonists; like Plato and Aristotle, they have thought that something besides pleasure and pain has intrinsic value. One of the most comprehensive lists of intrinsic goods that anyone has suggested is that given by William Frankena (Frankena 1973, pp. 87-88): life, consciousness, and activity; health and strength; pleasures and satisfactions of all or certain kinds; happiness, beatitude, contentment, etc.; truth; knowledge and true opinions of various kinds, understanding, wisdom; beauty, harmony, proportion in objects contemplated; aesthetic experience; morally good dispositions or virtues; mutual affection, love, friendship, cooperation; just distribution of goods and evils; harmony and proportion in one's own life; power and experiences of achievement; self-expression; freedom; peace, security; adventure and novelty; and good reputation, honor, esteem, etc. (Presumably a corresponding list of intrinsic evils could be provided.) Almost any philosopher who has ever addressed the question of what has intrinsic value will find his or her answer represented in some way by one or more items on Frankena's list. (Frankena himself notes that he does not explicitly include in his list the communion with and love and knowledge of God that certain philosophers believe to be the highest good, since he takes them to fall under the headings of "knowledge" and "love.") One conspicuous omission from the list, however, is the increasingly popular view that certain environmental entities or qualities have intrinsic value (although Frankena may again assert that these are implicitly represented by one or more items already on the list). Some find intrinsic value, for example, in certain "natural" environments (wildernesses untouched by human hand); some find it in certain animal species; and so on. Suppose that you were confronted with some proposed list of intrinsic goods. It would be natural to ask how you might assess the accuracy of the list. How can you tell whether something has intrinsic value or not? On one level, this is an epistemological question about which this article will not be concerned. (See the entry in this encyclopedia on moral epistemology.) On another level, however, this is a conceptual question, for we cannot be sure that something has intrinsic value unless we understand what it is for something to have intrinsic value. ## 2. What Is Intrinsic Value? The concept of intrinsic value has been characterized above in terms of the value that something has "in itself," or "for its own sake," or "as such," or "in its own right." The custom has been not to distinguish between the meanings of these terms, but we will see that there is reason to think that there may in fact be more than one concept at issue here. For the moment, though, let us ignore this complication and focus on what it means to say that something is valuable for its own sake as opposed to being valuable for the sake of something else to which it is related in some way. Perhaps it is easiest to grasp this distinction by way of illustration. Suppose that someone were to ask you whether it is good to help others in time of need. Unless you suspected some sort of trick, you would answer, "Yes, of course." If this person were to go on to ask you why acting in this way is good, you might say that it is good to help others in time of need simply because it is good that their needs be satisfied. If you were then asked why it is good that people's needs be satisfied, you might be puzzled. You might be inclined to say, "It just is." Or you might accept the legitimacy of the question and say that it is good that people's needs be satisfied because this brings them pleasure. But then, of course, your interlocutor could ask once again, "What's good about that?" Perhaps at this point you would answer, "It just is good that people be pleased," and thus put an end to this line of questioning. Or perhaps you would again seek to explain the fact that it is good that people be pleased in terms of something else that you take to be good. At some point, though, you would have to put an end to the questions, not because you would have grown tired of them (though that is a distinct possibility), but because you would be forced to recognize that, if one thing derives its goodness from some other thing, which derives its goodness from yet a third thing, and so on, there must come a point at which you reach something whose goodness is not derivative in this way, something that "just is" good in its own right, something whose goodness is the source of, and thus explains, the goodness to be found in all the other things that precede it on the list. It is at this point that you will have arrived at intrinsic goodness (cf. Aristotle, Nicomachean Ethics, 1094a). That which is intrinsically good is nonderivatively good; it is good for its own sake. That which is not intrinsically good but extrinsically good is derivatively good; it is good, not (insofar as its extrinsic value is concerned) for its own sake, but for the sake of something else that is good and to which it is related in some way. Intrinsic value thus has a certain priority over extrinsic value. The latter is derivative from or reflective of the former and is to be explained in terms of the former. It is for this reason that philosophers have tended to focus on intrinsic value in particular. The account just given of the distinction between intrinsic and extrinsic value is rough, but it should do as a start. Certain complications must be immediately acknowledged, though. First, there is the possibility, mentioned above, that the terms traditionally used to refer to intrinsic value in fact refer to more than one concept; again, this will be addressed later (in this section and the next). Another complication is that it may not in fact be accurate to say that whatever is intrinsically good is nonderivatively good; some intrinsic value may be derivative. This issue will be taken up (in Section 5) when the computation of intrinsic value is discussed; it may be safely ignored for now. Still another complication is this. It is almost universally acknowledged among philosophers that all value is "supervenient" or "grounded in" on certain nonevaluative features of the thing that has value. Roughly, what this means is that, if something has value, it will have this value in virtue of certain nonevaluative features that it has; its value can be attributed to these features. For example, the value of helping others in time of need might be attributed to the fact that such behavior has the feature of being causally related to certain pleasant experiences induced in those who receive the help. Suppose we accept this and accept also that the experiences in question are intrinsically good. In saying this, we are (barring the complication to be discussed in Section 5) taking the value of the experiences to be nonderivative. Nonetheless, we may well take this value, like all value, to be supervenient on, or grounded in, something. In this case, we would probably simply attribute the value of the experiences to their having the feature of being pleasant. This brings out the subtle but important point that the question whether some value is derivative is distinct from the question whether it is supervenient. Even nonderivative value (value that something has in its own right; value that is, in some way, not attributable to the value of anything else) is usually understood to be supervenient on certain nonevaluative features of the thing that has value (and thus to be attributable, in a different way, to these features). To repeat: whatever is intrinsically good is (barring the complication to be discussed in Section 5) nonderivatively good. It would be a mistake, however, to affirm the converse of this and say that whatever is nonderivatively good is intrinsically good. As "intrinsic value" is traditionally understood, it refers to a particular way of being nonderivatively good; there are other ways in which something might be nonderivatively good. For example, suppose that your interlocutor were to ask you whether it is good to eat and drink in moderation and to exercise regularly. Again, you would say, "Yes, of course." If asked why, you would say that this is because such behavior promotes health. If asked what is good about being healthy, you might cite something else whose goodness would explain the value of health, or you might simply say, "Being healthy just is a good way to be." If the latter were your response, you would be indicating that you took health to be nonderivatively good in some way. In what way, though? Well, perhaps you would be thinking of health as intrinsically good. But perhaps not. Suppose that what you meant was that being healthy just is "good for" the person who is healthy (in the sense that it is in each person's interest to be healthy), so that John's being healthy is good for John, Jane's being healthy is good for Jane, and so on. You would thereby be attributing a type of nonderivative interest-value to John's being healthy, and yet it would be perfectly consistent for you to deny that John's being healthy is intrinsically good. If John were a villain, you might well deny this. Indeed, you might want to insist that, in light of his villainy, his being healthy is intrinsically bad, even though you recognize that his being healthy is good for him. If you did say this, you would be indicating that you subscribe to the common view that intrinsic value is nonderivative value of some peculiarly moral sort.[2] Let us now see whether this still rough account of intrinsic value can be made more precise. One of the first writers to concern himself with the question of what exactly is at issue when we ascribe intrinsic value to something was G. E. Moore [1873-1958]. In his book Principia Ethica, Moore asks whether the concept of intrinsic value (or, more particularly, the concept of intrinsic goodness, upon which he tended to focus) is analyzable. In raising this question, he has a particular type of analysis in mind, one which consists in "breaking down" a concept into simpler component concepts. (One example of an analysis of this sort is the analysis of the concept of being a vixen in terms of the concepts of being a fox and being female.) His own answer to the question is that the concept of intrinsic goodness is not amenable to such analysis (Moore 1903, ch. 1). In place of analysis, Moore proposes a certain kind of thought-experiment in order both to come to understand the concept better and to reach a decision about what is intrinsically good. He advises us to consider what things are such that, if they existed by themselves "in absolute isolation," we would judge their existence to be good; in this way, we will be better able to see what really accounts for the value that there is in our world. For example, if such a thought-experiment led you to conclude that all and only pleasure would be good in isolation, and all and only pain bad, you would be a hedonist.[3] Moore himself deems it incredible that anyone, thinking clearly, would reach this conclusion. He says that it involves our saying that a world in which only pleasure existed--a world without any knowledge, love, enjoyment of beauty, or moral qualities--is better than a world that contained all these things but in which there existed slightly less pleasure (Moore 1912, p. 102). Such a view he finds absurd. Regardless of the merits of this isolation test, it remains unclear exactly why Moore finds the concept of intrinsic goodness to be unanalyzable. At one point he attacks the view that it can be analyzed wholly in terms of "natural" concepts--the view, that is, that we can break down the concept of being intrinsically good into the simpler concepts of being A, being B, being C..., where these component concepts are all purely descriptive rather than evaluative. (One candidate that Moore discusses is this: for something to be intrinsically good is for it to be something that we desire to desire.) He argues that any such analysis is to be rejected, since it will always be intelligible to ask whether (and, presumably, to deny that) it is good that something be A, B, C,..., which would not be the case if the analysis were accurate (Moore 1903, pp. 15-16). Even if this argument is successful (a complicated matter about which there is considerable disagreement), it of course does not establish the more general claim that the concept of intrinsic goodness is not analyzable at all, since it leaves open the possibility that this concept is analyzable in terms of other concepts, some or all of which are not "natural" but evaluative. Moore apparently thinks that his objection works just as well where one or more of the component concepts A, B, C,..., is evaluative; but, again, many dispute the cogency of his argument. Indeed, several philosophers have proposed analyses of just this sort. For example, Roderick Chisholm [1916-1999] has argued that Moore's own isolation test in fact provides the basis for an analysis of the concept of intrinsic value. He formulates a view according to which (to put matters roughly) to say that a state of affairs is intrinsically good or bad is to say that it is possible that its goodness or badness constitutes all the goodness or badness that there is in the world (Chisholm 1978). Eva Bodanszky and Earl Conee have attacked Chisholm's proposal, showing that it is, in its details, unacceptable (Bodanszky and Conee 1981). However, the general idea that an intrinsically valuable state is one that could somehow account for all the value in the world is suggestive and promising; if it could be adequately formulated, it would reveal an important feature of intrinsic value that would help us better understand the concept. We will return to this point in Section 5. Rather than pursue such a line of thought, Chisholm himself responded (Chisholm 1981) in a different way to Bodanszky and Conee. He shifted from what may be called an ontological version of Moore's isolation test--the attempt to understand the intrinsic value of a state in terms of the value that there would be if it were the only valuable state in existence--to an intentional version of that test--the attempt to understand the intrinsic value of a state in terms of the kind of attitude it would be fitting to have if one were to contemplate the valuable state as such, without reference to circumstances or consequences. This new analysis in fact reflects a general idea that has a rich history. Franz Brentano [1838-1917], C. D. Broad [1887-1971], W. D. Ross [1877-1971], and A. C. Ewing [1899-1973], among others, have claimed, in a more or less qualified way, that the concept of intrinsic goodness is analyzable in terms of the fittingness of some "pro" (i.e., positive) attitude (Brentano 1969, p. 18; Broad 1930, p. 283; Ross 1939, pp. 275-76; Ewing 1948, p. 152). Such an analysis, which has come to be called "the fitting attitude analysis" of value, is supported by the mundane observation that, instead of saying that something is good, we often say that it is valuable, which itself just means that it is fitting to value the thing in question. It would thus seem very natural to suppose that for something to be intrinsically good is simply for it to be such that it is fitting to value it for its own sake. ("Fitting" here is often understood to signify a particular kind of moral fittingness, in keeping with the idea that intrinsic value is a particular kind of moral value. The underlying point is that those who value for its own sake that which is intrinsically good thereby evince a kind of moral sensitivity.) Though undoubtedly attractive, this analysis can be and has been challenged. Brand Blanshard [1892-1987], for example, argues that the analysis is to be rejected because, if we ask why something is such that it is fitting to value it for its own sake, the answer is that this is the case precisely because the thing in question is intrinsically good; this answer indicates that the concept of intrinsic goodness is more fundamental than that of the fittingness of some pro attitude, which is inconsistent with analyzing the former in terms of the latter (Blanshard 1961, pp. 284-86). Ewing and others have resisted Blanshard's argument, maintaining that what grounds and explains something's being valuable is not its being good but rather its having whatever non-value property it is upon which its goodness supervenes; they claim that it is because of this underlying property that the thing in question is "both" good and valuable (Ewing 1948, pp. 157 and 172. Cf. Lemos 1994, p. 19). Thomas Scanlon calls such an account of the relation between valuableness, goodness, and underlying properties a buck-passing account, since it "passes the buck" of explaining why something is such that it is fitting to value it from its goodness to some property that underlies its goodness (Scanlon 1998, pp. 95 ff.). Whether such an account is acceptable has recently been the subject of intense debate. Many, like Scanlon, endorse passing the buck; some, like Blanshard, object to doing so. If such an account is acceptable, then Ewing's analysis survives Blanshard's challenge; but otherwise not. (Note that one might endorse passing the buck and yet reject Ewing's analysis for some other reason. Hence a buck-passer may, but need not, accept the analysis. Indeed, there is reason to think that Moore himself is a buck-passer, even though he takes the concept of intrinsic goodness to be unanalyzable; cf. Olson 2006). Even if Blanshard's argument succeeds and intrinsic goodness is not to be analyzed in terms of the fittingness of some pro attitude, it could still be that there is a strict correlation between something's being intrinsically good and its being such that it is fitting to value it for its own sake; that is, it could still be both that (a) it is necessarily true that whatever is intrinsically good is such that it is fitting to value it for its own sake, and that (b) it is necessarily true that whatever it is fitting to value for its own sake is intrinsically good. If this were the case, it would reveal an important feature of intrinsic value, recognition of which would help us to improve our understanding of the concept. However, this thesis has also been challenged. Krister Bykvist has argued that what he calls solitary goods may constitute a counterexample to part (a) of the thesis (Bykvist 2009, pp. 4 ff.). Such (alleged) goods consist in states of affairs that entail that there is no one in a position to value them. Suppose, for example, that happiness is intrinsically good, and good in such a way that it is fitting to welcome it. Then, more particularly, the state of affairs of there being happy egrets is intrinsically good; so too, presumably, is the more complex state of affairs of there being happy egrets but no welcomers. The simpler state of affairs would appear to pose no problem for part (a) of the thesis, but the more complex state of affairs, which is an example of a solitary good, may pose a problem. For if to welcome a state of affairs entails that that state of affairs obtains, then welcoming the more complex state of affairs is logically impossible. Furthermore, if to welcome a state of affairs entails that one believes that that state of affairs obtains, then the pertinent belief regarding the more complex state of affairs would be necessarily false. In neither case would it seem plausible to say that welcoming the state of affairs is nonetheless fitting. Thus, unless this challenge can somehow be met, a proponent of the thesis must restrict the thesis to pro attitudes that are neither truth- nor belief-entailing, a restriction that might itself prove unwelcome, since it excludes a number of favorable responses to what is good (such as promoting what is good, or taking pleasure in what is good) to which proponents of the thesis have often appealed. As to part (b) of the thesis: some philosophers have argued that it can be fitting to value something for its own sake even if that thing is not intrinsically good. A relatively early version of this argument was again provided by Blanshard (1961, pp. 287 ff. Cf. Lemos 1994, p. 18). Recently the issue has been brought into stark relief by the following sort of thought-experiment. Imagine that an evil demon wants you to value him for his own sake and threatens to cause you severe suffering unless you do. It seems that you have good reason to do what he wants--it is appropriate or fitting to comply with his demand and value him for his own sake--even though he is clearly not intrinsically good (Rabinowicz and Ronnow-Rasmussen 2004, pp. 402 ff.). This issue, which has come to be known as "the wrong kind of reason problem," has attracted a great deal of attention. Some have been persuaded that the challenge succeeds, while others have sought to undermine it. One final cautionary note. It is apparent that some philosophers use the term "intrinsic value" and similar terms to express some concept other than the one just discussed. In particular, Immanuel Kant [1724-1804] is famous for saying that the only thing that is "good without qualification" is a good will, which is good not because of what it effects or accomplishes but "in itself" (Kant 1785, Ak. 1-3). This may seem to suggest that Kant ascribes (positive) intrinsic value only to a good will, declaring the value that anything else may possess merely extrinsic, in the senses of "intrinsic value" and "extrinsic value" discussed above. This suggestion is, if anything, reinforced when Kant immediately adds that a good will "is to be esteemed beyond comparison as far higher than anything it could ever bring about," that it "shine[s] like a jewel for its own sake," and that its "usefulness...can neither add to, nor subtract from, [its] value." For here Kant may seem not only to be invoking the distinction between intrinsic and extrinsic value but also to be in agreement with Brentano et al. regarding the characterization of the former in terms of the fittingness of some attitude, namely, esteem. (The term "respect" is often used in place of "esteem" in such contexts.) Nonetheless, it becomes clear on further inspection that Kant is in fact discussing a concept quite different from that with which this article is concerned. A little later on he says that all rational beings, even those that lack a good will, have "absolute value"; such beings are "ends in themselves" that have a "dignity" or "intrinsic value" that is "above all price" (Kant 1785, Ak. 64 and 77). Such talk indicates that Kant believes that the sort of value that he ascribes to rational beings is one that they possess to an infinite degree. But then, if this were understood as a thesis about intrinsic value as we have been understanding this concept, the implication would seem to be that, since it contains rational beings, ours is the best of all possible worlds.[4] Yet this is a thesis that Kant, along with many others, explicitly rejects elsewhere (Kant, Lectures in Ethics). It seems best to understand Kant, and other philosophers who have since written in the same vein (cf. Anderson 1993), as being concerned not with the question of what intrinsic value rational beings have--in the sense of "intrinsic value" discussed above--but with the quite different question of how we ought to behave toward such creatures (cf. Bradley 2006). ## 3. Is There Such a Thing As Intrinsic Value At All? In the history of philosophy, relatively few seem to have entertained doubts about the concept of intrinsic value. Much of the debate about intrinsic value has tended to be about what things actually do have such value. However, once questions about the concept itself were raised, doubts about its metaphysical implications, its moral significance, and even its very coherence began to appear. Consider, first, the metaphysics underlying ascriptions of intrinsic value. It seems safe to say that, before the twentieth century, most moral philosophers presupposed that the intrinsic goodness of something is a genuine property of that thing, one that is no less real than the properties (of being pleasant, of satisfying a need, or whatever) in virtue of which the thing in question is good. (Several dissented from this view, however. Especially well known for their dissent are Thomas Hobbes [1588-1679], who believed the goodness or badness of something to be constituted by the desire or aversion that one may have regarding it, and David Hume [1711-1776], who similarly took all ascriptions of value to involve projections of one's own sentiments onto whatever is said to have value. See Hobbes 1651, Hume 1739.) It was not until Moore argued that this view implies that intrinsic goodness, as a supervening property, is a very different sort of property (one that he called "nonnatural") from those (which he called "natural") upon which it supervenes, that doubts about the view proliferated. One of the first to raise such doubts and to press for a view quite different from the prevailing view was Axel Hagerstrom [1868-1939], who developed an account according to which ascriptions of value are neither true nor false (Hagerstrom 1953). This view has come to be called "noncognitivism." The particular brand of noncognitivism proposed by Hagerstrom is usually called "emotivism," since it holds (in a manner reminiscent of Hume) that ascriptions of value are in essence expressions of emotion. (For example, an emotivist of a particularly simple kind might claim that to say "A is good" is not to make a statement about A but to say something like "Hooray for A!") This view was taken up by several philosophers, including most notably A. J. Ayer [1910-1989] and Charles L. Stevenson [1908-1979] (see Ayer 1946, Stevenson 1944). Other philosophers have since embraced other forms of noncognitivism. R. M. Hare [1919-2002], for example, advocated the theory of "prescriptivism" (according to which moral judgments, including judgments about goodness and badness, are not descriptive statements about the world but rather constitute a kind of command as to how we are to act; see Hare 1952) and Simon Blackburn and Allan Gibbard have since proposed yet other versions of noncognitivism (Blackburn 1984, Gibbard 1990). Hagerstrom characterized his own view as a type of "value-nihilism," and many have followed suit in taking noncognitivism of all kinds to constitute a rejection of the very idea of intrinsic value. But this seems to be a mistake. We should distinguish questions about value from questions about evaluation. Questions about value fall into two main groups, conceptual (of the sort discussed in the last section) and substantive (of the sort discussed in the first section). Questions about evaluation have to do with what precisely is going on when we ascribe value to something. Cognitivists claim that our ascriptions of value constitute statements that are either true or false; noncognitivists deny this. But even noncognitivists must recognize that our ascriptions of value fall into two fundamental classes--ascriptions of intrinsic value and ascriptions of extrinsic value--and so they too must concern themselves with the very same conceptual and substantive questions about value as cognitivists address. It may be that noncognitivism dictates or rules out certain answers to these questions that cognitivism does not, but that is of course quite a different matter from rejecting the very idea of intrinsic value on metaphysical grounds. Another type of metaphysical challenge to intrinsic value stems from the theory of "pragmatism," especially in the form advanced by John Dewey [1859-1952] (see Dewey 1922). According to the pragmatist, the world is constantly changing in such a way that the solution to one problem becomes the source of another, what is an end in one context is a means in another, and thus it is a mistake to seek or offer a timeless list of intrinsic goods and evils, of ends to be achieved or avoided for their own sakes. This theme has been elaborated by Monroe Beardsley, who attacks the very notion of intrinsic value (Beardsley 1965; cf. Conee 1982). Denying that the existence of something with extrinsic value presupposes the existence of something else with intrinsic value, Beardsley argues that all value is extrinsic. (In the course of his argument, Beardsley rejects the sort of "dialectical demonstration" of intrinsic value that was attempted in the last section, when an explanation of the derivative value of helping others was given in terms of some nonderivative value.) A quick response to Beardsley's misgivings about intrinsic value would be to admit that it may well be that, the world being as complex as it is, nothing is such that its value is wholly intrinsic; perhaps whatever has intrinsic value also has extrinsic value, and of course many things that have extrinsic value will have no (or, at least, neutral) intrinsic value. Far from repudiating the notion of intrinsic value, though, this admission would confirm its legitimacy. But Beardsley would insist that this quick response misses the point of his attack, and that it really is the case, not just that whatever has value has extrinsic value, but also that nothing has intrinsic value. His argument for this view is based on the claim that the concept of intrinsic value is "inapplicable," in that, even if something had such value, we could not know this and hence its having such value could play no role in our reasoning about value. But here Beardsley seems to be overreaching. Even if it were the case that we cannot know whether something has intrinsic value, this of course leaves open the question whether anything does have such value. And even if it could somehow be shown that nothing does have such value, this would still leave open the question whether something could have such value. If the answer to this last question is "yes," then the legitimacy of the concept of intrinsic value is in fact confirmed rather than refuted. As has been noted, some philosophers do indeed doubt the legitimacy, the very coherence, of the concept of intrinsic value. Before we turn to a discussion of this issue, however, let us for the moment presume that the concept is coherent and address a different sort of doubt: the doubt that the concept has any great moral significance. Recall the suggestion, mentioned in the last section, that discussions of intrinsic value may have been compromised by a failure to distinguish certain concepts. This suggestion is at the heart of Christine Korsgaard's "Two Distinctions in Goodness" (Korsgaard 1983). Korsgaard notes that "intrinsic value" has traditionally been contrasted with "instrumental value" (the value that something has in virtue of being a means to an end) and claims that this approach is misleading. She contends that "instrumental value" is to be contrasted with "final value," that is, the value that something has as an end or for its own sake; however, "intrinsic value" (the value that something has in itself, that is, in virtue of its intrinsic, nonrelational properties) is to be contrasted with "extrinsic value" (the value that something has in virtue of its extrinsic, relational properties). (An example of a nonrelational property is the property of being round; an example of a relational property is the property of being loved.) As an illustration of final value, Korsgaard suggests that gorgeously enameled frying pans are, in virtue of the role they play in our lives, good for their own sakes. In like fashion, Beardsley wonders whether a rare stamp may be good for its own sake (Beardsley 1965); Shelly Kagan says that the pen that Abraham Lincoln used to sign the Emancipation Proclamation may well be good for its own sake (Kagan 1998); and others have offered similar examples (cf. Rabinowicz and Ronnow-Rasmussen 1999 and 2003). Notice that in each case the value being attributed to the object in question is (allegedly) had in virtue of some extrinsic property of the object. This puts the moral significance of intrinsic value into question, since (as is apparent from our discussion so far) it is with the notion of something's being valuable for its own sake that philosophers have traditionally been, and continue to be, primarily concerned. There is an important corollary to drawing a distinction between intrinsic value and final value (and between extrinsic value and nonfinal value), and that is that, contrary to what Korsgaard herself initially says, it may be a mistake to contrast final value with instrumental value. If it is possible, as Korsgaard claims, that final value sometimes supervenes on extrinsic properties, then it might be possible that it sometimes supervenes in particular on the property of being a means to some other end. Indeed, Korsgaard herself suggests this when she says that "certain kinds of things, such as luxurious instruments, ... are valued for their own sakes under the condition of their usefulness" (Korsgaard 1983, p. 185). Kagan also tentatively endorses this idea. If the idea is coherent, then we should in principle distinguish two kinds of instrumental value, one final and the other nonfinal.[5] If something A is a means to something else B and has instrumental value in virtue of this fact, such value will be nonfinal if it is merely derivative from or reflective of B's value, whereas it will be final if it is nonderivative, that is, if it is a value that A has in its own right (due to the fact that it is a means to B), irrespective of any value that B may or may not have in its own right. Even if it is agreed that it is final value that is central to the concerns of moral philosophers, we should be careful in drawing the conclusion that intrinsic value is not central to their concerns. First, there is no necessity that the term "intrinsic value" be reserved for the value that something has in virtue of its intrinsic properties; presumably it has been used by many writers simply to refer to what Korsgaard calls final value, in which case the moral significance of (what is thus called) intrinsic value has of course not been thrown into doubt. Nonetheless, it should probably be conceded that "final value" is a more suitable term than "intrinsic value" to refer to the sort of value in question, since the latter term certainly does suggest value that supervenes on intrinsic properties. But here a second point can be made, and that is that, even if use of the term "intrinsic value" is restricted accordingly, it is arguable that, contrary to Korsgaard's contention, all final value does after all supervene on intrinsic properties alone; if that were the case, there would seem to be no reason not to continue to use the term "intrinsic value" to refer to final value. Whether this is in fact the case depends in part on just what sort of thing can be valuable for its own sake--an issue to be taken up in the next section. In light of the matter just discussed, we must now decide what terminology to adopt. It is clear that moral philosophers since ancient times have been concerned with the distinction between the value that something has for its own sake (the sort of nonderivative value that Korsgaard calls "final value") and the value that something has for the sake of something else to which it is related in some way. However, given the weight of tradition, it seems justifiable, perhaps even advisable, to continue, despite Korsgaard's misgivings, to use the terms "intrinsic value" and "extrinsic value" to refer to these two types of value; if we do so, however, we should explicitly note that this practice is not itself intended to endorse, or reject, the view that intrinsic value supervenes on intrinsic properties alone. Let us now turn to doubts about the very coherence of the concept of intrinsic value, so understood. In Principia Ethica and elsewhere, Moore embraces the consequentialist view, mentioned above, that whether an action is morally right or wrong turns exclusively on whether its consequences are intrinsically better than those of its alternatives. Some philosophers have recently argued that ascribing intrinsic value to consequences in this way is fundamentally misconceived. Peter Geach, for example, argues that Moore makes a serious mistake when comparing "good" with "yellow."[6] Moore says that both terms express unanalyzable concepts but are to be distinguished in that, whereas the latter refers to a natural property, the former refers to a nonnatural one. Geach contends that there is a mistaken assimilation underlying Moore's remarks, since "good" in fact operates in a way quite unlike that of "yellow"--something that Moore wholly overlooks. This contention would appear to be confirmed by the observation that the phrase "x is a yellow bird" splits up logically (as Geach puts it) into the phrase "x is a bird and x is yellow," whereas the phrase "x is a good singer" does not split up in the same way. Also, from "x is a yellow bird" and "a bird is an animal" we do not hesitate to infer "x is a yellow animal," whereas no similar inference seems warranted in the case of "x is a good singer" and "a singer is a person." On the basis of these observations Geach concludes that nothing can be good in the free-standing way that Moore alleges; rather, whatever is good is good relative to a certain kind. Judith Thomson has recently elaborated on Geach's thesis (Thomson 1997). Although she does not unqualifiedly agree that whatever is good is good relative to a certain kind, she does claim that whatever is good is good in some way; nothing can be "just plain good," as she believes Moore would have it. Philippa Foot, among others, has made a similar charge (Foot 1985). It is a charge that has been rebutted by Michael Zimmerman, who argues that Geach's tests are less straightforward than they may seem and fail after all to reveal a significant distinction between the ways in which "good" and "yellow" operate (Zimmerman 2001, ch. 2). He argues further that Thomson mischaracterizes Moore's conception of intrinsic value. According to Moore, he claims, what is intrinsically good is not "just plain good"; rather, it is good in a particular way, in keeping with Thomson's thesis that all goodness is goodness in a way. He maintains that, for Moore and other proponents of intrinsic value, such value is a particular kind of moral value. Mahrad Almotahari and Adam Hosein have revived Geach's challenge (Almotahari and Hosein 2015). They argue that if, contrary to Geach, "good" could be used predicatively, we would be able to use the term predicatively in sentences of the form 'a is a good K' but, they argue, the linguistic evidence indicates that we cannot do so (Almotahari and Hosein 2015, 1493-4). ## 4. What Sort of Thing Can Have Intrinsic Value? Among those who do not doubt the coherence of the concept of intrinsic value there is considerable difference of opinion about what sort or sorts of entity can have such value. Moore does not explicitly address this issue, but his writings show him to have a liberal view on the matter. There are times when he talks of individual objects (e.g., books) as having intrinsic value, others when he talks of the consciousness of individual objects (or of their qualities) as having intrinsic value, others when he talks of the existence of individual objects as having intrinsic value, others when he talks of types of individual objects as having intrinsic value, and still others when he talks of states of individual objects as having intrinsic value. Moore would thus appear to be a "pluralist" concerning the bearers of intrinsic value. Others take a more conservative, "monistic" approach, according to which there is just one kind of bearer of intrinsic value. Consider, for example, Frankena's long list of intrinsic goods, presented in Section 1 above: life, consciousness, etc. To what kind(s) of entity do such terms refer? Various answers have been given. Some (such as Panayot Butchvarov) claim that it is properties that are the bearers of intrinsic value (Butchvarov 1989, pp. 14-15). On this view, Frankena's list implies that it is the properties of being alive, being conscious, and so on, that are intrinsically good. Others (such as Chisholm) claim that it is states of affairs that are the bearers of intrinsic value (Chisholm 1968-69, 1972, 1975). On this view, Frankena's list implies that it is the states of affairs of someone (or something) being alive, someone being conscious, and so on, that are intrinsically good. Still others (such as Ross) claim that it is facts that are the bearers of intrinsic value (Ross 1930, pp. 112-13; cf. Lemos 1994, ch. 2). On this view, Frankena's list implies that it is the facts that someone (or something) is alive, that someone is conscious, and so on, that are intrinsically good. (The difference between Chisholm's and Ross's views would seem to be this: whereas Chisholm would ascribe intrinsic value even to states of affairs, such as that of everyone being happy, that do not obtain, Ross would ascribe such value only to states of affairs that do obtain.) Ontologists often divide entities into two fundamental classes, those that are abstract and those that are concrete. Unfortunately, there is no consensus on just how this distinction is to be drawn. Most philosophers would classify the sorts of entities just mentioned (properties, states of affairs, and facts) as abstract. So understood, the claim that intrinsic value is borne by such entities is to be distinguished from the claim that it is borne by certain other closely related entities that are often classified as concrete. For example, it has recently been suggested that it is tropes that have intrinsic value.[7] Tropes are supposed to be a sort of particularized property, a kind of property-instance (rather than simply a property). (Thus the particular whiteness of a particular piece of paper is to be distinguished, on this view, from the property of whiteness.) It has also been suggested that it is states, understood as a kind of instance of states of affairs, that have intrinsic value (cf. Zimmerman 2001, ch. 3). Those who make monistic proposals of the sort just mentioned are aware that intrinsic value is sometimes ascribed to kinds of entities different from those favored by their proposals. They claim that all such ascriptions can be reduced to, or translated into, ascriptions of intrinsic value of the sort they deem proper. Consider, for example, Korsgaard's suggestion that a gorgeously enameled frying pan is good for its own sake. Ross would say that this cannot be the case. If there is any intrinsic value to be found here, it will, according to Ross, not reside in the pan itself but in the fact that it plays a certain role in our lives, or perhaps in the fact that something plays this role, or in the fact that something that plays this role exists. (Others would make other translations in the terms that they deem appropriate.) On the basis of this ascription of intrinsic value to some fact, Ross could go on to ascribe a kind of extrinsic value to the pan itself, in virtue of its relation to the fact in question. Whether reduction of this sort is acceptable has been a matter of considerable debate. Proponents of monism maintain that it introduces some much-needed order into the discussion of intrinsic value, clarifying just what is involved in the ascription of such value and simplifying the computation of such value--on which point, see the next section. (A corollary of some monistic approaches is that the value that something has for its own sake supervenes on the intrinsic properties of that thing, so that there is a perfect convergence of the two sorts of values that Korsgaard calls "final" and "intrinsic". On this point, see the last section; Zimmerman 2001, ch. 3; Tucker 2016; and Tucker (forthcoming).) Opponents argue that reduction results in distortion and oversimplification; they maintain that, even if there is intrinsic value to be found in such a fact as that a gorgeously enameled frying pan plays a certain role in our lives, there may yet be intrinsic, and not merely extrinsic, value to be found in the pan itself and perhaps also in its existence (cf. Rabinowicz and Ronnow-Rasmussen 1999 and 2003). Some propose a compromise according to which the kind of intrinsic value that can sensibly be ascribed to individual objects like frying pans is not the same kind of intrinsic value that is the topic of this article and can sensibly be ascribed to items of the sort on Frankena's list (cf. Bradley 2006). (See again the cautionary note in the final paragraph of Section 2 above.) ## 5. How Is Intrinsic Value to Be Computed? In our assessments of intrinsic value, we are often and understandably concerned not only with whether something is good or bad but with how good or bad it is. Arriving at an answer to the latter question is not straightforward. At least three problems threaten to undermine the computation of intrinsic value. First, there is the possibility that the relation of intrinsic betterness is not transitive (that is, the possibility that something A is intrinsically better than something else B, which is itself intrinsically better than some third thing C, and yet A is not intrinsically better than C). Despite the very natural assumption that this relation is transitive, it has been argued that it is not (Rachels 1998; Temkin 1987, 1997, 2012). Should this in fact be the case, it would seriously complicate comparisons, and hence assessments, of intrinsic value. Second, there is the possibility that certain values are incommensurate. For example, Ross at one point contends that it is impossible to compare the goodness of pleasure with that of virtue. Whereas he had suggested in The Right and the Good that pleasure and virtue could be measured on the same scale of goodness, in Foundations of Ethics he declares this to be impossible, since (he claims) it would imply that pleasure of a certain intensity, enjoyed by a sufficient number of people or for a sufficient time, would counterbalance virtue possessed or manifested only by a small number of people or only for a short time; and this he professes to be incredible (Ross 1939, p. 275). But there is some confusion here. In claiming that virtue and pleasure are incommensurate for the reason given, Ross presumably means that they cannot be measured on the same ratio scale. (A ratio scale is one with an arbitrary unit but a fixed zero point. Mass and length are standardly measured on ratio scales.) But incommensurability on a ratio scale does not imply incommensurability on every scale--an ordinal scale, for instance. (An ordinal scale is simply one that supplies an ordering for the quantity in question, such as the measurement of arm-strength that is provided by an arm-wrestling competition.) Ross's remarks indicate that he in fact believes that virtue and pleasure are commensurate on an ordinal scale, since he appears to subscribe to the arch-puritanical view that any amount of virtue is intrinsically better than any amount of pleasure. This view is just one example of the thesis that some goods are "higher" than others, in the sense that any amount of the former is better than any amount of the latter. This thesis can be traced to the ancient Greeks (Plato, Philebus, 21a-e; Aristotle, Nicomachean Ethics, 1174a), and it has been endorsed by many philosophers since, perhaps most famously by Mill (Mill 1863, paras. 4 ff). Interest in the thesis has recently been revived by a set of intricate and intriguing puzzles, posed by Derek Parfit, concerning the relative values of low-quantity/high-quality goods and high-quantity/low-quality goods (Parfit 1984, Part IV). One response to these puzzles (eschewed by Parfit himself) is to adopt the thesis of the nontransitivity of intrinsic betterness. Another is to insist on the thesis that some goods are higher than others. Such a response does not by itself solve the puzzles that Parfit raises, but, to the extent that it helps, it does so at the cost of once again complicating the computation of intrinsic value. To repeat: contrary to what Ross says, the thesis that some goods are higher than others implies that such goods are commensurate, and not that they are incommensurate. Some people do hold, however, that certain values really are incommensurate and thus cannot be compared on any meaningful scale. (Isaiah Berlin [1909-1997], for example, is often thought to have said this about the values of liberty and equality. Whether he is best interpreted in this way is debatable. See Berlin 1969.) This view constitutes a more radical threat to the computation of intrinsic value than does the view that intrinsic betterness is not transitive. The latter view presupposes at least some measure of commensurability. If A is better than B and B is better than C, then A is commensurate with B and B is commensurate with C; and even if it should turn out that A is not better than C, it may still be that A is commensurate with C, either because it is as good as C or because it is worse than C. But if A is incommensurate with B, then A is neither better than nor as good as nor worse than B. (Some claim, however, that the reverse does not hold and that, even if A is neither better than nor as good as nor worse than B, still A may be "on a par" with B and thus be roughly comparable with it. Cf. Chang 1997, 2002.) If such a case can arise, there is an obvious limit to the extent to which we can meaningfully say how good a certain complex whole is (here, "whole" is used to refer to whatever kind of entity may have intrinsic value); for, if such a whole comprises incommensurate goods A and B, then there will be no way of establishing just how good it is overall, even if there is a way of establishing how good it is with respect to each of A and B. There is a third, still more radical threat to the computation of intrinsic value. Quite apart from any concern with the commensurability of values, Moore famously claims that there is no easy formula for the determination of the intrinsic value of complex wholes because of the truth of what he calls the "principle of organic unities" (Moore 1903, p. 96). According to this principle, the intrinsic value of a whole must not be assumed to be the same as the sum of the intrinsic values of its parts (Moore 1903, p. 28) As an example of an organic unity, Moore gives the case of the consciousness of a beautiful object; he says that this has great intrinsic value, even though the consciousness as such and the beautiful object as such each have comparatively little, if any, intrinsic value. If the principle of organic unities is true, then there is scant hope of a systematic approach to the computation of intrinsic value. Although the principle explicitly rules out only summation as a method of computation, Moore's remarks strongly suggest that there is no relation between the parts of a whole and the whole itself that holds in general and in terms of which the value of the latter can be computed by aggregating (whether by summation or by some other means) the values of the former. Moore's position has been endorsed by many other philosophers. For example, Ross says that it is better that one person be good and happy and another bad and unhappy than that the former be good and unhappy and the latter bad and happy, and he takes this to be confirmation of Moore's principle (Ross 1930, p. 72). Broad takes organic unities of the sort that Moore discusses to be just one instance of a more general phenomenon that he believes to be at work in many other situations, as when, for example, two tunes, each pleasing in its own right, make for a cacophonous combination (Broad 1985, p. 256). Others have furnished still further examples of organic unities (Chisholm 1986, ch. 7; Lemos 1994, chs. 3 and 4, and 1998; Hurka 1998). Was Moore the first to call attention to the phenomenon of organic unities in the context of intrinsic value? This is debatable. Despite the fact that he explicitly invoked what he called a "principle of summation" that would appear to be inconsistent with the principle of organic unities, Brentano appears nonetheless to have anticipated Moore's principle in his discussion of Schadenfreude, that is, of malicious pleasure; he condemns such an attitude, even though he claims that pleasure as such is intrinsically good (Brentano 1969, p. 23 n). Certainly Chisholm takes Brentano to be an advocate of organic unities (Chisholm 1986, ch. 5), ascribing to him the view that there are many kinds of organic unity and building on what he takes to be Brentano's insights (and, going further back in the history of philosophy, the insights of St. Thomas Aquinas [1225-1274] and others). Recently, a special spin has been put on the principle of organic unities by so-called "particularists." Jonathan Dancy, for example, has claimed (in keeping with Korsgaard and others mentioned in Section 3 above), that something's intrinsic value need not supervene on its intrinsic properties alone; in fact, the supervenience-base may be so open-ended that it resists generalization. The upshot, according to Dancy, is that the intrinsic value of something may vary from context to context; indeed, the variation may be so great that the thing's value changes "polarity" from good to bad, or vice versa (Dancy 2000). This approach to value constitutes an endorsement of the principle of organic unities that is even more subversive of the computation of intrinsic value than Moore's; for Moore holds that the intrinsic value of something is and must be constant, even if its contribution to the value of wholes of which it forms a part is not, whereas Dancy holds that variation can occur at both levels. Not everyone has accepted the principle of organic unities; some have held out hope for a more systematic approach to the computation of intrinsic value. However, even someone who is inclined to measure intrinsic value in terms of summation must acknowledge that there is a sense in which the principle of organic unities is obviously true. Consider some complex whole, W, that is composed of three goods, X, Y, and Z, which are wholly independent of one another. Suppose that we had a ratio scale on which to measure these goods, and that their values on this scale were 10, 20, and 30, respectively. We would expect someone who takes intrinsic value to be summative to declare the value of W to be (10 + 20 + 30 =) 60. But notice that, if X, Y, and Z are parts of W, then so too, presumably, are the combinations X-and-Y, X-and-Z, and Y-and-Z; the values of these combinations, computed in terms of summation, will be 30, 40, and 50, respectively. If the values of these parts of W were also taken into consideration when evaluating W, the value of W would balloon to 180. Clearly, this would be a distortion. Someone who wishes to maintain that intrinsic value is summative must thus show not only how the various alleged examples of organic unities provided by Moore and others are to be reinterpreted, but also how, in the sort of case just sketched, it is only the values of X, Y, and Z, and not the values either of any combinations of these components or of any parts of these components, that are to be taken into account when evaluating W itself. In order to bring some semblance of manageability to the computation of intrinsic value, this is precisely what some writers, by appealing to the idea of "basic" intrinsic value, have tried to do. The general idea is this. In the sort of example just given, each of X, Y, and Z is to be construed as having basic intrinsic value; if any combinations or parts of X, Y, and Z have intrinsic value, this value is not basic; and the value of W is to be computed by appealing only to those parts of W that have basic intrinsic value. Gilbert Harman was one of the first explicitly to discuss basic intrinsic value when he pointed out the apparent need to invoke such value if we are to avoid distortions in our evaluations (Harman 1967). However, he offers no precise account of the concept of basic intrinsic value and ends his paper by saying that he can think of no way to show that nonbasic intrinsic value is to be computed in terms of the summation of basic intrinsic value. Several philosophers have since tried to do better. Many have argued that nonbasic intrinsic value cannot always be computed by summing basic intrinsic value. Suppose that states of affairs can bear intrinsic value. Let X be the state of affairs of John being pleased to a certain degree x, and Y be the state of affairs of Jane being displeased to a certain degree y, and suppose that X has a basic intrinsic value of 10 and Y a basic intrinsic value of [?]20. It seems reasonable to sum these values and attribute an intrinsic value of [?]10 to the conjunctive state of affairs X&Y. But what of the disjunctive state of affairs XvY or the negative state of affairs ~X? How are their intrinsic values to be computed? Summation seems to be a nonstarter in these cases. Nonetheless, attempts have been made even in such cases to show how the intrinsic value of a complex whole is to be computed in a nonsummative way in terms of the basic intrinsic values of simpler states, thus preserving the idea that basic intrinsic value is the key to the computation of all intrinsic value (Quinn 1974, Chisholm 1975, Oldfield 1977, Carlson 1997). (These attempts have generally been based on the assumption that states of affairs are the sole bearers of intrinsic value. Matters would be considerably more complicated if it turned out that entities of several different ontological categories could all have intrinsic value.) Suggestions as to how to compute nonbasic intrinsic value in terms of basic intrinsic value of course presuppose that there is such a thing as basic intrinsic value, but few have attempted to provide an account of what basic intrinsic value itself consists in. Fred Feldman is one of the few (Feldman 2000; cf. Feldman 1997, pp. 116-18). Subscribing to the view that only states of affairs bear intrinsic value, Feldman identifies several features that any state of affairs that has basic intrinsic value in particular must possess. He maintains, for example, that whatever has basic intrinsic value must have it to a determinate degree and that this value cannot be "defeated" by any Moorean organic unity. In this way, Feldman seeks to preserve the idea that intrinsic value is summative after all. He does not claim that all intrinsic value is to be computed by summing basic intrinsic value, but he does insist that the value of entire worlds is to be computed in this way. Despite the detail in which Feldman characterizes the concept of basic intrinsic value, he offers no strict analysis of it. Others have tried to supply such an analysis. For example, by noting that, even if it is true that only states have intrinsic value, it may yet be that not all states have intrinsic value, Zimmerman suggests (to put matters somewhat roughly) that basic intrinsic value is the intrinsic value had by states none of whose proper parts have intrinsic value (Zimmerman 2001, ch. 5). On this basis he argues that disjunctive and negative states in fact have no intrinsic value at all, and thereby seeks to show how all intrinsic value is to be computed in terms of summation after all. Two final points. First, we are now in a position to see why it was said above (in Section 2) that perhaps not all intrinsic value is nonderivative. If it is correct to distinguish between basic and nonbasic intrinsic value and also to compute the latter in terms of the former, then there is clearly a respectable sense in which nonbasic intrinsic value is derivative. Second, if states with basic intrinsic value account for all the value that there is in the world, support is found for Chisholm's view (reported in Section 2) that some ontological version of Moore's isolation test is acceptable. ## 6. What Is Extrinsic Value? At the beginning of this article, extrinsic value was said simply--too simply--to be value that is not intrinsic. Later, once intrinsic value had been characterized as nonderivative value of a certain, perhaps moral kind, extrinsic value was said more particularly to be derivative value of that same kind. That which is extrinsically good is good, not (insofar as its extrinsic value is concerned) for its own sake, but for the sake of something else to which it is related in some way. For example, the goodness of helping others in time of need is plausibly thought to be extrinsic (at least in part), being derivative (at least in part) from the goodness of something else, such as these people's needs being satisfied, or their experiencing pleasure, to which helping them is related in some causal way. Two questions arise. The first is whether so-called extrinsic value is really a type of value at all. There would seem to be a sense in which it is not, for it does not add to or detract from the value in the world. Consider some long chain of derivation. Suppose that the extrinsic value of A can be traced to the intrinsic value of Z by way of B, C, D... Thus A is good (for example) because of B, which is good because of C, and so on, until we get to Y's being good because of Z; when it comes to Z, however, we have something that is good, not because of something else, but "because of itself," i.e., for its own sake. In this sort of case, the values of A, B, ..., Y are all parasitic on the value of Z. It is Z's value that contributes to the value there is in the world; A, B, ..., Y contribute no value of their own. (As long as the value of Z is the only intrinsic value at stake, no change of value would be effected in or imparted to the world if a shorter route from A to Z were discovered, one that bypassed some letters in the middle of the alphabet.) Why talk of "extrinsic value" at all, then? The answer can only be that we just do say that certain things are good, and others bad, not for their own sake but for the sake of something else to which they are related in some way. To say that these things are good and bad only in a derivative sense, that their value is merely parasitic on or reflective of the value of something else, is one thing; to deny that they are good or bad in any respectable sense is quite another. The former claim is accurate; hence the latter would appear unwarranted. If we accept that talk of "extrinsic value" can be appropriate, however, a second question then arises: what sort of relation must obtain between A and Z if A is to be said to be good "because of" Z? It is not clear just what the answer to this question is. Philosophers have tended to focus on just one particular causal relation, the means-end relation. This is the relation at issue in the example given earlier: helping others is a means to their needs being satisfied, which is itself a means to their experiencing pleasure. The term most often used to refer to this type of extrinsic value is "instrumental value," although there is some dispute as to just how this term is to be employed. (Remember also, from Section 3 above, that on some views "instrumental value" may refer to a type of intrinsic, or final, value.) Suppose that A is a means to Z, and that Z is intrinsically good. Should we therefore say that A is instrumentally good? What if A has another consequence, Y, and this consequence is intrinsically bad? What, especially, if the intrinsic badness of Y is greater than the intrinsic goodness of Z? Some would say that in such a case A is both instrumentally good (because of Z) and instrumentally bad (because of Y). Others would say that it is correct to say that A is instrumentally good only if all of A's causal consequences that have intrinsic value are, taken as a whole, intrinsically good. Still others would say that whether something is instrumentally good depends not only on what it causes to happen but also on what it prevents from happening (cf. Bradley 1998). For example, if pain is intrinsically bad, and taking an aspirin puts a stop to your pain but causes nothing of any positive intrinsic value, some would say that taking the aspirin is instrumentally good despite its having no intrinsically good consequences. Many philosophers write as if instrumental value is the only type of extrinsic value, but that is a mistake. Suppose, for instance, that the results of a certain medical test indicate that the patient is in good health, and suppose that this patient's having good health is intrinsically good. Then we may well want to say that the results are themselves (extrinsically) good. But notice that the results are of course not a means to good health; they are simply indicative of it. Or suppose that making your home available to a struggling artist while you spend a year abroad provides him with an opportunity he would otherwise not have to create some masterpieces, and suppose that either the process or the product of this creation would be intrinsically good. Then we may well want to say that your making your home available to him is (extrinsically) good because of the opportunity it provides him, even if he goes on to squander the opportunity and nothing good comes of it. Or suppose that someone's appreciating the beauty of the Mona Lisa would be intrinsically good. Then we may well want to say that the painting itself has value in light of this fact, a kind of value that some have called "inherent value" (Lewis 1946, p. 391; cf. Frankena 1973, p. 82). ("Inherent value" may not be the most suitable term to use here, since it may well suggest intrinsic value, whereas the sort of value at issue is supposed to be a type of extrinsic value. The value attributed to the painting is one that it is said to have in virtue of its relation to something else that would supposedly be intrinsically good if it occurred, namely, the appreciation of its beauty.) Many other instances could be given of cases in which we are inclined to call something good in virtue of its relation to something else that is or would be intrinsically good, even though the relation in question is not a means-end relation. One final point. It is sometimes said that there can be no extrinsic value without intrinsic value. This thesis admits of several interpretations. First, it might mean that nothing can occur that is extrinsically good unless something else occurs that is intrinsically good, and that nothing can occur that is extrinsically bad unless something else occurs that is intrinsically bad. Second, it might mean that nothing can occur that is either extrinsically good or extrinsically bad unless something else occurs that is either intrinsically good or intrinsically bad. On both these interpretations, the thesis is dubious. Suppose that no one ever appreciates the beauty of Leonardo's masterpiece, and that nothing else that is intrinsically either good or bad ever occurs; still his painting may be said to be inherently good. Or suppose that the aspirin prevents your pain from even starting, and hence inhibits the occurrence of something intrinsically bad, but nothing else that is intrinsically either good or bad ever occurs; still your taking the aspirin may be said to be instrumentally good. On a third interpretation, however, the thesis might be true. That interpretation is this: nothing can occur that is either extrinsically good or extrinsically neutral or extrinsically bad unless something else occurs that is either intrinsically good or intrinsically neutral or intrinsically bad. This would be trivially true if, as some maintain, the nonoccurrence of something intrinsically either good or bad entails the occurrence of something intrinsically neutral. But even if the thesis should turn out to be false on this third interpretation, too, it would nonetheless seem to be true on a fourth interpretation, according to which the concept of extrinsic value, in all its varieties, is to be understood in terms of the concept of intrinsic value.
knowledge-value	## 1. Value problems In Plato's Meno, Socrates raises the question of why knowledge is more valuable than mere true belief. Call this the Meno problem or, anticipating distinctions made below, the primary value problem. Initially, we might appeal to the fact that knowledge appears to be of more practical use than true belief in order to mark this difference in value. But, as Socrates notes, this could be questioned, because a true belief that this is the way to Larissa will get you to Larissa just as well as knowledge that this is the way to Larissa. Plato's own solution was that knowledge is formed in a special way distinguishing it from belief: knowledge, unlike belief, must be 'tied down' to the truth, like the mythical tethered statues of Daedalus. As a result, knowledge is better suited to guide action. For example, if one knows, rather than merely truly believes, that this is the way to Larissa, then one might be less likely to be perturbed by the fact that the road initially seems to be going in the wrong direction. Mere true belief at this point might be lost, since one might lose all confidence that this is the right way to go. The primary value problem has been distinguished from the secondary value problem (Pritchard 2007: SS2). The secondary value problem pertains to why knowledge is more valuable, from an epistemic point of view, than any proper subset of its parts. Put otherwise, why is knowledge better than any epistemic standing falling short of knowing? This includes, but is not restricted to, mere true belief. To illustrate the distinction, consider a possible solution to the primary value problem: knowledge is justified true belief, and justified true belief is better than mere true belief, which explains why knowledge is better than true belief. If correct, this hypothesis successfully answers the primary value problem. However, it requires further development to answer the secondary value problem. For example, it requires further development to explain why knowledge is better than justified belief. Of course, on many standard theories of knowledge, knowledge is not defined as justified true belief. For instance, according to some theorists, knowledge is undefeated justified true belief (Lehrer & Paxson 1969); on other widely discussed accounts, knowledge is true belief that is non-accidental (Unger 1968), sensitive (Nozick 1981), safe (Sosa 1999), appropriately caused (Goldman 1967), or produced by intellectual virtue (Zagzebski 1996). This puts us in a position to appreciate what some theorists call the tertiary value problem. The tertiary value problem pertains to why knowledge is qualitatively better than any epistemic standing falling short of knowledge. Consider that if knowledge were only quantitatively better than that which falls just short--for instance, on an envisioned continuum of epistemic value--then it would be mysterious why epistemologists have given such attention to this particular point on the continuum. Why does knowledge have this "distinctive value" not shared by that which falls just short of knowledge (Pritchard 2009: 14)? Not all theorists accept that the value problems are genuine. For example, in light of the literature on the Gettier problem, some theorists deny that the secondary value problem is genuine. On this approach, whatever is added to justified true belief to rule out Gettier cases does not increase the value of the agent's intellectual state: it is of no consequence whether we have Gettier-proof justified true belief rather than mere justified true belief (Kaplan 1985). Of course, Gettier cases are peculiar and presumably rare, so in practice having Gettier-proof justified true belief is almost invariably confounded with having mere justified true belief. This could lead some theorists to mistake the value of the latter for that of the former. Other theorists deny that the primary value problem is genuine. For example, on one approach, knowledge just is true belief (Sartwell 1991). If knowledge is true belief, then knowledge cannot be better than true belief, because nothing can be better than itself. However, the definition of knowledge as true belief has not been widely accepted. ## 2. Reliabilism and the Meno Problem The first contemporary wave of work on the value problem largely concerned whether this problem raised a distinctive difficulty for reliabilist accounts of knowledge--i.e., those views which essentially define knowledge as reliably-formed true belief. In particular, the claim was that reliabilism was unable to offer an answer even to the primary value problem. A fairly clear statement of what is at issue here is given in a number of places by Linda Zagzebski (e.g., 2003a; cf. DePaul 1993; Zagzebski 1996; Jones 1997; Swinburne 1999, 2000; Riggs 2002a; Kvanvig 2003; Sosa 2007: ch. 4; Carter & Jarvis 2012). To begin with, Zagzebski argues that the reliability of the process by which something is produced does not automatically add value to that thing, and thus that it cannot be assumed that the reliability of the process by which a true belief is produced will add value to that true belief. In defense of this claim, she offers the analogy of a cup of coffee. She claims that a good cup of coffee which is produced by a reliable coffee machine--i.e., one that regularly produces good cups of coffee--is of no more value than an equally good cup of coffee that is produced by an unreliable coffee machine. Furthermore, as this line of objection goes, true belief is in the relevant respects like coffee: a true belief formed via a reliable belief-forming process is no more valuable than a true belief formed via an unreliable belief-forming process. In both cases, the value of the reliability of the process accrues in virtue of its tendency to produce a certain valuable effect (good coffee/true belief), but this means that where the effect has been produced--where one has a good cup of coffee or a true belief--then the value of the product is no greater for having been produced in a reliable way. Elsewhere in the literature (e.g., Kvanvig 2003), this problem has been called the "swamping problem", on account of how the value of true belief 'swamps' the value of the true belief being produced in a reliable (i.e., truth-conducive) way. So expressed, the moral of the problem seems to be that where reliabilists go awry is by treating the value of the process as being solely captured by the reliability of the process--i.e., its tendency to produce the desired effect. Since the value of the effect swamps the value of the reliability of the process by which the effect was achieved, this means that reliabilism has no resources available to it to explain why knowledge is more valuable than true belief. It's actually not clear that this is a problem that is specific to reliabilism. That is, it seems that if this is a bona fide problem, then it will affect any account of the value of knowledge which has the same relevant features as reliabilism--i.e., which regards the greater value of knowledge over true belief as instrumental value, where the instrumental value in question is relative to the valuable good of true belief. In particular, it will affect veritist proposals about epistemic value which treat truth as the fundamental epistemic good. See Kvanvig (2003: Ch. 3) for discussion of how internalist approaches to epistemic justification interface with the swamping problem; see Pettigrew (2018) and Pritchard (2019) for responses to the swamping argument on behalf of the veritist. Furthermore, as J. Adam Carter and Benjamin Jarvis (2012) have argued, there are reasons to be suspicious of a key premise driving the swamping argument. The premise in question, which has been referred to as the "Swamping Thesis" (Pritchard 2011), states that if the value of a property possessed by an item is instrumentally valuable only relative to a further good, and that good is already present in that item, then it can confer no additional value. Carter and Jarvis contend that one who embraces the Swamping Thesis should also, by parity of reasoning, embrace a corollary thesis which they call the Swamping Thesis Complement, according to which, if the value of a property possessed by an item is instrumentally valuable only relative to a further good, and that good has already failed to be present in that item, then it can confer no additional value. However, as they argue, the Swamping Thesis and the Swamping Thesis Complement, along with other plausible premises, jointly entail the unpalatable conclusion that non-factive epistemic properties--most notably, justification--are never epistemically valuable properties of a belief. See Dutant (2013) and Bjelde (2020) for critical responses to Carter and Jarvis' line of reasoning and Sylvan (2018) for a separate challenge to the swamping argument, which rejects its tacit commitment to epistemic instrumentalism (cf., Bjelde 2020). For an overview of the key moves of the argument, see Pritchard (2011). However, even granting the main elements of the swamping argument, there are moves that the reliabilist can make in response (see, e.g., Goldman & Olsson 2009; Olsson 2011; Bates 2013; Roush 2010; cf. Brown 2012; Davis, & Jager 2012; Hovarth 2009; Piller 2009). For example, it is surely open to the reliabilist to argue that the greater instrumental value of reliable true belief over mere true belief does not need to be understood purely in terms of instrumental value relative to the good of true belief. There could, for instance, be all sorts of practical benefits of having a reliable true belief which generate instrumental value. Indeed, it is worth noting that the line of response to the Meno problem sketched by Plato, which we noted above, seems to specifically appeal to the greater practical instrumental value of knowledge over mere true belief. Moreover, there is reason to think that this objection will only at best have an impact on process reliabilist proposals--i.e., those views that treat all reliable belief-forming processes as conferring a positive epistemic standing on the beliefs so formed. For example, agent reliabilism (e.g., Greco 1999, 2000) might be thought to be untouched by this sort of argument. This is because, according to agent reliabilism, it is not any sort of reliable process that confers positive epistemic status to belief, but only those processes that are stable features of the agent's "cognitive character". The main motivation for this restriction on reliable processes is that it excludes certain kinds of reliable but nonetheless strange and fleeting processes which notoriously cause problems for the view (such as processes where the reliability is due to some quirk in the subject's environment, rather than because of any cognitive trait possessed by the agent herself). Plausibly, however, one might argue that the reliable traits that make up an agent's cognitive character have some value independently of the instrumental value they possess in virtue of being reliable--i.e., that they have some final or intrinsic value. If this is right, then this opens up the possibility that agent-reliabilists can evade the problem noted for pure reliabilists. Zagzebski's diagnosis of what is motivating this problem for reliabilism seems, however, explicitly to exclude such a counter-response. She argues that what gives rise to this difficulty is the fact that the reliabilist has signed up to a "machine-product model of belief"--see especially, Zagzebski (2003a)--where the product is external to the cause. It is not clear what exactly Zagzebski means by this point, but she thinks it shows that even where the reliable process is independently valuable--i.e., independently of its being reliable--it still doesn't follow that the value of the cause will transfer to add value to the effect. Here again the coffee analogy is appealed to: even if a reliable coffee machine were independently valuable, it would not thereby confer additional value on a good cup of coffee. Perhaps the best way to evaluate the above line of argument is to consider what is required in order to resolve the problem it poses. Perhaps what is needed is an 'internal' connection between product and cause, such as the kind of internal connection that exists between an act and its motive which is highlighted by how we explicitly evaluate actions in terms of the motives that led to them (Zagzebski 2003a). On this picture, then, we are not to understand knowledge as a state consisting of a known belief, but rather as a state which consists of both the true belief and the source from which that true belief was acquired. In short, then, the problem with the machine-product model of belief is that it leads us to evaluate the state of the knowledge independently of the means by which the knowledge was acquired. If, in contrast, we have a conception of knowledge that incorporates into the very state of knowledge the way that the knowledge was acquired, we can avoid this problem. Once one effects this transition away from the machine-product model of belief, one can allow that the independent value of the reliable process can ensure that knowledge, by being produced in this way, is more valuable than mere true belief (Zagzebski 2003a). In particular, if the process by which one gained the true belief is an epistemic virtue--a character trait which is both reliable and intrinsically valuable--then this can ensure that the value of the knowing state in this case is more valuable than any corresponding state which simply consisted of a true belief. Other commentators in the virtue epistemology camp, broadly conceived, have put forward similar suggestions. For example, Wayne Riggs (2002a) and Greco (e.g., 2003) have argued for a 'credit' version of virtue epistemology, according to which the agent, in virtue of bringing about the positively valuable outcome of a true belief, is due credit as a result. Rather than treating the extra value of knowledge over true belief as deriving simply from the agent's attainment of the target true belief, however, Riggs and Greco instead argue that we should regard the agent's knowing as the state the agent is in when she is responsible for her true belief. Only in so doing, they claim, can we answer the value problem. Jason Baehr (2012), by contrast with Riggs and Greco, has argued that credit theories of knowledge do not answer the value problem but, rather, 'provide grounds for denying' (2012: 1) that knowledge has value over and above the value of true belief. Interestingly, however, other virtue epistemologists, most notably Ernest Sosa (2003), have also advocated a 'credit' view, yet seem to stay within the machine-product picture of belief. That is, rather than analyze the state of knowing as consisting of both the true belief and its source, they regard the state of knowing as distinct from the process, yet treat the fact that the process is intrinsically valuable as conferring additional value on any true belief so produced. With Sosa's view in mind, it is interesting to ask just why we need to analyze knowledge in the way that Zagzebski and others suggest in order to get around the value problem. The most direct way to approach this question is by considering whether it is really true that a valuable cause cannot confer value on its effect where cause and effect are kept separate in the way that Zagzebski claims is problematic in the case of knowledge. One commentator who has objected to Zagzebski's argument by querying this claim on her part is Berit Brogaard (2007; cf. Percival 2003; Pritchard 2007: SS2), who claims that a valuable cause can indeed confer value on its effect in the relevant cases. Brogaard claims that virtue epistemologists like Zagzebski and Riggs endorse this claim because they adhere to what she calls a "Moorean" conception of value, on which if two things have the same intrinsic properties, then they are equally valuable. Accordingly, if true belief and knowledge have the same intrinsic properties (which is what would be the case on the view of knowledge that they reject), it follows that they must have the same value. Hence, it is crucial to understand knowledge as having distinct intrinsic properties from true belief before one can hope to resolve the value problem. If one holds that there is only intrinsic and instrumental value, then this conception of value is compelling, since objects with the same intrinsic properties trivially have the same amount of intrinsic value, and they also plausibly have the same amount of instrumental value as well (at least in the same sort of environment). However, the Moorean conception of value is problematic because--as Wlodek Rabinowicz & Toni Ronnow-Rasmussen (1999, 2003) have pointed out--there seem to be objects which we value for their own sake but whose value derives from their being extrinsically related to something else that we value. That is, such objects are finally--i.e., non-instrumentally--valuable without thereby being intrinsically valuable. For criticism of this account of final value, see Bradley (2002). The standard example in this regard is Princess Diana's dress. This would be regarded as more valuable than an exact replica simply because it belonged to Diana, which is clearly an extrinsic property of the object. Even though the extra value that accrues to the object is due to its extrinsic properties, however, it is still the case that this dress is (properly) valued for its own sake, and thus valued non-instrumentally. Given that value of this sort is possible, then it follows that it could well be the case that we value one true belief over another because of its extrinsic features--i.e., that the one true belief, but not the other, was produced by a reliable cognitive trait that is independently valuable. For example, it could be that we value forming a true belief via a reliable cognitive trait more than a mere true belief because the former belief is produced in such a way that it is of credit to us that we believe the truth. There is thus a crucial lacuna in Zagzebski's argument. A different response to the challenge that Zagzebski raises for reliabilism is given by Michael Brady (2006). In defense of reliabilism, Brady appeals to the idea that to be valuable is to be a fitting or appropriate object of positive evaluative attitudes, such as admiration or love (e.g., Brentano 1889 [1969]; Chisholm 1986; Wiggins 1987; Gibbard 1990; Scanlon 1998). That one object is more valuable than another is thus to be understood, on this view, in terms of the fact that that object is more worthy of positive evaluation. Thus, the value problem for reliabilism on this conception of value comes down to the question why knowledge is more worthy of positive evaluation on this view than mere true belief. Brady's contention is that, at least within this axiological framework, it is possible for the reliabilist to offer a compelling story about why reliable true belief--and thus knowledge--is more valuable than mere true belief. Central to Brady's argument is his claim that there are many ways one can positively evaluate something, and thus many different ways something can be valuable. Moreover, Brady argues that we can distinguish active from passive evaluative attributes, where the former class of attitudes involve pursuit of the good in question. For example, one might actively value the truth, where this involves, for instance, a striving to discover the truth. In contrast, one might at other times merely passively value the truth, such as simply respecting or contemplating it. With this point in mind, Brady's central thesis is that on the reliabilist account knowledge is more valuable than true belief because certain active positive evaluative attitudes are fitting only with regard to the former (i.e., reliable true belief). In particular, given its intrinsic features, reliable true belief is worthy of active love, whereas an active love of unreliable (i.e., accidental) true belief because of its intrinsic features would be entirely inappropriate because there is nothing that we can do to attain unreliable true belief that wouldn't conflict with love of truth. This is an intriguing proposal, which opens up a possible avenue of defense against the kind of machine-product objection to reliabilism considered. One problem that such a move faces, however, is that it is unclear whether we can make sense of the distinction Brady draws between active and passive evaluative attitudes, at least in the epistemic sphere. When Brady talks of passive evaluative attitudes towards the truth, he gives examples like contemplating, accepting, embracing, affirming, and respecting. Some of these attitudes are not clearly positive evaluative attitudes, however. Moreover, some of them are not obviously passive either. For example, is to contemplate the truth really to evaluate it positively, rather than simply to consider it? Furthermore, in accepting, affirming or embracing the truth, isn't one actively positively evaluating the truth? Wouldn't such evaluative attitudes manifest themselves in the kind of practical action that Brady thinks is the mark of active evaluative attitudes? More needs to be said about this distinction before it can do the philosophical work that Brady has in mind. A further, albeit unorthodox, recent approach to the swamping problem is due to Carter and Rupert (2020). Carter and Rupert point out that extant approaches to the swamping problem suppose that if a solution is to be found, it will be at the personal level of description, the level at which states of subjects or agents, as such, appear. They take exception to this orthodoxy, or at least to its unquestioned status. They maintain that from the empirically justified premise that subpersonal states play a significant role in much epistemically relevant cognition, we should expect that they constitute a domain in which we might reasonably expect to locate the "missing source" of epistemic value, beyond the value attached to mere true belief. ## 3. Virtue Epistemology and the Value Problem So far this discussion has taken it as given that whatever problems reliabilism faces in this regard, there are epistemological theories available--some form of virtue epistemology, for example--that can deal with them. But not everyone in the contemporary debate accepts this. Perhaps the best known sceptic in this respect is Jonathan Kvanvig (2003), who in effect argues that while virtue epistemology (along with a form of epistemic internalism) can resolve the primary value problem (i.e., the problem of explaining why knowledge is more valuable than mere true belief), the real challenge that we need to respond to is that set by the secondary value problem (i.e., the problem of explaining why knowledge is more valuable than that which falls short of knowledge); and Kvanvig says that there is no solution available to that. That is, Kvanvig argues that there is an epistemic standing--in essence, justified true belief--which falls short of knowledge but which is no less valuable than knowledge. He concludes that the focus of epistemology should not be on knowledge at all, but rather on understanding, an epistemic standing that Kvanvig maintains is clearly of more value than knowledge and those epistemic standings that fall short of knowledge, such as justified true belief. What Kvanvig says about understanding will be considered below. First though, let us consider the specific challenge that he poses for virtue epistemology. In essence, Kvanvig's argument rests on the assumption that it is essential to any virtue-theoretic account of knowledge--and any internalist account of knowledge as well, for that matter (i.e., an account that makes a subjective justification condition necessary for knowledge possession)--that it also includes an anti-Gettier condition. If this is right, then it follows that even if virtue epistemology has an answer to the primary value problem--and Kvanvig concedes that it does--it will not thereby have an answer to the secondary value problem since knowledge is not simply virtuous true belief. Moreover, Kvanvig argues that once we recognize what a gerrymandered notion a non-Gettierized account of knowledge is, it becomes apparent that there is nothing valuable about the anti-Gettier condition on knowledge that needs to be imposed. But if that is right, then it follows by even virtue epistemic lights that knowledge--i.e., non-Gettierized virtuous true believing--is no more valuable than one of its proper sub-sets--i.e., mere virtuous true believing. There are at least two aspects of Kvanvig's argument that are potentially problematic. To begin with, it isn't at all clear why the anti-Gettier condition on knowledge fails to add value, something that seems to be assumed here. More generally, Kvanvig seems to be implicitly supposing that if an analysis of knowledge is ugly and gerrymandered then that is itself reason to doubt that knowledge is particularly valuable, at least assuming that there are epistemic standings that fall short of knowledge which can be given an elegant analysis. While a similar assumption about the relationship between the elegance (or otherwise) of the analysis of knowledge and the value of the analysandum is commonplace in the contemporary epistemological literature--see, for example, Zagzebski (1999) and Williamson (2000: chapter 1)--this assumption is contentious. For critical discussion of this assumption, see DePaul (2009). In any case, a more serious problem is that many virtue epistemologists--among them Sosa (1988, 1991, 2007), Zagzebski (e.g., 1996, 1999) and Greco (2003, 2007, 2008, 2009)--hereafter, 'robust virtue epistemologists'--think that their view can deal with Gettier problems without needing to add an additional anti-Gettier condition on knowledge. The way this is achieved is by making the move noted above of treating knowledge as a state that includes both the truly believing and the virtuous source by which that true belief was acquired. However, crucially, for robust virtue epistemologists, there is an important difference between (i) a belief's being true and virtuously formed, and (ii) a belief's being true because virtuously formed. Formulating knowledge along the latter lines, they insist, ensures that the target belief is not Gettiered. Even more, robust virtue epistemologists think the latter kind of formulation offers the resources to account for why knowledge is distinctively valuable. To appreciate this point about value, consider the following 'performance normativity framework' which robust virtue epistemologists explicitly or implicitly embrace when accounting for the value of knowledge as a true belief because of virtue. Performance Normativity Framework Dimensions of evaluation thesis Any performance with an aim can be evaluated along three dimensions: (i) whether it is successful, (ii) whether it is skillful, and (iii) thirdly, whether the success is because of the skill. Achievement thesis If and only if the success is because of the skill, the performance is not merely successful, but also, an achievement. Value thesis Achievements are finally valuable (i.e., valuable for their own sake) in a way that mere lucky successes are not. Notice that, if knowledge is a cognitive performance that is an achievement, then with reference to the above set of claims, the robust virtue epistemologist can respond to not only the secondary value problem but also the tertiary value problem (i.e., the problem of explaining why knowledge is more valuable, in kind and not merely in degree, than that which falls short of knowledge). This is because knowledge, on this view, is simply the cognitive aspect of a more general notion, that of achievement, and this is the case even if mere successes that are produced by intellectual virtues but which are not because of them, are not achievements. (Though, see Kim 2021 for a reversal of the idea that knowledge involves achievement; according to Kim, all achievements, in any domain of endeavour, imply knowledge). As regards the value thesis, one might object that some successes that are because of ability--i.e., achievements, on this view--are too trivial or easy or wicked to count as finally valuable. This line of objection is far from decisive. After all, it is open to the proponent of robust virtue epistemology to argue that the claim is only that all achievements qua achievements are finally valuable, not that the overall value of every achievement is particularly high. It is thus consistent with the proposal that some achievements have a very low--perhaps even negative, if that is possible--value in virtue of their other properties (e.g., their triviality). Indeed, a second option in this regard is to allow that not all achievements enjoy final value whilst nevertheless maintaining that it is in the nature of achievements to have such value (e.g., much in the way that one might argue that it is in the nature of pleasure to be a good, even though some pleasures are bad). Since, as noted above, all that is required to meet the (tertiary) value problem is to show that knowledge is generally distinctively valuable, this claim would almost certainly suffice for the robust virtue epistemologist's purposes. In any case, even if the value thesis is correct--and indeed, even if the achievement and dimensions of evaluation theses are also correct--the robust virtue epistemologist has not yet satisfactorily vindicated any of the aforementioned value problems for knowledge unless knowledge is itself a kind of achievement--and that is the element of the proposal that is perhaps the most controversial. There are two key problems with the claim that knowledge involves cognitive achievement. The first is that there sometimes seems to be more to knowledge than a cognitive achievement; the second is that there sometimes seems to be less to knowledge than a cognitive achievement. As regards the first claim, notice that achievements seem to be compatible with at least one kind of luck. Suppose that an archer hits a target by employing her relevant archery abilities, but that the success is 'gettierized' by luck intervening between the archer's firing of the arrow and the hitting of the target. For example, suppose that a freak gust of wind blows the arrow off-course, but then a second freak gust of wind happens to blow it back on course again. The archer's success is thus lucky in the sense that it could very easily have been a failure. When it comes to 'intervening' luck of this sort, Greco's account of achievements is able to offer a good explanation of why the success in question does not constitute an achievement. After all, we would not say that the success was because of the archer's ability in this case. Notice, however, that not all forms of luck are of this intervening sort. Consider the following case offered by Pritchard (2010a: ch. 2). Suppose that nothing intervenes between the archer's firing of the arrow and the hitting of the target. However, the success is still lucky in the relevant sense because, unbeknownst to the archer, she just happened to fire at the only target on the range that did not contain a forcefield which would have repelled the arrow. Is the archer's success still an achievement? Intuition would seem to dictate that it is; it certainly seems to be a success that is because of ability, even despite the luckiness of that success. Achievements, then, are, it seems, compatible with luck of this 'environmental' form even though they are not compatible with luck of the standard 'intervening' form. The significance of this conclusion for our purposes is that knowledge is incompatible with both forms of luck. In order to see this, one only needs to note that an epistemological analogue of the archer case just given is the famous barn facade example (e.g., Ginet 1975; Goldman 1976). In this example, we have an agent who forms a true belief that there is a barn in front of him. Moreover, his belief is not subject to the kind of 'intervening' luck just noted and which is a standard feature of Gettier-style cases. It is not as if, for example, he is looking at what appears to be a barn but which is not in fact a barn, but that his belief is true nonetheless because there is a barn behind the barn shaped object that he is looking at. Nevertheless, his belief is subject to environmental luck in that he is, unbeknownst to him, in barn facade county in which every other barn-shaped object is a barn facade. Thus, his belief is only luckily true in that he could very easily have been mistaken in this respect. Given that this example is structurally equivalent to the 'archer' case just given, it seems that just as we treat the archer as exhibiting an achievement in that case, so we should treat this agent as exhibiting a cognitive achievement here. The problem, however, is that until quite recently many philosophers accepted that the agent in the barn facade case lacks knowledge. Knowledge, it seems, is incompatible with environmental luck in a way that achievements, and thus cognitive achievements, are not (see, for example, Pritchard, e.g., 2012). Robust virtue epistemologists have made a number of salient points regarding this case. For example, Greco (2010, 2012) has argued for a conception of what counts as a cognitive ability according to which the agent in the barn facade case would not count as exhibiting the relevant cognitive ability (see Pritchard 2010a: ch. 2 for a critical discussion of this claim). Others, such as Sosa (e.g., 2007, 2015) have responded by questioning whether the agent in the barn facade case lacks knowledge, albeit, in a qualified sense. While Sosa's distinctive virtue epistemology allows for the compatibility of barn facade cases with animal knowledge (roughly: true belief because of ability), Sosa maintains that the subject in barn facade cases lacks reflective knowledge (roughly: a true belief whose creditability to ability or virtue is itself creditable to a second-order ability or virtue of the agent). Other philosophers (e.g., Hetherington (1998) have challenged the view that barn facade protagonists in fact lack (any kind of) knowledge. In a series of empirical studies, most people attributed knowledge in barn facade cases and related cases (Colaco, Buckwalter, Stich & Machery 2014; Turri, Buckwalter & Blouw 2015; Turri 2016a). In one study, over 80% of participants attributed knowledge (Turri 2016b). In another study, most professional philosophers attributed knowledge (Horvath & Wiegmann 2016). At least one theory of knowledge has been defended on the grounds that it explains why knowledge is intuitively present in such cases (Turri 2016c). Even setting that issue aside, however, there is a second problem on the horizon, which is that it seems that there are some cases of knowledge which are not cases of cognitive achievement. One such case is offered by Jennifer Lackey (2007), albeit to illustrate a slightly different point. Lackey asks us to imagine someone arriving at the train station in Chicago who, wishing to obtain directions to the Sears Tower, approaches the first adult passer-by she sees. Suppose the person she asks is indeed knowledgeable about the area and gives her the directions that she requires. Intuitively, any true belief that the agent forms on this basis would ordinarily be counted as knowledge. Indeed, if one could not gain testimonial knowledge in this way, then it seems that we know an awful lot less than we think we know. However, it has been argued, in such a case the agent does not have a true belief because of her cognitive abilities but, rather, because of her informant's cognitive abilities. If this is correct, then there are cases of knowledge which are not also cases of cognitive achievement. It is worth being clear about the nature of this objection. Lackey takes cases like this to demonstrate that one can possess knowledge without it being primarily creditable to one that one's belief is true. Note though that this is compatible, as Lackey notes, with granting that the agent is employing her cognitive abilities to some degree, and so surely deserves some credit for the truth of the belief formed (she would not have asked just anyone, for example, nor would she have simply accepted just any answer given by her informant). The point is thus rather that whatever credit the agent is due for having a true belief, it is not the kind of credit that reflects a bona fide cognitive achievement because of how this cognitive success involves 'piggy-backing' on the cognitive efforts of others. ## 4. Understanding and Epistemic Value As noted above, the main conclusion that Kvanvig (2003) draws from his reflections on the value problem is that the real focus in epistemology should not be on knowledge at all but on understanding, an epistemic standing that Kvanvig does think is especially valuable but which, he argues, is distinct from knowing--i.e., one can have knowledge without the corresponding understanding, and one can have understanding without the corresponding knowledge. (Pritchard [e.g., 2010a: chs 1-4] agrees, though his reasons for taking this line are somewhat different to Kvanvig's). It is perhaps this aspect of Kvanvig's book that has prompted the most critical response, so it is worth briefly dwelling on his claims here in a little more detail. To begin with, one needs to get clear what Kvanvig has in mind when he talks of understanding, since many commentators have found the conception of understanding that he targets problematic. The two usages of the term 'understanding' in ordinary language that Kvanvig focuses on--and which he regards as being especially important to epistemology--are > > > when understanding is claimed for some object, such as some subject > matter, and when it involves understanding that something is the case. > (Kvanvig 2003: 189) > > > The first kind of understanding he calls "objectual understanding", the second kind "propositional understanding". In both cases, understanding requires that one successfully grasp how one's beliefs in the relevant propositions cohere with other propositions one believes (e.g., Kvanvig 2003: 192, 197-8). This requirement entails that understanding is directly factive in the case of propositional understanding and indirectly factive in the case of objectual understanding--i.e., the agent needs to have at least mostly true beliefs about the target subject matter in order to be truly said to have objectual understanding of that subject matter. Given that understanding--propositional understanding at any rate--is factive, Kvanvig's argument for why understanding is distinct from knowledge does not relate to this condition (as we will see in a moment, it is standard to argue that understanding is distinct from knowledge precisely because only understanding is non-factive). Instead, Kvanvig notes two key differences between understanding and knowledge: that understanding, unlike knowledge, admits of degrees, and that understanding, unlike knowledge, is compatible with epistemic luck. Most commentators, however, have tended to focus not on these two theses concerning the different properties of knowledge and understanding, but rather on Kvanvig's claim that understanding is (at least indirectly) factive. For example, Elgin (2009; cf. Elgin 1996, 2004; Janvid 2014) and Riggs (2009) argue that it is possible for an agent to have understanding and yet lack true beliefs in the relevant propositions. For example, Elgin (2009) argues that it is essential to treat scientific understanding as non-factive. She cites a number of cases in which science has progressed from one theory to a better theory where, we would say, understanding has increased in the process even though the theories are, strictly speaking at least, false. A different kind of case that Elgin offers concerns scientific idealizations, such as the ideal gas law. Scientists know full well that no actual gas behaves in this way, yet the introduction of this useful fiction clearly improved our understanding of the behavior of actual gasses. For a defense of Kvanvig's view in the light of these charges, see Kvanvig (2009a, 2009b; Carter & Gordon 2014). A very different sort of challenge to Kvanvig's treatment of understanding comes from Brogaard (2005, Other Internet Resources). She argues that Kvanvig's claim that understanding is of greater value than knowledge is only achieved because he fails to give a rich enough account of knowledge. More specifically, Brogaard claims that we can distinguish between objectual and propositional knowledge just as we can distinguish between objectual and propositional understanding. Propositional understanding, argues Brogaard, no more requires coherence in one's beliefs than propositional knowledge, and so the difference in value between the two cannot lie here. Moreover, while Brogaard grants that objectual understanding does incorporate a coherence requirement, this again fails to mark a value-relevant distinction between knowledge and understanding because the relevant counterpart--objectual knowledge (i.e., knowledge of a subject matter)--also incorporates a coherence requirement. So provided that we are consistent in our comparisons of objectual and propositional understanding on the one hand, and objectual and propositional knowledge on the other, Kvanvig fails to make a sound case for thinking that understanding is of greater value than knowledge. Finally, a further challenge to Kvanvig's treatment of knowledge and understanding focuses on his claims regarding epistemic luck, and in particular, his insistence that luck cases show how understanding and propositional knowledge come apart from one another. In order to bring the luck-based challenge into focus, we can distinguish three kinds of views about the relationship between understanding and epistemic luck that are found in the literature: strong compatibilism (e.g., Kvanvig 2003; Rohwer 2014), moderate compatibilism (e.g., Pritchard 2010a: ch. 4) and incompatibilism (e.g., Grimm 2006; Sliwa 2015). Strong compatibilism is the view that understanding is compatible with the varieties of epistemic luck that are generally taken to undermine propositional knowledge. In particular, incompatibilists maintain that understanding is undermined by neither (i) the kind of luck that features in traditional Gettier-style cases (1963) cases, nor with (ii) purely 'environmental luck (e.g., Pritchard 2005) of the sort that features in 'fake barn' cases (e.g., Goldman 1979) where the fact that one's belief could easily be incorrect is a matter of being in an inhospitable epistemic environment. Moderate compatibilism, by contrast, maintains that while understanding is like propositional knowledge in that it is incompatible with the kind of luck that features in traditional Gettier cases, it is nonetheless compatible with environmental epistemic luck. Incompatibilism rejects that either kind of epistemic luck case demonstrates that understanding and propositional knowledge come apart, and so maintains that understanding is incompatible with epistemic luck to the same extent that propositional knowledge is. ## 5. The Value of Knowledge-How The received view in mainstream epistemology, at least since Gilbert Ryle (e.g., 1949), has been to regard knowledge-that and knowledge-how as different epistemic standings, such that knowing how to do something is not simply a matter of knowing propositions, viz., of knowledge-that. If this view--known as anti-intellectualism--is correct, then the value of knowledge-how needn't be accounted for in terms of the value of knowing propositions. Furthermore, if anti-intellectualism is assumed, then--to the extent that there is any analogous 'value problem' for knowledge-how--such a problem needn't materialize as the philosophical problem of determining what it is about knowledge-how that makes it more valuable than mere true belief. Jason Stanley & Timothy Williamson (2001) have, however, influentially resisted the received anti-intellectualist thinking about knowledge-how. On Stanley & Williamson's view--intellectualism--knowledge-how is a kind of propositional knowledge, i.e., knowledge-that, such that (roughly) S knows how to ph iff there is a way w such that S knows that w is a way for S to ph. Accordingly, if Hannah knows how to ride a bike, then this is in virtue of her propositional knowledge--viz., her knowing of some way w that w is the way for her (Hannah) to ride a bike. By reducing in this manner knowledge--how to a kind of knowledge--that, intellectualists such as Stanley have accepted that knowledge-how should have properties characteristic of propositional knowledge, (see, for example, Stanley 2011: 215), of which knowledge-how is a kind. Furthermore, the value of knowledge-how should be able to be accounted for, on intellectualism, with reference to the value of the propositional knowledge that the intellectualist identifies with knowledge-how. In recent work, Carter and Pritchard (2015) have challenged intellectualism on this point. One such example they offer to this end involves testimony and skilled action. For example, suppose that a skilled guitarist tells an amateur how to play a very tricky guitar riff. Carter and Pritchard (2015: 801) argue that though the amateur can uncontroversially acquire testimonial knowledge from the expert that, for some way w that w is the way to play the riff, it might be that the expert, but not novice, knows how to play the riff. Further, they suggest that whilst the amateur is better off, with respect to the aim of playing the riff, than he was prior to gaining the testimonial knowledge he did, he would likewise be better off further--viz., he would have something even more valuable--if he, like the expert, had the lick down cold (something the amateur does not have simply on the basis of his acquired testimonial knowledge) (Ibid: 801). The conclusion Carter and Pritchard draw from this and other similar cases (e.g., 2015: SS3; see also Poston 2016) is that the value of knowledge-how cannot be accounted for with reference to the value of the items of knowledge-that which the intellectualist identifies with knowledge-how If this is right, then if there is a 'value problem' for knowledge-how, we shouldn't expect it to be the problem of determining what is it about certain items of propositional knowledge that makes these more valuable than corresponding mere true beliefs. A potential area for future research is to consider what an analogue value problem for knowledge-how might look like, on an anti-intellectualist framework. According to Carter and Pritchard's diagnosis, the underlying explanation for this difference in value is that knowledge-how (like understanding, as discussed in SS4) essentially involves a kind of cognitive achievement, unlike propositional knowledge, for reasons discussed in SS4. If this diagnosis is correct, then further pressure is arguably placed on the robust virtue epistemologist's 'achievement' solution to the value problems for knowledge-that, as surveyed in SS3. Recall that, according to robust virtue epistemology, the distinctive value of knowledge-that is accounted for in terms of the value of cognitive achievement (i.e., success because of ability) which robust virtue epistemologists take to be essential to propositional knowledge. But, if the presence of cognitive achievement is what accounts for why knowledge-how has a value that is not present in the items of knowledge-that the intellectualist identifies with knowledge-how, this result would seem to stand in tension with the robust virtue epistemologist's insistence that what affords propositional knowledge a value lacked by mere true belief is that the former essentially involves cognitive achievement. ## 6. Other Accounts of the Value of Knowledge John Hawthorne (2004; cf. Stanley 2005; Fantl & McGrath 2002) has argued that knowledge is valuable because of the role it plays in practical reasoning. More specifically, Hawthorne (2004: 30) argues for the principle that one should use a proposition p as a premise in one's practical reasoning only if one knows p. Hawthorne primarily motivates this line of argument by appeal to the lottery case. This concerns an agent's true belief that she holds the losing ticket for a fair lottery with long odds and a large cash prize, a belief that is based solely on the fact that she has reflected on the odds involved. Intuitively, we would say that such an agent lacks knowledge of what she believes, even though her belief is true and even though her justification for what she believes--assessed in terms of the likelihood, given this justification, of her being right--is unusually strong. Moreover, were this agent to use this belief as a premise in her practical reasoning, and so infer that she should throw the ticket away without checking the lottery results in the paper for example, then we would regard her reasoning as problematic. Lottery cases therefore seem to show that justified true belief, no matter how strong the degree of justification, is not enough for acceptable practical reasoning--instead, knowledge is required. Moreover, notice that we can alter the example slightly so that the agent does possess knowledge while at the same time having a weaker justification for what she believes (where strength of justification is again assessed in terms of the likelihood, given this justification, that the agent's belief is true). If the agent had formed her true belief by reading the results in a reliable newspaper, for example, then she would count as knowing the target proposition and can then infer that she should throw the ticket away without criticism. It is more likely, however, that the newspaper has printed the result wrongly than that she should win the lottery. This sort of consideration seems to show that knowledge, even when accompanied by a relatively weak justification, is better (at least when it comes to practical reasoning) than a true belief that is supported by a relatively strong justification but does not amount to knowledge. If this is the right way to think about the connection between knowledge possession and practical reasoning, then it seems to offer a potential response to at least the secondary value problem. A second author who thinks that our understanding of the concept of knowledge can have important ramifications for the value of knowledge is Edward Craig (1990). Craig's project begins with a thesis about the value of the concept of knowledge. Simplifying somewhat, Craig hypothesises that the concept of knowledge is important to us because it fulfills the valuable function of enabling us to identify reliable informants. The idea is that it is clearly of immense practical importance to be able to recognize those from whom we can gain true beliefs, and that it was in response to this need that the concept of knowledge arose. As with Hawthorne's theory, this proposal, if correct, could potentially offer a resolution of at least the secondary value problem. Recently, there have been additional attempts to follow--broadly speaking--Craig's project, for which the value of knowledge is understood in terms of the functional role that 'knowledge' plays in fulfilling our practical needs. The matter of how to identify this functional role has received increasing recent attention. For example, David Henderson (2009), Robin McKenna (2013), Duncan Pritchard (2012) and Michael Hannon (2015) have defended views about the concept of knowledge (or knowledge ascriptions) that are broadly inspired by Craig's favored account of the function of knowledge as identifying reliable informants. A notable rival account, defended by Klemens Kappel (2010), Christoph Kelp (2011, 2014) and Patrick Rysiew (2012; cf. Kvanvig 2012) identifies closure of inquiry as the relevant function. For Krista Lawlor (2013) the relevant function is identified (a la Austin) as that of providing assurance, and for James Beebe (2012), it's expressing epistemic approval/disapproval. In one sense, such accounts are in competition with one another, in that they offer different practical explications of 'knowledge'. However, these accounts all accept (explicitly or tacitly) a more general insight, which is that considerations about the function that the concept of knowledge plays in fulfilling practical needs should inform our theories of the nature and corresponding value of knowledge. This more general point remains controversial in contemporary metaepistemology. For some arguments against supposing that a practical explication of 'knowledge', in terms of some need-fulfilling function, should inform our accounts of the nature or knowledge, see for example Gerken (2015). For a more extreme form of argument in favor of divorcing considerations to do with how and why we use 'knows' from epistemological theorizing altogether, see Hazlett (2010; cf. Turri 2011b). A further and more recent practically oriented approach to the value of knowledge is defended by Grindrod (2019), who considers specifically the ramifications of epistemic contextualism for the value of knowledge. Contextualists maintain that knowledge attributing sentences can vary in truth value across different contexts of utterance. This kind of position about the semantics of knowledge attributions is often motivated by context-shifting cases, such as DeRose's (1992) bank case, which seem to suggest that the a knowledge attribution is truedepends on the epistemic standards (as fixed by practical stakes) of the attributor of the knowledge ascription (see entry on Epistemic Contextualism). Grindrod maintains that if epistemic contextualism is true, then epistemic value(including whatever epistemic value might separate knowledge from mere true belief) should be contextualised. ## 7. Weak and Strong Conceptions of Knowledge Laurence BonJour argues that reflecting on the value of knowledge leads us to reject a prevailing trend in epistemology over the past several decades, namely, fallibilism, or what BonJour calls the "weak conception" of knowledge. BonJour outlines four traditional assumptions about knowledge, understood as roughly justified true belief, which he "broadly" endorses (BonJour 2010: 58-9). First, knowledge is a "valuable and desirable cognitive state" indicative of "full cognitive success". Any acceptable theory of knowledge must "make sense of" knowledge's important value. Second, knowledge is "an all or nothing matter, not a matter of degree". There is no such thing as degrees of knowing: either you know or you don't. Third, epistemic justification comes in degrees, from weak to strong. Fourth, epistemic justification is essentially tied to "likelihood or probability of truth", such that the strength of justification covaries with how likely it makes the truth of the belief in question. On this traditional approach, we are invited to think of justification as measured by how probable the belief is given the reasons or evidence you have. One convenient way to measure probability is to use the decimals in the interval [0, 1]. A probability of 0 means that the claim is guaranteed to be false. A probability of 1 means that the claim is guaranteed to be true. A probability of .5 means that the claim is just as likely to be true as it is to be false. The question then becomes, how probable must your belief be for it to be knowledge? Obviously it must be greater than .5. But how much greater? Suppose we say that knowledge requires a probability of 1--that is, knowledge requires our justification or reasons to guarantee the truth of the belief. Call such reasons conclusive reasons. The strong conception of knowledge says knowledge requires conclusive reasons. We can motivate the strong conception as follows. If the aim of belief is truth, then it makes sense that knowledge would require conclusive reasons, because conclusive reasons guarantee that belief's aim is achieved. The three components of the traditional view of knowledge thus fit together "cohesively" to explain why knowledge is valued as a state of full cognitive success. But all is not well with the strong conception, or so philosophers have claimed over the past several decades. The strong conception seems to entail that we know nearly nothing at all about the material world outside of our own minds or about the past. For we could have had all the reasons we do in fact have, even if the world around us or the past had been different. (Think of Descartes's evil genius.) This conflicts with commonsense and counts against the strong conception. But what is the alternative? The alternative is that knowledge requires reasons that make the belief very likely true, but needn't guarantee it. This is the weak conception of knowledge. Most epistemologists accept the weak conception of knowledge. But BonJour asks a challenging question: what is the "magic" level of probability required by knowledge? BonJour then argues that a satisfactory answer to this question isn't forthcoming. For any point short of 1 would seem arbitrary. Why should we pick that point exactly? The same could be said for a vague range that includes points short of 1--why, exactly, should the vague range extend roughly that far but not further? This leads to an even deeper problem for the weak conception. It brings into doubt the value of knowledge. Can knowledge really be valuable if it is arbitrarily defined? A closely related problem for the weak conception presents itself. Suppose for the sake of argument that we settle on .9 as the required level of probability. Suppose further that you believe Q and you believe R, that Q and R are both true, and that you have reached the .9 threshold for each. Thus the weak conception entails that you know Q, and you know R. Intuitively, if you know Q and you also know R, then you're automatically in a position to know the conjunction Q & R. But the weak conception cannot sustain this judgment. For the probability of the conjunction of two independent claims, such as Q and R, equals the product of their probabilities. (This is the special conjunction rule from probability theory.) In this case, the probability of Q = .9 and the probability of R = .9. So the probability of the conjunction (Q & R) = .9 x .9 = .81, which falls short of the required .9. So the weak conception of knowledge along with a law of probability entail that you're automatically not in a position to know the conjunction (Q & R). BonJour considers this to be "an intuitively unacceptable result", because after all, > > > what is the supposed state of knowledge really worth, if even the > simplest inference from two pieces of knowledge [might] not lead to > further knowledge? (BonJour 2010: 63) > > > BonJour concludes that the weak conception fails to explain the value of knowledge, and thus that the strong conception must be true. He recognizes that this implies that we don't know most of the things we ordinarily say and think that we know. He explains this away, however, partly on grounds that knowledge is the norm of practical reasoning, which creates strong "practical pressure" to confabulate or exaggerate in claiming to know things, so that we can view ourselves as reasoning and acting appropriately, even though usually the best we can do is to approximate appropriate action and reasoning. (BonJour 2010: 75). ## 8. The Value of True Belief So far, in common with most of the contemporary literature in this regard, we have tended to focus on the value of knowledge relative to other epistemic standings. A related debate in this respect, however--one that has often taken place largely in tandem with the mainstream debate on the value of knowledge--has specifically concerned itself with the value of true belief and we will turn now to this issue. Few commentators treat truth or belief as being by themselves valuable (though see Kvanvig 2003: ch. 1), but it is common to treat true belief as valuable, at least instrumentally. True beliefs are clearly often of great practical use to us. The crucial caveat here, of course, concerns the use of the word 'often'. After all, it is also often the case that a true belief might actually militate against one achieving one's goals, as when one is unable to summon the courage to jump a ravine and thereby get to safety, because one knows that there is a serious possibility that one might fail to reach the other side. In such cases it seems that a false belief in one's abilities--e.g., the false belief that one could easily jump the ravine--would be better than a true belief, if the goal in question (jumping the ravine) is to be achieved. Moreover, some true beliefs are beliefs in trivial matters, and in these cases it isn't at all clear why we should value such beliefs at all. Imagine someone who, for no good reason, concerns herself with measuring each grain of sand on a beach, or someone who, even while being unable to operate a telephone, concerns herself with remembering every entry in a foreign phone book. Such a person would thereby gain lots of true beliefs but, crucially, one would regard such truth-gaining activity as rather pointless. After all, these true beliefs do not seem to serve any valuable purpose, and so do not appear to have any instrumental value (or, at the very least, what instrumental value these beliefs have is vanishingly small). It would, perhaps, be better--and thus of greater value--to have fewer true beliefs, and possibly more false ones, if this meant that the true beliefs that one had concerned matters of real consequence. At most, then, we can say that true beliefs often have instrumental value. What about final (or intrinsic) value? One might think that if the general instrumental value of true belief was moot then so too would be the intuitively stronger thesis that true belief is generally finally valuable. Nevertheless, many have argued for such a claim. One condition that seems to speak in favor of this thesis is that as truth seekers we are naturally curious about what the truth is, even when that truth is of no obvious practical import. Accordingly, it could be argued that from a purely epistemic point of view, we do regard all true belief as valuable for its own sake, regardless of what further prudential goals we might have (e.g., Goldman 1999: 3; Lynch 2004: 15-16; Alston 2005: 31; Pritchard 2019; cf. Baehr 2012: 5). Curiosity will only take you so far in this regard, however, since we are only curious about certain truths, not all of them. To return to the examples given a moment ago, no fully rational agent is curious about the measurements of every grain of sand on a given beach, or the name of every person in a random phone book--i.e., no rational person wants to know these truths independently of having some prudential reason for knowing them. Still, one could argue for a weaker claim and merely say that it is prima facie or pro tanto finally good to believe the truth (cf. David 2005; Lynch 2009), where cases of trivial truths such as those just given are simply cases where, all things considered, it is not good to believe the truth. After all, we are familiar with the fact that something can be prima facie or pro tanto finally good without being all-things-considered good. For example, it may be finally good to help the poor and needy, but not all-things-considered good given that helping the poor and needy would prevent you from doing something else which is at present more important (such as saving that child from drowning). At this point one might wonder why it matters so much to (some) epistemologists that true belief is finally valuable. Why not instead just treat true belief as often of instrumental value and leave the matter at that? The answer to this question lies in the fact that many want to regard truth--and thereby true belief--as being the fundamental epistemic goal, in the sense that ultimately it is only truth that is epistemically valuable (so, for example, while justification is epistemically valuable, it is only epistemically valuable because of how it is a guide to truth). Accordingly, if true belief is not finally valuable--and only typically instrumentally valuable--then this seems to downplay the status of the epistemological project. There are a range of options here. The conservative option is to contend that truth is the fundamental goal of epistemology and also contend that true belief is finally valuable--at least in some restricted fashion. Marian David (2001, 2005) falls into this category. In contrast, one might argue that truth is the fundamental goal while at the same time claiming that true belief is not finally valuable. Sosa (see especially 2004, but also 2000a, 2003) seems (almost) to fall into this camp, since he claims that while truth is the fundamental epistemic value, we can accommodate this thought without having to thereby concede that true belief is finally valuable, a point that has been made in a similar fashion by Alan Millar (2011: SS3). Sosa often compares the epistemic domain to other domains of evaluation where the fundamental good of that domain is not finally valuable. So, for example, the fundamental goal of the 'coffee-production' domain may be great tasting coffee, but no-one is going to argue that great tasting coffee is finally valuable. Perhaps the epistemic domain is in this respect like the coffee-production domain? Another line of response against the thesis that true belief is finally valuable is to suggest that this thesis leads to a reductio. Michael DePaul (2001) has notably advanced such an argument. According to DePaul, the thesis that true belief is finally valuable implies that all true beliefs are equally epistemically valuable. Though this latter claim, DePaul argues, is false, as is illustrated by cases where two sets each containing an equal number of true beliefs intuitively differ in epistemic value. Ahlstrom-Vij and Grimm (2013) have criticized DePaul's claim that the thesis that true belief is finally valuable implies that two sets each containing an equal number of true beliefs must not differ in epistemic value. Additionally, Nick Treanor (2014) has criticized the argument for a different reason, which is that (contra DePaul) there is no clear example of two sets which contain the same number of true beliefs. More recently, Xingming Hu (2017) has defended the final value of true belief against DePaul's argument, though Hu argues further that neither Ahlstrom-Vij and Grimm's (2013) nor Treanor's (2014) critique of DePaul's argument is compelling. Another axis on which the debate about the value of true belief can be configured is in terms of whether one opts for an epistemic-value monism or an epistemic-value pluralism--that is, whether one thinks there is only one fundamental epistemic goal, or several. Kvanvig (e.g., 2005) endorses epistemic-value pluralism, since he thinks that there are a number of fundamental epistemic goals, with each of them being of final value. Crucial to Kvanvig's argument is that there are some epistemic goals which are not obviously truth-related--he cites the examples of having an empirically adequate theory, making sense of the course of one's experience, and inquiring responsibly, and more recently, Brent Madison (2017) has argued by appealing to a new evil demon thought experiment, that epistemic justification itself should be included in such a list. This is important because if the range of goals identified were all truth-related, then it would prompt the natural response that such goals are valuable only because of their connection to the truth, and hence not fundamental epistemic goals at all. Presumably, though, it ought also to be possible to make a case for an epistemic-value pluralism where the fundamental epistemic goals were not finally valuable (or, at least, a la Sosa, where one avoided taking a stance on this issue). More precisely, if an epistemic-value monism that does not regard the fundamental epistemic goal as finally valuable can be made palatable, then there seems no clear reason why a parallel view that opted for pluralism in this regard could not similarly be given a plausible supporting story. ## 9. The Value of Extended Knowledge In his essay, "Meno in a Digital World", Pascal Engel (2016) questions whether the original value problem applies to the kind of knowledge or pseudo-knowledge that we get from the internet? (2016: 1). One might initially think that internet and/or digitally acquired knowledge raises no new issues for the value problem. On this line of thought, if digitally acquired (e.g., Googled knowledge, information stored in iPhone apps, etc.) is genuine knowledge, then whatever goes for knowledge more generally, vis-a-vis the value problems surveyed in SSSS1-2, thereby goes for knowledge acquired from our gadgets. However, recent work at the intersection of epistemology and the philosophy of mind suggests there are potentially some new and epistemologically interesting philosophical problems associated with the value of technology-assisted knowledge. These problems correspond with two ways of conceiving of knowledge as extending beyond traditional, intracranial boundaries (e.g., Pritchard 2018). In particular, the kinds of 'extended knowledge' which have potential import for the value of knowledge debate correspond with the extended mind thesis (for discussion on how this thesis interfaces with the hypothesis of extended cognition, see Carter, Kallestrup, Pritchard, & Palermos 2014) and cases involving what Michael Lynch (2016) calls 'neuromedia' intelligence augmentation. According to the extended mind thesis (EMT), mental states (e.g., beliefs) can supervene in part on extra-organismic elements of the world, such as laptops, phones and notebooks, that are typically regarded as 'external' to our minds. This thesis, defended most notably by Andy Clark and David Chalmers (1998), should not be conflated with comparatively weaker and less controversial thesis of content externalism (e.g., Putnam 1975; Burge 1986), according to which the meaning or content of mental states can be fixed by extra-organismic features of our physical or social-linguistic environments. What the proponent of EMT submits is that mental states themselves can partly supervene on extracranial artifacts (e.g., notebooks, iPhones) provided these extracranial artifacts play kinds of functional roles normally played by on-board, biological cognitive processes. For example, to borrow an (adapted) case from Clark and Chalmers (1998), suppose an Alzheimer's patient, 'Otto', begins to outsource the task of memory storage and retrieval to his iPhone, having appreciated that his biological memory is failing. Accordingly, when Otto acquires new information, he automatically records it in his phone's 'memory app', and when he needs old information, he (also, automatically and seamlessly) opens his memory app and looks it up. The iPhone comes to play for Otto the functionally isomorphic role that biological memory used to play for him vis-a-vis the process of memory storage and retrieval. Just as we attribute to normally functioning agents knowledge in virtue of their (non-occurrent) dispositional beliefs stored in biological memory (for example, five minutes ago, you knew that Paris is the capital of France), so, with EMT in play, we should be prepared to attribute knowledge to Otto in virtue of the 'extended' (dispositional) beliefs which are stored in his notebook, provided Otto is as epistemically diligent in encoding and retrieving information as he was before (e.g., Pritchard 2010b). The import EMT has for the value of knowledge debate now takes shape: whatever epistemically valuable properties (if any) are distinctively possessed by knowledge, they must be properties that obtain in Otto's case so as to add value to what would otherwise be mere true (dispositional) beliefs that are stored, extracranially, in Otto's iPhone. But it is initially puzzling just why, and how, this should be. After all, even if we accept the intuition that the epistemic value of traditional (intracranial) knowledge exceeds the value of corresponding true opinion, it is, as Engel (2016), Lynch (2016) and Carter (2017) have noted, at best not clear that this comparative intuition holds in the extended case, where knowledge is possessed simply by virtue of information persisting in digital storage. For example, consider again Plato's solution to the value problem canvassed in SS1: knowledge, unlike true belief, must be 'tied-down' to the truth. Mere true belief is more likely to be lost, which makes it less valuable than knowledge. One potential worry is that extended knowledge, as per EMT--literally, often times, knowledge stored in the cloud--is by its very nature not 'tethered', or for that matter even tetherable, in a way that corresponding items of accurate information which fall short of knowledge are not. Nor arguably does this sort of knowledge in the cloud clearly have the kind of 'stability' that Olsson (2009) claims is what distinguishes knowledge from true opinion (cf., Walker 2019). Perhaps even less does it appear to constitute a valuable cognitive 'achievement', as per robust virtue epistemologists such as Greco and Sosa. EMT is of course highly controversial, (see, for example, Adams & Aizawa 2008), and so one way to sidestep the implications for the value of knowledge debate posed by the possibility of knowledge that is extended via extended beliefs, is to simply resist EMT as a thesis about the metaphysics of mind. However, there are other ways in which the technology-assisted knowledge could have import for the traditional value problems. In recent work, Michael P. Lynch (2016) argues that, given the increase in cognitive offloading coupled with evermore subtle and physically smaller intelligence-augmentation technologies (e.g., Bostrom & Sandberg 2009), it is just a matter of time before the majority of the gadgetry we use for cognitive tasks will be by and large seamless and 'invisible'. Lynch suggests that while coming to know via such mechanisms can make knowledge acquisition much easier, there are epistemic drawbacks. He offers the following thought experiment: > > > NEUROMEDIA: Imagine a society where smartphones are miniaturized and > hooked directly into a person's brain. With a single mental > command, those who have this technology--let's call it > neuromedia--can access information on any subject [...] Now > imagine that an environmental disaster strikes our invented society > after several generations have enjoyed the fruits of neuromedia. The > electronic communication grid that allows neuromedia to function is > destroyed. Suddenly no one can access the shared cloud of information > by thought alone. [...] for the inhabitants of this society, > losing neuromedia is an immensely unsettling experience; it's > like a normally sighted person going blind. They have lost a way of > accessing information on which they've come to rely [...] > Just as overreliance on one sense can weaken the others, so > overdependence on neuromedia might atrophy the ability to access > information in other ways, ways that are less easy and require more > creative effort. (Lynch 2016: 1-6) > > > One conclusion Lynch has drawn from such thought experiments is that understanding has a value that mere knowledge lacks, a position we've seen has been embraced for different reasons in SS4 by Kvanvig and others. A further conclusion, advanced by Pritchard (2013) and Carter (2017), concerns the extent to which the acquisition of knowledge involves 'epistemic dependence'--viz., dependence on factors outwith one's cognitive agency. They argue that the greater the scope of epistemic dependence, the more valuable it becomes to cultivate virtues like intellectual autonomy that regulate the appropriate reliance and outsourcing (e.g., on other individuals, technology, medicine, etc.) while at the same time maintaining one's intellectual self-direction.
value-pluralism	## 1. Some Preliminary Clarifications ### 1.1 Foundational and Non-foundational Pluralism It is important to clarify the levels at which a moral theory might be pluralistic. Let us distinguish between two levels of pluralism: foundational and non-foundational. Foundational pluralism is the view that there are plural moral values at the most basic level--that is to say, there is no one value that subsumes all other values, no one property of goodness, and no overarching principle of action. Non-foundational pluralism is the view that there are plural values at the level of choice, but these apparently plural values can be understood in terms of their contribution to one more fundamental value.[3] Judith Jarvis Thomson, a foundational pluralist, argues that when we say that something is good we are never ascribing a property of goodness, rather we are always saying that the thing in question is good in some way. If we say that a fountain pen is good we mean something different from when we say that a logic book is good, or a film is good. As Thomson puts it, all goodness is a goodness in a way. Thomson focusses her argument on Moore, who argues that when we say 'x is good' we do not mean 'x is conducive to pleasure', or 'x is in accordance with a given set of rules' and nor do we mean anything else that is purely descriptive. As Moore points out, we can always query whether any purely descriptive property really is good--so he concludes that goodness is simple and unanalyzable.[4] Moore is thus a foundational monist, he thinks that there is one non-natural property of goodness, and that all good things are good in virtue of having this property. Thomson finds this preposterous. In Thomson's own words: > > Moore says that the question he will be addressing himself to in what > follows is the question 'What is good?', and he rightly > thinks that we are going to need a bit of help in seeing exactly what > question he is expressing in those words. He proposes to help us by > drawing attention to a possible answer to the question he is > expressing--that is, to something that would be an answer to it, > whether or not it is the correct answer to it. Here is what he offers > us: "Books are good." Books are good? What would you mean > if you said 'Books are good'? Moore, however, goes > placidly on: "though [that would be] an answer obviously false; > for some books are very bad indeed". Well some books are bad to > read or to look at, some are bad for use in teaching philosophy, some > are bad for children. What sense could be made of a person who said, > "No. no. I meant that some books are just plain bad > things"? (Thomson 1997, pp. 275-276) > According to Thomson there is a fundamental plurality of ways of being good. We cannot reduce them to something they all have in common, or sensibly claim that there is a disjunctive property of goodness (such that goodness is 'goodness in one of the various ways'. Thomson argues that that could not be an interesting property as each disjunct is truly different from every other disjunct. Thomson (1997), p. 277). Thomson is thus a foundational pluralist--she does not think that there is any one property of value at the most basic level. W.D. Ross is a foundational pluralist in a rather complex way. Most straightforwardly, Ross thinks that there are several prima facie duties, and there is nothing that they all have in common: they are irreducibly plural. This is the aspect of Ross's view that is referred to with the phrase, 'Ross-style pluralism'. However, Ross also thinks that there are goods in the world (justice and pleasure, for example), and that these are good because of some property they share. Goodness and rightness are not reducible to one another, so Ross is a pluralist about types of value as well as about principles. Writers do not always make the distinction between foundational and other forms of pluralism, but as well as Thomson and Ross, at least Bernard Williams (1981), Charles Taylor (1982), Stuart Hampshire (1983), Charles Larmore (1987), John Kekes (1993), Michael Stocker (1990 and 1997), David Wiggins (1997), Christine Swanton (2001), and Jonathan Dancy (2004) are all committed to foundational pluralism. Non-foundational pluralism is less radical--it posits a plurality of bearers of value. In fact, almost everyone accepts that there are plural bearers of value. This is compatible with thinking that there is only one ultimate value. G.E. Moore (1903), Thomson's target, is a foundational monist, but he accepts that there are non-foundational plural values. Moore thinks that there are many different bearers of value, but he thinks that there is one property of goodness, and that it is a simple non-natural property that bearers of value possess in varying degrees. Moore is clear that comparison between plural goods proceeds in terms of the amount of goodness they have. This is not to say that the amount of goodness is always a matter of simple addition. Moore thinks that there can be organic unities, where the amount of goodness contributed by a certain value will vary according to the combination of values such as love and friendship. Thus Moore's view is pluralist at the level of ordinary choices, and that is not without interesting consequences. (We return to the issue of how a foundational monist like Moore can account for organic unities in section 3.) Mill, a classic utilitarian, could be and often has been interpreted as thinking that there are irreducibly different sorts of pleasure. Mill argues that there are higher and lower pleasures, and that the higher pleasures (pleasures of the intellect as opposed to the body) are superior, in that higher pleasures can outweigh lower pleasures regardless of the quantity of the latter. As Mill puts it: "It is quite compatible with the principle of utility to recognize the fact, that some kinds of pleasure are more desirable and more valuable than others." (2002, p. 241). On the foundational pluralist interpretation of Mill, there is not one ultimate good, but two (at least): higher and lower pleasures. Mill goes on to give an account of what he means: > > If I am asked, what I mean by difference in quality in pleasures, or > what makes one pleasure more valuable than another, merely as a > pleasure, except its being greater in amount, there is but one > possible answer. Of two pleasures, if there be one to which all or > almost all who have experience of both give a decided preference, > irrespective of any feeling of moral obligation to prefer it, that is > the more desirable pleasure. (2002, p. 241). > The passage is ambiguous, it is not clear what role the expert judges play in the theory. On the pluralist interpretation of this passage we must take Mill as intending the role of the expert judges as a purely heuristic device: thinking about what such people would prefer is a way of discovering which pleasures are higher and which are lower, but the respective values of the pleasure is independent of the judges' judgment. On a monist interpretation we must understand Mill as a preference utilitarian: the preferences of the judges determine value. On this interpretation there is one property of value (being preferred by expert judges) and many bearers of value (whatever the judges prefer).[5] Before moving on, it is worth noting that a theory might be foundationally monist in its account of what values there are, but not recommend that people attempt to think or make decisions on the basis of the supervalue. A distinction between decision procedures and criteria of right has become commonplace in moral philosophy. For example, a certain form of consequentialism has as its criterion of right action: act so as to maximize good consequences. This might invite the complaint that an agent who is constantly trying to maximize good consequences will often, in virtue of that fact, fail to do so. Sometimes concentrating too hard on the goal will make it less likely that the goal is achieved. A distinction between decision procedure and right action can provide a response--the consequentialist can say that the criterion of right action, (act so as to maximize good consequences) is not intended as a decision procedure--the agent should use whichever decision procedure is most likely to result in success. If, then, there is some attraction or instrumental advantage from the point of view of a particular theory to thinking in pluralist terms, then it is open to that theory to have a decision procedure that deals with apparently plural values, even if the theory is monist in every other way. [6] ### 1.2 A Purely Verbal Dispute? One final clarification about different understandings of pluralism ought to be made. There is an ambiguity between the name for a group of values and the name for one unitary value. There are really two problems here: distinguishing between the terms that refer to groups and the terms that refer to individuals (a merely linguistic problem) and defending the view that there really is a candidate for a unitary value (a metaphysical problem). The linguistic problem comes about because in natural language we may use a singular term as 'shorthand': conceptual analysis may reveal that surface grammar does not reflect the real nature of the concept. For example, we use the term 'well-being' as if it refers to one single thing, but it is not hard to see that it may not. 'Well-being' may be a term that we use to refer to a group of things such as pleasure, health, a sense of achievement and so on. A theory that tells us that well-being is the only value may only be nominally monist. The metaphysical question is more difficult, and concerns whether there are any genuinely unitary values at all. The metaphysical question is rather different for naturalist and non-naturalist accounts of value. On Moore's non-naturalist account, goodness is a unitary property but it is not a natural property: it is not empirically available to us, but is known by a special faculty of intuition. It is very clear that Moore thinks that goodness is a genuinely unitary property: > > 'Good', then, if we mean by it that quality which we > assert to belong to a thing, when we say that the thing is good, is > incapable of any definition, in the most important sense of that word. > The most important sense of 'definition' is that in which > a definition states what are the parts which invariably compose a > certain whole; and in this sense 'good' has no definition > because it is simple and has no parts. (Moore, 1903, p. 9) > The question of whether there could be such a thing is no more easy or difficult than any question about the existence of non-natural entities. The issue of whether the entity is genuinely unitary is not an especially difficult part of that issue. By contrast, naturalist views do face a particular difficulty in giving an account of a value that is genuinely unitary. On the goods approach, for example, the claim must be that there is one good that is genuinely singular, not a composite of other goods. So for example, a monist hedonist must claim that pleasure really is just one thing. Pleasure is a concept we use to refer to something we take to be in the natural world, and conceptual analysis may or may not confirm that pleasure really is one thing. Perhaps, for example, we refer both to intellectual and sensual experiences as pleasure. Or, take another good often suggested by proponents of the goods approach to value, friendship. It seems highly unlikely that there is one thing that we call friendship, even if there are good reasons to use one umbrella concept to refer to all those different things. Many of the plausible candidates for the good seem plausible precisely because they are very broad terms. If a theory is to be properly monist then, it must have an account of the good that is satisfactorily unitary. The problem applies to the deontological approach to value too. It is often relatively easy to determine whether a principle is really two or more principles in disguise--the presence of a conjunction or a disjunction, for example, is a clear giveaway. However, principles can contain terms that are unclear. Take for example a deontological theory that tells us to respect friendship. As mentioned previously, it is not clear whether there is one thing that is friendship or more than one, so it is not clear whether this is one principle about one thing, or one principle about several things, or whether it is really more than one principle. Questions about what makes individuals individuals and what the relationship is between parts and wholes have been discussed in the context of metaphysics but these issues have not been much discussed in the literature on pluralism and monism in moral philosophy. However, these issues are implicit in discussions of the well-being, nature of friendship and pleasure, and in the literature on Kant's categorical imperative, or on Aristotelian accounts of eudaimonia. Part of an investigation into the nature of these things is an investigation into whether there really is one thing or not. [7] The upshot of this brief discussion is that monists must be able to defend their claim that the value they cite is genuinely one value. There may be fewer monist theories than it first appears. Further, the monist must accept the implications of a genuinely monist view. As Ruth Chang points out, (2015, p. 24) the simpler the monist's account of the good is, the less likely it is that the monist will be able to give a good account of the various complexities in choice that seem an inevitable part of our experience of value. But on the other hand, if the monist starts to admit that the good is complex, the view gets closer and closer to being a pluralist view. However, the dispute between monists and pluralists is not merely verbal: there is no prima facie reason to think that there are no genuinely unitary properties, goods or principles. ## 2. The Attraction of Pluralism If values are plural, then choices between them will be complex. Pluralists have pressed the point that choices are complex, and so we should not shy away from the hypothesis that values are plural. In brief, the attraction of pluralism is that it seems to allow for the complexity and conflict that is part of our moral experience. We do not experience our moral choices as simple additive puzzles. Pluralists have argued that there are incommensurabilities and discontinuities in value comparisons, value remainders (or residues) when choices are made, and complexities in appropriate responses to value. Recent empirical work confirms that our ethical experience is of apparently irreducible plural values. (See Gill and Nichols, 2008.) ### 2.1 Discontinuities John Stuart Mill suggested that there are higher and lower pleasures (Mill 2002, p. 241), the idea being that the value of higher and lower pleasures is measured on different scales. In other words, there are discontinuities in the measurement of value. As mentioned previously, it is unclear whether we should interpret Mill as a foundational pluralist, but the notion of higher and lower pleasures is a very useful one to illustrate the attraction of thinking that there are discontinuities in value. The distinction between higher and lower pleasures allows us to say that no amount of lower pleasures can outweigh some amount of higher pleasures. As Mill puts it, it is better to be an unhappy human being than a happy pig. In other words, the distinction allows us to say that there are discontinuities in value addition. As James Griffin (1986, p. 87) puts it: "We do seem, when informed, to rank a certain amount of life at a very high level above any amount of life at a very low level." Griffin's point is that there are discontinuities in the way we rank values, and this suggests that there are different values.[8] The phenomenon of discontinuities in our value rankings seems to support pluralism: if higher pleasures are not outweighed by lower pleasures, that suggests that they are not the same sort of thing. For if they were just the same sort of thing, there seems to be no reason why lower pleasures will not eventually outweigh higher pleasures. Ruth Chang argues that values form organic unities, which is to say that adding more of one value to a situation may not make it better overall, the balance of values in the situation is also important. For example, a life where extra pleasure is added, and all other value contributions left untouched, may not be better overall, because now the pleasure is unbalanced (Chang 2002). As Chang puts it, the Pareto principle of improvement (if something is better along one dimension and at least as good on all the others it will be better overall) does not hold here. The most extreme form of discontinuity is incommensurability or incomparability, when two values cannot be ranked at all. Pluralists differ on whether pluralism entails incommensurabilities, and on what incommensurability entails for the possibility of choice. Griffin denies that pluralism entails incommensurability (Griffin uses the term incomparability) whereas other pluralists embrace incommensurability, but deny that it entails that rational choice is impossible. Some pluralists accept that there are sometimes cases where incommensurability precludes rational choice. We shall return to these issues in Section 4. ### 2.2 Value Conflicts and Rational Regret Michael Stocker (1990) and Bernard Williams (1973 and 1981) and others have argued that it can be rational to regret the outcome of a correct moral choice. That is, even when the right choice has been made, the rejected option can reasonably be regretted, and so the choice involves a genuine value conflict. This seems strange if the options are being compared in terms of a supervalue. How can we regret having chosen more rather than less of the same thing? Yet the phenomenon seems undeniable, and pluralism can explain it. If there are plural values, then one can rationally regret not having chosen something which though less good, was different. It is worth noting that the pluralist argument is not that all cases of value conflict point to pluralism. There may be conflicts because of ignorance, for example, or because of irrationality, and these do not require positing plural values. Stocker argues that there are (at least) two sorts of value conflict that require plural values. The first is conflict that involves choices between doing things at different times. Stocker argues that goods become different values in different temporal situations, and the monist cannot accommodate this thought. The other sort of case (which Williams also points to) is when there is a conflict between things that have different advantages and disadvantages. The better option may be better, but it does not 'make up for' the lesser option, because it isn't the same sort of thing. Thus there is a remainder--a moral value that is lost in the choice, and that it is rational to regret. Both Martha Nussbaum (1986) and David Wiggins (1980) have argued for pluralism on the grounds that only pluralism can explain akrasia, or weakness of will. An agent is said to suffer from weakness of will when she knowingly chooses a less good option over a better one. On the face of it, this is a puzzling thing to do--why would someone knowingly do what they know to be worse? A pluralist has a plausible answer--when the choice is between two different sorts of value, the agent is preferring A to B, rather than preferring less of A to more of A. Wiggins explains the akratic choice by suggesting that the agent is 'charmed' by some aspect of the choice, and is swayed by that to choose what she knows to be worse overall (Wiggins 1980, p. 257). However, even Michael Stocker, the arch pluralist, does not accept that this argument works. As Stocker points out, Wiggins is using a distinction between a cognitive and an affective element to the choice, and this distinction can explain akrasia on a monist account of value too. Imagine that a monist hedonist agent is faced with a choice between something that will give her more pleasure and something that will give her less pleasure. The cognitive aspect to the choice is clear--the agent knows that one option is more pleasurable than the other, and hence on her theory better. However, to say that the agent believes that more pleasure is better is not to say that she will always be attracted to the option that is most pleasurable. She may, on occasion, be attracted to the option that is more unusual or interesting. Hence she may act akratically because she was charmed by some aspect of the less good choice--and as Stocker says, there is no need to posit plural values to make sense of this--being charmed is not the same as valuing. (Stocker 1990, p.219). ### 2.3 Appropriate Responses to Value Another argument for pluralism starts from the observation that there are many and diverse appropriate responses to value. Christine Swanton (2003, ch. 2) and Elizabeth Anderson (1993) both take this line. As Swanton puts it: > > According to value centered monism, the rightness of moral > responsiveness is determined entirely by degree or strength of > value...I shall argue, on the contrary, that just how things are > to be pursued, nurtured, respected, loved, preserved, protected, and > so forth may often depend on further general features of those things, > and their relations to other things, particularly the moral agent. > (Swanton 2003, p. 41). > The crucial thought is that there are various bases of moral responsiveness, and these bases are irreducibly plural. A monist could argue that there are different appropriate responses to value, but the monist would have to explain why there are different appropriate responses to the same value. Swanton's point is that the only explanation the monist has is that different degrees of value merit different responses. According to Swanton, this does not capture what is really going on when we appropriately honor or respect a value rather than promoting it. Anderson and Swanton both argue that the complexity of our responses to value can only be explained by a pluralistic theory. Elizabeth Anderson argues that it is a mistake to understand moral goods on the maximising model. She uses the example of parental love (Anderson 1997, p. 98). Parents should not see their love for their children as being directed towards an "aggregate child collective". Such a view would entail that trade offs were possible, that one child could be sacrificed for another. On Anderson's view we can make rational choices between conflicting values without ranking values: "...choices concerning those goods or their continued existence do not generally require that we rank their values on a common scale and choose the more valuable good; they require that we give each good its due" (Anderson 1997, p. 104). ## 3. Monist Solutions The last section begins by saying that if foundational values are plural, then choices between them will be complex. It is clear that our choices are complex. However, it would be invalid to conclude from that that values are plural--the challenge for monists is to explain how they too can make sense of the complexity of our value choices. ### 3.1 Different Bearers of Value One way for monists to make sense of complexity in value choice is to point out that there are different bearers of value, and this makes a big difference to the experience of choice. (See Hurka, 1996; Schaber, 1999; Klocksiem 2011). Here is the challenge to monism in Michael Stocker's words (Stocker, 1990, p. 272): "[if monism is true] there is no ground for rational conflict because the better option lacks nothing that would be made good by the lesser." In other words, there are no relevant differences between the better and worse options except that the better option is better. Thomas Hurka objects that there can be such differences. For example, in a choice between giving five units of pleasure to A and ten units to B, the best option (more pleasure for B) involves giving no pleasure at all to A. So there is something to rationally regret, namely, that A had no pleasure. The argument can be expanded to deal with all sorts of choice situation: in each situation, a monist can say something sensible about an unavoidable loss, a loss that really is a loss. If, of two options one will contribute more basic value, the monist must obviously choose that one. But the lesser of the options may contribute value via pleasure, while the superior option contributes value via knowledge, and so there is a loss in choosing the option with the greater value contribution--a loss in pleasure-- and it is rational for us to regret this. There is one difficulty with this answer. The loss described by Hurka is not a moral loss, and so the regret is not moral regret. In Hurka's example, the relevant loss is that A does not get any pleasure. The agent doing the choosing may be rational to regret this if she cares about A, or even if she just feels sorry for A, but there has been no moral loss, as 'pleasure for A' as opposed to pleasure itself is not a moral value. According to the view under consideration, pleasure itself is what matters morally, and so although A's pleasure matters qua pleasure, the moral point of view takes B's pleasure into account in just the same way, and there is nothing to regret, as there is more pleasure than there would otherwise have been. Stocker and Williams would surely insist that the point of their argument was not just that there is a loss, but that there is a moral loss. The monist cannot accommodate that point, as the monist can only consider the quantity of the value, not its distribution, and so we are at an impasse. However, the initial question was whether the monist has succeeded in explaining the phenomenon of 'moral regret', and perhaps Hurka has done that by positing a conflation of moral and non-moral regret in our experience. From our point of view, there is regret, and the monist can explain why that is without appealing to irrationality. On the other hand the monist cannot appeal to anything other than quantity of value in appraising the morality of the situation. So although Hurka is clearly right in so far as he is saying that a correct moral choice can be regretted for non-moral reasons, he can go no further than that. ### 3.2 Diminishing Marginal Value Another promising strategy that the monist can use in order to explain the complexity in our value choices is the appeal to 'diminishing marginal value'. The value that is added to the sum by a source of value will tend to diminish after a certain point--this phenomenon is known as diminishing marginal value (or, sometimes, diminishing marginal utility). Mill's higher and lower pleasures, which seem to be plural values, might be accommodated by the monist in this way. The monist makes sense of discontinuities in value by insisting on the distinction between sources of value, which are often ambiguously referred to as 'values', and the super value. Using a monist utilitarian account of value, we can distinguish between the non-evaluative description of options, the intermediate description, and the evaluative description as follows: \| \| \| \| \| --- \| --- \| --- \| \| Non-evaluative description of option \| Intermediate description of option \| Evaluative description of option \| \| Painting a picture \| - \| Producing x units of beauty \| - \| Producing y units of value \| \| Reading a book \| - \| Producing x units of knowledge \| - \| Producing y units of value \| On this account, painting produces beauty, and beauty (which is not a value but the intermediate source of value) produces value. Similarly, reading a book produces knowledge, and gaining knowledge produces value. Now it should be clear how the monist can make sense of phenomena like higher and lower pleasures. The non-evaluative options (e.g. eating donuts) have diminishing marginal non-basic value. On top of that, the intermediate effect, or non-basic value, (e.g. experiencing pleasure) can have a diminishing contribution to value. Varying diminishing marginal value in these cases is easily explained psychologically. It is just the way we are--we get less and less enjoyment from donuts as we eat more and more (at least in one sitting). However, we may well get the same amount of enjoyment from the tenth Johnny Cash song that we did from the first. In order to deal with the higher and lower pleasures case the monist will have to argue that pleasures themselves can have diminishing marginal utility--the monist can argue that gustatory pleasure gets boring after a while, and hence contributes less and less to the super value--well being, or whatever it is.[9] This picture brings us back to the distinction between foundational and non-foundational pluralism. Notice that the monist theories being imagined here are foundationally monist, because they claim that there is fundamentally one value, such as pleasure, and they are pluralist at the level of ordinary choice because they claim that there are intermediate values, such as knowledge and beauty, which are valuable because of the amount of pleasure they produce (or realize, or contain--the exact relationship will vary from theory to theory). ### 3.3 Theoretical Virtues The main advantage of pluralism is that it seems true to our experience of value. We experience values as plural, and pluralism tells is that values are indeed plural. The monist can respond, as we have seen, that there are ways to explain the apparent plurality of values without positing fundamentally plural values. Another, complementary strategy that the monist can pursue is to argue that monism has theoretical virtues that pluralism lacks. In general, it seems that theories should be as simple and coherent as possible, and that other things being equal, we should prefer a more coherent theory to a less coherent one. Thus so long as monism can make sense of enough of our intuitive judgments about the nature of value, then it is to be preferred to pluralism because it does better on the theoretical virtue of coherence. Another way to put this point is in terms of explanation. The monist can point out that the pluralist picture lacks explanatory depth. It seems that a list of values needs some further explanation: what makes these things values? (See Bradley 2009, p.16). The monist picture is superior, because the monist can provide an explanation for the value of the (non-foundational) plurality of values: these things are values because they contribute to well-being, or pleasure, or whatever the foundational monist value is. (See also the discussion of this in the entry on value theory). Patricia Marino argues against this strategy (2015). She argues that 'systematicity' (the idea that it is better to have fewer principles) is not a good argument in favour of monism. Marino points out that explanation in terms of fewer fundamental principles is not necessarily better explanation. If there are plural values, then the explanation that appeals to plural values is a better one, in the sense that it is the true one: it doesn't deny the plurality of values. Even if we could give a monist explanation without having to trade off against our pluralist intuitions, Marino argues, we have no particular reason to think that explanations appealing to fewer principles are superior. ### 3.4 Preference Satisfaction Views There is a different account of value that we ought to consider here: the view that value consists in preference or desire satisfaction. On this view, knowledge and pleasure and so on are valuable when they are desired, and if they are not desired anymore they are not valuable anymore. There is no need to appeal to complicated accounts of diminishing marginal utility: it is uncontroversial that we sometimes desire something and sometimes don't. Thus complexities in choices are explained by complexities in our desires, and it is uncontroversial that our desires are complex. Imagine a one person preference satisfaction account of value that says simply that what is valuable is what P desires. Apparently this view is foundationally monist: there is only one thing that confers value (being desired by P), yet at the non-foundational level there are many values (whatever P desires). Let us say that P desires hot baths, donuts and knowledge. The structure of P's desires is such that there is a complicated ranking of these things, which will vary from circumstance to circumstance. The ranking is not explained by the value of the objects,rather, her desire explains the ranking and determines the value of the objects. So it might be that P sometimes desires a hot bath and a donut equally, and cannot choose between them; it might be that sometimes she would choose knowledge over a hot bath and a donut, but sometimes she would choose a hot bath over knowledge. On James Griffin's slightly more complex view, well-being consist in the fulfillment of informed desire, and Griffin points out that his view can explain discontinuities in value without having to appeal to diminishing marginal utility: > > there may well turn out to be cases in which, when informed, I want, > say, a certain amount of one thing more than any amount of another, > and not because the second thing cloys, and so adding to it merely > produces diminishing marginal values. I may want it even though the > second thing does not, with addition, lose its value; it may be that I > think that no increase in that kind of value, even if constant and > positive, can overtake a certain amount of this kind of value. (1986, > p. 76). > This version of foundational monism/normative pluralism escapes some of the problems that attend the goods approach. First, this view can account for deep complexities in choice. The plural goods that P is choosing between do not seem merely instrumental. Donuts are not good because they contribute to another value, and P does not desire donuts for any reason other than their donuty nature. On this view, if it is hard to choose between donuts and hot baths it is because of the intrinsic nature of the objects. The key here is that value is conferred by desire, not by contribution to another value. Second, this view can accommodate incomparabilities: if P desires a hot bath because of its hot bathy nature, and a donut because of its donuty nature, she may not be able to choose between them. However, it is not entirely clear that a view like Griffin's is genuinely monist at the foundational level: the question arises, what is constraining the desires that qualify as value conferring? If the answer is 'nothing', then the view seems genuinely monist, but is probably implausible. Unconstrained desire accounts of value seem implausible because our desires can be for all sorts of things--we may desire things that are bad for us, or we may desire things because of some mistake we have made. If the answer is that there is something constraining the desires that count as value conferring, then of course the question is, 'what?' Is it the values of the things desired? A desire satisfaction view that restricts the qualifying desires must give an account of what restricts them, and obviously, the account may commit the view to foundational pluralism. Griffin addresses this question at the very beginning of his book on well being (Griffin, 1986, ch.2).[10] As he puts it, > > The danger is that desire accounts get plausible only by, in effect, > ceasing to be desire accounts. We had to qualify desire with informed, > and that gave prominence to the features or qualities of the objects > of desire, and not to the mere existence of desire. (1986, p. 26). > Griffin's account of the relationship between desire and value is subtle, and (partly because Griffin himself does not distinguish between foundational and normative pluralism) it is difficult to say whether his view is foundationally pluralist or not. Griffin argues that it is a mistake to see desire as a blind motivational force--we desire things that we perceive in a favorable light- we take them to have a desirability feature. When we try to explain what involved in seeing things in a favorable light, we cannot, according to Griffin, separate understanding from desire: > > ...we cannot, even in the case of a desirability feature such as > accomplishment, separate understanding and desire. Once we see > something as 'accomplishment', as 'giving weight and > substance to our lives', there is no space left for desire to > follow along in a secondary subordinate position. Desire is not blind. > Understanding is not bloodless. Neither is the slave of the other. > There is no priority. (1986, p. 30) > This suggests that the view is indeed pluralist at the foundation--values are not defined entirely by desire, but partly by other features of the situation, and so at the most fundamental level there is more than one value making feature. Griffin himself says that "the desire account is compatible with a strong form of pluralism about values" (p. 31). The question whether or not Griffin is a foundational pluralist is not pursued further here. The aim in this section is to show first, that monist preference satisfaction accounts of value may have more compelling ways of explaining complexities in value comparison than monist goods approaches, but second, to point out that any constrained desire account may well actually be foundationally pluralist. As soon as something is introduced to constrain the desires that qualify as value conferring, it looks as though another value is operating. ## 4. Pluralism and Rational Choice The big question facing pluralism is whether rational choices can be made between irreducibly plural values. Irreducible plurality appears to imply incommensurability--that is to say, that there is no common measure which can be used to compare two different values. (See the entry on incommensurable values.) Value incommensurability seems worrying: if values are incommensurable, then either we are forced into an ad hoc ranking, or we cannot rank the values at all. Neither of these are very appealing options. However, pluralists reject this dilemma. Bernard Williams argues that it is a mistake to think that pluralism implies that comparisons are impossible. He says: > > There is one motive for reductivism that does not operate simply on > the ethical, or on the non-ethical, but tends to reduce every > consideration to one basic kind. This rests on an assumption about > rationality, to the effect that two considerations cannot be > rationally weighed against each other unless there is a common > consideration in terms of which they can be compared. This assumption > is at once very powerful and utterly baseless. Quite apart from the > ethical, aesthetic considerations can be weighed against economic ones > (for instance) without being an application of them, and without their > both being an example of a third kind of consideration. (Williams > 1985, p. 17) > Making a similar point, Ruth Chang points out that incommensurability is often conflated with incomparability. She provides clear definitions of each: incommensurability is the lack of a common unit of value by which precise comparisons can be made. Two items are incomparable, if there is no possible relation of comparison, such as 'better than', or 'as good as' (1997, Introduction). Chang points out that incommensurability is often thought to entail incomparability, but it does not. Defenders of pluralism have used various strategies to show that it is possible to make rational choices between plural values. ### 4.1 Practical Wisdom The pluralist's most common strategy in the face of worries about choices between incommensurable values is to appeal to practical wisdom--the faculty described by Aristotle--a faculty of judgment that the wise and virtuous person has, which enables him to see the right answer. Practical wisdom is not just a question of being able to see and collate the facts, it goes beyond that in some way--the wise person will see things that only a wise person could see. So plural values can be compared in that a wise person will 'just see' that one course of action rather than another is to be taken. This strategy is used (explicitly or implicitly) by McDowell (1979), Nagel (1979), Larmore (1987), Skorupski (1996), Anderson (1993 and 1997) Wiggins (1997 and 1998), Chappell (1998), Swanton (2003). Here it is in Nagel's words: > > Provided one has taken the process of practical justification as far > as it will go in the course of arriving at the conflict, one may be > able to proceed without further justification, but without > irrationality either. What makes this possible is > judgment--essentially the faculty Aristotle described as > practical wisdom, which reveals itself over time in individual > decisions rather than in the enunciation of general principles. (1979, > p. 135) > The main issue for this solution to the comparison problem is to come up with an account of what practical wisdom is. It is not easy to understand what sort of thing the faculty of judgment might be, or how it might work. Obviously pluralists who appeal to this strategy do not want to end up saying that the wise judge can see which of the options has more goodness, as that would constitute collapsing back into monism. So the pluralist has to maintain that the wise judge makes a judgment about what the right thing to do is without making any quantitative judgment. The danger is that the faculty seems entirely mysterious: it is a kind of magical vision, unrelated to our natural senses. As a solution to the comparison problem, the appeal to practical wisdom looks rather like way of shifting the problem to another level. Thus the appeal to practical wisdom cannot be left at that. The pluralist owes more explanation of what is involved in practical wisdom. What follows below are various pluralists' accounts of how choice between plural values is possible, and whether such choice is rational. ### 4.2 Super Scales One direction that pluralists have taken is to argue that although values are plural, there is nonetheless an available scale on which to rank them. This scale is not rationalized by something that the values have in common (that would be monism), but by something over and above the values, which is not itself a super value. Williams sometimes writes as if this is his intention, as do Griffin (1986 and 1997), Stocker (1990), Chang (1997 and 2004), Taylor (1982 and 1997). James Griffin (1986) develops this suggestion in his discussion of plural prudential values. According to Griffin, we do not need to have a super-value to have super-scale. Griffin says: > > ...it does not follow from there being no super-value that there > is no super-scale. To think so would be to misunderstand how the > notion of 'quantity' of well-being enters. It enters > through ranking; quantitative differences are defined on qualitative > ones. The quantity we are talking about is 'prudential > value' defined on informed rankings. All that we need for the > all-encompassing-scale is the possibility of ranking items on the > basis of their nature. And we can, in fact, rank them in that way. We > can work out trade-offs between different dimensions of pleasure or > happiness. And when we do, we rank in a strong sense: not just choose > one rather than the other, but regard it as worth more. That is the > ultimate scale here: worth to one's life. (Griffin 1986, p. 90) > This passage is slightly hard to interpret (for more on why see my earlier discussion of Griffin in the section on preference satisfaction accounts). On one interpretation, Griffin is in fact espousing a sophisticated monism. The basic value is 'worth to one's life', and though it is important to talk about non-basic values, such as the different dimensions of pleasure and happiness, they are ultimately judged in terms of their contribution to the worth of lives. The second possible interpretation takes Griffin's claim that worth to life is not a supervalue seriously. On this interpretation, it is hard to see what worth to life is, if not a supervalue. Perhaps it is only a value that we should resort to when faced with incomparabilities. However, this interpretation invites the criticism that Griffin is introducing a non-moral value, perhaps prudential value, to arbitrate when moral values are incommensurable. In other words, we cannot decide between incommensurable values on moral grounds, so we should decide on prudential grounds. This seems reasonable when applied to incommensurabilities in aesthetic values. One might not be able to say whether Guernica is better than War and Peace, but one might choose to have Guernica displayed on the wall because it will impress one's friends, or because it is worth more money, or even because one just enjoys it more. In the case of moral choices this is a less convincing strategy: it introduces a level of frivolity into morality that seems out of place. Stocker's main strategy is to argue that values are plural, and comparisons are made, so it must be possible to make rational comparisons. He suggests that a "higher level synthesizing category" can explain how comparisons are made (1990, p. 172). According to Stocker these comparisons are not quantitative, they are evaluative: > > Suppose we are trying to choose between lying on a beach and > discussing philosophy--or more particularly, between the pleasure > of the former and the gain in understanding from the latter. To > compare them we may invoke what might be called a higher-level > synthesizing category. So, we may ask which will conduce to a more > pleasing day, or to a day that is better spent. Once we have fixed > upon the higher synthesizing category, we can often easily ask which > option is better in regard to that category and judge which to choose > on the basis of that. Even if it seems a mystery how we might > 'directly' compare lying on the beach and discussing > philosophy, it is a commonplace that we do compare them, e.g. in > regard to their contribution to a pleasing day. (Stocker 1990, p. 72) > Stocker claims that goodness is just the highest level synthesizing category, and that lower goods are constitutive means to the good. Ruth Chang's approach to comparisons of plural values is very similar (Chang 1997 (introduction) and 2004). Chang claims that comparisons can only be made in terms of a covering value--a more comprehensive value that has the plural values as parts. A recent suggestion by Brian Hedden and Daniel Munoz is that by definition, overall value supervenes on the dimensions of value (Hedden and Munoz 2023). The basic point is that if overall value is affected, then there must be a relevant and distinct way that goodness or badness has been contributed. So, for example, in Chang's case of adding more pleasure disrupting balance (see section 2.1), Hedden and Munoz argue that balance is a dimension of value, and it is because balance has been adversely affected that there is less value overall. There is a problem in understanding quite what a 'synthesizing category' or 'covering value' is. How does the covering value determine the relative weightings of the constituent values? One possibility is that it does it by pure stipulation--as a martini just is a certain proportion of gin and vermouth. However, stipulation does not have the right sort of explanatory power. On the other hand, if a view is to remain pluralist, it must avoid conflating the super scale with a super value. Chang argues that her covering values are sufficiently unitary to provide a basis for comparison, and yet preserve the separateness of the other values. Chang's argument goes as follows: the values at stake in a situation (for example, prudence and morality) cannot on their own determine how heavily they weigh in a particular choice situation--the values weigh differently depending on the circumstances of the choice. However, the values plus the circumstances cannot determine relevant weightings either--because (simplifying here) the internal circumstances of the choice will affect the weighting of the values differently depending on the external circumstances. To use Chang's own example, when the values at stake are prudence and morality (specifically, the duty to help an innocent victim), and the circumstances include the fact that the victim is far away, the effect this circumstance will have on the weighting of the values depends on external circumstances, which fix what matters in the choice. So, as Chang puts it, "'What matters' must therefore have content beyond the values and the circumstances of the choice" (2004, p. 134). Stocker is aware of the worry that appeal to something in terms of which comparisons can be made reduces the view to monism: Stocker insists that the synthesizing category (such as a good life) is not a unitary value--it is at most 'nominal monism' in my terminology. Stocker argues that it is a philosophical prejudice to think that rational judgment must be quantitative, and so he claims that he does not need to give an account of how we form and use the higher level synthesizing categories. ### 4.3 Basic Preferences Another approach to the comparison problem appeals to basic preferences. Joseph Raz takes the line that we can explain choice between irreducibly plural goods by talking about basic preferences. Raz approaches the issue of incommensurability by talking about the nature of agency and rationality instead of about the nature of value. He distinguishes between two conceptions of human agency: the rationalist conception, and the classical conception. The rationalist conception corresponds to what we have called the stronger use of the term rational. According to the rationalist conception, reasons require action. The classical conception, by contrast, "regards reasons as rendering options eligible" (Raz 1999, p. 47). Raz favors the classical conception, which regards the will as something separate from desire: > > The will is the ability to choose and perform intentional actions. We > exercise our will when we endorse the verdict of reason that we must > perform an action, and we do so, whether willingly, reluctantly, or > regretting the need, etc. According to the classical conception, > however, the most typical exercise or manifestation of the will is in > choosing among options that reason merely renders eligible. Commonly > when we so choose, we do what we want, and we choose what we want, > from among the eligible options. Sometimes speaking of wanting one > option (or its consequences) in preference to the other eligible ones > is out of place. When I choose one tin of soup from a row of identical > tins in the shop, it would be wrong and misleading to say that I > wanted that tin rather than, or in preference to, the others. > Similarly, when faced with unpalatable but unavoidable and > incommensurate options (as when financial need forces me to give up > one or another of incommensurate goods), it would be incorrect to say > that I want to give up the one I choose to give up. I do not want to > do so. I have to, and I would equally have regretted the loss of > either good. I simply choose to give up one of them. (Raz, 1999, p. > 48) > Raz's view about the nature of agency is defended in great detail over the course of many articles, and all of those arguments cannot be examined in detail here. What is crucial in the context of this discussion of pluralism is whether Raz gives us a satisfactory account of the weaker sense of rational. Raz's solution to the problem of incommensurability hangs on the claim that it can be rational (in the weak sense) to choose A over B when there are no further reasons favouring A over B. We shall restrict ourselves to mentioning one objection to the view in the context of moral choices between plural goods. Though Raz's account of choice may seem plausible in cases where we choose between non-moral values, it seems to do violence to the concept of morality. Consider one of Raz's own examples, the choice between a banana and a pear. It may be that one has to choose between them, and there is no objective reason to choose one or the other. In this case, it seems Raz's account of choice is plausible. If one feels like eating a banana, then in this case, desire does provide a reason. As Raz puts it, "A want can never tip the balance of reasons in and of itself. Rather, our wants become relevant when reasons have run their course." In the example where we choose between a banana and a pear, this sounds fine. However, if we apply it to a moral choice it seems a lot less plausible. Raz admits that "If of the options available to agents in typical situations of choice and decision, several are incommensurate, then reason can neither determine nor completely explain their choices or actions" (Raz, 1999, p. 48). Thus many moral choices are not directed by reason but by a basic preference. It is not fair to call it a desire, because on Raz's account we desire things for reasons--we take the object of our desire to be desirable. On Raz's picture then, when reasons have run their course, we are choosing without reasons. It doesn't matter hugely whether we call that 'rational' (it is not rational in the strong sense, but it is in the weak sense). What matters is whether this weak sense of rational is sufficient to satisfy our concept of moral choice as being objectively defensible. The problem is that choosing without reasons look rather like plumping. Plumping may be an intelligible form of choice, but it is questionable whether it is a satisfactory account of moral choice. ### 4.4 Accepting Incomparability One philosopher who is happy to accept that there may be situations where we just cannot make reasoned choices between plural values is Isaiah Berlin, who claimed that goods such as liberty and equality conflict at the fundamental level. Berlin is primarily concerned with political pluralism, and with defending political liberalism, but his views about incomparability have been very influential in discussions on moral pluralism. Bernard Williams (1981), Charles Larmore (1987), John Kekes (1993), Michael Stocker (1990 and 1997), David Wiggins (1997) have all argued that there are at least some genuinely irresolvable conflicts between values, and that to expect a rational resolution is a mistake. (See also the entry on moral dilemmas). For Williams this is part of a more general mistake made by contemporary moral philosophers--he thinks that philosophy tries to make ethics too easy, too much like arithmetic. Williams insists throughout his writings that ethics is a much more complex and multi-faceted beast than its treatment at the hands of moral philosophers would suggest, and so it is not surprising to him that there should be situations where values conflict irresolvably. Stocker (1990) discusses the nature of moral conflict at great length, and although he thinks that many apparent conflicts can be dissolved or are not serious, like Williams, he argues that much of contemporary philosophy's demand for simplicity is mistaken. Stocker argues that ethics need not always be action guiding, that value is much more complex than Kantians and utilitarians would have us think, and that as the world is complicated we will inevitably face conflicts. Several pluralists have argued that accepting the inevitability of value conflicts does not result in a breakdown of moral argument, but rather the reverse. Kekes (1993), for example, claims that pluralism enables us to see that irresolvable disagreements are not due to wickedness on the part of our interlocutor, but may be due to the plural nature of values. ## 5. Conclusion The battle lines in the debate between pluralism and monism are not always clear. This entry has outlined some of them, and discussed some of the main arguments. Pluralists need to be clear about whether they are foundational or non-foundational pluralists. Monists must defend their claim that there really is a unitary value. Much of the debate between pluralists and monists has focussed on the issue of whether the complexity of moral choice implies that values really are plural--a pattern emerges in which the monist claims to be able to explain the appearance of plurality away, and the pluralist insists that the appearance reflects a pluralist reality. Finally, pluralists must explain how comparisons between values are made, or defend the consequence that incommensurability is widespread.
value-theory	## 1. Basic Questions The theory of value begins with a subject matter. It is hard to specify in some general way exactly what counts, but it certainly includes what we are talking about when we say any of the following sorts of things (compare Ziff [1960]): > > "pleasure is good/bad"; "it would be good/bad if you > did that"; "it is good/bad for him to talk to her"; > "too much cholesterol is good/bad for your health"; > "that is a good/bad knife"; "Jack is a good/bad > thief"; "he's a good/bad man"; > "it's good/bad that you came"; "it would be > better/worse if you didn't"; "lettuce is > better/worse for you than Oreos"; "my new can opener is > better/worse than my old one"; "Mack is a better/worse > thief than Jack"; "it's better/worse for it to end > now, than for us to get caught later"; "best/worst of all, > would be if they won the World Series and kept all of their > players for next year"; "celery is the best/worst thing > for your health"; "Mack is the best/worst thief > around" > The word "value" doesn't appear anywhere on this list; it is full, however, of "good", "better", and "best", and correspondingly of "bad", "worse", and "worst". And these words are used in a number of different kinds of constructions, of which we may take these four to be the main exemplars: 1. Pleasure is good. 2. It is good that you came. 3. It is good for him to talk to her. 4. That is a good knife. Sentences like 1, in which "good" is predicated of a mass term, constitute a central part of traditional axiology, in which philosophers have wanted to know what things (of which there can be more or less) are good. I'll stipulatively call them value claims, and use the word "stuff" for the kind of thing of which they predicate value (like pleasure, knowledge, and money). Sentences like 2 make claims about what I'll (again stipulatively) call goodness simpliciter; this is the kind of goodness appealed to by traditional utilitarianism. Sentences like 3 are good for sentences, and when the subject following "for" is a person, we usually take them to be claims about welfare or well-being. And sentences like 4 are what, following Geach [1956], I'll call attributive uses of "good", because "good" functions as a predicate modifier, rather than as a predicate in its own right. Many of the basic issues in the theory of value begin with questions or assumptions about how these various kinds of claim are related to one another. Some of these are introduced in the next two sections, focusing in 1.1 on the relationship between our four kinds of sentences, and focusing in 1.2 on the relationship between "good" and "better", and between "good" and "bad". ### 1.1 Varieties of Goodness Claims about good simpliciter are those which have garnered the most attention in moral philosophy. This is partly because as it is usually understood, these are the "good" claims that consequentialists hold to have a bearing on what we ought to do. Consequentialism, so understood, is the view that you ought to do whatever action is such that it would be best if you did it. This leaves, however, a wide variety of possible theories about how such claims are related to other kinds of "good" claim. #### 1.1.1 Good Simpliciter and Good For For example, consider a simple point of view theory, according to which what is good simpliciter differs from what is good for Jack, in that being good for Jack is being good from a certain point of view -- Jack's -- whereas being good simpliciter is being good from a more general point of view -- the point of view of the universe (compare Nagel [1985]). The point of view theory reduces both good for and good simpliciter to good from the point of view of, and understands good simpliciter claims as about the point of view of the universe. One problem for this view is to make sense of what sort of thing points of view could be, such that Jack and the universe are both the kinds of thing to have one. According to a different sort of theory, the agglomerative theory, goodness simpliciter is just what you get by "adding up" what is good for all of the various people that there are. Rawls [1971] attributes this view to utilitarians, and it fits with utilitarian discussions such as that of Smart's contribution to Smart and Williams [1973], but much more work would have to be done in order to make it precise. We sometimes say things like, "wearing that outfit in the sun all day is not going to be good for your tan line", but your tan line is not one of the things whose good it seems plausible to "add up" in order to get what is good simpliciter. Certainly it is not one of the things whose good classical utilitarians would want to add up. So the fact that sapient and even sentient beings are not the only kinds of thing that things can be good or bad for sets an important constraint both on accounts of the good for relation, and on theories about how it is related to good simpliciter. Rather than accounting for either of goodness simpliciter or goodness-for in terms of the other, some philosophers have taken one of these seriously at the expense of the other. For example, Philippa Foot [1985] gives an important but compressed argument that apparent talk about what is good simpliciter can be made sense of as elliptical talk about what is good for some unmentioned person, and Foot's view can be strengthened (compare Shanklin [2011], Finlay [2014]) by allowing that apparent good simpliciter claims are often generically quantified statements about what is, in general, good for a person. Thomson [2008] famously defends a similar view. G.E. Moore [1903], in contrast, struggled to make sense of good-for claims. In his refutation of egoism, Moore attributed to ethical egoists the theory that what is good for Jack (or "in Jack's good") is just what is good and in Jack's possession, or alternatively, what it is good that Jack possesses. Moore didn't argue against these theses directly, but he did show that they cannot be combined with universalizable egoism. It is now generally recognized that to avoid Moore's arguments, egoists need only to reject these analyses of good for, which are in any case unpromising (Smith [2003]). #### 1.1.2 Attributive Good Other kinds of views understand good simpliciter in terms of attributive good. What, after all, are the kinds of things to which we attribute goodness simpliciter? According to many philosophers, it is to propositions, or states of affairs. This is supported by a cursory study of the examples we have considered, in which what is being said to be good appears to be picked out by complementizers like "if", "that", and "for": "it would be good if you did that"; "it's good that you came"; "it's better for it to end now". If complementizer phrases denote propositions or possible states of affairs, then it is reasonable to conjecture, along with Foot [1985] that being good simpliciter is being a good state of affairs, and hence that it is a special case of attributive good (if it makes sense at all -- Geach and Foot both argue that it does not, on the ground that states of affairs are too thin of a kind to support attributive good claims). See the > > Supplement on Four Complications about Attributive Good > for further complications that arise when we consider the attributive sense of "good". Some philosophers have used the examples of attributive good and good for in order to advance arguments against noncognitivist metaethical theories (See the entry cognitivism and non-cognitivism). The basic outlines of such an argument go like this: noncognitivist theories are designed to deal with good simpliciter, but have some kind of difficulties accounting for attributive good or for good for. Hence, there is a general problem with noncognitivist theories, or at least a significant lacuna they leave. It has similarly been worried that noncognitivist theories will have problems accounting for so-called "agent-relative" value [see section 4], again, apparently, because of its relational nature. There is no place to consider this claim here, but note that it would be surprising if relational uses of "good" like these were in fact a deep or special problem for noncognitivism; Hare's account in The Language of Morals (Hare [1952]) was specifically about attributive uses of "good", and it is not clear why relational noncognitive attitudes should be harder to make sense of than relational beliefs. #### 1.1.3 Relational Strategies In an extension of the strategies just discussed, some theorists have proposed views of "good" which aspire to treat all of good simpliciter, good for, and attributive good as special cases. A paradigm of this approach is the "end-relational" theory of Paul Ziff [1960] and Stephen Finlay [2004], [2014]. According to Ziff, all claims about goodness are relative to ends or purposes, and "good for" and attributive "good" sentences are simply different ways of making these purposes (more or less) explicit. Talk about what is good for Jack, for example, makes the purpose of Jack's being happy (say) explicit, while talk about what is a good knife makes our usual purposes for knives (cutting things, say) explicit. The claim about goodness is then relativized accordingly. Views adopting this strategy need to develop in detail answers to just what, exactly, the further, relational, parameter on "good" is. Some hold that it is ends, while others say things like "aims". A filled-out version of this view must also be able to tell us the mechanics of how these ends can be made explicit in "good for" and attributive "good" claims, and needs to really make sense of both of those kinds of claim as of one very general kind. And, of course, this sort of view yields the prediction that non-explicitly relativized "good" sentences -- including those used throughout moral philosophy -- are really only true or false once the end parameter is specified, perhaps by context. This means that this view is open to the objection that it fails to account for a central class of uses of "good" in ethics, which by all evidence are non-relative, and for which the linguistic data do not support the hypothesis that they are context-sensitive. J.L. Mackie held a view like this one and embraced this result -- Mackie's [1977] error theory about "good" extended only to such putative non-relational senses of "good". Though he grants that there are such uses of "good", Mackie concludes that they are mistaken. Finlay [2014], in contrast, argues that he can use ordinary pragmatic effects in order to explain the appearances. The apparently non-relational senses of "good", Finlay argues, really are relational, and his theory aspires to explain why they seem otherwise. #### 1.1.4 What's Special About Value Claims The sentences I have called "value claims" present special complications. Unlike the other sorts of "good" sentences, they do not appear to admit, in a natural way, of comparisons. Suppose, for example, with G.E. Moore, that pleasure is good and knowledge is good. Which, we might ask, is better? This question does not appear to make very much sense, until we fix on some amount of pleasure and some amount of knowledge. But if Sue is a good dancer and Huw is a good dancer, then it makes perfect sense to ask who is the better dancer, and without needing to fix on any particular amount of dancing -- much less on any amount of Sue or Huw. In general, just as the kinds of thing that can be tall are the same kinds of thing as can be taller than each other, the kinds of thing that can be good are the same kinds of thing as can be better than one another. But the sentences that we are calling "value claims", which predicate "good" of some stuff, appear not to be like this. One possible response to this observation, if it is taken seriously, is to conclude that so-called "value claims" have a different kind of logical form or structure. One way of implementing this idea, the good-first theory, is to suppose that "pleasure is good" means something roughly like, "(other things equal) it is better for there to be more pleasure", rather than, "pleasure is better than most things (in some relevant comparison class)", on a model with "Sue is a good dancer", which means roughly, "Sue is a better dancer than most (in some relevant comparison class)". According to a very different kind of theory, the value-first theory, when we say that pleasure is good, we are saying that pleasure is a value, and things are better just in case there is more of the things which are values. These two theories offer competing orders of explanation for the same phenomenon. The good-first theory analyzes value claims in terms of "good" simpliciter, while the value-first theory analyzes "good" simpliciter in terms of value claims. The good-first theory corresponds to the thesis that states of affairs are the "primary bearers" of value; the value-first theory corresponds to the alternative thesis that it is things like pleasure or goodness (or perhaps their instances) that are the "primary bearers" of value. According to a more skeptical view, sentences like "pleasure is good" do not express a distinctive kind of claim at all, but are merely what you get when you take a sentence like "pleasure is good for Jill to experience", generically quantify out Jill, and ellipse "to experience". Following an idea also developed by Finlay [2014], Robert Shanklin [2011] argues that in general, good-for sentences pattern with experiencer adjectives like "fun", which admit of these very syntactic transformations: witness "Jack is fun for Jill to talk to", "Jack is fun to talk to", "Jack is fun". This view debunks the issue over which the views discussed in the last paragraph disagree, for it denies that there is any such distinct topic for value claims to be about. (It may also explain the failures of comparative forms, above, on the basis of differences in the elided material.) ### 1.2 Good, Better, Bad #### 1.2.1 Good and Better On a natural view, the relationship between "good", "better", and "best" would seem to be the same as that between "tall", "taller", and "tallest". "Tall" is a gradable adjective, and "taller" is its comparative form. On standard views, gradable adjectives are analyzed in terms of their comparative form. At bottom is the relation of being taller than, and someone is the tallest woman just in case she is taller than every woman. Similarly, someone is tall, just in case she is taller than a contextually appropriate standard (Kennedy [2005]), or taller than sufficiently many (this many be vague) in some contextually appropriate comparison class. Much moral philosophy appears to assume that things are very different for "good", "better", and "best". Instead of treating "better than" as basic, and something as being good just in case it is better than sufficiently many in some comparison class, philosophers very often assume, or write as if they assume, that "good" is basic. For example, many theorists have proposed analyses of what it is to be good which are incompatible with the claim that "good" is to be understood in terms of "better". In the absence of some reason to think that "good" is very different from "tall", however, this may be a very peculiar kind of claim to make, and it may distort some other issues in the theory of value. #### 1.2.2 Value Moreover, it is difficult to see how one could do things the other way around, and understand "better" in terms of "good". Jon is a better sprinter than Jan not because it is more the case that Jon is a good sprinter than that Jan is a good sprinter -- they are both excellent sprinters, so neither one of these is more the case than the other. It is, however, possible to see how to understand both "good" and "better" in terms of value. If good is to better as tall is to taller, then the analogue of value should intuitively be height. One person is taller than another just in case her height is greater; similarly, one state of affairs is better than another just in case its value is greater. If we postulate something called "value" to play this role, then it is natural (though not obligatory) to identify value with amounts of values -- amounts of things like pleasure or knowledge, which "value" claims claim to be good. But this move appears to be implausible or unnecessary when applied to attributive "good". It is not particularly plausible that there is such a thing as can-opener value, such that one can-opener is better than another just in case it has more can-opener value. In general, not all comparatives need be analyzable in terms of something like height, of which there can be literally more or less. Take, for example, the case of "scary". The analogy with height would yield the prediction that if one horror film is scarier than another, it is because it has more of something -- scariness -- than the other. This may be right, but it is not obviously so. If it is not, then the analogy need not hold for "good" and its cognates, either. In this case, it may be that being better than does not merely amount to having more value than. #### 1.2.3 Good and Bad These questions, moreover, are related to others. For example, "better" would appear to be the inverse relation of "worse". A is better than B just in case B is worse than A. So if "good" is just "better than sufficiently many" and "bad" is just "worse than sufficiently many", all of the interesting facts in the neighborhood would seem to be captured by an assessment of what stands in the better than relation to what. The same point goes if to be good is just to be better than a contextually set standard. But it has been held by many moral philosophers that an inventory of what is better than what would still leave something interesting and important out: what is good. If this is right, then it is one important motivation for denying that "good" can be understood in terms of "better". But it is important to be careful about this kind of argument. Suppose, for example, that, as is commonly held about "tall", the relevant comparison class or standard for "good" is somehow supplied by the context of utterance. Then to know whether "that is good" is true, you do need to know more than all of the facts about what is better than what -- you also need to know something about the comparison class or standard that is supplied by the context of utterance. The assumption that "good" is context-dependent in this way may therefore itself be just the kind of thing to explain the intuition which drives the preceding argument. ## 2. Traditional Questions Traditional axiology seeks to investigate what things are good, how good they are, and how their goodness is related to one another. Whatever we take the "primary bearers" of value to be, one of the central questions of traditional axiology is that of what stuffs are good: what is of value. ### 2.1 Intrinsic Value #### 2.1.1 What is Intrinsic Value? Of course, the central question philosophers have been interested in, is that of what is of intrinsic value, which is taken to contrast with instrumental value. Paradigmatically, money is supposed to be good, but not intrinsically good: it is supposed to be good because it leads to other good things: HD TV's and houses in desirable school districts and vanilla lattes, for example. These things, in turn, may only be good for what they lead to: exciting NFL Sundays and adequate educations and caffeine highs, for example. And those things, in turn, may be good only for what they lead to, but eventually, it is argued, something must be good, and not just for what it leads to. Such things are said to be intrinsically good. Philosophers' adoption of the term "intrinsic" for this distinction reflects a common theory, according to which whatever is non-instrumentally good must be good in virtue of its intrinsic properties. This idea is supported by a natural argument: if something is good only because it is related to something else, the argument goes, then it must be its relation to the other thing that is non-instrumentally good, and the thing itself is good only because it is needed in order to obtain this relation. The premise in this argument is highly controversial (Schroeder [2005]), and in fact many philosophers believe that something can be non-instrumentally good in virtue of its relation to something else. Consequently, sometimes the term "intrinsic" is reserved for what is good in virtue of its intrinsic properties, or for the view that goodness itself is an intrinsic property, and non-instrumental value is instead called "telic" or "final" (Korsgaard [1983]). I'll stick to "intrinsic", but keep in mind that intrinsic goodness may not be an intrinsic property, and that what is intrinsically good may turn out not to be so in virtue of its intrinsic properties. See the > > Supplement on Atomism/Holism about Value > for further discussion of the implications of the assumption that intrinsic value supervenes on intrinsic properties. Instrumental value is also sometimes contrasted with "constitutive" value. The idea behind this distinction is that instrumental values lead causally to intrinsic values, while constitutive values amount to intrinsic values. For example, my giving you money, or a latte, may causally result in your experiencing pleasure, whereas your experiencing pleasure may constitute, without causing, your being happy. For many purposes this distinction is not very important and often not noted, and constitutive values can be thought, along with instrumental values, as things that are ways of getting something of intrinsic value. I'll use "instrumental" in a broad sense, to include such values. #### 2.1.2 What is the Intrinsic/Instrumental Distinction Among? I have assumed, here, that the intrinsic/instrumental distinction is among what I have been calling "value claims", such as "pleasure is good", rather than among one of the other kinds of uses of "good" from part 1. It does not make sense, for example, to say that something is a good can opener, but only instrumentally, or that Sue is a good dancer, but only instrumentally. Perhaps it does make sense to say that vitamins are good for Jack, but only instrumentally; if that is right, then the instrumental/intrinsic distinction will be more general, and it may tell us something about the structure of and relationship between the different senses of "good", to look at which uses of "good" allow an intrinsic/instrumental distinction. It is sometimes said that consequentialists, since they appeal to claims about what is good simpliciter in their explanatory theories, are committed to holding that states of affairs are the "primary" bearers of value, and hence are the only things of intrinsic value. This is not right. First, consequentialists can appeal in their explanatory moral theory to facts about what state of affairs would be best, without holding that states of affairs are the "primary" bearers of value; instead of having a "good-first" theory, they may have a "value-first" theory (see section 1.1.4), according to which states of affairs are good or bad in virtue of there being more things of value in them. Moreover, even those who take a "good-first" theory are not really committed to holding that it is states of affairs that are intrinsically valuable; states of affairs are not, after all, something that you can collect more or less of. So they are not really in parallel to pleasure or knowledge. For more discussion of intrinsic value, see the entry on intrinsic vs. extrinsic value. ### 2.2 Monism/Pluralism One of the oldest questions in the theory of value is that of whether there is more than one fundamental (intrinsic) value. Monists say "no", and pluralists say "yes". This question only makes sense as a question about intrinsic values; clearly there is more than one instrumental value, and monists and pluralists will disagree, in many cases, not over whether something is of value, but over whether its value is intrinsic. For example, as important as he held the value of knowledge to be, Mill was committed to holding that its value is instrumental, not intrinsic. G.E. Moore disagreed, holding that knowledge is indeed a value, but an intrinsic one, and this expanded Moore's list of basic values. Mill's theory famously has a pluralistic element as well, in contrast with Bentham's, but whether Mill properly counts as a pluralist about value depends on whether his view was that there is only one value -- happiness -- but two different kinds of pleasure which contribute to it, one more effectively than the other, or whether his view was that each kind of pleasure is a distinctive value. This point will be important in what follows. #### 2.2.1 Ontology and Explanation At least three quite different sorts of issues are at stake in this debate. First is an ontological/explanatory issue. Some monists have held that a plural list of values would be explanatorily unsatisfactory. If pleasure and knowledge are both values, they have held, there remains a further question to be asked: why? If this question has an answer, some have thought, it must be because there is a further, more basic, value under which the explanation subsumes both pleasure and knowledge. Hence, pluralist theories are either explanatorily inadequate, or have not really located the basic intrinsic values. This argument relies on a highly controversial principle about how an explanation of why something is a value must work -- a very similar principle to that which was appealed to in the argument that intrinsic value must be an intrinsic property [section 2.1.1]. If this principle is false, then an explanatory theory of why both pleasure and knowledge are values can be offered which does not work by subsuming them under a further, more fundamental value. Reductive theories of what it is to be a value satisfy this description, and other kinds of theory may do so, as well (Schroeder [2005]). If one of these kinds of theory is correct, then even pluralists can offer an explanation of why the basic values that they appeal to are values. #### 2.2.2 Revisionary Commitments? Moreover, against the monist, the pluralist can argue that the basic posits to which her theory appeals are not different in kind from those to which the monist appeals; they are only different in number. This leads to the second major issue that is at stake in the debate between monists and pluralists. Monistic theories carry strong implications about what is of value. Given any monistic theory, everything that is of value must be either the one intrinsic value, or else must lead to the one intrinsic value. This means that if some things that are intuitively of value, such as knowledge, do not, in fact, always lead to what a theory holds to be the one intrinsic value (for example, pleasure), then the theory is committed to denying that these things are really always of value after all. Confronted with these kinds of difficulties in subsuming everything that is pre-theoretically of value under one master value, pluralists don't fret: they simply add to their list of basic intrinsic values, and hence can be more confident in preserving the pre-theoretical phenomena. Monists, in contrast, have a choice. They can change their mind about the basic intrinsic value and try all over again, they can work on developing resourceful arguments that knowledge really does lead to pleasure, or they can bite the bullet and conclude that knowledge is really not, after all, always good, but only under certain specific conditions. If the explanatory commitments of the pluralist are not different in kind from those of the monist, but only different in number, then it is natural for the pluralist to think that this kind of slavish adherence to the number one is a kind of fetish it is better to do without, if we want to develop a theory that gets things right. This is a perspective that many historical pluralists have shared. #### 2.2.3 Incommensurability The third important issue in the debate between monists and pluralists, and the most central over recent decades, is that over the relationship between pluralism and incommensurability. If one state of affairs is better than another just in case it contains more value than the other, and there are two or more basic intrinsic values, then it is not clear how two states of affairs can be compared, if one contains more of the first value, but the other contains more of the second. Which state of affairs is better, under such a circumstance? In contrast, if there is only one intrinsic value, then this can't happen: the state of affairs that is better is the one that has more of the basic intrinsic value, whatever that is. Reasoning like this has led some philosophers to believe that pluralism is the key to explaining the complexity of real moral situations and the genuine tradeoffs that they involve. If some things really are incomparable or incommensurable, they reason, then pluralism about value could explain why. Very similar reasoning has led other philosophers, however, to the view that monism has to be right: practical wisdom requires being able to make choices, even in complicated situations, they argue. But that would be impossible, if the options available in some choice were incomparable in this way. So if pluralism leads to this kind of incomparability, then pluralism must be false. In the next section, we'll consider the debate over the comparability of values on which this question hinges. But even if we grant all of the assumptions on both sides so far, monists have the better of these two arguments. Value pluralism may be one way to obtain incomparable options, but there could be other ways, even consistently with value monism. For example, take the interpretation of Mill on which he believes that there is only one intrinsic value -- happiness -- but that happiness is a complicated sort of thing, which can happen in each of two different ways -- either through higher pleasures, or through lower pleasures. If Mill has this view, and holds, further, that it is in some cases indeterminate whether someone who has slightly more higher pleasures is happier than someone who has quite a few more lower pleasures, then he can explain why it is indeterminate whether it is better to be the first way or the second way, without having to appeal to pluralism in his theory of value. The pluralism would be within his theory of happiness alone. See a more detailed discussion in the entry on value pluralism. ### 2.3 Incommensurability/Incomparability We have just seen that one of the issues at stake in the debate between monists and pluralists about value turns on the question (vaguely put) of whether values can be incomparable or incommensurable. This is consequently an area of active dispute in its own right. There are, in fact, many distinct issues in this debate, and sometimes several of them are run together. #### 2.3.1 Is there Weak Incomparability? One of the most important questions at stake is whether it must always be true, for two states of affairs, that things would be better if the first obtained than if the second did, that things would be better if the second obtained than if the first did, or that things would be equally good if either obtained. The claim that it can sometimes happen that none of these is true is sometimes referred to as the claim of incomparability, in this case as applied to good simpliciter. Ruth Chang [2002] has argued that in addition to "better than", "worse than", and "equally good", there is a fourth "positive value relation", which she calls parity. Chang reserves the use of "incomparable" to apply more narrowly, to the possibility that in addition to none of the other three relations holding between them, it is possible that two states of affairs may fail even to be "on a par". However, we can distinguish between weak incomparability, defined as above, and strong incomparability, further requiring the lack of parity, whatever that turns out to be. Since the notion of parity is itself a theoretical idea about how to account for what happens when the other three relations fail to obtain, a question which I won't pursue here, it will be weak incomparability that will interest us here. It is important to distinguish the question of whether good simpliciter admits of incomparability from the question of whether good for and attributive good admit of incomparability. Many discussions of the incomparability of values proceed at a very abstract level, and interchange examples of each of these kinds of value claims. For example, a typical example of a purported incomparability might compare, say, Mozart to Rodin. Is Mozart a better artist than Rodin? Is Rodin a better artist than Mozart? Are they equally good? If none of these is the case, then we have an example of incomparability in attributive good, but not an example of incomparability in good simpliciter. These questions may be parallel or closely related, and investigation of each may be instructive in consideration of the other, but they still need to be kept separate. For example, one important argument against the incomparability of value was mentioned in the previous section. It is that incomparability would rule out the possibility of practical wisdom, because practical wisdom requires the ability to make correct choices even in complicated choice situations. Choices are presumably between actions, or between possible consequences of those actions. So it could be that attributive good is sometimes incomparable, because neither Mozart nor Rodin is a better artist than the other and they are not equally good, but that good simpliciter is always comparable, so that there is always an answer as to which of two actions would lead to an outcome that is better. #### 2.3.2 What Happens when there is Weak Incomparability? Even once it is agreed that good simpliciter is incomparable in this sense, many theories have been offered as to what that incomparability involves and why it exists. One important constraint on such theories is that they not predict more incomparabilities than we really observe. For example, though Rodin may not be a better or worse artist than Mozart, nor equally good, he is certainly a better artist than Salieri -- even though Salieri, like Mozart, is a better composer than Rodin. This is a problem for the idea that incomparability can be explained by value pluralism. The argument from value pluralism to incomparability suggested that it would be impossible to compare any two states of affairs where one contained more of one basic value and the other contained more of another. But cases like that of Rodin and Salieri show that the explanation of what is incomparable between Rodin and Mozart can't simply be that since Rodin is a better sculptor and Mozart is a better composer, there is no way of settling who is the better artist. If that were the correct explanation, then Rodin and Salieri would also be incomparable, but intuitively, they are not. Constraints like these can narrow down the viable theories about what is going on in cases of incomparability, and are evidence that incomparability is probably not going to be straightforwardly explained by value pluralism. There are many other kinds of theses that go under the title of the incomparability or incommensurability of values. For example, some theories which posit lexical orderings are said to commit to "incomparabilities". Kant's thesis that rational agents have a dignity and not a price is often taken to be a thesis about a kind of incommensurability, as well. Some have interpreted Kant to be holding simply that respect for rational agents is of infinite value, or that it is to be lexically ordered over the value of anything else. Another thesis in the neighborhood, however, would be somewhat weaker. It might be that a human life is "above price" in the sense that killing one to save one is not an acceptable exchange, but that for some positive value of $n$, killing one to save $n$ would be an acceptable exchange. On this view, there is no single "exchange value" for a life, because the value of a human life depends on whether you are "buying" or "selling" -- it is higher when you are going to take it away, but lower when you are going to preserve it. Such a view would intelligibly count as a kind of "incommensurability", because it sets no single value on human lives. A more detailed discussion of the commensurability of values can be found in the entry on incommensurable values. ## 3. Relation to the Deontic One of the biggest and most important questions about value is the matter of its relation to the deontic -- to categories like right, reason, rational, just, and ought. According to teleological views, of which classical consequentialism and universalizable egoism are classic examples, deontic categories are posterior to and to be explained in terms of evaluative categories like good and good for. The contrasting view, according to which deontic categories are prior to, and explain, the evaluative categories, is one which, as Aristotle says, has no name. But its most important genus is that of "fitting attitude" accounts, and Scanlon's [1998] "buck-passing" theory is another closely related contemporary example. ### 3.1 Teleology Teleological theories are not, strictly speaking, theories about value. They are theories about right action, or about what one ought to do. But they are committed to claims about value, because they appeal to evaluative facts, in order to explain what is right and wrong, and what we ought to do -- deontic facts. The most obvious consequence of these theories, is therefore that evaluative facts must not then be explained in terms of deontic facts. The evaluative, on such views, is prior to the deontic. #### 3.1.1 Classical Consequentialism The most familiar sort of view falling under this umbrella is classical consequentialism, sometimes called (for reasons we'll see in section 3.3) "agent-neutral consequentialism". According to classical consequentialism, every agent ought always to do whatever action, out of all of the actions available to her at that time, is the one such that if she did it, things would be best. Not all defenders of consequentialism interpret it in such classical terms; other prominent forms of consequentialism focus on rules or motives, and evaluate actions only derivatively. Classical consequentialism is sometimes supported by appeal to the intuition that one should always do the best action, and then the assumption that actions are only instrumentally good or bad -- for the sake of what they lead to (compare especially Moore [1903]). The problem with this reasoning is that non-consequentialists can agree that agents ought always to do the best action. The important feature of this claim to recognize is that it is a claim not about intrinsic or instrumental value, but about attributive good. And as noted in section 2.1, "instrumental" and "intrinsic" don't really apply to attributive good. Just as how good of a can opener something is or how good of a torturer someone is does not depend on how good the world is, as a result of the fact that they exist, how good of an action something is need not depend on how good the world is, as a result that it happens. Indeed, if it did, then the evaluative standards governing actions would be quite different from those governing nearly everything else. #### 3.1.2 Problems in Principle vs. Problems of Implementation Classical consequentialism, and its instantiation in the form of utilitarianism, has been well-explored, and its advantages and costs cannot be surveyed here. Many of the issues for classical consequentialism, however, are issues for details of its exact formulation or implementation, and not problems in principle with its appeal to the evaluative in order to explain the deontic. For example, the worry that consequentialism is too demanding has been addressed within the consequentialist framework, by replacing "best" with "good enough" -- substituting a "satisficing" conception for a "maximizing" one (Slote [1989]). For another example, problems faced by certain consequentialist theories, like traditional utilitarianism, about accounting for things like justice can be solved by other consequentialist theories, simply by adopting a more generous picture about what sort of things contribute to how good things are (Sen [1982]). In section 3.3 we'll address one of the most central issues about classical consequentialism: its inability to allow for agent-centered constraints. This issue does pose an in-principle general problem for the aspiration of consequentialism to explain deontic categories in terms of the evaluative. For more, see the entry on consequentialism and utilitarianism. #### 3.1.3 Other Teleological Theories Universalizable egoism is another familiar teleological theory. According to universalizable egoism, each agent ought always to do whatever action has the feature that, of all available alternatives, it is the one such that, were she to do it, things would be best for her. Rather than asking agents to maximize the good, egoism asks agents to maximize what is good for them. Universalizable egoism shares many features with classical consequentialism, and Sidgwick found both deeply attractive. Many others have joined Sidgwick in holding that there is something deeply attractive about what consequentialism and egoism have in common -- which involves, at minimum, the teleological idea that the deontic is to be explained in terms of the evaluative (Portmore [2005]). Of course, not all teleological theories share the broad features of consequentialism and egoism. Classical Natural Law theories (Finnis [1980], Murphy [2001]) are teleological, in the sense that they seek to explain what we ought to do in terms of what is good, but they do so in a very different way from consequentialism and egoism. According to an example of such a Natural Law theory, there are a variety of natural values, each of which calls for a certain kind of distinctive response or respect, and agents ought always to act in ways that respond to the values with that kind of respect. For more on natural law theories, see the entry on the natural law tradition in ethics. And Barbara Hermann has prominently argued for interpreting Kant's ethical theory in teleological terms. For more on Herman's interpretation of Kant, see the entry on Kant's Moral Philosophy, especially section 13. Philip Pettit [1997] prominently distinguishes between values that we are called to "promote" and those which call for other responses. As Pettit notes, classical consequentialists hold that all values are to be promoted, and one way of thinking of some of these other kinds of teleological theories is that like consequentialism they explain what we ought to do in terms of what is good, but unlike consequentialism they hold that some kinds of good call for responses other than promotion. ### 3.2 Fitting Attitudes In contrast to teleological theories, which seek to account for deontic categories in terms of evaluative ones, Fitting Attitudes accounts aspire to account for evaluative categories -- like good simpliciter, good for, and attributive good -- in terms of the deontic. Whereas teleology has implications about value but is not itself a theory primarily about value, but rather about what is right, Fitting Attitudes accounts are primarily theses about value -- in accounting for it in terms of the deontic, they tell us what it is for something to be good. Hence, they are theories about the nature of value. The basic idea behind all kinds of Fitting Attitudes account is that "good" is closely linked to "desirable". "Desireable", of course, in contrast to "visible" and "audible", which mean "able to be seen" and "able to be heard", does not mean "able to be desired". It means, rather, something like "correctly desired" or "appropriately desired". If being good just is being desirable, and being desirable just is being correctly or appropriately desired, it follows that being good just is being correctly or appropriately desired. But correct and appropriate are deontic concepts, so if being good is just being desirable, then goodness can itself be accounted for in terms of the deontic. And that is the basic idea behind Fitting Attitudes accounts (Ewing [1947], Rabinowicz and Ronnow-Rasmussen [2004]). #### 3.2.1 Two Fitting Attitudes Accounts Different Fitting Attitudes accounts, however, work by appealing to different deontic concepts. Some of the problems facing Fitting Attitudes views can be exhibited by considering a couple exemplars. According to a formula from Sidgwick, for example, the good is what ought to be desired. But this slogan is not by itself very helpful until we know more: desired by whom? By everyone? By at least someone? By someone in particular? And for which of our senses of "good" does this seek to provide an account? Is it an account of good simpliciter, saying that it would be good if $p$ just in case \_\_\_\_ ought to desire that $p$, where "\_\_\_\_" is filled in by whoever it is, who is supposed to have the desire? Or is it an account of "value" claims, saying that pleasure is good just in case pleasure ought to be desired by \_\_\_\_? The former of these two accounts would fit in with the "good-first" theory from section 1.1.4; the latter would fit in with the "value-first" theory. We observed in section 1.1.4 that "value" claims don't admit of comparatives in the same way that other uses of "good" do; this is important here because if "better" simpliciter is prior to "good" simpliciter, then strictly speaking a "good-first" theorist needs to offer a Fitting Attitudes account of "better", rather than of "good". Such a modification of the Sidgwickian slogan might say that it would be better if $p$ than if $q$ just in case \_\_\_\_ ought to desire that $p$ more than that $q$ (or alternatively, to prefer $p$ to $q$). In What We Owe to Each Other, T.M. Scanlon offered an influential contemporary view with much in common with Fitting Attitudes accounts, which he called the Buck-Passing theory of value. According to Scanlon's slogan, "to call something valuable is to say that it has other properties that provide reasons for behaving in certain ways with respect to it." One important difference from Sidgwick's view is that it appeals to a different deontic concept: reasons instead of ought. But it also aspires to be more neutral than Sidgwick's slogan on the specific response that is called for. Sidgwick's slogan required that it is desire that is always relevant, whereas Scanlon's slogan leaves open that there may be different "certain ways" of responding to different kinds of values. But despite these differences, the Scanlonian slogan shares with the Sidgwickian slogan the feature of being massively underspecified. For which sense of "good" does it aspire to provide an account? Is it really supposed to be directly an account of "good", or, if we respect the priority of "better" to "good", should we really try to understand it as, at bottom, an account of "better than"? And crucially, which are the "certain ways" that are involved? It can't just be that the speaker has to have some certain ways in mind, because there are some ways of responding such that reasons to respond in that way are evidence that the thing in question is bad rather than that it is good -- for example, the attitude of dread. So does the theory require that there is some particular set of certain ways, such that a thing is good just in case there are reasons to respond to it in any of those ways? Scanlon's initial remarks suggest rather that for each sort of thing, there are different "certain ways" such that when we say that that thing is good, we are saying that there are reasons to respond to it in those ways. This is a matter that would need to be sorted out by any worked out view. A further complication with the Scanlonian formula, is that appealing in the analysis to the bare existential claim that there are reasons to respond to something in one of these "certain ways" faces large difficulties. Suppose, for example, that there is some reason to respond in one of the "certain ways", but there are competing, and weightier, reasons not to, so that all things considered, responding in any of the "certain ways" would be a mistake. Plausibly, the thing under consideration should not turn out to be good in such a case. So even a view like Scanlon's, which appeals to reasons, may need, once it is more fully developed, to appeal to specific claims about the weight of those reasons. #### 3.2.2 The Wrong Kind of Reason Even once these kinds of questions are sorted out, however, other significant questions remain. For example, one of the famous problems facing such views is the Wrong Kind of Reasons problem (Crisp [2000], Rabinowicz and Ronnow-Rasmussen [2004]). The problem arises from the observation that intuitively, some factors can affect what you ought to desire without affecting what is good. It may be true that if we make something better, then other things being equal, you ought to desire it more. But we can also create incentives for you to desire it, without making it any better. For example, you might be offered a substantial financial reward for desiring something bad, or an evil demon might (credibly) threaten to kill your family unless you do so. If these kinds of circumstances can affect what you ought to desire, as is at least intuitively plausible, then they will be counterexamples to views based on the Sidgwickian formula. Similarly, if these kinds of circumstances can give you reasons to desire the thing which is bad, then they will be counterexamples to views based on the Scanlonian formula. It is in the context of the Scanlonian formula that this issue has been called the "Wrong Kind of Reasons" problem, because if these circumstances do give you reasons to desire the thing that is bad, they are reasons of the wrong kind to figure in a Scanlon-style account of what it is to be good. This issue has recently been the topic of much fruitful investigation, and investigators have drawn parallels between the kinds of reason to desire that are provided by these kinds of "external" incentives and familiar issues about pragmatic reasons for belief and the kind of reason to intend that exists in Gregory Kavka's Toxin Puzzle (Hieronymi [2005]). Focusing on the cases of desire, belief, and intention, which are all kinds of mental state, some have claimed that the distinction between the "right kind" and "wrong kind" of reason can be drawn on the basis of the distinction between "object-given" reasons, which refer to the object of the attitude, and "state-given" reasons, which refer to the mental state itself, rather than to its object (Parfit [2001], Piller [2006]). But questions have also been raised about whether the "object-given"/"state-given" distinction is general enough to really explain the distinction between reasons of the right kind and reasons of the wrong kind, and it has even been disputed whether the distinction tracks anything at all. One reason to think that the distinction may not be general enough, is that situations very much like Wrong Kind of Reasons situations can arise even where no mental states are in play. For example, games are subject to norms of correctness. External incentives to cheat -- for example, a credible threat from an evil demon that she will kill your family unless you do so -- can plausibly not only provide you with reasons to cheat, but make it the case that you ought to. But just as such external incentives don't make it appropriate or correct to desire something bad, they don't make it a correct move of the game to cheat (Schroeder [2010]). If this is right, and the right kind/wrong kind distinction among reasons really does arise in a broad spectrum of cases, including ones like this one, it is not likely that a distinction that only applies to reasons for mental states is going to lie at the bottom of it. Further discussion of fitting attitudes accounts of value and the wrong kind of reasons problem can be found in the entry on fitting attitude theories of value. #### 3.2.4 Application to the Varieties of Goodness One significant attraction to Fitting Attitudes-style accounts, is that they offer prospects of being successfully applied to attributive good and good for, as well as to good simpliciter (Darwall [2002], Ronnow-Rasmussen [2009], Suikkanen [2009]). Just as reasons to prefer one state of affairs to another can underwrite one state of affairs being better than another, reasons to choose one can-opener over another can underwrite its being a better can opener than the other, and reasons to prefer some state of affairs for someone's sake can underwrite its being better for that person than another. For example, here is a quick sketch of what an account might look like, which accepts the good-first theory from section 1.1.4, holds as in section 1.1.2 that good simpliciter is a special case of attributive good, and understands attributive "good" in terms of attributive "better" and "good for" in terms of "better for": > > > Attributive better: For all kinds K, and things > A and B, for A to be a better K > than B is for the set of all of the right kind of reasons to > choose A over B when selecting a K to be > weightier than the set of all of the right kind of reasons to choose > B over A when selecting a K. > > > > Better for: For all things A, B, and > C, A is better for C than B is > just in case the set of all of the right kind of reasons to choose > A over B on C's behalf is weightier > than the set of all of the right kind of reasons to choose B > over A on C's behalf. > > > If being a good K is just being a better K than most (in some comparison class), and "it would be good if $p$" just means that $p$'s obtaining is a good state of affairs, and value claims like "pleasure is good" just mean that other things being equal, it is better for there to be more pleasure, then this pair of accounts has the right structure to account for the full range of "good" claims that we have encountered. But it also shows how the various senses of "good" are related, and allows that even attributive good and good for have, at bottom, a common shared structure. So the prospect of being able to offer such a unified story about what the various senses of "good" have in common, though not the exclusive property of the Fitting Attitudes approach, is nevertheless one of its attractions. ### 3.3 Agent-Relative Value? #### 3.3.1 Agent-Centered Constraints The most central, in-principle problem for classical consequentialism is the possibility of what are called agent-centered constraints (Scheffler [1983]). It has long been a traditional objection to utilitarian theories that because they place no intrinsic disvalue on wrong actions like murder, they yield the prediction that if you have a choice between murdering and allowing two people to die, it is clear that you should murder. After all, other things being equal, the situation is stacked 2-to-1 -- there are two deaths on one side, but only one death on the other, and each death is equally bad. Consequentialists who hold that killings of innocents are intrinsically bad can avoid this prediction. As long as a murder is at least twice as bad as an ordinary death not by murder, consequentialists can explain why you ought not to murder, even in order to prevent two deaths. So there is no in-principle problem for consequentialism posed by this sort of example; whether it is an issue for a given consequentialist depends on her axiology: on what she thinks is intrinsically bad, and how bad she thinks it is. But the problem is very closely related to a genuine problem for consequentialism. What if you could prevent two murders by murdering? Postulating an intrinsic disvalue to murders does nothing to account for the intuition that you still ought not to murder, even in this case. But most people find it pre-theoretically natural to assume that even if you should murder in order to prevent thousands of murders, you shouldn't do it in order to prevent just two. The constraint against murdering, on this natural intuition, goes beyond the idea that murders are bad. It requires that the badness of your own murders affects what you should do more than it affects what others should do in order to prevent you from murdering. That is why it is called "agent-centered". #### 3.3.2 Agent-Relative Value The problem with agent-centered constraints is that there seems to be no single natural way of evaluating outcomes that yields all of the right predictions. For each agent, there is some way of evaluating outcomes that yields the right predictions about what she ought to do, but these rankings treat that agent's murders as contributing more to the badness of outcomes than other agents' murders. So as a result, an incompatible ranking of outcomes appears to be required in order to yield the right predictions about what some other agent ought to do -- namely, one which rates his murders as contributing more to the badness of outcomes than the first agent's murders. (The situation is slightly more complicated -- Oddie and Milne [1991] prove that under pretty minimal assumptions there is always some agent-neutral ranking that yields the right consequentialist predictions, but their proof does not show that this ranking has any independent plausibility, and Nair [2014] argues that it cannot be an independently plausible account of what is a better outcome.) As a result of this observation, philosophers have postulated a thing called agent-relative value. The idea of agent-relative value is that if the better than relation is relativized to agents, then outcomes in which Franz murders can be worse-relative-to Franz than outcomes in which Jens murders, even though outcomes in which Jens murders are worse-relative-to Jens than outcomes in which Franz murders. These contrasting rankings of these two kinds of outcomes are not incompatible, because each is relativized to a different agent -- the former to Franz, and the latter to Jens. The idea of agent-relative value is attractive to teleologists, because it allows a view that is very similar in structure to classical consequentialism to account for constraints. According to this view, sometimes called Agent-Relative Teleology or Agent-Centered Consequentialism, each agent ought always to do what will bring about the results that are best-relative-to her. Such a view can easily accommodate an agent-centered constraint not to murder, on the assumption that each agent's murders are sufficiently worse-relative-to her than other agent's murders are (Sen [1983], Portmore [2007]). Some philosophers have claimed that Agent-Relative Teleology is not even a distinct theory from classical consequentialism, holding that the word "good" in English picks out agent-relative value in a context-dependent way, so that when consequentialists say, "everyone ought to do what will have the best results", what they are really saying is that "everyone ought to do what will have the best-relative-to-her results" (Smith [2003]). And other philosophers have suggested that Agent-Relative Teleology is such an attractive theory that everyone is really committed to it (Dreier [1996]). These theses are bold claims in the theory of value, because they tell us strong and surprising things about the nature of what we are talking about, when we use the word, "good". #### 3.3.3 Problems and Prospects In fact, it is highly controversial whether there is even such a thing as agent-relative value in the first place. Agent-Relative Teleologists typically appeal to a distinction between agent-relative and agent-neutral value, but others have contested that no one has ever successfully made such a distinction in a theory-neutral way (Schroeder [2007]). Moreover, even if there is such a distinction, relativizing "good" to agents is not sufficient to deal with all intuitive cases of constraints, because common sense allows that you ought not to murder, even in order to prevent yourself from murdering twice in the future. In order to deal with such cases, "good" will need to be relativized not just to agents, but to times (Brook [1991]). Yet a further source of difficulties arises for views according to which "good" in English is used to make claims about agent-relative value in a context-dependent way; such views fail ordinary tests for context-dependence, and don't always generate the readings of sentences which their proponents require. One of the motivations for thinking that there must be such a thing as agent-relative value comes from proponents of Fitting Attitudes accounts of value, and goes like this: if the good is what ought to be desired, then there will be two kinds of good. What ought to be desired by everyone will be the "agent-neutral" good, and what ought to be desired by some particular person will be the good relative-to that person. Ancestors of this idea can be found in Sidgwick and Ewing, and it has found a number of contemporary proponents. Whether it is right will turn not only on whether Fitting Attitudes accounts turn out to be correct, but on what role the answer to the questions, "who ought?" or "whose reasons?" plays in the shape of an adequate Fitting Attitudes account. All of these issues remain unresolved. The questions of whether there is such a thing as agent-relative value, and if so, what role it might play in an agent-centered variant on classical consequentialism, are at the heart of the debate between consequentialists and deontologists, and over the fundamental question of the relative priority of the evaluative versus the deontic. These are large and open questions, but as I hope I've illustrated here, they are intimately interconnected with a very wide range of both traditional and non-traditional questions in the theory of value, broadly construed.
vasubandhu	## 1. Biography and Works ### 1.1 Biography (Disputed) Vasubandhu (4th century C.E.) is dated to the height of the Gupta period by the fact that, according to Paramartha, he provided instruction for the crown prince, and queen, of King "Vikramaditya"--a name for the great Chandragupta II (r. 380-415).[6] Vasubandhu lived his life at the center of controversy, and he won fame and patronage through his acumen as an author and debater. His writings, packed as they are with criticisms of his contemporaries' traditions and views, are an unparalleled resource for understanding the debates alive among Buddhist and Orthodox (Hindu) schools of his time. As a young Buddhist scholar monk, Vasubandhu is said to have traveled from his home in Gandhara to Kashmir, then the heartland of the Vaibhasika philosophical system, to study. There, students were sworn to keep the tradition secret amongst themselves. Vasubandhu questioned Vaibhasika orthodoxy and made enemies there, but managed to return home an expert in the system. Immediately, he violated Kashmiri Vaibhasika norms by giving public teachings, going so far as to dare all comers to debate him at the end of each day. Vasubandhu's Kashmiri colleagues were livid until they received a copy of the text from which he was teaching, the root verses of his Treasury of the Abhidharma. This brilliant summary of their system pleased them immensely. But then, Vasubandhu published his auto-commentary, which criticized the system in various ways, primarily from a Sautrantika perspective. Livid again. Years later, Vasubandhu's Vaibhasika contemporary Sanghabhadra composed two treatises purporting to refute the Commentary, and traveled to Ayodhya, where Vasubandhu was living under royal patronage, to challenge the master to a public debate (Vasubandhu refused). This narrative, reported in Chinese and Tibetan sources, accounts for the unusual fact that Vasubandhu's Commentary often disagrees with positions affirmed in the Treasury's root verses. Vasubandhu's elder half-brother was Asanga, the monk who meditated in solitude for twelve years until he was able to meet with and receive teachings directly from the future Buddha, Maitreya. Asanga thereby became the preeminent expounder of the Yogacara synthesis of "Great Vehicle" (Mahayana) Buddhism, writing some texts himself, and transmitting the so-called "Five Treatises" revealed by Maitreya. When he was advanced in years, Asanga became worried about his younger brother's karmic outlook (Vasubandhu had publicly insulted Asanga's works). So he sent his students to teach Vasubandhu and convince him to adopt the Yogacara system. It worked. This narrative provides a reason for Vasubandhu's having written many texts from the Yogacara perspective after his prior advocacy of so-called "Sravaka" positions. It explains why the author of the Treasury and its Commentary might have gone on to write several commentaries on Mahayana scriptures, a treatise that defends the legitimacy of Mahayana scriptures, and a series of concise Yogacara syntheses. The mythologized, hagiographic character of the traditional biographies, together with the (disputed) belief that they contradict one another, has led a number of modern scholars to doubt their veracity. Many have expressed doubts that the mass of extant texts attributed to Vasubandhu in the Tibetan and Chinese canons could have been written by a single individual. In the 1950s Erich Frauwallner proposed a thesis that there were two separate Vasubandhus, whom the traditional biographies mistakenly combined into one figure.[7] Versions of this theory hold sway in some areas of the academy. Some scholars have called into question the story of Vasubandhu's conversion from Sravaka (non-Mahayana) to Yogacara under the influence of his brother Asanga, by noting a more gradual set of transitions and commonalities across works previously thought to sit on opposite sides of this "conversion."[8] An extension of this argument, advocated today by Robert Kritzer and colleagues, holds that Vasubandhu was secretly an advocate of Yogacara all along.[9] ### 1.2 Doctrinal Positions and Works Vasubandhu composed works from the perspectives of several different philosophical schools. In addition, his works often state doctrinal positions and arguments with which he disagrees, in order to refute them. For this reason, scholars have used his works, especially his Commentary on the Treasury of the Abhidharma to explicate a range of Buddhist and non-Buddhist positions prevalent during his time. The main Buddhist schools represented in his works are the Vaibhasikas or Sarvastivadins of Kashmir and Gandhara, the Vatsiputriyas, the Sautrantikas (also called Darstantikas), and the Yogacarins. He also argues with non-Buddhist, Orthodox (Hindu) positions, including especially Samkhya and Nyaya-Vaisesika views. It would appear that the only Buddhist school with which he always expresses disagreement is the Vatsiputriyas, also called the Pudgalavadins or "Personalists." Vasubandhu's works are most often said to have been written either from the Sautrantika perspective or from the Yogacara perspective, depending upon the work. Yet the verses of his Treasury of the Abhidharma summarize the doctrines of the Kashmiri Vaibhasika school. It is only in the commentary that the Sautrantika system tends to win the day, through numerous arguments.[10] And, it is quite often the case, even in the Commentary, that Vaibhasika views are left unchallenged or unharmed. A great number of texts of the Theravada Abhidharma tradition are extant in Pali, and a great number of Sarvastivada Abhidharma texts exist in their Chinese translations. In Sanskrit and Tibetan, however, nearly all of the "Sravaka" Abhidharma texts that remain are the works of Vasubandhu and the commentarial traditions stemming from them. It is tempting to take this as evidence of Vasubandhu's philosophical mastery, to have so comprehensively defeated his foes that his tradition dominated from the 9th century on. Yet we may equally take this as evidence not of the victory of Sautrantika, but of the influence of the rising popularity of Yogacara in India and Tibet. Vasubandhu, a great systematizer of mainstream Abhidharma, provided arguments and doctrines, and a life story, that paved the way to, and justified, the later dominance of Mahayana. Vasubandhu wrote in Sanskrit, but many of his works are known from their Chinese and Tibetan translations alone. It is beyond the purposes of this essay to summarize all of the works attributed to Vasubandhu, or to take into consideration the relevant criteria for determining his authorship. A quick mention of well-known works must suffice.[11] Vasubandhu's most important work of Abhidharma is his Treasury of the Abhidharma (Abhidharmakosakarika), with its Commentary (Abhidharmakosabhasya).[12] After these, his best known works are his concise, Yogacara syntheses, the Twenty Verses (Vimsikakarika) with its Commentary (Vimsikavrtti), the Thirty Verses (Trimsika), and the Three Natures Exposition (Trisvabhavanirdesa).[13] The majority of the arguments discussed below are taken from these works. A number of Vasubandhu's shorter independent works are also available. The Explanation of the Establishment of Action (Karmasiddhiprakarana) draws together, synthesizes and advances somewhat the many discussions of karma from the Treasury of the Abhidharma--adding a defense of the Yogacara concept of the "storehouse consciousness" (alayavijnana). The Explanation of the Five Aggregates (Pancaskandhaprakarana) is a summary of central Abhidharma terminology that includes some Yogacara cross-referencing as well. The Rules for Debate (Vadavidhi) is lost, but numerous fragments preserved in later texts show it to be foundational for the development of formal Buddhist logic and epistemology.[14] Vasubandhu wrote a number of commentaries on Buddhist scriptures, primarily (but not exclusively) those that are classified as Mahayana Sutras. He also wrote a tremendously influential work on how to interpret and teach Buddhist scriptures, entitled the Proper Mode of Exposition (Vyakhyayukti), in which (among other things) he argued for the legitimacy of the Mahayana Sutras (which he called "Vaipulya Sutras").[15] Also quite influential, and still regularly studied in Tibetan educational settings, are Vasubandhu's commentaries on major Yogacara works. These include commentaries on three texts said to have been revealed to Asanga by the bodhisattva Maitreya, his Commentary on Distinguishing Elements from Reality (Dharmadharmatavibhagavrtti), Commentary on Distinguishing the Middle from the Extremes (Madhyantavibhagabhasya), and Commentary on the Ornament to the Great Vehicle Discourses (Mahayanasutralamkarabhasya). Finally, Vasubandhu is credited with an important commentary on Asanga's Yogacara summa, the Commentary on the Summary of the Great Vehicle (Mahayanasamgrahabhasya). Among these works, and numerous others attributed to Vasubandhu in the Tibetan and Chinese canons, one finds countless perspectives and approaches represented, and many scholars are skeptical as to the unity of the authorship even of the works listed. Many, however, still find it sensible to assume that Vasubandhu worked in various genres and schools, and adopted the norms of each, developing his own thinking along the way. Yet no narrative of Vasubandhu's intellectual development has achieved scholarly consensus. ## 2. Major Arguments from the Treasury of the Abhidharma ### 2.1 Disproof of the Self The Buddhist approach to the question of personal continuity attracts perhaps more attention from contemporary philosophy than any other aspect of the tradition.[16] Vasubandhu's Commentary on the Treasury of the Abhidharma includes, as its final chapter, an extended defense of the Buddhist doctrine of no-self, entitled, "The Refutation of the Person." The majority of the argument assumes a Buddhist interlocutor, and is intended to prove that no Buddhist ought to accept the reality of a so-called "self" (atman) or "person" (pudgala) over and above the five aggregates (skandhas) in which, the Buddha said, the person consists. (The aggregates, as their name suggests, are themselves constantly-changing collections of entities of five categories: the physical, feelings, ideas, dispositions, and consciousness.) Vasubandhu apparently believed that the Vatsiputriyas, also called Pudgalavadins or "Personalists," represented a significant, Buddhist rival. Late in the chapter, Vasubandhu also provides some arguments that are targeted at a Vaisesika (Hindu) view of the self. The chapter begins with a brief statement of the soteriological purpose of the treatise, and something of a definition of the Buddha's teachings. Vasubandhu says that for outsiders (meaning outsiders to Buddhism), there is no possibility of liberation, because all other systems impose a false construct of a self upon what is really only the continuum of aggregates (skandha). Since grasping the self generates the mental afflictions (klesa), liberation from suffering is impossible for one who holds onto the false, non-Buddhist view that the self has independent reality. Vasubandhu does not attempt, here, to prove the karmic causality that justifies his soteriological exclusivism. Instead, he moves directly to prove the non-existence of the self. What is real, he says, is known by one of two means: perception or inference.[17] Seven things are known directly, by perception. They include the five objects of the senses (visual forms, sounds, smells, tastes, and touchables), mental objects (mental images or ideas), and the mind itself.[18] What is not known directly can only be known indirectly, by inference. As an example, Vasubandhu provides an argument that the five sense organs (eye, ear, nose, tongue, skin) can each be inferred from the awareness of their respective sensory objects. But, he says, there is no such inference for the self. Since Vasubandhu believes in universal momentariness (see the section on Momentariness and Continuity), he would reject Descartes' deduction of "I am," as an enduring self, from the perception of the momentary mental event, "I think." Vasubandhu does not list the five aggregates here, but his discussion of perception and inference stands in for having done so. Any Buddhist scholastic would be able to see that he has claimed to have proven the reality of the twelve sense bases (ayatana), and these twelve are easy to correlate with the five aggregates. The first aggregate, the physical (rupa), includes the five sensory organs and their five objects. The second, third, and fourth aggregates--feelings (vedana), thoughts (samjna), and dispositions (samskara)--are kinds of mental objects. The fifth aggregate, consciousness (vijnana), is equivalent to the twelfth sense base, the mind.[19] Earlier in the Treasury Vasubandhu had included other elements within the five aggregates (including, for instance, avijnapti-rupa, which is imperceptible, within the physical), but since he had also provided arguments disproving those elements (see the section on Disproof of Invisible Physicality), perhaps he felt no compulsion to include them here. Focusing on the twelve sense bases may be Vasubandhu's way of providing as clean and parsimonious a Buddhist ontology as he can. \| 12 Sense Bases \| 5 Aggregates \| \| --- \| --- \| \| Eye \| Physical \| \| Visual Form \| \| Ear \| \| Sound \| \| Nose \| \| Smell \| \| Tongue \| \| Taste \| \| Skin/Body \| \| Touchable \| \| Mind \| Consciousness \| \| Mental Object \| Feelings \| \| Thoughts \| \| Dispositions \| In any event, Vasubandhu is affirming the mainstream Buddhist view that the constantly-changing elements, which seem to be the possessions of, or the components of, the so-called "self," are real, while the self is unreal. Buddhists readily admit that this view is counterintuitive (if it were intuitive we would all be liberated), and many doubts may arise when it is posited: If there is no self, then what is the same about me as a child and as an adult? What is named by my name? What perceives perceptions, or experiences experiences, or enacts actions, if not the self? What is memory, without a self? And crucially, from a Buddhist perspective, what transmigrates from one body to the next, and reaps the karmic fruit of good and evil deeds, if not a self? These are all questions of great importance in Buddhist philosophical treatments of the doctrine of no-self, and Vasubandhu will address them each in due course. Vasubandhu begins, however, by focusing on certain commonsense advantages to the no-self view, and exposing the difficulties in holding to the reality of the self. His first target, then, is Buddhist "Personalists." For Vasubandhu, everything that is real or substantial (dravya) is causally efficient, having specifiable cause-and-effect relations with other entities. Everything that does not have such a causal basis is unreal, and if anything, it is merely a conceptual construct, a mere convention (prajnapti). The so-called "person," like everything else, must be one or the other--real or unreal, causally conditioned or conceptually constructed. Vasubandhu places it in the latter category. Notice that, to call something a "conceptual construction" (parikalpita) is, for Vasubandhu, to remove it from the flow of causality. Abstract entities have no causal impetus and are unreal. This is Vasubandhu's formalization, and expansion, of the Buddhist doctrine that all conditioned things are impermanent. Some Buddhists accept (for instance) that space and nirvana are real, though unconditioned. Vasubandhu says, instead, that all things that are engaged with the causal world must be, themselves, conditioned (this is a corollary of his proof of momentariness), so he rejects the causal capacity of eternal, unchanging entities. But even setting aside Vasubandhu's expanded view, it is clearly out of the question for Buddhists to say that the self is unconditioned and eternal. That is a non-Buddhist view, which Vasubandhu treats later on. Vasubandhu thus begins his argument by posing a dilemma for the Personalists, between saying that the "person" is uncaused--which would be to adopt a non-Buddhist view--and specifying the cause. Since the Buddha explained the aggregates to be the psychophysical elements that make up the person, any posited "self" must be in some way related to those entities. But the Personalists will not wish to say simply that the "person" is caused by the aggregates. For, the aggregates are temporary and impermanent in the extreme. The whole issue at hand is how to account for the continuity of the person as the aggregates change. If the aggregates are the cause(s) of the person, then the person, too, must change as they change. As Vasubandhu explains in criticizing a parallel context, "No unchanging quality is seen to inhere in changing substances." (AKBh 159.20)[20] If the causal basis is accepted as changing, the "person" is no longer continuous across the aggregates and through time. So a "person" caused by the aggregates provides no answer to the doubts about personal identity raised above. Yet for Vasubandhu, if the cause cannot be specified, then the person must be conceptually constructed. He adduces the following as an example of conceptual construction: When we see, smell, and taste milk, we have distinct sensory impressions, which are combined in our awareness. The "milk," then, is a mental construct--a concept built out of discrete sensory impressions. The sensory impressions are real, but the milk is not. In the same way, the "self" is made up of constantly-changing sensory organs, sensory impressions, ideas, and mental events. These separate, momentary elements are real, but their imagined unity--as an enduring "I"--is a false projection. The Personalists are willing to accept that the person is a conceptual construction, but they do not accept that this makes it unreal, or causally inert. They believe in a perceptible "person," who is ineffably neither the same as nor distinct from the aggregates, but who comes into existence in a particular lifetime "depending upon" the aggregates. Vasubandhu considers this a muddled attempt to have one's cake and eat it too. He focuses in on this term, "depending upon." If, by this, the Personalists mean "dependent as a conceptual object," he says, then really they are conceding that the term "person" actually just refers to the aggregates. It is to admit that the aggregates are understood as unified through a mere conceptual construction, "person," just as the taste, touch, and smell of milk generate the conceptual construction, "milk." If, on the other hand, they are saying the "person" is causally dependent upon the aggregates, then they are saying that the "person" is caused by the aggregates, which is to specify its causes among impermanent entities. There is, for Vasubandhu, no acceptable explanation of conceptual constructs as real entities. His exclusive dichotomy (real & causal vs. unreal & conceptual) serves not only as a tool for refusing a separate self, but also as a method of denying apparent subject/object relations or apparent substance/quality relations, and translating them into linear, causal series. For instance, he says that objects of awareness do not exist as causally significant entities distinct from consciousness; rather, consciousness is caused by its apparent objects, from which it takes on a particular shape (akara). This is how Vasubandhu resolves some of the standard challenges to the no-self view, such as the question of how experience can exist without an experiencer. The answer is that any given awareness only appears to be made up of two parts, subject and object, a "cognizer" acting upon a cognitive object. In reality, any given moment of awareness consists in a full, constructed appearance of subject/object duality without there actually being any separate subject. The object is an aspect or shape of the consciousness itself. Similarly, when asked with regard to memory "who" remembers if not the self, Vasubandhu rejects the subject/object structure. What is really occurring is only a series of discrete elements in the continuum of aggregates, which arise successively, causally linked to one another. The reason I remember my past is because my aggregates now are the causal result of the past aggregates whose actions I remember. My consciousness today arises with the shape of my past aggregates imprinted upon it. Moving on, the "dependence" relation that the Personalists are attempting to defend resists none of Vasubandhu's arguments (a casualty, perchance, of its appearing as a counterargument within his text), but the Personalists persist by stating that the relation between the person and the aggregates is "ineffable." To this, Vasubandhu argues, interestingly, that ineffability in regard to an entity that is ostensibly perceived undermines the functionality of perception as a source of knowledge. Without any way of distinguishing, clearly, what is perceived directly and what is perceived "ineffably," all of perception becomes potentially ineffable. And, perhaps more importantly, not everything is said by the Buddha to be ineffable, or at least not equally so. For instance, when the Buddha denies the reality of the self, he does so in contradistinction to the sense bases. If the "person" also exists as an object of sensory perception, how can this distinction be upheld? Vasubandhu's dialogue with the Personalists takes some rather technical twists and turns, which I will not attempt to cover here. See the section on scripture for a summary of Vasubandhu's methods of scriptural interpretation, which he employs decisively to defend against the Personalist adducement of the Buddha's silence on the nature of the person. I will here mention only one further argument against the Personalists, before treating Vasubandhu's arguments with non-Buddhists. The Personalist challenges Vasubandhu to explain how it can be, if everything is only a constantly-changing fluctuation of causally-connected separate entities, that things do not arise in predictable patterns. Why is there so much unpredictable change, the Personalists are asking, if there is no independent person who enacts those changes? This is a question that seems to be affirming a greater degree of freedom of the will than would appear possible were there not some outside, uncaused intervener in the causal flow. If I am reading it correctly, it is something like an incredulous denial that we could be "mere automatons," given our capacities for innovation and creativity. Vasubandhu's answer is to emphasize the possibility of change in a self-regulating system.[21] Vasubandhu points out that in fact, there are predictable patterns to the causal fluctuations of thoughts, so it is a mistake to call them unpredictable. Seeing a woman, he says, causes you to think thoughts that you tend to think when you look at women. (If we needed it, here's confirmation that the implied reader is a heterosexual male--though probably a celibate one.) If you practice leaving a woman alone in your thoughts (for instance, by practicing thinking of the woman's overprotective father, as monks are taught to do), then when you see a woman you'll reject her in your thoughts. If you do not practice this, other thoughts arise: "...then those [thoughts] which are most frequent [or clearest], or most proximate arise, because they have been most forcefully cultivated, except when there are simultaneous special conditions external to the body." (Kapstein 2001a: 370) This is an argument that undermines a naive trust in the independence of the will. But of course, Vasubandhu adds, just because the mind is subject to conditions does not mean that it will always be the same. On the contrary, it is fundamental to the definition of conditioned things that they are always changing. It is with this setup that Vasubandhu turns to address the non-Buddhist sectarians, asking them to account as elegantly as his causal, no-self view does, for the changing flux of mental events. For, the non-Buddhist opponent believes in an independent self who is the agent and controller of mental actions. If that is so, Vasubandhu challenges them, why do mental events arise in such disorder? Just what kind of a controller is the "self"? This is an important point, because it implies that Buddhists might use as evidence what is introspectively obvious to any meditator--namely, the difficulty of controlling the mind. If the "self" is an uncaused agent, why can't I, for instance, focus my concentration indefinitely? Non-Buddhist (orthodox/Hindu) traditions generally affirm that the mind must be controlled, and harnessed, by the self; this is one of the central meanings of the term yoga (lit. "yoking") in classical Hinduism. And, of course, the reason that such "yoking" is necessary is that the mind has its own causal impetus, which overwhelms the self unless and until the self attains liberation. Vasubandhu asks why, if changes in experience must be admitted to originate in the fluctuations of the mind, it should be believed that the "yoking" and other willful, agent-driven events originate outside of the mind in an unchanging "self." The need for such an explanation is evidence that the "self" requires a more burdensome (less simple) theoretical structure than might appear at first blush. The Buddhist view has an in-built explanation for the difficulty of changing the mind's dispositions. Each temporary collection of psychophysical entities generates, through causal regularities, another collection in the next moment. The mind's present dispositions are conditioned by its past, and its every experience is only an expression of its internal transformations, triggered of course by influences coming from outside the individual. It is widely considered a difficulty for the Buddhist no-self view that it has trouble accounting for our intuitive senses of continuity and control. But here we see Vasubandhu turning a disadvantage into an advantage. The intuitive sense of control is a mistake. This is introspectively obvious and is implicit in the non-Buddhist's admission that a mental series, distinct from the "self," is dependent upon its own causal impetus. Of course, the non-Buddhists have a panoply of theories explaining the relation between the self and the mental events with which it is complexly entwined. This is territory over which Indian thought contemplates free will and determinism. Vasubandhu does not enter into these debates in any detail. His interest is only to indicate the Buddhist position's comparative simplicity and causal parsimony. In the last stages of the chapter, Vasubandhu's non-Buddhist believer in a permanent self proposes several potential counter-arguments. Here Vasubandhu addresses a number of common worries about the no-self view, including the questions of experience and agency without a self. With regard to experience, Vasubandhu simply makes reference to the argument above in which the subject/object relation was described in a causal line, with the object causing a consciousness to take on the "shape" or appearance of an object. An individual, momentary consciousness does not need to be possessed by a self in order to have a certain appearance. With regard to agency, Vasubandhu again redescribes the structure under consideration--in this case, the act/agent structure--as a unitary, causal line: "From recollection there is interest; from interest consideration; from consideration willful effort; from willful effort vital energy; and from that, action. So what does the self do here?"[22] (Kapstein 2001a: 373) The point, again, is not that there is no action, but that action takes place as one event in a causal series, no element of which requires an independent agent. The last argument in the work addresses the crucial, Buddhist concern about karmic fruits. If there is no self, how can there be karmic results accruing to the agent of karmic acts? What can it mean to say that, if I kill someone, I go to hell, when there is no "I"? The way the question is phrased is in terms of a technical question about the nature of karmic retribution. The way that karma works in the non-Buddhist schools is, naturally, via the self. In one way or another, karmic residue adheres to the soul/self. If there is no soul that is affected by the karmic residue, how can you be affected at a later time by a previous cause? How can something I do now affect me at the time of my death, if at that time the deed will be in the past? Vasubandhu gives a clear, if quick reply. A complete answer to this question would require a fuller discussion of the question of continuity.[23] (See the sections Momentariness and Continuity and Disproof of Invisible Physicality below.) The short answer is simply that the past action inaugurates a causal series, which eventuates in the result at a later time via a number of intermediate steps. When I act now, it does not alter some eternal soul, but it does alter the future of my aggregates by sparking a causal series. As he does many times throughout his disproof of the person, Vasubandhu encourages his opponents to recognize the term "I" as simply referring to the continuum of aggregates. The conceptual construction "I" is then understood to be only a manner of speaking, a useful shorthand. He mentions that when I say "I am pale," I know that it is only my body, not my eternal self, that is pale. Why not apply such figurative use to the term overall? Then, when I say that "I" experience the result of "my" actions, it can be seen to be both clear and accurate. Granted, it seems like there is a real "self." But it only looks that way, just as a line of ants looks like a brown stripe on the ground. Close philosophical and introspective attention reveals that what seemed like a solid, coherent whole is in fact a false mental construction based upon a failure to notice its countless, fluctuating parts. ### 2.2 Momentariness and Continuity For centuries before Vasubandhu, Buddhist philosophers of the Sarvastivadin tradition had believed in, and argued for, the doctrine of universal momentariness.[24] This view was a Buddhist scholastic interpretation and expression of the Buddha's doctrine that all things in the world of ordinary beings were subject to causes and conditions, and therefore impermanent. Buddhists rejected the notion of substances with changing qualities, and affirmed instead that change was logically impossible. To be, for Buddhists, is to express certain inalienable characteristics, whereas to change must be to exhibit the nature of a different being. Vasubandhu adopts this view, employing it in many contexts: "As for something that becomes different: that very thing being different is not accepted, for it is not acceptable that it differs from itself." (AKBh 193.9-10) One can see how the impossibility of change, coupled with the doctrine of impermanence, served to prove that all things persisted for only a moment. Vasubandhu certainly shared this view, and he drew upon the premises of impermanence and the impossibility of change to establish momentariness in his own works. Yet, as von Rospatt (1995) has shown, Vasubandhu added a new twist to the argument. What he added was that things must self-destruct, for destruction cannot be caused. And why not? Because a cause and a result are real entities, and the ostensible object of a destruction is a non-existent. How, he asks, can a non-existent be a result? Given that things must bring about their own destruction, then, Vasubandhu needs only to recall the impossibility of change to establish momentariness. If things have it as part of their nature to self-destruct, they must do so immediately upon coming into being. If they do not have it as part of their nature, it can never become so.[25] The great difficulty for such a position is accounting for apparent continuities. The standard Buddhist explanation is that usually, when things go out of existence, they are replaced in the next moment by new elements of the same kind, and these streams of entities cause the appearance of continuity. Modern interpreters often illustrate the point with the example of the apparent motion on a movie screen being caused by a quick succession of stills. This is said to be the case with the many entities that appear to make up the continuous self, and of course this was the main reason the Buddha affirmed his doctrine of impermanence in the first place. Yet for some phenomena, to call their continuity merely apparent causes philosophical problems, even for Buddhists. Consequently, Vasubandhu, like his Sarvastivada forbears, was repeatedly preoccupied with the need to account for continuity. Many of the more prominent problems in continuity have already been mentioned in the discussion of Vasubandhu's disproof of the self (see above), including the problem of memory, the problem of the reference of a name, and the problem of karmic causality across multiple lifetimes. See also the section on the disproof of invisible physicality for Vasubandhu's rejection of the Vaibhasika response to a specific problem of continuity. One issue of great importance for the Vaibhasika and Yogacara traditions, and consequently of interest to current scholarship,[26] is a problem specific to Buddhist meditation theory. The problem comes from an apparent inconsistency among well-founded early Buddhist scriptural positions. On the one hand, there was the orthodox belief that the body was kept alive by consciousness. Even in deep sleep, it was believed that there was some form of subtle consciousness that was keeping the body alive. On the other hand, there was the very old belief, possibly articulated by the Buddha himself, that there are six kinds of consciousness, and that each of them is associated with one of the six senses--the five traditional senses, plus the mental sense (which observes mental objects). The problem was that there are some meditative states that are defined as being completely free of all six sensory consciousnesses. So the question becomes, What keeps the meditator's body alive when all consciousness is cut off? Related to this is the problem that, given that each element can be caused only by a previous element of a corresponding kind (in an immediately preceding moment), there does not seem to be any way for the consciousness, once cut off, to restart. The distinctive, early Yogacara doctrine of the "store consciousness" (alayavijnana) or the "hidden consciousness"--the consciousness that is tucked away in the body--was first introduced to solve these continuity problems. Equally problematic is the same issue in reverse: Without some doctrinal shift there does not seem to be any way that beings born into formless realms, with no bodies, could be reborn among those with physical form. The later, Yogacara view that everything is only appearance takes care of this, too, by eliminating the need for real physicality. (See the section concerning disproof of invisible physicality.) Vasubandhu sets out these problems in the Treasury of the Abhidharma without resolving them; when he writes from a Yogacara perspective, he resolves them with recourse to the store consciousness and appearance-only. ### 2.3 Disproof of Invisible Physicality In a complex but significant passage from the Treasury of the Abhidharma, Vasubandhu provides refutations to arguments that defend the necessity of there being a special kind of physicality, called "unperceived physicality," or "invisible physicality," (avijnaptirupa).[27] As is made clear throughout these arguments, this entity (dharma) was affirmed by the Kashmiri Vaibhasikas in order to account for the karmic effects of our physical and vocal actions. The need for such an entity reflects a conundrum unique to Buddhist morality, premised as it is upon the lawlike, causal regularity of karma operating in the absence of any divine intervention. The issue at hand is not in itself the question of karmic continuity in the absence of a self, but rather a particular aspect of that larger problem. (See the sections concerning disproof of the self and momentariness and continuity.) For this argument, then, let us agree that the continuum of consciousness carries on after death, and that one's future rebirth is therefore determined by the shape and character of one's consciousness at death. Given that, we can see how a mental action--say, holding a wrong view, or slandering the Buddha in one's thoughts--can have consequences for one's future, by directly affecting the shape of one's consciousness. But how do actions of speech and body--such as insulting, or hitting someone--bear their karmic fruit in the mental continuum? Surely forming the intention to hit someone is a consequential mental act. We might think that Buddhists would simply say that it is karmically equivalent, then, to intend to do something (which is a mental act) and to actually do it. Although there is some blurring of the line here (the Buddha does give great moral weight to merely intending to act), intentions and the bodily (or vocal) acts that follow from them are understood as distinct types of actions with distinct karmic results. Furthermore, the Buddha is said to have indicated that the karmic effect--and so the moral significance--of doing something is additionally dependent upon the success of the action. Attempted murder is not punished to the same degree as murder, even karmically. These distinctions accord with widely held moral instincts, but the Abhidharma philosophers needed a full, causal explanation for why the successful results of physical actions bring about their distinctive effects in one's consciousness. I can see how my plunging a knife into someone affects him; but how his dying (which may happen later, in the hospital) affects me is invisible, or uncognized. How does what goes around, come around? The Vaibhasika answer is that, since his death is a physical event, there must be another physical event--the invisible physicality--which impinges upon me. Vasubandhu rejects this entity, and provides a compelling explanation of the Buddha's differentiation between intentions, acts, and success that does not need to appeal to invisible physicality. Although here Vasubandhu answers through the mouth of another (and so, presents this position without explicitly endorsing it), he advocates something like this position more forcefully in his later, Explanation of the Establishment of Action.[28] He admits that there must be a karmic difference between (1) merely intending, (2) intending and acting, and (3) intending, acting and accomplishing the deed. But the difference does not come from the accomplishment of the result itself--that is, it does not come from the death of the murder victim. Rather, the karmic effect is the result of my experience and my beliefs. If I experience myself stabbing someone, and I know that the person died (even if by reading the newspaper the next day), I gain a fuller karmic result than I would have by simply intending to act. The Vaibhasikas present a number of other arguments that suggest the necessity of invisible physicality, all of which follow a similar logic. A close parallel to the example above is a murder by proxy; why do I experience a karmic result if the hit man I've hired does his job, but not if he doesn't? My action was the same either way. The Vaibhasikas also point to the enduring, karmic benefits of having taken vows to refrain from negative actions such as killing or stealing. These vows generate a karmic benefit in spite of their being fulfilled through not acting. How is this non-action different from that of someone who just happens not to be killing or stealing? Another argument suggests that advanced practitioners in refined meditative states cannot be said to be practicing the eight-fold path of the Buddha's teachings, since it includes "right speech," "right action" and "right livelihood"--but no action at all can be taken while in meditative equipoise. Finally, another argument points out that a donor gains great karmic merit when (obviously, later) the monk who receives and eats her gift of food uses its energy to attain a great meditative accomplishment. Such arguments draw upon numerous scriptures to indicate that morally significant effects seem to take place in the world even when physical actions are no longer possible or relevant. In each case, Vasubandhu reinterprets the scripture, or the situation, or both, in such a way as to make it sensible to describe the effect without appealing to invisible physicality. In the case of the vow, Vasubandhu says that a vow helps a vow-taker to remember not to do immoral deeds, and the disciplinary vows of monks have the effect of transforming the whole character of the mind. Therefore a disciplined vow-taker does not act without the thought of the vow arising. The vow does not cause a magical imprint on some invisible entity; it transforms the dispositions of the vow-taker. The same is true of the meditating practitioner; there is no lack of "right action" during meditation, because the meditator's continuum of aggregates is still disposed to act properly after the meditation session is complete. Such arguments exhibit Vasubandhu's general pattern of reducing apparent continuities to causal chains of momentary events, and eliminating as many extraneous elements from the system as possible. ### 2.4 Disproof of a Creator God In the Treasury, Vasubandhu provides an argument against the existence of Isvara, a God who is the sovereign creator of everything.[29] For Vasubandhu, theism is essentially the absurd notion that all of creation might be the result of a single cause. He begins his disproof with what has been called verbal equivocation (Katsura 2003, p. 113), and might also be taken as a joke or a pun: > > > Living beings proceed from conditions; nor is it the case that > "the cause (karana) of all the world is a > single Isvara, Purusa or Pradhana [i.e., > God]." > > > > What is the reason (hetu) for this? > > > > If you think it is established upon the provision of a reason > (hetu), this is a refusal of the statement, "the cause > (karana) of all is a single > Isvara." (AKBh 101.19-21) > > > The pun is in the fact that the word for "reason," hetu, also means "cause." It is as if Vasubandhu is saying, "If you're looking for my [just] cause for believing this, then clearly you don't believe God is the cause of everything! If I believe it, it must be 'cause God made me believe it!" This is somewhat silly, and apparently an equivocation between reasons and causes. A more serious reading, though, rises to the surface when we notice that Vasubandhu does not, in fact, verbally equivocate. He uses the word hetu to refer to reasons, and another word, karana, to refer to causes. In his own philosophy, Vasubandhu separates these two regions of reality quite strictly: "Causes" are real, substantial entities, whereas "reasons" are conceptual constructs devoid of causal capacity. (Cf. Davidson's "Anomalism of the Mental"; see the entries on anomalous monism and Donald Davidson.) This is not to say that Vasubandhu would endorse Sellars's notion that ideas which inhabit the normative "space of reasons" are irreducible to those within the "space of causes." (See the discussion of the myth of the given, in the entry on Wilfrid Sellars.) He would not; on the contrary, for Vasubandhu, to say that something is a conceptual construct is to say that it is caused, but in a way quite different from how it appears. This is developed explicitly in his Yogacara theory of the Three Natures of all entities, where one of the natures is said to be how things appear, and another is how things are caused. (See the discussion of three natures and non-duality.) Yet here in the Treasury Vasubandhu is only gesturing in this direction. His non-Buddhist opponents would also like to distinguish reasons from causes, so instead of attempting to unify their causal basis himself, Vasubandhu suggests that commitment to the unitary nature of divine causality commits them to conflating them. The serious point behind the opening joke, then, is that philosophy requires that we make distinctions among ideas and entities, and debate assumes differences of opinion and perspective. For Vasubandhu, such differences are evidence that it is impossible that all things have issued from a single cause. He begins, then, with a shot across the bow, which expresses just how seriously he will hold the theists to their claim that all things have one cause. The majority of Vasubandhu's argument consists in his support for the notion that the diversity of the world--especially its changes through time--are inconsistent with the notion of a single creator. An important, implicit premise in this argument is the notion that things do not change their defining, characteristic natures. Everything is what it is: "As for something that becomes different: that very thing being different is not accepted, for it is not acceptable that it differs from itself." (AKBh 193.9-10) Thus, if God is the cause of everything, God is always the cause of everything--or, if God's nature at time t is to be the cause of everything, then God must cause everything at time t. In response, the theist proposes a variety of predictable alternatives. One option is that God is posited as the creator of each specific thing at its specific time due to God's having unique, temporally-indexed desires. One person is caused by God's desire for a person to be born in Gandhara in the fourth century, and another person is caused by God's desire for a person to be born in New York City in 1969. Vasubandhu responds that if that is the case, then things are not caused by a unitary God, but rather each thing is caused by one of God's countless, distinct desires. Furthermore, the theist is then responsible to account for the causes of those specific desires. In addition, specific temporally-indexed desires imply that something is interfering with God's creative action--namely, the particular temporal location. If God is capable of--not only capable of, but having it as His inherent nature to be--creating something, what could possibly prevent that thing's occurrence at every moment in which God is expressing God's own nature (which should be every moment forever)? If God has the desire for a tree in Washington Square Park in 2010, it should always be 2010, with a tree in Washington Square Park. Why should God wait? (This argument relies upon a typical Buddhist penchant for particulars and moments as the touchstone of the real.[30]) One unwise option the theist proposes is that God's creation changes through time because God is using other elements--material causes, for instance--to bring things into being, and those elements follow their own causal laws. That, of course, makes God's actions subject to other causes, and amounts to the Buddhist view that all things are the results of multiple causes and conditions. Furthermore, when it is proposed that God engages in creation for his own pleasure (priti), Vasubandhu replies that this means that God is not sovereign because He requires a means (upaya) to bring about a cause, namely his own pleasure. As a cap to this argument, Vasubandhu mocks the idea that a praiseworthy God should be satisfied with the evident suffering of sentient beings. Thus, along with his causal argument, Vasubandhu throws in a barb from the argument from evil. Vasubandhu's final argument in this section is to say that theists are, or ought to be, committed to the denial of apparent causes and conditions. If you think that God is the only cause, then you must deny that seeds cause sprouts. This may seem like another overly literalistic reading of the nature of God as the only cause, but is God really deserving of the name "cause," if He has intermediary causes do His work of creation? For, Vasubandhu points out, ordinarily we do not require that something be an uncaused cause in order to call it a "cause." We call a seed a cause of a sprout, even if it is itself caused by a previous plant. So to say that God is the "cause" of the sprout whereas the seed is not amounts to denying a visible cause and replacing it with an invisible one. To avoid this, the theist might say that to call God the "cause" is to say that God is the original cause of creation. But if this is what God is doing to gain the name "cause," and God is beginningless (uncaused), then creation too must be beginningless--which means that, once again, God has nothing to do. ## 3. Approaches to Scriptural Interpretation Vasubandhu was a master of Buddhist scripture. A great number of scriptural commentaries are attributed to him, and we find Vasubandhu citing scripture literally hundreds of times throughout his philosophical works. (Pasadika 1989) Furthermore, Vasubandhu is one of very few Indian or Tibetan authors to have written a treatise dedicated to the interpretation and exposition of Buddhist scripture, called the Proper Mode of Exposition (Vyakhyayukti).[31] Here is not the place for a thorough survey of this material, which awaits future research. It is, however, worth sketching a characteristic pattern in Vasubandhu's citation of scripture, and noting its relation to his view of scripture in the Proper Mode of Exposition. Like other scholastic philosophers, Vasubandhu often cites scriptures in support of an argument. Unlike many others, however, Vasubandhu regularly presents a scriptural passage as a potential counter-argument, only to refute the traditional interpretation of the passage. The opponent's interpretation, moreover, is quite often the most literal or direct reading of the passage in question. One of Vasubandhu's characteristic philosophical moves, then, is to argue in favor of a secondary interpretation of a given scriptural passage that might, on its face, be thought to disprove his view. A prominent, paradigmatic example of this use of scriptural citation takes place in the Twenty Verses Commentary. As will be discussed below (in the section concerning defense of appearance only), this text is dedicated to proving the central Yogacara thesis of "appearance only" (vijnaptimatra)--that is, that everything in the many worlds of living beings is only apparent, with no real existence outside of perception. A central challenge to this idealistic view, then, comes from the objector's simple statement that, "If the images of physical forms, and so on, were just consciousness, not physical things, then the Buddha would not have spoken of the existence of the sense bases of physical form, and so on." (Vims 5) Vasubandhu thus adduces the Buddha's word as the basis of the objection. Then, he turns to further scriptures to account for, and explain away, this extremely well-founded Buddhist doctrine. Vasubandhu's response to his internal objector is to explain that the passages in question need to be understood as spoken with a "special intention" (abhipraya--on which, see Broido 1985, Collier 1998 and Tzohar 2018). Furthermore, in addition to defending the idea of reinterpreting scripture in general, Vasubandhu defends the act of reinterpretation in this case. In fact, we may distinguish four separate elements of the argument: (1) Vasubandhu explains the Buddha's general practice of speaking words that are not technically true, but are beneficial to his listeners. (2) He proves that such interpretations are sometimes necessary, by providing two passages that without such a reading would prove the Buddha to have contradicted himself. (3) He provides an explanation of the Buddha's "special intention" in the passage under discussion--that is, he declares what the Buddha actually believes on the topic. And, (4) he suggests why, in this case, the Buddha might have chosen to speak the words he did, instead of simply saying what he really believed. For (1) and (2), Vasubandhu explains that the Buddha's statement that "there is a self-generated living being (satva)" must be taken as having a "special intention," so as to prevent its contradicting his statement that "there is no self or living being (satva), only entities with causes." (Vims 5) Vasubandhu says that the Buddha spoke the first statement in the context of explaining karma, in order to indicate that the continuity of the mental series was not "cut off" (uccheda). If the Buddha had taught about the no-self doctrine at the time he delivered the discourse on "the self-generated living being," his listeners would have been led to this false view that the mental stream comes to an end at death. Such a false view is often translated as "nihilism," and among Buddhists it is considered far more dangerous than the false belief in a self, because it entails the further false view that our moral actions have no afterlife consequences.[32] If you believe that, then you are liable to act in a way that will lead you to hell. The danger of nihilism is perhaps the primary justification for the Buddha's having spoken words that were technically untrue. Vasubandhu provides a similar example in the "Disproof of the Person" section of the Treasury. (See the section concerning disproof of person.) There, Vasubandhu was arguing with the Personalists, who pointed to a scripture in which the Buddha remained silent when asked whether or not the person existed after death. The Personalists, of course, saw this as evidence that the Buddha affirmed the ineffability of the person. Vasubandhu says, in response, that the Buddha explained quite clearly (to Ananda, his disciple, after the questioner left) why there had been no good way to answer. To affirm a self would be to imply a false doctrine, but to deny the self (in this case, stating the truth) would cause the confused questioner to fall into a still greater falsehood: Namely, the thought of formerly having had a self, but now having none. This is the view, once again, of the self being "cut off" or "destroyed" (uccheda), the view that leads to moral nihilism. The danger of the false view of "destruction" is an extreme case, in which the Buddha is silent so as not to have to lead his disciple into a dangerous view. To what degree can this paradigm be extended to suggest other, perhaps less dire, cases in which the Buddha may have neglected to speak frankly? For Vasubandhu, apparently, the presence of a contradiction between two statements from the Buddha is one key. The Buddha cannot have literally meant both that there was no living being (satva) and that living beings (satva) came into being in a particular bodily form. A similar contradiction is brought to bear in the discussion with the Personalists, during their argument over the functionality of memory in the absence of an enduring self. There, Vasubandhu cites a scripture in which the Buddha says that whoever claims to remember past lives is only remembering past aggregates in his own continuum--a quote which vitiates the Personalists' citation of the Buddha having said of his own past life, "I was such-and-such a person, with such-and-such a form." To return to the case at hand from the Twenty Verses Commentary, although Vasubandhu does not indicate a direct contradiction in the Buddha's words, he does explain (3) what he takes to be the truth behind the Buddha's statements about the "sense bases of physical form, and so on." The literal meaning of the twelve sense bases, recall, is simply the listing of the six sensory organs and their six sensory objects. The truth behind this, for Vasubandhu, is the Yogacara causal story, which tells how conscious events arise from karmic seeds within the mind's own continuum. Each consciousness comes into being with a particular subject/object appearance imprinted upon it by its karmic seed. There are no subjects and no objects, only "dual" appearances. It is this causal story that was the "special intention"--the real truth--behind the Buddha's words affirming the sense bases. Of course, the Yogacara causal story denies the functionality of the senses, and so the "special intention" is a direct contradiction of the literal meaning of the sense bases. In response to this supposed explanation, the opponent asks for the Buddha's justification for speaking in this way, and (4) Vasubandhu provides two separate reasons, targeted at two different levels of listeners. For students who do not yet understand the truth of appearance-only, the Buddha taught the sense bases in order to help them to understand the absence of self in persons. This seems to cohere with Vasubandhu's Treasury argument that uses the sense bases as a way to explain away the apparent self. For students who do understand the truth of appearance-only, the doctrine of the sense bases is useful for helping them to understand the absence of self in the elements--that is, in the sense bases themselves.[33] Thus, for Vasubandhu, the Buddha's teachings are still useful, even when they are understood to have been spoken from a provisional standpoint. Perhaps this point can be extended still further. For Vasubandhu, while scriptures are always valid sources of knowledge, to excavate their meaning often requires in-depth, sophisticated, philosophical reasoning. We find Vasubandhu mocking his opponents, in one place in the Treasury, for understanding "the words" rather than "the meaning" of a scriptural passage in question. (204.1) Vasubandhu often argues against his opponents' overly literal interpretations of scripture as a crucial part of his argument against their philosophical positions. When he turns to a formal analysis of expository method and theory, Vasubandhu defends this broad, rationalist approach to scripture. In the Proper Mode of Exposition, Vasubandhu defends the legitimacy of the Mahayana Sutras against a "Sravaka" opponent, and defends their figurative, Yogacara interpretation against a Madhyamika opponent. Cabezon (1992) usefully summarizes Vasubandhu's arguments here in three general categories: Arguments based on the structure of the canon; arguments based on doctrinal contents; and arguments based upon "intercanonical criteria for authenticity." As we have seen in his Treasury and Twenty Verses arguments, Vasubandhu argues that naively to require that all scriptures be interpreted literally is to insist that the Buddha repeatedly contradicted himself. He cites many internal references to lost or unknown texts, and argues that this shows that no lineage or school can claim to have a complete canon. Unlike his Madhyamika opponents, Vasubandhu believes that the Mahayana Sutras must be read under a "special intention," so as to prevent the danger of nihilism. As Cabezon writes, both the Sravaka and Madhyamaka position share an over-emphasis on literal readings. Vasubandhu's emphasis on "correct exegesis" (a literal translation of Vyakhyayukti) shows the Yogacara to be the school of greatest hermeneutical subtlety and sophistication. ## 4. Major Yogacara Arguments and Positions ### 4.1 Defense of Appearance Only The Twenty Verses is Vasubandhu's most readable, and evidently analytic, philosophical text, and it has consequently drawn a significant degree of modern scholarly attention.[34] It begins with a simple statement, which Vasubandhu defends throughout the brief text and commentary: "Everything in the three realms is only appearance (vijnaptimatra)."[35] The three realms are the three states into which Buddhists believe living beings may be reborn. For all beings except for Buddhas and advanced bodhisattvas, the three realms make up the universe. Vasubandhu glosses this statement with citations from scripture that make it clear that he means to say that there no things (artha), only minds (citta) and mental qualities (caitta). He says that the experience of the three realms is like the appearance of hairs in front of someone with eye disease. It is the experience of something that does not exist as it appears. Although the term has a history of controversy among interpreters of Yogacara, it seems safe to say that the Twenty Verses defends at least some form of idealism. (See the section on the controversy over Vasubandhu as idealist.) After stating his thesis that everything is "only apparent," Vasubandhu immediately voices a potential counter-argument, which consists in four reasons that the three realms being appearance only is impossible: First and second, why are things restricted to specific places and times, respectively? Apparent objects can appear anywhere, at any time. Third, why do beings in a given place and time experience the same objects, and not different objects according to their distinct continua? And fourth, why do objects perform causal functions in the real world, when merely apparent mental objects do not? These objections aim to prove the impossibility that the world is merely apparent by arguing that the elements of ordinary experience behave in ways that what is merely apparent does not. Vasubandhu sets up, and meets, these objections in order to prove the possibility, and hence the viability of the theory, that everything in experience is appearance only. He shows, essentially, that the objector has a narrow view of what is possible for merely apparent things; appearances are not necessarily limited in the ways the objector thinks. Notice that it is not incumbent upon Vasubandhu here to prove that all instances of the merely apparent transcend the limitations assumed by the objector. He has not set up this objection in order to try to prove, absurdly, that all mental images, for instance, are spatially restricted, or that all mere appearances perform observable causal functions in the real world. On the contrary, one upshot of the recognition that things are appearance only will be that things do not have the cause-and-effect structure that we ordinarily take them to have. In any event, the positive argument that ordinary experience is, in fact, illusory, comes later and takes the form of an argument to the best explanation. Before an explanation can be the best, though, it must be a possibility. So here, all Vasubandhu must do to counter these initial objections is provide, for each, a single example of a mental event that exemplifies the behavior that the objector claims is only available to physical objects.[36] To defeat the objections that mental objects are not restricted in space and time, Vasubandhu provides the counterexample that in dreams objects often appear to exist in one place and time, as they do in ordinary waking reality. In a dream, I can be looking at shells on a beach on Long Island, during the summer of my eighth year. It is only upon waking that I come to realize that the dream objects (the shells, the beach) were only mental fabrications, temporally dislocated, with no spatial reality. Thus, what is merely apparent can sometimes have the character of appearing in a particular place and time. To say they do not is to misremember the experience.[37] Next, to defeat the objection that unlike ordinary physical objects, merely apparent objects are not intersubjectively shared by different beings, Vasubandhu provides the counterexample that in hell, demonic entities appear to torment groups of hell beings. This is a case of a shared hallucination. When the objector wonders why the demons might not in fact be real, Vasubandhu appeals to karma theory: Any being with sufficient merit--sufficient "good karma"--to generate a body capable of withstanding the painful fires of hell would never be born into hell in the first place. Any creature in hell that is not suffering must be an apparition generated by the negative karma of the tormented. Finally, to defeat the objection that merely apparent objects do not produce functional causal results, Vasubandhu provides the memorable counterexample of a wet dream, in which an evident, physical result is produced by an imagined sexual encounter with a nonexistent lover. The initial objection to the mere possibility of the "appearance only" theory quelled, Vasubandhu turns to his main positive proof. This consists in a systematic evaluation of every possible account of sensory objects as physical, which ultimately leads Vasubandhu to conclude that no account of physicality makes more sense, or is more parsimonious, than the theory that it is appearance only. Before turning to Vasubandhu's treatment of physics, it is worth stopping to note the crucial importance that Buddhist karma theory plays in Vasubandhu's argument, overall. First, the proof of shared hallucinations in hell depends upon the particulars of the Buddhist belief in the hells. Of course, we might have believed in shared hallucinations even without believing in karma. But the tormenters in hell play an important, double role in Vasubandhu's argument. He has the objector raise the question again, and suggest as a last-ditch effort that perhaps, the tormenters are physical entities generated and controlled by the karmic energies of the tormented. At this, Vasubandhu challenges his objector: If you're willing to admit that karma generates physical entities, and makes them move around (pick up swords and saws, etc.), so that they might create painful results in the mental streams of the tormented, why not just eliminate the physical? Isn't it simpler to say that the mind generates mental images that torment the mind?[38] This a crucial question, because it resonates beyond hell beings, across Buddhist karmic theory. In the Treasury of the Abhidharma, Vasubandhu expressed the widely-held Buddhist view that in addition to causing beings' particular rebirths, karma also shapes the realms into which beings are reborn and the non-sentient contents of those realms. Such a belief provides Indian religions with answers to questions often thought unanswerable by western theisms, such as why the mudslide took out my neighbor's house, but not mine. But this view of karmic causality requires that the physical causes of positive or negative experiences are linked back to our intentional acts. (For more on the continuity problems associated with karmic causality, see the discussion of the disproof of invisible physicality.) Vasubandhu does not say so explicitly, but if it is easier to imagine the causes of a mind-only hell demon than a physical one, it should also be easier to imagine the causes of a mind-only mudslide--assuming that both are generated as a karmic repercussion for the beings that encounter them. The background assumption that any physical world must be subject to karma, therefore places the realist on the defensive from the start. Can the external realist adduce a theory intellectually satisfying enough to counter Vasubandhu's suggestion that we throw up our hands and admit that what appears to be out there is only in our minds? It is in response to this that the objector cites scripture, saying that the Buddha did after all, teach of the sense bases, saying that eyes perceive physical forms, and so on. As discussed above, Vasubandhu reinterprets this scripture as having a "special intention" (abhipraya). (See the section concerning scriptures.) The Buddha did not intend to affirm the ultimate reality of such entities. Yet the objector is not satisfied with Vasubandhu's alternate reading of the Buddha's words. In order to doubt such an explicit, repeated, statement, Vasubandhu needs to prove that it could not be interpreted directly. Even if the appearance-only world is possible, and even if it accounts better for karmic causality than the physical world, a direct reading of scripture is still supported, unless and until its doctrines are shown to be internally inconsistent. ### 4.2 Disproof of Sensory Objects In order to disprove the possibility of external objects, Vasubandhu delves into the atomic theory of the Kashimiri Vaibhasikas as well as that of the ("orthodox"/Hindu) Vaisesikas.[39] The purpose here is to undermine every possible theory that might account for perception as caused by non-mental entities. Vasubandhu's argument at this stage is entirely based in mereology--the study of the relations between parts and wholes. He argues first that atomism--the view that things are ultimately made up of parts that are themselves partless--cannot work. Then, he argues that any reasonable explanation of objects of perception must be atomic, by arguing that the alternative--an extended, partless whole--is incoherent. Vasubandhu takes it that together, these conclusions prove that external objects must be unreal appearances. I can only summarize Vasubandhu's sophisticated mereological arguments. He begins with the assertion that anything that serves as a sensory object must be a whole made up of basic parts, a bare multiplicity of basic parts, or an aggregate. But none of these can work. A whole made up of parts is rejected on the grounds that things are not perceived over and above their parts. This is a well-developed Buddhist Abhidharmika view, which coheres with the rejection of a self over and above the aggregates. A bare multiplicity of partless parts is rejected on the grounds that separate atoms are not perceived separately. Thus the only sensible option is a grouping of parts--an aggregate--that somehow becomes perceptible by being joined together. The combination of partless entities, however, is conceptually impossible. Vasubandhu points out that if they if they combine on one "side" with one atom and another "side" with another--those "sides" are parts. The opponent must account for the relation between those parts and the whole, and we are brought back to the beginning. Furthermore, if they are infinitesimal, they cannot be combined into larger objects. It is proposed, instead, that perhaps a partless entity may be extended in space, and so perceived. But perception is generated by contact between a sensory organ and its object. This requires the object to put up some kind of resistance. But if a thing has no parts, then its near side is its far side, which means that to be adjacent to it is to have passed it by. Partless atoms are therefore logically incapable of providing the resistance that is definitive of physicality and the basis for sensory contact (Vasubandhu says that they cannot produce light on one side and shade on the other). This confirms that entities must be combined into larger groupings in order to be functional and perceptible, which has already been shown impossible. If atoms and extended partless entities are impossible, so also are unitary sensory objects. Vasubandhu is well practiced in the Abhidharma arguments reducing apparent wholes to their composite entities. Suppose we say I see something that has both blue and yellow in it. How, if it is one unitary thing, can I see these two colors at different points? What makes one point in an object different from another point, if the object is of a single nature? This is clearly parallel to the argument against Isvara as a single cause of all things. A similar case is movement across an extended thing. As with a partless atom, to step on the near side of a partless extended singular entity would be to step on its far side, too. So there can be no gradual movement across a singular entity. As Vasubandhu concludes, one cannot but derive the need for partless, atomic units from such reasoning. But partless atomic units are imperceptible. So, perception is impossible; apparently perceived objects are only apparent. The main part of his argument settled, Vasubandhu entertains another set of counterarguments, which are the charges of solipsism and moral nihilism: How, the internal objector asks, can beings interact with one another? What is the karmic benefit of helping or harming, when we are all merely apparent beings? Vasubandhu's direct, candid response to these challenges may be viewed as a failure of imagination, or a surprising willingness to bite the bullet of anti-realism. Mental streams, he says, interact in essentially the same way as we imagine physical objects to interact. Minds affect minds directly. When you speak to me, and I hear you, we ordinarily think that your mind causes your mouth to produce sounds that my ears pick up and transform into mental events in my mind. Vasubandhu takes Occam's razor to this account and says that--given that we have no sensible account of physicality, let alone mental causation of a physical event and physical causation of a mental event--it makes more sense if we eliminate everything but the evident cause and the result: Your mind and mine. Note that Vasubandhu is not saying that nothing in our appearances exists; he is saying, on the contrary, that mere appearances bear all the reality that we need for full intersubjectivity and moral responsibility. It seems worth noting in this context that something similar to Vasubandhu's option just may be the only option available for a modern physicalist who believes that human minds are constituted by, or out of, neurons. Human persons, for such a modern, do not exist as we imagine them to, and are not causally constituted in ways that we can intuit or even comprehend, except vaguely. Does this mean that we must advert to moral nihilism or solipsism? Vasubandhu's alternative is to affirm our pragmatic acceptance of our own, and others' experiences and causal responsibilities, and not to imagine that we must await an account of physicality before trusting the effectiveness of our mental, moral, causality. Read in this way, Vasubandhu seems much closer to Madhyamaka philosophy than is ordinarily assumed; see Nagao (1991). ### 4.3 Three Natures and Non-Duality Vasubandhu did the work of defending the controversial, Yogacara thesis of appearance-only (vijnaptimatra) in his Twenty Verses. In his short, poetic works the Thirty Verses and the Three Natures Exposition, he does not entertain objections.[40] Instead, he draws together the basic doctrinal and conceptual vocabulary of the Yogacara tradition and forms an elegant, coherent system. The Thirty Verses includes a complete treatment of the Buddhist path from a Yogacara perspective. In order to understand Vasubandhu's contribution in this text, it would be necessary to place it against the wide backdrop of previous Yogacara doctrine. This scholarly work is ongoing. Here, let us focus instead on introducing the basics of the intricate doctrinal structure formulated in the Three Natures Exposition. The Three Natures Exposition takes as its topic, and its title, the three natures of reality from a Yogacara perspective: These are the fabricated nature (parikalpita-svabhava), the dependent nature (paratantra-svabhava) and the perfected nature (parinispanna-svabhava). In Yogacara metaphysics, things no longer have only one nature, one svabhava; rather, things have three different, if interconnected, natures. We may note immediately how fundamentally this differs from the Treasury, in which the apparent purpose is, at all times, to determine the singular nature of each entity (dharma). In that context, to discover that an entity must be admitted to have more than one, distinct nature, would be to discover that that entity is unreal. Something that changes, for instance, is internally inconsistent because of exhibiting multiple natures. Vasubandhu employs exactly such arguments to deny the ultimate reality of many specific entities (dharma) in the Treasury. Such entities, he says, are unreal; like a "self," they are only conceptual fabrications (parikalpita). One of the basic distinguishing features of Great Vehicle (Mahayana) metaphysics is the affirmation that not only the person, but all entities (dharma) are empty of a "self" or an inherent nature (svabhava). That means, in the terms laid out in the Treasury, that all dharmas are only conceptual fabrications. But that is not the only way of looking at things. To say that all things have three inherent natures is not to take back the Great Vehicle's denial of inherent natures, but rather to explain it. It is to say that, what we ordinarily take to be a thing's individual character or identity is best understood from three angles, so as to explode its unity. The first nature is the fabricated nature, which is the thing as it appears to be. Of course, to use this term is to indicate the acceptance that things do not really exist the way they appear. This is a thing's nature as it might be defined and explained in ordinary Abhidharma philosophy, but with the added proviso that we all know that this is not really how things work. The second nature is the dependent nature, which Vasubandhu defines as the causal process of the thing's fabrication, the causal story that brings about the thing's apparent nature. The third nature, finally, is the emptiness of the first nature--the fact that it is unreal, that it does not exist as it appears. Let's take the most important example, for Buddhists, of a thing that appears real but has no nature: the self. With this as our example of something mistakenly thought to have a nature (svabhava), we may run through its three natures. The self as it appears is just my self. I seem to be here, as a living being, typing on a keyboard, thinking thoughts. That is the first nature. The second nature is the causal story that brings about this seeming self, which is the cycle of dependent origination. We might appeal to the standard Abhidharma causal story, and say that the twelve sense bases or the five aggregates, causally conjoined, bring about the conceptually-fabricated self. In the Yogacara, this causal story is entirely mental, of course, so we cannot appeal to the sense bases themselves as the real cause. Rather, the sense bases only appear to be there due to karmic conditioning in my mental stream. For this reason, the second nature should instead be said to be the causal series according to which the mental seeds planted by previous deeds ripen into the appearance of the sense bases, so that I think I am perceiving things--which, in turn, makes me think I have a self. Finally, the third nature is the non-existence of the self, the fact that it does not exist where it appears. Of course, if there was a real self, I could not have provided my explanation of how it comes to appear to be there. In Vasubandhu's telling, the three natures are all one reality viewed from three distinct angles. They are the appearance, the process, and the emptiness of that same apparent entity. With this in mind, we can read the opening verses of the Three Natures Exposition: > > > 1. Fabricated, dependent and perfected: So the wise understand, in > depth, the three natures. > > > > 2. What appears is the dependent. How it appears is the fabricated. > Because of being dependent on conditions. Because of being only > fabrication. > > > > 3. The eternal non-existence of the appearance as it is appears: That > is known to be the perfected nature, because of being always the > same. > > > > 4. What appears there? The unreal fabrication. How does it appear? As > a dual self. What is its nonexistence? That by which the nondual > reality is > there.[41] > > > These verses emphasize the crystalline, internal structure to the three natures. "What appears" is one nature (the dependent), whereas "how it appears" is the fabricated. "The eternal non-existence of the appearance as it appears" is the perfected. These are ways of talking of the same entity, or event, from different perspectives. Also evident here is the crucial Yogacara concept of "duality." The fabricated is said to appear as a "dual self," whereas reality (dharmata) is nondual. Since, as we have seen above, Vasubandhu believes in the selflessness of all dharmas, the "dual self" is the false appearance of a self that is attributed to any entity. Things and selves mistakenly appear "dual." What does this mean? The twosome denied by the denial of "duality" has many interpretations across Yogacara thought, but for Vasubandhu the most important kinds of duality are conceptual duality and perceptual duality.[42] Conceptual duality is the bifurcation of the universe that appears necessary in the formation of any concept. When we say of any given thing that it is "physical," we are effectively saying, at the same time, that all things are either "physical" or "not physical." We create a "duality" according to which we may understand all things as falling into either one or the other category. This is what it is to define something, and to ascribe it its characteristic nature (svabhava). This is why every "self" is "dual." To say something is what it is imposes a duality upon the world. And, given that the world is in causal flux and does not accord with any conceptual construct, every duality imposes a false construct upon the world. To illuminate this with regard to the person, when I say that "I" exist, I am dividing the universe into "self" and "other"--me and not-me. Since, for Buddhists, the self is unreal, this is a mistaken imposition, and of course many ignorant, selfish actions follow from this fundamental conceptual error. In addition, but closely related to conceptual duality, is perceptual duality. This is the distinction between sensory organs and their objects that appears in perception. When I see a tree, I have the immediate impression that there is a distinction between that tree, which I see, and my eye organs, which see the tree. I take it that my eyes are "grasping" the tree, and furthermore I understand that the eyes (my eyes) are part of me, whereas the tree is not part of me. The same is true of all of the sensory organs and their objects. The Abhidharma system calls the sensory organs the "internal" sense bases, and the sensory objects the "external" ones. It is the combination of the "internal" sense bases upon which I impose the false construction of self. Thus perception provides the basis for the conceptual distinction between self and other. For Yogacara Buddhists, however, sense perception is just as false a "duality" as are false conceptual constructions. Given that the external world must be only mind, the sensory experience that we have of a world "out there" is only a figment of our imagination. The self/other distinction implicit in perception is as much a false imposition as is the self/other distinction implicit in the concept, "self." Perception, like conceptualization, is only a matter of the mind generating "dual" images. As counterintuitive as this sounds, it may be clarified by analogy to the idea of a multi-player virtual reality game. In a shared virtual reality experience, the first thing the computer system must be able to track is where, objectively, everything is (where the various players are, where the castle with the hidden jewels is, where the dragon is hiding, etc.). Then, when any new player logs in, the system can place that player somewhere in the multi-dimensional, virtual world. At that point, the computer must generate a sensory perspective for that individual. Immediately that person experiences herself to exist in a world of a certain kind, with certain abilities to move around, and fight, and so on. But this is only a trick of the software. The world is unreal, and so is the player's subjective perspective on that world. For the Yogacara, our sensory experience is something like this. We are all only mental, and so the apparent physicalities that intervene between us are merely false constructions of our deluded minds. Furthermore, the physical body that we take ourselves to have, as we negotiate the world, making use of our senses, is also not there. Both external and internal realities are mistaken to the degree that we see them as "two" and not "nondual" constructions arising out of a single mental stream. Vasubandhu's clearest, most evocative, and most famous explanation of the three natures appears later on, when they are analogized to a magician's production of an illusion of an elephant, using a piece of wood and a magical spell: > > > 27. It is just as [something] made into a magical illusion with the > power of an incantation (mantra) appears as the self of an > elephant. A mere appearance (akaramatra) is > there, but the elephant does not exist at all. > > > > 28. The fabricated nature is the elephant; the dependent is its > appearance (akrti); and the nonexistence of an > elephant there is the perfected. > > > > 29. In the same way, the fabrication of what does not exist appears as > the self of duality from the root mind. The duality is utterly > nonexistent. A mere appearance (akrtimatraka) > is there. > > > > 30. The root-consciousness is like the incantation. Suchness > (tathata) is understood to be like the piece of wood. > Construction (vikalpa) should be accepted to be like the > appearance of the elephant. The duality is like the > elephant.[43] > > > Notice the two different elements here in the ending of Verse 30: The "appearance of the elephant" is "construction," that is, the ongoing process of imagination that arises from the karmic causal stream. The real problem, though, "duality," the dangerous, false distinction between self and other, is analogized to the illusory elephant itself. If I am at a magic show, I should expect to see the "appearance of an elephant." That will make the show worth attending. But if I am not a fool, I will not imagine myself to actually see an elephant. That would be to construct duality. For Vasubandhu, this is a crucial distinction. The basic problem for living beings in his view is not that we experience an illusory world; the problem is that we are fooled into accepting the reality of our perceptions and our concepts. Once we no longer believe, once we see the falsity of the illusion, the illusion goes away and we can come to experience the truth that lies behind the illusion--the ineffable "suchness" (tathata), the piece of wood. ## 5. Controversy over Vasubandhu as "Idealist" Let us posit that metaphysical idealism need not be the counterintuitive view that all that exists is, simply, mental. To call a view idealist it is sufficient merely that it hold that everything is dependent upon mind--a mind, or many minds. Idealists hold that the natures or forms of things are mind-dependent. Of course, such dependence is, in itself, counterintuitive, given that we generally take the physical world to have existence independent of minds. Contemporary prevailing wisdom holds that the direction of causal dependence goes in the opposite direction; the physical world in fact generates and sustains minds (using neurons to do so). Idealism says that ideas, or minds, are in one way or other behind, or prior to, all other forms of existence. Given this definition, is Vasubandhu, the author of the Twenty Verses and the Thirty Verses and probably the Three Natures Exposition, a metaphysical idealist? At first blush, it seems quite sensible to say that he is. The evident purpose of the Twenty Verses is to argue that the core background assumptions of Mahayana Buddhism (especially its views on karma) suggest the idealistic thesis that is asserted in the text's opening: "In the Great Vehicle, the threefold world is only appearance."[44] Vasubandhu argues throughout the Twenty Verses that whatever aspects of the world appear to be independent of mind, they are better explained as mental constructions. What that means is, apparently, that not only is Vasubandhu an idealist in the sense that he believes that everything in experience is dependent upon mind, but he also believes that everything simply is mind. This is the radical "mind only" (cittamatra) position that he believes is necessitated by adherence to a Mahayana worldview. Things are not quite this simple, however. First, it is important to note two levels to Vasubandhu's "idealism." The first level points to an analysis of what appears in perception. Vasubandhu says that everything that we ordinarily ascribe existence in the world--such as a self or an object--is actually only mental. Tables and nebulae and eyes are mental fabrications that have no existence apart from their "shape" (akara), the way they appear.[45] The second level points to the causal story that brings about such appearances. Along with other Yogacarins, Vasubandhu argues that that causal story too is only mental. Perception, in the form of active sensory and mental consciousnesses, is caused by the "storehouse consciousness" (alayavijnana), which stores, in mental form, the seeds of all of our future experiences. The storehouse is constantly replenished, because new "seeds" (bija) are created every time we enact a morally significant action (karma). It is mental causation, then, that brings about our mental experiences. It is this second level, then, that confirms Vasubandhu as an idealist. Not only the reality that we inhabit (which we ordinarily think of as at least partly non-mental), but also the causal supports upon which such a reality depends, are only mental. Even if the storehouse consciousnesses of living beings provided ongoing support to a real, physical world, we would still want to call that view "idealism" for its insistence on the causal priority of the mental. Since Vasubandhu does not say as much, but instead says that everything that is created by minds is itself mental, we might say that in addition to a causal idealism, Vasubandhu's is a universal, ontological idealism. Behind both of these levels, however, there is an important sense in which Vasubandhu is not really an idealist. Like all Mahayana Buddhists, Vasubandhu believes that whatever can be stated in language is only conventional, and therefore, from an ultimate perspective, it is mistaken. Ultimate reality is an inconceivable "thusness" (tathata) that is perceived and known only by enlightened beings. Ultimately, therefore, the idea of "mind" is just as mistaken as are ordinary "external objects." For this reason, we can say that Vasubandhu is an idealist, but only in the realm of conventions. Ultimately, he affirms ineffability. \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| Level 1 \| Experience \| False Appearances (self, objects, concepts) \| Idealism \| \| Level 2 \| Causality \| Storehouse Consciousness (produces consciousness) \| Idealism \| \| Level 3 \| Ultimate Reality \| Only Buddhas Know (tathata) \| Ineffability \| Levels of Analysis of Vasubandhu It is extremely important to keep these levels apart, in order to prevent Vasubandhu from committing an evident self-contradiction. At the end of his commentary to verse 10 of the Twenty Verses, Vasubandhu imagines an interlocutor challenging the appearance-only view by saying that if no dharmas exist at all, then so also appearance-only doesn't exist. How, then, can it be established? This charge of self-referential incoherence is faced by anti-realists of all stripes, and it is familiar to Buddhist philosophers from Nagarjuna's Vigrahavyavartani. (See the entry on Nagarjuna in this encyclopedia.) Vasubandhu's reply in the Twenty Verses Commentary is that appearance-only does not mean the nonexistence of dharmas. It only means the absence of a nature (svabhava) that is falsely attributed to a constructed self (kalpitatmana), and Vasubandhu is quick to say that when one establishes appearance-only, appearance-only must also be understood to be itself a construction, without a self. Otherwise, there would be something other than an appearance, which would by definition contradict appearance-only. All there is, is appearance, and everything that appears is a false construction of self. We can take two important lessons from this. First, representation-only cannot be the ultimate truth. If Vasubandhu were affirming the ultimate reality of the "mind-only" causal story that brings about appearances, or the appearances themselves (levels 2 and 1, respectively), then he would surely have said so in response to the charge of self-referential, nihilistic incoherence. Instead, his argument adverts to another level for ultimate reality. Madhyamaka critics of Yogacara have made much of the notion that the latter have reified the mind and made it a new "ultimate."[46] Yet Vasubandhu's position is clearly intended to elude such a critique. Second, for Vasubandhu the lack of a self-nature (svabhava) just is, precisely, what he means by "appearance-only." This is also contrary to a common understanding of Yogacara which arises from the equation of "appearance-only" with "mind-only." This equation is not, technically, a mistake in the case of Vasubandhu. He says, specifically, that "mind" (citta) and "appearance" (vijnapti) are synonyms. The mistake is what we moderns tend to think, based on a background assumption of mind/matter Cartesian dualism--namely, that the expression "mind only" is used to affirm the reality of the mental at the expense of the physical. Here, reality is certainly not physical, but that is not Vasubandhu's point. His point is to equate the lack of independent nature (svabhava) with the product of mental construction. This is a simple statement of idealism, as mentioned above: Everything that exists must either have independent existence, or be dependent upon mind, a mental construction; and since, for the Great Vehicle, nothing has independent existence, everything is only a mental construction. What it means to be "mind only," then, is not simply to be made of mental stuff rather than physical stuff. It is to be a false construction by a deluded mind. Again, we can see that there is no possibility that Vasubandhu is talking about the "ultimate" here. These points, and the above table, may clarify some of the difficulties in the ongoing, contentious debates over whether Vasubandhu and other Yogacarins should be classified as idealists or mystics. A quarter century ago, Thomas A. Kochumuttom (1982) argued quite powerfully and sensibly for his time that Vasubandhu should not be called an idealist, but he was arguing primarily against authors such as Theodore Stcherbatsky and S. N. Dasgupta, about whom he wrote (200): > > All of the above quoted passages clearly show that their authors > almost unanimously accept vijnapti-matrata > or prajnapti-matrata or > citta-matrata as the Yogacarin's > description of the absolute, undefiled, undifferentiated, non-dual, > transcendent, pure, ultimate, permanent, unchanging, eternal, > supra-mundane, unthinkable, Reality, which, according to them, is the > same as Parnispanna-svabhava, or > Nirvana, or Pure Consciousness, or > Dharma-dhatu, or Dharma-kaya, or the > Absolute Idea of Hegel, or the Brahman of Vedanta. > One can appreciate Kochumuttom's frustration. Vasubandhu is cautious not to conflate a description of perceived entities as reducible to only mind (Level 1) with a description of the ineffable ultimate reality (Level 3). Vasubandhu explicitly denies that "mind" has ultimate reality. He is not a Hegelian idealist. But not all idealists are Hegelian, absolute idealists. Among idealisms, Vasubandhu's is more closely aligned with Kant's, in that both assert that the objects of our experience are only representations, while both also affirm the reality of unknowable things in themselves (Ding an sich).[47] Kochumuttom was aware of this, and even in the course of his extended argument against using the term "idealism" for Vasubandhu, he allowed that the term might be appropriate if understood in this way: > > This should not be understood to mean that there are no things other > than consciousness. On the contrary, it means only that what falls > within the range of experience are different forms of consciousness, > while the things-in-themselves remain beyond the limits of experience. > (48) > > > The form in which a thing is thought to be grasped is purely imagined > (parikalpita), and therefore is no sure guide to the > thing-in-itself. It is in this sense, and only in this sense, that > Vasubandhu's system can be called idealism. It by no means > implies that there is nothing apart from ideas or consciousness. (53) > Is the term "idealism" acceptable for the view that, even if nothing expressible in language has ultimate reality, mental events are still more real than physical events, which do not have even conventional reality? Perhaps "conventionalist idealism" would be a good term for this view, but surely it sits squarely within the range of idealisms. To say, with emphasis, as some do, that this view is not idealism, may be to privilege a narrow understanding of the term. In addition, the issue may be more than semantic. It will strike some as overstating the case to say, as Kochumuttom does, that the fact that everything within perception is only mind "by no means implies that there is nothing apart from ideas or consciousness." Vasubandhu's arguments in the Twenty Verses are intended to exclude the possibility of anything at all that may be coherently asserted to be non-mental.[48] So the doctrine of appearance-only does imply that there is nothing apart from ideas or consciousness, at least, as far as logic and implications are allowed to go. The only exceptions would be things about which we can say nothing. So what could allow us to say about them that they are exceptions?[49] Whether or not one wants to use the word "idealism" for Vasubandhu thus depends to a large degree upon just how strongly one wants to lean upon the idea that idealists must affirm the ultimate reality of mind. The main arguments that remain against viewing Vasubandhu as an idealist, exemplified prominently today by Dan Lusthaus, emphasize this idea. Given that many Buddhists in India, Tibet and East Asia have understood the Yogacara to be affirming the ultimate reality of the mental, the denial of such a view on Vasubandhu's behalf is an important and useful corrective. Lusthaus believes that Vasubandhu is not a "metaphysical idealist" or an "ontological idealist" but rather an "epistemological idealist," meaning that Vasubandhu is concerned only about awareness and its contents, and is not in the business of affirming or denying anything about ultimate reality. For this reason, Lusthaus refers to Yogacara as "phenomenology."[50] A final complaint is that the use of the term "idealism" is anachronistic or for some other reason inappropriate in its application to pre-modern India. Kochummutom's point that the term encompasses too much is well taken. It is one thing (perhaps already stretching the meaning) to apply the term "idealism," as R.P. Srivastava (1973) does, to modern thinkers whose views have been influenced by European idealists, such as Aurobindo, Vivekananda and Radhakrishnan. But it is anachronistic if we are to impose upon Vasubandhu the modern associations and baggage that come along with the term, which would include an advocacy of private, personal, sacred, "religious experience." Vasubandhu may be an idealist, in a strict sense, but he is not what 20th century religious studies textbooks call a "mystic philosopher." Vasubandhu does not, generally, appeal to his own, individual personal experience as evidence or confirmation of his positions. On the contrary, Vasubandhu repeatedly employs reasoning to cast doubts upon the apparent realities evident through perception, whether external or internal. This is, perhaps, a reason to avoid describing Vasubandhu as a "phenomenologist." His work does not take the direct examination of "experience" as its theme, but, rather, it draws upon scripture and rational argumentation for its critiques of the available accounts of reality.
vegetarianism	## 1. Terminology and Overview of Positions Moral vegetarianism is opposed by moral omnivorism, the view according to which it is permissible to consume meat (and also animal products, fungi, plants, etc.). Moral veganism accepts moral vegetarianism and adds to it that consuming animal products is wrong. Whereas in everyday life, "vegetarianism" and "veganism" include claims about what one may eat, in this entry, the claims are simply about what one may not eat. They agree that animals are among those things. In this entry, "animals" is used to refer to non-human animals. For the most part, the animals discussed are the land animals farmed for food in the West, especially cattle, chicken, and pigs. There will be some discussion of insects and fish but none of dogs, dolphins, or whales. Primarily, this entry concerns itself with whether moral vegetarians are correct that eating meat is wrong. Secondarily--but at greater length--it concerns itself with whether the production of meat is permissible. Primarily, this entry concerns itself with eating in times of abundance and abundant choices. Moral vegans need not argue that it is wrong to eat an egg if that is the only way to save your life. Moral vegetarians need not argue it is wrong to eat seal meat if that is the only food for miles. Moral omnivores need not argue it is permissible to eat the family dog. These cases raise important issues, but the arguments in this entry are not about them. Almost exclusively, the entry concerns itself with contemporary arguments.[1] Strikingly, many historical arguments and most contemporary arguments against the permissibility of eating meat start with premises about the wrongness of producing meat and move to conclusions about the wrongness of consuming it. That is, they argue that It is wrong to eat meat By first arguing that It is wrong to produce meat. The claim about production is the topic of SS2. ## 2. Meat Production The vast majority of animals humans eat come from industrial animal farms that are distinguished by their holding large numbers of animals at high stocking density. We raise birds and mammals this way. Increasingly, we raise fish this way, too. ### 2.1 Animal Farming Raising large numbers of animals enables farmers to take advantage of economies of scale but also produces huge quantities of waste, greenhouse gas, and, generally, environmental degradation (FAO 2006; Hamerschlag 2011; Budolfson 2016). There is no question of whether to put so many animals on pasture--there is not enough of it. Plus, raising animals indoors, or with limited access to the outdoors, lowers costs and provides animals with protection from weather and predators. Yet when large numbers of animals live indoors, they are invariably tightly packed, and raising them close together risks the development and quick spread of disease. To deal with this risk, farmers intensively use prophylactic antibiotics. Tight-packing also restricts species-typical behaviors, such as rooting (pigs) or dust-bathing (chickens), and makes it so that animals cannot escape each other, leading to stress and to antisocial behaviors like tail-biting in pigs or pecking in chickens. To deal with these, farmers typically dock tails and trim beaks, and typically (in the U.S., at least) do so without anesthetic. Animals are bred to grow fast on a restricted amount of antibiotics, food, and hormones, and the speed of growth saves farmers money, but this breeding causes health problems of its own. Chickens, for example, have been bred in such a way that their bodies become heavier than their bones can support. As a result, they "are in chronic pain for the last 20% of their lives" (John Webster, quoted in Erlichman 1991). Animals are killed young--they taste better that way--and are killed in large-scale slaughterhouses operating at speed. Animal farms have no use for, e.g., male chicks on egg-laying farms, are killed at birth or soon after.[2] Raising animals in this way has produced low sticker prices (BLS 2017). It enables us to feed our appetite for meat (OECD 2017). Raising animals in this way is also, in various ways, morally fraught. It raises concerns about its effects on humans. Slaughterhouses, processing this huge number of animals at high speed, threaten injury and death to workers. Slaughterhouse work is exploitative. Its distribution is classist, racist, and sexist with certain jobs being segmented as paupers' work or Latinx work or women's (Pachirat 2011). Industrial meat production poses a threat to public health through the creation and spread of pathogens resulting from the overcrowding of animals with weakened immune systems and the routine use of antibiotics and attendant creation of antibiotic-resistant bacteria. Anomaly (2015) and Rossi & Garner (2014) argue that these risks are wrongful because unconsented to and because they are not justified by the benefits of assuming those risks. Industrial meat production directly produces waste in the form of greenhouse gas emissions from animals and staggering amounts of waste, waste that, concentrated in those quantities, can contaminate water supplies. The Boll Foundation (2014) estimates that farm animals contribute between 6 and 32% of greenhouse gas emissions. The range is due partly to different ideas about what to count as being farm animals' contributions: simply what comes out of their bodies? Or should we count, too, what comes from deforestation that's done to grow crops to feed them and other indirect emissions? Industrial animal farming raises two concerns about wastefulness. One is that it uses too many resources and produces too much waste for the amount of food it produces. The other is that feeding humans meat typically requires producing crops, feeding them to animals, and then eating the animals. So it typically requires more resources and makes for more emissions than simply growing and feeding ourselves crops (PNAS 2013. Industrial animal farming raises concerns about the treatment of animals. Among others, we raise cattle, chickens, and pigs. Evidence from their behavior, their brains, and their evolutionary origins, adduced in Allen 2004, Andrews 2016, and Tye 2016, supports the view that they have mental lives and, importantly, are sentient creatures with likes and dislikes. Even chickens and other "birdbrains" have interesting mental lives. The exhaustive Marino 2017 collects evidence that chickens can adopt others' visual perspectives, communicate deceptively, engage in arithmetic and simple logical reasoning, and keep track of pecking orders and short increments of time. Their personalities vary with respect to boldness, self-control, and vigilance. We farm billions of these animals industrially each year (Boll Foundation 2014: 15). We also raise a much smaller number on freerange farms. In this entry "freerange" is not used in its tightly-defined, misleading, legal sense according to which it applies only to poultry and simply requires "access" to the outdoors. Instead, in the entry, freerange farms are farms that that, ideally, let animals live natural lives while offering some protection from predators and the elements and some healthcare. These lives are in various ways more pleasant than lives on industrial farms but involve less protection while still involving control and early death. These farms are designed, in part, to make animal lives go better for them, and their design assumes that a natural life is better, other things equal, than a non-natural life. The animal welfare literature converges on this and also on other components of animal well-being. Summarizing some of that literature, David Fraser writes, > > > [A]s people formulated and debated various proposals about what > constitutes a satisfactory life for animals in human care, three main > concerns emerged: (1) that animals should feel well by being spared > negative affect (pain, fear, hunger etc.) as much as possible, and by > experiencing positive affect in the form of contentment and normal > pleasures; (2) that animals should be able to lead reasonably natural > lives by being able to perform important types of normal behavior and > by having some natural elements in their environment such as fresh air > and the ability to socialize with other animals in normal ways; and > (3) that animals should function well in the sense of good health, > normal growth and development, and normal functioning of the body. > (Fraser 2008: 70-71) > > > In this light, it is clear why industrial farming seems to do less for animal welfare than freerange farming: The latter enables keeping animals healthy. It enables happy states ("positive affect") and puts up some safeguards against the infliction of suffering. There is no need, for example, to dock freerange pigs' tails or to debeak freerange chickens, if they have enough space to stay out of each other's way. It enables animals to socialize and to otherwise lead reasonably natural lives. A freerange's pig's life is in those ways better than an industrially-farmed pig's. Yet because freerange farming involves being outdoors, it involves various risks: predator- and weather-related risks, for example. These go into the well-being calculus, too. Animals in the wild are subjected to greater predator- and weather-related risks and have no health care. Yet they score very highly with regard to expressing natural behavior and are under no one's control. How well they do with regard to positive and negative affect and normal growth varies from case to case. Some meat is produced by hunting such animals. In practice, hunting involves making animals suffer from the pain of errant shots or the terror of being chased or wounded, but, ideally, it involves neither pain nor confinement. Of course, either way, it involves death.[3] ### 2.2 The Schematic Case Against Meat Production Moral vegetarian arguments about these practices follow a pattern. They claim that certain actions--killing animals for food we do not need, for example--are wrong and then add that some mode of meat production--recreational hunting, for example--does so. It follows that the mode of meat-production is wrong. Schematically X is wrong. Y involves X. Hence, Y is wrong. Among the candidate values of X are: * Causing animals pain for the purpose of producing food when there are readily available alternatives. * Killing animals for the purpose of... * Controlling animals... * Treating animals as mere tools... * Ontologizing animals as food... * Harming humans.... * Harming the environment... And among the candidate values of Y are: * Industrial animal farming * Freerange farming * Recreational hunting Space is limited and cranking through many instances of the schema would be tedious. This section focuses on causing animals pain, killing them, and harming the environment in raising them. On control, see Francione 2009, DeGrazia 2011, and Bok 2011. On treating animals as mere tools, see Kant's Lectures on Ethics, Korsgaard 2011 and 2015, and Zamir 2007. On ontologizing, see Diamond 1978, Vialles 1987 [1994], and Gruen 2011, Chapter 3. On harming humans, see Pachirat 2011, Anomaly 2015, and Doggett & Holmes 2018. #### 2.2.1 Suffering Some moral vegetarians argue: Causing animals pain while raising them for food when there are readily available alternatives is wrong. Industrial animal farming involves causing animals pain while raising them for food when there are readily available alternatives. Hence, Industrial animal farming is wrong. The "while raising them for food when there are readily available alternatives" is crucial. It is sometimes permissible to cause animals pain: You painfully give your cat a shot, inoculating her, or painfully tug your dog's collar, stopping him from attacking a toddler. The first premise is asserting that causing pain is impermissible in certain other situations. The "when there are readily available alternatives" is getting at the point that there are substitutes available. We could let the chickens be and eat rice and kale. The first premise asserts it is wrong to cause animals pain while raising them for food when there are readily available substitutes. It says nothing about why that is wrong. It could be that it is wrong because it would be wrong to make us suffer to raise us for food and there are no differences between us and animals that would justify making them suffer (Singer 1975 and the enormous literature it generated). It could, instead, be that it is wrong because impious (Scruton 2004) or cruel (Hursthouse 2011). So long as we accept that animals feel--for an up-to-date philosophical defense of this, see Tye 2016--it is uncontroversial that industrial farms do make animals suffer. No one in the contemporary literature denies the second premise, and Norwood and Lusk go so far as to say that > > > it is impossible to raise animals for food without some form of > temporary pain, and you must sometimes inflict this pain with your own > hands. Animals need to be castrated, dehorned, branded, and have other > minor surgeries. Such temporary pain is often required to produce > longer term benefits...All of this must be done knowing that > anesthetics would have lessened the pain but are too expensive. (2011: > 113) > > > There is the physical suffering of tail-docking, de-beaking, de-horning, and castrating, all without anesthetic. Also, industrial farms make animals suffer psychologically by crowding them and by depriving them of interesting environments. Animals are bred to grow quickly on minimal food. Various poultry industry sources acknowledge that this selective breeding has led to a significant percentage of meat birds walking with painful impairments (see the extensive citations in HSUS 2009). This--and much more like it that is documented in Singer & Mason 2006 and Stuart Rachels 2011--is the case for the second premise, namely, that industrial farming causes animals pain while raising them for food when there are readily available alternatives. The argument can be adapted to apply to freerange farming and hunting. Freerange farms ideally do not hurt, but, as the Norwood and Lusk quotation implies, they actually do: For one thing, animals typically go to the same slaughterhouses as industrially-produced animals do. Both slaughter and transport can be painful and stressful. The same goes for hunting: In the ideal, there is no pain, but, really, hunters hit animals with non-lethal and painful shots. These animals are often--but not always--killed for pleasure or for food hunters do not need.[4] Taken together the arguments allege that all manners of meat production in fact produce suffering for low-cost food and typically do so for food when we don't need to do so and then allege that that justification for producing suffering is insufficient. Against the arguments, one might accept that farms hurt animals but deny that it is even pro tanto wrong to do so (Carruthers 1992 and 2011; Hsiao 2015a and 2015b) on the grounds that animals lack moral status and, because of this, it is not intrinsically wrong to hurt them (or kill or control them or treat them like mere tools). One challenge for such views is to explain what, if anything, is wrong with beating the life out of a pet. Like Kant, Carruthers and Hsiao accept that it might be wrong to hurt animals when and because doing so leads to hurting humans. This view is discussed in Regan 1983: Chapter 5. It faces two distinct challenges. One is that if the only reason it is wrong to hurt animals is because of its effects on humans, then the only reason it is wrong to hurt a pet is because of its effects on humans. So there is nothing wrong with beating pets when that will have no bad effects on humans. This is hard to believe. Another challenge for such views, addressed at some length in Carruthers 1992 and 2011, is to explain whether and why humans with mental lives like the lives of, say, pigs have moral status and whether and why it is wrong to make such humans suffer. #### 2.2.2 Killing Consider a different argument: Killing animals while raising them for food when there are readily available alternatives is wrong. Most forms of animal farming and all recreational hunting involve killing animals while raising them for food when there are readily available alternatives. Hence, Most forms of animal farming and all recreational hunting are wrong. The second premise is straightforward and uncontroversial. All forms of meat farming and hunting require killing animals. There is no form of farming that involves widespread harvesting of old bodies, dead from natural causes. Except in rare farming and hunting cases, the meat produced in the industrialized world is meat for which there are ready alternatives. The first premise is more controversial. Amongst those who endorse it, there is disagreement about why it is true. If it is true, it might be true because killing animals wrongfully violates their rights to life (Regan 1975). It might be true because killing animals deprives them of lives worth living (McPherson 2015). It might be true because it treats animals as mere tools (Korsgaard 2011). There is disagreement about whether the first premise is true. The "readily available alternatives" condition matters: Everyone agrees that it is sometimes all things considered permissible to kill animals, e.g., if doing so is the only way to save your child's life from a surprise attack by a grizzly bear or if doing so is the only way to prevent your pet cat from a life of unremitting agony. (Whether it is permissible to kill animals in order to cull them or to preserve biodiversity is a tricky issue that is set aside here. It--and its connection to the permissibility of hunting--is discussed in Scruton 2006b.) At any rate, animal farms are in the business of killing animals simply on the grounds that we want to eat them and are willing to pay for them even though we could, instead, eat plants. The main objection to the first premise is that animals lack the mental lives to make killing them wrong. In the moral vegetarian literature, some argue that the wrongness of killing animals depends on what sort of mental life they have and that while animals have a mental life that suffices for hurting them being wrong, they lack a mental life that suffices for killing them being wrong (Belshaw 2015 endorses this; McMahan 2008 and Harman 2011 accept the first and reject the second; Velleman 1991 endorses that animal mental lives are such that killing them does not harm them). Animals could lack a mental life that makes killing them wrong because it is a necessary condition for killing a creature being wrong that that creature have long-term goals and animals don't or that it is a necessary condition that that creature have the capacity to form such goals and animals don't or that it is a necessary condition that the creature's life have a narrative structure and animals' lives don't or...[5] Instead, the first premise might be false and killing animals we raise for food might be permissible because > > > [t]he genesis of domestic animals is...a matter...of an > implicit social contract--what Stephen Budiansky...calls > 'a covenant of the wild.'...Humans could protect such > animals as the wild ancestors of domestic cattle and swine from > predation, shelter them from the elements, and feed them when > otherwise they might starve. The bargain from the animal's point > of view, would be a better life as the price of a shorter life... > (Callicott 2015: 56-57) > > > The idea is that we have made a "bargain" with animals to raise them, to protect them from predators and the elements, and to tend to them, but then, in return, to kill them. Moreover, the "bargain" renders killing animals permissible (defended in Hurst 2009, Other Internet Resources, and described in Midgley 1983). Such an argument might render permissible hurting animals, too, or treating them merely as tools. Relatedly, even conceding that it is pro tanto wrong to kill animals, it might be all things considered permissible to kill farm animals for food even when there are ready alternatives because and when their well-being is replaced by the well-being of a new batch of farmed animals (Tannsjo 2016). Farms kill one batch of chickens and then bring in a batch of chicks to raise (and then kill) next. The total amount of well-being is fixed though the identities of the receptacles of that well-being frequently changes. Anyone who endorses the views in the two paragraphs above needs to explain whether and then why their reasoning applies to animals but not humans. It would not be morally permissible to create humans on organ farms and harvest those organs, justifying this with the claim that these humans wouldn't exist if it weren't for the plan to take their organs and so part of the "deal" is that those humans are killed for their organs. Neither would it be morally permissible to organ-farm humans, justifying it with the claim that they will be replaced by other happy humans.[6] #### 2.2.3 Harming the Environment Finally, consider: Harming the environment while producing food when there are readily available alternatives is wrong. Industrial animal farming involves harming the environment while producing food when there are readily available alternatives. Hence, Industrial animal farming is wrong. A more plausible premise might be "egregiously harming the environment..." The harms, detailed in Budolfson 2018, Hamerschlag 2011, Rossi & Garner 2014, and Ranganathan et al. 2016, are egregious and include deforestation, greenhouse gas emission, soil degradation, water pollution, water and fossil fuel depletion. The argument commits to it being wrong to harm the environment. Whether this is because those harms are instrumental in harming sentient creatures or whether it is intrinsically wrong to harm the environment or ecosystems or species or living creatures regardless of sentience is left open.[7] The argument does not commit to whether these harms to the environment are necessary consequences of industrial animal farming. There are important debates, discussed in PNAS 2013, about whether, and how easily, these harms can be stripped off industrial animal production. There is an additional important debate, discussed in Budolfson 2018, about whether something like this argument applies to freerange animal farming. Finally, there is a powerful objection to the first premise from the claim that these harms are part of a package that leaves sentient creatures better off than they would've been under any other option. #### 2.2.4 General Moral Theories Nothing has been said so far about general moral theories and meat production. There is considerable controversy about what those theories imply about meat production. So, for example, utilitarians agree that we are required to maximize happiness. They disagree about which agricultural practices do so. One possibility is that because it brings into existence many trillions of animals that, in the main, have lives worth living and otherwise would not exist, industrial farming maximizes happiness (Tannsjo 2016). Another is that freerange farming maximizes happiness (Hare 1999; Crisp 1988). Instead, it could be that no form of animal agriculture does (Singer 1975 though Singer 1999 seems to agree with Hare). Kantians agree it is wrong to treat ends in themselves merely as means. They disagree about which agricultural practices do so. Kant (Lectures on Ethics) himself claims that no farming practice does--animals are mere means and so treating them as mere means is fine. Some Kantians, by contrast, claim that animals are ends in themselves and that typically animal farming treats them as mere means and, hence, is wrong (Korsgaard 2011 and 2015; Regan 1975 and 1983). Contractualists agree that it is wrong to do anything that a certain group of people would reasonably reject. (They disagree about who is in the group.) They disagree, too, about which agricultural practice contractualism permits. Perhaps it permits any sort of animal farming (Carruthers 2011; Hsiao 2015a). Perhaps it permits none (Rowlands 2009). Intermediate positions are possible. Virtue ethicists agree that it is wrong to do anything a virtuous person would not do or would not advise. Perhaps this forbids hurting and killing animals, so any sort of animal farming is impermissible and so is hunting (Clark 1984; Hursthouse 2011). Instead, perhaps it merely forbids hurting them, so freerange farming is permissible and so is expert, pain-free hunting (Scruton 2006b). Divine command ethicists agree that it is wrong to do anything forbidden by God. Perhaps industrial farming, at least, would be (Halteman 2010; Scully 2002). Lipscomb (2015) seems to endorse that freerange farming would not be forbidden by God. A standard Christian view is that no form of farming would be forbidden, that because God gave humans dominion over animals, we may treat them in any old way. Islamic and Jewish arguments are stricter about what may be eaten and about how animals may be treated though neither rules out even industrial animal farming (Regenstein, et al. 2003). Rossian pluralists agree it is prima facie wrong to harm. There is room for disagreement about which agricultural practices--controlling, hurting, killing--do harm and so room for disagreement about which farming practices are prima facie wrong. Curnutt (1997) argues that the prima facie wrongness of killing animals is not overridden by typical justifications for doing so. ## 3. Fish and Insects In addition to pork and beef, there are salmon and crickets. In addition to lamb and chicken, there are mussels and shrimp. There is little in the philosophical literature about insects and sea creatures and their products, and this entry reflects that.[8] Yet the topics are important. The organization Fish Count estimates that at least a trillion sea creatures are wild-caught or farmed each year (Mood & Brooke 2010, 2012, in Other Internet Resources). Globally, humans consume more than 20 kg of fish per capita annually (FAO 2016). In the US, we consume 1.5 lbs of honey per capita annually (Bee Culture 2016). Estimates of insect consumption are less sure. The UN FAO estimates that insects are part of the traditional diets of two billion humans though whether they are eaten--whether those diets are adhered to--and in what quantity is unclear (FAO 2013). Seafood is produced by farming and by fishing. Fishing techniques vary from a person using a line in a boat to large trawlers pulling nets across the ocean floor. The arguments for and against seafood production are much like the arguments for and against meat production: Some worry about the effects on humans of these practices. (Some workers, for example, are enslaved on shrimpers.) Some worry about the effects on the environment of these practices. (Some coral reefs, for example, are destroyed by trawlers.) Some worry about the permissibility of killing, hurting, or controlling sea creatures or treating them merely as tools. This last worry should not be undersold: Again, Mood and Brooke (2010, 2012, in Other Internet Resources) estimate that between 970 billion and 2.7 trillion fish are wild-caught yearly and between 37 and 120 billion farmed fish are killed. If killing, hurting, or controlling these creatures or treating them as mere tools is wrong, then the scale of our wrongdoing with regard to sea creatures beggars belief. Are these actions wrong? Complicating the question is that there is significantly more doubt about which sea creatures have mental lives at all and what those mental lives are like. And while whether shrimp are sentient is clearly irrelevant to the permissibility of enslaving workers who catch them, it does matter to the permissibility of killing shrimp. This doubt is greater still with regard to insect mental lives. In conversation, people sometimes say that bee mental life is such that nothing wrong is done to bees in raising them. Nothing wrong is done to bees in killing them. Because they are not sentient, there is no hurting them. Because of these facts about bee mental life, the argument goes, "taking" their honey need be no more morally problematic than "taking" apples from an apple tree. (There is little on the environmental impact of honey production or (human) workers and honey. So it is unclear how forceful environment- and human-based worries about honey are.) This argument supporting honey production hinges on some empirical claims about bee mental life. For an up-to-date assessment of bee mental life, see Tye 2016, which argues that bees "have a rich perceptual consciousness" and "can feel some emotions" and that "the most plausible hypothesis overall...is that bees feel pain" (2016: 158-159) and see, too, Barron & Klein 2016, which argues that insects, generally, have a capacity for consciousness. The argument supporting honey production might be objected to on those empirical grounds. It might, instead, be objected to on the grounds that we are uncertain what the mental lives of bees are like. It could be that they are much richer than we realize. If so, killing them or taking excessive honey--and thereby causing them significant harms--might well be morally wrong. And, the objection continues, the costs of not doing so, of just letting bees be, would be small. If so, caution requires not taking any honey or killing bees or hurting them. Arguments like this are sometimes put applied to larger creatures. For discussion of such arguments, see Guerrero 2007. ## 4. From Production to Consumption None of the foregoing is about consumption. The moral vegetarian arguments thus far have, at most, established that it is wrong to produce meat in various ways. Assuming that some such argument is sound, how to get from the wrongness of producing meat to the wrongness of consuming that meat? This question is not always taken seriously. Classics of the moral vegetarian literature like Singer 1975, Regan 1975, Engel 2000, and DeGrazia 2009 do not give much space to it. (C. Adams 1990 is a rare canonical vegetarian text that devotes considerable space to consumption ethics.) James Rachels writes, > > > Sometimes philosophers explain that [my argument for vegetarianism] is > unconvincing because it contains a logical gap. We are all opposed to > cruelty, they say, but it does not follow that we must become > vegetarians. It only follows that we should favor less cruel methods > of meat production. This objection is so feeble it is hard to believe > it explains resistance to the basic argument [for vegetarianism]. > (2004: 74) > > > Yet if the objection is that it does not follow from the wrongness of producing meat that consuming meat is wrong, then the objection is not feeble and is clearly correct. In order to validly derive the vegetarian conclusion, additional premises are needed. Rachels, it turns out, has some, so perhaps it is best to interpret his complaint as that it is obvious what the premises are. Maybe so. But there is quite a bit of disagreement about what those additional premises are and plausible candidates differ greatly from one another. ### 4.1 Bridging the Gap Consider a productivist idea about the connection between production and consumption according to which consumption of wrongfully-produced goods is wrong because it produces more wrongful production. The idea issues an argument that, in outline, is: Consuming some product P produces production of Q. Production of Q is wrong. It is wrong to produce wrongdoing. Hence, Consuming P is wrong. Or never mind actual production. A productivist might argue: Consuming some product P is reasonably expected to produce production of Q. Production of Q is wrong. It is wrong to do something that is reasonably expected to produce wrongdoing. Hence, Consuming P is wrong. (Singer 1975; Norcross 2004; Kagan 2011) (The main ideas about connecting consumption and production that follow can--but won't --be put in terms of expectation, too.) The moral vegetarian might then argue that meat is among the values of both P and Q: consuming meat is reasonably expect to produce production of meat. Or the moral vegetarian might argue that consuming meat produces more normalization of bad attitudes towards animals and that is wrong. There are various possibilities. Just consider the first, the one about meat consumption producing meat production. It is most plausible with regard to buying. It is buying the wrongfully-produced good that produces more of it. Eating meat produces more production, if it does, by producing more buying. When Grandma buys the wrongfully produced delicacy, the idea goes, she produces more wrongdoing. The company she buys from produces more goods whether you eat the delicacy or throw it out. These arguments hinge on an empirical claim about production and a moral claim about the wrongfulness of producing wrongdoing. The moral claim has far-reaching implications (DeGrazia 2009 and Warfield 2015). Consider this rent case: > > > You pay rent to a landlord. You know that he takes your rent and uses > the money to buy wrongfully-produced meat. > > > If buying wrongfully-produced meat is wrong because it produces more wrongfully-produced meat, is it wrong to pay rent in the rent case? Is it wrong to buy a vegetarian meal at a restaurant that then takes your money and uses it to buy wrongfully-produced steak? These are questions for productivists' moral claim. There are further, familiar questions about whether it is wrong to produce wrongdoing when one neither intends to nor foresees it and whether it is wrong to produce wrongdoing when one does not intend it but does foresee it and then about whether what is wrong is producing wrongdoing or, rather, simply producing a bad effect (see entries on the doctrine of double effect and doing vs. allowing harm). An objection to productivist arguments denies the empirical claim and, instead, accepting that because the food system is so enormous, fed by so many consumers, and so stuffed with money, our eating or buying typically has no effect on production, neither directly nor even, through influencing others, indirectly (Budolfson 2015; Nefsky 2018). The idea is that buying a burger at, say, McDonald's produces no new death nor any different treatment of live animals. McDonald's will produce the same amount of meat--and raise its animals in exactly the same way--regardless of whether one buys a burger there. Moreover, the idea goes, one should reasonably expect this. Whether or not this is a good account of how food consumption typically works, it is an account of a possible system. Consider the Chef in Shackles case, a modification of a case in McPherson 2015: > > > Alma runs Chef in Shackles, a restaurant at which the chef is known to > be held against his will. It's a vanity project, and Alma will > run the restaurant regardless of how many people come. In fact, Alma > just burns the money that comes in. The enslaved chef is superb; the > food is delicious. > > > The productivist idea does not imply it is wrong to buy food from or eat at Chef in Shackles. If that is wrong, a different idea needs to explain its wrongness. So consider instead an extractivist idea according to which consumption of wrongful goods is wrong because it is a benefiting from wrongdoing (Barry & Wiens 2016). This idea can explain why it is wrong to eat at Chef in Shackles--when you enjoy a delicious meal there, you benefit from the wrongful captivity of the chef. In outline, the extractivist argument is: Consuming some product P extracts benefit from the production of P. Production of P is wrong. It is wrong to extract benefit from wrongdoing. Hence, Consuming P is wrong. Moral vegetarians would then urge that meat is among the values of P. Unlike the productivist argument, this one is more plausible with regard to eating than buying. It's the eating, typically, that produces the benefit and not the buying. Unlike the productivist argument, it does not seem to have any trouble explaining what is wrong in the Chef in Shackles case. Unlike the productivist argument, it doesn't seem to imply that paying a landlord who pays for wrongfully produced food is wrong--paying a landlord is not benefiting from wrongdoing. Like the productivist argument, the extractivist argument hinges on an empirical claim about consumer benefits and a moral claim about the ethics of so benefiting. The notion of benefiting, however, is obscure. Imagine you go to Chef in Shackles, have a truly repulsive meal, and become violently ill afterwards. Have you benefit ted from wrongdoing? If not, the extractivist idea cannot explain what is wrong with going to the restaurant. Put so plainly, the extractivist's moral claim is hard to believe. Consider the terror-love case, a modification of a case Barry & Wiens 2016 credits to Garrett Cullity: > > > A terrorist bomb grievously injures Bob and Cece. They attend a > support group for victims, fall in love, and live happily ever after, > leaving them significantly better off than they were before the > attack. > > > Bob and Cece seem to benefit from wrongdoing but seem not to be doing anything wrong by being together. Whereas the productivist struggles to explain why it is wrong to patronize Chef in Shackles, the extractivist struggles to explain why it is permissible for Bob and Cece to benefit from wrongdoing. A participatory idea has no trouble with the terror-love case. According to it, consuming wrongfully-produced goods is wrong because it cooperates with or participates in or, in Hursthouse's phrase, is party to wrongdoing (2011). Bob and Cece do not participate in terror, so the idea does not imply they do wrong. The idea issues an argument that, in outline, goes: Consuming some product P is participating in the production of P. Production of P is wrong. It is wrong to participate in the production of wrongful things. Hence, Consuming P is wrong. (Kutz 2000; Lepora & Goodin 2013) Moral vegetarians would then urge that meat is among the values of P. Unlike the productivist or extractivist ideas, the participatory idea seems to as easily cover buying and eating for each is plausibly a form of participating in wrongdoing. Unlike the productivist idea, it has no trouble explaining why it is wrong to patronize Chef in Shackles and does not imply it is wrong to pay rent to a landlord who buys wrongfully-produced meat. Unlike the extractivist idea, whether or not you get food poisoning at Chef in Shackles has no moral importance to it. Unlike the extractivist idea, the participatory idea does not falsely imply that the Bob and Cece do wrong in benefiting from wrongdoing--after all, their failing in love is not a way of participating in wrongdoing. Yet it is not entirely clear what it is to participate in wrongdoing. Consider the Jains who commit themselves to lives without himsa (violence). Food production causes himsa. So Jains try to avoid eating many plants, uprooted to be eaten, and even drinking untreated water, filled with microorganisms, to minimize lives taken. Yet Jaina monastics are supported by Jaina laypersons. The monastic can't boil his own water--that would be violent--but the water needs boiling so he depends on a layperson to boil. He kills no animals but receives alms, including meat, from a layperson. Is the monastic participating in violence? Is he participating because he is complicit in this violence (Kutz 2000; Lepora & Goodin 2013)? Is he part of a group that together does wrong (Parfit 1984: Chapter 3)? When Darryl refuses to buy wrongfully-produced meat but does no political work with regard to ending its production is he party to the wrongful production? Does he participate in it or cooperate with its production? Is he a member of a group that does wrong? If so, what are the principles of group selection? As a matter of contingent fact, failing to politically protest meat exhibits no objectionable attitudes in contemporary US society. Yet it might be that consuming certain foods insults or otherwise disrespects creatures involved in that food's production (R.M. Adams 2002; Hill 1979). Hurka (2003) argues that virtue requires exhibiting the right attitude towards good or evil, and so if consuming exhibits an attitude towards production, it is plausible that eating wrongfully produced foods exhibits the wrong attitude towards them. These are all attitudinal ideas about consumption. They might issue in an argument like this: Consuming some product P exhibits a certain attitude towards production of P. Production of P is wrong. It is wrong to exhibit that attitude towards wrongdoing. Hence, Consuming P is wrong. Moral vegetarians would then urge that meat is among the values of P. Like the participatory idea, the attitudinal idea explains the wrongness of eating and buying various goods--both are ways of exhibiting attitudes. Like the participatory idea, it has no trouble with Chef in Shackles, the rent case, the food poisoning case, or the terror-love case. It does hinge on an empirical claim about exhibition--consuming certain products exhibits a certain attitude--and then a moral claim about the impermissibility of that exhibition. One might well wonder about both. One might well wonder why buying meat exhibits support for that enterprise but paying rent to someone who will buy that meat does not. One might well wonder whether eating wrongfully-produced meat in secret exhibits support and whether such an exhibition is wrong. Also, there are attitudes other than attitudes towards production to consider. Failing to offer meat to a guest might exhibit a failure of reverence (Fan 2010). In contemporary India, in light of the "meat murders" committed by Hindus against Muslims nominally for the latter group's consumption of beef, refusing to eat meat might exhibit support for religious discrimination (Doniger 2017). The productivist, extractivist, participatory, and attitudinal ideas are not mutually exclusive. Someone might make use of a number of them. Driver, for example, writes, > > > [E]ating [wrongfully produced] meat is supporting the > industry in a situation where there were plenty of other, better, > options open...What makes [the eater] complicit is that > she is a participant. What makes that participation morally > problematic...is that the eating of meat displays a > willingness to cooperate with the producers of a product that > is produced via huge amounts of pain and suffering. (2015: 79; all > italics mine) > > > This seems to at least incorporate participatory and attitudinal ideas. Lawford-Smith (2015) combines attitudinal and productivist ideas. McPherson (2015) combines extractivist and participatory ideas. James Rachels (2004) combines participatory and productivity ideas. And, of course, there are ideas not discussed here, e.g., that it is wrong to reward wrongdoers for wrongdoing and buying wrongfully produced meat does so. The explanation of why it is wrong to consume certain goods might be quite complex. ### 4.2 Against Bridging the Gap Driver, Lawford-Smith, McPherson, and James Rachels argue that it is wrong to consume wrongfully produced food and try to explain why this is. The productivist, extractivist, participatory, and attitudinal ideas, too, try to explain it. But it could be that there is nothing to explain. It could be that certain modes of production are wrong yet consuming their products is permissible. We might assume that if consumption of certain goods is wrong, then that wrongness would have to be partly explained in terms of the wrongness of those goods' production and then argue that there are no sound routes from a requirement not to produce a food to a requirement not to consume it (Frey 1983). This leaves open the possibility that consumers might be required to do something--for example, work for political changes that end the wrongful system--but permitted to eat wrongfully-produced food. As SS4.1 discusses, Warfield raises a problem for productivist accounts that they seem to falsely imply that morally permissible activities like paying rent to meat-eaters or buying salad at a restaurant serving meat are morally wrong (2015). Add the assumption that if consumption is wrong, it is wrong because some productivist view is true, and it follows that consumption of wrongful goods need not be wrongful. (Warfield does not assume this but instead says that "the best discussion" of the connection between production and consumption is "broadly consequentialist" (ibid., 154).) Instead, we might assume that an extractivist or participatory or attitudinal view is correct if any is and then argue no such view is correct. We might, for example, argue that these anti-consumption views threaten to forbid too much. If the wrongness of producing and wrongness of consuming are connected, what else is connected? If buying meat is wrong because it exhibits the wrong attitude towards animals, is it permissible to be friends with people who buy that meat--or does this, too, evince the wrong attitudes towards animals? If killing animals for food is wrong, is it permissible merely to abstain from consuming them or must one do more work to stop their killing? The implications of various arguments against consuming animals and animal products might be far-reaching. Some will see this as an acknowledgment that something is wrong with moral vegetarian arguments. As Gruen and Jones (2015) note, the lifestyle some such arguments point to might not be enactable by creatures like us. Yet they see this not as grounds for rejection of the argument but, rather, as acknowledgment that the argument sets out an aspiration that we can orient ourselves towards (cf. SS4 of Curtin 1991 on "contextual vegetarianism"). A different sort of argument in favor of the all things considered permissibility of consuming meat comes from the idea that eating and buying animals actually makes for a great cultural good (Lomasky 2013). Even if we accept that the production of those animals is wrong, it could be that the great good of consuming justifies doing so. (Relatedly, it could be that the bad of refusing to consume justifies consumption as in a case in which a host has labored over barbequed chicken for hours and your refusing to eat it would devastate him.) Yet this seems to leave open the possibility that all sorts of awful practices might be permissible because they are essential parts of great cultural goods. It threatens to permit too much. ## 5. Extending Moral Vegetarian Arguments: Animal Products and Plants Moral veganism accepts moral vegetarianism and adds to it that consuming animal products is wrong. Mere moral vegetarians deny this and add to moral vegetarianism that it is permissible to consume animal products. An additional issue that divides some moral vegans and moral vegetarians is whether animal product production is wrong. This raises a general question: If it is wrong to produce meat on the grounds adduced in SS2, what other foods are wrongfully produced? If it is wrong to hurt chickens for meat, isn't it wrong to hurt them for eggs? If it is wrong to harm workers in the production of meat, isn't it wrong to harm workers in the production of animal products? If it is wrong to produce huge quantities of methane for meat, isn't it wrong to produce it for milk? These are challenges posed by moral veganism. But various vegan diets raise moral questions. If it is wrong to hurt chickens for meat, is it wrong to hurt mice and moles while harvesting crops? If it is wrong to harm workers in the production of meat, isn't it wrong to harm workers in the production of tomatoes? If it is wrong to use huge quantities of water for meat, isn't it wrong to use huge quantities of water for almonds? ### 5.1 Animal Products As it might be that meat farming wrong, it might be that animal product farming is wrong for similar reasons. These reasons stem from concerns about plants, animals, humans, and the environment. This entry will focus on the first, second, and fourth and will consider eggs and dairy. #### 5.1.1 Eggs Like meat birds, egg layers on industrial farms are tightly confined, given on average a letter-sized page of space. Their beaks are seared off. They are given a cocktail of antibiotics. Males, useless as layers, are killed right away: crushed, dehydrated, starved, suffocated. As they age and their laying-rate slows, females are starved so as to force them to shed feathers and induce more laying. They are killed within a couple years (HSUS 2009; cf. Norwood & Lusk 2011: 113-127, which rates layer hen lives as not worth living). Freerange egg farming ideally avoids much of this. Yet it still involves killing off young but spent hens and also baby roosters. It often involves painful, stressful trips to industrial slaughterhouses. So, as it is plausible that industrially and freerange farming chickens for meat makes them suffer, so too is it plausible that industrially and freerange farming them for eggs does. The same goes for killing. The threat to the environment, too, arises from industrial farming itself rather than whether it produces meat or eggs. Chickens produce greenhouse gas and waste regardless of whether they are farmed for meat or eggs. Land is deforested to grow food for them and resources are depleted to care for them regardless of whether they are farmed for meat or eggs. In sum, arguments much like arguments against chicken production seem to apply as forcefully to egg production. Arguments from premises about killing, hurting, and harming the environment seem to apply to typical egg production as they do to typical chicken production. #### 5.1.2 Dairy Like beef cattle, dairy cows on industrial farms are tightly confined and bereft of much stimulation. As dairy cows, however, they are routinely impregnated and then constantly milked. Males, useless as milkers, are typically turned to veal within a matter of months. Females live for maybe five years. (HSUS 2009; cf. Norwood & Lusk 2011: 145-150). Freerange milk production does not avoid very much of this. Ideally, it involves less pain and suffering but it typically involves forced impregnation, separation of mother and calf, and an early death, typically in an industrial slaughterhouse. So far as arguments against raising cows for meat on the basis that doing so kills them and makes them suffer are plausible, so are analogous arguments against raising cows for dairy. The threat to the environment is also similar regardless of whether cattle are raised for meat or milk. So far as arguments against raising cows for meat on the basis that doing so harms the environment are plausible, so are analogous arguments against raising cows for milk. Raising cows for meat and for milk produces greenhouse gas and waste; it deforests and depletes resources. In fact, to take just one example, the greenhouse-gas-based case against dairy is stronger than the greenhouse-gas-based case against poultry and pork (Hamerschlag 2013). In sum, arguments much like arguments against beef production seem to apply as forcefully to dairy production. Arguments from premises about killing, hurting, and harming the environment seem to apply to typical dairy production as they do to typical beef production. ### 5.2 Plants As it might be that animal, dairy, and egg farming are wrong, it might be that plant farming is wrong for similar reasons. These reasons stem from concerns about plants, animals, humans, and the environment. This entry will focus on the first, second, and fourth. #### 5.2.1 Plants Themselves Ed drenches Fatima's prized cactus in pesticides without permission. This is uncontroversially wrongful but only uncontroversial because the cactus is Fatima's. If a cactus grows in Ed's yard and, purely for fun, she drenches it in pesticides, killing it, is that wrong? There is a family of unorthodox but increasingly common ideas about the treatment of plants according to which any killing of plants is at least pro tanto wrongful and that treating them as mere tools is too (Marder 2013; Stone 1972, Goodpaster 1978, and Varner 1998 are earlier discussions and Tinker 2015 discusses much earlier discussions). One natural way to develop this thought is that it is wrong to treat plants this way simply because of the effects on plants themselves. An alternative is wrong to treat the plants this way simply because of its effects on the biosphere. In both cases, we can do intrinsic wrong to non-sentient creatures. The objection raises an important issue about interests. Singer, following Porphyry and Bentham, assumes that all and only sentient creatures have interests. The challenge that Marder, et al. raise is that plants at least seem to do better or worse, to flourish or founder, because they seem to have interests in a certain amount of light, nutrients, and water. One way to interpret the position of Porphyry, et al. is that things are not as they seem here and, in fact, plants, lacking sentience, have no interests. This invites the question of why sentience is necessary for interests (Frey 1980 and 1983). Another way to interpret the position of Porphyry, et al. is that plants do have interests but they have no moral import. This invites the questions of when and why is it permissible to deprive plants of what they have interests in. Marder's view is that plants have interests and that these interests carry significantly more moral weight than one might think. So, for example, as killing a dog for fun is wrong, so, too, is killing a dandelion. If killing a chicken for food we don't need is wrong, so, too, is killing some carrots. If it is impermissible to kill plants to provide ourselves food we don't need, how far does the restriction on killing extend: To bacteria? Pressed about this by Gary Francione, Marder is open-minded: "We should not reject the possibility of respecting communities of bacteria without analyzing the issue seriously" (2016: 179). #### 5.2.2 Plant Production and Animals Marder's view rests on a controversial interpretation of plant science and, in particular, on a controversial view that vegetal responses to stimuli--for example that "roots...are capable of altering their growth pattern in moving toward resource-rich soil or away from nearby roots of other members of the same species" (2016: 176)--suffice to show that plants have interests, are ends in themselves, and it is pro tanto wrong to kill them and treat them as tools. Uncontroversially, much actual plant production does have various bad consequences for animals. Actual plant production in the US is largely large scale. Large-scale plant production involves--intentionally or otherwise--killing a great many sentient creatures. Animals are killed by tractors and pesticides. They are killed or left to die by loss of habitat (Davis 2003; Archer 2011). The scope of the killing is disputed in Lamey 2007 and Matheny 2003 but all agree it is vast (cf. Saja 2013 on the moral imperative to kill large animals). The "intentionally or otherwise" is important to some. While these harms are foreseen consequences of farming, they are unintended. To some, that animals are harmed but not intentionally harmed in producing corn in Iowa helps to make those harms permissible (see entry on doctrine of double effect). Pigs farmed in Iowa, by contrast, are intentionally killed. Chickens and cows, too. (Are any intentionally hurt? Not typically. Farming is not sadistic.) The scale is important, too. Davis (2003) and Archer (2011) argue that some forms of meat production kill fewer animals than plant production and, because of that, are preferable to plant production. The idea is that if animal farming is wrong because it kills animals simply in the process of producing food we don't need, then some forms of plant farming are wrong for the same reason. More weakly, if animal farming is wrong because it kills very large numbers of animals in the process of producing food we don't need, then some forms of plant farming are wrong for the same reason. An outstanding issue is whether these harms are necessary components of plant production or contingent. A further issue is how easy it would be to strip these harms off of plant production while still producing foods humans want to eat at prices they are willing to pay. #### 5.2.3 Plant Production and the Environment A final objection to the permissibility of plant production: There are clearly environmental costs of plant production. Indeed, the environmental costs of plant farming are large: topsoil loss; erosion; deforestation; run-off; resource-depletion; greenhouse gas emissions. To take just the last two examples, Budolfson (2016: 169) estimates that broccoli produces more kilograms of CO2 per thousand calories than pork and that almonds use two and a half times the water per thousand calories that chicken does. If some forms of animal farming are wrong for those environmental reasons, then some forms of plant farming are wrong for those reasons (Budolfson 2018). Again, an outstanding issue is whether these harms are necessary components of plant production or contingent. A further issue is how easy it would be to strip these harms off of plant production while still producing foods people want to eat at prices they are willing to pay. ### 5.3 Summary of Animal Product and Plant Subsections Moral vegetarian arguments standardly oppose treating animals in various ways while raising them for food that we do not need to eat to survive. This standardly makes up part of the arguments that it is wrong to eat animals. These arguments against meat production can be extended mutatis mutandis to animal product production.[9] They can be extended, too, to some forms of plant production. This suggests: The arguments against industrial plant production and animal product production are as strong as the arguments against meat production. The arguments against meat production show that meat production is wrong. Hence, The arguments against industrial plant production and animal product production show that those practices are wrong. One possibility is that the first premise is false and that some of the arguments are stronger than others. Another possibility is that the first premise is true and all these arguments are equally strong. We would then have to choose between accepting the second premise--and thereby accepting the conclusion--or denying that meat production is wrong. Another possibility is that the argument is sound but of limited scope, there being few if any alternatives in the industrialized West to industrialized plant, animal product, and meat production. A final possibility is that the parity of these arguments and evident unsoundness of an argument against industrial plant production show that the ideas behind those arguments are being misexpressed. Properly understood, they issue not in a directive about the wrongness of this practice or that. Rather, properly understood, they just show that various practices are bad in various ways. If so, we can then ask: Which are worse? And in which ways? The literature typically ranks factory farming as worse for animals than industrial plant farming if only because the former requires the latter and produces various harms--the suffering of billions of chickens--that the latter does not. Or consider the debate in the literature about the relative harmfulness to animals of freerange farming and industrial plant farming. Which produces more animal death or more animal suffering? Ought we minimize that suffering? Or consider the relative harmfulness of freerange and industrial animal farming. Some argue that the former is worse for the environment but better for animals. If so, there is a not-easy question about which, if either, to go in for. ## 6. Conclusion: Where the Debate About Vegetarianism Stands and Is Going Given length requirements, this entry cannot convey the vastness of the moral vegetarian literature. There is some excellent work in the popular press. Between the Species, Journal of Agricultural and Environmental Ethics, Journal of Animal Ethics, Environmental Ethics, and Journal of Food Ethics publish articles yearly. Dozens of good articles have been omitted from discussion. This entry has omitted quite direct arguments against consuming meat, arguments that do not derive from premises about the wrongness of producing this or that. Judeo-Islamic prohibitions on pork, for example, derive from the uncleanliness of the product rather than the manner of its production. Rastafari prohibitions on eating meat, for another example, derive in part from the view that meat consumption is unnatural. Historically, such prohibitions and justifications for them have not been limited to prohibitions on consuming meat. The Laws of Manu's prohibition on onion-eating derives from what consuming onion will do to the consumer rather than the manner of onion-production (Doniger & Smith (trans.) 1991: 102). The Koran's prohibition on alcohol-drinking derives from what consuming alcohol will do to the consumer rather than the manner of alcohol-production (5:90-91). Arguments like this, arguments against consumption that start from premises about intrinsic features of the consumed or about the consumed's effects on consumers, largely do not appear in the contemporary philosophical literature. What we have now are arguments according to which certain products are wrongfully produced and consumption of such products bears a certain relation to that wrongdoing and, ipso facto, is wrong. Moral vegetarians then argue that meat is such a product: It is typically wrongfully produced and consuming it typically bears a certain relation to that wrongdoing. This then leaves the moral vegetarian open to two sorts of objections: objections to the claims about production--is meat produced that way? Is such production wrongful?--and objections to the claims connecting consumption to production--is consuming meat related to wrongful production in the relevant way? Is being so related wrong? Explaining moral vegetarian answers to these questions was the work of SS2 and SS4. There are further questions. If moral vegetarian arguments against meat-consumption are sound, then are arguments against animal product consumption also sound? Might dairy, eggs, and honey be wrongfully produced as moral vegetarians argue meat is? Might consuming them wrongfully relate the consumer to that production? Explaining the case for "yes" was some of the work of SS5. Relatedly, some plants, fruit, nuts, and other putatively vegetarian foods might be wrongfully produced. Some tomatoes are picked by workers working in conditions just short of slavery (Bowe 2007); industrial production of apples sucks up much water (Budolfson 2016); industrial production of corn crushes numerous small animals to death (Davis 2003). Are these food wrongfully produced? Might consuming them wrongfully relate the consumer to that production? Explaining the case for "yes" here, too, was some of the work of SS5. Fischer (2018) suggests that the answers to some of the questions noted in the previous two paragraphs support a requirement to "eat unusually" and, one might add, to produce unusually. If meat, for example, is usually wrongfully produced, it must be produced unusually for that production to stand a chance of being permissible, perhaps as faultless roadkill (Koelle 2012; Bruckner 2015) or as the corpse of an animal dead from natural causes (Foer 2009) or as a test-tube creation (Milburn 2016; Pluhar 2010; see the essays in Donaldson & Carter (eds.) 2016 for discussion of plant-based "meat"). If consuming meat is usually wrong because it usually bears a certain relation to production, it must be consumed unusually to stand a chance of being permissible. Some people eat only food they scavenge from dumpsters, food that would otherwise go to waste. Some people eat only food that is given to them without asking for any food in particular. If consuming is wrong only because it produces more production, neither of these modes of consumption would be wrongful. As some unusual consumption might, by lights of the arguments considered in this entry, turn out to be morally unobjectionable, some perfectly usual practices having nothing to do with consumption might turn out, by those same lights, to be morally objectionable. Have you done all you are required to do by moral vegetarian lights if you stop eating, for example, factory-farmed animals? Clearly not. If it is wrong to eat a factory-farmed cow, it is for very similar reasons wrong to wear the skin of that cow. Does the wrongful road stop at consumption, broadly construed to include buying, eating, or otherwise using? Or need consumers do more than not consume wrongfully-produced goods? Need they be pickier in how they spend their money than simply not buying meat, e.g., not going to restaurants that serve any meat? Need they protest or lobby? Need they take more direct action against farms? Or more direct action against the government? Need they refuse to pay rent to landlords who buy wrongfully-produced meat? Is it permissible for moral vegetarians to befriend--or to stay friends with--meat-eaters? As there are questions about whether the moral road gets from production to consumption, there are questions about whether the road stops at consumption or gets much farther. As discussed in SS5, the moral vegetarian case against killing, hurting, or raising animals for food might well be extended to killing, hurting, or raising animals in other circumstances. What, if anything, do those cases show about the ethical treatment of pets (Bok 2011; Overall (ed.) 2017; Palmer 2010 and 2011)? Of zoo creatures (DeGrazia 2011; Gruen 2011: Chapter 5; Gruen 2014)? What, if anything, do they show about duties regarding wild animals? Palmer 2010 opens with two cases from 2007, one of which involved the accidental deaths of 10,000 wildebeest in Kenya, the other involving the mistreatment and death of 150 horses in England. As Palmer notes, it is plausible that we are required to care for and help domesticated animals--that's why it is plausibly wrong to let horses under our care suffer--but permissible to let similar harms befall wild animals--that's why it is plausibly permissible to let wildebeest suffer and die. And yet, Palmer continues, it is also plausible that animals with similar capacities--animals like horses and wildebeest--should be treated similarly. So is the toleration of 10,000 wildebeest deaths permissible? Or do we make a moral mistake in not intervening in such cases? Relatedly, moral vegetarians oppose chicken killing and consumption and yet some of them aid and abet domestic cats in the killings of billions of birds each year in the United States alone (Loss, et al. 2013; Pressler 2013). Is this permissible? If so, why (Cohen 2004; Milburn 2015; Sittler-Adamczewski 2016)? McMahan (2015) argues that standard moral vegetarian arguments against killing and suffering lead (eventually) to the conclusion that we ought to reduce predation in the wild. What, if anything do moral vegetarian arguments show about duties regarding fetuses? There are forceful arguments that if abortion is wrong, then so is killing animals for food we don't need (Scully 2013). The converse is more widely discussed but less plausible (Abbate 2014; Colb & Dorf 2016; Nobis 2016). Finally, in the food ethics literature, questions of food justice are among the most common questions about food consumption. Sexism, racism, and classism, are unjust. Among the issues of food justice, then, are how, if at all, the practices of vegetarianism and omnivorism or encouragement of them are sexist (C. Adams 1990) or racist (Alkon & Agyeman (eds.) 2011) or classist (Guthman 2011). Industrial animal agriculture raises a pair of questions of justice: It degrades the environment--is this unjust to future generations who will inherit this degraded environment? Also, what makes it so environmentally harmful is the scale of it. That scale is driven, in part, by demand for meat among the increasingly affluent in developing countries (Herrero & Thornton 2013). Is refusing to meet that demand--after catering to wealthy Western palates for a long stretch--a form of classism or racism? The animals we eat dominate the moral vegetarian literature and have dominated it ever since there has been a moral vegetarian literature. A way to think about these last few paragraphs is that questions about what we eat lead naturally to questions about other, quite different topics: the animals we eat but also the animals we don't; eating those animals but also eating plants; refusing to eat those animals but also raising pets and refusing to intervene with predators and prey in the wild; refusing to eat but also failing to protest or rectify various injustices. Whereas the questions about animals--and the most popular arguments about them--are very old, these other questions are newer, and there is much progress to be made in answering them.
original-position	## 1. Historical Background: The Moral Point of View The idea of the moral point of view can be traced back to David Hume's account of the "judicious spectator." Hume sought to explain how moral judgments of approval and disapproval are possible given that people normally are focused on achieving their own interests and concerns. He conjectured that in making moral judgments individuals abstract in imagination from their own interests and adopt an impartial point of view from which they assess the effects of their own and others' actions on the interests of everyone. Since, according to Hume, we all can adopt this impartial perspective in imagination, it accounts for our agreement in moral judgments (see Hume 1739 [1978, 581]; Rawls, LHMP 84-93, LHPP 184-187). Subsequently, philosophers posited similar perspectives for moral reasoning designed to yield impartial judgments once individuals abstract from their own aims and interests and assess situations from an impartial point of view. But rather than being mainly explanatory of moral judgments like Hume's "judicious spectator," the role of these impartial perspectives is to serve as a basis from which to assess and justify moral rules and principles. Kant's categorical imperative procedure, Adam Smith's "impartial spectator,", and Sidgwick's "point of view of the universe" are all different versions of the moral point of view. An important feature of the moral point of view is that it is designed to represent what is essential to the activity of moral reasoning. For example, Kant's categorical imperative is envisioned as a point of view any reasonable morally motivated person can adopt in deliberating about what they ought morally to do (Rawls, CP 498ff; LHMP). When joined with the common assumption that the totality of moral reasons is final and override non-moral reasons, the moral point of view might be regarded as the most fundamental perspective that we can adopt in our reasoning about justice and what we morally ought to do. Rawls's idea of the original position, as initially conceived, is his account of the moral point of view regarding matters of justice. The original position is a hypothetical perspective that we can adopt in our moral reasoning about the most basic principles of social and political justice. What primarily distinguishes Rawls's impartial perspective from its antecedents (in Hume, Smith, Kant, etc.) is that, rather than representing the judgment of one person, it is conceived socially, as a general agreement by (representatives of all adult) members of an ongoing society. The point of view of justice is then represented as a general "social contract" or agreement by free and equal persons on the basic terms of cooperation for their society. ## 2. The Original Position and Social Contract Doctrine Historically the idea of a social contract had a more limited role than Rawls assigns to it. In Thomas Hobbes and John Locke the social contract serves as an argument for the legitimacy of political authority. Hobbes argues that in a pre-social state of nature it would be rational for all to agree to authorize one person to exercise the absolute political power needed to maintain peace and enforce laws necessary for productive social cooperation. (Hobbes, 1651) By contrast, Locke argued against absolute monarchy by contending that no existing political constitution is legitimate or just unless it could be contracted into starting from a position of equal right within a (relatively peaceful) state of nature, and without violating any natural rights or duties. (Locke, 1690) For Rousseau and perhaps Kant too, the idea of a social contract plays a different role: It is an "idea of reason" (Kant) depicting a point of view that lawmakers and citizens should adopt in their reasoning to ascertain the "general will," which enables them to assess existing laws and decide upon measures that promote justice and citizens' common good. (Rousseau, 1762; Kant, 1793, 296-7; Kant 1797, 480) Rawls generalizes on Locke's, Rousseau's and Kant's natural rights theories of the social contract (TJ vii/xviii rev.; 32/28 rev.): The purpose of his original position is to yield principles to determine and assess the justice of political constitutions and of economic and social arrangements and the laws that sustain them. To do so, he seeks in the original position "to combine into one conception the totality of conditions which we are ready upon due reflection to recognize as reasonable in our conduct towards one another" (TJ 587/514 rev.). Why does Rawls represent principles of justice as originating in a kind of social contract? Rawls says that "justice as fairness assigns a certain primacy to the social" (CP 339). Unlike Kant's categorical imperative procedure, the original position is designed to represent the predominantly social bases of justice. To say that justice is predominantly social does not mean that people do not have "natural" moral rights and duties outside society or in non-cooperative circumstances--Rawls clearly thinks there are human rights (see LP, SS10) and certain "natural duties" (TJ, SSSS19, 51) that apply to all human beings as such. But whatever our natural or human rights and duties may be, they do not provide an adequate basis for ascertaining the rights and duties of justice that we owe one another as members of the same ongoing political society. It is in large part due to "the profoundly social nature of human relationships" (PL 259) that Rawls sees political and economic justice as grounded in social cooperation on terms of reciprocity and mutual respect. For this reason Rawls eschews the idea of a state of nature where pre-social but fully rational individuals agree to cooperative terms (as in Hobbesian views), or where pre-political persons with antecedent natural rights agree on the form of a political constitution (as in Locke). Rawls regards us as social beings in the sense that in the absence of society and social development we have but inchoate and unrealized capacities, including our capacities for rationality, morality, even language itself. As Rousseau says, outside society we are but "stupid and shortsighted animals" (Rousseau, 1762, bk.I, ch.8, par. 1). This draws into question the main point of the idea of a state of nature in Hobbesian and Lockean views, which is to distinguish the rights, claims, duties, powers and competencies we have prior to membership in society from those we acquire as members of society. Not being members of some society is not an option for us. In so far as we are rational and reasonable beings at all, we have developed as members of some society, within its social framework and institutions. Accordingly Rawls says that no sense can be made of the notion of that part of an individual's social benefits that exceed what would have been that person's situation in a state of nature (PL 278). The traditional idea of pre-social or even pre-political rational moral agents thus plays no role in Rawls's account of justice and the social contract; for him the state of nature is an idea without moral significance (PL 278-280). The original position is set forth largely as an alternative to the state of nature and is regarded by Rawls as the appropriate initial situation for a social contract. (Below we consider a further reason behind Rawls's rejection of the state of nature: it does not adequately allow for impartial judgment and the equality of persons.) Another way Rawls represents the "profoundly social" bases of principles of justice is by focusing on "the basic structure of society." The "first subject of justice," Rawls says, is principles that regulate the basic social institutions that constitute the "basic structure of society" (TJ sect.2). These basic institutions include the political constitution, which specifies political offices and procedures for legislating and enforcing laws and the system of trials for adjudicating disputes; the bases and organization of the economic system, including laws of property, its transfer and distribution, contractual relations, etc. which are all necessary for economic production, exchange, and consumption; and finally norms that define and regulate permissible forms of the family, which is necessary to reproduce and perpetuate society from one generation to the next. It is the role of principles of justice to specify and assess the system of rules that constitute these basic institutions, and determine the fair distribution of rights, duties, opportunities, powers and positions of office, and income and wealth realized within them. What makes these basic social institutions and their arrangement the first subject for principles of social justice is that they are all necessary to social cooperation and have such profound influences on our circumstances, aims, characters, and future prospects. No stable, enduring society could exist without certain rules of property, contract, and transfer of goods and resources, for they make economic production, trade, and consumption possible. Nor could a society long endure without some political mechanism for resolving disputes and making, revising, interpreting, and enforcing its economic and other cooperative norms; nor without some form of the family, to reproduce, sustain, and nurture members of its future generations. This is what distinguishes the social institutions constituting the basic structure from other profoundly influential social institutions, such as religion; religion and other social institutions are not basic in Rawls's sense because they are not generally necessary to social cooperation among members of society. (Even if certain religions have been ideologically necessary to sustain the norms of particular societies, many societies can and do exist without the involvement or support of religious institutions). Another reason Rawls regards the original position as the appropriate setting for a social contract is implicit in his stated aim in A Theory of Justice: it is to discover the most appropriate moral conception of justice for a democratic society wherein persons regard themselves as free and equal citizens (TJ viii/xviii rev.). Here he assumes an ideal of citizens as "moral persons" who regard themselves as free and equal, have a conception of their rational good, and have a "sense of justice." "Moral persons" (an 18th century term) are not all necessarily morally good persons. Instead moral persons are persons who are capable of being rational since they have the capacities to form, revise and pursue a rational conception of their good; moreover, moral persons also are capable of being reasonable since they have a moral capacity for a sense of justice--to cooperate with others on terms that are fair and to understand, apply, and act upon principles of justice and their requirements. Because people have these capacities, or "moral powers," (as Rawls calls them, following Kant) we hold them responsible for their actions, and they are regarded as capable of freely pursuing their interests and engaging in social cooperation. Rawls's idea is that, being reasonable and rational, moral persons (like us) who regard ourselves as free and equal should be in a position to accept and endorse as both rational and morally justifiable the principles of justice regulating our basic social institutions and individual conduct. Otherwise, our conduct is coerced or manipulated for reasons we cannot (reasonably or rationally) accept and we are not fundamentally free persons. Starting from these assumptions, Rawls construes the moral point of view from which to decide moral principles of justice as a social contract in which (representatives of) free and equal persons are given the task of coming to an agreement on principles of justice that are to regulate their social and political relations in perpetuity. How otherwise, Rawls contends, should we represent the justification of principles of justice for free and equal persons who have different conceptions of their good, as well as different religious, philosophical, and moral views? There is no commonly accepted moral or religious authority or doctrine to which they could appeal in order to discover principles of justice that all could agree to and accept. Rawls contends that, since his aim is to discover a conception of justice appropriate for a democratic society, it should be justifiable to free and equal persons in their capacity as citizens on terms which all can endorse and accept. The role of the social contract is to represent this idea, that the basic principles of social cooperation are justifiable hence acceptable to all reasonable and rational members of society, and that they are principles which all can commit themselves to support and comply with. How is this social contract to be conceived? It is not an historical event that must actually take place at some point in time (TJ 120/104 rev.ed.). It is rather a hypothetical situation, a kind of "thought experiment" (JF 17), that is designed to uncover the most reasonable principles of justice. Rawls maintains (in LHPP, cf. p.15) that the major advocates of social contract doctrine--Hobbes, Locke, Rousseau, and Kant--all regarded the social contract, as a hypothetical event. Hobbes and Locke thus posited a hypothetical state of nature in which there is no political authority, and where people are regarded as rational and (for Locke) also reasonable. The purpose of this hypothetical social contract is to demonstrate what types of political constitutions and governments are politically legitimate, and determine the nature of individuals' political obligations (LHPP p.16). The presumption is that if a constitution or form government could be agreed to by rational persons subject to it according to principles and terms they all accept, then it should be acceptable to rational persons generally, including you and me, and hence is legitimate and is the source of our political obligations. Thus, Hobbes argues that all rational persons in a state of nature would agree to authorize an absolute sovereign to enforce the "laws of nature" necessary for society; whereas Locke comes to the opposite conclusion, contending that absolutism would be rejected in favor of constitutional monarchy with a representative assembly. Similarly, in Rousseau and Kant, the social contract is a way to reason about the General Will, including the political constitution and laws that hypothetical moral agents would all agree to in order to promote the common good and realize the freedom and equality of citizens. (Rousseau, 1762, I:6, p.148; II:1, p.153; II:11, p.170; Kant, 1793, 296-7; Kant 1797, 480; cf. Rawls, LHPP, 214-48). Rawls employs the idea of a hypothetical social contract for more general purposes than his predecessors. He aims to provide principles of justice that can be applied to determine not only the justice of political constitutions and the laws, but also the justice of the institution of property and of social and economic arrangements for the production and distribution of income and wealth, as well as the distribution of educational and work opportunities, and of powers and positions of office and responsibility. Some have objected that hypothetical agreements cannot bind or obligate people; only actual contracts or agreements can impose obligations and commitments (Dworkin, 1977, 150ff). But the original position is not intended to impose new obligations on us; rather it is a device for discovery and justification: It is to be used, as Rawls says, "to help us work out what we now think" (CP 402); it incorporates "conditions...we do in fact accept" (TJ 587/514 rev.) and is a kind of "thought experiment for the purpose of public- and self-clarification" (JF, p.17). Hypothetical agreement in the original position does not then bind anyone to duties or commitments they do not already have. Its point rather is to help discover and explicate the requirements of our moral concepts of justice and enable us to draw the consequences of considered moral convictions of justice that we all presumably share. Whether we in turn consciously accept or agree to these consequences and the principles and duties they implicate once brought to our awareness does not undermine their moral justification. The point rather of conjecturing the outcome of a hypothetical agreement is that, if the premises underlying the original position correctly represent our most deeply held considered moral convictions and concepts of justice, then we are morally and rationally committed to endorsing the resulting principles and duties whether or not we actually accept or agree to them. Not to do so implies a failure to accept and live up to the consequences of our own moral convictions about justice. Here critics may deny that the original position incorporates all the relevant reasons and considered moral convictions for justifying principles of justice (e.g. it omits beneficence, or the parties' knowledge of their final ends), and/or that some reasons it incorporates are not relevant to moral justification to begin with (such as the publicity of fundamental principles, as utilitarians argue, Sidgwick, 1907, or the separateness or persons, temporal neutrality and rationality of the parties in promoting their own conception of the good). (Parfit, 1985, 163, 336; Cohen, G.A., 2009; Cohen, J., 2015,). Or they may argue that the state or nature, not the original position, is the appropriate perspective from which to ascertain fundamental principles of justice, since individuals moral and property rights are pre-social and not dependent upon social cooperation. (Nozick, 1974, 183-231). ## 3. The Veil of Ignorance Rawls calls his conception "justice as fairness." His aim in designing the original position is to describe an agreement situation that is fair among all the parties to the hypothetical social contract. He assumes that if the parties are fairly situated and take all relevant information into account, then the principles they agree to are also fair. The fairness of the original agreement situation transfers to the principles everyone agrees to; furthermore, whatever laws or institutions are required by the principles of justice are also fair. The principles of justice chosen in the original position are in this way the result of a choice procedure designed to "incorporate pure procedural justice at the highest level" (CP, 310, cf. TJ 120/104). This feature of Rawls's original position is closely related to his constructivism, and his subsequent understanding of the original position as a "procedure of construction"; see the supplementary section: > > Constructivism, Objectivity, Autonomy, and the Original Position, > in the supplementary document Further Topics on the Original Position. There are different ways to define a fair agreement situation depending on the purpose of the agreement and the description of the parties to it. For example, certain facts are relevant to entering a fair employment contract - knowledge of a prospective employee's talents, skills, prior training, experience, motivation, and reliability - that may not be relevant to other fair agreements. What is a fair agreement situation among free and equal persons when the purpose of the agreement is fundamental principles of justice for the basic structure of society? What sort of facts should the parties to such a fundamental social contract know, and what sort of facts are irrelevant or even prejudicial to a fair agreement? Here it is helpful to compare Rawls's and Locke's social contracts. A feature of Locke's social contract is that it transpires in a state of nature among free and equal persons who know everything about themselves that you and I know about ourselves and each other. Thus, Locke's parties know their natural talents, skills, education, and other personal characteristics; their racial and ethnic group, gender, social class, and occupations; their level of wealth and income, their religious and moral beliefs, and so on. Given this knowledge, Locke assumes that, while starting from a position of equal political right, the great majority of free and equal persons in a state of nature - including all women and racial minorities, and all other men who do not meet a rigid property qualification - could and most likely would rationally agree to alienate their natural rights of equal political jurisdiction in order to gain the benefits of political society. Thus, Locke envisions as legitimate a constitutional monarchy that is in effect a gender-and-racially biased class state wherein a small restricted class of amply propertied white males exercise political rights to vote, hold office, exercise political and social influence, and enjoy other important benefits and responsibilities to the exclusion of everyone else (see Rawls, LHPP, 138-139). The problem with this arrangement, of course, is that gender and racial classifications, social class, wealth and lack thereof, are, like absence of religious belief, not good reasons for depriving free and equal persons of their equal political rights or opportunities to occupy social and political positions. Knowledge of these and other facts are not then morally relevant for deciding who should qualify to vote, hold office, and actively participate in governing and administering society. Rawls suggests that the reason Locke's social contract results in this unjust outcome is that it transpires (hypothetically) under unfair conditions of a state of nature, where the parties have complete knowledge of their circumstances, characteristics and social situations. Socially powerful and wealthy parties then have access to and can unfairly benefit from their knowledge of their "favorable position and exercise their threat advantage" to extract favorable terms of cooperation for themselves from those in less favorable positions (JF 16). Consequently, the parties' judgments regarding constitutional provisions are biased by their knowledge of their particular circumstances and their decisions are insufficiently impartial. The remedy for such biases of judgment is to redefine the initial situation within which the social contract transpires. Rather than a state of nature Rawls situates the parties to the social contract so that they do not have access to factual knowledge that can distort their judgments and result in unfair principles. Rawls's original position is an initial agreement situation wherein the parties are without information enabling them to tailor principles of justice favorable to their personal circumstances and interests. Among the essential features of the original position are that no one knows their place in society, class position, wealth, or social status, nor does anyone know their race, gender, fortune or misfortune in the distribution of natural assets and abilities, level of intelligence, strength, education, and the like. Rawls even assumes that the parties do not know their values or "conceptions of the good," their religious or philosophical convictions, or their special psychological propensities. The principles of justice are chosen behind a "veil of ignorance" (TJ 12/11). This veil of ignorance deprives the parties of all knowledge of particular facts about themselves, about one another, and even about their society and its history. The parties are not however completely ignorant of facts. They know all kinds of general facts about persons and societies, including knowledge of relatively uncontroversial scientific laws and generalizations accepted within the natural and social sciences - economics, psychology, political science, biology, and other natural sciences (including applications of Darwinian evolutionary theory that are generally accepted by scientists, however controversial they may be among religious fundamentalists). They know then about the general tendencies of human behavior and psychological development, about neuropsychology and biological evolution, and about how economic markets work, including neo-classical price theory of supply and demand. As discussed below, they also know about the circumstances of justice--moderate scarcity and limited altruism--as well as the desirability of the "primary social goods" that are needed by anyone in modern society to live a good life and to develop their "moral powers" and other capacities. What the parties lack however is knowledge of any particular facts about their own and other persons' lives, as well as knowledge of any historical facts about their society and its population, its level of wealth and resources, religious institutions, etc.. Rawls thinks that since the parties are required to come to an agreement on objective principles that supply universal standards of justice applying across all societies, knowledge of particular and historical facts about any person or society is morally irrelevant and potentially prejudicial to their decision. Another reason Rawls gives for such a "thick" veil of ignorance is that it is designed to be a strict "position of equality" (TJ 12/11) that represents persons purely in their capacity as free and equal moral persons. The parties in the original position do not know any particular facts about themselves or society; they all have the same general information. They are then situated equally in a very strong way, "symmetrically" (JF 18) and purely as free and equal moral persons. They know only characteristics and interests they share in their capacity as free and equal moral persons--their "higher-order interests" in developing the moral powers of justice and rationality, their need for the primary social goods, and so on. The moral powers, Rawls contends, are the "basis of equality, the features of human beings in virtue of which they are to be treated in accordance with the principles of justice" (TJ, 504/441). Knowledge of the moral powers and their essential role in social cooperation, along with knowledge of other general facts, is all that is morally relevant, Rawls believes, to a decision on principles of justice that are to reflect people's status as free and equal moral persons. A thick veil of ignorance thus is designed to represent the equality of persons purely as moral persons, and not in any other contingent capacity or social role. In this regard the veil of ignorance interprets the Kantian idea of equality as equal respect for moral persons (cf. CP 255). Many criticisms have been leveled against Rawls's veil of ignorance. Among the more common criticisms are that the parties' choice in the original position is indeterminate (Sen, 2009, 11-12, 56-58), or would result in choice of the principle of (average) utility (Harsanyi, 1975), or a principle of relative prioritarianism that gives greater weight to but does not maximize the least advantaged position (Buchak, 2017) (The argument for the choice of the principle of average utility is discussed below.) Among reasons given for the indeterminacy of decision in the original position are that the parties are deprived of so much information about themselves that they are psychologically incapable of making a choice; or they cannot decide between a plurality of reasonable principles. (Sen 2009, 56-58). Or they are incapable of making a rational choice, since we cannot decide upon ethical principles without knowing our primary purposes in life, the values of community, or certain other final ends and commitments. (MacIntyre, 1981; Sandel 1982) One answer to to the criticism of inability to make a rational choice due to ignorance of our final ends is that we do not need to know everything about ourselves, including these primary purposes, to make rational decisions about the background social conditions needed to pursue these primary purposes. For example, whatever our ends, we know that personal security and an absence of social chaos are conditions of most anyone's living a good life (as Hobbes contends). Similarly, though Rawls's parties do not know their own values and commitments, they do know that as free and equal persons they require an adequate share of primary social goods (rights and liberties, powers and opportunities, income and wealth, and the social bases of self-respect) to effectively pursue their purposes, whatever they may be. They also know they have a "higher-order interest" in adequately developing and exercising their "moral powers" - the capacities to be rational and reasonable - which are conditions of responsible agency, effectively pursuing one's purposes, and engaging in social cooperation. Rawls contends that knowledge of these "essential goods" is sufficient for a rational choice on principles of justice by the parties in the original position. To the objection that choice behind the veil of ignorance is psychologically impossible, Rawls says that it is important not to get too caught up in the theoretical fiction of the original position, as if it were some historical event among real people who are being asked to do something impossible. The original position is not supposed to be realistic but is a "device of representation" (PL 27), or a "thought experiment," (JF, 83), that is designed to organize our considered convictions of justice and clarify their implications. The parties in it are not real but are "artificial persons" who have a role to play in this thought experiment. They represent an ideal of free and equal reasonable and rational moral persons that Rawls assumes is implicit in our reasoning about justice. The veil of ignorance is a representation of the kinds of reasons and information that are relevant to a decision on principles of justice for the basic structure of a society of free and equal moral persons (TJ 17/16). Many kinds of reasons and facts are not morally relevant to that kind of decision (e.g., information about people's race, gender, religious affiliation, wealth, and even, Rawls says more controversially, their conceptions of their good), just as many different kinds of reasons and facts are irrelevant to mathematicians' ability to work out the formal proof of a theorem. As a mathematician, scientist, or musician exercise their expertise by ignoring knowledge of particular facts about themselves, presumably we can do so too in reasoning about principles of justice for the basic structure of society. Rawls says we can "enter the original position at any time simply by reasoning in accordance with the enumerated restrictions on information," (PL 27) and considering general facts about persons, their needs, and social and economic cooperation that are provided to the parties (TJ 120/104, 587/514). A related criticism of Rawls's "thick" veil of ignorance is that even if the parties can make certain rational decisions in their interest without knowledge of their final ends, still they cannot come to a decision about principles of justice without knowing the desires and interests of people. For justice consists of the measures that most effectively promote good consequences, and these ultimately reflect facts about individuals' utility or welfare. This criticism is mirrored in utilitarian versions of the moral point of view, which incorporate a "thin" veil of ignorance that represents a different idea of impartiality. The impartial sympathetic spectator found in David Hume and Adam Smith, or the self-interested rational chooser in John Harsanyi's average utilitarian account, all have complete knowledge of everyone's desires, interests and purposes as well as knowledge of particular facts about people and their historical situations. Impartiality is achieved by depriving the impartial observer or rational chooser of any knowledge of its own identity. This leads it to give equal consideration to everyone's desires and interests, and impartially take everyone's desires and interests into account. Since rationality is presumed to involve maximizing something - or taking the most effective means to promote the greatest realization of one's ends - the impartial observer/chooser rationally chooses the rule or course of action that maximizes the satisfaction of desires, or utility (aggregate or average), summed across all persons. (See TJ, SS30) Rawls's original position with its "thick" veil of ignorance represents a different conception of impartiality than the utilitarian requirement that equal consideration be given to everyone's desires, preferences, or interests. The original position abstracts from all information about current circumstances and the status quo, including everyone's desires and particular interests. Utilitarians assume peoples' desires and interests are given by their circumstances and seek to maximize their satisfaction; in so doing utilitarians suspend judgment regarding the moral permissibility of peoples' desires, preferences, and ends and of the social circumstances and institutions within which these are shaped and cultivated. For Rawls, a primary reason for a thick veil of ignorance is to enable an unbiased assessment of the justice of existing social and political institutions and of existing desires, preferences, and conceptions of the good that they sustain. People's desires and purposes are not then assumed to be given, whatever they are, and then promoted and fulfilled. On Rawls's Kantian view, principles of right and justice are designed to put limits on what satisfactions and purposes have value and impose restrictions on what are reasonable conceptions of persons' good. This basically is what Rawls means by "the priority of right over the good." People's desires and aspirations are constrained from the outset by principles of justice, which specify the criteria for determining permissible ends and conceptions of the good. (TJ 31-32/27-28) If the parties to Rawls's original position had knowledge of peoples' beliefs and desires, as well as knowledge of the laws, institutions and circumstances of their society, then this knowledge would influence their decisions on principles of justice. The principles agreed to would then not be sufficiently detached from the very desires, circumstances, and institutions these principles are to critically assess. Since utilitarians take peoples' desires, preferences, and/or ends as given under existing circumstances, any principles, laws, or institutions chosen behind their thin veil of ignorance will reflect and be biased by the status quo. To take an obvious counterexample, there is little if any justice in laws approved from a utilitarian impartial perspective when these laws take into account racially prejudiced preferences which are cultivated by grossly unequal, racially discriminatory and segregated social conditions. To impartially give equal consideration to everyone's desires formed under such under unjust conditions is hardly sufficient to meet requirements of justice. This illustrates some of the reasons for a "thick" as opposed to a "thin" veil of ignorance. ## 4. Description of the Parties: Rationality and the Primary Social Goods Rawls says that in the original position, "the Reasonable frames the Rational" (CP 319). He means the OP is a situation where rational choice of the parties is made subject to reasonable (or moral) constraints. In what sense are the parties and their choice and agreement rational? Philosophers have different understandings of practical rationality. Rawls seeks to incorporate a relatively uncontroversial account of rationality into the original position, one that he thinks most any account of practical rationality would endorse as at least necessary for rational decision. The parties are then described as rational in a formal or "thin" sense that is characteristic of the theories of rational and social choice. They are resourceful, take effective means to their ends, and seek to make their preferences consistent. They also take the course of action that is more likely to achieve their ends (other things being equal). And they choose courses of action that satisfy more rather than fewer of their purposes. Rawls calls these principles of rational choice the "counting principles" (TJ SSSS25, 63; JF 87). More generally, for Rawls rational persons upon reflection can formulate a conception of their good, or of their primary values and purposes and the best way of life for themselves to live given their purposes. This conception incorporates their primary aims, ambitions, and commitments to others, and is informed by the conscientious moral, religious, and philosophical convictions that give meaning for them to their lives. Ideally, rational persons have carefully thought about these things and their relative importance, and they can coherently order their purposes and commitments into a "rational plan of life," which extends over their lifetimes (TJ SSSS63-64). For Rawls, rational persons regard life as a whole, and do not give preference to any particular period of it. Rather in drawing up their rational plans, they are equally concerned with their (future) good at each part of their lives. In this regard, rational persons are prudent--they care for their future good, and while they may discount the importance of future purposes based on probability assessments, they do not discount the achievement of their future purposes simply because they are in the future (TJ, SS45). (For a different view, see Parfit, 1984) These primary aims, convictions, ambitions, and commitments are among the primary motivations of the parties in the original position. The parties want to provide favorable conditions for the pursuit of the various elements of the rational plan of life that defines a good life for them. This is ultimately what the parties are trying to accomplish in their choice of principles of justice. In this sense they are rational. Rawls says the parties in the original position are "mutually disinterested," in the sense that "they take no interest in each other's interests" (TJ 110/[omitted in rev. ed.]). This does not mean that they are self-interested or selfish persons, indifferent to the welfare of others. The interests advanced by the parties' life plans, Rawls says, "are not assumed to be interests in the self, they are interests of a self that regards its conception of the good as worthy of satisfaction..." (TJ 127/110) Most people are concerned, not just with their own happiness or welfare, but with others as well, and have all kinds of commitments, including other-regarding, beneficent, and moral purposes, that are part of their conceptions of the good. But in the original position itself the parties are not altruistically motivated to benefit each other, in their capacity as contracting parties. They try to do as best as they can for themselves and for those persons and causes that they care for. Their situation is comparable, Rawls says, to that of "trustees or guardians" acting to promote the interests of the beneficiaries they represent. (JF, 84-85) Trustees cannot sacrifice the well-being of the beneficiary they represent to benefit other trustees or individuals. If they did, they would be derelict in their duties. It is perhaps to address the common criticism that the parties to the original position are self-interested that Rawls in the revised edition (TJ 110 rev.) omitted the phrase from the 1st edition, cited above, that "the parties take no interest in each other's interest." Moreover in later writings increasingly he says that we should imagine that the parties are "representatives" of free and equal citizens and their interests and "act as guardians or trustees," seeking to do as best as they can for the particular individuals that each trustee represents. (PL SS4, JFSS24) In either case, Rawls believes this account of the parties' motivations promotes greater clarity, and that to attribute to the parties moral motivations or benevolence towards each other would not result in definite choice of a conception of justice (TJ, 148-9/128-9; 584/512). (For example, how much benevolence should the parties have towards one another or towards people in general? Surely not impartial benevolence towards everyone, for then we might as well dispense with the social contract and rely on a disinterested impartial spectator point of view. It is one of the "circumstances of justice" that people have different and conflicting values, and they value their own purposes and special commitments to others more than they value others' purposes and special commitments, This is a good thing, not to be discouraged or undermined by justice, but rather regulated by it, since special obligations and commitments to specific others give meaning to people's lives. (cf. Scheffler, 2001, chs.3, 4, 6) But if not equal concern for other parties and/or persons including themselves (and perhaps other animals), then how much care and concern should the parties in the original position exhibit towards others generally, as compared with concern for themselves and their own good? (Half as much concern for others' good as for their own? One-fifth as much? There is no clear answer.) Rawls's thought is that, so far as justice is concerned, fair regard for others' interests is best represented by each party's rational choice behind a thick veil of ignorance; for each party has to be equally concerned with the consequences of their choice of principles for each position in society, since they could end up in that same position. Mutual disinterest of the parties also means they are not moved by envy or rancor towards each other or others generally. This implies that the parties do not strive to be wealthier or better off than others for its own sake, and thus do not sacrifice advantages to prevent others from having more than they do. Instead, each party in the original position is motivated to do as well as they can in promoting the optimal achievement of the many purposes that constitute their rational conception of the good, without regard to how much or how little others may have. For this reason they strive to guarantee themselves a share of primary social goods that is at least sufficient to enable them each to effectively pursue their (unknown) conception of the good. Another feature of the parties is that they represent not just themselves, but also family lines, including their descendants, or at least their own children. This assumption is needed, Rawls says, to include representation of "the interests of all," including children and future generations. In the first edition of Theory Rawls says. "For example, we may think of the parties as heads of families and therefore as having a desire to further the welfare of their nearest descendants" (Rawls 1971, 128). Because of criticisms of the heads of families assumption, (by English, 1977 and others) Rawls said in the revised edition that the problem of future generations can be addressed by the parties assuming that all preceding generations have followed the same principles that the parties choose to apply to future generations. (Rawls 1999a, 111 rev.). The "heads of families" assumption is discussed further in connection with feminist criticisms of Rawls in the supplementary section: > > A Liberal Feminist Critique of the Original Position and Justice within the Family > in the supplementary document Further Topics on the Original Position. Though the parties are not motivated by beneficence or even a concern for justice, still they have a moral capacity for reasonableness and a sense of justice (TJ, 145/125 rev.). Rawls distinguishes between the requirements of rationality and reasonableness; both are part of practical reasoning about what we ought to do (JF 6-7; 81-2). The concept of "the Rational" concerns a person's good--hence Rawls refers to his account of the good as "goodness as rationality." A person's good for Rawls is the rational plan of life they would choose under hypothetical conditions of "deliberative rationality," where there is full knowledge of one's circumstances, capacities, and interests, as well as knowledge of the likelihood of succeeding at alternative life plans one may be drawn to (TJ, SS64). "The Reasonable" on the other hand addresses the concept and principles of right, including individual moral duties and obligations as well as moral requirements of right and justice that apply to institutions and society. Both rationality and reasonableness are independent aspects of practical reason for Rawls. They are independent in that Rawls, unlike Hobbes and other interest-based social contract views, does not regard justice and the reasonable as simply principles of prudence that are beneficial for a person to comply with in order to successfully pursue their purposes in social contexts. (Cf. Gauthier, 1984) Unlike Hobbes, Rawls does not argue that an immoral or unjust person is irrational, or that morality is necessarily required by rationality in the narrow sense of maximizing individual utility or taking effective means to realize one's purposes. But rational persons who violate demands of justice are unreasonable in so far as they infringe upon moral principles and requirements of practical reasoning. Being reasonable, even if not required by rationality, is still an independent aspect of practical reason. Rawls resembles Kant in this regard (PL 25n); his distinction between the reasonable and rational parallels Kant's distinction between categorical and hypothetical imperatives. Essential to being reasonable is having a sense of justice with the capacities to understand and reason about and act upon what justice requires. The sense of justice is a normally effective desire to comply with duties and obligations required by justice; it includes a willingness to cooperate with others on terms that are fair and that reasonable persons can accept and endorse. Rawls sees a sense of justice as an attribute people normally have; it "would appear to be a condition for human sociability" (TJ, 495/433 rev.). He rejects the idea that people are motivated only by self-interest in all that they do; he also rejects the Hobbesian assumption that a willingness to do justice must be grounded in enlightened self-interest. It is essential to Rawls's argument for the feasibility and stability of justice as fairness that the parties upon entering society have an effective sense of justice, and that they are capable of doing what justice requires of them for its own sake, or at least because they believe this is what morality requires of them. An amoralist, Rawls believes, is largely a philosophical construct; amoralists who actually exist Rawls regards as sociopaths. "A capacity for a sense of justice ... would appear to be a condition of sociability" (TJ 495/433). Subsequent to A Theory of Justice, beginning in 'Kantian Constructivism in Moral Theory,' (1980) (CP 303ff.) Rawls says that the parties to the original position have a "highest-order interest" in the development and full and informed exercise of their two "moral powers": their capacity for a sense of justice as well as in their capacity for a rational conception of the good. Fulfilling these interests in the moral powers is one of the main aims behind their agreement on principles of justice. Subsequently in Political Liberalism (1993) Rawls changed this to the parties' "higher order interests" in development and exercise of the two moral powers (to avoid giving the appearance that the moral powers were final ends for free and equal moral persons, as was argued in TJ). The parties' interest in developing these two moral powers is a substantive feature of Rawls's account of the rationality of free and equal persons in the original position itself. (In this regard, his account of goodness as rationality is not as "thin" as in social theory; cf. TJ 143/124 rev.) Here Rawls is still not attributing specifically moral motives--a desire to be reasonable and do what is right and just for their own sake--to the parties in the original position. The idea behind the parties' rationality in cultivating their sense of justice is that, since being reasonable and exercising one's sense of justice by complying with fair terms is a condition of human sociability and social cooperation, then it is in people's rational interest--part of their good--that they normally develop their capacities for justice under social conditions. Otherwise they will not be in a position to cooperate with others and benefit from social life. A person who is without a sense of justice is wholly unreasonable and as a result is normally eschewed by others, for they are not trustworthy or reliable or even safe to interact with. Since having a sense of justice is a condition of taking part in social cooperation, the parties have a "higher-order interest" in establishing conditions for the development and full exercise of their capacity for a sense of justice. The parties' interest in developing their capacity for a sense of justice is then a rational interest in being reasonable; justice is then regarded by the parties as instrumental to their realizing their conception of the good. (Here again, it is important to distinguish the purely rational motivation of the parties or their trustees in the original position from that of free and equal citizens in a well-ordered society, who are normally morally motivated by their sense of justice to do what is right and just for its own sake.) Three factors then play a role in motivating the parties in the original position: (1) First, they aim to advance their determinate conception of the good, or rational plan of life, even though they do not know what that conception is. Moreover, they also seek conditions that enable them to exercise and develop their "moral powers," namely (2) their rational capacities to form, revise and rationally pursue a conception of their good, and (3) their capacity to be reasonable and to have an effective sense of justice. These are the three "higher-order interests" the parties to Rawls's original position aim to promote in their agreement on principles of justice. The three higher-order interests provide the basis for Rawls's account of primary social goods. (TJ SS15, PL 178-190) The primary goods are the all-purpose social means that are necessary to the exercise and development of the moral powers and to pursue a wide variety of conceptions of the good. Rawls describes them initially in Theory as goods that any rational person should want, whatever their rational plan of life. The primary social goods are basically: rights and liberties; powers and diverse opportunities; income and wealth; and the social bases of self-respect. 'Powers' refer not (simply) to a capacity to effect outcomes or influence others' behavior. Rawls rather uses the term 'powers' to refer to the legal and other institutional abilities and prerogatives that attend offices and social position. Hence, he sometimes refers to the primary goods of "powers and prerogatives of offices and positions of authority and responsibility" (JF 58). Members of various professions and trades have institutional powers and prerogatives that are characteristic of their position and which are necessary if they are to carry out their respective roles and responsibilities. By income and wealth Rawls says he intends "all-purpose means" that have an exchange value, which are generally needed to achieve a wide range of ends (JF 58-59). Finally, "the social bases of self-respect" are features of institutions that are needed to enable people to have the confidence that they and their position in society are respected and that their conception of the good is worth pursuing and achievable by themselves. These features depend upon history and culture. Primary among these social bases of self respect in a democratic society, Rawls will contend, are equal recognition of persons as citizens, and hence the institutional conditions needed for equal citizenship, including equality of basic rights and liberties with equal political rights; fair equality of opportunities; and personal independence guaranteed by adequate material means for achieving it. The social bases of self-respect are crucial to Rawls's argument for equal basic liberties, especially political equality and equal rights of political participation. The parties to the original position are motivated to achieve a fully adequate share of primary goods so they can achieve their higher-order interests in pursuing their rational plans of life and exercising their moral powers. "They assume that they normally prefer more primary social goods rather than less" (TJ, 142/123 rev.). This too is part of being rational. Because they are not envious, their concern is with the absolute level of primary goods, not their share relative to other persons. To sum up, the parties in the original position are formally rational in that they are assumed to have and to effectively pursue a rational plan of life with a schedule of coherent purposes and commitments that they find valuable and give their lives meaning. As part of their rational plans, they have a substantive interest in the adequate development and full exercise of their capacities to be rational and to be reasonable. These "higher-order interests" together with their rational life plans provide them with sufficient reason to procure for themselves in their choice of principles of justice an adequate share of the primary social goods that enable them to achieve these higher-order ends and effectively pursue their conceptions of the good. A final feature of Rawls's account of rationality is a normal human tendency he calls "the Aristotelian principle" (TJ sect.65). This "deep psychological fact" says that, other things being equal, people normally find activities that call upon the exercise of their developed capacities to be more interesting and preferable to engaging in simpler tasks, and their enjoyment increases the more the capacity is developed and realized and the greater the complexity of activities (TJ, 426/374). Humans enjoy doing something as they become more proficient at it, and of two activities they perform equally well, they normally prefer the one that calls upon a larger repertoire of more intricate and subtler discriminations. Rawls's examples: someone who does both activities well generally prefers playing chess to checkers, and studying algebra to arithmetic. (TJ 426/374) Moreover Rawls, citing J.S. Mill believes that development at least some of our "higher capacities" (Mill's term) is normally important to our sense of self-respect. These general facts imply that rational people should incorporate into their life plans activities that call upon the exercise and development of their talents and skills and distinctly human capacities (TJ 432/379). This motivation becomes especially relevant to Rawls's argument for the stability of justice as fairness, the good of social union, and the good of justice (TJ SS79, SS86; see below, SS5.3). The important point here is that the Aristotelian principle is taken into account by the parties in their decision on principles of justice. They want to choose principles that maintain their sense of self-respect and enable them to freely develop their human capacities and pursue a wide range of activities, as well as engage their capacities for a sense of justice. ## 5. Other Conditions on Choice in the Original Position The veil of ignorance is the primary moral constraint upon the rational choice of the parties in the original position. There are several other conditions imposed on their agreement. ### 5.1 The Circumstances of Justice (TJ SS22) Among the general facts the parties know are "the circumstances of justice." Rawls says these are "conditions under which human cooperation is both possible and necessary" (TJ 126/109 rev.). Following Hume, Rawls distinguishes two general kinds: the objective and subjective circumstances of justice. The former include physical facts about human beings, such as their rough similarity in mental and physical faculties, and vulnerability to the united force of others. Objective circumstances also include conditions of moderate scarcity of resources: there are not enough resources to satisfy everyone's desires, but there are enough to provide all with adequate satisfaction of their basic needs; unlike conditions of extreme scarcity (e.g. famine), cooperation then seems productive and worthwhile for people. Among the subjective circumstances of justice are the parties' mutual disinterestedness, which reflects the "limited altruism" (TJ 146/127) of persons in society.. Free and equal persons have their own plans of life and special commitments to others, as well as different philosophical and religious beliefs and moral doctrines (TJ 127/110). Hume says that if humans were impartially benevolent, equally concerned with everyone's welfare, then justice would be unnecessary. People then would willingly sacrifice their interests for the greater advantage of other. They would not be concerned about their personal rights or possessions, and property would be unnecessary (Hume 1777 [1970, 185-186]). But we are more concerned with our own aims and interests--which include our interests in the interests of those nearer and dearer to us--than we are with the interests of strangers with whom we have few if any interactions. This implies a potential conflict of human interests. Rawls adds that concern for our interests and plans of life does not mean we are selfish or have interests only in ourselves--again, interests of a self should not be confused with interests in oneself; we have interests in others and in all kinds of causes, ends, and commitments to other persons (TJ 127/110). But, as history shows, our benevolent interests in others and in religious and philosophical doctrines are at least as often the cause of social and international conflict as is self-interest. The subjective circumstances of justice also include limitations on human knowledge, thought, and judgment, as well as emotional influences and great diversity of experiences. These lead to biases and inevitable disagreements in factual and other judgments, as well as to differences in religious, philosophical, and moral convictions. In Political Liberalism, Rawls highlights these subjective circumstances, calling them "the burdens of judgment" (PL 54-58). They imply, significantly, that regardless how impartial and altruistic people are, they still will disagree in their factual judgments and in religious, philosophical and moral doctrines. Disagreements in these matters are inevitable even among fully rational and reasonable people. This is "the fact of reasonable pluralism" (PL 36), which is another general fact known to the parties in the original position. Reasonable pluralism of doctrines lends significant support to Rawls's arguments for the first principle of justice, especially to equal basic liberties of conscience, expression, and association. ### 5.2 Publicity and other Formal Constraints of Right (TJ SS23) There are five "formal constraints" associated with the concept of right that Rawls says the parties must take into account in coming to agreement on principles of justice. The more a conception of justice satisfies these formal constraints of right, the more reason the parties have to choose that conception. The formal constraints of right are: generality, universality in application, ordering of conflicting claims, publicity, and finality. The ordering condition says that a conception of justice should aspire to completeness: it should be able to resolve conflicting claims and order their priority. Ordering implies a systematicity requirement: principles of justice should provide a determinate resolution to problems of justice that arise under them; and to the degree that a conception of justice is not able to order conflicting claims and resolve problems of justice, that gives greater reason against choosing it in the original position compared with those that do. The ordering condition is important in Rawls's argument against pluralist moral doctrines he calls "Intuitionism." Sidgwick attaches a great deal of importance to the ordering condition, and contends that "Universal Hedonism" is the only reasonable moral doctrine that can satisfy it (Sidgwick 1907 [1981], 406). Rawls would have to concede that justice as fairness does not possess, at least theoretically, the same degree of systematic ordering of claims as does hedonistic utilitarianism which has cardinal measures of utility. For example, Rawls's priority principles can resolve conflicting claims regarding the priority of basic liberties over fair equality of opportunity, fair opportunity over the difference principle, the difference principle over the principle of efficiency and the general welfare, as well as many disputes arising within the difference principle itself regarding measures that maximally promote the position of the least advantaged. But there is no priority principle or algorithm to resolve many conflicts between basic liberties themselves (e.g. conflicts between freedom of speech vs. rights of security and integrity of persons in hate speech cases; or the conflict between free speech and the fair value of equal political liberties in restrictions on campaign finance contributions). Often in such conflicts we have to weigh competing considerations and come to a decision about where the greater balance of reasons lies, much like intuitionist views. (See Hart, 1973). Rawls in 'Basic Liberties and their Priority,' 1980, PL ch.VIII, addresses this problem to some degree with the idea of the significance of a basic liberty to the development and full and informed exercise of the moral powers.) The lack of a priority or algorithmic ordering principle does not mean the balance of reasons in such conflicts regarding basic liberties is indeterminate but rather that reasonable individuals will often disagree, and that final decisions practically will have to be made through the appropriate democratic, judicial, or other procedures (which of course can be mistaken). But for Rawls a moral conception's capacity to clearly order conflicting claims is not dispositive, but one among several formal and substantive moral conditions that a conception of justice should satisfy (ultimately in reflective equilibrium). The publicity condition says that the parties are to assume that the principles of justice they choose will be publicly known to members of society and recognized by them as the bases for their social cooperation. This implies that people will not be uninformed, manipulated, or otherwise have false beliefs about the bases of their social and political relations. There are to be no "noble lies", false ideologies, or "fake news" obscuring a society's principles of justice and the moral bases for its basic social institutions. The publicity of principles of justice is ultimately for Rawls a condition of respect for persons as free and equal moral persons. Rawls believes that individuals in a democratic society should know the bases of their social and political relations and not have to be deceived about them in order to cooperate and live together peacably and on fair terms. Publicity plays an important role in Rawls's arguments against utilitarianism and other consequentialist conceptions. The idea of publicity is further developed in Political Liberalism through the ideas of public justification and the role of public reason in political deliberation. Related to publicity is that principles should be universal in application. This implies not simply that "they hold for everyone in virtue of their being moral persons" (TJ 132/114 rev.). It also means that everyone can understand the principles of justice and use them in their deliberations about justice and its requirements. Universality in application then imposes a limit on how complex principles of justice can be--they must be understandable to common moral sense, and not so complicated that only experts can apply them in deliberations. For among other things, these principles are to guide democratic citizens in their judgments and shared deliberations about just laws and policies. Both publicity and universality in application (as Rawls defines it) are controversial conditions. Utilitarians, for example, have argued that the truth about morality and justice is so complicated and controversial that it might be necessary to keep fundamental moral principles (the principle of utility) hidden from most individuals' awareness. For morality and justice often require much that is contrary to peoples' beliefs and personal interests. Also sometimes it's just too complicated for people to understand the reasons for their moral duties. So long as they understand their individual duties, it may be better if they do not understand the principles and reasons behind them. So Sidgwick argues that the aims of utilitarianism might better be achieved if it remains an "esoteric morality," knowledge of which is confined to "an enlightened few" (Sidgwick 1907 [1981], 489-90). The reason Rawls sees publicity and universality as necessary relates to the conception of the person implicit in justice as fairness. If we conceive of persons as free and equal moral persons capable of political and moral autonomy, then they should not be under any illusions about the bases of their social relations, but should be able to understand, accept, and apply these principles in their deliberations about justice. These are important conditions Rawls contends for the freedom, equality, and autonomy (moral and political) of democratic citizens. Finally, the generality condition is straightforward in that it requires that principles of justice not contain any proper names or rigged definite descriptions, which Rawls says rules out free-rider and other forms of egoism together with the ordering condition. The finality condition says that moral principles of justice provide conclusive reasons for action, providing "the final court of appeal in practical reasoning." They override demands of law and custom, social rules, and reasons of personal prudence and self-interest. (TJ 135-36/116-17). Finality is one of several Kantian conditions Rawls imposes that have been questioned by critics on grounds that it underestimates inevitable and sometimes irresolvable conflicts of moral reasons with other values. For example, should reasons of justice always be given priority over special obligations owed to specific persons or associations? Should moral reasons always be given priority over reasons of love, prudence, or even self-interest? (See Williams 1981, chs.1, 5; Wolf, 2014, chs.2, 3, 9) ### 5.3 The Stability Requirement Rawls says, "An important feature of a conception of justice is that it should generate its own support. Its principles should be such that when they are embodied in the basic structure of society, people tend to acquire the corresponding sense of justice and develop a desire to act in accordance with its principles. In this case a conception of justice is stable" (TJ, 138/119). The parties in the original position are to take into account the "relative stability" of a conception of justice and the society that institutes it. The stability of a just society does not mean that it must be unchanging. It means rather that in the face of inevitable change members of a society should be able to maintain their allegiance to principles of justice and the institutions they support. When disruptions to society do occur (via economic crises, war, natural catastrophes, etc.) and/or society departs from justice, citizens' commitments to principles of justice are sufficiently robust that just institutions are eventually restored. The role of the stability requirement for Rawls is twofold: first, to test whether potential principles of justice are compatible with human natural propensities, or our moral psychology and general facts about social and economic institutions; and second, to determine whether acting on and from principles of justice are conducive and even essential to realizing the human good. To be stable principles of justice should be realizable in a feasible and enduring social world, the ideal of which Rawls calls a "well-ordered society." (See below, SS6.3.) They need to be practicably possible given the limitations of the human condition. Moreover, this feasible social world must be one that can endure over time, not by just any means, but by gaining the willing support of people who live in it. People should knowingly want to uphold and maintain society's just institutions not just because they benefit from them, but on grounds of their sense of justice. In choosing principles of justice, the parties in the original position must take into account their "relative stability" (TJ SS76). They have to consider the degree to which a conception (in comparison with other conceptions) describes an achievable and sustainable system of social cooperation, and whether the institutions and demands of such a society will attract people's willing compliance and generally engage their sense of justice. For example, suppose principles of justice were to impose a duty to practice impartial benevolence towards all people, and thus a duty to show no greater concern for the welfare of ourselves and loved ones than we do towards billions of others. This principle demands too much of human nature and would not be sustainable or even feasible--people simply would reject its onerous demands. But Rawls's stability requirement implies more than just 'ought implies can.' It says that principles of justice and the scheme of social cooperation they describe should evince "stability for the right reasons" ((as Rawls later says in PL xli, 143f., 459f.,). Recall here the higher-order interests of the parties in development and exercise of their capacities for justice. A just society should be able to endure not simply as a modus vivendi, or compromise among conflicting interests; nor simply endure by promoting the majority of peoples' interests and/or coercive enforcement of its provisions. Stability "for the right reasons," as conceived in Theory, requires that people support society for moral reasons of justice. Society's basic principles must respond to reasonable persons' capacities for justice and engage their sense of justice. Rawls regards our moral capacities for justice as an integral part of our nature as sociable beings. He believes that one role of a conception of justice is to accommodate human capacities for sociability, the capacities for justice that enable us to be cooperative social beings. So not only should a conception of justice advance human interests, but it should also answer to our moral psychology by enabling us to knowingly and willingly exercise our moral capacities and sensibilities, which are among the moral powers to be reasonable. This is one way that Rawls's conception of justice is "ideal-based" (CP 400-401 n.): it is based in an ideal of human beings as free and equal moral persons and an ideal of their social relations as generally acceptable and justifiable to all reasonable persons whatever their circumstances (the ideal of a well-ordered society). This relates to the second ground for the stability condition, which can only be mentioned here: it is that the correct principles of justice should be compatible with, and even integral to realizing the human good. It speaks strongly in favor of a conception of justice that it is compatible with and promotes the human good. First, if a conception of justice requires of many reasonable people that they change their conscientious philosophical or religious convictions for the sake of satisfying a majority's beliefs, or abandon their pursuit of the important interests that constitute their plan of life, this conception could not gain their willing support and would not be stable over sustained periods of time. Moreover, Rawls contends that a conception of justice should enable citizens to fully exercise and adequately develop their moral powers, including their capacities for justice. It must then engage their sense of justice in such a way that they do not regard justice as a burden but should come to experience that acting on and from principles of justice is worth doing for its own sake. For Rawls, it speaks strongly in favor of a conception of justice that acting for the sake of its principles is experienced as an activity that is good in itself (as Rawls contends in Theory of Justice); or at least that willing compliance with requirements of justice is an essential part of the reasonable comprehensive philosophical, religious, or moral doctrines that reasonable persons affirm (as Rawls contends later in Political Liberalism). For then justice and the full and informed exercise of the sense of justice are for reasonable and rational persons essential goods, preconditions for their living a good life, as that is defined by their rational conception of the good. ## 6. The Arguments for the Principles of Justice from the Original Position The original position is not a bargaining situation where the parties make proposals and counterproposals and negotiate over different principles of justice. Nor is it a wide ranging discussion where the parties debate, deliberate, and design their own conception of justice (unlike, for example, Habermas's discourse ethics; see Habermas, 1995). Instead, the parties' deliberations are much more constrained and regulated. They are presented with a list of conceptions of justice taken from the tradition of western political philosophy. These include different versions of utilitarianism, perfectionism, and intuitionism (or pluralist views), rational egoism, justice as fairness, and a group of "mixed conceptions" that combine elements of these. (For Rawls's initial list see TJ 124/107) Rawls later says libertarian entitlement principles should also be added to the list, and contends the principles of justice are still preferable. (JF 83). (Nozick agrees and says the OP is incapable of yielding historical entitlement principles, but only patterned end-state principles instead. Nozick 1974, 198-204. Rawls replies that the difference principle does not conform to any observable pattern but grounds fair distributions in a fair social process that must actually be carried out. PL, 282-83) The parties' deliberations are confined to discussing and agreeing upon the conception that each finds most rational, given their specified interests. In a series of pairwise comparisons, they consider all the conceptions of justice made available to them and ultimately agree unanimously to accept the conception that survives this winnowing process. In this regard, the original position is best seen as a kind of selection process wherein the parties' deliberations are constrained by the background conditions imposed by the original position as well as the list of conceptions of justice provided to them. They are assigned the task of agreeing on principles for designing the basic structure of a self-contained society under the circumstances of justice. In making their decision, the parties are motivated only by their own rational interests. They do not take moral considerations of justice into account except in so far as these considerations bear on their achieving their interests within society. Their interests again are defined in terms of their each acquiring an adequate share of primary social goods (rights and liberties, powers and opportunities, income and wealth, etc.) and achieving the background social conditions enabling them to effectively pursue their conception of the good and realize their higher-order interests in developing and exercising their moral powers. Since the parties are ignorant of their particular conceptions of the good and of all other particular facts about their society, they are not in a position to engage in bargaining. In effect they all have the same general information and are motivated by the same interests. Rawls makes four arguments in Theory, Part I for the principles of justice. The main argument for the difference principle is made later in TJ SS49, and is amended and clarified in Justice as Fairness: A Restatement. The common theme throughout the original position arguments is that it is more rational for the parties to choose the principles of justice over any other alternative. Rawls devotes most of his attention to the comparison of justice as fairness with classical and average utilitarianism, with briefer discussions of perfectionism (TJ, SS50) and intuitionism (TJ 278-81) Here I'll focus discussion primarily on Rawls's comparison between justice as fairness and utilitarianism. ### 6.1 The Principles of Justice Before turning to Rawls's arguments from the original position, it is helpful to have available the principles of justice and other principles that constitute Justice as Fairness. First Principle: "Each person has an equal right to the most extensive total system of equal basic liberties compatible with a similar system of liberty for all." (TJ 266) The first principle was revised in 1982 to say "Each person has an equal right to a fully adequate scheme of equal basic liberties ..." (PL, 291) replacing "... the most extensive scheme of equal basic liberties.") Notably, Rawls also introduces in Political Liberalism, almost in passing, a principle of basic needs that precedes the first principle and requires that citizens' basic needs be met at least to the extent that they can understand and fruitfully exercise their basic rights and liberties. (PL 7; JF 79n.) This social minimum is also said in Political Liberalism to be a "constitutional essential" for any reasonable liberal conception of justice. (PL 166, 228ff.; JF 47, n.7) The basic rights and liberties protected by the first principle are specified by a list (see TJ 53f., PL 291): liberty of conscience and freedom of association, (TJ SSSS33-4); freedom of thought and freedom of speech and expression (PL, pp.340-363); the integrity and freedom of the person and the right to hold personal property; equal rights of political participation and their fair value (TJ SSSS36-37); and the rights and liberties protected by the rule of law (due process, freedom from arbitrary arrest, etc. TJ SS38). (Rawls says the right to ownership of means of production and laissez faire freedom of contract are not included among the basic liberties. TJ, 54 rev. Also freedom of movement and free choice of occupation are said to be primary goods protected by fair equality of opportunity principle. PL 76, JF 58f.)) Second Principle: "Social and economic inequalities are to satisfy two conditions. First they must attach to offices and positions open to all under conditions of fair equality of opportunity; and second they must be to the greatest advantage of the least advantaged members of society [the difference principle]" consistent with the just savings principle. (PL 281, JF 42-43, TJ 301/266 rev.) Just Savings Principle: Each generation should save for future generations at a savings rate that they could rationally expect past generations to have saved for them. (TJ SS44; JF 159-160) Principles for individuals, include (a) the natural duties to uphold justice, mutual respect, mutual aid, and not to injure or harm the innocent (TJ SSSS19, 51); and (b) the principle of fairness, to do one's fair share in just or nearly just practices and institutions from which one accepts their benefits, (which grounds the principle of fidelity, to keep one's promises and commitments. (TJ SSSS18, 52). The Priority Principles: the principles of justice are ranked in lexical order. (a) The priority of liberty requires that basic liberties can only be restricted to strengthen the system of liberties shared by all. (b) Fair equality of opportunity is lexically prior to the difference principle. (c) The second principle is prior to the principle of efficiency and maximizing the sum of advantages. (TJ 302/266 rev.) The General Conception of Justice: "All social goods--liberty and opportunity, income and wealth, and the bases of self-respect, are to be distributed equally unless an unequal distribution of any or all of these goods is to the advantage of the least favored." TJ 1971, 302. Note: The general conception is the difference principle generalized to all primary goods (TJ 1971, 83); it applies in non-ideal conditions where the priority of liberty and opportunity is not sustainable. ### 6.2 The Argument from the Maximin Criterion (TJ, SSSS26-28) Describing the parties' choice as a rational choice subject to the reasonable constraints imposed by the original position allows Rawls to invoke the theory of rational choice and decision under conditions of uncertainty. In rational choice theory there are a number of potential "strategies" or rules of choice that are more or less reliably used depending on the circumstances. One rule of choice--called "maximin"--directs that we play it as safe as possible by choosing the alternative whose worst outcome leaves us better off than the worst outcome of all other alternatives. The aim is to "maximize the minimum" regret or loss to one's position (measured in terms of welfare or, for Rawls, one's share of primary social goods). To follow this strategy, Rawls says you should choose as if your enemy were to assign your social position in whatever kind of society you end up in. By contrast another strategy leads us to focus on the most advantaged position and says we should "maximize the maximum" potential gain--"maximax"--and choose the alternative whose best outcome leaves us better off than all other alternatives. Which, if either, of these strategies is more sensible to use depends on the circumstances and many other factors. A third strategy advocated by orthodox Bayesian decision theory, says we should always choose to directly maximize expected utility. To do so under conditions of uncertainty of outcomes, the degree of uncertainty should be factored into one's utility function, with probability estimates assigned to alternatives based on the limited knowledge that one has. Given these subjective estimates of probability incorporated into one's utility function, one can always choose the alternative that maximizes expected utility. Since it simplifies matters to apply the same rule of choice to all decisions this is a highly attractive idea, so long as one can accept that it is normally safe to assume that that the maximization of expected utility leads over time to maximizing actual utility. What about those extremely rare instances where there is absolutely no basis upon which to make probability estimates? Suppose you don't even have a hunch regarding the greater likelihood of one alternative over another. According to orthodox Bayesian decision theory, the "principle of insufficient reason" should then be observed; it says that when there is no reason to assign a greater likelihood to one alternative rather than another, then an equal probability is to be assigned to each potential outcome. This makes sense on the assumption that if you have no more premonition of the likelihood of one option rather than another, they are for all you know equally likely to occur. By observing this rule of choice consistently over time, a rational chooser presumably should maximize expected individual utility, and hopefully actual utility as well. What now is the appropriate decision rule to be used to choose principles of justice under conditions of complete uncertainty of probabilities in Rawls's original position? Rawls argues that, given the enormous gravity of choice in the original position, plus the fact that the choice is not repeatable (there's no opportunity to renegotiate or revise one's decision), it is rational for the parties to follow the maximin strategy when choosing between the principles of justice and principles of average or aggregate utility (or any other principles that do not guarantee basic rights, liberties, opportunities, and a social minimum). Not surprisingly, following the maximin rule of choice results in choice of the principles of justice over the principles of utility (average or aggregate); for unlike utilitarianism, justice as fairness guarantees equal basic liberties, fair equal opportunities, and an adequate social minimum for all citizens. Why does Rawls think maximin is the rational choice rule? Recall what is at stake in choice from the original position. The decision is not an ordinary choice. It is rather a unique and irrevocable choice where the parties decide the basic structure of their society, or the kind of social world they will live in and the background conditions against which they will develop and pursue their aims. It is a kind of superchoice--an inimitable choice of the background conditions for all one's future choices. Rawls argues that because of the unique importance of the choice in the original position--including the gravity of the choice, the fact that it is not renegotiable or repeatable, and the fact that it determines all one's future prospects--it is rational to follow the maximin rule and choose the principles of justice. For should even the worst transpire, the principles of justice guarantee an adequate share of primary goods enabling one to maintain one's conscientious convictions and sincerest affections and pursue a wide range of permissible ends by protecting equal basic liberties and fair equal opportunities and guaranteeing an adequate social minimum of income and wealth. The principles of utility, by contrast, provide no guarantee of any of these benefits. Rawls says that in general there are three conditions that must be met in order to make it rational to follow the maximin rule (TJ 154-55/134 rev.). First, there should be no basis or at most a very insecure basis upon which to make estimates of probabilities. Second, the choice singled out by observing the maximin rule is an acceptable alternative we can live with, so that one cares relatively little by comparison for what is to be gained above the minimum conditions secured by the maximin choice. When this condition is satisfied, then no matter what position one eventually ends up in, it is at least acceptable. The third condition for applying the maximin rule is that all the other alternatives have worse outcomes that we could not accept and live with. Of these three conditions Rawls later says that the first plays a minor role, and that it is the second and third conditions that are crucial to the maximin argument for justice as fairness (JF 99). This seems to suggest that, even if the veil of ignorance were not as thick and parties did have some degree of knowledge of the likelihood of ending up in one social position rather than another, still it would be more rational to choose the principles of justice over the principle of utility. Rawls contends all three conditions for the maximin strategy are satisfied in the original position when choice is made between the principles of justice and the principle of utility (average and aggregate). Because all one's values, commitments, and future prospects are at stake in the original position, and there is no hope of renegotiating the outcome, a rational person would agree to the principles of justice instead of the principle of utility. For the principles of justice imply that no matter what position you occupy in society, you will have the rights and resources needed to maintain your valued commitments and purposes, to effectively exercise your capacities for rational and moral deliberation and action, and to maintain your sense of self-respect as an equal citizen. With the principle of utility there is no such guarantee; everything is "up for grabs" (so to speak) and subject to loss if required by the greater sum of utilities. Conditions (2) and (3) for applying maximin are then satisfied in the comparison of justice as fairness with the principle of (average or aggregate) utility. It is often claimed that Rawls's parties are "risk-averse;" otherwise they would never follow the maximin rule but would take a chance on riskier but more rewarding outcomes provided by the principle of utility. Thus, John Harsanyi contends that it is more rational under conditions of complete uncertainty always to choose according to the principle of insufficient reason and assume an equal probability of occupying any position in society. When the equiprobability assumption is made, the parties in the original position would choose the principle of average utility instead of the principles of justice (Harsanyi 1975). Rawls denies that the parties have a psychological disposition to risk-aversion. They have no knowledge of their attitudes towards risk. He argues however that it is rational to choose as if one were risk averse under the highly exceptional circumstances of the original position. His point is that, while there is nothing rational about a fixed disposition to risk aversion, it is nonetheless rational in some circumstances to choose conservatively to protect certain fundamental interests against loss or compromise. It does not make one a risk averse person, but instead it is normally rational to purchase auto liability, health, and home insurance against accident or calamity (assuming it is affordable). The original position is such a situation writ large. Even if one knew in the original position that the citizen one represents enjoys gambling and taking great risks, this would still not be a reason to gamble with their rights, liberties and starting position in society. For if the high risktaker were born into a traditional, repressive, or fundamentalist society, they might never have an opportunity for gambling and taking other risks they normally enjoy. It is rational then even for high risktakers to choose conservatively in the original position and guarantee their future opportunities to gamble or otherwise take risks. Harsanyi and other orthodox Bayesians contend that maximin is an irrational decision rule, and they provide ample examples. To take Rawls' own example, in a lottery where the loss and gain alternatives are either (0, n) or (1/n, 1) for all natural numbers n, maximin says choose the latter alternative (1/n, 1). This is clearly irrational for almost any number n except very small numbers. (TJ 136 rev.). But such examples do not suffice here; simply because maximin is under most circumstances irrational does not mean that it is never rational. For example, suppose n>1 and you must have 1/n to save you own life. Given the gravity of the circumstances, it would be rational to choose conservatively since you are guaranteed 1/n according to the maximin strategy, and there is no guarantee you will survive if you choose according to the principle of insufficient reason. No doubt maximin is an irrational strategy under most circumstances of choice uncertainty, particularly under circumstances where we will have future opportunities to recoup our losses and choose again. But these are not the circumstances of the original position. Once the principles of justice are decided, they apply in perpetuity, and there is no opportunity to renegotiate or escape the situation. One who relies on the equiprobability assumption in choosing principles of justice in the original position is being foolishly reckless given the gravity of choice at stake. It is not being risk-averse, but rather entirely rational to refuse to gamble with one's basic liberties, fair equal opportunities and adequate resources needed to pursue one's most cherished ends and commitments, simply for the unknown chance of gaining the marginally greater social powers, income and wealth that might be available to some in a society governed entirely by the principle of utility. Rawls exhibits the force of the maximin argument in discussing liberty of conscience. He says (TJ, sect. 33) that a person who is willing to jeopardize their right to hold and practice their conscientious religious, philosophical and moral convictions, all for the sake of gaining uncertain added benefits via the principle of utility, does not know what it means to have conscientious beliefs, or at least does not take such beliefs seriously (TJ 207-08/181-82 rev.). A rational person with convictions about what gives life meaning is not willing to negotiate with and gamble away the right to hold and express those convictions and the freedom to act on them. After all what could be the basis for negotiation, for what could matter more than the objects of one's most sincere convictions and commitments? Some people (e.g. some nihilists) may not have any conscientious convictions (except the belief that nothing is worthwhile) and are simply willing to act on impulse or on whatever thoughts and desires they happen to have at the moment. But behind the veil of ignorance no one knows whether they are such a person, and it would be foolish to make this assumption. Knowing general facts about human propensities and sociability, the parties must take into account that people normally have conscientious convictions and values and commitments they are unwilling to compromise. (Besides, even the nihilist should want to protect the freedom to be a nihilist, to avoid ending up in an intolerant religious society.) Thus it remains irrational to jeopardize basic liberties by choosing the principle of utility instead of the principles of justice. None of this is to say that maximin is normally a rational choice strategy. Rawls himself says it "is not, in general, a suitable guide for choices under uncertainty" (TJ 153). It is not even a rational strategy in the original position when the alternatives for choice guarantee basic liberties, equal opportunities, and a social minimum guaranteed by the principle of average utility - see the discussion in the supplementary section: > > The Argument for the Difference Principle > in the supplementary document The Argument for the Difference Principle and the Four Stage Sequence. Rawls relies upon the maximin argument mainly to argue for the first principle of justice and fair equality of opportunity. Other arguments are needed to support his claim that justice requires the social minimum be determined by the difference principle. ### 6.3 The Strains of Commitment There are three additional arguments Rawls makes to support justice as fairness (all in TJ, sect. 29). Each of these depends upon the concept of a "well-ordered society." The parties in the original position are to choose principles that are to govern a well-ordered society where everyone agrees, complies with, and wants to comply with its principles of justice. The ideal of a well-ordered society is Rawls's development of social contract doctrine. It is a society in which (1) everyone knows and willingly accepts and affirms the same public principles of justice and everyone knows this; (2) these principles are successfully realized in basic social institutions, including laws and conventions, and are generally complied with by citizens; and (3) reasonable persons are morally motivated to comply by their sense of justice - they want to do what justice requires of them (TJ 4-5, SS69). There are then two sides to Rawls's social contract. The parties in the original position have the task of agreeing to principles that all can rationally accept behind the veil of ignorance under the conditions of the original position. But their rational choice is partially determined by the principles that free and equal moral persons in a well ordered society who are motivated by their sense of justice reasonably can accept, agree to.and comply with, as the basic principles governing their social and political relations. The parties are to assess principles according to the relative stability of the well ordered societies into which they are incorporated. Thus a well-ordered society of justice as fairness is to be compared with a well-ordered society whose basic structure is organized according to the average utility principle, aggregate utility, perfectionism, intuitionism, libertarianism, and so on. They are to consider which of these societies' basic struture is relatively more stable and likely to endure over time from one generation to the next, given natural and socially influenced psychological propensities and conditions of social cooperation as they interact with alternative principles of justice. Now to return to Rawls's arguments for his principles of justice. The first of Rawls's three arguments highlights the idea that choice in the original position is an agreement, and involves certain "strains of commitment." It is assumed by all the parties that all will comply with the principles they agree to once the veil is lifted and they are members of a well-ordered society (TJ 176f./153f. and CP 250ff). Knowing that they will be held to their commitment and expected to comply with principles for a well-ordered society, the parties must choose principles that they sincerely believe they will be able to accept, endorse and willingly observe under conditions where these principles are generally accepted and enforced. For reasons to be discussed shortly, Rawls says this condition favors agreement on the principles of justice over utilitarianism and other alternatives. But first, consider the frequent objection that there is no genuine agreement in the original position, for the thick veil of ignorance deprives the parties of all bases for bargaining (cf. TJ, 139-40/120-21 rev.). In the absence of bargaining, it is said, there can be no contract. For contracts must involve a quid pro quo--something given for something received (called 'consideration' at common law). The parties in the OP cannot bargain without knowing what they have to offer or to gain in exchange. So (the objection continues) Rawls's original position does not involve a real social contract, unlike those that transpire, say, in a state of nature. Rather, since the parties are all "described in the same way," there is no need for multiple parties but simply the rational choice of one person in the original position (see Hampton, 1980, 334; see also Gauthier, 1974 and 1985, 203). In response, not all contracts involve bargaining or are of the nature of economic transactions. Some involve a mutual pledge and commitment to shared purposes and principles. Marriage contracts, or agreements among friends or the members of a religious, benevolent, or political association are often of this nature. For example, the Mayflower Compact was a "covenant" to "combine ourselves together into a civil body politic" charged with making and administering "just and equal laws...for the general good." Likewise the U.S. Constitution represents itself as a commitment wherein "We The People ... ordain and establish this Constitution" in order "to establish justice," "promote the general welfare," "secure the blessings of liberty," and so on. The agreement in Rawls's original position is more of this nature. Even though ignorant of particular facts about themselves, the parties in fact do give something in exchange for something received: they all exchange their mutual commitment to accept and abide by the principles of justice and to uphold just institutions once they enter their well-ordered society. Each agrees only on condition others do too, and all tie themselves into social and political relations in perpetuity. Their agreement is final, and they will not permit its renegotiation should circumstances turn out to be different than some had hoped for. Their mutual commitment to justice is reflected by the fact that once these principles become embodied in institutions there are no legitimate means that permit anyone to depart from the terms of their agreement. As a result, the parties have to take seriously the moral and legal obligations and potential social sanctions they will incur as a result of their agreement, for there is no going back to the initial situation. So if they do not sincerely believe that they can accept the requirements of a conception of justice and voluntarily conform their actions and life plans accordingly, then these are strong reasons to avoid choosing those principles. It would not be rational for the parties to take risks, falsely assuming that if they end up badly, they can violate at will the terms of agreement or later regain their initial situation and renegotiate terms of cooperation (see Freeman, 1990; Freeman, 2007b, 180-182). Rawls gives special poignancy to this mutual commitment of the parties by making it a condition that the parties cannot choose and agree to principles in bad faith; they have to be able, not simply to live with and grudgingly accept, but instead to willingly endorse the principles of justice as members of society. Essential to Rawls's argument for stability is the assumption of everyone's willing compliance with requirements of justice. This is a feature of a well-ordered society. The parties are assumed to have a sense of justice; indeed the development and exercise of it is one of their fundamental interests. Hence they must choose principles that that they can not only accept and live with, but which are responsive to their sense of justice and they can unreservedly endorse. Given these conditions on choice, the parties cannot take risks with principles they know they will have difficulty complying with voluntarily. They would be making an agreement in bad faith, and this is ruled out by the conditions of the original position. Rawls contends that these "strains of commitment" created by the parties' agreement strongly favor the principles of justice over the principles of utility and other teleological (and most consequentialist) views. For everyone's freedom, basic rights and liberties, and basic needs are met by the principles of justice because of their egalitarian nature. Given the lack of these guarantees under the principle of utility, it is much more difficult for those who end up worse off in a utilitarian society to willingly accept their situation and commit themselves to the utility principle. It is a rare person indeed who can freely and without resentment sacrifice their life prospects so that those who are better off can have even greater comforts, privileges, and powers. This is too much to demand of our capacities for human benevolence. It requires a kind of commitment that people cannot make in good faith, for who could willingly support laws that are so detrimental to oneself and the people one cares about most that they must sacrifice their fundamental interests for the sake of those more advantaged? Besides, why should we encourage such subservient dispositions and the accompanying lack of self-respect? The principles of justice, by contrast, conform better with everyone's interests, their desire for self-respect and their natural moral capacities to reciprocally recognize and respect others' legitimate interests while freely promoting their own good. The strains of commitment incurred by agreement in the original position provide strong reasons for the parties to choose the principles of justice and reject the risks involved in choosing the principles of average or aggregate utility. ### 6.4 Stability, Publicity, and Self-Respect Rawls's strains-of-commitment argument explicitly relies upon a rarely noted feature of his argument: as mentioned earlier, there are in effect two social contracts. First, hypothetical agents situated equally in the original position unanimously agree to principles of justice. This agreement has attracted the most attention from Rawls's critics. But the parties' hypothetical agreement in the original position is patterned on the general acceptability of a conception of justice by free and equal persons in a well-ordered society. Rawls says, "The reason for invoking the concept of a contract in the original position lies in its correspondence with the features of a well-ordered society [which] require...that everyone accepts, and knows that the others accept, the same principles of justice" (CP 250). In order for the hypothetical parties in the original position to agree on principles of justice, there must be a high likelihood that real persons, given human nature and general facts about social and economic cooperation, can also agree and act on the same principles, and that a society structured by these principles is feasible and can endure. This is the stability requirement referred to earlier. One conception of justice is relatively more stable than another the more willing people are to observe its requirements under conditions of a well-ordered society. Assuming that each conception of justice has a corresponding society that is as well-ordered as can be according to its terms, the stability question raised in Theory is: Which conception of justice is more likely to engage the moral sensibilities and sense of justice of free and equal persons as well as affirm their good? This requires an inquiry into moral psychology and the human good, which takes up most of Part III of A Theory of Justice. Rawls makes two arguments in Theory from the original position that invoke the stability requirement, the arguments (1) from publicity and (2) from self-respect (see TJ, SS29) (1) The argument from publicity: Rawls contends that utilitarianism, perfectionism, and other "teleological" conceptions are unlikely to be freely acceptable to many citizens when made fully public under the conditions of a well-ordered society. Recall the publicity condition discussed earlier: A feature of a well-ordered society is that its regulative principles of justice are publicly known and appealed to as a basis for deciding laws and justifying basic institutions. Because all reasonable members of society accept the public conception of justice, there is no need for the illusions and delusions of ideology for a society to function properly and citizens to accept its laws and institutions willingly. In this sense a well-ordered society lacks false consciousness about the bases of social and political relations. (PL 68-69n.) A conception of justice that satisfies the publicity condition but that cannot maintain the stability of a well-ordered society is to be rejected by the parties in the original position. Rawls contends that under the publicity condition justice as fairness generally engages citizens' sense of justice and remains more stable than utilitarianism (TJ 177f./154f.rev.) For public knowledge that reasons of maximum average (or aggregate) utility determine the distribution of benefits and burdens would lead those worse-off to object to and resent their situation, and reject the principle of utility as the basic principle governing social institutions. After all, the well-being and interests of the least advantaged, perhaps even their basic liberties, are being sacrificed for the greater happiness of those who are already more fortunate and have a greater share of primary social goods. It is too much to expect of human nature that people should freely acquiesce in and embrace such publicly known terms of cooperation. By contrast, the principles of justice are designed to advance reciprocally everyone's position; those who are better off do not achieve their gains at the expense of the less advantaged. "Since everyone's good is affirmed, all acquire inclinations to uphold the scheme" (TJ, 177/155). It is a feature of our moral psychology, Rawls contends, that we normally come to form attachments to people and institutions that are concerned with our good; moreover we tend to resent those persons and institutions that take unfair advantage of us and act contrary to our good. Rawls argues at length in chapter 8 of Theory, SSSS70-75, that justice as fairness accords better than alternative principles with the reciprocity principles of moral psychology that are characteristic of human beings' moral development. In Political Liberalism, Rawls expands the publicity condition to include three levels: First, the principles of justice governing a well-ordered society are publicly known and appealed to in political debate and deliberation; second, so too are the general beliefs in light of which society's conception of justice is generally accepted--including beliefs about human nature and the way political and social institutions generally work--and citizens generally agree on these beliefs that support society's onception of justice. Finally the full justification of the public conception of justice is also publicly known (or at least publicly available to any who are interested) and is reflected in society's system of law, judicial decisions and other political institutions, as well as its system of education. (2) The argument from the social bases of self-respect: The publicity condition is also crucial to Rawls's fourth argument for the principles of justice, from the social bases of self-respect (TJ, 178-82/155-59 rev.). These principles, when publicly known, give greater support to citizens' sense of self-respect than do utilitarian and perfectionist principles. Rawls says self-respect is "perhaps the most important primary good," (TJ, 440/386 rev.) since few things seem worth doing if a person has little sense of their own worth or no confidence in their abilities to execute a worthwhile life plan or fulfill the duties and expectations in their role as citizens. The parties in the original position will then aim to choose principles that best secure their sense of self-respect. Now being regarded by others as a free and independent person of equal status with others is crucial to the self-respect of persons who regard themselves as free and equal members of a democratic society. Justice as fairness, by affording and protecting the priority of equal basic liberties and fair equal opportunities for all, secures the status of each as free and equal citizens. For example, because of equal political liberties, there are no "passive citizens" who must depend on others to politically protect their rights and interests; and with fair equal opportunities no one has grounds to experience the resentment that inevitably arises in societies where social positions are effectively closed off to those less advantaged or less powerful. Moreover, the second principle secures adequate social powers and economic resources for all so that they find the effective exercise of their equal basic liberties to be worthwhile. The second principle has the effect of making citizens socially and economically independent, so that no one need be subservient to the will of another. Citizens then can regard and respect one another as equals, and not as masters or subordinates. ("Non-domination," an idea central to contemporary Republicanism, is then essential to citizens' sense of self-respect in Rawls's sense. See Pettit 1997.) Equal basic liberties, fair equal opportunities, and political and economic independence are primary among the social bases of self-respect in a democratic society. The parties in the original position should then choose the principles of justice over utilitarianism and other teleological views both to secure their sense of self-respect, and to procure the same for others, thereby guaranteeing greater overall stability. In connection with Rawls's argument for the greater stability of principles of justice on grounds of their publicity and the bases of self-respect, Rawls provides a Kantian interpretation of difference principle. He says: "[T]he difference principle interprets the distinction between treating men as means only and treating them as ends in themselves. To regard persons as ends in themselves in the basic design of society is to agree to forgo those gains that do not contribute to everyone's expectations. By contrast to regard persons as means is to be prepared to impose on those already less favored still lower prospects of life for the sake of the higher expectations of others" (TJ 157 rev.). Rawls says the principle of utility does just this; it treats the less fortunate as means since it requires them to accept even lower life prospects for the sake of others who are more fortunate and already better off. This exhibits a lack of respect for the less advantaged and in turn has the effect of undermining their sense of self respect. (TJ 158 rev.) The difference principle, by contrast, does not treat people as means or undermine their sense of self respect, and this adds to the reasons the parties have for choosing the principles of justice instead of the principle of utility. Rawls substantially relies on the publicity condition to argue against utilitarianism and perfectionism. He says publicity "arises naturally from a contractarian standpoint" (TJ, 133/115 rev.). In Theory he puts great weight on publicity ultimately because he thinks that giving people knowledge of the moral bases of coercive laws and the principles governing society is a condition of fully acknowledging and respecting them as free and responsible rational moral agents. With publicity of principles of justice, people have knowledge of the real reasons for their social and political relations and the formative influences of the basic structure on their characters, plans and prospects. In a well-ordered society with a public conception of justice, there is no need for an "esoteric morality" that must be confined "to an enlightened few" (as Sidgwick says of utilitarianism, Sidgwick 1907 [1981], 490). Moreover, public principles of justice can serve agents in their practical reasoning and provide democratic citizens a common basis for political argument and justification. These considerations underlie Rawls's later contention that having knowledge of the principles that determine the bases of social relations is a precondition of individuals' freedom.(CP 325f.) Rawls means in part that publicity of society's fundamental principles is a condition of citizens' exercise of the powers and abilities that enable them to take full responsibility for their lives. Full publicity is then a condition of the political and (in TJ) moral autonomy of persons, which are significant values according to justice as fairness. (TJ SS78, PL 68, CP 325-26) Utilitarians often regard Rawls's emphasis on the publicity of the fundamental principles underlying social cooperation as unwarranted. They contend that publicity of laws is of course important for them to be effective, but there's no practical need for the publicity of the fundamental principles (such as the principles of efficiency and utility) that govern political decisions, the economy, and society, much less so for the publicity of the full justification of these principles. Most people are not interested and have little understanding of the complex often technical details that must go into deciding laws and social policies. Moreover, as Sidgwick claimed, utilitarianism functions better as an "esoteric morality" that is not generally incorporated into the public justification of laws and institutions. Others claim that Rawls's arguments from publicity are exaggerated. If people were properly educated to believe that promoting greater overall happiness or welfare is the ultimate requirement of justice and more generally of morality, then just as they have for centuries constrained their conduct and their self-interests and accepted political constraints on their own liberties for the sake of their religious beliefs, so too could they be educated to accept the promotion of social utility and the general welfare as the fundamental bases for social and political cooperation. ## Supplementary Documents on Other Topics Additional topics concerining the original position are discussed in the following supplementary documents: * The Argument for the Difference Principle. Explains the Difference Principle and the least advantaged class. Comparison of the difference principle with mixed conceptions, including restricted utility. Arguments from reciprocity, stability and self-respect, and the strains of commitment. Rawls's reasons why the difference principle supports property owning democracy rather than welfare-state capitalism. * The Four Stage Sequence. How principles chosen in OP (first stage) apply to choice of political constitution (second-stage), democratic legislation (third stage), and application of laws to particular circumstances (fourth stage). * Ideal Theory, Strict Compliance and the Well-Ordered Society. Why strict compliance is said to be necessary to justification of universal principles. Sen's, Mills's, and others' criticisms of ideal theory. Rawls's contention that ideal theory is necessary to determine injustice in non-ideal conditions. Role of non-ideal theory. * A Liberal Feminist Critique of the Original Position and Justice within the Family. Criticism of "heads of families" assumption in OP and Rawls's response to criticisms that principles do not secure equal justice for women and children. Rawls's discussion of justice within the family. * The Original Position and the Law of Peoples. Rawls's extension of OP to decisions on the Law of Peoples governing relations among liberal and decent societies. Human rights, the duty to assist burdened peoples, oulaw societies, and Rawls's rejection of a global principle of distributive justice. * Constructivism, Objectivity, Autonomy, and the Original Position. Kantian Interpretation of the OP and Constructivism. OP as a procedure of construction and objective point of view. Response to Humean argument that social agreements cannot justify. Role of OP in reflective equilibrium. * Is the Original Position Necessary or Relevant? Reply to claims that OP is superfluous or irrelevant. Why Rawls thinks rational acceptance of principles in OP and congruence of Right and Good is essential to justice.
intensional-trans-verbs	## 1. Some groups of ITV's and their behavior Search verbs and desire verbs manifest all three of the behaviors listed in the prologue as "marks" or effects of intensionality. Thus, Lois Lane may be seeking Superman. But it does not seem to follow that she is seeking Clark, even though Superman is Clark, and so we have an example of the first kind of anomaly mentioned in the prologue: substitution of one name with another for the same person leads to a change in truth-value for the embedding sentence (here "Lois is seeking Superman"). Similarly, a thirsty person who believes that water quenches thirst and that H2O is a kind of rat poison may want some water but not some H2O. [According to some, this alleged failure of substitution to preserve truth-value is an illusion; but for reasons of space I do not pursue this theory -- its locus classicus is (Salmon 1986).] Second, both search verbs and desire verbs create specific-unspecific ambiguities in their containing VPs when the syntactic object of the verb consists in a determiner followed by a nominal (this ambiguity is also known as the relational/notional ambiguity, following Quine 1956, where it was first studied, at least in the modern period). For example, 'Oedipus is seeking a member of his family' could be true because Oedipus is seeking specifically Jocasta, who is a member of his family, though he doesn't realize it. On such an occasion, what is true can be more carefully stated as 'there is a member of his family such that Oedipus is seeking that person'. The alternative, unspecific or notional reading, is forced by adding 'but no particular one': 'Oedipus is seeking a member of his family, but no particular one'. Here Oedipus is implied just to have a general intention to find some member or other of his family. Contrast the extensional 'embrace': Oedipus cannot embrace a member of his family, but no particular one. Third, it is obvious that it is possible both to want, and to search for, that which does not exist, for instance, a fountain of eternal youth. But it isn't possible to, say, stumble across such a thing, unless it exists. Depiction verbs, such as 'draw', 'sculpt', and 'imagine', resist substitution in their syntactic objects, at least if the clausal 'imagine' does: if imagining that Superman rescues you is not the same thing as imagining that Clark rescues you, it is hard to see why imagining Superman would be the same thing as imagining Clark. A specific/unspecific ambiguity is also possible, as is attested by the wall label for Guercino's The Aldrovandi Dog (ca. 1625) in the Norton Simon Museum, which states 'this must be the portrait of a specific dog', thereby implicating an alternative, that 'Guercino drew a dog' could be taken to mean that he drew a dog, but no particular dog -- he just made one up. And we can clearly draw or imagine that which does not exist (as opposed to, say, photographing it). Braque's Little Harbor in Normandy (1909) is an example, according to the curators of the Art Institute of Chicago: 'it appears this work was painted from imagination, since the landscape depicted cannot be identified.' However, whether or not a notional reading of a depiction VP is possible depends on which quantificational determiner occurs in the noun phrase complement. If we say 'Guercino drew every dog', 'Guercino drew most dogs', or 'Guercino drew the dog' (non-anaphoric 'the dog'), we seem to advert to some antecedent domain with respect to which 'drew every/most/the dog(s)' are to be evaluated. So specific readings are required.[1] This resistance to unspecific construal is robust across languages and is typical of those quantificational determiners which do not occur naturally in existential contexts such as 'there is': contrast 'there is a dog in the garden' with 'there is every dog in the garden', 'there are most dogs in the garden', or 'there is the dog in the garden'. An account of what is wrong with 'there is every dog in the garden' (see Keenan 2003) might well contain the materials for explaining the lack of unspecific readings of depiction VPs with determiners like 'every', 'most' and 'the'; see further (Forbes 2006:142-150). It should be emphasized that depiction verbs are special in this respect, for there is no problem getting unspecific readings with 'every', 'most' and 'the' using desire verbs or search verbs. Guercino might be looking for every dog on Aldrovandi's estate, though there are no particular dogs he is looking for; the reader might be driving around an unfamiliar airport rental car lot, looking for the exit, and in this case there is no exit such that it is being sought. ('look' isn't really a transitive verb, but when the following preposition is 'for', a search activity is denoted, so it is customary to count the likes of 'looking for' as a "transitive verb" in contexts where intensionality is under discussion.) Mixed behavior is also manifested by evaluative verbs, for example 'respect', 'admire', 'disdain', 'worship', including emotion verbs such as 'lust (after)' and 'fear'. Lex Luthor might fear Superman, but not Clark, and Lois might disdain Clark, but not Superman. However, unspecific readings of VPs with quantified complements are harder to hear, at least when the quantifier is existential. 'Lois admires an extraterrestrial' can be heard in two ways: there is the 'admires a particular extraterrestrial' reading, and there is a generic reading, which means that among the kinds of thing she admires are extraterrestrials in general. Generic readings of evaluative VPs attribute dispositions, and are not the same as unspecific or notional readings (see Cohen 1999, 2008, and the entry on generics). There does not seem to be a sensible non-generic construal of 'Lois admires an extraterrestrial, but no particular one'. The verb 'need' is an interesting case. A sports team might need a better coach, though no specific better coach, and might need a better coach even if there are none to be had. So two out of three marks of intensionality are present. However, 'need' contrasts with 'want' as regards substitution: our dehydrated subject who does not want H2O because he believes it to be a kind of rat poison, nevertheless needs H2O. It seems that co-denoting terms may be interchanged in the complement of 'need'. But merely accidentally co-extensive ones cannot be: Larson (2001, 232) gives the example of Max the theater impresario, who needs more singers but not more dancers, even though all who sing dance, and vice-versa. The property singer and the property dancer are different properties, so expressions for them cannot be exchanged in the complement of 'need'. Similar restricted substitutivity is observed with transaction verbs such as 'wager', 'owe', 'buy', 'sell', 'reserve', and perhaps the transaction resultative 'own'. One may reserve a table at a restaurant, though there need not be a specific table one has reserved in making the reservation, since the restaurant may be expecting a slow night. But these verbs do allow interchange of co-referential expressions (a purchase of water-rights is a purchase of H2O-rights) though not (Zimmerman 1993, 151) of accidentally co-extensive ones. For 'own' see (Zimmerman 2001, passim). Indeed, it is even arguable that some marks of intensionality are present with verbs that allow interchange of accidentally co-extensive expressions. A case in point is verbs of absence, such as 'omit' and 'lack'. If it so happens that all and only the physicists on the faculty are the faculty's Nobel prize winners, then a faculty committee that lacks a physicist lacks a Nobel prize winner. However, not too much weight can be put on this case, since it may be that at some level, 'lack' should be analyzed (possibly in a complicated way) in terms of not having, in which case it would not really be an intensional verb at all; though so far as the author knows, no convincing analysis of this type has been formulated. See (Zimmerman 2001, 516-20) for further discussion of relationships among the marks of intensionality. ## 2. How many mechanisms for how many marks? We have distinguished three "marks" or effects of intensionality: substitution-resistance, the availability of unspecific readings, and existence-neutrality. A natural question is whether one and the same semantic mechanism underlies all three effects, whether they are entirely independent, or whether two have a common source distinct from the third's. In the context of a discussion of propositional attitude verbs, that is, verbs which take clausal or clause-embedding noun-phrase complements rather than simple noun-phrase (NP) objects, one hypothesis which keeps explanatory apparatus to the minimum is that all three effects of intensionality arise from the possibility of the complement having narrow scope with respect to the attitude verb. Thus we might distinguish two readings of (1) Lex Luthor fears that Superman is nearby, namely (2a) Lex Luthor fears-true the proposition that Superman is nearby[2] and (2b) Superman is someone such that Lex Luthor fears(-true the proposition) that he is nearby. In (2a) we make the clause 'that Superman is nearby' the complement of 'proposition' to guarantee that 'Superman' is within the scope of 'fears' (the resulting NP 'the proposition...' is a "scope island"). And in (2b) we use a form of words that encourages an audience to process 'Superman' ahead of 'fears'. We can associate substitution-resistance with (2a), while allowing substitution in (2b). (2b) ascribes a complex property to Superman, and therefore also to Clark; namely, being an x such that Lex fears x to be nearby. (2a), on the other hand, puts Lex in the fearing-true attitude relation to a certain proposition, with essentially no implications about what other propositions he may fear-true. So, provided the proposition that Superman is nearby is distinct from the proposition that Clark Kent is nearby, substitution-failure in (1) construed as (2a) is explicable. As for taking these two propositions to be distinct, there are many serviceable accounts, mostly involving some variation of the original proposal in modern philosophical semantics, found in (Frege 1892, 1970). According to Frege, every meaningful expression or phrase has both a customary reference that it denotes, and a customary sense that it expresses. In the case of clauses, the customary reference would be a truth-value which is compositionally derived from the denotations of the words of the clause, and the customary sense would be a way of thinking of that truth-value, compositionally derived from the senses of the words of the clause. Therefore, provided that 'Superman' and 'Clark Kent' have different though coreferential senses (i.e., provided they express different ways of thinking of the same individual) we will get different propositions. (However, it's highly non-trivial to find an adequate account of the senses of names, given the critique of the most straightforward accounts in (Kripke 1972).) However, on the face of it, this only has (1), intended in the (2a) manner, expressing a different proposition from (2c) Lex Luthor fears that Clark is nearby. Since truth-value is on the level of reference, and the corresponding words of (2a) and (2c) all have the same referents, the resulting truth-values for (2a) and (2c) will be the same; but they are supposed to be different. So Frege makes the ingenious suggestion that it is an effect of embedding in intensional contexts (he only considered clausal verbs) that expressions in such contexts no longer denote their customary references, but rather their customary senses. Using 'iff' henceforth to abbreviate 'if and only if', then (1) intended as (2a) is true iff the reference of 'Lex' stands in the fearing-true relation to the switched reference of 'Superman is nearby', namely, its customary sense. Now we have our explanation of why merely interchanging customarily co-referential expressions in (1) can produce falsehood from a truth: the substitution does not preserve reference, since the names are now denoting their customary senses. However, if (1) is intended as (2b), there will be no truth-value switch, since there is no name-reference switch: in (2b) 'Superman' is not in the scope of 'fears', therefore it denotes its customary reference, and interchange with any other expression denoting that same referent must of necessity be truth-value preserving. (Readers in search of more detailed discussion of Frege's notion of sense might consult the section on Frege's philosophy of language in the entry on Frege. See also the entry on propositional attitude reports, and for other uses of Frege-style "switcher semantics", Gluer and Pagin 2012.) A view like this has the capacity to explain the other intensionality effects. The specific-unspecific ambiguity in 'Lex fears that an extraterrestrial is nearby' is explained in terms of scope ambiguity, the notional or unspecific reading corresponding to (3a) Lex Luthor fears-true the proposition that an extraterrestrial is nearby and the relational or specific reading to (3b) An extraterrestrial is such that Lex Luthor fears that it is nearby. Existence-neutrality is also explained, since the proposition that an extraterrestrial is nearby is available for its truth to be feared, believed, doubted or denied, whether or not there are extraterrestrials. There are other accounts of substitution failure, but details are incidental at this point. For there are real issues about (A) whether a single mechanism could be responsible for all three effects, and (B) whether an account of any effect in terms of a scope mechanism is workable for transitive, as opposed to clausal, verbs. (A) The behaviors cited in the previous section suggest that substitution-resistance and the availability of an unspecific reading have different explanations. For we saw that the verb 'need' contrasts with the verb 'want' as regards substitution-resistance, but is similar as regards the availability of unspecific readings of embedding VPs. So it seems that there is a mechanism that blocks substitution, perhaps the Fregean reference-switch one, perhaps something else more compatible with what Davidson calls "semantic innocence" (Davidson 1969, 172--a semantically innocent account of substitution-failure is one that does not alter the semantics of the substitution-resisting expressions themselves for the special case in which they occur in intensional contexts). And this mechanism cannot occur with 'need', but can with 'want' ('can' rather than 'does' because it is optional; this is to allow for "transparent" or substitution-permitting readings of the likes of 'Lex fears Superman' analogous to (3b)). On the other hand, whatever accounts for notional readings is evidently available to both verbs, and therefore it is not the same mechanism as underlies the substitution-resistance of 'wants'. However, this reasoning is not conclusive, since the substitution-resistance mechanism may be present with 'needs' (and transaction verbs) but somehow rendered ineffective (see Parsons 1997, 370). One would need to hear how the ineffectiveness comes about. Evaluative verbs present the converse challenge: substitution-resistance but apparently no unspecific readings of embedding VPs, certainly not existential ones. It is less clear how a defender of a 'single explanation' theory would handle this, at least if the single explanation is a scope mechanism, since it appears from the other cases that occurrence within the scope of the intensional verb immediately produces an unspecific reading. Suspension of existential commitment may group with availability of unspecific readings for explanatory purposes. There appear to be no cases of intensional transitives which allow notional readings of embedded VPs, but where those VPs have the same existential consequences as ones which differ just by the substitution of an extensional for the intensional verb. (B) The scope account is the only real contender for a single explanation of the intensionality effects. But there is a major question about whether it can be transferred at all from clausal to transitive verbs. For the intensionality effects would all be associated with narrow-scope occurrences of noun phrases (NPs), and with a transitive verb such syntactic configuration is problematic when the NP is quantified. This is because, in standard first-order syntax, a quantifier must have a sentence within its scope (an open sentence with a free variable the NP binds, if redundant quantification is ruled out in the syntax). We can provide this for relational or wide-scope readings, for example (4) An extraterrestrial is such that Lois is looking for it in which 'Lois is looking for it' is the scope of 'an extraterrestrial'. But if 'an extraterrestrial' is supposed to be within the scope of 'looking for' there is no clause to be its scope; it has to be an argument of the relation, which is not allowed in first-order language. As Kaplan says, 'without an inner sentential context...distinctions of scope disappear' (Kaplan 1986, 266). (Though readers who have taught symbolic logic will be very familiar with the student who, having symbolized 'Jack hit Bill' as 'Hjb', then offers something like 'Hj([?]x)' as the symbolization of 'Jack hit someone.') The description of the problem suggests two forms of solution. One is to preserve first-order syntax by uncovering hidden material to be the scope of a quantified NP even when the latter is within the scope of the intensional verb. The other is to drop first-order syntax in favor of a formalism which permits the meanings of quantified NPs to be arguments of intensional relations such as fearing and seeking. We consider these options in turn in the following two sections. ## 3. Propositionalism The idea of uncovering hidden material to provide NPs in notional readings of intensional VPs with sentential scope was prominently endorsed in (Quine 1956), where the proposal is to paraphrase search verbs with 'endeavor to find'. So for (5a) we would have (5b): (5a) Lois is looking for an extraterrestrial (5b) Lois is endeavoring to find an extraterrestrial Partee (1974, 97) objects that this cannot be the whole story, since search verbs are not all synonyms ('groping for' doesn't mean exactly the same as 'rummaging about for'), but den Dikken, Larson, & Ludlow (1996) and Parsons (1997, 381) suggest that the search verb itself be used in place of 'endeavor'. So we get (6a) Lois is looking to find an extraterrestrial or in somewhat non-Quinean lingo, (6b) Lois is looking in order to make true the proposition that an extraterrestrial is such that she herself finds it.[3] Here 'an extraterrestrial' is within the scope of 'looking' but has the open sentence 'she herself finds it' as its own scope. It may or may not be meaning-preserving to replace (5a)'s prepositional phrase with (6a)'s purpose clause, but even if it is meaning-preserving, that is insufficient to show that (6a) or (6b) articulates the semantics of (5a); it may merely be a synonym. However, with 'need' and desire verbs, evidence for the presence of a hidden clause is strong. For example, in (7) Physics needs some new computers soon it makes little sense to construe 'soon' as modifying 'needs'; it seems rather to modify a hidden 'get' or 'have', as is explicit in 'Physics needs to get some new computers soon', i.e., 'Physics needs it to be the case that, for some new computers, it gets them soon'. (For 'have' versus 'get', see (Harley 2004).) Secondly, there is the phenomenon of propositional anaphor (den Dikken, Larson, & Ludlow 2018, 52-3), illustrated in (8) Physics needs some new computers, but its budget won't allow it. What is not allowed is the truth of the proposition that Physics gets some new computers. Third, attachment ambiguities suggest there is more than one verb present for modifiers to attach to (Dikken, Larson, & Ludlow 1996, 332): (9) Physics will need some new computers next year could mean that a need for new computers will arise in the department next year, but could also mean that next year is when Physics should get new computers, if its need (which may arise later this year) is to be met. Finally, ellipsis generates similar ambiguities: (10) Physics will need some new computers before Chemistry could mean that the need will arise in Physics before it does in Chemistry, but could also mean that Physics will need to get some new computers before Chemistry gets any. However, the strength of the case for a hidden 'get' with 'need' or 'want' contrasts with the case for propositionalism about search verbs. As observed by Partee (1974, 99), for the latter there are no attachment ambiguities like those in (9). For example, (11) Physics will shop for some new computers next year can only mean that the shopping will occur next year. There is no second reading, corresponding to the other reading of (9), in which 'next year' attaches to a hidden 'find/buy'. The phenomena in (8) and (10) also lack parallels with search verbs; for example, 'Physics will shop for some new computers before Chemistry' lacks a reading that has Physics shopping with the following goal: to find/buy new computers before Chemistry finds/buys any. And although the propositionalist might offer something like (12) Physics will seek more office space by noon as an analog of (7), it is not easy to decide if (12) genuinely has a reading of the 'seek to find more office space by noon' sort, or whether the hint of such a reading is just an echo of (7). Other groups of intensional transitives, such as depiction verbs and evaluative verbs, raise the problem that there is no evident propositional paraphrase in the first place. For psychological depiction verbs such as 'fantasize' and 'imagine', Parsons (1997, 376) proposes what he calls "Hamlet ellipsis": for 'Mary imagined a unicorn' we would have the clausal 'Mary imagined a unicorn to be'. Larson (2001, 233) suggests that the complement is a "small" or "verbless" clause, and for 'Max visualizes a unicorn' proposes 'Max visualizes a unicorn in front of him'. This is too specific, for we can understand 'Max visualizes a unicorn' without knowing whether he visualizes it in front of him, above him or below him, but even if we change the paraphrase to 'Max visualizes a unicorn spatially related to him', this proposal, as well as Parsons', have problems with negation: 'Mary didn't imagine a unicorn' is not synonymous with either 'Mary didn't imagine a unicorn to be' or with 'Mary didn't imagine a unicorn spatially related to her', since the first of these allows for her to imagine a unicorn but not imagine it to be, the second, for her to imagine a unicorn but not as spatially related to her. There may be philosophical arguments that exclude these options,[4] but the very fact that a philosophical argument is needed makes the proposals unsatisfactory as semantics. Clausal paraphrases for verbs like 'fear' are even less likely, since the extra material in the paraphrase can be read as the focus of the fear, making the paraphrase insufficient. For example, fearing x is not the same as fearing encountering x, since it may be the encounter that is feared, say if x is an unfearsome individual with a dangerous communicable disease. In the same vein, fearing x is not the same as fearing that x will hurt you; for instance, you may fear that your accident-prone dentist will hurt you, without fearing the dentist. We conclude that if any single approach to intensional transitives is to cover all the ground, it will have to be non-propositionalist. But it is also possible, perhaps likely, that intensional transitives are not a unitary class, and that propositionalism is correct for some of these verbs but not for others (see further Schwarz 2006, Montague 2007). ## 4. Montague's semantics The main non-propositionalist approaches to ITVs begin from the work of Richard Montague, especially his paper "The Proper Treatment of Quantification in Ordinary English" (Montague 1973), usually referred to as PTQ in the literature (Montague's condition (9) (1974, 264) defines 'seek' as 'try to find', but this is optional). Montague developed a systematic semantics of natural language based on higher-order intensional type-theory. We explain this term right-to-left. Type theory embodies a specific model of semantic compositionality in terms of functional application. According to this model, if two expressions x and y can concatenate into a meaningful expression xy, then (i), the meaning of one of these expressions is taken to be a function, (ii) the meaning of the other is taken to be an item of the kind that the function in question is defined for, and (iii) the meaning of xy is the output of the function when applied to the input. The type-theoretic representation of this meaning is written x(y) or y(x), depending on which expression is taken to be the function and which the input or argument. A functional application such as x(y) is said to be well-typed iff the input that y denotes is the type of input for which the function x is defined. For example, in the simple theory of types, a common noun such as 'sweater' is assigned a meaning of the following type: a function from individuals to truth-values (a function of type ib, for short; b for 'boolean'). For 'sweater' the function in question is the one which maps all sweaters to the truth-value TRUE, and all other individuals to the truth-value FALSE. On the other hand, an ("intersective") adjective such as 'woollen' would be assigned a meaning of the following type: a function from (functions from individuals to truth-values) to (functions from individuals to truth-values), or a function of type (ib)(ib) for short. Thus the meaning of 'woollen' can take the meaning of 'sweater' (an ib) as input and produce the meaning of 'woollen sweater' (another ib) as output; this is why the meaning of 'woollen' has the type (ib)(ib). woollen(sweater) is the specific function of type ib that maps sweaters made of wool to TRUE, and all other individuals to FALSE. In this framework, a quantified NP such as 'every sweater' has a meaning which can take the meaning of an intransitive verb (e.g., 'unravelled'), or more generally, a Verb Phrase (VP), as input and produce the meaning (truth-value) of a sentence (e.g. 'every sweater unravelled') as output. Intransitive verbs and VPs are like common nouns in being of type ib. For example, the VP quickly(unravelled) is of type ib, mapping all and only individuals that unravelled quickly to TRUE. So a quantified NP is a function from inputs of type ib to outputs of type b, and is thus of type (ib)b. 'Every sweater unravelled quickly' would be represented as (every(sweater))(quickly(unravelled)), and would denote the truth-value that is the result of applying a meaning of type (ib)b, that of every(sweater), to a meaning of type ib, that of quickly(unravelled) (the adverb quickly itself is of type (ib)(ib), like the adjective woollen). Rules specific to every guarantee that every(sweater) maps quickly(unravelled) to TRUE iff quickly(unravelled) maps to TRUE everything that sweater maps to TRUE. So far, the apparatus is extensional, which, besides providing only two possible sentence-meanings, TRUE and FALSE, imposes severe limitations on the range of concepts we can express. Suppose that the Scottish clothing company Pringle has a monopoly on the manufacture of woollen sweaters, and makes sweaters of no other material. Then a garment is a woollen sweater iff it is a Pringle sweater, meaning that woollen(sweater) and pringle(sweater) are the same function of type ib, and these two terms for that function are everywhere interchangeable in the type-theoretic language. Then modal operators such as 'it is contingent that' cannot be in the language, since interchanging woollen(sweater) and pringle(sweater) within their scope should sometimes lead to change of truth-value, but cannot if the two expressions receive the same meaning in the semantics. For example, 'it is contingent that every Pringle sweater is woollen' is true, but 'it is contingent that every woollen sweater is woollen' is false. Therefore the concept of contingency has no adequate representation in the type-theoretic (boldface) language. Shifting to intensional type theory deals with this difficulty. The intension of any expression X is a function from possible worlds to an extension of the type which that expression has in the extensional theory just sketched, if it has such an extension, otherwise to something appropriate for intensional vocabulary such as 'it is contingent that'. An intension which is a function from possible worlds to items of type *t* is said to be of type st. sweater*, for instance, will have as its intension a function from possible worlds to functions of type ib, providing for each possible world a function that specifies the individuals which are sweaters at that world; so sweater's intension is of type s(ib). However, a modal sentential operator such as 'it is contingent that' will have as its intension a function that, for each possible world, produces the very same function, which takes as input functions of type sb* and produces truth-values as output. So the extension of 'it is contingent that' at each world is the same function, of type (sb)b. (The operator is said to be intensional because its extension at each world is a function taking intensions, such as functions of type sb, as input.) A function of type sb is sometimes called a possible-worlds proposition, since it traces the truth-value of a sentence across worlds. For example, with appropriate assignments to the constituents, (13) (every(woollen(sweater)))(woollen) should be true, that is, refer to TRUE, at every world.[5] So the intension of (13) is the function *f* of type sb such that for every world w, *f(w) = TRUE. This is a constant* intension. On the other hand, (14) (every(pringle(sweater)))(woollen) is true at some worlds but false at others, those where Pringle makes non-woollen sweaters; so its intension is non-constant. We define the intension of contingent to be the function which, for each world w as input, produces as output the function *c* of type (sb)b such that for any function *p* of type sb, *c(p) is true at w iff there are worlds u* and v such that *p(u) = TRUE and p(v) = FALSE (this is the meaning of 'contingent that' in the sense 'contingent whether'). So the intension of 'contingent' is also constant, since the same function c* is the output at every world. Since contingent expects an input of type sb, we cannot write (15) contingent((every(woollen(sweater)))(woollen)) since in evaluating this formula at a world w we would find ourselves trying to apply the reference of contingent at w, namely, the function *c* just defined, to the reference of (every(woollen(sweater)))(woollen) at w, namely, the truth-value TRUE. But *c* requires an input of type sb, not b. So we introduce a new operator, written ^, such that if X is an expression and *t* is the type of X's reference at each w, then at each w, the reference of ^X is of type st. ^X may be read as 'the intension of X', since the rule for ^ is that at each world, ^X refers to that function which for each world w, outputs the reference of X at w. If we now evaluate (16) contingent^((every(woollen(sweater)))(woollen)) at a world w, the result will be FALSE. This is because the function p** of type sb that is the reference of ^((every(woollen(sweater)))(woollen)) at every world, maps every world to TRUE. So there is no u such that *p(u) = FALSE. But there is such a u* for ^((every(pringle(sweater)))(woollen)), and so (17) contingent^((every(pringle(sweater)))(woollen)) is TRUE at w. Note that choice of w doesn't matter, since the intension of contingent produces the same function *c* at every world, and the reference of, e.g., ^((every(pringle(sweater)))(woollen)), is the same function of type sb at every world. Finally, our type-theory, intensional or extensional, is higher-order, because the semantics makes available higher-order domains of quantification and reference. sweater refers to a property of individuals, a first-order property. (every(sweater)) refers to a property of properties of individuals, a second-order property. For just as sweater(my favorite garment) attributes the property of being a sweater to a certain individual, so we can think of (every(sweater))(woollen) as attributing a property to the property of being woollen. Which property is attributed to being woollen? The rules governing every ensure that (every(F)) is truly predicated of G iff G is a property of every F. In that case, G has the property of being a property of every F. So (every(F)) stands for the property of being a property of every F. Treating quantified NPs as terms for properties of properties means they can occur as arguments to any expression defined for properties of properties. We can even rescue the uncomprehending student's attempt at 'Jack hit someone', for provided 'hit' is of the right type -- which is easily arranged -- we can have (hit(someone))(jack). Here hit accepts the property of being a property of at least one person, and produces the first-order property of hitting someone, which is then attributed to Jack. In extensional type-theory, hit has the type ((ib)b)(ib) if (hit(someone))(jack) is well-typed and jack is of type i.[6] The significance of this for the semantics of intensional transitives is that we now have a way of representing a reading of, say, (18) Jack wants a woollen sweater in which the quantified NP is within the semantic scope of the verb without having scope over a hidden subsentence with a free variable for the NP to bind: the quantified NP 'a woollen sweater' can just be the argument to the verb. To allow for the intensionality of the transitive verb, Montague adopts the rule that if x and y can concatenate into a meaningful expression xy, the reference of the functional expression is a function which operates on the intension of the argument expression. Suppressing some irrelevant detail, this means that if 'wants' syntactically combines with 'a woollen sweater' to produce the VP 'wants a woollen sweater', then in its semantics, want applies to the intension of (a(woollen (sweater))), resulting in the following semantics for (18): (19) want(^(a(woollen(sweater))))(jack).[7] In (19), a(woollen((sweater)) is within the scope of want. So if we take (19) to represent the notional reading of (18), the idea that notional readings are readings in which the quantified NP has narrow scope with respect to the intensional verb is sustained. ## 5. Revisions and refinements How does Montague's account of intensional transitives fare vis a vis the three marks of intensionality? The existence-neutrality of existential NPs is clearly supported by (19), for there is nothing to prevent the application at a world w of want to ^(a(woollen(sweater))) from producing a function mapping Jack to TRUE even if, at the same w, (woollen(sweater)) maps every individual to FALSE (woollen sweaters don't exist). Substitution-failure is supported for contingently coextensive expressions. For instance, (19) does not entail want(^(a(pringle(sweater))))(jack) even if there are worlds where all and only woollen sweaters are Pringle sweaters, so long as there are other worlds where this is not so. Let u be a world of the latter sort. Then a(pringle(sweater)) at u and a(woollen(sweater)) at u are different functions of type (ib)b, making ^(a(pringle(sweater))) and ^(a(woollen(sweater))) different at every world. Therefore want(^(a(pringle(sweater)))) may map Jack to FALSE at worlds where want(^(a(woollen(sweater)))) maps Jack to TRUE: since want is applied to different inputs here, the outputs may also be different. But this result depends on the fact that (pringle(sweater)) and (woollen(sweater)) are merely contingently coextensive. If 'water' and 'H2O' are necessarily co-extensive, then wanting a glass of water and wanting a glass of H2O will be indistinguishable in higher-order intensional type theory. This failure to make a distinction, however, traces to the intensionality of the semantics -- to its being (merely) a possible-worlds semantics -- not to its being higher-order or type-theoretic. So possible solutions include (i) augmenting higher-order intensional type theory with extra apparatus, or (ii) employing a different kind of higher-order type theory. In both cases the aim is to mark distinctions like that between wanting a glass of water and wanting a glass of H2O. A solution of the first kind, following (Carnap 1947), is proposed in (Lewis 1972:182-6); the idea is that the meaning of a complex expression is not its intension, but rather a tree that exhibits the expression's syntactic construction bottom-up, with each node in the tree decorated by an appropriate syntactic category label and semantic intension. But as Lewis says (p. 182), for non-compound lexical constituents, sameness of intension implies sameness of meaning. So although his approach will handle the water/H2O problem if we assume the term 'H2O' has structure that the term 'water' lacks, it will not without such an assumption. For the same reason, it cannot explain substitution-failure involving unstructured proper names, on the usual view (deriving from Kripke 1972) that identity of extension (at any world) for such names implies identity of intension. So we have no account of why admiring Cicero isn't the same thing as admiring Tully. A solution of the second kind, employing a different kind of higher-order type theory, is pursued in (Thomason 1980). In Thomason's "intentional" logic, propositions are taken as a primitive category, instead of being analyzed as intensions of type sb. In turns out that a somewhat familiar higher-order type theory can be built on this basis, in which, roughly, the type of propositions plays a role analogous to the type of truth-values in extensional type theory. A property such as orator, for example, is a function of type ip (as opposed to ib), where p is the type of propositions: given an individual as input, orator will produce the proposition that that individual is an orator as output. Proper names, however, are not translated as terms of type i, for then cicero and tully would present the same input to orator, resulting in the same proposition as output: orator(cicero) = orator(tully). So there would be no believing Cicero is an orator without believing Tully is an orator. Instead, Thomason assigns proper names the type (ip)p, functions from properties to propositions. And merely the fact that Cicero and Tully are the same individual does not require us to say that cicero and tully must produce the same propositional output given the same property input. Instead, we can have cicero(orator) and tully(orator) distinct (see Muskens 2005 for further development of Thomason's approach). Applying this to intensional transitives is just a matter of assigning appropriate types so that the translations of, say, 'Lucia seeks Cicero' and 'Lucia seeks Tully', are different propositions (potentially with different truth-values). We need to keep the verb as function, and we already have the types of cicero and tully set to (ip)p. The translations of 'seeks Cicero' and 'seeks Tully' should be functions capable of accepting inputs of type (ip)p, such as lucia, and producing propositions as output. seeks therefore accepts input of type (ip)p and produces output that accepts input of type (ip)p and produces output of type p. Thus seeks is of type ((ip)p)(((ip)p)p), and we get substitution-failure because seeks(cicero) and seeks(tully) can be different functions of type ((ip)p)p so long as cicero and tully are different functions of type (ip)p (as we already said they should be). seeks(cicero) can therefore map lucia to one proposition while seeks(tully) maps it (not 'her') to another; and these propositions can have different truth-values.[8] Finally, there is the question whether (19) shows that Montague's semantics supports notional readings. One problem is that Montague's semantics for extensional verbs such as 'get' is exactly the same as for intensional verbs, and it takes an extra stipulation, or meaning-postulate, for 'get', to guarantee that the extension of get(^(a(woollen(sweater)))) at w maps Jack to TRUE only if the extension of (woollen(sweater)) at w maps some individual to TRUE (you can want a golden fleece even if there aren't any, but you can't get one if there aren't any). So apparently (19)'s pattern embodies something in common to the notional meaning of 'want a woollen sweater' and the meaning of 'gets a woollen sweater', something which is neutral on the existence of woollen sweaters. This is unintuitive, but is perhaps not a serious problem, since it can be avoided by a different treatment of extensional transitives. A more pressing question is what justification we have for thinking that (19) captures the notional, 'no particular one', reading of (18).[9] On the face of it, (19) imputes to Jack the wanting attitude towards the property of being a property of a woollen sweater. This is the same attitude as Jack may stand in to a particular woollen sweater, say that one. But it is not at all clear that we have any grasp of what a single attitude with such diverse objects could be, and the difficulty seems to lie mainly with the proposed semantics for notional readings. What does it mean to have the attitude of desire towards the property of being a property of a woollen sweater? Two ways of dealing with this suggest themselves. First, we might supplement the formal semantics with an elucidation of what it is to stand in a common-or-garden attitude to a property of properties. Second, we might revise the analysis to eliminate this counterintuitive aspect of it, but without importing the propositionalist's hidden sentential contexts. Both (Moltmann 1997) and (Richard 2001) can be read as providing, within Montague's general approach, an account of what it is to stand in an attitude relation to a property of properties. Both accounts are modal, having to do with the nature of possible situations in which the attitude is in some sense "matched" by the situation: an attitude-state of need or expectation is matched if the need or expectation is met, an attitude-state of desire is matched if the desire is satisfied, an attitude-event of seeking is met if the search concludes successfully, and so on. According to Moltmann's account (1997, 22-3) one stands in the attitude relation of seeking to ^(a(woollen(sweater))) iff, in every minimal situation s in which that search concludes successfully, you find a woollen sweater in s. Richard (2001, p. 116) offers a more complex analysis that is designed to handle negative quantified NPs as well ('no woollen sweater', 'few woollen sweaters', etc.). On this account, a search p demands ^(a(P)) iff for every relevant success-story m = <w, s> for p, things in s with a property entailing P are in the extension of ^(a(P)) at w. Here s is the set of things that are found when the search concludes successfully in w. By contrast, (Zimmerman 1993) and (Forbes 2000, 2006) propose revisions in (19) itself and its ilk. Zimmerman (161-2) replaces the quantifier intension with a property intension, since he holds that (i) unspecific readings are restricted to "broadly" existential quantified NPs, and (ii) the property corresponding to the nominal in the existential NP (e.g., 'woollen sweater') can do duty for the NP itself. Of course, the proposed restriction of unspecific readings to existentials is controversial (cf. our earlier example, 'Guercino is looking for every dog on Aldrovandi's estate'). It may also be wondered whether there is any less of a need to explain what it is to stand in the seeking relation to a property of objects than to a property of properties (but for a response to this kind of objection, see Grzankowski 2018, 146-9). According to (Forbes 2000) the need for such an explanation already threatens the univocality of a verb such as 'look for' as it occurs in 'look for that woollen sweater' and 'look for some woollen sweater'. Observing that search verbs are action verbs, Forbes applies Davidson's event semantics to them (Davidson 1967). In this semantics, as developed in (Parsons 1990), search verbs become predicates of events, and in relational (specific) readings, the object searched for is said to be in a thematic relation to the event, one denoted by 'for'; thus 'some search e is for that woollen sweater'. But in unspecific readings, no thematic relation is invoked; rather, the quantified NP is used to characterize the search. So we would have 'some search e is characterized by ^(a(woollen(sweater)))', i.e., e is an a-woollen-sweater search (Forbes 2000, 174-6; 2006, 77-84). What it is for a search to be characterized by a quantifier, say ^(a(woollen(sweater))), is explained in terms of 'outcome postulates'. For the current example, as a first approximation, a search is characterized by ^(a(woollen(sweater))) iff any course of events in which that search culminates successfully includes an event of finding a woollen sweater whose agent is the agent of the search. Similar postulates can be given for, e.g., the meeting of expectations and the satisfaction of desires: a state of desire is characterized by ^(a(woollen(sweater))) iff any course of events in which that desire is satisfied includes an event of getting a woollen sweater whose recipient is the agent of the search (Forbes 2006, 94-129). There is therefore a range of different non-propositionalist approaches to intensional transitives. As we already remarked, one possibility is that propositionalism is correct for some verbs and non-propositionalism correct for others. However, there is also the option that non-propositionalism is correct for all. A non-propositionalist who makes this claim will have to explain the phenomena illustrated in (7)-(10), without introducing degrees of freedom that make it unintelligible that these phenomena do not arise for all intensional transitives. ## 6. Prior's Puzzle Intensional transitive verbs are also involved in another puzzle of substitution-resistance besides the one already discussed. In the literature on propositional attitude reports, it's the received view that the complement clauses in such reports refer to propositions. So, for example, in 'Holmes believes that Moriarty has returned', the clause 'that Moriarty has returned' is taken to stand for the proposition that Moriarty has returned. The whole ascription is then understood to have the form Rab, which in terms of the example means that Holmes (a) stands in the relation of belief (R) to the proposition that Moriarty has returned (b). However, as well as being denoted by that-clauses, it seems that propositions are also denoted by proposition descriptions, noun phrases that explicitly use 'the proposition', such as, in the previous sentence, 'the proposition that Moriarty has returned'. So, if the clause and the description co-denote, we have the following truth: (20) that Moriarty has returned is the proposition that Moriarty has returned. But then we should be able to substitute proposition-description for that-clause, which, indeed, works well enough for 'believes': from 'Holmes believes that Moriarty has returned' it does seem to follow that Holmes believes the proposition that Moriarty has returned, even though a side-effect of the substitution is to change the clausal 'believes' into its transitive form. However, 'believes' is rather special in this respect. Despite (20), examples (21a) and (21b) below appear to have very different meanings: (21a) Holmes {fears/suspects} that Moriarty has returned. (21b) Holmes {fears/suspects} the proposition that Moriarty has returned. (21a) may well be true, but it is unlikely that Holmes fears a proposition, or that some proposition is a thing of which he is suspicious. (This phenomenon seems first to have been noted in print by A. N. Prior (1963).) That the meaning of the minor premise does not survive substitution is the rule rather than the exception: we get a similar outcome with 'announce', 'anticipate', 'ask', 'boast', 'calculate', 'caution', 'complain', 'conclude', 'crow', 'decide', 'detect', 'discover', 'dream', 'estimate', 'forget', 'guess', 'hope', 'insinuate', 'insist', 'interrogate' (literary theory), 'judge', 'know', 'love', 'mention', 'notice', 'observe', 'prefer', 'pretend', 'question','realize', 'rejoice', 'require', 'see', 'suggest', 'surmise', 'suspect', 'trust', 'understand', 'vote', 'wish', and various cognates of these. In some cases, substitution fails because it is meaning-altering, in others because it is meaning-dissolving (a selection constraint is violated, or the purported transitive verb simply doesn't exist in the language). The verbs for which substitution is acceptable are thinner on the ground: inference verbs such as 'conclude', 'deduce', 'entail' and 'establish', along with a few others like 'accept','believe', 'doubt', 'state' and 'verify' (but for 'believe' see King 2002, 359-60; Forbes 2018, 118; and Nebel 2019, 97-9). The puzzle doesn't depend on the credentials of (20) as an identity sentence. Even if, for whatever reason, it isn't, it is still hard to see how truth-value can change going from (21a) to (21b), granted that the proposition that Moriarty has returned and that Moriarty has returned are co-denoting. There also doesn't seem to be a useful application of a traditional account of referential opacity, such as Frege's (discussed in connection with (2c) above). Substitution-failure is explained in Fregean terms by co-denoting expressions having different senses, but it's unclear that adding or deleting 'the proposition' is a significant enough difference to change sense. We also expect that examples of substitution-failure become unrealistic if the subject is attributed explicit belief in the identity premise. However, adding that Holmes is absolutely clear about the truth of (20) and has it at the forefront of his consciousness doesn't make it any more likely that (21b) is true, even though (21a) is. Consequently, there is some appeal to solutions of Prior's Puzzle that discern an equivocation in the inference, or propose that the crucial terms, the proposition that Moriarty has returned and that Moriarty has returned, aren't really coreferential as they occur in the inference. An approach of the first kind, in (King 2002), argues that the transitive and clausal forms of the intensional verb are polysemous, that is, weakly ambiguous (the two senses are related). This has been criticized on the basis of ellipsis examples, such as 'Bob didn't even mention the proposition that first-order logic is undecidable, let alone that it is provable' (Boer 2009, 552), and 'the Soviet authorities genuinely fear a religious revival, and that the contagion of religion will spread' (after Nebel 2019, 77). Since these examples don't strike one as amusingly incongruous, in the way that typical examples of zeugma do ('All over Ireland the farmers grew potatoes, barley, and bored'), they are in tension with a postulation of polysemy. Consequently, Boer (2009) and Nebel (2019) propose that the problem lies in an equivocation in the terms the proposition that Moriarty has returned and that Moriarty has returned. On both their accounts, it is the propositional description 'the proposition that...' which fails to denote what one might expect. On the other hand, as evidence in favor of polysemy, there is the fact that some cases of ellipsis seem in some way anomalous, such as 'John heard thunder and that a storm was rolling in'. Here the anomaly can be explained in terms of how different the two senses of 'heard' are. Event semantics (see section 5 above) provides an alternative which may avoid the need for any kind of equivocation. An initial proposal is made in (Pietroski 2000) for 'explain', which behaves like 'fear' and 'suspect', as in (22a) Martin explained that the nothing itself nothings. (22b) Martin explained the proposition that the nothing itself nothings. (22a) may well be true but (22b) is unlikely. The idea is then that the clausal and transitive forms of 'explain' take different thematic complements, content versus theme: with the clausal verb of (22a), a 'that'-clause provides a content, while with the transitive verb of (22b), a proposition description provides a theme, the thing that gets explained. Since, as already noted, a side-effect of the substitution is to change the verb from its clausal to its transitive form, a consequent side-effect is to change the role ascribed to the proposition from content to theme. This account is developed in (Pietroski 2005) and also, as an extension of the event semantics of (Forbes 2006), in (Forbes 2018). ## 7. The logic of intensional transitives There may be no such topic as the logic of propositional attitudes: it may be doubted whether 'Mary wants to meet a man who has read Proust and a man who has read Gide' logically entails 'Mary wants to meet a man who has read Proust'. Even if standing in the wanting-to-be-true attitude to I meet a man who has read Proust and a man who has read Gide somehow necessitates standing in the wanting-to-be-true attitude to I meet a man who has read Proust, the necessitation appears to be more psychological than logical. On the other hand, what we might call 'objectual attitudes', the non-propositional attitudes ascribed by intensional transitives, seem to have a logic (for a compendium of examples, see Richard 2000, 105-7): 'Mary seeks a man who has read Proust and a man who has read Gide' does seem to entail 'Mary seeks a man who has read Proust'. Yet, as Richard notes (Richard 2001, 107-8), the inferential behavior of quantified complements of intensional transitives is still very different from the extensional case. For example (his 'Literary Example'), even if it is true that Mary seeks a man who has read Proust and a man who has read Gide, it may be false both that she seeks at most one man and that she seeks at least two men; for she may be indifferent between finding a man who has read both, versus finding two men, one a reader of Proust but not Gide, the other of Gide but not Proust. Contrast 'saw', 'photographed', or 'met': if she met a man who has read Proust and a man who has read Gide, it cannot be false both that she met at most at one man and also that she met at least two men. As Richard insists, it is a constraint on any semantics of intensional transitives that they get this type of case right. By contrast, in other cases, even very simple ones, it is controversial exactly what inferences intensional transitives in unspecific readings support. If Mary seeks a man who has read Proust, does it follow that she seeks a man who can read? After all, it is unlikely that a comic-book reader will satisfy her tastes in men.[10] If Perseus seeks a mortal gorgon, does it follow that he seeks a gorgon?[11] After all, if he finds an immortal gorgon, he is in trouble. Zimmerman (1993, 173) takes it to be a requirement on accounts of notional readings that they validate these deletion or "weakening" inferences. But there are characterizations of unspecific readings on which these inferences are in fact invalid, for instance, a characterization in terms of indifference towards which object of the relevant kind is found (Lewis uses such an 'any one would do' characterization in Lewis 1972, p. 199). For even if it is true that Mary seeks a man who has read Proust, and any male Proust-reader would do, it does not follow that Mary seeks a man who can read, and any man who can read would do. For not every man who can read has read Proust. Is there anything intrinsic to the indifference characterization of unspecificity ('any one would do') that we can object to while leaving the status of weakening inferences open? One objection is that the characterization does not work for every verb or quantifier: 'Guercino painted a dog, any dog would do' makes little sense, and 'the police are looking for everyone who was in the room, any people who were in the room would do' is not much better. More importantly, the characterization seems to put warranted assertibility out of reach, since the grounds which we normally take to justify ascribing an existentially quantified objectual attitude will rarely give reason to think that the agent has absolutely no further preferences going beyond the characterization of the object-kind given in the ascription (probably Mary would pass on meeting a male psychopathic killer who has read Proust; see further Graff Fara 2013 for analogous discussion of desire). Still, this is not to validate weakening inferences; for that we would need to show that the more usual gloss of the unspecific reading, using a 'but no particular one' rider, supports the inferences as strongly as the indifference characterization refutes them. And it is unclear how such an argument would proceed (see further Forbes 2006, 94-6). In addition, making a good case that the inferences are intuitively valid is one thing, getting the semantics to validate them is another. Both the propositionalist and the Montagovian need to add extra principles, since there is nothing in their bare semantics to compel these inferences. For even if Jack stands in the wanting relation to ^(a(woollen(sweater))), that by itself is silent on whether he also stands in the wanting relation to ^(a(sweater)). The accounts of Moltmann and Richard both decide the matter positively, however (e.g., if in every minimal situation in which Jack's desire is satisfied, he gets a woollen sweater, then in every such situation, he gets a sweater; so he wants a sweater). The intuitive validity of weakening can also be directly challenged. For example, via weakening we can infer that if A is looking for a cat and B is looking for a dog, then A and B are looking for the same thing (an animal). For discussion of this kind of example, see (Zimmerman 2006), and for the special use of 'the same thing', (Moltmann 2008). Asher (1987, 171) proposes an even more direct counterexample. Suppose you enter a competition whose prize is a free ticket on the Concorde to New York. So presumably you want a free ticket on the Concorde. But you don't want a ticket on the Concorde, since you know that normally these are very expensive, you are poor, and you strictly resist desiring the unattainable. Asher is here assimilating notional uses of indefinites to generics, which on his account involve quantification over normal worlds. So if for some bizarre reason you want a sloop, but one whose hull is riddled with holes, it will not be literally true to say you want a sloop. Undeniably there is a real phenomenon here, but perhaps it belongs to pragmatics rather than semantics. If I say 'I want a sloop', someone who offers to buy me any sloop floating in the harbor could reasonably complain 'You should have said that' if I decline the offer on the grounds that none of those sloops meets my unstated requirement of having a hull riddled with holes. But my aspirational benefactor's complaint might be justified because normality is a default implicature or presupposition that a co-operative speaker is under some obligation to let her audience know isn't in force, when it isn't. It is still literally true, on this view, that you want a sloop, despite the idiosyncrasy of the details of your desire. However, this is far from the end of the story. Those with doubts about weakening will find the discussion in (Sainsbury 2018, 129-133) congenial. Another interesting logical problem concerns the "conjunctive force" of disjoined quantified NPs in objectual ascriptions. There is a large literature on the conjunctive force of disjunction in many other contexts (e.g., Kamp 1973, Loewer 1976, Makinson 1984, Jennings 1994, Zimmerman 2000, Simons 2005, Fox 2007), for instance as exhibited in 'x is larger than y or z' and 'John can speak French or Italian to you'. In these cases the conjunctive force is easily captured by simple distribution: 'x is larger than y and larger than z', 'John can speak French to you and can speak Italian to you'.[12] However, with intensional transitives we find the same conjunctive force, but no distributive articulation. If we say that Jack needs a woollen sweater or a fleece jacket, we say something to the effect that (i) his getting a woollen sweater is one way his need could be met, and (ii) his getting a fleece jacket is another way his need could be met. But 'Jack needs a woollen sweater or a fleece jacket' does not mean that Jack needs a woollen sweater and needs a fleece jacket. This last conjunction ascribes two needs, only one of which is met by getting a satisfactory woollen sweater. But the latter acquisition by itself meets the disjunctive need for a woollen sweater or a fleece jacket. So there is a challenge to explain the semantics of the disjunctive ascription, while at the same time remaining within a framework that can accommodate all cases of conjunctive force -- comparatives, various senses of 'can', counterfactuals with disjunctive antecedents ('if Jack were to put on a woollen sweater or a fleece jacket he'd be warmer') and so on; see further (Forbes 2006, 97-111). A penultimate type of inference we will mention is one in which intensional and extensional verbs both occur, and the inference seems valid even when the intensional VPs are construed unspecifically. An example: (23a) Jack wants a woollen sweater (23b) Whatever Jack wants, he gets (23c) Therefore, Jack will get a woollen sweater Obviously, (23a,b) entail (23c) when (23a) is understood specifically. But informants judge that the inference is also valid when (23a) is understood unspecifically, with 'but no particular one' explicitly appended. If we seek a validation of the inference that hews to surface form, Montague's uniform treatment of intensional and extensional verbs has its appeal: (23b) will say that for whatever property of properties *P* Jack stands in the wanting relation to, he stands in the getting relation to. So the inference is portrayed as the simple modus ponens it seems to be. It would then be the task of other meaning-postulates to carry us from his standing in the getting relation to ^(a(woollen(sweater))) to there being a woollen sweater such that he gets it. The final example to be considered here involves fictional or mythical names in the scope of intensional transitives. An example (Zalta 1988, 128) of the puzzles these can lead to is: (24a) The ancient Greeks worshipped Zeus. (24b) Zeus is a mythical character. (24c) Mythical characters do not exist. (24d) Therefore, the ancient Greeks worshipped something that does (did) not exist. Or even: (24e) Therefore, there is something that doesn't exist such that the ancient Greeks worshipped it. One thing the example shows is that specific/unspecific is not to be confused with real/fictional. (24a) is a true specific ascription, just as 'the ancient Greeks worshipped Ahura Mazda' is a false one. (24b) is also true. So the ancient Greeks, who would not have knowingly worshipped a mythical character, were making a rather large mistake, if one of a familiar sort. (24c) is also true, if we are careful about what 'do not exist' means in this context. It is contingent that Zeus-myths were ever formulated, and one sense in which we might mean (24c) turns on the assumption that fictional and mythical characters exist iff fictions and myths about them do. In this sense, (24c) is false, though an actual fictional character such as Zeus would not have existed if there had been no Zeus-myths. This also explains why (24a) and (24b) can both be true: 'Zeus' refers to the mythical character, a contingently existing abstract object. However, by far the more likely reading of (24c) is one on which it means that mythical characters are not real. Zeus is not flesh and blood, not even immaterial flesh and blood. With this in mind, (24d) and (24e) are both true. The quantifier 'something' ranges over a domain that includes both real and fictional or mythical entities, and there is something in that domain, the mythical character, which was worshipped by the ancient Greeks and which is not in the subdomain of real items. This gets the right truth-values for the statements in (24), but might be thought to run into trouble with the likes of 'Zeus lives on Mt. Olympus': if 'Zeus' refers to an abstract object, how can Zeus live anywhere? One way of dealing with this kind of case is to suppose, evidently plausibly, that someone who says 'Zeus lives on Mt. Olympus' and knows the facts means (25) According to the myth, Zeus lives on Mt. Olympus. On the other hand, if an ancient Greek believer says 'Zeus lives on Mt. Olympus', he or she says something false, there being no reason to posit a covert 'according to the myth' in this case. However, even the covert operator theory might be contested, on the grounds that within its scope we are still unintelligibly predicating 'lived on Mt. Olympus' of an abstract object. Simply prefixing 'according to the myth' to the unintelligible cannot render it intelligible. But there is the evident fact that (25) is both intelligible and true. So either prefixing 'according to the myth' can render the unintelligible intelligible, or what is going on in the embedded sentence is not to be construed as standard predication. For further discussion of these matters, see, for example, van Inwagen 1977, Parsons 1980, Zalta 1988, Thomasson 1998, and Salmon 2002.
truthlikeness	## 1. The Logical Problem Truth, perhaps even more than beauty and goodness, has been the target of an extraordinary amount of philosophical dissection and speculation. By comparison with truth, the more complex and much more interesting concept of truthlikeness has only recently become the subject of serious investigation. The logical problem of truthlikeness is to give a consistent and materially adequate account of the concept. But first, we have to make it plausible that there is a coherent concept in the offing to be investigated. ### 1.1 What's the problem? Suppose we are interested in what is the number of planets in the solar system. With the demotion of Pluto to planetoid status, the truth of this matter is that there are precisely 8 planets. Now, the proposition the number of planets in our solar system is 9 may be false, but quite a bit closer to the truth than the proposition that the number of planets in our solar system is 9 billion. (One falsehood may be closer to the truth than another falsehood.) The true proposition the number of the planets is between 7 and 9 inclusive is closer to the truth than the true proposition that the number of the planets is greater than or equal to 0. (So a truth may be closer to the truth than another truth.) Finally, the proposition that the number of the planets is either less than or greater than 9 may be true but it is arguably not as close to the whole truth as its highly accurate but strictly false negation: that there are 9 planets. This particular numerical example is admittedly simple, but a wide variety of judgments of relative likeness to truth crop up both in everyday parlance as well as in scientific discourse. While some involve the relative accuracy of claims concerning the value of numerical magnitudes, others involve the sharing of properties, structural similarity, or closeness among putative laws. Consider a non-numerical example, also highly simplified but quite topical in the light of the recent rise in status of the concept of fundamentality. Suppose you are interested in the truth about which particles are fundamental. At the outset of your inquiry all you know are various logical truths, like the tautology either electrons are fundamental or they are not. Tautologies are pretty much useless in helping you locate the truth about fundamental particles. Suppose that the standard model is actually on the right track. Then, learning that electrons are fundamental edges you a little bit closer to your goal. It is by no means the complete truth about fundamental particles, but it is a piece of it. If you go on to learn that electrons, along with muons and tau particles, are a kind of lepton and that all leptons are fundamental, you have presumably edged closer. If this is right, then some truths are closer to the truth about fundamental particles than others. The discovery that atoms are not fundamental, that they are in fact composite objects, displaced the earlier hypothesis that atoms are fundamental. For a while the proposition that protons, neutrons and electrons are the fundamental components of atoms was embraced, but unfortunately it too turned out to be false. Still, this latter falsehood seems closer to the truth than its predecessor (assuming, again, that the standard model is true). And even if the standard model contains errors, as surely it does, it is presumably closer to the truth about fundamental particles than these other falsehoods. So again, some falsehoods may be closer to the truth about fundamental particles than other falsehoods. As we have seen, a tautology is not a terrific truth locator, but if you moved from the tautology that electrons either are or are not fundamental to embrace the false proposition that electrons are not fundamental you would have moved further from your goal. So, some truths are closer to the truth than some falsehoods. But it is by no means obvious that all truths about fundamental particles are closer to the whole truth than any falsehood. The false proposition that electrons, protons and neutrons are the fundamental components of atoms, for instance, may well be an improvement over the tautology. If this is right, certain falsehoods are closer to the truth than some truths. Investigations into the concept of truthlikeness only began in earnest in the early nineteen sixties. Why was truthlikeness such a latecomer to the philosophical scene? It wasn't until the latter half of the twentieth century that mainstream philosophers gave up on the Cartesian goal of infallible knowledge. The idea that we are quite possibly, even probably, mistaken in our most cherished beliefs, that they might well be just false, was mostly considered tantamount to capitulation to the skeptic. By the middle of the twentieth century, however, it was clear that many of our commonsense beliefs, as well as previous scientific theories, are strictly speaking, false. Further, the increasingly rapid turnover of scientific theories suggested that, far from being certain, they are ever vulnerable to refutation, and typically are eventually refuted and replaced by some new theory. The history of inquiry is one of a parade of refuted theories, replaced by other theories awaiting their turn at the guillotine. (This is the "dismal induction", see also the entry on realism and theory change in science.) Realism holds that the constitutive aim of inquiry is the truth of some matter. Optimism holds that the history of inquiry is one of progress with respect to its constitutive aim. But fallibilism holds that our theories are false or very likely to be false, and to be replaced by other false theories. To combine these three ideas, we must affirm that some false propositions better realize the goal of truth - are closer to the truth - than others. We are thus stuck with the logical problem of truthlikeness. While a multitude of apparently different solutions to the problem have been proposed, they can be classified into three main approaches, each with its own heuristic - the content approach, the consequence approach and the likeness approach. Before exploring these possible solutions to the logical problem, it could be useful to dispel a couple of common confusions, since truthlikeness should not be conflated with either epistemic probability or with vagueness. We discuss this latter notion in the supplement Why truthlikeness is not probability or vagueness (see also the entry on vagueness); as for the former, we shall discuss the difference between (expected) truthlikeness and probability when discussing the epistemological problem (SS2). ### 1.2 The content approach Karl Popper was the first philosopher to take the logical problem of truthlikeness seriously enough to make an assay on it. This is not surprising, since Popper was also the first prominent realist to embrace a very radical fallibilism about science while also trumpeting the epistemic superiority of the enterprise. In his early work, he implied that the only kind of progress an inquiry can make consists in falsification of theories. This is a little depressing, to say the least. It is almost as depressing as the pessimistic induction. What it lacks is a positive account of how a succession of falsehoods might constitute positive cognitive progress. Perhaps this is why Popper's early work received a pretty short shrift from other philosophers. If a miss is as good as a mile, and all we can ever establish with confidence is that our inquiry has missed its target once again, then epistemic pessimism seems inevitable. Popper eventually realized that falsificationism is compatible with optimism provided we have an acceptable notion of verisimilitude (or truthlikeness). If some false hypotheses are closer to the truth than others, then the history of inquiry may turn out to be one of progress towards the goal of truth. Moreover, it may even be reasonable, on the basis of our evidence, to conjecture that our theories are in fact making such progress, even though we know they are all false or highly likely to be false. Popper saw clearly that the concept of truthlikeness should not be confused with the concept of epistemic probability, and that it has often been so confused. (See Popper 1963 for a history of the confusion and the supplement Why truthlikeness is not probability or vagueness for an explanation of the difference between the two concepts.) Popper's insight here was facilitated by his deep but largely unjustified antipathy to epistemic probability. He thought that his starkly falsificationist account favored bold, contentful theories. Degree of informative content varies inversely with probability - the greater the content the less likely a theory is to be true. So if you are after theories which seem, on the evidence, to be true, then you will eschew those which make bold - that is, highly improbable - predictions. On this picture, the quest for theories with high probability is simply misguided. To see this distinction between truthlikeness and probability clearly, and to articulate it, was one of Popper's most significant contributions, not only to the debate about truthlikeness, but to philosophy of science and logic in general. However, his deep antagonism to probability, combined with his love of boldness, was both a blessing and a curse. The blessing: it led him to produce not only the first interesting and important account of truthlikeness, but to initiate an approach to the problem in terms of content. The curse: content alone, as Popper envisaged it, is insufficient to characterize truthlikeness. Popper made the first attempt to solve the problem in his famous collection Conjectures and Refutations. As a great admirer of Tarski's assay on the concept of truth, he modelled his theory of truthlikeness on Tarski's theory. First, let a matter for investigation be circumscribed by a formalized language $L$ adequate for discussing it. Tarski showed us how each possible world, or model of the language, induces a partition of sentences of $L$ into those that are true and those that are false. The set of all sentences true in the actual world is thus a complete true account of the world, as far as that language goes. It is aptly called the Truth, $T$. $T$ is the target of the investigation couched in $L$. It is the theory (relative to the resources in $L)$ that we are seeking. If truthlikeness is to make sense, theories other than $T$, even false theories, come more or less close to capturing $T$. $T$, the Truth, is a theory only in the technical Tarskian sense, not in the ordinary everyday sense of that term. It is a set of sentences closed under the consequence relation: $T$ may not be finitely axiomatizable, or even axiomatizable at all. However, it is a perfectly good set of sentences all the same. In general, we will follow the Tarski-Popper usage here and call any set of sentences closed under consequence a theory, and we will assume that each proposition we deal with is identified with a theory in this sense. (Note that theory $A$ logically entails theory $B$ just in case $B$ is a subset of $A$.) The complement of $T$, the set of false sentences $F$, is not a theory even in this technical sense. Since falsehoods always entail truths, $F$ is not closed under the consequence relation. (This may be the reason why we have no expression like the Falsth: the set of false sentences does not describe a possible alternative to the actual world.) But $F$ too is a perfectly good set of sentences. The consequences of any theory $A$ that can be formulated in $L$ will thus divide between $T$ and $F$. Popper called the intersection of $A$ and $T$, the truth content of $A$ ($A\_T$), and the intersection of $A$ and $F$, the falsity content of $A$ ($A\_T$). Any theory $A$ is thus the union of its non-overlapping truth content and falsity content. Note that since every theory entails all logical truths, these will constitute a special set, at the center of $T$, which will be included in every theory, whether true or false. ![missing text, please inform](diagram1.png) Diagram 1. Truth and falsity contents of false theory $A$ A false theory will cover some of $F$, but because every false theory has true consequences, including all logical truths, it will also overlap with $T$ (Diagram 1). A true theory, however, will only overlap $T$ (Diagram 2): ![missing text, please inform](diagram2.png) Diagram 2. True theory $A$ is identical to its own truth content Amongst true theories, then, it seems that the more true sentences that are entailed, the closer we get to $T$, hence the more truthlike. Set theoretically that simply means that, where $A$ and $B$ are both true, $A$ will be more truthlike than $B$ just in case $B$ is a proper subset of $A$ (which for true theories means that $B\_T$ is a proper subset of $A\_T$). Call this principle: the value of content for truths. ![missing text, please inform](diagram3.png) Diagram 3. True theory $A$ has more truth content than true theory $B$ This essentially syntactic account of truthlikeness has some nice features. It induces a partial ordering of truths, with the whole Truth $T$ at the top of the ordering: $T$ is closer to the Truth than any other true theory. The set of logical truths is at the bottom: further from the Truth than any other true theory. In between these two extremes, true theories are ordered simply by logical strength: the more logical content, the closer to the Truth. Since probability varies inversely with logical strength, amongst truths the theory with the greatest truthlikeness $(T)$ must have the smallest probability, and the theory with the largest probability (the logical truth) is the furthest from the Truth. Popper made a simple and perhaps plausible generalization of this. Just as truth content (coverage of $T)$ counts in favor of truthlikeness, falsity content (coverage of $F)$ counts against. In general then, a theory $A$ is closer to the truth if it has more truth content without engendering more falsity content, or has less falsity content without sacrificing truth content (diagram 4): ![missing text, please inform](diagram4.png) Diagram 4. False theory $A$ closer to the Truth than false theory $B$ The generalization of the truth content comparison, by incorporating falsity content comparisons, also has some nice features. It preserves the comparisons of true theories mentioned above. The truth content $A\_T$ of a false theory $A$ (itself a theory in the Tarskian sense) will clearly be closer to the truth than $A$ (Diagram 1). And the whole truth $T$ will be closer to the truth than any falsehood $B$ because the truth content of $B$ must be contained within $T$, and the falsity content of $T$ (the empty class) must be properly contained within the non-empty falsity content of $B$. Despite its attractive features, the account has a couple of disastrous consequences. Firstly, since a falsehood has some false consequences, and no truth has any, it follows that no falsehood can be as close to the truth as a logical truth - the weakest of all truths. A logical truth leaves the location of the truth wide open, so it is rather worthless as an approximation to the whole truth. On Popper's account, no falsehood can ever be more worthwhile than a worthless logical truth. (We could call this result the absolute worthlessness of falsehoods). Furthermore, it is impossible to add a true consequence to a false theory without thereby adding additional false consequences (or subtract a false consequence without subtracting true consequences). So the account entails that no false theory is closer to the truth than any other. We could call this result the relative worthlessness of all falsehoods. These worthlessness results were proved independently by Pavel Tichy and David Miller (Miller 1974, and Tichy 1974) - for a proof, see the supplement on Why Popper's definition of truthlikeness fails: the Tichy-Miller theorem. It is tempting (and Popper was so tempted) to retreat in the face of these results to something like the comparison of truth contents alone. That is to say, $A$ is as close to the truth as $B$ if $B\_T$ is contained in $A\_T$, and $A$ is closer to the truth than $B$ just in case $B\_T$ is properly contained in $B\_T$. Call this the Simple Truth Content account. This Simple Truth Content account preserves what many consider to be the chief virtue of Popper's account: the value of content for truths. And while it delivers the absolute worthlessness of falsehoods (no falsehood is closer to the truth than a tautology) it avoids the relative worthlessness of falsehoods. If $A$ and $B$ are both false, then $A\_T$ may well properly contain $B\_T$. But that holds if and only if $A$ is logically stronger than $B$. That is to say, a false proposition is the closer to the truth the stronger it is. According to this principle - call it the value of content for falsehoods - the false proposition that there are nine planets, and all of them are made of green cheese is more truthlike than the false proposition there are nine planets. And so once one knows that a certain theory is false one can be confident that tacking on any old arbitrary proposition, no matter how inaccurate it is, will lead us inexorably closer to the truth. This is sometimes called the child's play objection. Among false theories, brute logical strength becomes the sole criterion of a theory's likeness to truth. Even though Popper's particular proposals were flawed, his idea of comparing truth-content and falsity-content is nevertheless worth exploring. Several philosophers have developed variations on the idea. Some stay within Popper's essentially syntactic paradigm, elucidating content in terms of consequence classes (e.g., Newton Smith 1981; Schurz and Weingartner 1987, 2010; Cevolani and Festa 2020). Others have switched to a semantic conception of content, construing semantic content in terms of classes of possibilities, and searching for a plausible theory of distance between those. A variant of this approach takes the class of models of a language as a surrogate for possible states of affairs (Miller 1978a). The other utilizes a semantics of incomplete possible states like those favored by structuralist accounts of scientific theories (Kuipers 1987b, Kuipers 2019). The idea which these accounts have in common is that the distance between two propositions $A$ and $B$ is measured by the symmetric difference $A\mathbin{\Delta} B$ of the two sets of possibilities: $(A - B)\cup(B - A)$. Roughly speaking, the larger the symmetric difference, the greater the distance between the two propositions. Symmetric differences might be compared qualitatively - by means of set-theoretic inclusion - or quantitatively, using some kind of probability measure. Both can be shown to have the general features of a measure of distance. The fundamental problem with the content approach lies not in the way it has been articulated, but rather in the basic underlying assumption: that truthlikeness is a function of just two variables - content and truth value. This assumption has several rather problematic consequences. Firstly, any given proposition $A$ can have only two degrees of verisimilitude: one in case it is false and the other in case it is true. This is obviously wrong. A theory can be false in very many different ways. The proposition that there are eight planets is false whether there are nine planets or a thousand planets, but its degree of truthlikeness is much higher in the first case than in the latter. Secondly, if we combine the value of content for truths and the value of content for falsehoods, then if we fix truth value, verisimilitude will vary only according to amount of content. So, for example, two equally strong false theories will have to have the same degree of verisimilitude. That's pretty far-fetched. That there are ten planets and that there are ten billion planets are (roughly) equally strong, and both are false in fact, but the latter seems much further from the truth than the former. Finally, how might strength determine verisimilitude amongst false theories? There seem to be just two plausible candidates: that verisimilitude increases with increasing strength (the principle of the value of content for falsehoods) or that it decreases with increasing strength (the principle of the disvalue of content for falsehoods). Both proposals are at odds with attractive judgements and principles, which suggest that the original content approach is in need of serious revision (see, e.g., Kuipers 2019 for a recent proposal). ### 1.3 The Consequence Approach Popper crafted his initial proposal in terms of the true and false consequences of a theory. Any sentence at all that follows from a theory is counted as a consequence that, if true, contributes to its overall truthlikeness, and if false, detracts from that. But it has struck many that this both involves an enormous amount of double counting, and that it is the indiscriminate counting of arbitrary consequences that lies behind the Tichy-Miller trivialization result. Consider a very simple framework with three primitive sentences: $h$ (for the state hot), $r$ (for rainy) and $w$ (for windy). This framework generates a very small space of eight possibilities. The eight maximal conjunctions (like $h \amp r \amp w, {\sim}h \amp r \amp w,$ etc.) of the three primitive sentences and of their negations express those possibilities. Suppose that in fact it is hot, rainy and windy (expressed by the maximal conjunction $h \amp r \amp w)$. Then the claim that it is cold, dry and still (expressed by the sentence ${\sim}h \amp{\sim}r \amp{\sim}w)$ is further from the truth than the claim that it is cold, rainy and windy (expressed by the sentence ${\sim}h \amp r \amp w)$. And the claim that it is cold, dry and windy (expressed by the sentence ${\sim}h \amp{\sim}r \amp w)$ is somewhere between the two. These kinds of judgements, which seem both innocent and intuitively correct, Popper's theory cannot accommodate. And if they are to be accommodated we cannot treat all true and false consequences alike. For the three false claims mentioned here have exactly the same number of true and false consequences (this is the problem we called the relative worthlessness of all falsehoods). Clearly, if we are going to measure closeness to truth by counting true and false consequences, some true consequences should count more than others. For example, $h$ and $r$ are both true, and ${\sim}h$ and ${\sim}r$ are false. The former should surely count in favor of a claim, and the latter against. But ${\sim}h\rightarrow{\sim}r$ is true and $h\rightarrow{\sim}r$ is false. After we have counted the truth $h$ in favor of a claim's truthlikeness and the falsehood ${\sim}r$ against it, should we also count the true consequence ${\sim}h\rightarrow{\sim}r$ in favor, and the falsehood $h\rightarrow{\sim}r$ against? Surely this is both unnecessary and misleading. And it is precisely counting sentences like these that renders Popper's account susceptible to the Tichy-Miller argument. According to the consequence approach, Popper was right in thinking that truthlikeness depends on the relative sizes of classes of true and false consequences, but erred in thinking that all consequences of a theory count the same. Some consequences are relevant, some aren't. Let $R$ be some criterion of relevance of consequences; let $A\_R$ be the set of relevant consequences of $A$. Whatever the criterion $R$ is it has to satisfy the constraint that $A$ be recoverable from (and hence equivalent to) $A\_R$. Popper's account is the limiting one - all consequences are relevant. (Popper's relevance criterion is the empty one, $P$, according to which $A\_P$ is just $A$ itself.) The relevant truth content of A (abbreviated $A\_R^T$) can be defined as $A\_R\cap T$ (or $A\cap T\_R$), and similarly the relevant falsity content of $A$ can be defined as $A\_R\cap F$. Since $A\_R = (A\_R\cap T)\cup(A\_R\cap F)$ it follows that the union of true and false relevant consequences of $A$ is equivalent to $A$. And where $A$ is true $A\_R\cap F$ is empty, so that A is equivalent to $A\_R\cap T$ alone. With this restriction to relevant consequences we can basically apply Popper's definitions: one theory is more truthlike than another if its relevant truth content is larger and its relevant falsity content no larger; or its relevant falsity content is smaller, and its relevant truth content is no smaller. This idea was first explored by Mortensen in his 1983, but he abandoned the basic idea as unworkable. Subsequent proposals within the broad program have been offered by Burger and Heidema 1994, Schurz and Weingartner 1987 and 2010, and Gemes 2007. (Gerla 2007 also uses the notion of the relevance of a "test" or factor, but his account is best located more squarely within the likeness approach.) One possible relevance criterion that the $h$-$r$-$w$ framework might suggest is atomicity. This amounts to identifying relevant consequences as basic ones, i.e., atomic sentences or their negations (Cevolani, Crupi and Festa 2011; Cevolani, Festa and Kuipers 2013). But even if we could avoid the problem of saying what it is for a sentence to be atomic, since many distinct propositions imply the same atomic sentences, this criterion would not satisfy the requirement that $A$ be equivalent to $A\_R$. For example, $(h\vee r)$ and $({\sim}h\vee{\sim}r)$, like tautologies, imply no atomic sentences at all. This latter problem can be solved by resorting to the notion of partial consequence; interestingly, the resulting account becomes virtually indentical to one version of the likeness approach (Cevolani and Festa 2020). Burger and Heidema 1994 compare theories by positive and negative sentences. A positive sentence is one that can be constructed out of $\amp,$ $\vee$ and any true basic sentence. A negative sentence is one that can be constructed out of $\amp,$ $\vee$ and any false basic sentence. Call a sentence pure if it is either positive or negative. If we take the relevance criterion to be purity, and combine that with the relevant consequence schema above, we have Burger and Heidema's proposal, which yields a reasonable set of intuitive judgments. Unfortunately purity (like atomicity) does not quite satisfy the constraint that $A$ be equivalent to the class of its relevant consequences. For example, if $h$ and $r$ are both true then $({\sim}h\vee r)$ and $(h\vee{\sim}r)$ both have the same pure consequences (namely, none). Schurz and Weingartner 2010 use the following notion of relevance $S$: being equivalent to a disjunction of atomic propositions or their negations. With this criterion they can accommodate a range of intuitive judgments in the simple weather framework that Popper's account cannot. For example, where $\gt\_S$ is the relation of greater S-truthlikeness we capture the following relations among false claims, which, on Popper's account, are mostly incommensurable: \[ (h \amp{\sim}r) \gt\_S ({\sim}r) \gt\_S ({\sim}h \amp{\sim}r). \] and \[ (h\vee r) \gt\_S ({\sim}r) \gt\_S ({\sim}h\vee{\sim}r) \gt\_S ({\sim}h \amp{\sim}r). \] The relevant consequence approach faces three major hurdles. The first is an extension problem: the approach does produce some intuitively acceptable results in a finite propositional framework, but it needs to be extended to more realistic frameworks - for example, first-order and higher-order frameworks (see Gemes 2007 for an attempt along these lines). The second is that, like Popper's original proposal, it judges no false proposition to be closer to the truth than any truth, including logical truths. Schurz and Weingartner (2010) have answered this objection by quantitatively extending their qualitative account by assigning weights to relevant consequences and summing; one problem with this is that it assumes finite consequence classes. The third involves the language-dependence of any adequate relevance criterion. This problem will be outlined and discussed below in connection with the likeness approach (SS1.4.3). ### 1.4 The Likeness Approach In the wake of the difficulties facing Popper's approach, two philosophers, working quite independently, suggested a radically different approach: one which takes the likeness in truthlikeness seriously (Tichy 1974, Hilpinen 1976). This shift from content to likeness was also marked by an immediate shift from Popper's essentially syntactic approach (something it shares with the consequence program) to a semantic approach. Traditionally the semantic contents of sentences have been taken to be non-linguistic, or rather non-syntactic, items - propositions. What propositions are is highly contested, but most agree that a proposition carves the class of possibilities into two sub-classes - those in which the proposition is true and those in which it is false. Call the class of worlds in which the proposition is true its range. Some have proposed that propositions be identified with their ranges (for example, David Lewis, in his 1986). This is implausible since, for example, the informative content of $7+5=12$ seems distinct from the informative content of $12=12$, which in turn seems distinct from the informative content of Godel's first incompleteness theorem - and yet all three have the same range: they are all true in all possible worlds. Clearly, if semantic content is supposed to be sensitive to informative content, classes of possible worlds will are not discriminating enough. We need something more fine-grained for a full theory of semantic content. Despite this, the range of a proposition is certainly an important aspect of informative content, and it is not immediately obvious why truthlikeness should be sensitive to differences in the way a proposition picks out its range. (Perhaps there are cases of logical falsehoods some of which seem further from the truth than others. For example $7+5=113$ might be considered further from the truth than $7+5=13$ though both have the same range - namely, the empty set of worlds; see Sorensen 2007.) But as a first approximation, we will assume that it is not hyperintensional and that logically equivalent propositions have the same degree of truthlikess. The proposition that the number of planets is eight for example, should have the same degree of truthlikeness as the proposition that the square of the number of the planets is sixty four. Leaving apart the controversy over the nature of possible worlds, we shall call the complete collection of possibilities, given some array of features, the logical space, and call the array of properties and relations which underlie that logical space, the framework of the space. Familiar logical relations and operations correspond to well-understood set-theoretic relations and operations on ranges. The range of the conjunction of two propositions is the intersection of the ranges of the two conjuncts. Entailment corresponds to the subset relation on ranges. The actual world is a single point in logical space - a complete specification of every matter of fact (with respect to the framework of features) - and a proposition is true if its range contains the actual world, false otherwise. The whole Truth is a true proposition that is also complete: it entails all true propositions. The range of the Truth is none other than the singleton of the actual world. That singleton is the target, the bullseye, the thing at which the most comprehensive inquiry is aiming. Without additional structure on the logical space we have just three factors for a theorist of truthlikeness to work with - the size of a proposition (content factor), whether it contains the actual world (truth factor), and which propositions it implies (consequence factor). The likeness approach requires some additional structure to the logical space. For example, worlds might be more or less like other worlds. There might be a betweenness relation amongst worlds, or even a fully-fledged distance metric. If that's the case, we can start to see how one proposition might be closer to the Truth - the proposition whose range contains just the actual world - than another. The core of the likeness approach is that truthlikeness supervenes on the likeness between worlds. The likeness theorist has two tasks: firstly, making it plausible that there is an appropriate likeness or distance function on worlds; and secondly, extending likeness between individual worlds to likeness of propositions (i.e., sets of worlds) to the actual world. Suppose, for example, that worlds are arranged in similarity spheres nested around the actual world, familiar from the Stalnaker-Lewis approach to counterfactuals. Consider Diagram 5. ![missing text, please inform](diagram5.png) Diagram 5. Verisimilitude by similarity circles The bullseye is the actual world and the small sphere which includes it is $T$, the Truth. The nested spheres represent likeness to the actual world. A world is less like the actual world the larger the first sphere of which it is a member. Propositions $A$ and $B$ are false, $C$ and $D$ are true. A carves out a class of worlds which are rather close to the actual world - all within spheres two to four - whereas $B$ carves out a class rather far from the actual world - all within spheres five to seven. Intuitively $A$ is closer to the bullseye than is $B$. The largest sphere which does not overlap at all with a proposition is plausibly a measure of how close the proposition is to being true. Call that the truth factor. A proposition $X$ is closer to being true than $Y$ if the truth factor of $X$ is included in the truth factor of $Y$. The truth factor of $A$, for example, is the smallest non-empty sphere, $T$ itself, whereas the truth factor of $B$ is the fourth sphere, of which $T$ is a proper subset: so $A$ is closer to being true than $B$. If a proposition includes the bullseye, then of course it is true simpliciter and it has the maximal truth factor (the empty set). So all true propositions are equally close to being true. But truthlikeness is not just a matter of being close to being true. The tautology, $D,$ $C$ and the Truth itself are equally true, but in that order they increase in their closeness to the whole truth. Taking a leaf out of Popper's book, Hilpinen argued that closeness to the whole truth is in part a matter of degree of informativeness of a proposition. In the case of the true propositions, this correlates roughly with the smallest sphere which totally includes the proposition. The further out the outermost sphere, the less informative the proposition is, because the larger the area of the logical space which it covers. So, in a way which echoes Popper's account, we could take truthlikeness to be a combination of a truth factor (given by the likeness of that world in the range of a proposition that is closest to the actual world) and a content factor (given by the likeness of that world in the range of a proposition that is furthest from the actual world): > > $A$ is closer to the truth than $B$ if and only if $A$ does as > well as $B$ on both truth factor and content factor, and better on > at least one of those. > Applying Hilpinen's definition we capture two more particular judgements, in addition to those already mentioned, that seem intuitively acceptable: that $C$ is closer to the truth than $A$, and that $D$ is closer than $B$. (Note, however, that we have here a partial ordering: $A$ and $D$, for example, are not ranked.) We can derive from this various apparently desirable features of the relation closer to the truth: for example, that the relation is transitive, asymmetric and irreflexive; that the Truth is closer to the Truth than any other theory; that the tautology is at least as far from the Truth as any other truth; that one cannot make a true theory worse by strengthening it by a truth (a weak version of the value of content for truths); that a falsehood is not necessarily improved by adding another falsehood, or even by adding another truth (a repudiation of the value of content for falsehoods). But there are also some worrying features here. While it avoids the relative worthlessness of falsehoods, Hilpinen's account, just like Popper's, entails the absolute worthlessness of all falsehoods: no falsehood is closer to the truth than any truth. So, for example, Newton's theory is deemed to be no more truthlike, no closer to the whole truth, than the tautology. Characterizing Hilpinen's account as a combination of a truth factor and an information factor seems to mask its quite radical departure from Popper's account. The incorporation of similarity spheres signals a fundamental break with the pure content approach, and opens up a range of possible new accounts: what such accounts have in common is that the truthlikeness of a proposition is a non-trivial function of the likeness to the actual world of worlds in the range of the proposition. There are three main problems for any concrete proposal within the likeness approach. The first concerns an account of likeness between states of affairs - in what does this consist and how can it be analyzed or defined? The second concerns the dependence of the truthlikeness of a proposition on the likeness of worlds in its range to the actual world: what is the correct function? (This can be called "the extension problem".) And finally, there is the famous problem of "translation variance" or "framework dependence" of judgements of likeness and of truthlikeness. This last problem will be taken up in SS1.4.3. #### 1.4.1 Likeness of worlds in a simple propositional framework One objection to Hilpinen's proposal (like Lewis's proposal for counterfactuals) is that it assumes the similarity relation on worlds as a primitive, there for the taking. At the end of his 1974 paper, Tichy not only suggested the use of similarity rankings on worlds, but also provided a ranking in propositional frameworks and indicated how to generalize this to more complex frameworks. Examples and counterexamples in Tichy 1974 are exceedingly simple, utilizing the little propositional framework introduced above, with three primitives - $h$ (for the state hot), $r$ (for rainy) and $w$ (for windy). Corresponding to the eight-members of the logical space generated by distributions of truth values through the three basic conditions, there are eight maximal conjunctions (or constituents): \| \| \| \| \| \| \| --- \| --- \| --- \| --- \| --- \| \| $w\_1$ \| $h \amp r \amp w$ \| \| $w\_5$ \| ${\sim}h \amp r \amp w$ \| \| $w\_2$ \| $h \amp r \amp{\sim}w$ \| \| $w\_6$ \| ${\sim}h \amp r \amp{\sim}w$ \| \| $w\_3$ \| $h \amp{\sim}r \amp w$ \| \| $w\_7$ \| ${\sim}h \amp{\sim}r \amp w$ \| \| $w\_4$ \| $h \amp{\sim}r \amp{\sim}w$ \| \| $w\_8$ \| ${\sim}h \amp{\sim}r \amp{\sim}w$ \| Worlds differ in the distributions of these traits, and a natural, albeit simple, suggestion is to measure the likeness between two worlds by the number of agreements on traits. This is tantamount to taking distance to be measured by the size of the symmetric difference of generating states - the so-called city-block measure. As is well known, this will generate a genuine metric, in particular satisfying the triangular inequality. If $w\_1$ is the actual world this immediately induces a system of nested spheres, but one in which the spheres come with numbers attached: ![missing text, please inform](diagram6.png) Diagram 6. Similarity circles for the weather space Those worlds orbiting on the sphere $n$ are of distance $n$ from the actual world. In fact, the structure of the space is better represented not by similarity circles, but rather by a three-dimensional cube: ![missing text](diagram7.png) Diagram 7. The three-dimensional weather space This way of representing the space makes a clearer connection between distances between worlds and the role of the atomic propositions in generating those distances through the city-block metric. It also eliminates inaccuracies in the relations between the worlds that are not at the center that the similarity circle diagram suggests. #### 1.4.2 The likeness of a proposition to the truth Now that we have numerical distances between worlds, numerical measures of propositional likeness to, and distance from, the truth can be defined as some function of the distances, from the actual world, of worlds in the range of a proposition. But which function is the right one? This is the extension problem. Suppose that $h \amp r \amp w$ is the whole truth about the weather. Following Hilpinen, we might consider overall distance of a propositions from the truth to be some function of the distances from actuality of two extreme worlds. Let truth$(A)$ be the truth value of $A$ in the actual world. Let min$(A)$ be the distance from actuality of that world in $A$ closest to the actual world, and max$(A)$ be the distance from actuality of that world in $A$ furthest from the actual world. Table 1 display the values of the min and max functions for some representative propositions. \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| $\boldsymbol{A}$ \| **truth$(\boldsymbol{A})$ \| min$(\boldsymbol{A})$ \| max$(\boldsymbol{A})$** \| \| $h \amp r \amp w$ \| true \| 0 \| 0 \| \| $h \amp r$ \| true \| 0 \| 1 \| \| $h \amp r \amp{\sim}$w \| false \| 1 \| 1 \| \| $h$ \| true \| 0 \| 2 \| \| $h \amp{\sim}r$ \| false \| 1 \| 2 \| \| ${\sim}h$ \| false \| 1 \| 3 \| \| ${\sim}h \amp{\sim}r \amp w$ \| false \| 2 \| 2 \| \| ${\sim}h \amp{\sim}r$ \| false \| 2 \| 3 \| \| ${\sim}h \amp{\sim}r \amp{\sim}w$ \| false \| 3 \| 3 \| Table 1. The min and max functions. The simplest proposal (made first in Niiniluoto 1977) would be to take the average of the min and the max (call this measure min-max-average). This would remedy a rather glaring shortcoming which Hilpinen's qualitative proposal shares with Popper's proposal, namely that no falsehood is closer to the truth than any truth (even the worthless tautology). This numerical equivalent of Hilpinen's proposal renders all propositions comparable for truthlikeness, and some falsehoods it deems more truthlike than some truths. But now that we have distances between all worlds, why take only the extreme worlds in a proposition into account? Why shouldn't every world in a proposition potentially count towards its overall distance from the actual world? A simple measure which does count all worlds is average distance from the actual world. Average delivers all of the particular judgements we used above to motivate Hilpinen's proposal in the first place, and in conjunction with the simple metric on worlds it delivers the following ordering of propositions in our simple framework: \| \| \| \| \| --- \| --- \| --- \| \| $\boldsymbol{A}$ \| **truth$(\boldsymbol{A})$ \| average$(\boldsymbol{A})$** \| \| $h \amp r \amp w$ \| true \| 0 \| \| $h \amp r$ \| true \| 0.5 \| \| $h \amp r \amp{\sim}$w \| false \| 1.0 \| \| $h$ \| true \| 1.3 \| \| $h \amp{\sim}r$ \| false \| 1.5 \| \| ${\sim}h$ \| false \| 1.7 \| \| ${\sim}h \amp{\sim}r \amp w$ \| false \| 2.0 \| \| ${\sim}h \amp{\sim}r$ \| false \| 2.5 \| \| ${\sim}h \amp{\sim}r \amp{\sim}w$ \| false \| 3.0 \| Table 2. The average function. This ordering looks promising. Propositions are closer to the truth the more they get the basic weather traits right, further away the more mistakes they make. A false proposition may be made either worse or better by strengthening $(h \amp r \amp{\sim}w$ is better than ${\sim}w$ while ${\sim}h \amp{\sim}r \amp{\sim}w$ is worse). A false proposition (like $h \amp r \amp{\sim}w)$ can be closer to the truth than some true propositions (like $h)$. And so on. These judgments may be sufficient to show that average is superior to min-max-average, at least on this group of propositions, but they are clearly not sufficient to show that averaging is the right procedure. What we need are some straightforward and compelling general desiderata which jointly yield a single correct function. In the absence of such a proof, we can only resort to case by case comparisons. Furthermore average has not found universal favor. Notably, there are pairs of true propositions such that average deems the stronger of the two to be the further from the truth. According to average, the tautology is not the true proposition furthest from the truth. Averaging thus violates the Popperian principle of the value of content for truths, see Table 3 for an example. \| \| \| \| \| --- \| --- \| --- \| \| $\boldsymbol{A}$ \| **truth$(\boldsymbol{A})$ \| average$(\boldsymbol{A})$** \| \| $h \vee{\sim}r \vee w$ \| true \| 1.4 \| \| $h \vee{\sim}r$ \| true \| 1.5 \| \| $h \vee{\sim}h$ \| true \| 1.5 \| Table 3. average violates the value of content for truths. Let's then consider other measures, like the sum function - the sum of the distances of worlds in the range of a proposition from the actual world (Table 4). \| \| \| \| \| --- \| --- \| --- \| \| $\boldsymbol{A}$ \| **truth$(\boldsymbol{A})$ \| sum$(\boldsymbol{A})$** \| \| $h \amp r \amp w$ \| true \| 0 \| \| $h \amp r$ \| true \| 1 \| \| $h \amp r \amp{\sim}$w \| false \| 1 \| \| $h$ \| true \| 4 \| \| $h \amp{\sim}r$ \| false \| 3 \| \| ${\sim}h$ \| false \| 8 \| \| ${\sim}h \amp{\sim}r \amp w$ \| false \| 2 \| \| ${\sim}h \amp{\sim}r$ \| false \| 5 \| \| $h \amp{\sim}r \amp{\sim}w$ \| false \| 3 \| Table 4. The sum function. The sum function is an interesting measure in its own right. While, like average, it is sensitive to the distances of all worlds in a proposition from the actual world, it is not plausible as a measure of distance from the truth, and indeed no one has proposed it as such a measure. What sum does measure is a certain kind of distance-weighted logical weakness. In general the weaker a proposition is, the larger its sum value. But adding words far from the actual world makes the sum value larger than adding worlds closer to it. This guarantees, for example, that of two truths the sum of the logically weaker is always greater than the sum of the stronger. Thus sum might play a role in capturing the value of content for truths. But it also delivers the implausible value of content for falsehoods. If you think that there is anything to the likeness program it is hardly plausible that the falsehood ${\sim}h \amp{\sim}r \amp{\sim}w$ is closer to the truth than its consequence ${\sim}h$. Niiniluoto argues that sum is a good likeness-based candidate for measuring Hilpinen's "information factor". It is obviously much more sensitive than is max to the proposition's informativeness about the location of the truth. Niiniluoto thus proposes, as a measure of distance from the truth, the average of this information factor and Hilpinen's truth factor: min-sum-average. Averaging the more sensitive information factor (sum) and the closeness-to-being-true factor (min) yields some interesting results (see Table 5). \| \| \| \| \| --- \| --- \| --- \| \| $\boldsymbol{A}$ \| **truth$(\boldsymbol{A})$ \| min-sum-average$(\boldsymbol{A})$** \| \| $h \amp r \amp w$ \| true \| 0 \| \| $h \amp r$ \| true \| 0.5 \| \| $h \amp r \amp{\sim}$w \| false \| 1 \| \| $h$ \| true \| 2 \| \| $h \amp{\sim}r$ \| false \| 2 \| \| ${\sim}h$ \| false \| 4.5 \| \| ${\sim}h \amp{\sim}r \amp w$ \| false \| 2 \| \| ${\sim}h \amp{\sim}r$ \| false \| 3.5 \| \| ${\sim}h \amp{\sim}r \amp{\sim}w$ \| false \| 3 \| Table 5. The min-sum-average function. For example, this measure deems $h \amp r \amp w$ more truthlike than $h \amp r$, and the latter more truthlike than $h$. And in general min-sum-average delivers the value of content for truths. For any two truths the min factor is the same (0), and the sum factor increases as content decreases. Furthermore, unlike the symmetric difference measures, min-sum-average doesn't deliver the objectionable value of contents for falsehoods. For example, ${\sim}h \amp{\sim}r \amp{\sim}w$ is deemed further from the truth than ${\sim}h$. But min-sum-average is not quite home free, at least from an intuitive point of view. For example, ${\sim}h \amp{\sim}r \amp{\sim}w$ is deemed closer to the truth than ${\sim}h \amp{\sim}r$. This is because what ${\sim}h \amp{\sim}r \amp{\sim}w$ loses in closeness to the actual world (min) it makes up for by an increase in strength (sum). In deciding how to proceed here we confront a methodological problem. The methodology favored by Tichy is very much bottom-up. For the purposes of deciding between rival accounts it takes the intuitive data very seriously. Popper (along with Popperians like Miller) favor a more top-down approach. They are suspicious of folk intuitions, and sometimes appear to be in the business of constructing a new concept rather than explicating an existing one. They place enormous weight on certain plausible general principles, largely those that fit in with other principles of their overall theory of science: for example, the principle that strength is a virtue and that the stronger of two true theories (and maybe even of two false theories) is the closer to the truth. A third approach, one which lies between these two extremes, is that of reflective equilibrium. This recognizes the claims of both intuitive judgements on low-level cases, and plausible high-level principles, and enjoins us to bring principle and judgement into equilibrium, possibly by tinkering with both. Neither intuitive low-level judgements nor plausible high-level principles are given advance priority. The protagonist in the truthlikeness debate who has argued most consistently for this approach is Niiniluoto. How might reflective equilibrium be employed to help resolve the current dispute? Consider a different space of possibilities, generated by a single magnitude like the number of the planets $(N).$ Suppose that $N$ is in fact 8 and that the further $n$ is from 8, the further the proposition that $N=n$ from the Truth. Consider the three sets of propositions in Table 6. In the left-hand column we have a sequence of false propositions which, intuitively, decrease in truthlikeness while increasing in strength. In the middle column we have a sequence of corresponding true propositions, in each case the strongest true consequence of its false counterpart on the left (Popper's "truth content"). Again members of this sequence steadily increase in strength. Finally on the right we have another column of falsehoods. These are also steadily increasing in strength, and like the left-hand falsehoods, seem (intuitively) to be decreasing in truthlikeness as well. \| \| \| \| \| --- \| --- \| --- \| \| Falsehood (1) \| Strongest True Consequence \| Falsehood (2) \| \| $10 \le N \le 20$ \| $N=8$ or $10 \le N \le 20$ \| $N=9$ or $10 \le N \le 20$ \| \| $11 \le N \le 20$ \| $N=8$ or $11 \le N \le 20$ \| $N=9$ or $11 \le N \le 20$ \| \| ...... \| ...... \| ...... \| \| $19 \le N \le 20$ \| $N=8$ or $19 \le N \le 20$ \| $N=9$ or $19 \le N \le 20$ \| \| $N= 20$ \| $N=8$ or $N= 20$ \| $N=9$ or $N= 20$ \| Table 6. Judgements about the closeness of the true propositions in the center column to the truth may be less intuitively clear than are judgments about their left-hand counterparts. However, it would seem highly incongruous to judge the truths in Table 6 to be steadily increasing in truthlikeness, while the falsehoods both to the left and the right, both marginally different in their overall likeness relations to truth, steadily decrease in truthlikeness. This suggests that that all three are sequences of steadily increasing strength combined with steadily decreasing truthlikeness. And if that's right, it might be enough to overturn Popper's principle that amongst true theories strength and truthlikeness must covary (even while granting that this is not so for falsehoods). If this argument is sound, it removes an objection to averaging distances, but it does not settle the issue in its favor, for there may still be other more plausible counterexamples to averaging that we have not considered. Schurz and Weingartner argue that this extension problem is the main defect of the likeness approach: > > the problem of extending truthlikeness from possible worlds to > propositions is intuitively underdetermined. Even if we are granted an > ordering or a measure of distance on worlds, there are many very > different ways of extending that to propositional distance, and > apparently no objective way to decide between them. (Schurz and > Weingartner 2010, 423) > One way of answering this objection head on is to identify principles that, given a distance function on worlds, constrain the distances between worlds and sets of worlds, principles perhaps powerful enough to identify a unique extension. Apart from the extension problem, two other issues affect the likeness approach. The first is how to apply it beyond simple propositional examples as the ones considered above (Popper's content approach, whatever else its shortcomings, can be applied in principle to theories expressible in any language, no matter how sophisticated). We discuss this in the supplement on Extending the likeness approach to first-order and higher-order frameworks. The second has to do with the fact that assessments of relative likeness are sensitive to how the framework underlying the logical space is defined. This "framework dependence" issue is discussed in the next section. #### 1.4.3 The framework dependence of likeness The single most powerful and influential argument against the whole likeness approach is the charge that it is "language dependent" or "framework dependent" (Miller 1974a, 197 a, 1976, and most recently defended, vigorously as usual, in his 2006). Early formulations of the likeness approach (Tichy 1974, 1976, Niiniluoto 1976) proceeded in terms of syntactic surrogates for their semantic correlates - sentences for propositions, predicates for properties, constituents for partitions of the logical space, and the like. The question naturally arises, then, whether we obtain the same measures if all the syntactic items are translated into an essentially equivalent language - one capable of expressing the same propositions and properties with a different set of primitive predicates. Newton's theory can be formulated with a variety of different primitive concepts, but these formulations are typically taken to be equivalent. If the degree of truthlikeness of Newton's theory were to vary from one such formulation to another, then while such a concept might still might have useful applications, it would hardly help to vindicate realism. Take our simple weather-framework above. This trafficks in three primitives - hot, rainy, and windy. Suppose, however, that we define the following two new weather conditions: > > > minnesotan $ =\_{df}$ hot if and only if > rainy > > > > arizonan $ =\_{df}$ hot if and only if > windy > > > Now it appears as though we can describe the same sets of weather states in the new $h$-$m$-$a$-ese language based on the above conditions. Table 7 shows the translations of four representative theories between the two languages. \| \| \| \| \| --- \| --- \| --- \| \| \| $h$-$r$-$w$-ese \| $h$-$m$-$a$-ese \| \| $T$ \| $h \amp r \amp w$ \| $h \amp m \amp a$ \| \| $A$ \| ${\sim}h \amp r \amp w$ \| ${\sim}h \amp{\sim}m \amp{\sim}a$ \| \| $B$ \| ${\sim}h \amp{\sim}r \amp w$ \| ${\sim}h \amp m \amp{\sim}a$ \| \| $C$ \| ${\sim}h \amp{\sim}r \amp{\sim}w$ \| ${\sim}h \amp m \amp a$ \| Table 7. If $T$ is the truth about the weather then theory $A$, in $h$-$r$-$w$-ese, seems to make just one error concerning the original weather states, while $B$ makes two and $C$ makes three. However, if we express these two theories in $h$-$m$-$a$-ese however, then this is reversed: $A$ appears to make three errors and $B$ still makes two and $C$ makes only one error. But that means the account makes truthlikeness, unlike truth, radically language-relative. There are two live responses to this criticism. But before detailing them, note a dead one: the likeness theorist cannot object that $h$-$m$-$a$ is somehow logically inferior to $h$-$r$-$w$, on the grounds that the primitives of the latter are essentially "biconditional" whereas the primitives of the former are not. This is because there is a perfect symmetry between the two sets of primitives. Starting within $h$-$m$-$a$-ese we can arrive at the original primitives by exactly analogous definitions: > > rainy $ =\_{df}$ hot if and only if > minnesotan > > > windy $ =\_{df}$ hot if and only if > arizonan > Thus if we are going to object to $h$-$m$-$a$-ese it will have to be on other than purely logical grounds. Firstly, then, the likeness theorist could maintain that certain predicates (presumably "hot", "rainy" and "windy") are primitive in some absolute, realist, sense. Such predicates "carve reality at the joints" whereas others (like "minnesotan" and "arizonan") are gerrymandered affairs. With the demise of predicate nominalism as a viable account of properties and relations this approach is not as unattractive as it might have seemed in the middle of the last century. Realism about universals is certainly on the rise. While this version of realism presupposes a sparse theory of properties - that is to say, it is not the case that to every definable predicate there corresponds a genuine universal - such theories have been championed both by those doing traditional a priori metaphysics of properties (e.g. Bealer 1982) as well as those who favor a more empiricist, scientifically informed approach (e.g. Armstrong 1978, Tooley 1977). According to Armstrong, for example, which predicates pick out genuine universals is a matter for developed science. The primitive predicates of our best fundamental physical theory will give us our best guess at what the genuine universals in nature are. They might be predicates like electron or mass, or more likely something even more abstruse and remote from the phenomena - like the primitives of String Theory. One apparently cogent objection to this realist solution is that it would render the task of empirically estimating degree of truthlikeness completely hopeless. If we know a priori which primitives should be used in the computation of distances between theories it will be difficult to estimate truthlikeness, but not impossible. For example, we might compute the distance of a theory from the various possibilities for the truth, and then make a weighted average, weighting each possible true theory by its probability on the evidence. That would be the credence-mean estimate of truthlikeness (see SS2). However, if we don't even know which features should count towards the computation of similarities and distances then it appears that we cannot get off first base. To see this consider our simple weather frameworks. Suppose that all I learn is that it is rainy. Do I thereby have some grounds for thinking $A$ is closer to the truth than $B$? I would if I also knew that $h$-$r$-$w$-ese is the language for calculating distances. For then, whatever the truth is, $A$ makes one fewer mistake than $B$ makes. $A$ gets it right on the rain factor, while $B$ doesn't, and they must score the same on the other two factors whatever the truth of the matter. But if we switch to $h$-$m$-$a$-ese then $A$'s epistemic superiority is no longer guaranteed. If, for example, $T$ is the truth then $B$ will be closer to the truth than $A$. That's because in the $h$-$m$-$a$ framework raininess as such doesn't count in favor or against the truthlikeness of a proposition. This objection would fail if there were empirical indicators not just of which atomic states obtain, but also of which are the genuine ones, the ones that really carve reality at the joints. Obviously the framework would have to contain more than just $h, m$ and $a$. It would have to contain resources for describing the states that indicate whether these were genuine universals. Maybe whether they enter into genuine causal relations will be crucial, for example. Once we can distribute probabilities over the candidates for the real universals, then we can use those probabilities to weight the various possible distances which a hypothesis might be from any given theory. The second live response is both more modest and more radical. It is more modest in that it is not hostage to the objective priority of a particular conceptual scheme, whether that priority is accessed a priori or a posteriori. It is more radical in that it denies a premise of the invariance argument that at first blush is apparently obvious. It denies the equivalence of the two conceptual schemes. It denies that $h \amp r \amp w$, for example, expresses the very same proposition as $h \amp m \amp a$ expresses. If we deny translatability then we can grant the invariance principle, and grant the judgements of distance in both cases, but remain untroubled. There is no contradiction (Tichy 1978). At first blush this response seems somewhat desperate. Haven't the respective conditions been defined in such a way that they are simple equivalents by fiat? That would, of course, be the case if $m$ and $a$ had been introduced as defined terms into $h$-$r$-$w$. But if that were the intention then the likeness theorist could retort that the calculation of distances should proceed in terms of the primitives, not the introduced terms. However, that is not the only way the argument can be read. We are asked to contemplate two partially overlapping sequences of conditions, and two spaces of possibilities generated by those two sequences. We can thus think of each possibility as a point in a simple three dimensional space. These points are ordered triples of 0s and 1s, the $n$th entry being 0 if the $n$th condition is satisfied and 1 if it isn't. Thinking of possibilities in this way, we already have rudimentary geometrical features generated simply by the selection of generating conditions. Points are adjacent if they differ on only one dimension. A path is a sequence of adjacent points. A point $q$ is between two points $p$ and $r$ if $q$ lies on a shortest path from $p$ to $r$. A region of possibility space is convex if it is closed under the betweenness relation - anything between two points in the region is also in the region (Oddie 1987, Goldstick and O'Neill 1988). Evidently we have two spaces of possibilities, S1 and S2, and the question now arises whether a sentence interpreted over one of these spaces expresses the very same thing as any sentence interpreted over the other. Does $h \amp r \amp w$ express the same thing as $h \amp m \amp a$? $h \amp r \amp w$ expresses (the singleton of) $u\_1$ (which is the entity $\langle 1,1,1\rangle$ in S1 or $\langle 1,1,1\rangle\_{S1})$ and $h \amp m \amp a$ expresses $v\_1$ (the entity $\langle 1,1,1\rangle\_{S2}). {\sim}h \amp r \amp w$ expresses $u\_2 (\langle 0,1,1\rangle\_{S1})$, a point adjacent to that expressed by $h \amp r \amp w$. However ${\sim}h \amp{\sim}m \amp{\sim}a$ expresses $v\_8 (\langle 0,0,0\rangle\_{S2})$, which is not adjacent to $v\_1 (\langle 1,1,1\rangle\_{S2})$. So now we can construct a simple proof that the two sentences do not express the same thing. * $u\_1$ is adjacent to $u\_2$. * $v\_1$ is not adjacent to $v\_8$. * Therefore, either $u\_1$ is not identical to $v\_1$, or $u\_2$ is not identical $v\_8$. * Therefore, either $h \amp r \amp w$ and $h \amp m \amp a$ do not express the same proposition, or ${\sim}h \amp r \amp w$ and ${\sim}h \amp{\sim}m \amp{\sim}a$ do not express the same proposition. Thus at least one of the two required intertranslatability claims fails, and $h$-$r$-$w$-ese is not intertranslatable with $h$-$m$-$a$-ese. The important point here is that a space of possibilities already comes with a structure and the points in such a space cannot be individuated without reference to rest of the space and its structure. The identity of a possibility is bound up with its geometrical relations to other possibilities. Different relations, different possibilities. This kind of response has also been endorsed in the very different truth-maker proposal put forward in Fine (2021, 2022). This kind of rebuttal to the Miller argument would have radical implications for the comparability of actual theories that appear to be constructed from quite different sets of primitives. Classical mechanics can be formulated using mass and position as basic, or it can be formulated using mass and momentum. The classical concepts of velocity and of mass are different from their relativistic counterparts, even if they were "intertranslatable" in the way that the concepts of $h$-$r$-$w$-ese are intertranslatable with $h$-$m$-$a$-ese. This idea meshes well with recent work on conceptual spaces in Gardenfors (2000). Gardenfors is concerned both with the semantics and the nature of genuine properties, and his bold and simple hypothesis is that properties carve out convex regions of an $n$-dimensional quality space. He supports this hypothesis with an impressive array of logical, linguistic and empirical data. (Looking back at our little spaces above it is not hard to see that the convex regions are those that correspond to the generating (or atomic) conditions and conjunctions of those. See Burger and Heidema 1994.) While Gardenfors is dealing with properties, it is not hard to see that similar considerations apply to propositions, since propositions can be regarded as 0-ary properties. Ultimately, however, this response may seem less than entirely satisfactory by itself. If the choice of a conceptual space is merely a matter of taste then we may be forced to embrace a radical kind of incommensurability. Those who talk $h$-$r$-$w$-ese and conjecture ${\sim}h \amp r \amp w$ on the basis of the available evidence will be close to the truth. Those who talk $h$-$m$-$a$-ese while exposed to the "same" circumstances would presumably conjecture ${\sim}h \amp{\sim}m \amp{\sim}a$ on the basis of the "same" evidence (or the corresponding evidence that they gather). If in fact $h \amp r \amp w$ is the truth (in $h$-$r$-$w$-ese) then the $h$-$r$-$w$ weather researchers will be close to the truth. But the $h$-$m$-$a$ researchers will be very far from the truth. This may not be an explicit contradiction, but it should be worrying all the same. Realists started out with the ambition of defending a concept of truthlikeness which would enable them to embrace both fallibilism and optimism. But what the likeness theorists seem to have ended up with here is something that suggests a rather unpalatable incommensurability of competing conceptual frameworks. To avoid this, the realist will need to affirm that some conceptual frameworks really are better than others. Some really do "carve reality at the joints" and others don't. But is that something the realist should be reluctant to affirm? ## 2. The Epistemological Problem The quest to nail down a viable concept of truthlikeness is motivated, at least in part, by fallibilism (SS1.1). It is certainly true that a viable notion of distance from the truth renders progress in an inquiry through a succession of false theories at least possible. It is also true that if there is no such viable notion, then truth can be retained as the goal of inquiry only at the cost of making partial progress towards it virtually impossible. But does the mere possibility of making progress towards the truth improve our epistemic lot? Some have argued that it doesn't (see for example Laudan 1977, Cohen 1980, Newton-Smith 1981). One common argument can be recast in the form of a simple dilemma. Either we can ascertain the truth, or we can't. If we can ascertain the truth then we do not need a concept of truthlikeness - it is an entirely useless addition to our intellectual repertoire. But if we cannot ascertain the truth, then we cannot ascertain the degree of truthlikeness of our theories either. So again, the concept is useless for all practical purposes. (See the entry on scientific progress, especially SS2.4.) Consider the second horn of this dilemma. Is it true that if we can't know what the (whole) truth of some matter is, we also cannot ascertain whether or not we are making progress towards it? Suppose you are interested in the truth about the weather tomorrow. Suppose you learn (from a highly reliable source) that it will be hot. Even though you don't know the whole truth about the weather tomorrow, you do know that you have added a truth to your existing corpus of weather beliefs. One does not need to be able to ascertain the whole truth to ascertain some less-encompassing truths. And it seems to follow that you can also know you have made at least some progress towards the whole weather truth. This rebuttal is too swift. It presupposes that the addition of a new truth $A$ to an existing corpus $K$ guarantees that your revised belief $K$\$A$ constitutes progress towards the truth. But whether or not $K$\$A$ is closer to the truth than $K$ depends not only on a theory of truthlikeness but also on a theory of belief revision. (See also the entry on the logic of belief revision.) Let's consider a simple case. Suppose $A$ is some newly discovered truth, and that $A$ is compatible with $K$. Assume that belief revision in such cases is simply a matter of so-called expansion - i.e., conjoining $A$ to $K$. Consider the case in which $K$ also happens to be true. Then any account of truthlikeness that endorses the value of content for truths (e.g. Niiniluoto's min-sum-average) guarantees that $K$\$A$ is closer to the truth than $K$. That's a welcome result, but it has rather limited application. Typically one doesn't know that $K$ is true: so even if one knows that $A$ is true, one cannot use this fact to celebrate progress. The situation is more dire when it comes to falsehoods. If $K$ is in fact false then, without the disastrous principle of the value of content for falsehoods, there is certainly no guarantee that $K$\$A$ will constitute a step toward the truth. (And even if one endorsed the disastrous principle one would hardly be better off. For then the addition of any proposition, whether true or false, would constitute an improvement on a false theory.) Consider again the number of the planets, $N$. Suppose that the truth is N$=8$, and that your existing corpus $K$ is ($N=7 \vee N=100)$. Suppose you somehow acquire the truth $A$: $N\gt 7$. Then $K$\$A$ is $N=100$, which (on average, min-max-average* and min-sum-average) is further from the truth than $K$. So revising a false theory by adding truths by no means guarantees progress towards the truth. For theories that reject the value of content for truths (e.g., the average proposal) the situation is worse still. Even if $K$ happens to be true, there is no guarantee that expanding $K$ with truths will constitute progress. Of course, there will be certain general conditions under which the value of content for truths holds. For example, on the average proposal, the expansion of a true $K$ by an atomic truth (or, more generally, by a convex truth) will guarantee progress toward the truth. So under very special conditions, one can know that the acquisition of a truth will enhance the overall truthlikeness of one's theories, but these conditions are exceptionally narrow and provide at best a very weak defense against the dilemma. (See Niiniluoto 2011. For rather more optimistic views of the relation between truthlikeness and belief revision see Kuipers 2000, Lavalette, Renardel & Zwart 2011, Cevolani, Crupi and Festa 2011, and Cevolani, Festa and Kuipers 2013.) A different tack is to deny that a concept is useless if there is no effective empirical decision procedure for ascertaining whether it applies. For even if we cannot know for sure what the value of a certain unobservable magnitude is, we might well have better or worse estimates of the value of the magnitude on the evidence. And that may be all we need for the concept to be of practical value. Consider, for example, the propensity of a certain coin-tossing set-up to produce heads - a magnitude which, for the sake of the example, we assume to be not directly observable. Any non-extreme value of this magnitude is compatible with any number of heads in a sequence of $n$ tosses. So we can never know with certainty what the actual propensity is, no matter how many tosses we observe. But we can certainly make rational estimates of the propensity on the basis of the accumulating evidence. Suppose one's initial state of ignorance of the propensity is represented by an even distribution of credences over the space of possibilities for the propensity (i.e., the unit interval). Using Bayes theorem and the Principal Principle, after a fairly small number of tosses we can become quite confident that the propensity lies in a small interval around the observed relative frequency. Our best estimate of the value of the magnitude is its expected value on the evidence. Similarly, suppose we don't and perhaps cannot know which constituent is in fact true. But suppose that we do have a good measure of distance between constituents (or the elements of some salient partition of the space) $C\_i$ and we have selected the right extension function. So we have a measure $TL(A\mid C\_i)$of the truthlikeness of a proposition $A$ given that constituent $C\_i$ is true. Provided we also have a measure of epistemic probability $P$ (where $P(C\_i\mid e)$ is the degree of rational credence in $C\_i$ given evidence $e)$ we also have a measure of the expected degree of truthlikeness of $A$ on the evidence (call this $\mathbf{E}TL(A\mid e))$ which we can identify with the best epistemic estimate of truthlikeness. (Niiniluoto, who first explored this concept in his 1977, calls the epistemic estimate of degree of truthlikeness on the evidence, or expected degree of truthlikeness, verisimilitude. Since verisimilitude is typically taken to be a synonym for truthlikeness, we will not follow him in this, and will stick instead with expected truthlikeness for the epistemic notion. See also Maher (1993).) \[ \mathbf{E}TL(A\mid e) = \sum\_i TL(A\mid C\_i)\times P(C\_i \mid e). \] Clearly, the expected degree of truthlikeness of a proposition $is$ epistemically accessible, and it can serve as our best empirical estimate of the objective degree of truthlikeness. Progress occurs in an inquiry when actual truthlikeness increases. And apparent progress occurs when the expected degree of truthlikeness increases. (See the entry on scientific progress, SS2.5.) This notion of expected truthlikeness is comparable to, but sharply different from, that of epistemic probability: the supplement on Expected Truthlikeness discusses some instructive differences between the two concepts. With this proposal, Niiniluoto also made a connection with the application of decision theory to epistemology. Decision theory is an account of what it is rational to do in light of one's beliefs and desires. One's goal, it is assumed, is to maximize utility. But given that one does not have perfect information about the state of the world, one cannot know for sure how to accomplish that. Given uncertainty, the rule to maximize utility or value cannot be applied in normal circumstances. So under conditions of uncertainty, what it is rational to do, according to the theory, is to maximize subjective expected utility. Starting with Hempel's classic 1960 essay, epistemologists conjectured that decision-theoretic tools might be applied to the problem of theory acceptance - which hypothesis it is rational to accept on the basis of the total available evidence to hand. But, as Hempel argued, the values or utilities involved in a decision to accept a hypothesis cannot be simply regular practical values. These are typically thought to be generated by one's desires for various states of affairs to obtain in the world. But the fact that one would very much like a certain favored hypothesis to be true does not increase the cognitive value of accepting that hypothesis. > > This much is clear: the utilities should reflect the value or disvalue > which the different outcomes have from the point of view of pure > scientific research rather than the practical advantages or > disadvantages that might result from the application of an accepted > hypothesis, according as the latter is true or false. Let me refer to > the kind of utilities thus vaguely characterized as purely scientific, > or epistemic, utilities. (Hempel 1960, 465) > If we had a decent theory of epistemic utility (also known as cognitive utility or cognitive value), perhaps what hypotheses one ought to accept, or what experiments one ought to perform, or how one ought to revise one's corpus of belief in the light of new information, could be determined by the rule: maximize expected epistemic utility (or maximize expected cognitive value). Thus decision-theoretic epistemology was born. Hempel went on to ask what epistemic utilities are implied in the standard conception of scientific inquiry - "...construing the proverbial 'pursuit of truth' in science as aimed at the establishment of a maximal system of true statements..." Hempel (1960), p 465) Already we have here the germ of the idea central to the truthlikeness program: that the goal of an inquiry is to end up accepting (or "establishing") the theory that yields the whole truth of some matter. It is interesting that, around the same time that Popper was trying to articulate a content-based account of truthlikeness, Hempel was attempting to characterize partial fulfillment of the goal (that is, of maximally contentful truth) in terms of some combination of truth and content. These decision-theoretic considerations lead naturally to a brief consideration of the axiological problem of truthlikeness. ## 3. The Axiological Problem Our interest in the concept of truthlikeness seems grounded in the value of highly truthlike theories. And that, in turn, seems grounded in the value of truth. Let's start with the putative value of truth. Truth is not, of course, a good-making property of the objects of belief states. The proposition $h \amp r \amp w$ is not a better proposition when it happens to be hot, rainy and windy than when it happens to be cold, dry and still. Rather, the cognitive state of believing $h \amp r \amp w$ is often deemed to be a good state to be in if the proposition is true, not good if it is false. So the state of believing truly is better than the state of believing falsely. At least some, perhaps most, of the value of believing truly can be accounted for instrumentally. Desires mesh with beliefs to produce actions that will best achieve what is desired. True beliefs will generally do a better job of this than false beliefs. If you are thirsty and you believe the glass in front of you contains safe drinking water rather than lethal poison then you will be motivated to drink. And if you do drink, you will be better off if the belief you acted on was true (you quench your thirst) rather than false (you end up dead). We can do more with decision theory utilizing purely practical or non-cognitive values. For example, there is a well-known, decision-theoretic, value-of-learning theorem, the Ramsey-Good theorem, which partially vindicates the value of gathering new information, and it does so in terms of "practical" values, without assuming any purely cognitive values. (Good 1967.) Suppose you have a choice to make, and you can either choose now, or choose after doing some experiment. Suppose further that you are rational (you always choose by expected value) and that the experiment is cost-free. It follows that performing the experiment and then choosing always has at least as much expected value as choosing without further ado. Further, doing the experiment has higher expected value if one possible outcome of the experiment would alter the relative values of some your options. So, you should do the experiment just in case the outcome of the experiment could make a difference to what you choose to do. Of course, the expected gain of doing the experiment has to be worth the expected cost. Not all information is worth pursuing when you have limited time and resources. David Miller notes the following serious limitation of the Ramsey-Good value-of-learning theorem: > > The argument as it stands simply doesn't impinge on the question > of why evidence is collected in those innumerable cases in theoretical > science in which no decisions will be, or anyway are envisaged to be, > affected. (Miller 1994, 141). > There are some things that we think are worth knowing about, the knowledge of which would not change the expected value of any action that one might be contemplating performing. We spent billions of dollars conducting an elaborate experiment to determine whether the Higgs boson exists, and the results were promising enough that Higgs was awarded the Nobel Prize for his prediction. The discovery may also yield practical benefits, but it is the cognitive change it induces that makes it worthwhile. It is valuable simply for what it did to our credal state - not just our credence in the existence of the Higgs, but our overall credal state. We may of course be wrong about the Higgs - the results might be misleading - but from our new epistemic viewpoint it certainly appears to us that we have made cognitive gains. We have a little bit more evidence for the truth, or at least the truthlikeness, of the standard model. What we need, it seems, is an account of pure cognitive value, one that embodies the idea that getting at the truth is itself valuable whatever its practical benefits. As noted this possibility was first explicitly raised by Hempel, and developed in various difference ways in the 1960s and 1970s by Levi (1967) and Hilpinen (1968), amongst others. (For recent developments see the entry epistemic utility arguments for probabilism, SS6.) We could take the cognitive value of believing a single proposition $A$ to be positive when $A$ is true and negative when $A$ is false. But acknowledging that belief comes in degrees, the idea might be that the stronger one's belief in a truth (or of one's disbelief in a falsehood), the greater the cognitive value. Let $V$ be the characteristic function of answers, both complete and partial, to the question $\{C\_1, C\_2 , \ldots ,C\_n , \ldots \}$. That is to say, where $A$ is equivalent to some disjunction of complete answers: * $V\_i (A) = 1$ if $A$ is entailed by the complete answer $C\_i$ (i.e $A$ is true according to $C\_i)$; * $V\_i (A) = 0$ if the negation of $A$ is entailed by the complete answer $C\_i$ (i.e., $A$ is false according to $C\_i)$. Then the simple view is that the cognitive value of believing $A$ to degree $P(A)$ (where $C\_i$ is the complete true answer) is greater the closer $P(A)$ is to the actual value of $V\_i (A)$- that is, the smaller $\|V\_i (A)-P(A)\|$ is. Variants of this idea have been endorsed, for example by Horwich (1982, 127-9), and Goldman (1999), who calls this "veristic value". $\|V\_i (A)-P(A)\|$ is a measure of how far $P(A)$ is from $V\_i (A)$, and $-\|V\_i (A)-P(A)\|$ of how close it is. So here is the simplest linear realization of this desideratum: \[ Cv^1 \_i (A, P) = - \|V\_i (A)-P(A)\|. \] But there are of course many other measures satisfying the basic idea. For example we have the following quadratic measure: \[ Cv^2 \_i (A, P) = -((V\_i (A)- P(A))^2. \] Both $Cv^1$ and $Cv^2$ reach a maximum when $P(A)$ is maximal and $A$ is true, or $P(A)$ is minimal and $A$ is false; and a minimum when $P(A)$ is maximal and $A$ is false, or $P(A)$ is minimal and $A$ is true. These are measures of local value - the value of investing a certain degree of belief in a single answer $A$ to the question $Q$. But we can agglomerate these local values into the value of a total credal state $P$. This involves a substantive assumption: that the value of a credal state is some additive function of the values of the individual beliefs states it underwrites. A credal state, relative to inquiry $Q$, is characterized by its distribution of credences over the elements of the partition $Q$. An opinionated credal state is one that assigns credence 1 to just one of the complete answers. The best credal state to be in, relative to the inquiry $Q$, is the opinionated state that assigns credence 1 to the complete correct answer. This will also assign credence 1 to every true answer, and 0 to every false answer, whether partial or complete. In other words, if the correct answer to $Q$ is $C\_i$ then the best credal state to be in is the one identical to $V\_i$: for each partial or complete answer $A$ to $Q, P(A) = V\_i (A)$. This credal state is the state of believing the truth, the whole truth and nothing but the truth about $Q$. Other total credal states (whether opinionated or not) should turn out to be less good than this perfectly accurate credal state. But there will, of course, have to be more constraints on the accuracy and value of total cognitive states. Assuming additivity, the value of the credal state $P$ can be taken to be the (weighted) sum of the cognitive values of local belief states. > > $CV\_i(P) = \sum\_A \lambda\_A Cv\_i(A,P)$ (where $A$ ranges over all > the answers, both complete and incomplete, to question $Q)$, and the > $\lambda$-terms assign a fixed (non-contingent) weight to the > contribution of each answer $A$ to overall accuracy. > Plugging in either of the above local cognitive value measures, total cognitive value is maximal for an assignment of maximal probability to the true complete answer, and falls off as confidence in the correct answer falls away. Since there are many different measures that reflect the basic idea of the value of true belief how are we to decide between them? We have here a familiar problem of underdetermination. Joyce (1998) lays down a number of desiderata, most of them very plausible, for a measure of what he calls the accuracy of a cognitive state. These desiderata are satisfied by the $Cv^2$ but not by $Cv^1$. In fact all the members of a family of closely related measures, of which $Cv^2$ is a member, satisfy Joyce's desiderata: \[ CV\_i(P) = \sum\_A -\lambda\_A [(V\_i (A) - P(A))^2]. \] Giving equal weight to all propositions $(\lambda\_A =$ some non-zero constant $c$ for all $A)$, this is equivalent to the Brier measure (see the entry on epistemic utility arguments for probabilism, SS6). Given the $\lambda$-weightings can vary, there is considerable flexibility in the family of admissible measures. Our main interest here is whether any of these so-called scoring rules (rules which score the overall accuracy of a credal state) might constitute an acceptable measure of the cognitive value of a credal state. Absent specific refinements to the $\lambda$-weightings, the quadratic measures seem unsatisfactory as a solution either to the problem of accuracy or to the problem of cognitive value. Suppose you are inquiring into the number of the planets and you end up fully believing that the number of the planets is 9. Given that the correct answer is 8, your credal state is not perfect. But it is pretty good, and it is surely a much better credal state to be in than the opinionated state that sets probability 1 on the number of planets being 9 billion. It seems rather natural to hold that the cognitive value of an opinionated credal state is sensitive to the degree of truthlikeness of the complete answer that it fixes on, not just to its truth value. But this is not endorsed by quadratic measures. Joyce's desiderata build in the thesis that accuracy is insensitive to the distances of various different complete answers from the complete true answer. The crucial principle is Extensionality. > > Our next constraint stipulates that the "facts" which a > person's partial beliefs must "fit" are exhausted by > the truth-values of the propositions believed, and that the only > aspect of her opinions that matter is their strengths. (Joyce 1998, > 591) > We have already seen that this seems wrong for opinionated states that confine all probability to a false hypothesis. Being convinced that the number of planets in the solar system is 9 is better than being convinced that it is 9 billion. But the same idea holds for truths. Suppose you believe truly that the number of the planets is either 7, 8, 9 or 9 billion. This can be true in four different ways. If the number of the planets is 8 then, intuitively, your belief is a little bit closer to the truth than if it is 9 billion. (This judgment is endorsed by both the average and the min-sum-average proposals.) So even in the case of a true answer to some query, the value of believing that truth can depend not just on its truth and the strength of the belief, but on where the truth lies. If this is right Extensionality is misguided. Joyce considers a variant of this objection: namely, that given Extensionality Kepler's beliefs about planetary motion would be judged to be no more accurate than Copernicus's. His response is that there will always be propositions which distinguish between the accuracy of these falsehoods: > > I am happy to admit that Kepler held more accurate beliefs than > Copernicus did, but I think the sense in which they were more accurate > is best captured by an extensional notion. While Extensionality rates > Kepler and Copernicus as equally inaccurate when their false beliefs > about the earth's orbit are considered apart from their effects > on other beliefs, the advantage of Kepler's belief has to do > with the other opinions it supports. An agent who strongly believes > that the earth's orbit is elliptical will also strongly believe > many more truths than a person who believes that it is circular (e.g., > that the average distance from the earth to the sun is different in > different seasons). This means that the overall effect of > Kepler's inaccurate belief was to improve the extensional > accuracy of his system of beliefs as a whole. Indeed, this is why his > theory won the day. I suspect that most intuitions about falsehoods > being "close to the truth" can be explained in this way, > and that they therefore pose no real threat to Extensionality. (Joyce > 1998, 592) > Unfortunately, this contention - that considerations of accuracy over the whole set of answers which the two theories give will sort them into the right ranking - isn't correct (Oddie 2019). Wallace and Greaves (2005) assert that the weighted quadratic measures "can take account of the value of verisimilitude." They suggest this can be done "by a judicious choice of the coefficients" (i.e., the $\lambda$-values). They go on to say: "... we simply assign high $\lambda\_A$ when $A$ is a set of 'close' states." (Wallace and Greaves 2005, 628). 'Close' here presumably means 'close to the actual world'. But whether an answer $A$ contains complete answers that are close to the actual world - that is, whether $A$ is truthlike - is clearly a world-dependent matter. The coefficients $\lambda\_A$ were not intended by the authors to be world-dependent. (But whether or not the co-efficients of world-dependent or world-independent the quadratic measures cannot capture truthlikeness. See Oddie 2019.) One defect of the quadratic class of measures, combined with world-independent weights, corresponds to a problem familiar from the investigation of attempts to capture truthlikeness simply in terms of classes of true and false consequences, or in terms of truth value and content alone. Any two complete false answers yield precisely the same number of true consequences and the same number of false consequences. The local quadratic measure accords the same value to each true answer and the same disvalue to each false answer. So if, for example, all answers are given the same $\lambda$-weighting in the global quadratic measure, any two opinionated false answers will be accorded the same degree of accuracy by the corresponding global measure. It may be that there are ways of tinkering with the $\lambda$-terms to avoid the objection, but on the face of it the notions of local cognitive value embodied in both the linear and quadratic rules seem to be deficient because they omit considerations of truthlikeness even while valuing truth. It seems that any adequate measure of the cognitive value of investing a certain degree of belief in $A$ should take into account not just whether $A$ is true or false, but here the truth lies in relation to the worlds in $A$. Whatever our account of cognitive value, each credal state $P$ assigns an expected cognitive value to every credal state $Q$ (including $P$ itself of course). \[ \mathbf{E}CV\_P(Q) = \sum\_i P(C\_i) CV\_i(Q). \] Suppose we accept the injunction that one ought to maximize expected value as calculated from the perspective of one's current credal state. Let us say that a credal state $P$ is self-defeating if to maximize expected value from the perspective of $P$ itself one would have to adopt some distinct credal state $Q$, without the benefit of any new information: > > $P$ is self-defeating = for some $Q$ distinct from $P$, > $\mathbf{E}$CV$\_P$$(Q) \gt$ > $\mathbf{E}$CV$\_P$$(P)$. > The requirement of propriety requires that no credal state be self-defeating. No credal state demands that you shift to another credal state without new information. One feature of the quadratic family of proper scoring rules is that they guarantee propriety, and propriety is an extremely attractive feature of cognitive value. From propriety alone, one can construct arguments for the various elements of probabilism. For example, propriety effectively guarantees that the credence function must obey the standard axioms governing additivity (Leitgeb and Pettigrew 2010); propriety guarantees a version of the value-of-learning theorem in purely cognitive terms (Oddie, 1997); propriety provides an argument that conditionalization is the only method of updating compatible with the maximization of expected cognitive value (Oddie 1997, Greaves and Wallace 2005, and Leitgeb and Pettigrew 2010). Not only does propriety deliver these goods for probabilism, but any account of cognitive value that doesn't obey propriety entails that self-defeating probability distributions should be eschewed a priori. (And that might appear to violate a fundamental commitment of empiricism.) There is however a powerful argument that any measure of cognitive value that satisfies propriety cannot deliver a fundamental desideratum on an adequate theory of truthlikeness (or accuracy): the principle of proximity. Every serious account of truthlikeness satisfies proximity. But proximity and propriety turn out to be incompatible, given only very weak assumptions (see Oddie 2019 and, for an extended critique, also Schoenfield 2022). If this is right then the best account of genuine accuracy cannot provide a vindication of pure probabilism. ## 4. Conclusion We are all fallibilists now, but we are not all skeptics, or antirealists or nihilists. Most of us think inquiries can and do progress even when they fall short of their goal of locating the truth of the matter. We think that an inquiry can progress by moving from one falsehood to another falsehood, or from one imperfect credal state to another. To reconcile epistemic optimism with realism in the teeth of the dismal induction we need a viable concept of truthlikeness, a viable account of the empirical indicators of truthlikeness, and a viable account of the role of truthlikeness in cognitive value. And all three accounts must fit together appropriately. There are a number of approaches to the logical problem of truthlikeness but, unfortunately, there is as yet little consensus on what constitutes the best or most promising approach, and prospects for combining the best features of each approach do not at this stage seem bright. In fact, recent work on whether the three main approaches to the logical problem of truthlikeness presented above are compatible seems to point to a negative answer (see the supplement on The compatibility of the approaches). There is, however, much work to be done on both the epistemological and the axiological aspects of truthlikeness, and it may well be that new constraints will emerge from those investigations that will help facilitate a fully adequate solution to the logical problem as well.
vico	## 1. Vico's Life and Influence Giovanni Battista Vico was born in Naples, Italy, June 23 1668, to a bookseller and daughter of a carriage maker. He received his formal education at local grammar schools, from various Jesuit tutors, and at the University of Naples from which he graduated in 1694 as Doctor of Civil and Canon Law. As he reports in his autobiography (Vita di Giambattista Vico), however, Vico considered himself as "teacher of himself,"[1] a habit he developed under the direction of his father, after a fall at the age of 7 caused a three year absence from school. Vico reports that "as a result of this mischance he grew up with a melancholy and irritable temperament such as belongs to men of ingenuity and depth" (Vita, 111). Vico left Naples in 1686 for Vatolla where, with occasional returns to his home city, he remained for the next nine years as tutor to the sons of Domenico Rocca. Vico returned to Naples in 1695 and four years later married Teresa Caterina Destito with whom he had eight children. Of his surviving children (three of them died), his younger son Gennaro was a favorite and went on to an academic career; his daughter Lusia achieved success as a singer and minor poet, and Ignazio is known to have been a source of disappointment to him. Of the others (Angela Teresa and Filippo) little is known. Vico was never rich, though he made a living, and the legend of his poverty derives from Villarosa's embellishment of the autobiography. He did, however, suffer bouts of ill-health, and failed in his life-long ambition of succeeding to the chair of Jurisprudence at the University of Naples, having to settle instead for a lower and poorly paid professorship in Rhetoric. He retained this position until 1741 at which time he was succeeded by his son Gennaro. Vico died in Naples on January 22-23, 1744, aged 75. In his own time Vico's work was largely neglected and generally misunderstood-he describes himself living as a "stranger" and "quite unknown" in his native city (Vita, 134)-and it was not until the nineteenth century that his thought began to make a significant impression on the philosophical world. However, Vico's thought never suffered from complete obscurity and its influence can be discerned in various traditions of European thought from the mid-eighteenth century onwards.[2] In Italy, Vico's impact on aesthetic and literary criticism is evident in the writings of Francesco De Sanctis and Benedetto Croce, and in jurisprudence, economics, and political theory, his influence can be traced from Antonio Genovesi (one of Vico's own pupils), Ferdinando Galiani, and Gaetano Filangieri. In Germany, Vico's ideas were known to Johann-George Hamman and, via his disciple J.G. von Herder, to Johann Wolfgang von Goethe and Friedrich Heinrich Jacobi. Vico's ideas were sufficiently familiar to Friedrich August Wolf to inspire his article "G.B. Vico on Homer." In France, Vico's thought was likely known to Charles de Secondat, baron de Montesquieu, and Jean-Jacques Rousseau, and some have seen his influence in the writings of Denis Diderot, Etienne Bonnot, abbe de Condillac, and Joseph Marie, comte de Maistre. In Great Britain, although Vichian themes are intimated in the philosophical writings of the Empiricists and thinkers of the Scottish Enlightenment, there is no direct evidence that they knew of his writings. The earliest known disseminator of Vico's views in the English-speaking world is Samuel Taylor Coleridge, who was responsible for much of the interest in Vico in the second half of the nineteenth century. Vico's ideas reached a wider audience with a German translation of The New Science by W.E. Weber which appeared in 1822, and, more significantly, through a French version by Jules Michelet in 1824, which was reissued in 1835. Michelet's translation was widely read and was responsible for a new appreciation of Vico's work in France. Subsequently, Vico's views impacted the work of Wilhelm Dilthey, Karl Marx, R.G. Collingwood, and James Joyce, who used The New Science to structure Finnegans Wake. Twentieth century scholarship has established illuminating comparisons with the tradition of Hegelian idealism, and taken up the relationship between Vico's thought and that of philosophers in the western tradition and beyond, including Plato, Aristotle, Ibn Khaldun, Thomas Hobbes, Benedict de Spinoza, David Hume, Immanuel Kant, and Friedrich Nietzsche. Comparisons and connections have also been drawn between Vichean themes and the work of various modern and contemporary thinkers, inter alia W.B. Yeats, Friedrich Froebel, Max Horkheimer, Walter Benjamin, Martin Heidegger, Hans-Georg Gadamer, Jurgen Habermas, Paul Ricoeur, Jean-Francois Lyotard, and Alisdair MacIntyre. As a review of recent and current literature demonstrates, an appreciation of Vico's thought has spread far beyond philosophy, and his ideas have been taken up by scholars within a range of contemporary disciplines, including anthropology, cultural theory, education, hermeneutics, history, literary criticism, psychology, and sociology. Thus despite obscure beginnings, Vico is now widely regarded as a highly original thinker who anticipated central currents in later philosophy and the human sciences. ## 2. Vico's Early Works Vico's earliest publications were in poetry rather than philosophy-the Lucretian poem "Affetti di un disperato" ("The Feelings of One in Despair") (composed in 1692) and "Canzone in morte di Antonio Carafa" ("Ode on the Death of Antonio Carafa") both published in 1693. After his appointment to Professor of Rhetoric in 1699, Vico began to address philosophical themes in the first of six Orazioni Inaugurali (Inaugural Orations). In 1707 Vico gave a seventh oration-intended as an augmentation of the philosophy of Bacon that he subsequently revised and published in 1709 under the title De nostri temporis studiorum ratione (On the Study Methods of Our Time). Two years later his projected but uncompleted statement on metaphysics appeared, De antiquissima Italorum sapientia ex linguae latinae originibus eruenda libri tres (On the Most Ancient Wisdom of the Italians Unearthed from the Origins of the Latin Language). The years that followed saw the publication of various works, including two replies to critical reviews of De antiquissima (1711 and 1712), a work of royal historiography-De rebus gestis Antonii Caraphaei libri quattuor (The Life of Antonio Caraffa) (1716)-Il diritto universale (Universal Right) (1720-22), and in 1725 and 1731 the two parts of his autobiography which together compose the Vita di Giambattista Vico scritta da se medesimo (Life of Giambattista Vico Written by Himself). Over the course of his professional career, Vico also composed and delivered lectures on rhetoric to students preparing to enter into the study of jurisprudence. These survive as student transcriptions or copies of the original lectures, and are collected under the title Institutiones Oratoriae (1711-1741) (translated as The Art of Rhetoric) in which Vico's develops his views on rhetoric or eloquence, "the faculty of speaking appropriate to the purpose of persuading" through which the orator aims to "bend the spirit by his speech."[3] Although these early writings are increasingly studied as significant works in their right, they have generally been regarded as important for tracing the development of various themes and ideas which Vico came to refine and express fully in his major and influential work, The New Science. In De nostri temporis studiorum ratione, for example, Vico takes up the theme of modernity and raises the question of "which study method is finer or better, ours or the Ancients?" and illustrates "by examples the advantages and drawbacks of the respective methods."[4] Vico observes that the Moderns are equipped with the "instruments" of "philosophical 'critique'" and "analytic geometry" (especially in the form of Cartesian logic) (DN, 7-9), and that these open realms of natural scientific inquiry (chemistry, pharmacology, astronomy, geographical exploration, and mechanics) and artistic production (realism in poetry, oratory, and sculpture) which were unknown and unavailable to the Ancients (DN, 11-12). Although these bring significant benefits, Vico argues, modern education suffers unnecessarily from ignoring the ars topica (art of topics) which encourage the use of imagination and memory in organizing speech into eloquent persuasion. The result, Vico argues, is an undue attention to the "geometrical method" modeled on the discipline of physics (DN, 21ff.), and an emphasis on abstract philosophical criticism over poetry. This undermines the importance of exposition, persuasion, and pleasure in learning; it "benumbs...[the] imagination and stupefies...[the] memory" (DN, 42), both of which are central to learning, complex reasoning, and the discovery of truth. Combining the methods of both Ancients and Moderns, Vico argues that education should aim ideally at cultivating the "total life of the body politic" (DN, 36): students "should be taught the totality of the sciences and arts, and their intellectual powers should be developed to the full" so that they "would become exact in science, clever in practical matters, fluent in eloquence, imaginative in understanding poetry or painting, and strong in memorizing what they have learned in their legal studies" (DN, 19). This defense of humanistic education is expanded in the Orations, directed "to the flower and stock of well-born young manhood,"[5] where Vico makes a case for modern humanistic education and focuses on a certain kind of "practical wisdom" or prudentia which the human mind, with the appropriate discipline and diligence, is able to attain. This theme is continued in De Antiquissima, where Vico traces the consequences of his insight that language can be treated as a source of historical knowledge. Many words of the Latin language, Vico observes, appear to be "derived from some inward learning rather than from the vernacular usage of the people."[6] Treated as a repository of the past, Latin might be investigated as a way of "seek[ing] out the ancient wisdom of the Italians from the very wisdom of their words."(DA, 40). In the course of pursuing this task, Vico also develops two central themes of his mature philosophy: the outlines of a philosophical system in contrast to the then dominant philosophy of Descartes already criticized in De nostri temporis studiorum ratione, and his statement of the principle that verum et factum convertuntur, that "the true and the made are...convertible," or that "the true is precisely what is made" (verum esse ipsum factum). Vico emphasizes that science should be conceived as the "genus or mode by which a thing is made" so that human science in general is a matter of dissecting the "anatomy of nature's works" (DA, 48), albeit through the "vice" of human beings that they are limited to "abstraction" as opposed to the power of "construction" which is found in God alone (DA, 50-52). Given that "the norm of the truth is to have made it" (DA, 52), Vico reasons, Descartes' famous first principle that clear and distinct ideas are the source of truth must be rejected: "For the mind does not make itself as it gets to know itself," Vico observes, "and since it does not make itself, it does not know the genus or mode by which it makes itself" (DA, 52). Thus the truths of morality, natural science, and mathematics do not require "metaphysical justification" as the Cartesians held, but demand an analysis of the causes-the "activity"-through which things are made (DA, 64). Vico's Vita di Giambattista Vico is of particular interest, not only as a source of insight into the influences on his intellectual development, but as one of the earliest and most sophisticated examples of philosophical autobiography. Vico composed the work in response (and indeed the only response) to a proposal published by Count Gian Artico di Porcia to Italian scholars to write their biographies for the edification of students. Referring to himself in the third person, Vico records the course of his life and the influence of various thinkers which led him to develop the concepts central to his mature work. Vico reports on the importance of reading Plato, Aristotle, the Hellenics, Scotus, Suarez, and the Classical poets, and traces his growing interest in jurisprudence and the Latin language (Vita, 116ff. passim). Vico describes how he came to "meditate a principle of the natural law, which should be apt for the explanation of the origins of Roman law and every other gentile civil law in respect of history" (Vita, 119) and how he discovered that "an ideal eternal law...should be observed in a universal city after the idea or design of providence" (Vita, 122). According to Vico's own account, his studies culminated in a distinction between ideas and languages. The first, he says, "discovers new historical principles of geography and chronology, the two ideas of history, and thence the principles of universal history lacking hitherto" (Vita, 167), while the latter "discovers new principles of poetry, both of song and verse, and shows that both it and they sprang up by the same natural necessity in all the first nations" (Vita, 168). Taken together, these form the central doctrine of The New Science, namely, that there is a "philosophy and philology of the human race" which produces "an ideal eternal history based on the idea of...providence...[and] traversed in time by all the particular histories of the nations, each with its rise, development, acme, decline and fall" (Vita, 169). ## 3. The New Science Many themes of these early works-language, wisdom, history, truth, causality, philology, rhetoric, philosophy, poetry, and the relative strengths and weaknesses of ancient and modern learning-are incorporated into and receive their fullest treatment in Vico's major work, The New Science, first published in 1725 (an edition subsequently known as the Scienza Nuova Prima or First New Science) and again in a second and largely rewritten version five years later. Vico published a third edition in 1744, which was later edited by Fausto Nicolini and appeared in 1928. Nicoloni is responsible for updating Vico's punctuation and breaking the work up into its current structure of chapters, sections, and numbered paragraphs (1112 in all), not originally supplied by Vico himself, which have been reproduced in many though not all subsequent editions. Nicolini referred to his edition as the Scienza Nuova seconda-Second New Science (or simply The New Science), under which title the definitive version is known today. The text consists of an overview ("Idea of the Work") couched as an explication of a frontispiece depicting a female figure (Metaphysics), standing on a globe of the Earth and contemplating a luminous triangle containing the eye of God, or Providence. Below stands a statue of Homer representing the origins of human society in "poetic wisdom." This is followed by five Books and a Conclusion, the first of which ("Establishment of Principles") establishes the method upon which Vico constructs a history of civil society from its earliest beginnings in the state of nature (stato di natura) to its contemporary manifestation in seventeenth century Europe. In the work, Vico consciously develops his notion of scienza (science or knowledge) in opposition to the then dominant philosophy of Descartes with its emphasis on clear and distinct ideas, the most simple elements of thought from which all knowledge, the Cartesians held, could be derived a priori by way of deductive rules. As Vico had already argued, one consequence and drawback of this hypothetico-deductive method is that it renders phenomena which cannot be expressed logically or mathematically as illusions of one sort or another. This applies not only most obviously to the data of sense and psychological experience, but also to the non-quantifiable evidence that makes up the human sciences. Drawing on the verum factum principle first described in De Antiquissima, Vico argues against Cartesian philosophy that full knowledge of any thing involves discovering how it came to be what it is as a product of human action and the "principal property" of human beings, viz., "of being social" ("Idea of the Work," SS2, p.3).[7] The reduction of all facts to the ostensibly paradigmatic form of mathematical knowledge is a form of "conceit," Vico maintains, which arises from the fact that "man makes himself the measure of all things" (Element I, SS120, p.60) and that "whenever men can form no idea of distant and unknown things, they judge them by what is familiar and at hand" (Element II, SS122, p.60). Recognizing this limitation, Vico argues, is at once to grasp that phenomena can only be known via their origins, or per caussas (through causes). For "Doctrines must take their beginning from that of the matters of which they treat" (Element CVI, SS314, p.92), he says, and it is one "great labor of...Science to recover...[the] grounds of truth-truth which, with the passage of years and the changes in language and customs, has come down to us enveloped in falsehood" (Element XVI, SS150, pp.64-5). Unveiling this falsehood leads to "wisdom," which is "nothing but the science of making such use of things as their nature dictates" (Element CXIV, SS326, p.94). Given that verum ipsum factum-"the true is the made," or something is true because it is made-scienzia both sets knowledge per caussas as its task and as the method for attaining it; or, expressed in other terms, the content of scienza is identical with the development of that scienza itself. The challenge, however, is to develop this science in such a way as to understand the facts of the human world without either reducing them to mere contingency or explaining their order by way of speculative principles of the sort generated by traditional metaphysics. They must be rendered intelligible, that is, without reducing them, as did the Cartesians, to the status of ephemera. Vico satisfies this demand by distinguishing at the outset of The New Science between il vero and il certo, "the true" and "the certain." The former is the object of knowledge (scienza) since it is universal and eternal, whereas the latter, related as it is to human consciousness (coscienza), is particular and individuated. This produces two pairs of terms-il vero/scienza and il certo/coscienza-which constitute, in turn, the explananda of philosophy and philology ("history" broadly conceived), respectively. As Vico says, "philosophy contemplates reason, whence comes knowledge of the true; philology observes that of which human choice is author, whence comes consciousness of the certain" (Element X, SS138, p.63). These two disciplines combine in a method or "new critical art" (nuova'arte critica) where philosophy aims at articulating the universal forms of intelligibility common to all experience, while philology adumbrates the empirical phenomena of the world which arise from human choice: the languages, customs, and actions of people which make up civil society. Understood as mutually exclusive disciplines-a tendency evident, according to Vico, in the history of philosophy up to his time-philosophy and philology appear as empty and abstract (as in the rational certainty of Cartesian metaphysics) and merely empirical and contingent, respectively. Once combined, however, they form a doctrine which yields a full knowledge of facts where "knowledge" in the Vichean sense means to have grasped both the necessity of human affairs (manifest in the causal connections between otherwise random events) and the contingency of the events which form the content of the causal chains. Philosophy yields the universally true and philology the individually certain. The text of The New Science then constitutes Vico's attempt to develop a method which itself comes to be in the course of applying it to human experience, and this takes the form of a history of civil society and its development through the progress of war and peace, law, social order, commerce, and government. "Thus our Science," Vico says near the beginning of the work, "comes to be at once a history of the ideas, the customs, the deeds of mankind. From these three we shall derive the principles of the history of human nature, which we shall show to be the principles of universal history, which principles it seems hitherto to have lacked" ("Poetic Wisdom," SS368, p.112). Accomplishing this task involves tracing human society back to its origins in order to reveal a common human nature and a genetic, universal pattern through which all nations run. Vico sees this common nature reflected in language, conceived as a store-house of customs, in which the wisdom of successive ages accumulates and is presupposed in the form of a sensus communis or "mental dictionary" by subsequent generations. Vico defines this common sense as "judgment without reflection, shared by an entire class, an entire people, and entire nation, or the entire human race" (Element XII, SS145, pp.63-4). It is also available to the philosopher who, by deciphering and thus recovering its content, can discover an "ideal eternal history traversed in time by the histories of all nations" (Proposition XLII, SS114, p.57). The result of this, in Vico's view, is to appreciate history as at once "ideal"-since it is never perfectly actualized-and "eternal," because it reflects the presence of a divine order or Providence guiding the development of human institutions. Nations need not develop at the same pace-less developed ones can and do coexist with those in a more advanced phase-but they all pass through the same distinct stages (corsi): the ages of gods, heroes, and men. Nations "develop in conformity to this division," Vico says, "by a constant and uninterrupted order of causes and effects present in every nation" ("The Course the Nations Run," SS915, p.335). Each stage, and thus the history of any nation, is characterized by the manifestation of natural law peculiar to it, and the distinct languages (signs, metaphors, and words), governments (divine, aristocratic commonwealths, and popular commonwealths and monarchies), as well as systems of jurisprudence (mystic theology, heroic jurisprudence, and the natural equity of free commonwealths) that define them. In addition to specifying the distinct stages through which social, civil, and political order develops, Vico draws on his earlier writings to trace the origin of nations back to two distinct features of human nature: the ages of gods and heroes result from memory and creative acts of "imagination" (fantasia), while the age of men stems from the faculty of "reflection" (riflessione). Vico thus claims to have discovered two kinds of wisdom-"poetic" and "philosophical"-corresponding to the dual nature of human beings (sense and intellect), represented in the creations of theological poets and philosophers, respectively ("Poetic Wisdom," SS779, p.297). Institutions arise first from the immediacy of sense-experience, pure feeling, curiosity, wonder, fear, superstition, and the child-like capacity of human beings to imitate and anthropomorphize the world around them. Since "in the world's childhood men were by nature sublime poets" (Element XXXVII, SS187, p.71), Vico reasons, nations must be "poetic in their beginnings" (Element XLIV, SS200, p.73), so that their origin and course can be discovered by recreating or remembering the "poetic" or "metaphysical truth" which underlies them (Element XLVII, SS205, p.74). This is manifest primarily in fable, myth, the structure of early languages, and the formations of polytheistic religion. The belief systems of early societies are thus characterized by "poetic metaphysics" which "seeks its proofs not in the external world but within the modifications of the mind of him who meditates it" ("Poetic Wisdom," SS374, p.116), and "poetic logic," through which the creations of this metaphysics are signified. Metaphysics of this sort is "not rational and abstract like that of learned men now," Vico emphasizes, "but felt and imagined [by men] without power of ratiocination...This metaphysics was their poetry, a faculty born with them...born of their ignorance of causes, for ignorance, the mother of wonder, made everything wonderful to men who were ignorant of everything" ("Poetic Wisdom," SS375, p.116). Incapable of forming "intelligible class concepts of things"-a feature of human mind realized only in the age of men-people "had a natural need to create poetic characters; that is, imaginative class concepts or universals, to which, as to certain models or ideal portraits, to reduce all the particular species which resembled them" (Element XLIX, SS209, p.74). From this genus of poetic metaphysics, Vico then extrapolates the various species of wisdom born of it. "Poetic morals" have their source in piety and shame ("Poetic Wisdom," SS502, p.170), he argues, while "poetic economy" arises from the feral equality of human beings and the family relationships into which they were forced by need ("Poetic Wisdom," SS523, p.180). Similarly, "poetic cosmography" grows from the seeing "the world as composed of gods of the sky, of the underworld...and gods intermediate between earth and sky" ("Poetic Wisdom," SS710, p.269), "poetic astronomy" from raising the gods "to the planets and [assigning] the heroes to the constellations" ("Poetic Wisdom," SS728, p.277), "poetic chronology" out of the cycles of harvest and the seasons ("Poetic Wisdom," SS732, p.279), and "poetic geography" from naming the natural world through "the semblances of things known or near at hand" ("Poetic Wisdom," SS741, p.285). As the faculty of reason develops and grows, however, the power of imagination from which the earliest forms of human society grew weakens and gives way finally to the power of reflection; the cognitive powers of human beings gain ascendance over their creative capacity, and reason replaces poetry as the primary way of understanding the world. This defines the age of men which makes philosophy, Vico reasons, a relatively recent development in history, appearing as it did "some two thousand years after the gentile nations were founded" (Element CV, SS313, p.92). Since history itself, in Vico's view, is the manifestation of Providence in the world, the transition from one stage to the next and the steady ascendance of reason over imagination represent a gradual progress of civilization, a qualitative improvement from simpler to more complex forms of social organization. Vico characterizes this movement as a "necessity of nature" ("Idea of the Work," SS34, p.21) which means that, with the passage of time, human beings and societies tend increasingly towards realizing their full potential. From rude beginnings undirected passion is transformed into virtue, the bestial state of early society is subordinated to the rule of law, and philosophy replaces sentiments of religion. "Out of ferocity, avarice, and ambition, the three vices which run throughout the human race," Vico says, "legislation creates the military, merchant, and governing classes, and thus the strength, riches, and wisdom of commonwealths. Out of these three great vices, which could certainly destroy all mankind on the face of the earth, it makes civil happiness" (Element VII, SS132, p.62). In addition, the transition from poetic to rational consciousness enables reflective individuals-the philosopher, that is, in the shape of Vico-to recover the body of universal history from the particularity of apparently random events. This is a fact attested to by the form and content of The New Science itself. Although from a general point of view history reveals a progress of civilization through actualizing the potential of human nature, Vico also emphasizes the cyclical feature of historical development. Society progresses towards perfection, but without reaching it (thus history is "ideal"), interrupted as it is by a break or return (ricorso) to a relatively more primitive condition. Out of this reversal, history begins its course anew, albeit from the irreversibly higher point to which it has already attained. Vico observes that in the latter part of the age of men (manifest in the institutions and customs of medieval feudalism) the "barbarism" which marks the first stages of civil society returns as a "civil disease" to corrupt the body politic from within. This development is marked by the decline of popular commonwealths into bureaucratic monarchies, and, by the force of unrestrained passions, the return of corrupt manners which had characterized the earlier societies of gods and heroes. Out of this "second barbarism," however, either through the appearance of wise legislators, the rise of the fittest, or a the last vestiges of civilization, society returns to the "primitive simplicity of the first world of peoples," and individuals are again "religious, truthful, and faithful" ("Conclusion of the Work," SS1104-1106, pp.423-4). From this begins a new corso which Vico saw manifest in his own time as the "second age of men" characterized by the "true" Christian religion and the monarchical government of seventeenth century Europe. In addition to his account of the origins and history of civil society developed in The New Science, Vico also advances a highly original thesis about the origin and character of Homeric poetry, which he refers to as "The Discovery of the New Homer." Vico observes that the "vulgar feelings and vulgar customs provide the poets with their proper materials" ("Discovery of the True Homer," SS781, p.301), and, given that in the age of heroes these customs constituted a "savage" and "unreasonable" state of human nature, the poetry of Homer cannot be the esoteric wisdom or creative act of a single individual, as scholars have assumed, but represents the imaginative universals of the Greek people themselves. Homeric poetry thus contains the "models or ideal portraits" which form the mental dictionary of the Ancients, and explains the place Vico assigns to Homer in the frontispiece where his statue represents the origins of human society in poetic wisdom. Vico thus applies his doctrine of the imaginative universal to the case of Homer and concludes that though it "marks a famous epoch in history it never in the world took place." The historical Homer was "quite simply a man of the people" ("Discovery of the True Homer, SS806, p.308) and quite distinct from the ostensible author of the Odyssey and Iliad who was actually a "purely ideal poet who never existed in the world of nature...but was an idea or a heroic character of Grecian men insofar as they told their histories in song" ("Discovery of the True Homer, SS873, p.323).
vienna-circle	## 1. Introductory Remarks While it is in the nature of philosophical movements and their leading doctrines to court controversy, the Vienna Circle and its philosophies did so more than most. To begin with, its members styled themselves as conceptual revolutionaries who cleared the stables of academic philosophy by showing metaphysics not simply to be false, but to be cognitively empty and meaningless. In addition, they often associated their attempt to overcome metaphysics with their public engagement for scientific Enlightenment reason in the ever-darkening political situation of 1920s and 1930s central Europe. Small wonder then that the Vienna Circle has sharply divided opinion from the start. There is very little beyond the basic facts of membership and its record of publications and conferences that can be asserted about it without courting some degree of controversy. (For English-language survey monographs and articles on the Vienna Circle, see Kraft 1950, Jorgensen 1951, Ayer 1959b, Passmore 1967, Hanfling 1981, Stadler 1998, Richardson 2003. Particularly rich in background and bio-bibliographical materials is Stadler 1997 [2015]. The best short introductory book has remained untranslated: Haller 1993. For popular overviews see Sigmund 2015 [2018] and Edmonds [2020].) Fortunately, more than three decades worth of recent scholarship in history of philosophy of science now allows at least some disputes to be put into perspective. (See, e.g., the following at least in part English-language collections of articles and research monographs: Haller 1982, McGuinness 1985, Rescher 1985, Gower 1987, Proust 1989, Zolo 1989, Coffa 1991, Spohn 1991, Uebel 1991, Bell and Vossenkuhl 1992, Sarkar 1992, Uebel 1992, Oberdan 1993, Stadler 1993, Cirera 1994, Salmon and Wolters 1994, Cartwright, Cat, Fleck and Uebel 1996, Giere and Richardson 1996, Nemeth and Stadler 1996, Sarkar 1996, Richardson 1998, Friedman 1999, Wolenski and Kohler 1999, Fetzer 2000, Friedman 2000, Bonk 2003, Hardcastle and Richardson 2003, Parrini, Salmon and Salmon 2003, Stadler 2003, Awodey and Klein 2004, Reisch 2005, Galavotti 2006, Carus 2007, Creath and Friedman 2007, Nemeth, Schmitz and Uebel 2007, Richardson and Uebel 2007, Uebel 2007, Wagner 2009, Manninen and Stadler 2010, McGuinness 2011, Symons, Pombo and Torres 2011, Creath 2012, Wagner 2012, Dambock 2016, Pihlstrom, Stadler and Weidtmann 2017, Schiemer 2017, Tuboly 2017, Carus 2018, Cat and Tuboly 2019, Makovec and Shapiro 2019.) What distinguishes these works from valuable collections like Schilpp 1963, Hintikka 1975 and Achinstein and Barker 1979 is that the unspoken assumption, to have understood Vienna Circle philosophy correctly enough so as to consider its consequences straightforward, is questioned in the more recent scholarship. Many other pieces of new Vienna Circle scholarship are spread throughout philosophical journals and essay collections with more systematic or wider historical scope; important work has also been done in German, Italian and French language publications but here must remain unreferenced.) Two facts must be clearly recognized if a proper evaluation of the Vienna Circle is to be attempted. The first is, that, despite its relatively short existence, even some of the most central theses of the Vienna Circle underwent radical changes. The second is that its members were by no means of one mind in all important matters; occasionally they espoused perspectives so radically at variance with each other that even their ostensive agreements cannot remain wholly unquestioned. Behind the rather thin public front, then, quite different philosophical projects were being pursued by the leading participants with, moreover, changing alliances. One way of taking account of this is by speaking (as above) explicitly of the philosophies (in the plural) of the Vienna Circle (and to avoid the singular definite description) while using the expression "Vienna Circle philosophy" (without an article) in a neutral generic sense. Recent scholarship has provided what the received view of Viennese neopositivism lacks: recognition and documentation of the sometimes sharply differentiated positions behind the generic surface. This does not invalidate all previous scholarship, including some fundamental criticisms of its positions, but it restores a depth to Vienna Circle philosophy that was absent from the standard histories. The value of this development must not be underestimated, for the recognition of the Vienna Circle's sophisticated engagement with aspects of the philosophical tradition and contemporaneous challenges calls into question unwarranted certainties of our own self-consciously post-positivist era. While there remains support for the view that philosophical doctrines were held in the Vienna Circle that wholly merited many of the standard criticisms to be cited below, there is now also support for the view that in nearly all such cases, these doctrines were already in their day opposed within the Circle itself. While some of the Vienna Circle philosophies are dated and may even be, as John Passmore once put it, as dead as philosophies can be, others show signs of surprising vitality. Which ones these are, however, remains a matter of debate. The lead pursued in this article is provided by the comments of a long-time associate of the Vienna Circle, C.G. Hempel, made in 1991: > > When people these days talk about logical positivism or the Vienna > Circle and say that its ideas are passe, this is just wrong. > This overlooks the fact that there were two quite different schools of > logical empiricism, namely the one of Carnap and Schlick and so on and > then the quite different one of Otto Neurath, who advocates a > completely pragmatic conception of the philosophy of science.... > And this form of empiricism is in no way affected by any of the > fundamental objections against logical positivism.... (quoted in > Wolters 2003, 117) > While differing with Hempel's specific claim about how the two "schools" divide, the aim here is to fill out his suggestive picture by indicating what Schlick, Carnap and Neurath stand for philosophically and why the different wings of the Vienna Circle require differentiated assessments. After reviewing the basic facts and providing an overall outline of Vienna Circle philosophy (in sect. 2), this article considers various doctrines in greater detail by way of discussing standard criticisms with the appropriate distinctions in mind (in sect. 3). No comprehensive assessment of the Vienna Circle and the work of its members can be attempted here, but some basic conclusions will be drawn (in sect. 4). ## 2. The Basics: People, Activities and Overview of Doctrines ### 2.1 People The Vienna Circle was a group of scientifically trained philosophers and philosophically interested scientists who met under the (nominal) leadership of Moritz Schlick for often weekly discussions of problems in the philosophy of science during academic terms in the years from 1924 to 1936. As is not uncommon with such groups, its identity is blurred along several edges. Not all of those who ever attended the discussions can be called members, and not all who attended did so over the entire period. Typically, attention is focused on long-term regulars who gained prominence through their philosophical publications, but even these do not in all cases fall into the period of the Vienna Circle proper. It is natural, nevertheless, to consider under the heading "Vienna Circle" also the later work of leading members who were still active in the 1940s, 50s and 60s. Finally, there is the so-called periphery of international contacts and visitors that prefigured the post-World War II network of analytical philosophers of science. In the present article the emphasis will be placed on the long-term regulars whose contributions will be followed, selectively, into the post-Schlick era. According to its unofficial manifesto (see section 2.3 below), the Circle had "members" and recognized others as "sympathetic" to its aims. It included as members, besides Schlick who had been appointed to Mach's old chair in Philosophy of the Inductive Sciences at the University of Vienna in 1922, the mathematician Hans Hahn, the physicist Philipp Frank, the social scientist Otto Neurath, his wife, the mathematician Olga Hahn-Neurath, the philosopher Viktor Kraft, the mathematicians Theodor Radacovic and Gustav Bergmann and, since 1926, the philosopher and logician Rudolf Carnap. (Even before World War I, there existed a similarly oriented discussion circle that included Frank, Hahn and Neurath. During the time of the Schlick Circle, Frank resided in Prague throughout, Carnap did so from 1931.) Further members were recruited among Schlick's students, like Friedrich Waismann, Herbert Feigl and Marcel Natkin, others were recruited among Hahn's students, like Karl Menger and Kurt Godel. Though listed as members in the manifesto, Menger and Kraft later wanted to be known only as as sympathetic associates, like, all along, the mathematician Kurt Reidemeister and the philosopher and historian of science Edgar Zilsel. (Karl Popper was never a member or associate of the Circle, though he studied with Hahn in the 1920s and in the early 1930s discussed its doctrines with Feigl and Carnap.) Over the years, other participants (not listed in the manifesto) included other students of Schlick's and Hahn's like Bela von Juhos, Josef Schachter, Walter Hollitscher, Heinrich Neider, Rose Rand, Josef Rauscher and Kathe Steinhardt, a secondary teacher, Robert Neumann, and, as notable thinkers with independent connections, the jurist and philosopher Felix Kaufmann (also a member of F.A. von Hayek's "Geistkreis") and the innovative psychologist Egon Brunswik (coming, like the even more loosely associated sociologists Paul Lazarsfeld and Marie Jahoda, from the pioneering Karl Buhler's University Institute of Psychology). Despite its prominent position in the rich, if fragile, intellectual culture of inter-war Vienna and most likely due to its radical doctrines, the Vienna Circle found itself virtually isolated in most of German speaking philosophy. The one exception was its contact and cooperation with the Berlin Society for Empirical (later: Scientific) Philosophy (the other point of origin of logical empiricism). The members of the Berlin Society sported a broadly similar outlook and included, besides the philosopher Hans Reichenbach, the logicians Kurt Grelling and Walter Dubislav, the psychologist Kurt Lewin, the surgeon Friedrich Kraus and the mathematician Richard von Mises. (Its leading members Reichenbach, Grelling and Dubislav were listed in the Circle's manifesto as sympathisers.) At the same time, members of the Vienna Circle also engaged directly, if selectively, with the Warsaw logicians (Tarski visited Vienna in 1930, Carnap later that year visited Warsaw and Tarski returned to Vienna in 1935). Probably partly because of its firebrand reputation, the Circle attracted also a series of visiting younger researchers and students including Carl Gustav Hempel from Berlin, Hasso Harlen from Stuttgart, Ludovico Geymonat from Italy, Jorgen Jorgensen, Eino Kaila, Arne Naess and Ake Petzall from Scandinavia, A.J. Ayer from the UK, Albert Blumberg, Charles Morris, Ernest Nagel and W.V.O. Quine from the USA, H.A. Lindemann from Argentina and Tscha Hung from China. (The reports and recollections of these former visitors--e.g. Nagel 1936--are of interest in complementing the Circle's in-house histories and recollections which start with the unofficial manifesto--Carnap, Hahn and Neurath 1929--and extend through Neurath 1936, Frank 1941, 1949a and Feigl 1943 to the memoirs by Carnap 1963, Feigl 1969a, 1969b, Bergmann 1987, Menger 1994.) The aforementioned social and political engagement of members of the Vienna Circle and of Vienna Circle philosophy for Enlightenment reason had never made the advancement of its associates or protegees easy in Viennese academia. From 1934 onwards, with anti-semitism institutionalized and irrationalism increasingly dominating public discourse, this engagement began to cost the Circle still more dearly. Not only was the Verein Ernst Mach closed down early that year for political reasons, but the ongoing dispersal of Circle members by emigration, forced exile and death meant that after the murder of Schlick in 1936 only a small rump was able to continue meetings for another two years in Vienna. (1931: Feigl emigrated to USA; 1934: Hahn died, Neurath fled to Holland, 1940 to UK; 1935: Carnap emigrated to USA; 1936: Schlick murdered; 1937: Menger emigrated to USA, Waismann to UK; 1938: Frank, Kaufmann, Brunswik, Bergmann emigrated to USA; Zilsel, Rand to UK, later to USA; Hollitscher fled to Switzerland, later to UK; Schachter emigrated to Palestine; 1940: Godel emigrated to USA; see Dahms 1995 for a chronology of the exodus.) But the end of the Vienna Circle as such did not mean the end of its influence due to the continuing development of the philosophy of former members (and the work of Kraft in post-World War II Vienna; on this see Stadler 2008). Particularly through their work in American exile (esp. Carnap at Harvard, Chicago and UCLA; Feigl at Iowa and Minnesota; less so Frank at Harvard) and that of earlier American visitors (esp. Quine, Nagel), as well as through the work of fellow emigrees from the Berlin Society (esp. Reichenbach, Hempel) and their students (Hilary Putnam, Wesley Salmon), logical empiricism strongly influenced the post-World War II development of analytic philosophy of science. By contrast, Waismann had little influence in the UK where Neurath, already marginalized, had died in 1945. (The full story of logical empiricism's acculturation in the English speaking world remains to be written, but see Howard 2003, Reisch 2005, Uebel 2005a, 2010, Douglas 2009, and Romizi 2012 for considerations of aspects of Vienna Circle philosophy early and/or late that were neglected in the process and remained long forgotten.) ### 2.2 Activities After its formative phase which was confined to the Thursday evening discussions, the Circle went public in 1928 and 1929 when it seemed that the time had come for their emerging philosophy to play a distinctive role not only in the academic but also the public sphere. In November 1928, at its founding session, Schlick accepted the presidency of the newly formed Verein Ernst Mach (Association Ernst Mach), Hahn accepted one of its vice-presidencies and Neurath and Carnap joined its secretariat. Originally proposed by the Austrian Freidenkerbund (Free Thinker Association), the Verein Ernst Mach was dedicated to the dissemination of scientific ways of thought and so provided a forum for popular lectures on the new scientific philosophy. In the following year the Circle stepped out under its own name (invented by Neurath) with a manifesto and a special conference. The publication of "The Scientific World Conception: The Vienna Circle", signed by Carnap, Hahn and Neurath and dedicated to Schlick, coincided with the "First Conference for the Epistemology of the Exact Sciences" in mid-September 1929, organised jointly with the Berlin Society as an adjunct to the Fifth Congress of German Physicists and Mathematicians in Prague (where Frank played a prominent role in the local organising committee). (On the production history and early reception of the manifesto see Uebel 2008.) A distinct philosophical school appeared to be emerging, one that was dedicated to ending the previous disputes of philosophical schools by dismissing them, controversially, as strictly speaking meaningless. Throughout the early and mid-1930s the Circle kept a high and increasingly international profile with its numerous publications and conferences. In 1930, the Circle took over, again together with the Berlin Society, the journal Annalen der Philosophie and restarted it under the name of Erkenntnis with Carnap and Reichenbach as co-editors. (Besides publishing original articles and sustaining lengthy debates, this journal featured selected proceedings of their early conferences and documented the lecture series of the Verein Ernst Mach and the Berlin Society as well as their international congresses.) In addition, from 1928 until 1936, Schlick and Frank served as editors of their book series "Schriften zur wissenschaftlichen Weltauffassung" ("Writings on the Scientific World Conception"), which published major works by leading members and early critics like Popper, while Neurath edited, from 1933 until 1939, the series "Einheitswissenschaft" ("Unified Science"), which published more introductory essays by leading members and sympathisers. Conference-wise, the Circle organized, again with the Berlin Society, a "Second Conference for the Epistemology of the Exact Sciences" as an adjunct to the Sixth Congress of German Physicists and Mathematicians in Konigsberg in September 1930 (where Reidemeister played a prominent role in the organization and Godel first announced his incompleteness result in the discussion) and then began the series of International Congresses on the Unity of Science with a "Pre-Conference" just prior to the start of the Eighth International Congress of Philosophy in Prague in September 1934. This, their last conference in Central Europe, was followed by the International Congresses of various sizes in Paris (September 1935, July 1937), Copenhagen (June 1936), Cambridge, England (July 1938), Cambridge, Mass. (September 1939), all in the main organized by Neurath; a smaller last gathering was held in Chicago in September 1941. By 1938 their collective publication activity began to centre on a monumentally planned International Encyclopedia of Unified Science, with Neurath as editor-in-chief and Carnap and Charles Morris as co-editors; by the time of Neurath's death in 1945, only 10 monographs had appeared and the series was wound up in 1970 numbering 20 monographs under the title "Foundations of the Unity of Science" (notably containing Thomas Kuhn's Structure of Scientific Revolutions amongst them). Individually, the members of the Vienna Circle published extensively before, during and after the years of the Circle around Schlick. For some (Frank, Hahn, Menger, Neurath), philosophy was only part of their scientific output, with numerous monographs and articles in their respective disciplines (mathematics, physics and social science); others (Schlick, Carnap, Feigl, Waismann) concentrated on philosophy, but even their output cared relatively little for traditional concerns of the field. Here it must be noted that two early monographs by Schlick (1918/25) and Carnap (1928a), commonly associated with the Vienna Circle, mostly predate their authors' participation there and exhibit a variety of influences not typically associated with logical positivism (see section 3.7 below). Moreover, important monographs by Frank (1932), Neurath (1931a), Carnap (1934/37) and Menger (1934) in the first half of the 1930s represent moves away from positions that had been held in the Circle before and contradict its orthodox profile. Yet the Circle's orthodoxy, as it were, is not easily pinned down either. Schlick himself was critical of the manifesto of 1929 and gave a brief vision statement of his own in "The Turning Point in Philosophy" (1930). A long-planned book by Waismann of updates on Wittgenstein's thought, to which Schlick was extremely sympathetic, was never completed as originally planned and only appeared posthumously (Waismann 1965; for earlier material see Baker 2003). Comparison with rough transcripts of the Circle's discussions in the early 1930s (for transcripts from between December 1930 and July 1931 see Stadler 1997 [2015, 69-123]) suggest that Waismann's Wittgensteinian "Theses", dated to "around 1930" (Waismann 1967 [1979, Appendix B]), come closest to an elaboration of the orthodox Circle position at that time (but which remained not undisputed even then). Again, what needs to be stressed is that all of the Circle's publications are to be understood as contributions to ongoing discussions among its members and associates. ### 2.3 Overview of Doctrines Despite the pluralism of the Vienna Circle's views, there did exist a minimal consensus which may be put as follows. A theory of scientific knowledge was propagated which sought to renew empiricism by freeing it from the impossible task of justifying the claims of the formal sciences. It will be noted that this updating did not leave empiricism unchanged. Following the logicism of Frege and Russell, the Circle considered logic and mathematics to be analytic in nature. Extending Wittgenstein's insight about logical truths to mathematical ones as well, the Circle considered both to be tautological. Like true statements of logic, mathematical statements did not express factual truths: devoid of empirical content they only concerned ways of representing the world, spelling out implication relations between statements. The knowledge claims of logic and mathematics gained their justification on purely formal grounds, by proof of their derivability by stated rules from stated axioms and premises. (Depending on the standing of these axioms and premises, justification was conditional or unconditional.) Thus defanged of appeals to rational intuition, the contribution of pure reason to human knowledge (in the form of logic and mathematics) was thought easily integrated into the empiricist framework. (Carnap sought to accommodate Godel's incompleteness results by separating analyticity from effective provability and by postulating that arithmetic consisted of an infinite series of ever richer arithmetical languages; see the discussion and references in section 3.2 below.) The synthetic statements of the empirical sciences meanwhile were held to be cognitively meaningful if and only if they were empirically testable in some sense. They derived their justification as knowledge claims from successful tests. Here the Circle appealed to a criterion of meaningfulness (cognitive significance) the correct formulation of which was problematical and much debated (and will be discussed in greater detail in section 3.1 below). Roughly, if synthetic statements failed testability in principle they were considered to be cognitively meaningless and to give rise only to pseudo-problems. No third category of significance besides that of a priori analytical and a posteriori synthetic statements was admitted: in particular, Kant's synthetic a priori was banned as having been refuted by the progress of science itself. (The theory of relativity showed what had been held to be an example of the synthetic a priori, namely Euclidean geometry, to be false as the geometry of physical space.) Thus the Circle rejected the knowledge claims of metaphysics as being neither analytic and a priori nor empirical and synthetic. (On related but different grounds, they also rejected the knowledge claims of normative ethics: whereas conditional norms could be grounded in means-ends relations, unconditional norms remained unprovable in empirical terms and so depended crucially on the disputed substantive a priori intuition.) Given their empiricism, all of the members of the Vienna Circle also called into question the principled separation of the natural and the human sciences. They were happy enough to admit to differences in their object domains, but denied the categorical difference in both their overarching methodologies and ultimate goals in inquiry, which the historicist tradition in the still only emerging social sciences and the idealist tradition in philosophy insisted on. The Circle's own methodologically monist position was sometimes represented under the heading of "unified science". Precisely how such a unification of the sciences was to be effected or understood remained a matter for further discussion (see section 3.3 below). It is easy to see that, combined with the rejection of rational intuition, the Vienna Circle's exclusive apportionment of reason into either formal a priori reasoning, issuing in analytic truths (or contradictions), and substantive a posteriori reasoning, issuing in synthetic truths (or falsehoods), severely challenged the traditional understanding of philosophy. All members of the Circle hailed the end of distinctive philosophical system building. In line with the Tractatus claim that all philosophy is really a critique of language, the Vienna Circle took the so-called linguistic turn, the turn to representation as the proper subject matter of philosophy. Philosophy itself was denied a separate first-order domain of expertise and declared a second-order inquiry. Whether the once queen of the sciences was thereby reduced to the mere handmaiden of the latter was still left open. It remained a matter of disagreement what type of distinctively philosophical insight, if any, would remain legitimate. Just as importantly, however, the tools of modern logic were employed also for metatheoretical construction, not just for the reduction of empirical claims to their observational base or, more generally, for the derivation of their observational consequences. For the price of abandoning foundationalist certainty this allowed for an enormous expansion of the domain of empirical discourse. Ultimately, it opened the space for the still ongoing discussions of scientific realism and its alternatives (see section 3.4 below). The Circle's leading protagonists differed in how they conceptualized this reflexive second-order inquiry that the linguistic turn had inaugurated. Nevertheless, they all agreed broadly that the ways of representing the world were largely determined by convention. A multitude of ideas hide behind this invocation of conventionality. One particularly radical one is the denial of the apodicity of all apriority, the denial of the claim that knowledge justified through reason alone represents truths that are unconditionally necessary. Another one is the imputation of agency in the construction of the logico-linguistic frameworks that make human cognition possible, the denial that conventionality could only mean acquiescence in tradition. Whether such ideas were followed by individual members of the Circle, however, depended on their own interests and influences. It is these often overlooked or misunderstood differences that hold the key to understanding the interplay of occasionally incompatible positions that make up Vienna Circle philosophy. (As can be seen from some of their internal disputes, moreover, these differences were also not always obvious to the protagonists themselves.) To see a striking example, consider their overarching conceptions of philosophy itself. Some protagonists retained the idea that philosophy possessed a separate disciplinary identity from science and, like Schlick, turned philosophy into a distinctive, albeit non-formal activity of meaning determination. Others, like Carnap, agreed on the distinction between philosophy and science but turned philosophy into a purely formal enterprise, the so-called logic of science. Still others went even further and, like Neurath under the banner of "unified science", also rejected philosophy as a separate discipline and treated what remained of it, after the rejection of metaphysics, as its partly empirical meta-theory, to be set beside the logic of science. With Schlick, then, philosophy became the activity of achieving a much clarified and deepened understanding of the cognitive and linguistic practices actually already employed in science and everyday discourse. By contrast, for Carnap, philosophy investigated and reconstructed existing language fragments, developed new logico-linguistic frameworks and suggested possible formal conventions for science, while, for Neurath, the investigation of science was pursued by an interdisciplinary meta-theory that encompassed empirical disciplines, again with a pragmatic orientation. Thus we find in apparent competition different conceptions of post-metaphysical philosophy: the projects of experiential meaning determination, of formal rational reconstruction and of naturalistic explications of leading theoretical and methodological notions. (For roughly representative early essay-length statements of their positions see Schlick 1930, Carnap 1932a and Neurath 1932a; later restatements are given in Schlick 1937, Carnap 1936b and Neurath 1936b.) In the more detailed discussions below these differences of overall approach will figure repeatedly (see also section 3.6 below). Criticisms of the basic positions adopted in the Vienna Circle are legion, though it may be questioned whether most of them took account of the sophisticated variations on offer. (Sometimes the Circle's own writings are disregarded altogether and "logical positivism" is discussed only via the proxy of Ayer's popular exposition; see, e.g., Soames 2003.) But some Neo-Kantians like Ernst Cassirer may claim that they too accepted developments like the merely relative a priori and an appropriate conception of the historical development of science. Likewise, Wittgensteinians may claim that Wittgenstein's own opposition to metaphysics only concerned false attempts to render it intelligible: his merely ineffable but uneliminated metaphysics concerned precisely what for him were essentials of ways of representing the world. The commonest criticisms, however, concerned not the uniqueness of the Vienna Circle's doctrines, or their faithfulness to their supposed sources, but whether they were tenable at all. Prominent objects of this type of criticism include the verificationist theory of meaning and its claimed anti-metaphysical and non-cognitivist consequences as well as its own significance; the reductionism in phenomenalist or physicalist guises that appeared to attend the Circle's attempted operationalisation of the logical atomism of Russell and Wittgenstein; and the Circle's alleged scientism in general and their formalist and a-historical conception of scientific cognition in particular. These criticisms are discussed in some detail below in order to assess why which of the associated doctrines remain of what importance. As noted, the Vienna Circle did not last long: its philosophical revolution came at a cost. Yet what was so socially, indeed politically, explosive about what appears on first sight to be a particularly arid, if not astringent, doctrine of specialist scientific knowledge? To a large part, precisely what made it so controversial philosophically: its claim to refute opponents not by proving their statements to be false but by showing them to be (cognitively) meaningless. Whatever the niceties of their philosophical argument here, the socio-political impact of the Vienna Circle's philosophies of science was obvious and profound. All of them opposed the increasing groundswell of radically mistaken, indeed irrational, ways of thinking about thought and its place in the world. In their time and place, the mere demand that public discourse be perspicuous, in particular, that reasoning be valid and premises true--a demand implicit in their general ideal of reason--placed them in the middle of crucial socio-political struggles. Some members and sympathisers of the Circle also actively opposed the then increasingly popular volkisch supra-individual holism in social science as a dangerous intellectual aberration. Not only did such ideas support racism and fascism in politics, but such ideas themselves were supported only by radically mistaken arguments concerning the nature and explanation of organic and unorganic matter. So the first thing that made all of the Vienna Circle philosophies politically relevant was the contingent fact that in their day much political discourse exhibited striking epistemic deficits. That some of the members of the Circle went, without logical blunders, still further by arguing that socio-political considerations can play a legitimate role in some instances of theory choice due to underdetermination is yet another matter. This particular issue will not be pursued further here (see references at the end of section 2.1 above), nor the general topic of the Circle's embedding in modernism and the discourse of modernity (see Putnam 1981b for a reductionist, Galison 1990 for a foundationalist, Uebel 1996 for a constructivist reading of their modernism, Dahms 2004 for an account of personal relations with representatives of modernism in art and architecture). ## 3. Selected Doctrines and their Criticisms Given only the outlines of Vienna Circle philosophy, its controversial character is evident. The boldness of its claims made it attractive but that boldness also seemed to be its undoing. Turning to the questions of how far and, if at all, which forms of Vienna Circle philosophy stand up to some common criticisms, both the synchronic variations and the diachronic trajectories of its variants must be taken into account. This will be attempted in the sections below. Before expectations are raised too high, however, it must also remembered that in this article only the views of members of the Vienna Circle can be discussed, even though the problematic issues were pervasive in logical empiricism generally. (Compare the SEP article "Logical Empiricism"; for articles on Reichenbach see, e.g, Spohn 1991 and Salmon and Wolters 1994, Richardson 2005, 2006, and, for the Berlin Group as a whole, Milkov and Peckhaus 2013.) Moreover, here the emphasis must lie on the main protagonists: Schlick, Carnap and Neurath. (Neither Hahn or Frank, nor Waismann or Feigl, for instance, can be discussed here as extensively as their work deserves; see, e.g., Uebel 2005b, 2011b, McGuinness 2011, Haller 2003, respectively.) What will be noted, however, is that Vienna Circle philosophy was by no means identical with the post-World War II logical empiricism that has come to be known as the "received view" of scientific theories, even though it would be hard to imagine the latter without the former. (For a systematic if schematic critical discussion of the received view, see Suppe 1977, for a partial defense Mormann 2007a.) To deepen the somewhat cursory overview of Vienna Circle philosophy given above, we now turn to the discussion of the following issues: first, the viability of the conceptions of empirical significance employed by the Vienna Circle in its classical period; second, its uses of the analytic-synthetic distinction; third, its supposedly reductionist designs and foundationalist ambitions for philosophy; fourth, its stances in the debate about realism or anti-realism with regard to the theoretical terms in science; fifth, Carnap's later ideas in response to some of the problems encountered; sixth, the issue of the status of the meaning criterion itself (variably referred to also as "criterion of significance" or "criterion of empirical significance") and of the point of their critique of metaphysics; seventh, the Vienna Circle's attitude towards history and of their own place in the history of philosophy. These topics have been chosen for the light their investigation throws on the Circle's own agendas and the reception of its doctrines amongst philosophers at large, as well as for the relative ease with which their discussion allows its development and legacy to be charted. There can be little doubt about the enormous impact that the members of the Vienna Circle had on the development of twentieth-century philosophy. What is less clear is whether any of its distinctive doctrines are left standing once the dust of their discussion has settled or whether those of its teachings that were deemed defensible merged seamlessly into the broad church that analytic philosophy has become (and, if so, what those surviving doctrines and teachings may be). It must be noted, then, that the topics chosen for this article do not exhaust the issues concerning which the members of the Vienna Circle made significant contributions (which continue to stimulate work in the history of philosophy of science). Important topics like that of the theory and practice of unified science, of the nature of the empirical basis of science (the so-called protocol-sentence debate) and of the general structure of the theories of individual sciences can only be touched upon selectively. Likewise, while the general topic of ethical non-cognitivism receives only passing mentions, the Circle's varied approaches to value theory cannot be discussed here (for an overview see Rutte 1986, for detailed analyses see Siegetsleitner 2014). Other matters, like the contributions made by Vienna Circle members to the development of probability theory and inductive logic, the philosophy of logic and mathematics (apart from the guiding ideas of Carnap) and to the philosophy of individual empirical sciences (physics, biology, psychology, social science), cannot be discussed at all (see Creath and Friedman 2007 and Richardson and Uebel 2007 for relevant essays with further references). But it may be noted that with his "logic of science" Carnap counts among the pioneers of what nowadays is called "formal epistemology". ### 3.1 Verificationism and the Critique of Metaphysics Not surprisingly, it was the Circle's rejection of metaphysics by means of their seemingly devastating criterion of cognitive significance that attracted immediate opposition. (That they did not deny all meaning to statements thus ruled out of court was freely admitted from early on, but this "expressive" surplus was considered secondary to so-called "cognitive" meaning and discountable in science (see Carnap 1928b, 1932a).) Notwithstanding the metaphysicans' thunder, however, the most telling criticisms of the criterion came from within the Circle or broadly sympathetic philosophers. When it was protested that failure to meet an empiricist criterion of significance did not make philosophical statements meaningless, members of the Circle simply asked for an account of what this non-empirical and presumably non-emotive meaning consisted in and typically received no convincing answer. The weakness of their position was rather that their own criterion of empirical significance seemed to resist an acceptable formal characterization. To start with, it must be noted that long before the verification principle proper entered the Circle's discourse in the late 1920s, the thought expressed by Mach's dictum that "where neither confirmation nor refutation is possible, science is not concerned" (1883 [1960, 587]) was accepted as a basic precept of critical reflection about science. Responsiveness to evidence for and against a claim was the hallmark of scientific discourse. (Particularly the group Frank-Hahn-Neurath, who formed part of a pre-World War I discussion group (Frank 1941, 1949a) sometimes called the "First Vienna Circle" (Haller 1985, Uebel 2003), can be presumed to be familiar with Mach's criterion.) Beyond this, still in the 1920s, Schlick (1926) convicted metaphysics for falsely trying to express as logically structured cognition what is but the inexpressible qualitative content of experience. Already then, however, Carnap (1928b, SS7) edged towards a formal criterion by requiring empirically significant statements to be such that experiential support for them or for their negation is at least conceivable. Meaningfulness meant the possession of "factual content" which could not, on pain of rendering many scientific hypotheses meaningless, be reduced to actual testability. Instead, the empirical significance of a statement had to be conceived of as possession of the potential to receive direct or indirect experiential support (via deductive or inductive reasoning). In 1930, considerations of this sort appeared to receive a considerable boost due to Waismann's reports of Wittgenstein's meetings with him and Schlick. Wittgenstein had discussed the thesis "The sense of a proposition is [the method of] its verification" in conversations with Schlick and Waismann on 22 December 1929 and 2 January 1930 (see Waismann 1967 [1979, 47 and 79]) and Waismann elaborated it in his "Theses" which were presented to the Circle as Wittgenstein's considered views. While Wittgenstein appears to have thought of his dictum primarily as a constitutive principle of meaning, in the Circle it was put to work mainly as a demarcation criterion against metaphysics. Whether it was always noted that Wittgenstein's thesis and the criterion of cognitive significance must be distinguished (the former entails the latter, but not vice versa) may be doubted, but striking differences emerged all the same. Like Carnap's criterion of significance of 1928, the version of the criterion which followed from Wittgenstein's dictum also allowed for verifiability in principle (and did not demand actual verifiability): like Carnap's notion of experiential support, it worked with the mere conceivability of verifiability. Yet Wittgenstein's criterion required conclusive verifiability which Carnap's did not. (The demand for conclusive verifiability was discussed in the meetings with Wittgenstein.) By 1931, however, it had become clear to some that this would not do. What Carnap later called the "liberalization of empiricism" was underway and different camps became discernible within the Circle. It was over this issue that the so-called "left wing" with Carnap, Hahn, Frank and Neurath first distinguished itself from the "more conservative wing" around Schlick. (See Carnap 1936-37, 422 and 1963a, SS9. Carnap 1936-37, 37n dated the opposition to strict verificationism to "about 1931".) In the first place, this liberalization meant the accommodation of universally quantified statements and the return, as it were, to salient aspects of Carnap's 1928 conception. Everybody had noted that Wittgenstein's criterion rendered universally quantified statements meaningless. Schlick (1931) thus followed Wittgenstein's own suggestion to treat "hypotheses" instead as representing rules for the formation of verifiable singular statements. (His abandonment of conclusive verifiability is indicated only in Schlick 1936a.) By contrast, Hahn (1933, drawn from lectures in 1932) pointed out that hypotheses should be counted as properly meaningful as well and that the criterion be weakened to allow for less than conclusive verifiability. But other elements played into this liberalization as well. One that began to do so soon was the recognition of the problem of the irreducibility of disposition terms to observation terms (more on this presently). A third element was that disagreement arose as to whether the in-principle verifiability or support turned on what was merely logically possible or on what was nomologically possible, as a matter of physical law etc. A fourth element, finally, was that differences emerged as to whether the criterion of significance was to apply to all languages or whether it was to apply primarily to constructed, formal languages. Schlick retained the focus on logical possibility and natural languages throughout, but Carnap had firmly settled his focus on nomological possibility and constructed languages by the mid-thirties. Concerned with natural language, Schlick (1932, 1936a) deemed all statements meaningful for which it was logically possible to conceive of a procedure of verification; concerned with constructed languages only, Carnap (1936-37) deemed meaningful only statements for whom it was nomologically possible to conceive of a procedure of confirmation of disconfirmation. Many of these issues were openly discussed at the Paris congress in 1935. Already in 1932 Carnap had sought to sharpen his previous criterion by stipulating that those statements were meaningful that were syntactically well-formed and whose non-logical terms were reducible to terms occurring in the basic observational evidence statements of science. While Carnap's focus on the reduction of descriptive terms allows for the conclusive verification of some statements, it must be noted that his criterion also allowed universally quantified statements to be meaningful in general, provided they were syntactically and terminologically correct (1932a, SS2). It was not until one of his Paris addresses, however, that Carnap officially declared the criterion of cognitive significance to be mere confirmability. Carnap's new criterion required neither verification nor falsification but only partial testability so as now to include not only universal statements but also the disposition statements of science (see Carnap 1936-37; the English translation of the original Paris address (1936a [1949]) combines it with extraneous material). These disposition terms were thought to be linked to observation statements by a variety of reduction postulates or longer reduction chains, all of which provided only partial definitions (despite their name they provided no eliminative reductions). Though plausible initially, the device of introducing non-observational terms in this way gave rise to a number of difficulties which impugned the supposedly clear distinctions between logical and empirical matters and analytic and synthetic statements (Hempel 1951, 1963). Independently, Carnap himself (1939) soon gave up the hope that all theoretical terms of science could be related to an observational base by such reduction chains. This admission raised a serious problem for the formulation of a criterion of cognitive significance: how was one to rule out unwanted metaphysical claims while admitting as significant highly abstract scientific claims? Consider that Carnap (1939, 1956b) admitted as legitimate theoretical terms that are implicitly defined in calculi that are only partially interpreted by correspondence rules between some select calculus terms and expressions belonging to an observational language (via non-eliminative reductions). The problem was that mere confirmability was simply too weak a criterion to rule out some putative metaphysical claims. Moreover, this problem arose for both the statement-based approach to the criterion (taken by Carnap in 1928, by Wittgenstein in 1929/30, and by Ayer both in the first (1936) and the second editions (1946) of Language Truth and Logic) and for the term-based approach (taken by Carnap since 1932). For the former approach, the problem was that the empirical legitimacy of statements obtained via indirect testing also transferred to any expressions that could be truth-functionally conjoined to them (for instance, by the rule of 'or'-introduction). Statements thus became empirically significant, however vacuous they had been on their own. For the term-based approach, the problem was that, given the non-eliminability of dispositional and theoretical terms, empirical significance was no longer ascribable to individual expressions in isolation but became a holistic affair, with little guarantee in turn for the empiricist legitimacy of all the terms now involved. For most critics (even within the ranks of logical empiricism), the problem of ruling out metaphysical statements while retaining the terms of high theory remained unsolved. By 1950, in response to the troubles of Ayer's two attempts to account for the indirect testing of theoretical statements via their consequences, Hempel conceded that it was "useless to continue to search for an adequate criterion of testability in terms of deductive relationships to observation sentences" (1950 [1959, 116]). The following year, Hempel also abandoned the idea of using, as a criterion of empirical significance, Carnap's method of translatability into an antecedently determined empirical language consisting only of observational non-logical vocabulary. Precisely because it was suitably liberalized to allow abstract scientific theories with merely partial interpretations, its anti-metaphysical edge was blunted: it allowed for combination with "some set of metaphysical propositions, even if the latter have no empirical interpretation at all" (1951, 70). Hempel drew the holistic conclusion that the units of empirical significance were entire theories and that the measure of empirical significance itself was multi-criterial and, moreover, allowed for degrees of significance. To many, this amounted to the demise of the Circle's anti-metaphysical campaign. By contrast, Feigl's reaction (1956) was to reduce the ambition of the criterion of significance to the mere provision of necessary conditions. Some further work was undertaken on rescuing and, again, debunking a version of the statement-based criterion, but mostly not by (former) members of the Vienna Circle. However, in response to the problem of how to formulate a meaning criterion that suitably distinguished between empirically significant and insignificant non-observational terms, Carnap proposed a new solution in 1956. We will return to discuss it separately (see section 3.5 below); for now we need only note that these proposals were highly technical and applied only to axiomatized theories in formal languages. They too, however, found not much favor amongst philosophers. Yet whatever the problems that may or may not beset them, it would seem that far more general philosophical considerations contributed to the disappearance of the problem of the criterion of cognitive significance from most philosophical discussions since the early 1960s (other than as an example of mistaken positivism). These include the increasing opposition to the distinctions between analytic and synthetic statements and observational and theoretical terms as well as a general sense of dissatisfaction with Carnap's approach to philosophy which began to seem both too formalist in execution and too deflationary in ambition. The entire philosophical program of which the search for a precise criterion of empirical significance was a part had begun to fall out of favor (and with it technical discussions about the criterion's latest version). The widely perceived collapse of the classical Viennese project to find in an empiricist meaning criterion a demarcation criterion against metaphysics--we reserve judgement about Carnap's last two proposals here--can be interpreted in a variety of ways. It strongly suggests that cognitive significance cannot be reduced to what is directly observable, whether that be interpreted in phenomenalist or intersubjective, physicalist terms. In that important but somewhat subsidiary sense, the collapse spelt the failure of many of the reductivist projects typically ascribed to Viennese neopositivism (but see section 3.3 below). Beyond that, what actually had failed was the attempt to characterize for natural languages the class of cognitively significant propositions by recursive definitions in purely logical terms, either by relations of deducibility or translatability. What failed, in other words, was the attempt to apply a general conception of philosophical analysis as purely formal, pursued also in other areas, to the problem of characterizing meaningfulness. This general conception can be considered formalist in several senses. It was formalist, first, in demanding the analysis of the meaning of concepts and propositions in terms of logically necessary and sufficient conditions: it was precise and brooked no exceptions. And it was formalist, second, in demanding that such analyses be given solely in terms of the logical relations of these concepts and propositions to other concepts and propositions: it used the tools of formal logic. There is also a third sense that is, however, applicable predominantly to the philosophical project in Carnap's hands, in that it was formalist since it concentrated on the analysis of contested concepts via their explication in formal languages. (Discussion of its viability must be deferred until sections 3.5 and 3.6 below, since what's at issue currently is only the formalist project as applied to concepts in natural language.) The question arises whether all Vienna Circle philosophers concerned with empirical significance in natural language were equally affected, for the collapse of the formalist project may leave as yet untouched other ways of sustaining the objection that metaphysics is, in some relevant sense, cognitively insignificant. (Such philosophers in turn would have to answer the charge, of course, that only the formalist project of showing metaphysics strictly meaningless rendered the Viennese anti-metaphysics distinctive.) Even though the formalist project became identified with mainstream logical empiricism generally (consider its prominence in confirmation theory and in the theory of explanation), it was not universally subscribed to in the Vienna Circle itself. In different ways, neither Schlick nor Neurath or Frank adhered to it. As noted in the overview above, Schlick's attempts to exhibit natural language meaning abjured efforts to characterize it in explicitly formal terms, even though he accepted the demand for necessary and sufficient conditions of significance. In the end, moreover, Schlick turned away from his colleagues' search for a criterion of empirical significance. In allowing talk of life after death as meaningful (1936a), for the very reason that what speaks against it is only the empirical impossibility of verifying such talk, Schlick's final criterion clearly left empiricist strictures behind. By contrast, Neurath and Frank kept their focus on empirical significance. While they rarely discussed these matters explicitly, their writings give the impression that Neurath and Frank chose to adopt (if not retain) a contextual, exemplar-based approach to characterizing the criterion of meaninglessness and so decided to forego the enumeration of necessary and sufficient conditions. Mach's precept cited earlier is an example of such a pragmatic approach, as is, it should be noted, Peirce's criterion of significance, endorsed by Quine (1969), which claims that significance consists in making a discernible difference whether a proposition is likely to be true or false. Mach's pragmatic approach had been championed already before verificationism proper by Neurath, Frank and Hahn who became, like Carnap, early opponents of conclusive verifiability. (Indeed, it is doubtful whether Neurath's radical fallibilism, most clearly expressed already in 1913, ever wavered.) This pragmatic understanding found clear expression in Neurath's adoption (1935a, 1938) of K. Reach's formulation of metaphysical statements as "isolated" ones, as statements that do not derive from and hold no consequences for those statements that we do accept on the basis of empirical evidence or for logical reasons. (Hempel's dismissal, in 1951, of this pragmatic indicator presupposes the desiderata of the formalist project.) Finally, there is Frank's suggestion (1963), coupled with his longstanding advice to combine logical empiricism with pragmatism, that Carnap's purely logical critique of metaphysics in (1932a) was bound to remain ineffective as long as the actual use of metaphysics remained unexamined. It would be worth investigating whether--if the critique of the alleged reductionist ambitions of their philosophy could also be deflected (see section 3.3 below)--the impetus of the anti-verificationist critique can be absorbed by those with a pragmatic approach to the demarcation against metaphysics. Much as with Quine's Peirce, such a criterion rules out as without interest for epistemic activity all concepts and propositions whose truth or falsity make no appreciable difference to the sets of concepts and propositions we do accept already. An entirely different moral was drawn by Reichenbach (1938) and thinkers indebted to his probabilistic conception of meaning and his probabilistic version of verificationism, which escaped the criticisms surveyed above by vagaries of its own. Such theorists perceive the failure of the formalist model to accommodate the empirical significance of theoretical terms to stem from its so-called deductive chauvinism. In place of the exclusive reliance on the hypothetical-deductive method these theorists employ non-demonstrative analogical and causal inductive reasoning to ground theoretical statements empirically. Like Salmon, these theorists adopt a form of "non-linguistic empiricism" which they sharply differentiate from the empiricism of the Vienna Circle (Salmon 1985, 2003 and Parrini 1998). Now against both the pragmatic and the post-linguistic responses to the perceived failure of the attempt to provide a precise formal criterion of significance serious worries can be raised. Thus it must be asked whether without a precise way of determining when a statement 'makes an appreciable difference', criticism of metaphysics based on such a criterion may be not be considered as a biased dismissal rather than a demonstration of fact and so fall short of what is needed. Likewise in the case of the anti-deductivist response, it must be noted that a criterion based on analogical reasoning will only be as effective as the strength of the analogy which can always be criticized as inapt (and similarly for appeals to causal reasoning). The very point of exact philosophy in a scientific spirit--for many the very point of Vienna Circle philosophy itself--seems threatened by such maneuvres. Acquiescence in the perceived failure of the proposed criteria of significance thus comes with a price: if not that of abandoning Vienna Circle philosophy altogether, then at least that of formulating an alternative understanding to how some of its ambitions ought to be understood. (Recent reconstructive work on Carnap, Neurath and Frank may be regarded in this light.) A still different response--but one emblematic for the philosophical public at large--is that of another of Reichenbach's former students, Putnam, who came to reject the anti-metaphysical project that powered verificationism in its entirety. Repeatedly in his later years, Putnam called for a refashioning of analytic philosophy as such, providing, as it were, a philosophically conservative counterweight to Rorty's turn to postmodernism. Putnam's reasons (the alleged self-refutation of the meaning criterion) are still different from those surveyed above and will be discussed when we return to reconsider the very point of the Circle's campaign against metaphysics (see section 3.6 below). ### 3.2 The Analytic/Synthetic Distinction and the Relative A Priori Whether the verificationist agenda was pursued in a formalist or pragmatic vein, however, all members shared the belief that meaningful statements divided exclusively into analytic and synthetic statements which, when asserted, were strictly matched with a priori and a posteriori reasoning for their support. The Vienna Circle wielded this pairing of epistemic and semantic notions as a weapon not only against the substantive a priori of the Schoolmen but also against Kant's synthetic a priori. Moreover, their notion of analyticity comprized both logical and mathematical truths, thereby extending Wittgenstein's understanding of the former as "tautological" in support of a broadly logicist program. It is well known that this central component of the Vienna Circle's arsenal, the analytic/synthetic distinction, came under sharp criticism from Quine in his "Two Dogmas of Empiricism" (1951a), less so that the criticism can only be sustained by relying on objections of a type first published by Tarski. The argument is more complex, but here is a very rough sketch. So as to discard the analytic/synthetic distinction as an unwarranted dogma, Quine in "Two Dogmas" argued for the in-principle revisability of all knowledge claims and criticised the impossibility of defining analyticity in a non-circular fashion. The first argument tells against the apodictic a priori of old (the eternal conceptual verities), but, as we shall see, it is unclear whether it tells against at least some of the notions of the a priori held in the Vienna Circle. The second argument presupposes a commitment to extensionalism that likewise can be argued not to have been shared by all in the Circle. By contrast, Tarski had merely observed that, at a still more fundamental level, he knew of no basis for a sharp distinction between logical and non-logical terms. (For relevant primary source materials see also Quine 1935, 1951b, 1963, Carnap 1950, 1955, 1963b, their correspondence and related previously unpublished lectures and writings in Creath 1990, the memoir Quine 1991, and Tarski 1936.) The central role on the Vienna Circle's side in this discussion falls to Carnap and the reorientation of philosophy he sought to effect in Logical Syntax (1934/37). It was the notion of the merely relative and therefore non-apodictic a priori that deeply conditioned his notion of analyticity and allowed him to sidestep Quine's fallibilist argument in a most instructive fashion. In doing so Carnap built on an idea behind Reichenbach's early attempt (1920) to comprehend the general theory of relativity by means of the notion of a merely constitutive a priori. Now Schlick had objected to the residual idealism of this proposal (see Oberdan 2009) and prefered talk of conventions instead and Reichenbach soon followed him in this (see Coffa 1991, Ch. 10). Carnap too did not speak of the relative a priori as such (in returning to this terminology present discussions follow Friedman 1994), but his pluralism of logico-linguistic frameworks furnishes precisely that. First consider Schlick as a contrast class. Schlick (1934) appeared to show little awareness of the language-relativity of the analytic/synthetic distinction and spoke of analytic truths as conceptual necessities that can be conclusively surveyed. This would suggest that Schlick rejected Kant's apodictic synthetic a priori but not the apodicity of analytic statements. Clearly, if that were so, Quine's argument from universal fallibilism would find a target here. Matters are not quite so clear-cut, however. Schlick had long accepted the doctrine of semantic conventionalism that the same facts could be captured by different conceptual systems (1915): this would suggest that his analytic truths were conventions that were framework-relative and as such necessary only in the very frameworks they helped to constitute. Yet Schlick did not countenance the possibility of incommensurable conceptual frameworks: any fact was potentially expressible in any framework (1936b). As a result, Schlick did not accept the possibility that after the adoption of a new framework the analytic truths of the old one may be no longer assertable, that they could be discarded as no longer applicable even in translation, as it were. Herein lay a point that Quine's argument could exploit: Schlick's analytic truths remained unassailable despite their language-relativity. Now Carnap, under the banner of the principle of logical tolerance (1934/37, SS17), abandoned the idea of the one universal logic which had informed Frege, Russell and Wittgenstein before him. Instead, he recognized a plurality of logics and languages whose consistency was an objective matter even though axioms and logical rules were fixed entirely by convention. Already due to this logical pluralism, the framework-relativity of analytic statements went deeper for Carnap than it did for Schlick. But Carnap also accepted the possibility of incommensurability between seemingly similar descriptive terms and between entire conceptual systems (1936a). Accepting the analytic truths of the framework of our best physical theory may thus be incompatible with accepting those of an earlier one, even if the same logic is employed in both. Carnapian analyticities do not therefore express propositions that we hold to be true unconditionally, but only propositions true relative to their own framework: they are no longer held to be potentially translatable across all frameworks. Quine's charge of universal revisability (which itself needs some modification; see Putnam 1978) thus misses its mark against them. (Quine, of course, also rejected Carnap's ultimately intensionalist accommodation of radical fallibilism via the notion of language change.) Concerning the criticism of the circular nature of the definition of analyticity, Carnap responded that it pertained primarily to the idea of analyticity in natural language whereas he was interested in "explications" as provided by the logic of science (or better, a logic of science, since there existed no unique logic of science for Carnap). Explications are reconstructions in a formal language of selected aspects of complex terms that should not be expected to model the original in all respects (1950b, Ch.1). Moreover, Carnap held that explication of the notion of analyticity in formal languages yielded the kind of precision that rendered the complaint of circularity irrelevant: vague intuitions of meaning were no longer relied upon. Those propositions of a given language were analytic that followed from its axioms and, once the syntactic limitations of the Logical Syntax period had been left behind, from its definitions and meaning postulates, by application of its rules: no ambiguity obtained. So it may appear that the notion of analyticity is easily delimitable in Carnap's explicational approach: analytic propositions would be those that constitute a logico-linguistic framework. But complications arose from the fact that, on Carnap's understanding, not all propositions defining a logico-linguistic framework need be analytic ones (1934/37, SS51). It was possible for a framework to consist not only of L-rules, whose entirety determines the notion of logical consequence, but also of P-rules, which represent presumed physical laws. So let analytic propositions be those framework propositions whose negations are self-contradictory. Here a problem arose once the syntactic constraints were dropped by Carnap after Logical Syntax so as to allow semantic reasoning and the introduction of so-called meaning postulates: now the class of analytic propositions was widened to include not only logical and mathematical truths but also those obtained by substitution of semantically equivalent expressions. How was one now to explicate the idea that there can be non-analytic framework propositions (whose negations are not self-contradictory)? Consider that for opponents like Quine, responding that the negations of non-analytic framework propositions do not contradict meaning postulates, was merely to dress up a presupposed notion of meaning in pseudo-formal garb: while it provided what looked like formal criteria, Carnap's method did not leave the circle of intensional notions behind and so seemed to beg the question. Meaning postulates (Carnap 1952), after all, could only be identified by appearing on a list headed "meaning postulates" (as Quine added in reprints of 1951a). Here one must note that in Logical Syntax, Carnap also modified the thesis of extensionality he had previously defended alongside Russell and Wittgenstein: now it merely claimed the possibility of purely extensional languages and no longer demanded that intensional languages be reduced to them (ibid., SS67). Of course, the mere claim that the language of science can be extensional still proves troublesome enough, given that in such a language a distinction between laws and accidentally true universal propositions cannot be drawn (the notion of a counterfactual conditional, needed to distinguish the former, is an intensional one). Even so, this opening of Carnap's towards intensionalism already at the height of his syntacticism--to say nothing of his explicit intensionalism in Meaning and Necessity (1947)--seems enough to thwart Quine's second complaint in "Two Dogmas". Carnap did not share Quine's extensionalist agenda, so the need to break out of the circle of intensional notions once these were clearly defined in his formal languages did not apply. That theirs were in fact different empiricist research programmes was insufficiently stressed, it would appear, by Quine and Quinean critics of Carnap (as noted pointedly by Stein 1992; cf. Ricketts 1982, 2003, Creath 1991, 2004, Richardson 1997). To sustain his critique, Quine had to revive his and Tarski's early doubts about Carnap's methodological apparatus and dig even deeper. (Tarski also shared Quine's misgivings about analyticity when they discussed these issues with Carnap at Harvard; see Mancosu 2005, Frost-Arnold 2013.) Their scepticism found its target in Carnap's ingenious measures in Logical Syntax taken to preserve the thesis that mathematics is analytic from the ravages of Godel's incompleteness theorems. Godel proved that every formal system strong enough to represent number theory contains a formula that is true but neither itself or its negation is provable in that system; such formulae--known since as Godel sentences--are provable in a still stronger system which, however, also contains a formula of its own that is true but not provable in it (and neither is its negation). Commonly, Godel's proof is taken to have undermined the thesis of the analyticity of arithmetic. (For discussions of this challenge to Carnap's logical syntax project, see Friedman 1988, 1999a, 2009, Goldfarb and Ricketts 1992, Richardson 1994, Awodey and Carus 2003, 2004.) Carnap responded by stating that arithmetic demands an infinite sequence of ever richer languages and by declaring analytic statements to be provable by non-finite reasoning (1934/37, SS60a-d). This looked like fitting the bill on purely technical grounds, but it is questionable whether such reasoning may still count as syntactic. Nowadays, it is computational effectiveness that is taken to distinguish purely formal from non-formal, material reasoning. (Carnap's move highlights the tension within Logical Syntax between formal and crypto-semantic reasoning: that the rigid syntacticism officially advertised there was at the same time undermined as its failings were being compensated--illegitimately so by official standards, e.g., by considering translatability a syntactic notion--points ahead to his acceptance of semantics in 1935, only one year after the publication of Logical Syntax and contrary to his disparagement of the notion of synthetic truth there. For a discussion of Carnap's moves, see Coffa 1978, Ricketts 1996, Goldfarb 1997, Creath 1996, 1999.) It is no criticism that Carnap's reconstruction of arithmetic was not standard logicism, but that Carnap unduly stretched the idea of formal reasoning is. Was he saved by his shift to semantics? Tarski (1936) granted the language-relativity of the reconstructed notion of analyticity in Logical Syntax. He also did not object that Carnap's procedure of circumventing the problem which the Godel sentences presented to the thesis of the analyticity of arithmetic was illegitimate. Tarski rather questioned whether there were "objective reasons" for the sharp distinction between logical and non-logical terms and he pointed out that Carnap's distinction between the logical and the empirical was not a hard and fast one. Since noting that the distinction between logical and non-logical was not a sharp one and arguing that no principled distinction could be upheld between them are two quite different reactions, however, Tarski's point on its own does not fully support the Quinean critique. Quine's conclusion (1940, SS60) that the notion of logical truth itself is "informal" rather reflects the moral that he drew from Tarski's observation. It appears that what motivated him (after a nominalistic interlude in the 1940s) to develop his naturalistic alternative to Carnap's conception of philosophy was his considered rejection of Carnap's accommodation of the thesis that arithmetic is analytic to Godel's result. Different strands of Quine's criticism of the analytic/synthetic distinction must thus be distinguished. While Quine's criticisms in "Two Dogmas" on their own clearly did not undermine all forms of the distinction that were defended in the Vienna Circle--Carnap's reconstructions of the notion of analyticity did not express unconditionally necessary and unrevisable propositions--they do gain in plausibility even against Carnap's once it is recognized that the deepest ground of contestation lies elsewhere: not in the notion of analyticity widely understood but in that of logical truth narrowly understood. Read in this way, Quine can be seen to argue that the notion of L-consequence as explication of analytic truth--as opposed to P-consequence as non-analytic, mere framework entailments--traded not only on the idea of non-finitary notions of proof but also on a distinction of logical from descriptive expressions that itself only proceeded on the basis of a finite enumeration of the former (compare Carnap 1934/37, SSSS51-52). (This deficit was not repaired in Carnap's later work either; see Awodey 2007.) What Quine criticized was precisely the fact that Carnap could ground the distinction between logical and non-logical terms no deeper than by the enumeration of the former in a given framework: was the distinction therefore not quite arbitrary? Quine's direct arguments against the distinction between logical and empirical truth (1963) have been found to beg the question against Carnap and his way of conceiving of philosophy (Creath 2003). This way of responding to Quine's objection requires us to specify still more precisely just what Carnap thought he was doing when he employed the distinction between analytic and synthetic propositions. To be sure, in (1955) he gave broadly behavioural criteria for when meaning ascriptions could be deemed accepted in linguistic practice, but he also noted that this was not a general requirement for the acceptability of explicatory discourse. To repeat, explications did not seek to model natural language concepts in their tension-filled vividness, but to make proposals for future use and to extract and systematize certain aspects for constructive purposes. Thus Carnap clarified (1963b, SS32) that he regarded the distinction between analytic and synthetic statements--just like the distinction between descriptive or factual and prescriptive or evaluative statements--not as descriptive of natural language practices, but as a constructive tool for logico-linguistic analysis and theory construction. It is difficult to overstress the significance of this stance of Carnap's for the evaluation of his version of the philosophical project of the Vienna Circle. Carnap's understanding of philosophy has thus been aptly described as the "science of possibilities" (Mormann 2000). As Carnap understood the analytic/synthetic distinction, it was a distinction drawn by a logician to enable greater theoretical systematicity in the reconstructive understanding of a given symbol system, typically a fragment of a historically developed one. That fully determinative objective criteria of what to regard as a logical and what as a non-logical term cannot be assumed to be pre-given does not then in and of itself invalidate the use of that distinction by Carnap. On the contrary, it has been convincingly argued that Carnap himself did not hold to a notion of what is a factual and what is a formal expression or statement that was independent of the specification of the language in question (Ricketts 1994). The ultimate ungroundedness of his basic semantic explicatory categories, this suggests instead, was a fact that his own theories fully recognized and consciously exploited. (Somewhat analogously, that we cannot define science independently of the practice of scientists of "our culture" was admitted by Neurath 1932a, Carnap 1932d and Hempel 1935, much to the exasperation of Zilsel 1932 and Schlick 1935a.) It remained open for Carnap then to declare his notion of analyticity to be only operationally defined for constructed languages and to let that notion be judged entirely in terms of its utility for meta-theoretical reflection. Just on that account, however, a last hurdle remains: finding a suitable criterion of significance for theoretical terms that allows the distinction between analytic and synthetic statements to be drawn in the non-observational, theoretical languages of science. (This was a problem ever since the non-eliminative reducibility of disposition terms had been accepted and one that still held for Carnap's 1956 criterion; see section 3.5 below). Only if that can be done, we must therefore add, can Carnap claim his formalist explicationist project to emerge unscathed from the criticisms of both Tarski and Quine. An important related though independently pursued line of criticism may be noted here. It finds its origin in Saunders Mac Lane's review (1937) of Carnap's Logical Syntax and focusses less on the analytic/synthetic distinction than on Carnap's failure to give a formally correct definition of logic. It challenges the ambition to have accounted for the formal sciences but declines to embrace a naturalistic alternative. Further research along these lines has suggested to some that by extending Carnap's approach and framework it can be linked to attempts in category theory to provide the missing definition (see Awodey 2012, 2017), while a different response to Mac Lane's as well as Quine's criticisms defends Carnap's inability, frankly admitted in (1942, 57) to define logical terms as such in full generality (Creath 2017). None of the above considerations should lead one to deny, however, that one can find understandings of the term "analytic" by members of the Vienna Circle (like Schlick) that do fall victim to the criticisms of Quine more easily. Nor should one discount the fact that Carnap's logic of science emerges as wilfully ill-equipped to deal with the problems that exercise the traditional metaphysics or epistemologies that deal in analyticities. (Of course, unlike his detractors, Carnap considered this to be a merit of his approach.) Lastly, it must be noted that Carnap's intensionalist logic of science holds out the promise of practical utility only for the price of a pragmatic story that remains to be told. Of what nature are the practical considerations and decisions that, as Carnap so freely conceded (1950a), are called for when choosing logico-linguistic frameworks? (Such conventional choices do not respond to truth or falsity, but instead to whatever is taken to measure convenience.) That Carnap rightly may have considered such pragmatic questions beyond his own specific brief as a logician of science does not obviate the need for an answer to the question itself. (Carus 2007 argues that in this broadly pragmatic dimension lies the very point of Carnap's explicationism.) On this issue, too, it would have been helpful if there had been more collaboration with Neurath and Frank, who were sympathetic to Carnap's explication of analyticity but did not refer to it much in their own, more practice-oriented investigations (see section 3.6 below). ### 3.3 Reductionism and Foundationalism: Two Criticisms Partly Rebutted As it happens, anti-verificationism has two aspects: opposition to meaning reductionism and opposition to the formalist project. Turning to the former, we must distinguish two forms of reductionism, phenomenalist and physicalist reductionism. Phenomenalism holds statements to be cognitively significant if they can be reduced to statements about one's experience, be it outer (senses) or inner (introspection). Physicalism holds statements to be cognitively significant if they can be reduced or evidentially related to statements about physical states of affairs. Here it must be noted not only that the Vienna Circle is typically remembered in terms of the apparently phenomenalist ambitions of Carnap's Aufbau of 1928, but also that already by the early 1930s some form of physicalism was favoured by some leading members including Carnap (1932b) and that already in the Aufbau the possibility of a different basis for a conceptual genealogy than the phenomenal one actually chosen had been indicated. Thus one must not only ask about the reductionism in the Aufbau but also consider just how reductivist in intent the physicalism was meant to be. Considerations can begin with an early critique that has given rise in some quarters to a sharp distinction between Viennese logical positivism and German logical empiricism, with the former accused of reductionism and the latter praised for their anti-reductionism, a distinction which falsely discounts the changing nature and variety of Vienna Circle doctrines. Reichenbach's defense of empiricism (1938) turned on the replacement of the criterion of strict verifiability with one demanding only that the degree of probability of meaningful statements be determinable. This involved opposition also to demands for the eliminative reduction of non-observational to observational statements: both phenomenalism and reductive physicalism were viewed as untenable and a correspondentist realism was advanced in their stead. Now it is true that of the members of the Vienna Circle only Feigl ever showed sympathies for scientific realism, but it is incorrect that all opposition to it in the Circle depended on the naive semantics of early verificationism. Again, of course, some Vienna Circle positions were liable to Reichenbach's criticism. Another misunderstanding to guard against is that the Vienna Circle's ongoing concern with "foundational issues" and the "foundations of science" does in itself betoken foundationalism. (In the Vienna Circle's days, foundationalism had it that the basic items of knowledge upon which all others depended were independent of each other, concerned phenomenal states of affairs and were infallible; nowadays, foundationalists drop phenomenalism and infallibility.) Already the manifesto sought to make clear the Circle's opposition when it claimed that "the work of 'philosophic' or 'foundational' investigations remains important in accord with the scientific world conception. For the logical clarification of scientific concepts, statements and methods liberates one from inhibiting prejudices. Logical and epistemological analysis does not wish to set barriers to scientific enquiry; on the contrary, analysis provides science with as complete a range of formal possibilities as is possible, from which to select what best fits each empirical finding (example: non-Euclidean geometries and the theory of relativity)." (Carnap-Hahn-Neurath 1929 [1973, 316]) This passage can be read as an early articulation of the project of a critical-constructivist meta-theory of science that abjures a special authority of its own beyond that stemming from the application of the methods of the empirical and formal sciences to science itself, but instead remains open to what the actual practice of these sciences demands. How then can Vienna Circle philosophy be absolved of foundationalism? As noted, it is the Aufbau (and echoes of it in the manifesto) that invites the charge of phenomenalist reductionism. To begin with, one must distinguish between the strategy of reductionism and the ambition of foundationalism. Concerning the Aufbau it has been argued that its strategy of reconstructing empirical knowledge from the position of methodological solipsism (phenomenalism without its ontological commitments and some of its epistemological ambitions) is owed not to foundationalist aims but to the ease by which this position seemed to allow the demonstration of the interlocking and structural nature of our system of empirical concepts, a system that exhibited unity and afforded objectivity, which was Carnap's main concern. (See Friedman's path-breaking 1987 and 1992, and Richardson 1990, 1998, Ryckman 1991, Pincock 2002, 2005. For the wide variety of influences on the Aufbau more generally, see Dambock 2016.) However, it is hard to deny categorically that Carnap ever harbored foundationalist ambitions. Not only did Carnap locate his Aufbau very close to foundationalism in retrospect (1963a), but a passage in his (1930) led Uebel (2007, Ch.6) to claim that around 1929/30 Carnap was motivated by foundationalist principles and reinterpreted his own Aufbau along these lines (around the same time that Wittgenstein entertained a psychologistic reinterpretation of his own Tractatus that was reported back to the Circle by Waismann). To correct this foundationalist aberration was the task of the Circle's subsequent protocol-sentence debate about the content, form and status of the evidence statements of science. This concession to the foundationalist misinterpretation of Vienna Circle philosophies generally must not, however, be taken to tell against the new reading of the Aufbau or the epistemologies developed from 1930 onwards on the physicalist wing of the Circle. In fact, the Aufbau itself fails when it is read as a foundationalist project, as it was by Quine (1951a) who pointed out that no eliminative definition of the relation 'is at' was provided (required for locating objects of consciousness in physical space). Yet other failures of reduction were detected by Richardson (1998, Ch.3), further questioning the Aufbau's supposed foundationalist intent. Ultimately it was a still different failure of reduction that prompted Carnap to abandon as mistaken reconstructions of the scientific language on the basis of methodological solipsism (though not logical investigations of such languages for their own sake, as noted in his 1961a). Initially Carnap had not been prepared to draw this conclusion even though Neurath (1931b, 1932a) argued that such a type of rational reconstruction traded on objectionable counterfactual presuppositions (methodological solipsism did not provide a correct description of the reasoning involved in cognitive commerce with the world around us). At the time Carnap merely conceded that it was more "convenient" to reconstruct the language of science on a physicalistic basis (1932e). Only the failure of reducing dispositional predicates to observational ones convinced him to abandon the methodologically solipsist approach (1936-37, 464 and 10) and to adopt an exclusively physicalist basis for his reconstructions of the language of science from then on (a decision explicitly reaffirmed in 1963b). Carnap's initial reluctance to draw this conclusion is best interpreted, not as hankering after epistemological foundations, but as indicating the growing conviction that philosophical methodology had to be built on logical reasons and should not be "entangled with psychological questions" (1934/37, SS72). Moreover, already Carnap's initial hope for a phenomenalist reduction in the Aufbau as well as his decision in (1932e) to loosen the methodologically solipsist claim on epistemology were motivated by logical considerations, the former by mistaken ideas in Untersuchungen zur allgemeinen Axiomatik (2000, unpublished at the time) and the latter partly as a consequence of Godel's advice on a faulty definition of analyticity in a draft of Logical Syntax (see Ricketts 2010 and Awodey and Carus 2007 respectively). It could likewise be asked concerning physicalism whether it represented, on a different basis, the pursuit of a foundationalist agenda. But that Carnap later in (1936/37) called his non-eliminative definitions of disposition terms "reduction sentences" already indicates that it was enough for him to provide a basis for the applicability of these terms by merely sufficient but not necessary conditions. Likewise, Carnap's proposal (1939) to conceive of theoretical terms as defined by implicit definition in a non-interpreted language, selected terms of which were, however, linked via non-eliminative reductive chain to the observational language, suggests that what concerned him primarily was the capture of indicator relations to sustain in principle testability of statements containing the terms in question. This is best understood as an attempt to preserve the empirical applicability of the formal languages constructed for high-level theory, but not as reductivism with regard to some foundational given. By contrast, Neurath never advocated methodological solipsism. Consider that his complex conception of the form of protocol statements (1932b) explicated the concept of observational evidence in terms that expressly reflected debts to empirical assumptions which called for theoretical elaboration in turn. For unlike the logician of science Carnap, who left it to psychology and brain science to determine more precisely what the class observational predicates were that could feature in protocol statements (1936/37), the empirically oriented meta-theoretician of science Neurath was concerned to encompass and comprehend the practical complexities of reliance on scientific testimony: the different clauses (embeddings) of his proposed protocol statements stand for conditions on the acceptance of such testimony (see Uebel 2009). In addition, Neurath's theory of protocol statements also makes clear that his understanding of physicalism did not entail the eliminative reduction of the phenomenon of intentionality but, like Carnap (1932c), merely sought its integration into empiricist discourse. Given these different emphases of their respective physicalisms, mention must also be made of the significant differences between Carnap's and Neurath's conceptions of unified science: where the formalist Carnap once preferred a hierarchical ordering of finitely axiomatized theoretical languages that allowed at least partial cross-language definitions and derivations--these requirements were liberalized over the years (1936b), (1938), (1939)--the pragmatist Neurath opted from the start to demand only the interconnectability of predictions made in the different individual sciences (1935a), (1936c), (1944). (Metereology, botany and sociology must be combinable to predict the consequences of a forest fire, say, even though each may have its own autonomous theoretical vocabulary.) Here too it must be remembered that, unlike Carnap, Neurath only rarely addressed issues in the formal logic of science but mainly concerned himself with the partly contextually fixed pragmatics of science. (One exception is his 1935b, a coda to his previous contributions to the socialist calculation debate with Ludwig von Mises and others.) Not surprisingly, at times the priorities set by Neurath for the pragmatics of science seemed to conflict with those of Carnap's logic of science. (These tensions often were palpable in the grand publication project undertaken by Carnap and Neurath in conjunction with Morris, the International Encyclopedia of the Unity of Science; see Reisch 2003.) That said, however, note that Carnap's more hierarchical approach to the unity of science also does not support the attribution of foundationalist ambitions. But for a brief lapse around 1929/30 and perhaps in some pre-Vienna years, then, Carnap fully represents the position of Vienna Circle anti-foundationalism. In this he joined Neurath whose long-standing anti-foundationalism is evident from his famous simile, first used in 1913, that likens scientists to sailors who have to repair their boat without ever being able to pull into dry dock (1932b). Their positions contrasted at least prima facie with that of Schlick (1934) who explicitly defended the idea of foundations in the Circle's protocol-sentence debate. Even Schlick conceded, however, that all scientific statements were fallible ones, so his position on foundationalism was by no means the traditional one. The point of his "foundations" remained less than wholly clear and different interpretation of it have been put forward (e.g., Oberdan 1998, Uebel 1996, 2020). (On the protocol sentence debate as a whole, which included not only the debate between Carnap and Neurath but also debates between the physicalists and Schlick and other occasional participants, see, e.g., the differently centered accounts of Uebel 1992, 2007, Oberdan 1993, Cirera 1994.) While all in the Circle thus recognized as futile the attempt to restore certainty to scientific knowledge claims, not all members embraced positions that rejected foundationalism tout court. Clearly, however, attributing foundationalist ambitions to the Circle as a whole constitutes a total misunderstanding of its internal dynamics and historical development, if it does not bespeak wilfull ignorance. At most, a foundationalist faction around Schlick can be distinguished from the so-called left wing whose members pioneered anti-foundationalism with regard to both the empirical and formal sciences. ### 3.4 Scientific Theories, Theoretical Terms and the Problem of Realism Yet even if it be conceded that the members of the Vienna Circle did not harbour undue reductionist-foundationalist ambitions, the question remains open whether they were able to deal with the complexities of scientific theory building. Here the prominent role of Schlick must be mentioned, whose General Theory of Knowledge (1918, second edition 1925) was one of the first publications by (future) members of the Vienna Circle to introduce the so-called two-languages model of scientific theories. According to this model, scientific theories comprised an observational part formulated with observational predicates as customarily interpreted, in which observations and experiential laws were stated, and a theoretical part which consisted of theoretical laws the terms of which were merely implicitly defined, namely, in terms of the roles they played in the laws in which they figured. Both parts were connected in virtue of a correlation that could be established between selected terms of the theoretical part and observational terms. In the second half of the 1920s, however, Schlick's model, involving separate conceptual systems, was put aside in favor of a more streamlined conception of scientific theories along lines as suggested by the Aufbau. Clearly, however, Schlick's model represents an early form of the conception of scientific theories as uninterpreted calculi connected to observation by potentially complicated correspondence rules that Carnap reintroduced in (1939) and that became standard in the "received view". (Another, albeit faint precursor was the idea contained in a 1910 remark of Frank's pointing out the applicability of Hilbert's method of implicit definition to the reconstruction empirical scientific theories as conceived, also along the lines of two distinct languages, by the French conventionalists Rey and Duhem; see Uebel 2003.) Even granted the model in outline, questions arise both concerning its observational base as well as its theoretical superstructure. We already discussed one aspect of the former topic, the issue of protocols, in the previous section; let's here turn to the latter topic. Talk of correspondence rules only masks the problem that is raised by theoretical terms. One of the pressing issues concerns their so-called surplus meaning over and above their observational consequences. This issue is closely related to the problem of scientific realism: are there truth-evaluatable matters of fact for scientific theories beyond their empirical, observational adequacy? Even though the moniker "neo-positivism" would seem to prescribe an easy answer as to what the Vienna Circle's position was, it must be noted that just as there is no consensus discernible today there was none in the Circle beyond certain basics that left the matter undecided. All in the Vienna Circle followed Carnap's judgement in Pseudoproblems of Philosophy (1928b) and Schlick's contention in his response to Planck's renewal of anti-Machian polemics (1932) that questions like that of the reality of the external world were not well-formed ones but only constituted pseudo-questions. While this left the observables of empirical reality clearly in place, theoretical entities remained problematical: were they really only computational fictions introduced for the ease with which they they allowed complex predictive reasoning, as Frank (1932) held? This hardly seems to do justice to the surplus meaning of theoretical terms over and above their computational utility: theories employing them seem to tell us about non-observable features of the world. This indeed was Feigl's complaint (1950) in what must count as the first of very few forays into "semantical realism" (scientific realism by another word) by a former member of the Vienna Circle--and one that was quickly opposed by Frank's instrumentalist rejoinder (1950). Carnap sought to remain aloof on this as on other ontological questions. So while in the heyday of the Vienna Circle itself the issue had not yet come into clear focus, by mid-century one could distinguish amongst its surviving members both realists (Feigl) and anti-realists (Frank) as well as ontological deflationists (Carnap). Carnap's general recipe for avoiding undue commitments (while pursuing his investigations of various language forms, including the intensional ones Quine frowned upon) was given in terms of the distinction between so-called internal and external questions (1950a). Given the adoption of a logico-linguistic framework, we can state the facts in accordance with what that framework allows us to say. Given any of the languages of arithmetic, say, we can state as arithmetical fact whatever we can prove in them; to say that there are numbers, however, is at best to express the fact that numbers are a basic category of that framework (irrespective of whether they are logically derived from a still more basic category). As to whether certain special types of numbers exist (in the deflated sense), that depends on the expressive power of the framework at hand and on whether the relevant facts can be proven. Analogous considerations apply to the existence of physical things (the external world) given the logico-linguistic frameworks of everyday discourse and empirical science. (The near-tautologous nature of these categorical claims in Carnap's hands echoes his earlier diagnosis of metaphysical claims as pseudo-statements; see also 1934/37, Part V.A.) Unlike such internal questions, however, external questions, questions whether numbers or electrons "really exist" irrespective of any framework, are ruled out as illegitimate and meaningless. The only way in which sense could be given to them was to read them as pragmatic questions concerned with the utility of talk about numbers or electrons, of adopting certain frameworks. Carnap clearly retained his basic position: existence claims remain the province of science and there must be seen as mediated by the available conceptual tools of inquiry. Logicians of science are in no position to double-guess the scientists in their own proper domain. (Needless to say, Quine 1951b opposed the internal/external distinction as much as his 1951a opposed the analytic/synthetic distinction.) Matters came to a head with the discovery of a proof (see Craig 1956) that the theoretical terms of a scientific theory are dispensable in the sense of it being possible to devise a functionally equivalent theory that does not make use of them. Did this not rob theoretical terms of their distinctive role and so support instrumentalism? The negative answer was twofold. As regards defending their utility, Carnap (1963b, SS24) agreed with Hempel (1958) that in practice theoretical terms were indispensable in facilitating inductive relations between observational data. As regards the defense of their cognitive legitimacy, Carnap held that this demanded determining what he called their "experiential import", namely, determining what specifically their empirical significance consisted in. It was for this purpose that Carnap came to employ Ramsey's method of regimenting theoretical terms. Nowadays this so-called ramseyfication is often discussed as a means for expressing a position of "structural realism", a position midway between fully-blown scientific realism and anti-realism and so sometimes thought to be of interest to Carnap. Carnap's own concern with ramseyfication throws into relief not only the question of the viability of one of the Vienna Circle's most forward-looking stances in the debate about theoretical terms--intending to avoid both realism and anti-realism--but also several other issues that bear on the question of which, if any, forms of Vienna Circle philosophy remain viable. ### 3.5 Carnap's Later Meaning Criterion and the Problem of Ramseyfication Note that the issue of realism vis-a-vis theoretical terms is closely related to two other issues central to the development of Vienna Circle philosophy: Carnap's further attempts to develop a criterion of empiricist significance for the terms of the theoretical languages of science and his attempts to defend the distinction between synthetic and analytic statements with regard to such theoretical languages. In 1956 Carnap introduced a new criterion of significance specifically for theoretical terms (1956b). This criterion was explicitly theory-relative. Roughly, Carnap first defined the concept of the "relative significance" of a theoretical term. A term is relatively significant if and only if there exists a statement in the theoretical language that contains it as the only non-logical term and from which, in conjunction with another theoretical statement and the sets of theoretical postulates and correspondence rules, an observational statement is derivable that is not derivable from that other theoretical statement and the sets of theoretical postulates and correspondence rules alone. Then Carnap defined the "significance" of a theoretical term in terms of it belonging to a sequence of such terms such that each is relatively significant to the class of those terms that precede it in the sequence. Now those theoretical statements were legitimate and cognitively significant that were well-formed and whose descriptive constants were significant the sense just specified. It is clear that by the stepwise introduction of theoretical terms as specified, Carnap sought to avoid the deleterious situations that rendered Ayer's criterion false (and his own of 1928). Nevertheless, this proposal too was subjected to criticism (e.g., Rozeboom 1960, Kaplan 1975a). A common impression amongst philosophers appears to be that this criterion failed as well, but this judgement is by no means universally shared (for the majority view see Glymour 1980, for a contrary assessment see Sarkar 2001). Thus it has been argued that subject to some further refinements, Carnap's proposal can be made to work (see for discussion Creath 1976, Justus 2014, Lutz 2017)--as long as the sharp distinction between observational and theoretical terms can be sustained. (In light of the objections to the latter distinction one wants to add: or by a dichotomy of terms functionally equivalent to it.) Carnap's own position on his 1956 criterion appears somewhat ambiguous. While he is reported to have accepted one set of criticisms (Kaplan 1975b), he also asserted even after they had been put to him that he thought his 1956 criterion remained adequate (1963b, SS24b). Yet Carnap there also advised investigation of whether still another, then entirely new approach to theoretical terms that he was developing would allow for an improved criterion of significance for them. So when Carnap offered "the Ramsey method" as a method of characterizing the "empirical meaning of theoretical terms" it was not their empirical significance as such but the specific empirical import of theoretical terms that he considered (1966, Ch. 26). What prompted him to undertake his investigations of ramseyfications was not dissatisfaction with his 1956 proposal as a criterion of significance for theoretical terms, but the fact that it still proved impossible with this model to draw the distinction between synthetic and analytic statements in the theoretical language. The reason for this was that the postulates for the theoretical language also specify factual relations between phenomena that fall under the concepts that are implicitly defined by them. (As noted, a similar problem already had plagued Carnap's analyses of disposition terms ever since he allowed for non-eliminative reduction chains.) Carnap's attempt to address this problem by ramseyfication was published in several places from 1958 onwards. (See Carnap 1958, 1963, SS24C-D and the popular exposition 1966, Chs. 26 and 28; compare Ramsey 1929 and see Psillos 1999, Ch.3. This proposal and a variant of it (1961b) were both presented in his 1959 Santa Barbara lecture (published in Psillos 2000a); as it happened, Kaplan presented his criticism (1975a) of Carnap's 1956 criterion at the same conference.) With ramseyfication Carnap adverted again to entire theories as the unit of assessment. Ramseyfication consists in the replacement of the theoretical terms of a finitely axiomatized theory by bound higher-order variables. This involves combining all the theoretical postulates which define theoretical terms (call this conjunction T) and correspondence rules of a theory which link some of these theoretical terms with observational ones (call this C) in one long sentence (call this TC) and then replacing all the theoretical predicates that occur in it by bound higher-order variables (call this RTC). This is the so-called Ramsey-sentence of the entire theory; in it no theoretical terms appear, but it possesses the same explanatory and predictive power as the original theory: it has the same observational consequences. However, Carnap stressed that the Ramsey sentence cannot be said to be expressed in a "simple" but only an "extended" observational language, for due to its higher-order quantificational apparatus it includes "an advanced complicated logic embracing virtually the whole of mathematics" (1966, [1996, 253]). To distinguish between analytic and synthetic statements in the theoretical language Carnap made the following proposal. Let the Ramsey sentence of the conjunction of all theoretical postulates and the conjunction of all correspondence rules of that theory be considered as expressing the entire factual, synthetic content of the scientific theory and its terms in their entirety. By contrast, the statement "RTC - TC" expressed the purely analytic component of the theory, its "A-postulate" (or so-called Carnap sentence). This "A-postulate states that if entities exist (referred to by the existential quantifiers of the Ramsey sentence) that are of a kind bound together by all the relations expressed in the theoretical postulates of the theory and that are related to observational entities by all the relations specified by the correspondence postulates of the theory, then the theory itself is true." Or differently put, the A-postulate "says only that if the world is this way, then the theoretical terms must be understood as satisfying the theory" (1966 [1996, 271]). With RTC - TC expressing a meaning postulate Carnap claimed to have separated the analytic and synthetic components of a scientific theory. Carnap's adoption of the Ramsey method met mainly with criticism (Psillos 1999, ch.3, 2000b, Demopoulos 2003, 2017), even though ramseyfications continue to be discussed as a method of characterizing theoretical terms in a realist vein (albeit with conditions not yet introduced by Carnap, as in Lewis 1970, Papineau 1996). With Carnap's ramseyfications, however, we do not get the answer that what exists is the structure that the ramseyfication at hand identifies. Given the absence of a clause requiring unique realizability, ramseyfications counseled modesty: the structure that is identified remains indeterminate to just that degree to which theoretical terms remain incompletely interpreted (Carnap 1961b). To this we must add that for Carnap ramseyfications of theoretical terms can support only internal existence claims: he explicitly reaffirmed his confidence in the distinction between internal and external question to defuse the realism/anti-realism issue (1966 [1996, 256]). This strongly suggests that with these proposals Carnap did not intend to deviate from his deflationist approach to ontology. What must be considered, however, is that Carnap's proposal to reconstruct the contribution of theoretical terms by ramseyfication falls foul of arguments deriving from M.H.A. Newman's objection to Russell's structuralism in Analysis of Matter (see Demopoulos and Friedman 1985). This objection says that once they are empirically adequate, ramseyfied theories are trivially true, given the nature of their reconstruction of original theories. Russell held that "nothing but the structure of the external world is known". But if nothing is known about the generating relation that produces the structure, then the claim that there exists such a structure is vacuous, Newman claimed. "Any collection of things can be organised so as to have the structure W, provided there are the right number of them." (Newman 1928, 144) To see how this so-called cardinality constraint applies to ramseyfications of theories, note that in Carnap's hands, the non-observational part of reconstructed theories, their theoretical entities, were represented by "purely logico-mathematical entities, e.g. natural numbers, classes of such, classes of classes, etc." For him the Ramsey sentence asserted that "observable events in the world are such that there are numbers, classes of such, etc., which are correlated with events in a prescribed way and which have among themselves certain relations", this being "clearly a factual statement about the world" (Carnap 1963b, 963). Carnap here had mathematical physics in mind where space-time points are represented by quadruples of real numbers and physical properties like electrical charge-density or mass-density are represented as functions of such quadruples of real numbers. The problem that arises from this for Carnap is not the failure to single out the intended interpretation of the theory: as noted, Carnap clearly thought it an advantage of the method that it remained suitably indeterminate. The problem for Carnap is rather that, subject to its empirical adequacy, the truth conditions of the Ramsey-sentences are fulfilled trivially on logico-mathematical grounds alone. As he stated, Ramsey-sentences demand that there be a structure of entities that is correlated with observable events in the way described. Yet given the amount of mathematics that went into the ramseyfied theory--"virtually the whole of mathematics"--some such structure as demanded by the Ramsey-sentence is bound to be found among those entities presupposed by its representational apparatus. (Here the cardinality constraint is no constraint at all.) That any theory is trivially true for purely formal reasons (as long as it is empirically adequate) therefore is held against Carnap's proposal to use Ramsey sentences as reconstructions of the synthetic content of the theoretical part of empirical scientific theories. Given that with ramseyfications "the truth of physical theory reduces to the truth of its observational consequences" (Demopoulos and Friedman 1985, 635), this is a problem for Carnap's project on its own terms: any surplus empirical meaning of theoretical terms that Carnap sought to capture simply evaporates (Demopoulos 2003). This result casts its shadow over Carnap's last treatment of theoretical terms in its entirety and threatens further consequences. If the reconstruction of empirical theories by ramseyfication in Carnap's fashion is unacceptable, then all explications that build on this are called into question: explications of theoretical analyticity as much as explications of the experiential import of theories. If no justice has been done to the experiential import of theoretical terms, then one must ask whether the analytic components of a theoretical language have been correctly identified. If they have not, then the meta-theoretical utility of the synthetic/analytic distinction is once again be called into question. One is lead to wonder whether Carnap would not be well advised to return to his 1956 position. This allowed for a criterion of empirical significance for theoretical terms but not for the analytic/synthetic distinction to be sustained with regard to the theoretical language. According to Carnap's fall-back position before he hit upon ramseyfication, it was thought possible to distinguish narrow logical truth from factual truth in the theoretical language (1966, Ch. 28). Yet it is difficult to silence the suspicion that an analytic/synthetic distinction that applies only to observational languages--and admits inescapable semantical holism for theoretical languages--is not what the debate between Carnap and Quine was all about. Attempts have thus been undertaken to provide interpretations of Carnap's ramseyfications that contain or mitigate the effects of the Newman objection (Friedman 2008, 2011, Uebel 2011b, Creath 2012). What has become clear, in any case, is that much is at stake for Carnap's formal explicationism, indeed for the standard logical empiricist model of scientific theories (see below). ### 3.6 The Status of the Criterion of Significance and the Point of the Project of Explication We are now in a position to return to a final criticism of the search for a criterion of empiricist significance. Much has been made of the very status of the criterion itself (however it may be put in the end): was it empirically testable? It is common to claim that it is not and therefore to consign it to insignificance in turn, following Putnam (1981a, 1981b). The question arises whether this is to overlook the fact that the criterion of significance was put forward not as an empirical claim but as a meta-theoretical proposal for how to delimit empiricist languages from non-empiricist ones. Again, pursuing this line of inquiry is not to deny that the meaning criterion may have been understood by some members of the Circle in such a way that it became liable to charges of self-refutation. Even if that were the case, the reason for this may be found not in the very idea of such a criterion, but in the contradictory status of the Tractarian "elucidations" to which the criterion was likened. (The legitimacy of these elucidations was at issue already in the debates that divided the Circle in the early 1930s; see, e.g., Neurath 1932a.) What will be considered here is primarily the view of Carnap, who from 1932 onwards put his philosophical theses in the form of "proposals" for alternative language forms, but how the pragmatist alternative fares will also be considered. Finally, we will consider where this does leave neopositivist anti-metaphysics. For Carnap, the empiricist criterion of significance was an analytic principle, but in a very special sense. As a convention, the criterion had the standing of an analytic statement, but it was not a formally specifiable framework principle of the language Ln to which it pertained. Properly formulated, it was a semantic principle concerning Ln that was statable only in its meta-language Ln+1. To argue that the criterion itself is meaningless because it has no standing in Ln is to commit a category mistake, for meta-linguistic assertions need not have counterparts in their object languages (Goldfarb 1997, Creath 2004, Richardson 2004). Nor would it be correct to claim that the criterion hides circular reasoning, allegedly because its rejection of the meaningless depends on an unquestioned notion of experiential fact as self-explanatory (when such fact is still to be constituted). Importantly, Carnap's language constructor does not start with fixed notions of what is empirical (rather than formal) or what is given (rather than assumed or inferred), but from the beginning allows a plurality of perspectives on these distinctions (Ricketts 1994). Carnap's empiricist criterion of significance is precisely this: an explication, a proposal for how empiricists may wish to speak. It is not an explanation of how meaning arises from what is not meaningful in itself. Unlike theorists who wish to explain how meaning itself is constituted, explicationists can remain untroubled by the regress of formal semantics with Tarskian strictures. For them, the lack of formal closure (the incompleteness of arithmetic and the inapplicability of the truth predicate to its own language) only betokens the fact that our very own home languages cannot ever be fully explicated. It may be wondered whether such considerations have not become pointless, given the troubles that attempts to provide a criterion of significance ran into. However, as we saw, Carnap's 1956 criterion for constructed languages remains in play. Moreover, there also remains the informal, pragmatic approach that can be applied even more widely. Thus it is not without importance to see that pragmatic principles delineating empirical significance (like Mach's or Quine's Peircean insight) are not ruled out from the start either. The reason for this is different however. For pragmatists, the anti-metaphysical demarcation criterion is not strictly speaking a meaning criterion. The pragmatic criterion of significance is expressly epistemic, not semantic: it speaks of relevance with regard to an established cognitive practice, not in-principle truth-evaluability. This criterion is most easily expressed as a conditional norm, alongside other methodological maxims. (If you want your reasoning to be responsible to evidence, then avoid statements that experience can neither confirm or disconfirm, however indirectly.) So the suggestion that the criterion of empirical significance can be regarded as a proposal for how to treat the language of science cannot be brushed aside but for the persistent neglect of the philosophical projects of Carnap or the non-formalist left Vienna Circle. Still, some readers may wonder whether in the course of responding to the various counter-criticisms, the Vienna Circle's position has not shifted considerably. This indeed is true: the attempt to show metaphysics strictly meaningless once and for all did not succeed. For even if Carnap's 1956 criterion and the pragmatic approach work, they do not achieve that: Carnap's criterion only works for constructed languages and the pragmatic one does not address the semantic issue and only works case by case. But it can be argued that while this debilitates the Vienna Circle's most notorious claim (if understood without qualifications), it does not debilitate their entire program. That was, we recall, to defend Enlightenment reason and to counter the abuse of possibly empty but certainly ill-understood deep-sounding language in science and in public life. Their program was, to put it somewhat anachronistically, to promote epistemic empowerment. This program would have been helped by an across-the-board criterion to show metaphysics meaningless, but it can also proceed in its absence. But now the suspicion may be that if all that is meant to be excluded is speculative reason without due regard to empirical and logico-linguistic evidence, the program's success appears too easy. Few contemporary philosophers would confess to such reckless practices. Still, even the rejection of speculative reason is by no means uncontroversial, as shown by the unresolved status of the appeal to intuitions that characterizes much of contemporary analytical metaphysics and epistemology. Moreover, much depends on what's considered to be "due regard": is merely bad science "metaphysics"? Or only appeals to the supernatural? And what about de re necessities? Or the seeming commitments of existential quantification? The promotion of anti-metaphysics may be applauded in principle as an exercise in intellectual hygiene, the objection goes, but in practice it excludes either too much or too little: it either cripples our understanding of theoretical science or normalizes away the Vienna Circle's most notorious claim. In response it is helpful to consider the conception of metaphysics that can be seen to be motivating much of the Circle's ethical non-cognitivism. What did Carnap (1935) and Neurath (1932a) dismiss when they dismissed normative ethics as metaphysical and cognitively meaningless? One may concede that due to the brusque way in which they put their broadly Humean point, they opened themselves up to significant criticism, but it is very important to see also what they did not do. Most notably, they did not dismiss as meaningless all concern with how to live. Conditional prescriptions remained straight-forwardly truth-evaluable in instrumental terms and so cognitively meaningful. In addition, their own active engagement for Enlightenment values in public life showed that they took these matters very seriously themselves. (In fact, their engagement as public intellectuals compares strikingly with that of most contemporary philosophers of science.) But neither did they fall victim to the naturalistic fallacy nor were they simply inconsistent. In the determination of basic values they rather saw acts of personal self-definition, but, characteristically, Carnap showed a more individualistic and Neurath a more collectively oriented approach to the matter. What needs to be borne in mind, then, is the meaning that they attached to the epithet "metaphysical" in this and other areas: the arrogation of unique and fully determined objective insight into matters beyond scientific reason. It was in the ambition of providing such unconditional prescriptions that they saw philosophical ethics being the heir of theology. (Compare Carnap 1935 and 1963b, SS32 and Neurath 1932a and 1944, SS19.) Needless to say, it remains contentious to claim those types of philosophical ethics to be cognitively meaningless that seek to derive determinate sets of codes from some indisputable principle or other. But the ongoing discussion of non-cognitivism and its persistent defense in analytical ethics as "expressivism" suggest that, understood as outlined, the Circle's non-cognitivism was by no means absurd or contradictory. It may be noted here that a newly discovered fragment of Carnap's writing (2017) has given fresh impetus to explorations of the model of ethical reasoning in terms of optatives that Carnap outlined (1963b, SS32) in response to A. Kaplan's criticism (1963) of his earlier position. What emerges is that Carnap was prepared to integrate ethical desiderata among non-ethical ones within the network of means and ends that decision theory as a normative theory of rational action seeks to systematize and regiment. Moral reasoning is assimilated to practical reasoning and no longer suffers from a deficit of significance--albeit at the cost of not being able to exclude appeals to certain types of intrinsic value on the ground their being beyond the pale (see Carus 2017). Carnap may reasonably respond here that as a theorist of science he is not required to account for normative ethics beyond providing a framework for understanding its undeniable role in a generic theory of human behavior. What was rightly objected against his earlier position was that it made such understanding impossible. Whatever the details, their non-cognitivism supports the idea that the left wing's anti-metaphysics was primarily deflationist. They opposed all claims to have a categorically deeper insight into reality than either empirical or formal science, such that philosophy would stand in judgement of these sciences as to their reality content or that mere science would stand in need of philosophical interpretations. (Concerned with practical problems, they likewise opposed philosophical claims to stand above the contestations of mere mortals.) Importantly, such deflationism need not remain general and vague, but can be given precise content. For instance, it has been argued (Carus 1999) that Carnap was correct to understand Tarski's theory of truth not as a traditional correspondence theory such that truth consisted in some kind of agreement of statements or judgements and facts or the world where the latter make true the former. In Carnap's unchanged opposition to the classical correspondence theory of truth in turn lies not only the continuity between his own syntactic and semantic phases, but also the key to his and the entire left Vienna Circle's understanding of their anti-metaphysical campaign. (On various occasions in the early 30s, Hahn, Frank and Neurath opposed correspondence truth very explicitly, while, in later years, Neurath resisted Tarskian semantics precisely because he wrongly suspected it of resurrecting correspondentism and Frank continued to castigate correspondentism whenever required. On this tangled issue, see Mormann 1999, Uebel 2004, Mancosu 2008.) This suggests that a hard core of Viennese anti-metaphysics survives the criticism and subsequent qualifications of early claims made for their criteria of empirical significance, yet retains sufficient philosophical teeth to remain of contemporary interest. The metaphysics which the left wing attacked, besides everyday supernaturalism and the supra-scientific essentialism of old, was the correspondence conception of truth and associated realist conceptions of knowledge. These notions were deemed attackable directly on epistemological grounds, without any diversion through the theory of meaning: how could such correspondences or likenesses ever be established? As Neurath liked to put it (1930), we cannot step outside of our thinking to see whether a correspondence obtains between what we think and how the world is. (Against defenses of the correspondence theory by arguments from analogy it would be argued that the analogy is overextended.) Against the counter that this is merely an epistemic argument that does not touch the ontological issue Neurath is likely to have argued that doing without an epistemic account is a recipe for uncontrollable metaphysics. Importantly, the left wing's deflationary anti-metaphysics was accompanied by a distinctively constructivist attitude. Here one must hasten to add, of course, that what was constructed were not the objects of first-order discourse (tables, chairs, electrons and black holes) but concepts, be they concepts associated with technical terms for observables, theoretical terms or terms needed for reflection about the cognitive enterprise of science (ideas like evidence and its degrees and presuppositions). As meta-theorists of science they developed explications: different types of explications were envisaged, ranging from analytic definitions giving necessary and sufficient conditions in formal languages all the way to pragmatic, exemplar-based criterial delimitations of the central applications of contested concepts or practices. Two branches of the Circle's constructivist tendency can thus be distinguished: Carnap's rational reconstructions and formalist explications and Neurath's and Frank's empirically informed and practice-oriented reconceptualizations. The difference between these two approaches can be understood as a division of labor between the tasks of exploring logico-linguistic possibilities of conceptual reconstruction and considering the efficacy of particular scientific practices. In principle, the constructivist tendency in Vienna Circle philosophy was able to embrace both (compare Carnap 1934/37, SS72 and Neurath 1936b). However, in its own day, this two-track approach remained incompletely realized as philosophical relations between Carnap and Neurath soured over disputes stemming from mutual misunderstandings. Frank's final paper (1963) was a terse reminder that the logic of science was not the sole successor or replacement of traditional philosophy and Carnap's response (1963b, SS3) again acknowledged the compatibility in principle of his logic of science and what Frank called the "pragmatics of science" (1957). Considering the Vienna Circle as a whole in the light of this reading of its anti-metaphysical philosophy, we find the most striking division within it yet. Unlike Carnap and the left wing, Schlick had little problem with a correspondence theory of truth once it was cleansed of psychologistic and intuitive accretions and centered on the idea of unique coordination of statement and fact. In this lay the strongest sense of continuity between his pre-Vienna Circle General Theory of Knowledge (1918/25) and his post-Tractarian epistemology (1935a, 1935b). (Schlick also showed little enthusiasm for the constructivist tendencies which already the manifesto of 1929 had celebrated.) Allowing for some simplification, it must be noted that Schlick's attack on metaphysics (which gradually weakened anyway) presupposed a non-constructivist reading of the criterion of significance. Whether his conception can escape the charge self-refutation must be left open here. ### 3.7 The Vienna Circle and History Much confusion exists concerning the Vienna Circle and history, that is, both concerning the Vienna Circle's attitude towards the history of philosophy and science and concerning its own place in that history. As more has been learnt about the history of the Vienna Circle itself--the development and variety of its doctrines as well as its own prehistory as a philosophical forum--this confusion can be addressed more adequately. As the unnamed villain of the opening sentences of Kuhn's influential Structure of Scientific Revolutions (1962), logical empiricism is often accused of lacking historical consciousness and any sense of the embedding of philosophy and science in the wider culture of the day. Again it can hardly be denied that much logical empiricist philosophy, especially after World War II, was ahistorical in outlook and asocial in its orientation. Reichenbach's distinction (1938) between the contexts of discovery and justification--which echoed distinctions made since Kant (Hoyningen-Huene 1987) and was already observed under a different name by Carnap in the Aufbau--was often employed to shield philosophy not only from contact with the sciences as practiced but also culture at large. But this was not the case for the Vienna Circle generally. On the one hand, unlike Reichenbach, who drew a sharp break between traditional philosophy and the new philosophy of logical empiricism in his popular The Rise of Scientific Philosophy (1951), Schlick was very much concerned to stress the remaining continuities with traditional philosophy and its cultural mission in his last paper (1938). On the other hand, on the left wing of the Circle scientific meta-theory was opened to the empirical sciences. To be sure, Carnap for his own part was happy to withdraw to the "icy slopes" of the logic of science and showed no research interest of his own in the history of science or philosophy, let alone its social history. By way of the division of labor he left it to Neurath and Frank to pursue the historical and practice-related sociological questions that the pure logic of science had to leave unaddressed. (See, e.g., Neurath's studies of the history of optics (1915, 1916), Frank's homage to Mach (1917), his pedagogical papers in (1949b) and his concern with the practice of theory acceptance and change in (1956); cf. Uebel 2000 and Nemeth 2007.) Moreover, it must be noted that Neurath himself all along had planned a volume on the history of science for the International Encyclopedia of Science, a volume that in the end became Kuhn's Structure. Its publication in this series is often regarded as supremely ironical, given how Kuhn's book is commonly read. But this is not only to overlook that the surviving editors of that series, Carnap and Morris were happy to accept the manuscript, but also that Carnap found himself in positive agreement with Kuhn (Reisch 1991, Irzik and Grunberg 1995; cf. Friedman 2001). Finally, one look at the 1929 manifesto shows that its authors were very aware of and promoted the links between their philosophy of science and the socio-political and cultural issues of the day. Turning to the historical influences on the Vienna Circle itself, the scholarship of recent decades has unearthed a much greater variety than was previously recognized. Scientifically, the strongest influences belonged to the physicists Helmholtz, Mach and Boltzmann, the mathematicians Hilbert and Klein and the logicians Frege and Russell; amongst contemporaries, Einstein was revered above all others. The Circle's philosophical influences extend far beyond that of the British empiricists (especially Hume), to include the French conventionalists Henri Poincare, Pierre Duhem and Abel Rey, American pragmatists like James and, in German-language philosophy, the Neo-Kantianism of both the Heidelberg and the Marburg variety, even the early phenomenology of Husserl as well as the Austrian tradition of Bolzano's logic and the Brentano school. (See Frank 1949a for the influence of the French conventionalists; for the importance of Neo-Kantianism for Carnap, see Friedman 1987, 1992, Sauer 1989, Richardson 1998, Mormann 2007; for Neo-Kantianism in Schlick, see Coffa 1991, Ch. 9 and Gower 2000; for the significance of Husserl for Carnap, see Sarkar 2004, Ryckman 2007, Carus 2016, Dambock 2018; for the influence of and sympathies for pragmatism see Frank 1949a and Uebel 2016; the Bolzano-Brentano connection is explored in Haller 1986.) It is against this very wide background of influences that the seminal force must be assessed that their contemporary Wittgenstein exerted. The literature on the relation between Wittgenstein and the Vienna Circle is vast but very often suffers from an over-simplified conception of the latter. (See Stern 2007 for an attempt by a Wittgenstein scholar to redress the balance.) Needless to say, different wings of the Circle show these influences to different degrees. German Neo-Kantianism was important for Schlick and particularly so for Carnap, whereas the Austrian naturalist-pragmatist influences were particularly strong on Hahn, Frank and Neurath. Frege was of great importance for Carnap, less so for Hahn who looked to Russell. Most importantly, by no means all members of the Vienna Circle sought to emulate Wittgenstein--thus the division between the faction around Schlick and the left wing (see Uebel 2017). While these findings leave numerous questions open, they nevertheless refute the standard picture of Vienna Circle philosophy which confuses A.J. Ayer's Language, Truth and Logic with the real thing. Ayer offered a version of British empiricism (Berkeley's epistemic phenomenalism updated with Russellian logic) and paid no attention, for instance, to the Circle's overarching concern with establishing the objectivity claim of science. Ayer's remark in the preface to his later anthology Logical Positivism that his own Language, Truth and Logic "did something to popularize what may be called the classical position of the Vienna Circle" (1959b, 8) is highly misleading therefore. What he called "the classical position" was at best a partial characterisation of the starting position of some--by no means all--of its members, a position which by 1932 the left wing as a whole rejected and even Schlick had no reason to endorse any longer. All that said by way of embedding the Vienna Circle's philosophy in its time, one must also ask whether its members understood their own position correctly. Here one issue in particular has become increasingly prominent and raises questions that are of importance for philosophy of science still today. That is whether, after all, logical empiricism did have the resources to understand correctly the then paradigm modern science, the general theory of relativity. According to the standard logical empiricist story (Schlick 1915, 1917, 1921, 1922), their theory conclusively refuted the Kantian conception of the synthetic a priori: Euclidean geometry was not only one geometry amongst many, it also was not the one that characterized empirical reality. With one of its most prominent exemplars refuted, the synthetic a priori was deemed overthrown altogether. As noted, Schlick convinced the young Reichenbach to drop his residually Kantian talk of constitutive principles and speak of conventions instead. Likewise Schlick rejected efforts by Ernst Cassirer (see his 1921, developing themes from his 1910) to make do with a merely relative a priori in helping along scientific self-reflection. Even though much later, and on the independent grounds of quantum physics, Frank attested to the increased proximity of his and Cassirer's understanding of scientific theories (1938), Schlick's disregard of Cassirer's efforts remains notable. Most controversial is how the issue of general relativity as a touchstone for competing philosophies of science was framed: having dismissed Kant's own synthetic a priori for its mistaken apodicity, no time was spared for discussion of its then contemporary development in Neo-Kantianism as a merely relative but still constitutive a priori. Now in the philosophy of physics, this omission--committed both by Schlick and Reichenbach--has recently come back to haunt logical empiricists with considerable vengeance. Thus it has been argued that the Schlick-Reichenbach reading of general relativity as embodying the standard logical empiricist model of scientific theories, with high theory linked to its observational strata by purely conventional coordinative definitions, is deeply mistaken in representing the local metric of space-time not to be empirically but conventionally determined as in special relativity (Ryckman 1992) and that it is instead only the tradition of transcendental idealism that possesses the resources to understand the achievements of mathematical physics (Ryckman 2005; cf. Friedman 2001). It is tempting to speak of the return of the repressed Neo-Kantian opposition. But it is tempting too to note that Schlick's and Reichenbach's mistake was already corrected quietly and without fanfare by Carnap (see the example in his 1934/37, SS50). Clearly then, the mistake was not inevitable and inherent in logical empiricist theorizing about science as such. (For discussion of different criticism of the Circle's reading of Einstein, especially his relation to Mach, see DiSalle 2006, Ch.4.) The charge of a constitutive failing would rather seem to come from Demopoulos's challenge to the two-languages model (nearly) universally adopted in logical empiricism (forthcoming). Importantly, this challenge does not proceed, as some previous ones have, from the impossibility of drawing a sharp distinction between the observational and the theoretical (Putnam 1962). Rather, the two-languages model falsely supposes that the process of testing scientific hypotheses must only advert to theoretically uncontaminated facts and so results in misunderstanding the empirical import of theoretical claims (as in the Newman problem). Instead, a conception of theory-mediated measurement and testing is suggested that extends responsiveness to observational data to theoretical claims by showing them to be essentially implicated in the production of observed experimental consequences. Hoping to advance beyond the stalemate between realism and instrumentalism without appealing to question-begging semantics, Demopoulos here breaks with a supposition upheld by Carnap throughout, namely that the theoretical language be regarded as an essentially uninterpreted, at best partially interpreted calculus. Whatever the outcome of this challenge, it is remarkable how on this far-reaching and fundamental issue contemporary philosophy of science intersects with the history of philosophy of science. ## 4. Concluding Remarks In conclusion, the results of the discussions in section 3 can be briefly summarised. To start with, the dominant popular picture of the Vienna Circle as a monolithic group of simple-minded verificationists who pursued a blandly reductivist philosophy with foundationalist ambitions is widely off the mark. Instead, the Vienna Circle must be seen as a forum in which widely divergent ideas about how empiricism can cope with modern empirical and formal science were discussed. While by no means all of the philosophical initiatives started by members of the Vienna Circle have born fruit, it is neither the case that all of them have remained fruitless. Nor is it the case that everything once distinctive of Vienna Circle philosophy has to be discarded. Consider verificationism. While the idea to show metaphysics once-and-for-all and across-the-board to be not false but meaningless--arguably the most distinctive thesis associated with the Vienna Circle--did indeed have to be abandoned, two elements of that program remain so far unrefuted. On the one hand, it remains an option to pursue the search for a criterion of empirical significance in terms of constructed, formal languages further along the lines opened by Carnap with his theory-relative proposal of 1956 (and its later defense against critics). On the other hand--albeit at the cost of merging with the pragmatist tradition and losing the apparent Viennese distinctiveness--the option to neglect as cognitively irrelevant, and in this sense metaphysical, all assertions whose truth or falsity would not make a difference remains as open as it always was. In addition it must be noted that, properly formulated, neither the formalist version of the criterion of empirical significance for constructed languages nor the pragmatist version of the criterion for natural languages are threatened by self-refutation. Consider analyticity. Here again, the traditional idea--sometimes defended by some members--did show itself indefensible, but this leaves Carnap's framework-relative interpretation of analyticity and the understanding of the a priori as equally relative to be explored. As noted, if Carnap's ramseyfications can be defended, an analytic/synthetic distinction could be upheld also for the theoretical languages of science. In any case, however, the distinction between framework principles and content continues to be drawable on a case by case basis. Consider reductionism and foundationalism. While it cannot be denied that various reductionist projects were at one time or another undertaken by members of the Vienna Circle and that not all of its members were epistemological anti-foundationalists either from the start or at the end, it is clearly false to paint all of them with reductivist and/or foundationalist brushes. This is particularly true of the members of the so-called left wing of the Circle, all of whom ended up with anti-foundationalist and anti-reductionist positions (even though this did involve instrumentalism for some). Consider also, however, the challenges mentioned above to the fundamental tenets of logical empiricism that remain issues of intense discussion: challenges to its conception of the nature of empirical theory and of what is distinctive about the formal sciences. That to this day no agreement has been reached about how its proposals are to be replaced is not something that is unique to logical empiricism as a philosophical movement, but that they remain on the table, as it were, shows the ongoing relevance and centrality of its work for philosophy of science. Whether the indicated qualifications and/or modifications count as defeats of the original project depends at least in part on what precisely is meant to be rejected when metaphysics is rejected and that in turn depends on what the positive vision for philosophy consists in. Here again one must differentiate. While some members ended with considerable more sympathy for traditional philosophy than they displayed in the Circle's heyday--and may thus be charged with partial surrender--others stuck to their guns. For them, what remained of philosophy stayed squarely in the deflationist vein established by the linguistic turn. They offered explications of contested concepts or practices that, they hoped, would prove useful. Importantly, the explications given can be of two sorts: the formal explications of the logic of science by means of exemplary models of constructed languages, and the more informal explications of the empirical theory of science given by spelling out how certain theoretical desiderata can be attained more or less under practical constraints. This has been designated as the bipartite metatheory conception of scientific philosophy and ascribed to the left wing of the Circle as an ideal unifying its diverse methodologies (Uebel 2007, Ch. 12, 2015). Readers will note therefore that despite his enormous contribution to the development of Vienna Circle philosophy, it is not Schlick's version of it that appears to this reviewer to be of continuing relevance to contemporary philosophy--unlike, in their very different but not incompatible ways, Carnap's and Neurath's and Frank's. This may be taken as a partial endorsement of Hempel's 1991 judgement (quoted in sect. 1 above), against which, however, Carnap has here been re-claimed for the Neurathian wing. Needless to say, recent work on Vienna Circle philosophies continues to inspire a variety of approaches to the legacy they constitute (besides prompting continuing excavations of other members' non-standard variants; e.g., on Feigl see Neuber 2011). There is Michael Friedman's extremely wide-ranging project (2001, 2010, 2012) to use the shortcomings of Vienna Circle philosophies as a springboard for developing a renewed Kantian philosophy that also overcomes the failings of neo-Kantianism and provides a philosophy-cum-history fit for our post-Kuhnian times. Then there is Richardson's proposal (2008) to turn the ambition to develop a scientific philosophy into a research programme for the history of science, so as to reveal more clearly the real world dynamics and limitations of philosophy as a scientific metatheory. And there is Carus's suggestion (2007) that Carnap's minimalist explicationism be placed in the service of a renewed Enlightenment agenda (continuing the task of the "scientific world conception"). This connects with the current metaphilosophical interest in conceptual engineering, the relevance of Carnap's work to which becoming increasingly recognised (Justus 2012, Wagner 2012, Brun 2016, Reck and Dutilh Novaes 2017, Dutilh Novaes forthcoming, Lutz forthcoming). All along, of course, Vienna Circle philosophies also continue to serve as foils for alternative and self-consciously post-positivist programs, fruitfully so when informed by the results of recent scholarship (e.g. Ebbs 2017). It would appear then that despite continued resistance to recent revisionist scholarship--a resistance that consists not so much in contesting but in ignoring its results--the fortune of Vienna Circle philosophy has turned again. Restored from the numerous distortions of its teachings that accrued over generations of acolytes and opponents, the Vienna Circle is being recognized again as a force of considerable philosophical sophistication. Not only is it the case that its members profoundly influenced the actual development of analytical philosophy of science with conceptual initiatives that, typically, were seen through to their bitter end. It is also the case that some of its members offered proposals and suggested approaches that were not taken up widely at the time (if at all), but that are relevant again today. Much like its precursors Frege, Russell and Wittgenstein, the conventionalists Poincare and Duhem, the pragmatists Peirce and Dewey--and like its contemporaries from Reichenbach's Berlin Group and the Warsaw-Lvov school of logic to the Neo-Kantian Cassirer--the Vienna Circle affords a valuable vantage point on contemporary philosophy of empirical and formal science.
ethics-ancient	## 1. Introduction In their moral theories, the ancient philosophers depended on several important notions. These include virtue and the virtues, happiness (eudaimonia), and the soul. We can begin with virtue. Virtue is a general term that translates the Greek word arete. Sometimes arete is also translated as excellence. Many objects, natural or artificial, have their particular arete or kind of excellence. There is the excellence of a horse and the excellence of a knife. Then, of course, there is human excellence. Conceptions of human excellence include such disparate figures as the Homeric warrior chieftain and the Athenian statesman of the period of its imperial expansion. Plato's character Meno sums up one important strain of thought when he says that excellence for a man is managing the business of the city so that he benefits his friends, harms his enemies, and comes to no harm himself (Meno 71e). From this description we can see that some versions of human excellence have a problematic relation to the moral virtues. In the ancient world, courage, moderation, justice and piety were leading instances of moral virtue. A virtue is a settled disposition to act in a certain way; justice, for instance, is the settled disposition to act, let's say, so that each one receives their due. This settled disposition consists in a practical knowledge about how to bring it about, in each situation, that each receives their due. It also includes a strong positive attitude toward bringing it about that each receives their due. Just people, then, are not ones who occasionally act justly, or even who regularly act justly but do so out of some other motive; rather they are people who reliably act that way because they place a positive, high intrinsic value on rendering to each their due and they are good at it. Courage is a settled disposition that allows one to act reliably to pursue right ends in fearful situations, because one values so acting intrinsically. Moderation is the virtue that deals similarly with one's appetites and emotions. Human excellence can be conceived in ways that do not include the moral virtues. For instance, someone thought of as excellent for benefiting friends and harming enemies can be cruel, arbitrary, rapacious, and ravenous of appetite. Most ancient philosophers, however, argue that human excellence must include the moral virtues and that the excellent human will be, above all, courageous, moderate, and just. This argument depends on making a link between the moral virtues and happiness. While most ancient philosophers hold that happiness is the proper goal or end of human life, the notion is both simple and complicated, as Aristotle points out. It seems simple to say everyone wants to be happy; it is complicated to say what happiness is. We can approach the problem by discussing, first, the relation of happiness to human excellence and, then, the relation of human excellence to the moral virtues. It is significant that synonyms for eudaimonia are living well and doing well. These phrases imply certain activities associated with human living. Ancient philosophers argued that whatever activities constitute human living - e.g., those associated with pleasure - one can engage in those activities in a mediocre or even a poor way. One can feel and react to pleasures sometimes appropriately and sometimes inappropriately; or one might always act shamefully and dishonorably. However, to carry out the activities that constitute human living well over a whole lifetime, or long stretches of it, is living well or doing well. At this point the relation of happiness to human excellence should be clear. Human excellence is the psychological basis for carrying out the activities of a human life well; to that extent human excellence is also happiness. So described, human excellence is general and covers many activities of a human life. However, one can see how human excellence might at least include the moral virtues. The moral virtue relevant to fear, for instance, is courage. Courage is a reliable disposition to react to fear in an appropriate way. What counts as appropriate entails harnessing fear for good or honorable ends. Such ends are not confined to one's own welfare but include, e.g., the welfare of one's city. In this way, moral virtues become the kind of human excellence that is other-regarding. The moral virtues, then, are excellent qualities of character - intrinsically valuable for the one who has them; but they are also valuable for others. In rough outline, we can see one important way ancient moral theory tries to link happiness to moral virtue by way of human excellence. Happiness derives from human excellence; human excellence includes the moral virtues, which are implicitly or explicitly other-regarding. Since happiness plays such a vital role in ancient moral theory, we should note the difference between the Greek word eudaimonia and its usual translation as 'happiness'. Although its usage varies, most often the English word 'happiness' refers to a feeling. For example, we say, "You can tell he feels happy right now, from the way he looks and how he is behaving." The feeling is described as one of contentment or satisfaction, perhaps with the way one's life as a whole is going. While some think there is a distinction between feeling happy and feeling content, still happiness is a good and pleasant feeling. However, 'happiness' has a secondary sense that does not focus on feelings but rather on activities. For instance, one might say, "It was a happy time in my life; my work was going well." The speaker need not be referring to the feelings he or she was experiencing but just to the fact that some important activity was going well. Of course, if their work is going well, they might feel contentment. But in speaking of their happiness, they might just as well be referring to their absorption in some successful activity. For ancient philosophers eudaimonia is closer to the secondary sense of our own term. Happiness means not so much feeling a certain way, or feeling a certain way about how one's life as a whole is going, but rather carrying out certain activities or functioning in a certain way. This sort of happiness is an admirable and praiseworthy accomplishment, whereas achieving satisfaction or contentment may not be. In this way, then, ancient philosophers typically justify moral virtue. Being courageous, just, and moderate is valuable for the virtuous person because these virtues are inextricably linked with happiness. Everyone wants to be happy, so anyone who realizes the link between virtue and happiness will also want to be virtuous. This argument depends on two central ideas. First, human excellence is a good of the soul - not a material or bodily good such as wealth or political power. Another way to put this idea is to say happiness is not something external, like wealth or political power, but an internal, psychological good. The second central idea is that the most important good of the soul is moral virtue. By being virtuous one enjoys a psychological state whose value outweighs whatever other kinds of goods one might have by being vicious. Finally, a few words about the soul are in order since, typically, philosophers argue that virtue is a good of the soul. In some ways, this claim is found in many traditions. Many thinkers argue that being moral does not necessarily provide physical beauty, health, or prosperity. Rather, as something good, virtue must be understood as belonging to the soul; it is a psychological good. However, in order to explain virtue as a good of the soul, one does not have to hold that the soul is immortal. While Plato, for example, holds that the soul is immortal and that its virtue is a good that transcends death, his argument for virtue as a psychological good does not depend on the immortality of the soul. He argues that virtue is a psychological good in this life. To live a mortal human life with this good is in itself happiness. This position that links happiness and virtue is called eudaimonism - a word based on the principal Greek word for happiness, eudaimonia. By eudaimonism, we will mean one of several theses: (a) virtue, together with its active exercise, is identical with happiness; (b) virtue, together with its activities, is the most important and dominant constituent of happiness; (c) virtue is the only means to happiness. However, one must be cautious not to conclude that ancient theories in general attempt to construe the value of virtue simply as a means to achieving happiness. Each theory, as we shall see, has its own approach to the nature of the link between virtue and happiness. It would not be advisable to see ancient theories as concerned with such contemporary issues as whether moral discourse - i.e., discourse about what one ought to do - can or should be reduced to non-moral discourse - i.e., to discourse about what is good for one. These reflections on virtue can provide an occasion for contrasting ancient moral theory and modern. One way to put the contrast is to say that ancient moral theory is agent-centered while modern moral theory is action-centered. To say that it is action-centered means that, as a theory of morality, it explains morality, to begin with, in terms of actions and their circumstances, and the ways in which actions are moral or immoral. We can roughly divide modern thinkers into two groups. Those who judge the morality of an action on the basis of its known or expected consequences are consequentialist; those who judge the morality of an action on the basis of its conformity to certain kinds of laws, prohibitions, or positive commandments are deontologists. The former include, e.g., those utilitarians who say an action is moral if it provides the greatest good for the greatest number. Deontologists say an action is moral if it conforms to a moral principle, e.g., the obligation to tell the truth. While these thinkers are not uninterested in the moral disposition to produce such actions, or in what disposition is required if they are to show any moral worth in the persons who do them, their focus is on actions, their consequences, and the rules or other principles to which they conform. The result of these ways of approaching morality is that moral assessment falls on actions. This focus explains, for instance, contemporary fascination with such questions of casuistry as, e.g., the conditions under which an action like abortion is morally permitted or immoral. By contrast, ancient moral theory explains morality in terms that focus on the moral agent. These thinkers are interested in what constitutes, e.g., a just person. They are concerned about the state of mind and character, the set of values, the attitudes to oneself and to others, and the conception of one's own place in the common life of a community that belong to just persons simply insofar as they are just. A modern might object that this way of proceeding is backwards. Just actions are logically prior to just persons and must be specifiable in advance of any account of what it is to be a just person. Of course, the ancients had a rough idea of what just actions were; and this rough idea certainly contributed to the notion of a just person, and his motivation and system of values. Still, the notion of a just person is not exhausted by an account of the consequences of just actions, or any principle for determining which actions are and which are not just. For the ancients, the just person is compared to a craftsman, e.g., a physician. Acting as a physician is not simply a collection of medically effective actions. It is knowing when such actions are appropriate, among other things; and this kind of knowledge is not always definable. To understand what being a physician means one must turn to the physician's judgment and even motivation. These are manifested in particular actions but are not reducible to those actions. In the same way, what constitutes a just person is not exhausted by the actions he or she does nor, for that matter, by any catalogue of possible just actions. Rather, being a just person entails qualities of character proper to the just person, in the light of which they decide what actions justice requires of them and are inclined or disposed to act accordingly. ## 2. Socrates In this section we confine ourselves to the character Socrates in Plato's dialogues, and indeed to only certain ones of the dialogues in which a Socrates character plays a role. In those dialogues in which he plays a major role, Socrates varies considerably between two extremes. On the one hand, there is the Socrates who claims to know nothing about virtue and confines himself to asking other characters questions; this Socrates is found in the Apology and in certain dialogues most of which end inconclusively. These dialogues, e.g., Charmides, Laches, Crito, Euthydemus, and Euthyphro, are called aporetic. On the other hand, in other dialogues we find a Socrates who expounds positive teachings about virtue; this Socrates usually asks questions only to elicit agreement. These dialogues are didactic, and conclusive in tone, e.g., Republic, Phaedo, Phaedrus, and Philebus. However, these distinctions between kinds of dialogues and kinds of Socratic characters are not exclusive; there are dialogues that mix the aporetic and conclusive styles, e.g., Protagoras, Meno, and Gorgias. In observing these distinctions, we refer only to the characteristic style of the dialogue and leave aside controversies about the relative dates of composition of the dialogues. (See the entry on Plato, especially the section on Socrates and the section on the historical Socrates.) The significance of this distinction among dialogues is that one can isolate a strain of moral teaching in the aporetic and mixed dialogues. In spite of their inconclusive nature, in the aporetic dialogues the character Socrates maintains principles about morality that he seems to take to be fundamental. In the mixed dialogues we find similar teaching. This strain is distinct enough from the accounts of morality in the more didactic dialogues that it has been called Socratic, as opposed to Platonic, and associated with the historical personage's own views. In what follows we limit ourselves to this "Socratic" moral teaching - without taking a position about the relation of "Socratic" moral teaching to that of the historical Socrates. For our purposes it is sufficient to point out a distinction between kinds of moral teaching in the dialogues. We will focus on the aporetic dialogues as well as the mixed dialogues Protagoras, Gorgias, and Meno. The first feature of Socratic teaching is its heroic quality. In the Apology, Socrates says that a man worth anything at all does not reckon whether his course of action endangers his life or threatens death. He looks only at one thing - whether what he does is just or not, the work of a good or of a bad man (28b-c). Said in the context of his trial, this statement is both about himself and a fundamental claim of his moral teaching. Socrates puts moral considerations above all others. If we think of justice as, roughly, the way we treat others, the just actions to which he refers cover a wide range. It is unjust to rob temples, betray friends, steal, break oaths, commit adultery, and mistreat parents (Rep 443a-b). A similarly strong statement about wrong-doing is found in the Crito, where the question is whether Socrates should save his life by escaping from the jail in Athens and aborting the sentence of death. Socrates says that whether he should escape or not must be governed only by whether it is just or unjust to do so (48d). Obviously, by posing wrong-doing against losing one's life, Socrates means to emphasize that nothing outweighs in positive value the disvalue of doing unjust actions. In such passages, then, Socrates seems to be a moral hero, willing to sacrifice his very life rather than commit an injustice, and to recommend such heroism to others. However, this heroism also includes an important element of self-regard. In the passage from the Apology just quoted Socrates goes on to describe his approach to the citizens of Athens. He chides them for being absorbed in the acquisition of wealth, reputation, and honor while they do not take care for nor think about wisdom, truth, and how to make their souls better (Ap. 29d-e). As he develops this idea it becomes clear that the perfection of the soul, making it better, means acquiring and having moral virtue. Rather than heaping up riches and honor, Athenians should seek to perfect their souls in virtue. From this exhortation we can conclude that for Socrates psychological good outweighs material good and that virtue is a psychological good of the first importance. The Crito gives another perspective on psychological good. Socrates says (as something obvious to everyone) that life is not worth living if that which is harmed by disease and benefited by health - i.e., the body - is ruined. But even more so, he adds, life is not worth living if that which is harmed by wrong-doing (to adikon) and benefited by the right - sc. the soul - is ruined, insofar as the soul is more valuable than the body (47e-48a). We can understand this claim in positive terms. Virtue is the chief psychological good; wrong-doing destroys virtue. So Socrates' strong commitment to virtue reflects his belief in its value for the soul, as well as the importance of the soul's condition for the quality of our lives. A second feature of Socratic teaching is its intellectualism. Socratic intellectualism is usually expressed in the claim that virtue is knowledge, implying that if one knows what is good one will do what is good. We find a clear statement of the claim in Protagoras 352c; but it underlies a lot of Socratic teaching. The idea is paradoxical because it flies in the face of what seems to be the ordinary experience of knowingly doing what is not good, called akrasia or being overcome (sometimes anachronistically translated as weakness of will). However, Socrates defends the idea that akrasia is impossible. First, he argues that virtue is all one needs for happiness. In the Apology (41c), Socrates says that no evil at all can come to a good man either in living or in dying (...ouk esti andri agatho(i) kakon ouden oute zonti oute teleutesanti), implying the good man's virtue alone makes him proof against bad fortune. In the Crito (48b), he says living well and finely and justly are the same thing (to de eu kai kalos kai dikaios [zen] hoti tauton estin...). In turn, in the Meno (77c-78b), Socrates argues that everyone desires happiness, i.e., one's own welfare. So, in the first place, one always has a good reason for acting virtuously; doing so entails one's own happiness. However, even if the link between virtue and happiness is granted, another problem remains. It is possible that one can, knowingly, act against one's happiness, understood, as it is by Socrates, as one's own welfare. It seems that an individual can choose to do what is not in her own best interest. People can all too easily desire and go after the pastry that they know is bad for them. Socrates, however, seems to think that once one recognizes (i.e., really knows and fully appreciates) that the pastry is not good in this way, one will cease to desire it. There is no residual desire for, e.g., pleasure, that might compete with the desire for what is good. This position is called intellectualism because it implies that what ultimately motivates any action is some cognitive state, rational or doxastic: if you know what is good you will do it, and if you do an action, and it is bad, that is because you thought somehow that it was good. All error in such choices is due to ignorance. In support of the idea that if one knows what happiness is, one will pursue it, Socrates argues, in the Euthydemus, that wisdom is necessary and sufficient for happiness. While most of this dialogue is given over to Euthydemus' and Dionysiodorus' eristic display, there are two Socratic interludes. In the first of these - in a passage that has a parallel in Meno (88a ff) - Socrates helps the young Cleinias to see that wisdom is a kind of knowledge that infallibly brings happiness. He uses an analogy with craft (techne); a carpenter must not only have but know how to use his tools and materials to be successful (Euthyd. 280b-d). In turn, someone may have such goods as health, wealth, good birth, and beauty, as well as the virtues of justice, moderation, courage, and wisdom (279a-c). Wisdom is the most important, however, because, like carpentry, for example, it is a kind of knowledge, about how to use the other assets so that they are beneficial (281b-c). Moreover, all of these other so-called goods are useless - in fact, even harmful - without wisdom, because without it one will misuse any of the other assets one may possess, so as to act not well but badly. Wisdom is the only unconditional good (281d-e). Socrates' argument leaves it ambiguous whether wisdom (taken together with its exercise) is identical with happiness or whether it is the dominant and essential component of happiness (282a-b). In this account, the focus is on a kind of knowledge as the active ingredient in happiness. The other parts of the account are certain assets that seem as passive in relation to wisdom as wood and tools are to the carpenter. Socratic intellectualism has been criticized for either ignoring the non-rational, desiderative and volitional causes of human action or providing an implausibly rationalist account of them. In either case, the common charge is that it fails to account for or appreciate the apparent complexities of moral psychology. ## 3. Plato If the objections to intellectualism are warranted, Plato makes significant progress by having his character Socrates suppose that the soul has desires that are not always for what is good. This allows for the complexities of moral psychology to become an important issue in the account of virtue. That development is found in Plato's mature moral theory. In the Republic, especially in its first four books, Socrates presents the most thorough and detailed account of moral psychology and virtue in the dialogues. It all begins with the challenge to the very notion of morality, understood along traditional lines, mounted by Callicles in the second half of the Gorgias and by Thrasymachus in Republic I. Callicles thinks that moral convention is designed by the numerous weak people to intimidate the few strong ones, to keep the latter from taking what they could if they would only use their strength. No truly strong person should be taken in by such conventions (Gorg. 482e ff). Thrasymachus argues that justice is the advantage of the more powerful; he holds that justice is a social practice set up by the powerful, i.e., rulers who require their subjects through that practice to act against their own individual and group self-interest (Rep. 338d ff). No sensible person should, and no strong-willed person would, accept rules of justice as having any legitimate authority over them. In answer to the latter challenge, in Republic II, Glaucon and Adeimantus repeatedly urge Socrates to show what value justice has in itself, apart from its rewards and reputation. They gloss the intrinsic value of justice as what value it has in the soul (358b-c), what it does to the soul simply and immediately by its presence therein. Before giving what will be a new account of the soul, Socrates introduces his famous comparison between the soul and the city. As he develops his account of the city, however, it becomes clear that Socrates is talking about an ideal city, which he proceeds to construct in his discussion. This city has three classes (gene) of citizens. The rulers are characterized by their knowledge about and devotion to the welfare of the city. The auxiliaries are the warrior class that helps the rulers. These two are collectively the guardians of the city (413c-414b; 428c-d). Finally, there are the farmers, artisans, and merchants - in general those concerned with the production of material goods necessary for daily life (369b-371e). The importance of this structure is that it allows Socrates to define virtues of the city by relations among its parts. These virtues are justice, wisdom, courage, and moderation. For instance, justice in the city is each one performing that function for which he is suited by nature and not doing the work that belongs to others (433a-b). One's function, in turn, is determined by the class to which he belongs. So rulers should rule and not amass wealth, which is the function of the farmers, artisans, and merchants; if the rulers turn from ruling to money-making they are unjust. Completing the analogy, Socrates gives an account of the soul. He argues that it has three parts, each corresponding to one of the classes in the city. At this point, we should note the difficulty of talking about parts of the soul. In making his argument about conflict in the soul, Socrates does not usually use a word that easily corresponds to the English 'part.' Sometimes he uses 'form' (eidos) and 'kind' (genos) (435b-c), other times a periphrasis such as 'that in the soul that calculates' (439d-e), although in the subsequent account of virtue, we do find meros, which means part (442b-d). So, insofar as vocabulary is concerned, one should be cautious about taking the subdivisions of the soul to be independent agents. Perhaps the least misleading way of thinking about the parts is as distinct functions. Reason is the function of calculating, especially about what is good for the soul. The appetites for food, drink, and sex are like the producing class - they are necessary for bodily existence (439c-e). These two parts are familiar to the modern reader, who will recognize the psychological capacity for reasoning and calculating, on the one hand, and the bodily desires, on the other. Less familiar is the part of the soul that corresponds to the auxiliaries, the military class. This is the spirited part (thymos or thymoeides). Associated with the heart, it is an aggressive drive concerned with honor. Thumos is manifested as anger with those who attack one's honor. Perhaps more importantly, it is manifested as anger with oneself when failing to do what one knows he should do (439e-440d). The importance of this account is that it is a moral psychology, an account of the soul which serves as a basis for explaining the virtues. Socrates' account also introduces the idea that there is conflict in the soul. For instance, the appetites can lead one toward pleasure which reason recognizes is not good for the whole soul. In cases of conflict, Socrates says the spirited part sides with reason against appetite. Here we see in rough outline the chief characters in a well-known moral drama. Reason knows what is good both for oneself and in the treatment of others. The appetites, short-sighted and self-centered, pull in the opposite direction. The spirited part is the principle that sides with reason and enforces its decrees. While the opposition between reason and appetite establishes their distinctness, it has another, more profound consequence. Most commentators read this section of the argument as implying that reason looks out for what is good for the soul while appetite seeks food, drink, and sex, heedless of their benefit for the soul (437d ff). Although open to various interpretations, the difference between reason and appetite does seem aimed at one of the central paradoxes of Socratic intellectualism. Common experience seems to contradict the claim that if one knows what is good he will do it; there seem to be obvious instances where someone does what she, in some sense, knows is not good, while having options to act otherwise. By introducing non-rational elements in the soul, the argument in the Republic also introduces the possibility of doing what one knows, all things considered, not to be good. In such cases, one is motivated by appetite, which lacks the capacity to conceive of what is good, all things considered. Some interpret this heedlessness as appetite's being good-independent, whereas reason is good-dependent. Thus, appetite pursues what it pursues without reference to whether what it pursues is good; reason pursues what it pursues always understanding that what it pursues is good. In this kind of interpretation, Socrates in the Republic accepts the possibility of akrasia because some parts of the soul, which are indifferent to the good, can motivate actions that do not aim at what is good. Others interpret this heedlessness as appetite's operating on a constrained notion of good; for instance, for appetite only pleasure is good. By contrast, reason operates on a larger notion, i.e., what is good, all things considered. In this interpretation, akrasia is also possible, but now because some parts of the soul, which motivate action, do so with a constrained view of good. In any event, the stage is set for conflict in the soul between reason and appetite. If we assume that either can motive action on its own, the possibility exists for the soul's pursuing bodily pleasure in spite of reason's determination that doing so is not good, all things considered. This potential gives rise to a separate strain of thinking about virtue. While, for Socratic intellectualism, virtue just is knowledge, in the aftermath of the argument for subdividing the soul, virtue comes to have two aspects. The first is to acquire the knowledge which is the basis of virtue; the second is to instill in the appetites and emotions - which cannot grasp the knowledge - a docility so that they react in a compliant way to what reason knows to be the best thing to do. Thus, non-rational parts of the soul acquire reliable habits on which the moral virtues depend. Given the complexity of the differences among the parts, we can now understand how their relations to one another define virtue in the soul. Virtue reduces the potential for conflict to harmony. The master virtue, justice, is each part doing its function and not interfering with that of another (441d-e; 443d). Since the function of reason is to exercise forethought for the whole soul, it should rule. The appetites, which seek only their immediate satisfaction, should not rule. A soul ruled by appetites is the very picture of psychological injustice. Still, to fulfill its function of ruling, reason needs wisdom, the knowledge of what is beneficial for each of the parts of the soul and for the whole. Moderation is a harmony among the parts based on agreement that reason should rule. Courage is the spirited part carrying out the decrees of reason about what is to be feared (442b-d). Any attentive reader of the dialogues must feel that Socrates has now given an answer to the questions that started many of the aporetic dialogues. At this point, we have the fully developed moral psychology that allows the definition of the moral virtues. They fall into place around the tripartite structure of the soul. One might object, however, that all Socrates has accomplished is to define justice and the other virtues as they operate within the soul. While each part treats the others justly, so to speak, it is not clear what justice among the parts of someone's soul has to do with that person treating other people justly. Socrates at first addresses this issue rather brusquely, saying that someone with a just soul would not embezzle funds, rob temples, steal, betray friends, break oaths, commit adultery, neglect parents, nor ignore the gods. The reason for this is that each part in the soul does its own function in the matter of ruling and being ruled (443a-b). Socrates does not explain this connection between psychic harmony and moral virtue. However, if we assume that injustice is based in overweening appetite or unbridled anger, then one can see the connection between restrained appetites, well-governed anger and treating others justly. The man whose sexual appetite is not governed by reason, e.g., would commit the injustice of adultery. This approach to the virtues by way of moral psychology, in fact, proves to be remarkably durable in ancient moral theory. In one way or another, the various schools attempt to explain the virtues in terms of the soul, although there are, of course, variations in the accounts. Indeed, we can treat the theory of the Republic as one such variation. While the account in Republic IV has affinities with that of Aristotle in Nicomachean Ethics, for instance, further developments in Republic V-VII make Plato's overall account altogether unique. It is in these books that the theory of forms makes its appearance. In Republic V, Socrates returns to the issue of political rule by asking what change in actual cities would bring the ideal city closer to realization. The famous answer is that philosophers should rule as kings (473d). Trying to make the scandalous, even ridiculous, answer more palatable, Socrates immediately begins to explain what he means by philosophers. They are the ones who can distinguish between the many beautiful things and the one beautiful itself. The beautiful itself, the good itself, and the just itself are what he calls forms. The ability to understand such forms defines the philosopher (476a-c). Fully elaborated, this extraordinary theory holds that there is a set of unchanging and unambiguous entities, collectively referred to as being. These are known directly by reason in a way that is separate from the use of sensory perception. The objects of sensory perception are collectively referred to as becoming since they are changing and ambiguous (508d). This infallible grasp means that one knows what goodness is, what beauty is, and what justice is. Because only philosophers have this knowledge - an infallible grasp of goodness, beauty, and justice - they and only they are fit to be rulers in the city. While moral theory occupies a significant portion of the Republic, Socrates does not say a lot about its relation to the epistemology and metaphysics of the central books. For instance, one might have expected the account to show how, e.g., reason might use knowledge of the forms to govern the other parts of the soul. Indeed, in Book VI, Socrates does say that the philosopher will imitate in his own soul the order and harmony of the forms out of admiration (500c-d). Since imitation is the heart of the account of moral education in Books II-III, the idea that one might imitate forms is intriguing; but it is not developed. In fact, the relation between virtues in the soul and the metaphysics and epistemology of the forms is not an easy one. For instance, Socrates says that virtue in the soul is happiness (580b-c). However, he also says that knowledge and contemplation of the forms is happiness (517c-d). Since having or exercising the virtues of wisdom, moderation, courage, and justice is different from knowledge and contemplation of forms, the questions naturally arises about how to bring the two together into an integrated life. In the dialogues, the relation between knowledge of forms and virtues such as moderation, justice, and courage is an issue that never seems to be fully resolved because it receives different treatments. In RepublicIX, Socrates offers a sketch of one way to bring these two dimensions together. Toward the end of the book, Socrates conducts a three-part contest to show the philosophical, or aristocratic, man is the happiest. In the second of these contests, he claims that each part of the soul has its peculiar desire and corresponding pleasure. Reason, for instance, desires to learn; satisfying that desire is a pleasure distinct from the pleasure, e.g., of eating (580d-e). One can now see that the functions of each of the parts has a new, affective dimension. Then, in the last of the contests, Socrates makes an ontological distinction between true pleasure of the soul and less true pleasure of the body. The former is the pleasure of knowing being, i.e., the forms, and the latter is the pleasure of filling bodily appetite. The less true pleasures also give rise to illusions and phantoms of pleasure; these illusions and phantoms recall the images and shadows in the allegory of the cave (583b-586c). Clearly, then, distinctions from the metaphysics and epistemology of the central books are being made relevant to the divisions within the soul. Pleasures of reason are on a higher ontological level than those of the bodily appetites; this difference calls for a modification of the definitions of the virtues from Book IV. For instance, in Book IV, justice is each part fulfilling its own function; in Book IX, that function is specified in terms of the kinds of pleasure each part pursues. Reason pursues the true pleasure of knowing and the appetites the less true pleasure of the body. Appetite commits injustice if it pursues pleasure not proper to it, i.e., if it pursues bodily pleasures as though they were true pleasures (586d-587a). The result is an account of virtue which discriminates among pleasures, as any virtue ethics should; but the criterion of discrimination reflects the distinction between being and becoming. Rational pleasures are more real than bodily pleasures, although bodily pleasures are not negligible. Happiness, in turn, integrates the virtue of managing bodily appetites and pleasures with the pleasures of learning, but definitely gives greater weight to the latter. However, in the Phaedo, we find another approach to the relation between knowledge of forms and virtues in the soul. Socrates identifies wisdom with kowledge of such forms as beauty, justice, and goodness (65d-e). The philosopher can know these only by reason that is detached from the body as much as possible (65e-66a). In fact, pure knowledge is found only when the soul is completely detached from the body in death. With such a severe account of the knowledge of the forms it is not surprising that courage, justice, and moderation are subordinated to wisdom. When Socrates says the true exchange of pleasures, pains, and fears is for wisdom, he invokes moderation, justice, and courage as virtues that serve this exchange (68b-69e). This passage suggests, for instance, that moderation would control bodily pleasures and pains and fears so that reason could be free of these disturbances in order to pursue knowledge. This notion of virtues differs from the account in the first books of the Republic in that the latter presents a comprehensive picture of the welfare of all the parts of the soul whereas the Phaedo, by implication, subordinates to reason those parts characterized by pleasures, pain, and fears. In the Symposium, we find yet another configuration of the relation between virtue and the forms, in which the non-rational parts of the soul disappear - or are sublimated. In Diotima's discourse on eros, she argues that the real purpose of love is to give birth in beauty (206b-c). This idea implies two dimensions: inspiration by what is itself beautiful and producing beautiful objects. Although Diotima talks about the way these concepts work in sexual procreation, her real interest is in their psychological manifestations. For instance, the lover, inspired by the physical beauty of the beloved, will produce beautiful ideas. In turn, the beauty of a soul will inspire ideas that will improve young men (210a-d). Finally, Diotima claims that the true object of erotic love is not the beautiful body or even the beautiful soul - but beauty itself. Unlike the beauty of bodies and of souls, this beauty does not come to be nor pass away, neither increases nor decreases. Nor does it vary according to aspect or context. It is not in a face or hands but is itself by itself, one in form (211a-c). Since this paradigm of beauty is the true object of eros, it inspires its own particular product, i.e., true virtue, distinct from the images of virtue inspired by encounters with beauty in bodies and souls (212a-c). While this passage highlights the relation between eros and the form of beauty in motivating virtue, it is almost silent on what this virtue might be. We do not know whether it just is the expression of love of beauty in the lover's actions, or its concretization in the dispositions of the lover's soul, or another manifestation. Clearly, this account in the Symposium works without the elaborate theory about parts of the soul and their function in virtue, found in Republic I-IV. As with other ancient theories, this account of virtue can be called eudaimonist. Plato's theory is best represented as holding that virtue, together with its active exercise, is the most important and the dominant constituent of happiness (580b-c). One might object that eudaimonist theories reduce morality to self-interest. We should recall however that eudaimonia in this theory does not refer primarily to a feeling. In the Republic, it refers to a state of the soul, and the active life to which it leads, whose value is multifaceted. The order and harmony of the soul is, of course, good for the soul because it provides what is good for each of the parts and the whole, and so makes the parts function well, for the benefit of each and of the whole person. In this way, the soul has the best internal economy, so to speak. Still, we should not overlook the ways in which order and harmony in the soul are paired with order and harmony in the city. They are both modeled on the forms. As a consequence, virtue in the soul is not a private concern artificially joined to the public function of ruling. Rather, the philosopher who imitates forms in ruling her soul is equally motivated to imitate forms in ruling the city. So, insofar as virtue consists in imitating the forms, it is also a state of the soul best expressed by exercising rule in the city - or at least in the ideal city. Eudaimonia, then, includes looking after the welfare of others. Indeed, the very nature of Plato's account of virtue and happiness leaves some aspects of the link between the two unclear. While virtue is the dominant factor in happiness, we still cannot tell whether for Plato one can have a reason for being, e.g., courageous that does not depend directly on happiness. Plato, even though he has forged a strong link between virtue and happiness, has not addressed such issues. In his account, it is still possible that one might be courageous just for its own sake while at the same time believing courage is also reliably linked to happiness. In the Republic, Socrates tries to answer the question what value justice has by itself in the soul; it does not follow that he is trying to convince Glaucon and Adeimantus that the value of justice is exhausted by its connection to one's own happiness. (For further developments in Plato's moral theory in dialogues usually thought to postdate the Republic (especially Philebus), see Plato entry, especially the section on Socrates and the section on Plato's indirectness.) ## 4. Aristotle The moral theory of Aristotle, like that of Plato, focuses on virtue, recommending the virtuous way of life by its relation to happiness. His most important ethical work, Nicomachean Ethics, devotes the first book to a preliminary account of happiness, which is then completed in the last chapters of the final book, Book X. This account ties happiness to excellent activity of the soul. In subsequent books, excellent activity of the soul is tied to the moral virtues and to the virtue of "practical wisdom" - excellence in thinking and deciding about how to behave. This approach to moral theory depends on a moral psychology that shares a number of affinities with Plato's. However, while for Plato the theory of forms has a role in justifying virtue, Aristotle notoriously rejects that theory. Aristotle grounds his account of virtue in his theory about the soul - a topic to which he devotes a separate treatise, de Anima. Aristotle opens the first book of the Nicomachean Ethics by positing some one supreme good as the aim of human actions, investigations, and crafts (1094a). Identifying this good as happiness, he immediately notes the variations in the notion (1095a15-25). Some think the happy life is the life of enjoyment; the more refined think it is the life of political activity; others think it is the life of study or theoretical contemplation (1095b10-20). The object of the life of enjoyment is bodily pleasure; that of political activity is honor or even virtue. The object of the life of study is philosophical or scientific understanding. Arguing that the end of human life must be the most complete, he concludes that happiness is the most complete end. Whereas pleasure, honor, virtue, and understanding are choice-worthy in themselves, they are also chosen for the sake of happiness. Happiness is not chosen for the sake of anything else (1097a25-1097b5). That the other choice-worthy ends are chosen for the sake of happiness might suggest that they are chosen only as instrumental means to happiness, as though happiness were a separate state. However, it is more likely that the other choice-worthy ends are constituents of happiness. As a consequence, the happy life is composed of such activities as virtuous pursuits, honorable acts, and contemplation of truth. While conceiving these choice-worthy ends as constituents of happiness might be illuminating, it does, in turn, raise the issue of whether happiness is a jumble of activities or whether it requires organization - even prioritization - of the constituents. Next, Aristotle turns to his own account of happiness, the summit of Book I. The account depends on an analogy with the notion of function or characteristic activity or work (ergon). A flutist has a function or work, i.e., playing the flute. The key idea for Aristotle is that the good of flute players (as such) is found in their functioning as a flutist. By analogy, if there is a human function, the good for a human is found in this function (1097b20-30). Aristotle then turns to the human soul. He argues that, in fact, there is a human function, to be found in the human soul's characteristic activity, i.e., the exercise of reason (1098a1-20). Then, without explanation, he makes the claim that this rational function is expressed in two distinct ways: by deliberating and issuing commands, on the one hand, and by obeying such commands, on the other. The part that has reason in itself deliberates about decisions, both for the short term and the long. The part that obeys reason is that aspect of the soul, such as the appetites, that functions in a human being under the influence of reason. The appetites can fail to obey reason; but they at least have the capacity to obey (as, for example, such autonomic functions as nutritive and metabolic ones do not). Aristotle then argues that since the function of a human is to exercise the soul's activities according to reason, the function of a good human is to exercise well and finely the soul's activities according to reason. Given the two aspects of reason that Aristotle has distinguished, one can see that both can be well or badly done. On the one hand, one can reason well or badly - about what to do within the next five minutes, twenty-four hours, or ten years. On the other, actions motivated by appetites can be well or badly done, and likewise having an appetite at all can sometimes not be a well done, but a badly done, activity of the soul. Acting on the desire for a drink from the wine cooler at a banquet is not always a good idea, nor is having such a desire. According to Aristotle, the good human being has a soul in which these functions are consistently done well. Thus, good persons reason well about plans, short term or long; and when they satisfy their appetites, and even when they have appetites, it is in conformity with reason. Returning to the question of happiness, Aristotle says the good for a human is to live the way the good human lives, that is, to live with one's life aimed at and structured by the same thing that the good human being aims at in his or her life. So, his account of happiness, i.e., the highest good for a human, is virtuous or excellent activity of the soul. But he has not identified this virtuous activity with that of the moral virtues, at this point. In fact, he says if there are many kinds of excellence, then human good is found in the active exercise of the highest. He is careful to point out that happiness is not just the ability to function well in this way; it is the activity itself. Moreover, this activity must be carried out for a complete life. One swallow does not make a spring. Although the reference here to parts or aspects of the soul is cursory, it is influenced by Aristotle's theories in the de Anima. Fundamental to the human soul and to all living things, including plants, is nutrition and growth (415a20 ff). Next is sensation and locomotion; these functions are characteristic of animals (416b30 ff). Aristotle associates appetite and desire with this part of the soul (414b1-5). Thus we have a rough sketch of animal life: animals, moved by appetite for food, go toward the objects of desire, which are discerned by sensation. To these functions is added thought in the case of humans. Thought is both theoretical and practical (427a 15-20). The bulk of the de Anima is devoted to explaining the nutritive, sensory, and rational functions; Aristotle considers desire and appetite as the source of movement in other animals (432a15 ff), and these plus reason as its source in humans. In Nicomachean Ethics he focuses on the role that appetite and desire, together with reason, play in the moral drama of human life. In chapter 8 of Book I, Aristotle explicitly identifies human good with psychological good. Dividing goods into external goods, those of the body, and those of the soul, he states that his account of happiness agrees with those who hold it is a good of the soul. In fact, in this account, happiness is closely related to traditionally conceived psychological goods such as pleasure and moral virtue, although the nature of the relation has yet to be shown (1098b10-30). Still, in Book I Aristotle is laying the foundation in his moral psychology for showing the link between the moral virtues and happiness. In Book II he completes this foundation when he turns to the question of which condition of the soul is to be identified with (moral) virtue, or virtue of character. In II. 5, he says that conditions of the soul are either feelings (pathe), capacities for feeling (dynameis) or dispositions (hexeis). Feelings are such things as appetite, anger, fear and generally those conditions that are accompanied by pleasure and pain. Capacities are, for example, the simple capacity to have these feelings. Finally, disposition is that condition of the soul whereby we are well or badly off with respect to feelings. For instance, people are badly disposed with respect to anger who typically get angry violently or who typically get angry weakly (1105b20-30). Virtue, as a condition of the soul, will be one of these three. After arguing that virtue is neither feeling nor capacity, Aristotle turns to what it means to be well or badly off with respect to feelings. He says that in everything that is continuous and divisible it is possible to take more, less, or an equal amount (1106a25). This remark is puzzling until we realize that he is actually talking about feelings. Feelings are continuous and divisible; so one can take more, less, or an equal amount of them. Presumably, when it comes to feeling anger, e.g., one can feel too much, not enough, or a balanced amount. Aristotle thinks that what counts as too much, not enough, or a balanced amount can vary to some extent from individual to individual. At this point he is ready to come back to moral virtue for it is concerned with feelings and actions (to which feelings give rise), in which one can have excess, deficiency, or the mean. To have a feeling like anger at the right time, on the right occasion, towards the right people, for the right purpose and in the right manner is to feel the right amount, the mean between extremes of excess and deficiency; this is the mark of moral virtue (1106a15-20). Finally, virtue is not a question only of feelings since there is a mean between extremes of action. Presumably, Aristotle means that the appropriate feeling - the mean between the extremes in each situation - gives rise to the appropriate action. At last Aristotle is ready to discuss particular moral virtues. Beginning with courage, he mentions here two feelings, fear and confidence. An excessive disposition to confidence is rashness and an excessive disposition to fear and a deficiency in confidence is cowardice. When it comes to certain bodily pleasures and pains, the mean is moderation. While the excess is profligacy, deficiency in respect of pleasures almost never occurs. Aristotle gives a fuller account of both of these virtues in Book III; however, the basic idea remains. The virtue in each case is a mean between two extremes, the extremes being vices. Virtue, then, is a reliable disposition whereby one reacts in relevant situations with the appropriate feeling - neither excessive nor deficient - and acts in the appropriate way - neither excessively nor deficiently. To complete the notion of moral virtue we must consider the role reason plays in moral actions. Summing up at Book II.6, Aristotle says virtue is a disposition to choose, lying in the mean which is relative to us, determined by reason (1107a1). Since he is talking about choosing actions, he is focusing on the way moral virtue issues in actions. In turn, it is the role of practical wisdom (phronesis) to determine choice. While moral virtues, virtues of character, belong to the part of the soul which can obey reason, practical wisdom is a virtue of the part of the soul that itself reasons. The virtues of thought, intellectual virtues, are knowledge (episteme), comprehension (nous), wisdom (sophia), craft (techne), and practical wisdom (1139b15-25). The first three grasp the truth about what cannot be otherwise and is not contingent. A good example of knowledge about what cannot be otherwise is mathematics. Craft and practical wisdom pursue the truth that can be had about what can be otherwise and is contingent. What can be otherwise includes what is made - the province of craft - and what is done - the province of practical wisdom (1140a1). While Aristotle's account of practical wisdom raises several problems, we will focus on only two closely related issues. He says that it is the mark of someone with practical wisdom to deliberate well about what leads to the good for himself (to dunasthai kalos bouleusasthai peri ta hauto(i) agatha). This good is not specific, such as health and strength, but is living well in general (1140a25-30). This description of practical wisdom, first of all, implies that it deliberates about actions; it is a skill for discerning those actions which hit the mean between the two extremes. However, the ambiguity of the phrase 'what leads to the good' might suggest that practical wisdom deliberates only about instrumental means to living well. Still since practical wisdom determines which actions hit the mean between two extremes, such actions are not instrumental means to living well - as though living well were a separate state. Actions which hit the mean are parts of living well; the good life is composed of actions under the headings of, for instance, honor and pleasure, which achieve the mean. In addition, the deliberation of practical wisdom does not have to be confined to determining which actions hit the mean. While Aristotle would deny that anyone deliberates about whether happiness is the end of human life, we do deliberate about the constituents of happiness. So, one might well deliberate about the ways in which honor and pleasure fit into happiness. Now we can discern the link between morality and happiness. While happiness itself is excellent or virtuous activity of the soul, moral virtue is a disposition to achieve the mean between two extremes in feeling and in action. The missing link is that achieving the mean is also excellent activity of the soul. Activity that expresses the virtue of courage, for example, is also the best kind of activity when it comes to the emotion of fear. Activity that expresses the virtue of moderation is also excellent activity when it comes to the bodily appetites. In this way, then, the happy person is also the virtuous person. However, in Book I Aristotle has already pointed out the problem of bodily and external goods in relation to happiness. Even if happiness is virtuous activity of the soul, in some cases these goods are needed to be virtuous - for example, one must have money to be generous. In fact, the lack of good birth, good children, and beauty can mar one's happiness for the happy person does not appear to be one who is altogether ugly, low born, solitary, or childless, and even less so if he has friends and children who are bad, or good friends and children who then die (1099a30-1099b10). Aristotle is raising a problem that he does not attempt to solve in this passage. Even if happiness is virtuous activity of the soul, it does not confer immunity to the vicissitudes of life. Aristotle's moral psychology has further implications for his account of happiness. In Book I, chapter 7, he said that human good is virtuous activity of the soul but was indefinite about the virtues. In most of the Nicomachean Ethics he talks about the moral virtues, leaving the impression that virtuous activity is the same as activity associated with moral virtues. In Book X, however, Aristotle revisits the issue of virtuous activity. If happiness, he says, is activity in accordance with virtue, it will be activity in accordance with the highest. The highest virtue belongs to the best part of the soul, i.e., the intellect (nous) or the part that governs in the soul and contemplates the fine and godly, being itself the divine part of the soul or that which is closest to the divine (1177a10-20). Up to this point, Aristotle has apparently been talking about the man of political action and the happiness that is suitable to rational, embodied human beings. Active in the life of the city, this person exercises courage, moderation, liberality, and justice in the public arena. Now, instead of the life of an effective and successful citizen, Aristotle is holding up the life of study and contemplation as the one that achieves happiness - that is, the highest human good, the activity of the highest virtue. Such a life would achieve the greatest possible self-sufficiency and invulnerability (1177a30). Indeed, at first he portrays these two lives as so opposite that they seem incompatible. In the end, however, he palliates the differences, leaving the possibility for some way to harmonize the two (1178a30). The differences between the two lives are rooted in the different aspects of the soul. Moral virtues belong to the appetites and desires of the sensory soul - the part obviously associated with the active political life, when its activities are brought under the guidance and control of excellent practical thought and judgment. The "highest" virtues, those belonging to the scientific or philosophical intellect, belong to theoretical reason. To concentrate on these activities one must be appropriately disengaged from active political life. While the latter description leads Aristotle to portray as possible a kind of human life that partakes of divine detachment (1178b5 ff), finally human life is an indissolvable composite of intellect, reason, sensation, desires, and appetites. For Aristotle, strictly speaking, happiness simply is the exercise of the highest virtues, those of theoretical reason and understanding. But even persons pursuing those activities as their highest good, and making them central to their lives, will need to remain connected to daily life, and even to political affairs in the community in which they live. Hence, they will possess and exercise the moral virtues and those of practical thought, as well as those other, higher, virtues, throughout their lives. Clearly, this conception of happiness does not hold all virtue, moral and intellectual, to be of equal value. Rather, Aristotle means the intellectual virtue of study and contemplation to be the dominant part of happiness. However, problems remains since we can understand dominance in two ways. In the first version, the activity of theoretical contemplation is the sole, exclusive component of happiness and the exercise of the moral virtues and practical wisdom is an instrumental means to happiness, but not integral to it. The problem with this version of dominance is that it undermines what Aristotle has said about the intrinsic value of the virtuous activity of politically and socially engaged human beings, including friendship. In a second version of dominance, we might understand contemplation to be the principal, but not exclusive, constituent of happiness. The problem with this version of dominance lies in integrating such apparently incompatible activities into a coherent life. If we give the proper weight to the divine good of theoretical contemplation it may leave us little interest in the virtuous pursuits of the moral goods arising from our political nature, except, again, as means for establishing and maintaining the conditions in which we may contemplate. Like Plato, Aristotle is a eudaimonist in that he argues that virtue (including in some way the moral virtues of courage, justice and the rest) is the dominant and most important component of happiness. However, he is not claiming that the only reason to be morally virtuous is that moral virtue is a constituent of happiness. He says that we seek to have virtue and virtuous action for itself as well (Nicomachean Ethics, 1097b 1-10); not to do so is to fail even to be virtuous. In this regard, it is like pleasure, which is also a constituent of the happy life. Like pleasure, virtue is sought for its own sake. Still, as a constituent of happiness, virtuous action is grounded in the highest end for a human being. One can discern in the Nicomachean Ethics two different types of argument for the link between virtue and happiness. One is based on Aristotle's account of human nature and culminates in the so-called function argument of Book I. If happiness is excellent or virtuous activity of the soul, the latter is understood by way of the uniquely human function. If one understands the human function then one can understand what it is for that function to be done excellently (1098a5-15). This sort of argument has been criticized because it moves from a premise about what humans are to a conclusion about what they ought to be. Such criticism reflects the modern claim that there is a fact-value distinction. One defense of Aristotle's argument holds that his account of human nature is meant both to be objective and to offer the basis for an understanding of excellence. The difference, then, between modern moral theory and ancient is over what counts as an objective account of human nature. However, even if we accept this defense, we can still ask why a human would consider it good to achieve human excellence as it is defined in the function argument. At this point another argument for the link between happiness and virtue - one more dispersed in the text of the Nicomachean Ethics - becomes relevant; it is based on value terms only and appeals to what a human might consider it good to achieve. Aristotle describes virtuous activity of the soul as fine (kalos) and excellent (spoudaios). Finally, the link between virtue and happiness is forged if a human sees that it is good to live a life that one considers to be fine and excellent. (For further detailed discussion, see entry on Aristotle's ethics.) ## 5. Cynics Although the Cynics had an impact on moral thinking in Athens after the death of Socrates, it is through later, and highly controversial, reports of their deeds and sayings - rather than their writings - that we know of them. Diogenes the Cynic, the central figure, is famous for living in a wine jar (Diogenes Laertius [= DL] VI 23) and going about with a lantern looking for 'a man' - i.e., someone not corrupted (DL VI 41). He claimed to set courage over against fortune, nature against convention, and reason against passion (DL VI 38). Of this trio of opposites, the most characteristic for understanding the Cynics is nature against convention. Diogenes taught that a life according to nature was better than one that conformed to convention. First of all, natural life is simpler. Diogenes ate, slept, or conversed wherever it suited him and carried his food around with him (DL VI 22). When he saw a child drinking out of its hand, he threw away his cup, saying that a child had bested him in frugality (DL VI 37). He said the life of humans had been made easy by the gods but that humans had lost sight of this through seeking after honeyed cakes, perfumes, and similar things (DL VI 44). With sufficient training the life according to nature is the happy life (DL VI 71). Accordingly Diogenes became famous for behavior that flouted convention (DL VI 69). Still, he thought that the simple life not only freed one from unnecessary concerns but was essential to virtue. Although he says nothing specific about the virtues, he does commend training for virtuous behavior (DL VI 70). His frugality certainly bespeaks self-control. He condemned love of money, praised good men, and held love to be the occupation of the idle (DL VI 50-51). Besides his contempt for convention, what is most noteworthy about Diogenes as a moral teacher is his emphasis on detachment from those things most people consider good. In this emphasis, Diogenes seems to have intensified a tendency found in Socrates. Certainly Socrates could be heedless of convention and careless about providing for his bodily needs. To Plato, however, Diogenes seemed to be Socrates gone mad (DL VI 54). Still, in Diogenes' attitude, we can see at least the beginning of the idea that the end of life is a psychological state marked by detachment. Counseling the simple and uncomplicated satisfaction of one's natural instincts and desires, Diogenes urges detachment from those things held out by convention to be good. While he is not so explicit, others develop the theme of detachment into the notion of tranquility. The Stoics and Epicureans hold that happiness depends on detachment from vulnerable or difficult to obtain bodily and external goods and consists in a psychological state more under one's own direct control. In this way, happiness becomes associated (for the Epicureans) with tranquility (ataraxia). Finally, in Skepticism, suspension of judgment is a kind of epistemic detachment that provides tranquility. So in Diogenes we find the beginnings of an idea that will become central to later ancient moral theory. ## 6. Cyrenaics The first of the Cyrenaic school was Aristippus, who came from Cyrene, a Greek city on the north African coast. The account of his teachings, in Diogenes Laertius, can seem sometimes inconsistent. Nevertheless, Aristippus is interesting because, as a thorough hedonist, he is something of a foil for Epicurus. First of all, pleasure is the end or the goal of life - what everyone should seek in life. However, the pleasure that is the end is not pleasure in general, or pleasure over the long term, but immediate, particular pleasures. Thus the end varies situation by situation, action by action. The end is not happiness because happiness is the sum of particular pleasures (DL II 87-88). Accumulating the pleasures that produce happiness is tiresome (DL II 90). Particular pleasures are ones that are close-by or sure. Moreover, Aristippus said that pleasures do not differ from one another, that one pleasure is not more pleasant than another. This sort of thinking would encourage one to choose a readily available pleasure rather than wait for a "better" one in the future. This conclusion is reinforced by other parts of his teaching. His school says that bodily pleasures are much better than mental pleasures. While this claim would seem to contradict the idea that pleasures do not differ, it does show preference for the immediately or easily available pleasures of bodily gratification over, e.g., the mental pleasure of a self-aware just person. In fact, Aristippus' school holds that pleasure is good even if it comes from the most unseemly things (DL II 88). Aristippus, then, seems to have raised improvidence to the level of a principle. Still, it is possible that the position is more than an elaborate justification for short-sighted pleasure-seeking. Cyrenaics taught that a wise man (sophos) (one who always pursues immediate gratification) will in general live more pleasantly than a foolish man. That prudence or wisdom (phronesis) is good, not in itself but in its consequences, suggests that some balance, perhaps even regarding others, is required in choosing pleasures (DL II 91). The Cyrenaic attitude to punishment seems to be an example of prudence. They hold that nothing is just, fine, or base by nature but only by convention and custom; still a good man will do nothing out of line through fear of punishment (DL II 93). Finally, they hold that friendship is based in self-interest (DL II 91). These aspects of Cyrenaic teaching suggest they are egoist hedonists. If so, there are grounds for taking the interest of others into account as long as doing so is based on what best provides an individual pleasure. Nevertheless, Aristippus' school holds that the end of life is a psychological good, pleasure. Still, it is particular pleasures not the accumulation of these that is the end. As a consequence, their moral theory contrasts sharply with others in antiquity. If we take the claims about the wise man, prudence, and friendship to be references to virtue, then Aristippus' school denies that virtue is indispensable for achieving the end or goal of life. While they hold that virtue is good insofar as it leads to the end, they seem prepared to dispense with virtue in circumstances where it proves ineffective. Even if they held virtue in more esteem, the Cyrenaics would nonetheless not be eudaimonists since they deny that happiness is the end of life. ## 7. Epicurus Epicurean moral theory is the most prominent hedonistic theory in the ancient world. While Epicurus holds that pleasure is the sole intrinsic good and pain is what is intrinsically bad for humans, he is also very careful about defining these two. Aware of the Cyrenaics who hold that pleasures, moral and immoral, are the end or goal of all action, Epicurus presents a sustained argument that pleasure, correctly understood, will coincide with virtue. In the Letter to Menoeceus, Epicurus begins by making a distinction among desires. Some desires are empty or groundless and others are natural; the natural are further subdivided into the merely natural and the necessary. Finally, the necessary are those necessary for happiness, those necessary for the body's freedom from distress, and those necessary for life itself (Letter to Menoeceus 127). A helpful scholiast (cf. Principal Doctrines XXIX) gives us some examples; necessary desires are ones that bring relief from unavoidable pain, such as drinking when thirsty - if we don't drink when we need replenishment, we will just get thirstier and thirstier, a painful experience. The natural but not necessary are the ones that vary pleasure but are not needed in order to motivate us to remove or ward off pain, such as the desire for expensive food: we do not need to want, or to eat, expensive food in order to ward off the pain of prolonged hunger. Finally, the groundless desires are for such things as crowns and statues bestowed as civic honors - these are things that when desired at all are desired with intense and harmful cravings. Keeping these distinctions in mind is a great help in one's life because it shows us what we need to aim for. The aim of the blessed life is the body's health and the soul's freedom from disturbance (ataraxia) (128). After this austere introduction, Epicurus makes the bold claim that pleasure is the beginning and end of the blessed life. Then he makes an important qualification. Just because pleasure is the good, Epicureans do not seek every pleasure. Some lead to greater pain. Just so, they do not avoid all pains; some lead to greater pleasures (128-29). Such a position sounds, of course, like common-sense hedonism. If one's aim is to have as much pleasure as possible over the long term, it makes sense to avoid some smaller pleasures that will be followed by larger pains. If one wants, for example, to have as much pleasure from drinking wine as possible, then it would make sense to exercise some judgment about how much to drink on an occasion since the next morning's hangover will be very unpleasant, and might keep one from having wine the next day. However, his distinction among groundless, natural, and necessary desires should make us suspicious that Epicurus is no common-sense hedonist. The aim of life is not maximizing pleasures in the way the above example suggests. Rather, real pleasure, the aim of life, is what we experience through freedom from pain and distress. So it is not the pains of the hangover or the possible loss of further bouts of wine drinking that should restrain my drinking on this occasion. Rather one should be aiming at the pleasure given by freedom from bodily pain and mental distress (131-32). The usual way to understand the pleasure of freedom from pain and from distress is by way of the distinction between kinetic pleasures and katastematic, or what are, following Cicero, misleadingly called 'static' ones (Diogenes Laertius X 136). The name of the former implies motion and the name of the latter implies a state or condition. The reason the distinction is important is that freedom from pain and from distress is a state or condition, not a motion. Epicurus holds not only that this state is a kind of pleasure but that it is the most complete pleasure (Principal Doctrines III). Modern commentators have taken various approaches to explaining why this state should be considered a pleasure, as opposed to, e.g., the attitude of taking pleasure in the fact that one is free of pain and distress. After all, taking pleasure in a fact is not a feeling in the same sense as the pleasure of drinking when thirsty. Since the latter is usually taken to be an example of kinetic pleasure (and is associated with the pain of thirst), sometimes katastematic pleasure is said to be just kinetic pleasure free from pain and distress, e.g., the pleasure of satiety. A somewhat broader conception of katastematic pleasure holds that it is the enjoyment of one's natural constitution when one is not distracted by bodily pain or mental distress. Finally, some commentators hold the pleasure of freedom from pain and from distress to be a feeling available only to the wise person, who properly appreciates simple pleasures. Since eating plain bread and drinking water are usually not difficult to achieve, to the wise, their enjoyment is not overwhelmed by fear (130-31). Accordingly, the pleasure they take in these is free of pain and distress. Of course, the wise do not have to confine themselves to simple pleasures; they can enjoy luxurious ones as well - as long as they avoid needing them, which entails fear. At this point, we can see that Epicurus has so refined the account of pleasure and pain that he is able to tie them to virtue. In the Letter to Menoeceus, he claims, as a truth for which he does not argue, that virtue and pleasure are inseparable and that living a prudent, honorable, and just life is the necessary and sufficient means to the pleasure that is the end of life (132). An example of what he might mean is found in Principal Doctrines, where Epicurus holds that justice is a contract among humans to avoid suffering harm from one another. Then he argues that injustice is not bad per se but is bad because of the fear that arises from the expectation that one will be punished for his misdeeds. He reinforces this claim by arguing that it is impossible for someone who violates the compact to be confident that he will escape detection (Principal Doctrines XXXIV-V). While one might doubt this claim about the malfactor's state of mind, nevertheless, we can see that Epicurus means to ground justice, understood as the rules governing human intercourse, in his moral psychology, i.e., the need to avoid distress. Epicurus, like his predecessors in the ancient moral tradition, identified the good as something psychological. However, instead of, for example, the complex Aristotelian notion of excellent activity of the soul, Epicurus settled on the fairly obvious psychological good of pleasure. Of course, Aristotle argues that excellent activity of the soul is intrinsically pleasurable (Nicomachean Ethics 1099a5). Still, in his account pleasure seems something like a dividend of excellent activity (1175b30). By contrast, for Epicurus pleasure itself is the end of life. However, since Epicureans hold freedom from pain (aponia) and distress (ataraxia) gives the preferable pleasure, they emphasize tranquility (ataraxia) as the end of life. Modern utilitarians, for whom freedom from pain and distress is not paramount, would include a broader palate of pleasures. Epicurus' doctrine can be considered eudaimonist. While Plato and Aristotle maintain that virtue is constitutive of happiness, Epicurus holds that virtue is the only means to achieve happiness, where happiness is understood as a continuous experience of the pleasure that comes from freedom from pain and from mental distress. Thus, he is a eudaimonist in that he holds virtue is indispensable to happiness; but he does not identify virtuous activity, in whole or in part, with happiness. Finally, Epicurus is usually interpreted to have held a version of psychological hedonism - i.e., everything we do is done for the sake of pleasure - rather than ethical hedonism - i.e., we ought to do everything for the sake of pleasure. However, Principal Doctrine XXV suggests the latter position; and when in the Letter to Menoeceus he says that "we" do everything in order not to be in pain or in fear, he might mean to be referring to "we" Epicureans. If so, the claim would be normative. Still, once all disturbance of the soul is dispelled, he says, one is no longer in need nor is there any other good that could be added (128). Since this claim appears to be descriptive, Epicurus could be taken, as he usually has been, to be arguing that whatever we do is done for the sake of pleasure. In this account, that aspect of human nature on which virtue is based is fairly straightforward. The account is certainly less complex than, e.g., Aristotle's. In turn, Epicurus seems to have argued in such a way as to make pleasure the only reason for being virtuous. If psychological hedonism is true, then when one realizes the necessary link between virtue and pleasure, one has all the reason one needs to be virtuous and the only reason one can have. ## 8. Stoics The Stoics are well known for their teaching that the good is to be identified with virtue. Virtues include logic, physics, and ethics (Stoicorum Veterum Fragmenta [=SVF]II 35), as well as wisdom, moderation, justice, and courage. To our modern ears, the first three sound like academic subjects; but for the Stoics, they were virtues of thought. However, orthodox Stoics do not follow the Aristotelian distinction between intellectual and moral virtues because - as we shall see - they hold that all human psychological functions, including the affective and volitional, are rational in a single, unified sense. For them, consequently, all virtues form a unity around the core concept of knowledge. Finally, all that is required for happiness (i.e., the secure possession of the good, of what is needed to make one's life a thoroughly good one) - and the only thing - is to lead a virtuous life. In this teaching Stoics are addressing the problem of bodily and external goods raised by Aristotle. Their solution takes the radical course of dismissing such alleged goods from the account of happiness because they are not necessary for virtue, and are not, in fact, in any way good at all. They argue that health, pleasure, beauty, strength, wealth, good reputation, and noble birth are neither good nor bad. Since they can be used well or badly and the good is invariably good, these assets are not good. The virtues, however, are good (DL VII 102-103), since they are perfections of our rationality, and only rationally perfected thoughts and decisions can possibly have the features of harmony and order in which goodness itself consists. Since possessing and exercising virtue is happiness, happiness does not include such things as health, pleasure, and wealth. Still, the Stoics do not dismiss these assets altogether since they still have a kind of value. These things are indifferent to happiness in that they do not add to one's virtue nor detract from it, and so they do not add to or take away from one's possession of the good. One is not more virtuous because healthy nor less virtuous because ill. But being healthy generally conforms with nature's plans for the lives of animals and plants, so it is preferable to be healthy, and one should try to preserve and maintain one's health. Health is, then, the kind of value they call a preferred indifferent; but it is not in any way a good, and it makes no contribution to the quality of one's life as a good or a bad one, happy or miserable. In order to understand the Stoic claims about the relation of virtue to happiness, we can begin with virtue. Chrysippus says that virtue is a craft (techne) having to do with the things of life (SVF II 909). In other texts, we learn that the things of life include impulse (horme). Each animal has an impulse for self-preservation; it has an awareness of its constitution and strives to preserve its integrity. There is also a natural impulse to care for offspring. Humans, then, are naturally inclined toward preserving life, health, and children. But then grown-up humans also do these things under the guidance of reason; reason in the adult human case is the craftsman of impulse (DL VII 85-6). This latter phrase is significant because it implies that just following natural impulse is not enough. In fact, it is not even possible for an adult human, whose nature is such as to do everything they do by reason, even to follow the sort of natural impulses that animals and immature humans do in their actions. In order to lead a virtuous life, reason must shape our impulses and guide their expression in action. Impulse is the key to understanding the relation between virtue and happiness. An impulse is a propositional attitude that, when assented to, leads to action, e.g., 'I should eat this bread now.' However, impulses do not arise in a separate, non-rational part of the soul. Stoics deny there are any non-rational desires of appetite capable of impelling action. The soul, insofar as it provides motivations and is the cause of our actions, consists of the commanding faculty (hegemonikon) which is also reason (SVF I 202). We can distinguish correct impulses into those which treat virtue as the only good and those which treat things indifferent as indifferent. By contrast, emotions or passions (pathe) are incorrect impulses that treat what is indifferent as good. However, they do not come from a non-rational part of the soul but are false judgments about the good, where judgment is understood as assent to some impulse. Emotions, such as desire, fear, pleasure, and pain, embody such erroneous judgments (SVF III 391, 393, 394). For instance, the desire for health arises from assenting to the impulse that embodies the false judgment that health is good, instead of a preferred indifferent. The sage - someone perfected in virtue - would never assent to such false propositions and thus would never have emotions in this sense, no feelings that carried him beyond reason's true assessment. He would, however, experience feelings attuned to reason, eupatheiai -literally good emotions or feelings. For instance, he would feel joy over his virtue, but not pleasure - the latter being an emotion that treats the actual possession of an indifferent as a good. Knowledge, then, about what is good, bad, and indifferent is the heart of virtue. Courage, e.g., is simply knowledge of what is to be endured: the impulse to endure or not, and the only impulse that is needed by courage, then follows automatically, as a product or aspect of that knowledge. This tight unity in the soul is the basis for the Stoic teaching about the unity of the virtues. Zeno (the founder of the school) defines wisdom (phronesis), or rather practical knowledge, in matters requiring distribution as justice, in matters requiring choice as moderation, and in matters requiring endurance as courage (Plutarch On Moral Virtue 440E-441D). Practical knowledge, then, is a single, comprehensive knowledge of what is good and bad in each of these kinds of circumstance. Attending to this identification of virtue and practical knowledge is a good way to understand the central Stoic teaching that virtue is living in agreement with nature (SVF III 16). Nature includes not only what produces natural impulses but also the rest of the government of the cosmos, the natural world. The universe is governed by right reason that pervades everything and directs (causes) the way it functions - with the exception of the only rational animals there are, the adult human beings: their actions are governed by themselves, i.e., by assenting to, or withholding assent from particular impulses. Nature is even identified with Zeus, who is said to be the director of the administration of all that exists (DL VII 87-9). Since reason governs the universe for the good, everything happens of necessity and for the overall good. Virtue, then, includes understanding both one's individual nature as a human being and the way nature arranges the whole universe. At this point, we can appreciate the role of logic and physics as virtue since these entail knowledge of the universe. This understanding is the basis for living in agreement with the government of the universe, i.e., with nature, by making one's decisions and actions be such as to agree with Zeus's or nature's own plans, so far as one can understand what those are. It is in this context that we can best understand the Stoic teaching about indifferents, such as health and wealth. An individual's health is vulnerable to being lost if right reason that governs the universe requires it for the good of the whole. If happiness depended on having these assets and avoiding their opposites, then, in these cases, happiness would be impossible. However, if virtue is living in agreement with nature's government of the universe and if virtue is the only good, one's happiness is entirely determined by his patterns of assent and is therefore not vulnerable to being lost. If one understands that the good of the whole dictates that in a particular case one's health must be sacrificed, then one recognizes that his happiness does not require health. We should not, however, see this recognition as tantamount to renunciation. If the Stoic notion of happiness has any relation at all to the ordinary sense, renunciation cannot be a part of it. Rather, the Stoic view of living in accordance with nature should imply not only understanding the way right reason rules the universe but agreeing with it and even desiring that things happen as they do. We can best appreciate the notion that virtue is the good, then, if we take virtue as both acknowledging that the universe is well governed and adopting the point of view, so to speak, of the government (DL VII 87-9). A refinement of the Stoic approach to indifferents gives us a way of understanding what living in agreement with nature might look like. After all, such things as health and wealth cannot just be dismissed since they are something like the raw material of virtue. It is in pursuing and using them that one exercises virtue, for instance. In the attempt to integrate preferred indifferents into the pursuit of the good, Stoics used an analogy with archery (On Ends III.22). Since he aims to hit the target, the archer does everything in his power to hit the target. Trying everything in his power reflects the idea that such factors as a gust of wind - chance happenings that cannot be controlled or foreseen - can intervene and keep him from achieving the goal. To account for this type of factor, it is claimed that the goal of the archer is really trying everything in his power to attain the end. The analogy with the art of living focuses on this shift from the goal of hitting the target to the goal of trying everything in one's power to hit the target (On Ends V.17-20). It is the art of trying everything in one's power to attain such preferred indifferents as health and wealth. However, if right reason, which governs the universe, decides that one will not have either, then the sage follows right reason. At this point the analogy with archery breaks down since, for the sage, trying everything in one's power does not mean striving until one fails; rather, it means seeking preferred indifferents guided by right reason. By this reasoning, one should see that virtue and happiness are not identified with achieving heath and wealth but with the way one seeks them and the evaluative propositions one assents to. Finally, the art of living is best compared to such skills as acting and dancing (On Ends V. 24-5). This way of relating preferred indifferents to the end, i.e., to happiness, was challenged in antiquity as incoherent. Plutarch, for instance, argues that it is contrary to common understanding to say that the end is different from the reference point of all action. If the reference point of all action - what one does everything to achieve - is to have preferred indifferents such as health, then the end is to have preferred indifferents. However, if the end is not to have preferred indifferents (but, say, always to act prudently), then the reference point of all action cannot be the preferred indifferents (On Common Conceptions 1070F-1071E). The Stoics are presented with a dilemma: either preferred indifferents are integral to the end or they are not the object of choice. The Stoics are extreme eudaimonists compared to Plato or Aristotle, although they are clearly inspired by Socratic intellectualism. While Plato clearly associates virtue and happiness, he never squarely faces the issue whether happiness may require other goods, e.g., wealth and health. Aristotle holds happiness to be virtuous activity of the soul; but he raises - without solving - the problem of bodily and external goods and happiness. For these two, virtue, together with its active exercise, is the dominant and most important component of happiness, while Stoics simply identify virtue and the good, and so make it the only thing needed for a happy life. Still, Stoics do not reduce happiness to virtue, as though 'happiness' is just a name for being perfectly just, courageous, and moderate. Rather they have independent ways of describing happiness. Following Zeno, all the Stoics say it is a good flow of life. Seneca says the happy life is peacefulness and constant tranquility. However, we should keep in mind that, while they do not reduce happiness to virtue, their account of happiness is not that of the common person. So in recommending virtue because it secures happiness the Stoics are relying on happiness in a special, although not idiosyncratic, sense. In fact, their idea of happiness shares an important feature with the Epicurean, which puts a premium on tranquil pleasures. In Stoicism as well, deliverance from the vicissitudes of fate leads to a notion of happiness that emphasizes tranquility. And, as we shall see, tranquility is a value for the Skeptics. Clearly, the Stoic account of virtue and happiness depends on their theory about human nature. For Aristotle, virtue is perfection of the human function and the Stoics follow in this line of thinking. While their notion of virtue builds on their notion of the underlying human nature, their account of the perfection of human nature is more complex than Aristotle's. It includes accommodation to the nature of the universe. Virtue is the perfection of human nature that makes it harmonious with the workings of fate, i.e. with Zeus's overall plan, regarded as the ineluctable, though providential, cause of what happens in the world at large. ## 9. Pyrrhonian Skeptics Pyrrho, a murky figure, roughly contemporary with Epicurus and Zeno the Stoic, left no writings. In the late, anecdotal tradition he is credited with introducing suspension of judgment (DL IX 61). He became the eponymous hero for the founding of Pyrrhonian skepticism in the first century B.C. (See entry on Pyrrho and on Ancient Skepticism.) Having discovered that for every argument examined so far there is an opposing argument, Pyrrhonian Skeptics expressed no determinate opinions (DL IX 74). This attitude would seem to lead to a kind of epistemic paralysis. The Skeptics reply that they do not abolish, e.g., relying on sight. They do not say that it is unreliable and they do not refuse, personally, ever to rely on it. Rather, they have the impression that there is no reason why we are entitled to rely on it, in a given case or in general, even if we go ahead and rely on it anyhow (DL IX 103). For example, if one has the visual impression of a tower, that appearance is not in dispute. What is in dispute is whether the tower is as it appears to be. For Skeptics, claiming that the tower is as it appears to be is a dogmatic statement about the object or the causal history of the appearance. As far as the Skeptic is concerned, all such statements have failed to be adequately justified, and all supporting arguments can be opposed by equally convincing counter-arguments. Lacking any grounds on which to prefer one dogmatic statement or view over another, he suspends judgment. Such views have an obvious impact on practical and moral issues. First of all, the Skeptics argue that, so far as we have been given any compelling reason by philosophers to believe, there is nothing good or bad by nature (Sextus Empiricus, Outlines of Pyrrhonism [=PH] III 179-238, Adversus Mathematicos [=M] XI 42-109). And just in case he does find such reasons compelling, the Skeptic will resist the temptation to assent by posing a general counter-argument: what is good by nature would be recognized by everyone, but clearly not everyone agrees - for instance, Epicurus holds that pleasure is good by nature but Antisthenes holds that it is not (DL IX 101) - hence there is nothing good by nature (see section 4 of the entry on Moral Skepticism). The practical consequences of suspending judgment are illustrated by two traditions about the life of Pyrrho. In one, Pyrrho himself did not avoid anything, whether it was wagons, precipices, or dogs. It would appear that, suspending judgment about whether being hit by a wagon was naturally good or bad, Pyrrho might walk into its path, or not bother to get out of the way. His less skeptical friends kept him alive - presumably by guiding him away from busy roads, vicious dogs, and deep gorges. Another tradition, however, says that he only theorized about suspension of judgments, and took action to preserve himself and otherwise lead a normal life, but doing so without any judgments as to natural good and bad. Living providently, he reached ninety years of age (DL IX 62). In any event, Pyrrho's suspension of judgment led to a certain detachment. Not knowing what was good or bad by nature, he was indifferent where dogmatists would be unhappy or at least anxious. For instance, he performed household chores usually done by women (DL IX 63-66). Thus Pyrrho's skepticism detached him from the dogmatic judgments of a culture in which a man's performing women's work was considered demeaning. In turn, his skepticism and suspension of judgment led to freedom from disturbance (ataraxia) (DL IX 68). It is significant that this psychological state, so important for Epicureans as part of the end of life, should play a key role in Pyrrhonian skepticism at its beginning with Pyrrho, but certainly in its development with Sextus Empiricus. While suspension of judgment in moral matters brings freedom from disturbance, it does not lead to immoral behavior - anymore than it leads to the foolhardy behavior of the first tradition about Pyrrho's life. Sextus generalizes the Skeptic teaching about appearances to cover the whole area of practical activity. He says the rules of everyday conduct are divided into four parts: (1) the guidance from nature, (2) compulsion that comes from bodily states like hunger and thirst, (3) traditional laws and customs about pious and good living, (4) the teachings of the crafts (PH, I 21-24). The Skeptic is guided by all of these as by appearance, and thus undogmatically. For instance, he would follow the traditional laws about pious and good living, accepting these laws as the way things appear to him to be in matters of piety and goodness but claiming no knowledge and holding no beliefs. Sextus says that the end of life is freedom from disturbance in matters of belief, plus moderate states in matters of compulsion. Suspension of judgment provides the former in that one is not disturbed about which of two opposing claims is true, when (as always seems to happen) one cannot rationally decide between them. Matters of compulsion cover such things as bodily needs for food, drink, and warmth. While the Skeptic undeniably suffers when hungry, thirsty, or cold, he achieves a moderate state with respect to these sufferings when compared to the person who both suffers them and believes they are naturally bad. The Skeptic's suspension of judgment about whether his suffering is naturally bad gives him a certain detachment from the suffering, and puts him in a better condition than those who also believe their suffering is naturally bad (PH I 25-30). As a consequence, the Skeptic conception of the end of life is similar in some ways to Epicurean and Stoic beliefs. For Epicureans the end is freedom from pain and distress; the Stoic identification of virtue and the good promises freedom from disturbance. While Pyrrho and Sextus hold freedom from disturbance to be the end of life, they differ from the former over the means by which it is achieved. Both the Epicureans and Stoics, for example, hold that a tranquil life is impossible in the absence of virtue. By contrast, Sextus does not have a lot to say in a positive vein about virtue, although he does recommend following traditional laws about piety and goodness, and he indicates that the Skeptic may live virtuously (PH I 17). Rather, it is for the Skeptics an epistemic attitude embodied in a distinctive practice, not virtue, that leads to the desired state. This alone would seem to be enough to disqualify Pyrrhonism as a form of eudaimonism. Sextus does however offer his skeptical practice as a corrective to the dogmatists' misleading path to eudaimonia: it is not possible to be happy while supposing things are good or bad by nature (M XI 130, 144). It is the person who suspends judgment that enjoys the most complete happiness (M XI 160, 140). It is important to observe that Sextus proposes his skeptical practice as an alternative to dogmatic theories. For the Skeptic makes no commitments, or even assertions, regarding the supposedly objective conditions that are supposed by other philosophers to underlie our natural, human telos. Accordingly, some have objected that the kind of tranquility and happiness that the Skeptic enjoys could just as well be produced through pharmacology or, we might add, Experience Machines. More generally speaking, the objection is that skeptical tranquility and happiness comes at a cost we should not be willing to pay. By refusing to accept the world is ever as it appears, the Skeptic becomes a detached, passive spectator, undisturbed by any of the thoughts that pass through his mind. Such detachment might seem to threaten his ability to care about other people as he observes all that happens to them with the same tranquil indifference. Also, he would seem to be at best a moral conformist, unable or at least unwilling to engage in any moral deliberation, either hypothetically or as a means to directly address a pressing moral issue. Sextus provides an example of the latter situation when he imagines the Skeptic being compelled by a tyrant to commit some unspeakable deed, or be tortured (M XI 164-66). Critics propose that the Skeptic's choice will necessarily reveal his evaluative convictions, and thus his inconsistency in claiming to have no such convictions. But Sextus replies that he will simply opt for one or the other in accordance with his family's laws and customs. In other words, he will rely on whatever moral sentiments and dispositions he happens to have. These cognitive states will be neither the product of any rational consideration nor will they inform any premises for moral reasoning or philosophical deliberation. Like Sartre's existentialism, Sextus' skepticism offers a way to respond to moral dilemmas without supposing there is a correct, or even objectively better, choice. We should note that it is not entirely fair to ask whether the Pyrrhonist's moral life is one that we, non-Skeptics, would deem good or praise-worthy. For, as Sextus would say, we are part of the dispute regarding what kind of life is morally good and praise-worthy. To claim that nothing really matters to the Skeptic, for example, presupposes some account of the mental state of caring. In the end, the question of how or whether a Skeptic might be morally good is equivalent to asking what role belief, or knowledge, plays in the moral goodness of people and their actions. If we suppose that one must, at a minimum, have adequately justified evaluative beliefs informing her actions in order for those actions, and the underlying character, to count as morally good, then the Skeptic will be immoral, or at least amoral. On the other hand, if the Skeptic manages to undermine our confidence in that supposition, we will have to suspend judgment regarding whether the Skeptic, or anyone else for that matter, is in fact capable of performing morally good and praise-worthy action, or whether their reasoned account of eudaimonia is true. The question we should ask here is whether and in what respect those with strong, rational convictions will be better off than the Skeptic when confronting moral dilemmas, or even everyday moral choices. If, like the Skeptic, one suspects that the dogmatists' have no knowledge, let alone adequately justified beliefs to guide them, then it seems that they are no better or worse off than those who simply conform to moral customs and norms.
ethics-virtue	## 1. Preliminaries In the West, virtue ethics' founding fathers are Plato and Aristotle, and in the East it can be traced back to Mencius and Confucius. It persisted as the dominant approach in Western moral philosophy until at least the Enlightenment, suffered a momentary eclipse during the nineteenth century, but re-emerged in Anglo-American philosophy in the late 1950s. It was heralded by Anscombe's famous article "Modern Moral Philosophy" (Anscombe 1958) which crystallized an increasing dissatisfaction with the forms of deontology and utilitarianism then prevailing. Neither of them, at that time, paid attention to a number of topics that had always figured in the virtue ethics tradition--virtues and vices, motives and moral character, moral education, moral wisdom or discernment, friendship and family relationships, a deep concept of happiness, the role of the emotions in our moral life and the fundamentally important questions of what sorts of persons we should be and how we should live. Its re-emergence had an invigorating effect on the other two approaches, many of whose proponents then began to address these topics in the terms of their favoured theory. (One consequence of this has been that it is now necessary to distinguish "virtue ethics" (the third approach) from "virtue theory", a term which includes accounts of virtue within the other approaches.) Interest in Kant's virtue theory has redirected philosophers' attention to Kant's long neglected Doctrine of Virtue, and utilitarians have developed consequentialist virtue theories (Driver 2001; Hurka 2001). It has also generated virtue ethical readings of philosophers other than Plato and Aristotle, such as Martineau, Hume and Nietzsche, and thereby different forms of virtue ethics have developed (Slote 2001; Swanton 2003, 2011a). Although modern virtue ethics does not have to take a "neo-Aristotelian" or eudaimonist form (see section 2), almost any modern version still shows that its roots are in ancient Greek philosophy by the employment of three concepts derived from it. These are arete (excellence or virtue), phronesis (practical or moral wisdom) and eudaimonia (usually translated as happiness or flourishing). (See Annas 2011 for a short, clear, and authoritative account of all three.) We discuss the first two in the remainder of this section. Eudaimonia is discussed in connection with eudaimonist versions of virtue ethics in the next. ### 1.1 Virtue A virtue is an excellent trait of character. It is a disposition, well entrenched in its possessor--something that, as we say, goes all the way down, unlike a habit such as being a tea-drinker--to notice, expect, value, feel, desire, choose, act, and react in certain characteristic ways. To possess a virtue is to be a certain sort of person with a certain complex mindset. A significant aspect of this mindset is the wholehearted acceptance of a distinctive range of considerations as reasons for action. An honest person cannot be identified simply as one who, for example, practices honest dealing and does not cheat. If such actions are done merely because the agent thinks that honesty is the best policy, or because they fear being caught out, rather than through recognising "To do otherwise would be dishonest" as the relevant reason, they are not the actions of an honest person. An honest person cannot be identified simply as one who, for example, tells the truth because it is the truth, for one can have the virtue of honesty without being tactless or indiscreet. The honest person recognises "That would be a lie" as a strong (though perhaps not overriding) reason for not making certain statements in certain circumstances, and gives due, but not overriding, weight to "That would be the truth" as a reason for making them. An honest person's reasons and choices with respect to honest and dishonest actions reflect her views about honesty, truth, and deception--but of course such views manifest themselves with respect to other actions, and to emotional reactions as well. Valuing honesty as she does, she chooses, where possible to work with honest people, to have honest friends, to bring up her children to be honest. She disapproves of, dislikes, deplores dishonesty, is not amused by certain tales of chicanery, despises or pities those who succeed through deception rather than thinking they have been clever, is unsurprised, or pleased (as appropriate) when honesty triumphs, is shocked or distressed when those near and dear to her do what is dishonest and so on. Given that a virtue is such a multi-track disposition, it would obviously be reckless to attribute one to an agent on the basis of a single observed action or even a series of similar actions, especially if you don't know the agent's reasons for doing as she did (Sreenivasan 2002). Possessing a virtue is a matter of degree. To possess such a disposition fully is to possess full or perfect virtue, which is rare, and there are a number of ways of falling short of this ideal (Athanassoulis 2000). Most people who can truly be described as fairly virtuous, and certainly markedly better than those who can truly be described as dishonest, self-centred and greedy, still have their blind spots--little areas where they do not act for the reasons one would expect. So someone honest or kind in most situations, and notably so in demanding ones, may nevertheless be trivially tainted by snobbery, inclined to be disingenuous about their forebears and less than kind to strangers with the wrong accent. Further, it is not easy to get one's emotions in harmony with one's rational recognition of certain reasons for action. I may be honest enough to recognise that I must own up to a mistake because it would be dishonest not to do so without my acceptance being so wholehearted that I can own up easily, with no inner conflict. Following (and adapting) Aristotle, virtue ethicists draw a distinction between full or perfect virtue and "continence", or strength of will. The fully virtuous do what they should without a struggle against contrary desires; the continent have to control a desire or temptation to do otherwise. Describing the continent as "falling short" of perfect virtue appears to go against the intuition that there is something particularly admirable about people who manage to act well when it is especially hard for them to do so, but the plausibility of this depends on exactly what "makes it hard" (Foot 1978: 11-14). If it is the circumstances in which the agent acts--say that she is very poor when she sees someone drop a full purse or that she is in deep grief when someone visits seeking help--then indeed it is particularly admirable of her to restore the purse or give the help when it is hard for her to do so. But if what makes it hard is an imperfection in her character--the temptation to keep what is not hers, or a callous indifference to the suffering of others--then it is not. ### 1.2 Practical Wisdom Another way in which one can easily fall short of full virtue is through lacking phronesis--moral or practical wisdom. The concept of a virtue is the concept of something that makes its possessor good: a virtuous person is a morally good, excellent or admirable person who acts and feels as she should. These are commonly accepted truisms. But it is equally common, in relation to particular (putative) examples of virtues to give these truisms up. We may say of someone that he is generous or honest "to a fault". It is commonly asserted that someone's compassion might lead them to act wrongly, to tell a lie they should not have told, for example, in their desire to prevent someone else's hurt feelings. It is also said that courage, in a desperado, enables him to do far more wicked things than he would have been able to do if he were timid. So it would appear that generosity, honesty, compassion and courage despite being virtues, are sometimes faults. Someone who is generous, honest, compassionate, and courageous might not be a morally good person--or, if it is still held to be a truism that they are, then morally good people may be led by what makes them morally good to act wrongly! How have we arrived at such an odd conclusion? The answer lies in too ready an acceptance of ordinary usage, which permits a fairly wide-ranging application of many of the virtue terms, combined, perhaps, with a modern readiness to suppose that the virtuous agent is motivated by emotion or inclination, not by rational choice. If one thinks of generosity or honesty as the disposition to be moved to action by generous or honest impulses such as the desire to give or to speak the truth, if one thinks of compassion as the disposition to be moved by the sufferings of others and to act on that emotion, if one thinks of courage as mere fearlessness or the willingness to face danger, then it will indeed seem obvious that these are all dispositions that can lead to their possessor's acting wrongly. But it is also obvious, as soon as it is stated, that these are dispositions that can be possessed by children, and although children thus endowed (bar the "courageous" disposition) would undoubtedly be very nice children, we would not say that they were morally virtuous or admirable people. The ordinary usage, or the reliance on motivation by inclination, gives us what Aristotle calls "natural virtue"--a proto version of full virtue awaiting perfection by phronesis or practical wisdom. Aristotle makes a number of specific remarks about phronesis that are the subject of much scholarly debate, but the (related) modern concept is best understood by thinking of what the virtuous morally mature adult has that nice children, including nice adolescents, lack. Both the virtuous adult and the nice child have good intentions, but the child is much more prone to mess things up because he is ignorant of what he needs to know in order to do what he intends. A virtuous adult is not, of course, infallible and may also, on occasion, fail to do what she intended to do through lack of knowledge, but only on those occasions on which the lack of knowledge is not culpable. So, for example, children and adolescents often harm those they intend to benefit either because they do not know how to set about securing the benefit or because their understanding of what is beneficial and harmful is limited and often mistaken. Such ignorance in small children is rarely, if ever culpable. Adults, on the other hand, are culpable if they mess things up by being thoughtless, insensitive, reckless, impulsive, shortsighted, and by assuming that what suits them will suit everyone instead of taking a more objective viewpoint. They are also culpable if their understanding of what is beneficial and harmful is mistaken. It is part of practical wisdom to know how to secure real benefits effectively; those who have practical wisdom will not make the mistake of concealing the hurtful truth from the person who really needs to know it in the belief that they are benefiting him. Quite generally, given that good intentions are intentions to act well or "do the right thing", we may say that practical wisdom is the knowledge or understanding that enables its possessor, unlike the nice adolescents, to do just that, in any given situation. The detailed specification of what is involved in such knowledge or understanding has not yet appeared in the literature, but some aspects of it are becoming well known. Even many deontologists now stress the point that their action-guiding rules cannot, reliably, be applied without practical wisdom, because correct application requires situational appreciation--the capacity to recognise, in any particular situation, those features of it that are morally salient. This brings out two aspects of practical wisdom. One is that it characteristically comes only with experience of life. Amongst the morally relevant features of a situation may be the likely consequences, for the people involved, of a certain action, and this is something that adolescents are notoriously clueless about precisely because they are inexperienced. It is part of practical wisdom to be wise about human beings and human life. (It should go without saying that the virtuous are mindful of the consequences of possible actions. How could they fail to be reckless, thoughtless and short-sighted if they were not?) The second is the practically wise agent's capacity to recognise some features of a situation as more important than others, or indeed, in that situation, as the only relevant ones. The wise do not see things in the same way as the nice adolescents who, with their under-developed virtues, still tend to see the personally disadvantageous nature of a certain action as competing in importance with its honesty or benevolence or justice. These aspects coalesce in the description of the practically wise as those who understand what is truly worthwhile, truly important, and thereby truly advantageous in life, who know, in short, how to live well. ## 2. Forms of Virtue Ethics While all forms of virtue ethics agree that virtue is central and practical wisdom required, they differ in how they combine these and other concepts to illuminate what we should do in particular contexts and how we should live our lives as a whole. In what follows we sketch four distinct forms taken by contemporary virtue ethics, namely, a) eudaimonist virtue ethics, b) agent-based and exemplarist virtue ethics, c) target-centered virtue ethics, and d) Platonistic virtue ethics. ### 2.1 Eudaimonist Virtue Ethics The distinctive feature of eudaimonist versions of virtue ethics is that they define virtues in terms of their relationship to eudaimonia. A virtue is a trait that contributes to or is a constituent of eudaimonia and we ought to develop virtues, the eudaimonist claims, precisely because they contribute to eudaimonia. The concept of eudaimonia, a key term in ancient Greek moral philosophy, is standardly translated as "happiness" or "flourishing" and occasionally as "well-being." Each translation has its disadvantages. The trouble with "flourishing" is that animals and even plants can flourish but eudaimonia is possible only for rational beings. The trouble with "happiness" is that in ordinary conversation it connotes something subjectively determined. It is for me, not for you, to pronounce on whether I am happy. If I think I am happy then I am--it is not something I can be wrong about (barring advanced cases of self-deception). Contrast my being healthy or flourishing. Here we have no difficulty in recognizing that I might think I was healthy, either physically or psychologically, or think that I was flourishing but be wrong. In this respect, "flourishing" is a better translation than "happiness". It is all too easy to be mistaken about whether one's life is eudaimon (the adjective from eudaimonia) not simply because it is easy to deceive oneself, but because it is easy to have a mistaken conception of eudaimonia, or of what it is to live well as a human being, believing it to consist largely in physical pleasure or luxury for example. Eudaimonia is, avowedly, a moralized or value-laden concept of happiness, something like "true" or "real" happiness or "the sort of happiness worth seeking or having." It is thereby the sort of concept about which there can be substantial disagreement between people with different views about human life that cannot be resolved by appeal to some external standard on which, despite their different views, the parties to the disagreement concur (Hursthouse 1999: 188-189). Most versions of virtue ethics agree that living a life in accordance with virtue is necessary for eudaimonia. This supreme good is not conceived of as an independently defined state (made up of, say, a list of non-moral goods that does not include virtuous activity) which exercise of the virtues might be thought to promote. It is, within virtue ethics, already conceived of as something of which virtuous activity is at least partially constitutive (Kraut 1989). Thereby virtue ethicists claim that a human life devoted to physical pleasure or the acquisition of wealth is not eudaimon, but a wasted life. But although all standard versions of virtue ethics insist on that conceptual link between virtue and eudaimonia, further links are matters of dispute and generate different versions. For Aristotle, virtue is necessary but not sufficient--what is also needed are external goods which are a matter of luck. For Plato and the Stoics, virtue is both necessary and sufficient for eudaimonia (Annas 1993). According to eudaimonist virtue ethics, the good life is the eudaimon life, and the virtues are what enable a human being to be eudaimon because the virtues just are those character traits that benefit their possessor in that way, barring bad luck. So there is a link between eudaimonia and what confers virtue status on a character trait. (For a discussion of the differences between eudaimonists see Baril 2014. For recent defenses of eudaimonism see Annas 2011; LeBar 2013b; Badhwar 2014; and Bloomfield 2014.) ### 2.2 Agent-Based and Exemplarist Virtue Ethics Rather than deriving the normativity of virtue from the value of eudaimonia, agent-based virtue ethicists argue that other forms of normativity--including the value of eudaimonia--are traced back to and ultimately explained in terms of the motivational and dispositional qualities of agents. It is unclear how many other forms of normativity must be explained in terms of the qualities of agents in order for a theory to count as agent-based. The two best-known agent-based theorists, Michael Slote and Linda Zagzebski, trace a wide range of normative qualities back to the qualities of agents. For example, Slote defines rightness and wrongness in terms of agents' motivations: "[A]gent-based virtue ethics ... understands rightness in terms of good motivations and wrongness in terms of the having of bad (or insufficiently good) motives" (2001: 14). Similarly, he explains the goodness of an action, the value of eudaimonia, the justice of a law or social institution, and the normativity of practical rationality in terms of the motivational and dispositional qualities of agents (2001: 99-100, 154, 2000). Zagzebski likewise defines right and wrong actions by reference to the emotions, motives, and dispositions of virtuous and vicious agents. For example, "A wrong act = an act that the phronimos characteristically would not do, and he would feel guilty if he did = an act such that it is not the case that he might do it = an act that expresses a vice = an act that is against a requirement of virtue (the virtuous self)" (Zagzebski 2004: 160). Her definitions of duties, good and bad ends, and good and bad states of affairs are similarly grounded in the motivational and dispositional states of exemplary agents (1998, 2004, 2010). However, there could also be less ambitious agent-based approaches to virtue ethics (see Slote 1997). At the very least, an agent-based approach must be committed to explaining what one should do by reference to the motivational and dispositional states of agents. But this is not yet a sufficient condition for counting as an agent-based approach, since the same condition will be met by every virtue ethical account. For a theory to count as an agent-based form of virtue ethics it must also be the case that the normative properties of motivations and dispositions cannot be explained in terms of the normative properties of something else (such as eudaimonia or states of affairs) which is taken to be more fundamental. Beyond this basic commitment, there is room for agent-based theories to be developed in a number of different directions. The most important distinguishing factor has to do with how motivations and dispositions are taken to matter for the purposes of explaining other normative qualities. For Slote what matters are this particular agent's actual motives and dispositions. The goodness of action A, for example, is derived from the agent's motives when she performs A. If those motives are good then the action is good, if not then not. On Zagzebski's account, by contrast, a good or bad, right or wrong action is defined not by this agent's actual motives but rather by whether this is the sort of action a virtuously motivated agent would perform (Zagzebski 2004: 160). Appeal to the virtuous agent's hypothetical motives and dispositions enables Zagzebski to distinguish between performing the right action and doing so for the right reasons (a distinction that, as Brady (2004) observes, Slote has trouble drawing). Another point on which agent-based forms of virtue ethics might differ concerns how one identifies virtuous motivations and dispositions. According to Zagzebski's exemplarist account, "We do not have criteria for goodness in advance of identifying the exemplars of goodness" (Zagzebski 2004: 41). As we observe the people around us, we find ourselves wanting to be like some of them (in at least some respects) and not wanting to be like others. The former provide us with positive exemplars and the latter with negative ones. Our understanding of better and worse motivations and virtuous and vicious dispositions is grounded in these primitive responses to exemplars (2004: 53). This is not to say that every time we act we stop and ask ourselves what one of our exemplars would do in this situations. Our moral concepts become more refined over time as we encounter a wider variety of exemplars and begin to draw systematic connections between them, noting what they have in common, how they differ, and which of these commonalities and differences matter, morally speaking. Recognizable motivational profiles emerge and come to be labeled as virtues or vices, and these, in turn, shape our understanding of the obligations we have and the ends we should pursue. However, even though the systematising of moral thought can travel a long way from our starting point, according to the exemplarist it never reaches a stage where reference to exemplars is replaced by the recognition of something more fundamental. At the end of the day, according to the exemplarist, our moral system still rests on our basic propensity to take a liking (or disliking) to exemplars. Nevertheless, one could be an agent-based theorist without advancing the exemplarist's account of the origins or reference conditions for judgments of good and bad, virtuous and vicious. ### 2.3 Target-Centered Virtue Ethics The touchstone for eudaimonist virtue ethicists is a flourishing human life. For agent-based virtue ethicists it is an exemplary agent's motivations. The target-centered view developed by Christine Swanton (2003), by contrast, begins with our existing conceptions of the virtues. We already have a passable idea of which traits are virtues and what they involve. Of course, this untutored understanding can be clarified and improved, and it is one of the tasks of the virtue ethicist to help us do precisely that. But rather than stripping things back to something as basic as the motivations we want to imitate or building it up to something as elaborate as an entire flourishing life, the target-centered view begins where most ethics students find themselves, namely, with the idea that generosity, courage, self-discipline, compassion, and the like get a tick of approval. It then examines what these traits involve. A complete account of virtue will map out 1) its field, 2) its mode of responsiveness, 3) its basis of moral acknowledgment, and 4) its target. Different virtues are concerned with different fields. Courage, for example, is concerned with what might harm us, whereas generosity is concerned with the sharing of time, talent, and property. The basis of acknowledgment of a virtue is the feature within the virtue's field to which it responds. To continue with our previous examples, generosity is attentive to the benefits that others might enjoy through one's agency, and courage responds to threats to value, status, or the bonds that exist between oneself and particular others, and the fear such threats might generate. A virtue's mode has to do with how it responds to the bases of acknowledgment within its field. Generosity promotes a good, namely, another's benefit, whereas courage defends a value, bond, or status. Finally, a virtue's target is that at which it is aimed. Courage aims to control fear and handle danger, while generosity aims to share time, talents, or possessions with others in ways that benefit them. A virtue, on a target-centered account, "is a disposition to respond to, or acknowledge, items within its field or fields in an excellent or good enough way" (Swanton 2003: 19). A virtuous act is an act that hits the target of a virtue, which is to say that it succeeds in responding to items in its field in the specified way (233). Providing a target-centered definition of a right action requires us to move beyond the analysis of a single virtue and the actions that follow from it. This is because a single action context may involve a number of different, overlapping fields. Determination might lead me to persist in trying to complete a difficult task even if doing so requires a singleness of purpose. But love for my family might make a different use of my time and attention. In order to define right action a target-centered view must explain how we handle different virtues' conflicting claims on our resources. There are at least three different ways to address this challenge. A perfectionist target-centered account would stipulate, "An act is right if and only if it is overall virtuous, and that entails that it is the, or a, best action possible in the circumstances" (239-240). A more permissive target-centered account would not identify 'right' with 'best', but would allow an action to count as right provided "it is good enough even if not the (or a) best action" (240). A minimalist target-centered account would not even require an action to be good in order to be right. On such a view, "An act is right if and only if it is not overall vicious" (240). (For further discussion of target-centered virtue ethics see Van Zyl 2014; and Smith 2016). ### 2.4 Platonistic Virtue Ethics The fourth form a virtue ethic might adopt takes its inspiration from Plato. The Socrates of Plato's dialogues devotes a great deal of time to asking his fellow Athenians to explain the nature of virtues like justice, courage, piety, and wisdom. So it is clear that Plato counts as a virtue theorist. But it is a matter of some debate whether he should be read as a virtue ethicist (White 2015). What is not open to debate is whether Plato has had an important influence on the contemporary revival of interest in virtue ethics. A number of those who have contributed to the revival have done so as Plato scholars (e.g., Prior 1991; Kamtekar 1998; Annas 1999; and Reshotko 2006). However, often they have ended up championing a eudaimonist version of virtue ethics (see Prior 2001 and Annas 2011), rather than a version that would warrant a separate classification. Nevertheless, there are two variants that call for distinct treatment. Timothy Chappell takes the defining feature of Platonistic virtue ethics to be that "Good agency in the truest and fullest sense presupposes the contemplation of the Form of the Good" (2014). Chappell follows Iris Murdoch in arguing that "In the moral life the enemy is the fat relentless ego" (Murdoch 1971: 51). Constantly attending to our needs, our desires, our passions, and our thoughts skews our perspective on what the world is actually like and blinds us to the goods around us. Contemplating the goodness of something we encounter--which is to say, carefully attending to it "for its own sake, in order to understand it" (Chappell 2014: 300)--breaks this natural tendency by drawing our attention away from ourselves. Contemplating such goodness with regularity makes room for new habits of thought that focus more readily and more honestly on things other than the self. It alters the quality of our consciousness. And "anything which alters consciousness in the direction of unselfishness, objectivity, and realism is to be connected with virtue" (Murdoch 1971: 82). The virtues get defined, then, in terms of qualities that help one "pierce the veil of selfish consciousness and join the world as it really is" (91). And good agency is defined by the possession and exercise of such virtues. Within Chappell's and Murdoch's framework, then, not all normative properties get defined in terms of virtue. Goodness, in particular, is not so defined. But the kind of goodness which is possible for creatures like us is defined by virtue, and any answer to the question of what one should do or how one should live will appeal to the virtues. Another Platonistic variant of virtue ethics is exemplified by Robert Merrihew Adams. Unlike Murdoch and Chappell, his starting point is not a set of claims about our consciousness of goodness. Rather, he begins with an account of the metaphysics of goodness. Like Murdoch and others influenced by Platonism, Adams's account of goodness is built around a conception of a supremely perfect good. And like Augustine, Adams takes that perfect good to be God. God is both the exemplification and the source of all goodness. Other things are good, he suggests, to the extent that they resemble God (Adams 1999). The resemblance requirement identifies a necessary condition for being good, but it does not yet give us a sufficient condition. This is because there are ways in which finite creatures might resemble God that would not be suitable to the type of creature they are. For example, if God were all-knowing, then the belief, "I am all-knowing," would be a suitable belief for God to have. In God, such a belief--because true--would be part of God's perfection. However, as neither you nor I are all-knowing, the belief, "I am all-knowing," in one of us would not be good. To rule out such cases we need to introduce another factor. That factor is the fitting response to goodness, which Adams suggests is love. Adams uses love to weed out problematic resemblances: "being excellent in the way that a finite thing can be consists in resembling God in a way that could serve God as a reason for loving the thing" (Adams 1999: 36). Virtues come into the account as one of the ways in which some things (namely, persons) could resemble God. "[M]ost of the excellences that are most important to us, and of whose value we are most confident, are excellences of persons or of qualities or actions or works or lives or stories of persons" (1999: 42). This is one of the reasons Adams offers for conceiving of the ideal of perfection as a personal God, rather than an impersonal form of the Good. Many of the excellences of persons of which we are most confident are virtues such as love, wisdom, justice, patience, and generosity. And within many theistic traditions, including Adams's own Christian tradition, such virtues are commonly attributed to divine agents. A Platonistic account like the one Adams puts forward in Finite and Infinite Goods clearly does not derive all other normative properties from the virtues (for a discussion of the relationship between this view and the one he puts forward in A Theory of Virtue (2006) see Pettigrove 2014). Goodness provides the normative foundation. Virtues are not built on that foundation; rather, as one of the varieties of goodness of whose value we are most confident, virtues form part of the foundation. Obligations, by contrast, come into the account at a different level. Moral obligations, Adams argues, are determined by the expectations and demands that "arise in a relationship or system of relationships that is good or valuable" (1999: 244). Other things being equal, the more virtuous the parties to the relationship, the more binding the obligation. Thus, within Adams's account, the good (which includes virtue) is prior to the right. However, once good relationships have given rise to obligations, those obligations take on a life of their own. Their bindingness is not traced directly to considerations of goodness. Rather, they are determined by the expectations of the parties and the demands of the relationship. ## 3. Objections to virtue ethics A number of objections have been raised against virtue ethics, some of which bear more directly on one form of virtue ethics than on others. In this section we consider eight objections, namely, the a) application, b) adequacy, c) relativism, d) conflict, e) self-effacement, f) justification, g) egoism, and h) situationist problems. a) In the early days of virtue ethics' revival, the approach was associated with an "anti-codifiability" thesis about ethics, directed against the prevailing pretensions of normative theory. At the time, utilitarians and deontologists commonly (though not universally) held that the task of ethical theory was to come up with a code consisting of universal rules or principles (possibly only one, as in the case of act-utilitarianism) which would have two significant features: i) the rule(s) would amount to a decision procedure for determining what the right action was in any particular case; ii) the rule(s) would be stated in such terms that any non-virtuous person could understand and apply it (them) correctly. Virtue ethicists maintained, contrary to these two claims, that it was quite unrealistic to imagine that there could be such a code (see, in particular, McDowell 1979). The results of attempts to produce and employ such a code, in the heady days of the 1960s and 1970s, when medical and then bioethics boomed and bloomed, tended to support the virtue ethicists' claim. More and more utilitarians and deontologists found themselves agreed on their general rules but on opposite sides of the controversial moral issues in contemporary discussion. It came to be recognised that moral sensitivity, perception, imagination, and judgement informed by experience--phronesis in short--is needed to apply rules or principles correctly. Hence many (though by no means all) utilitarians and deontologists have explicitly abandoned (ii) and much less emphasis is placed on (i). Nevertheless, the complaint that virtue ethics does not produce codifiable principles is still a commonly voiced criticism of the approach, expressed as the objection that it is, in principle, unable to provide action-guidance. Initially, the objection was based on a misunderstanding. Blinkered by slogans that described virtue ethics as "concerned with Being rather than Doing," as addressing "What sort of person should I be?" but not "What should I do?" as being "agent-centered rather than act-centered," its critics maintained that it was unable to provide action-guidance. Hence, rather than being a normative rival to utilitarian and deontological ethics, it could claim to be no more than a valuable supplement to them. The rather odd idea was that all virtue ethics could offer was, "Identify a moral exemplar and do what he would do," as though the university student trying to decide whether to study music (her preference) or engineering (her parents' preference) was supposed to ask herself, "What would Socrates study if he were in my circumstances?" But the objection failed to take note of Anscombe's hint that a great deal of specific action guidance could be found in rules employing the virtue and vice terms ("v-rules") such as "Do what is honest/charitable; do not do what is dishonest/uncharitable" (Hursthouse 1999). (It is a noteworthy feature of our virtue and vice vocabulary that, although our list of generally recognised virtue terms is comparatively short, our list of vice terms is remarkably, and usefully, long, far exceeding anything that anyone who thinks in terms of standard deontological rules has ever come up with. Much invaluable action guidance comes from avoiding courses of action that would be irresponsible, feckless, lazy, inconsiderate, uncooperative, harsh, intolerant, selfish, mercenary, indiscreet, tactless, arrogant, unsympathetic, cold, incautious, unenterprising, pusillanimous, feeble, presumptuous, rude, hypocritical, self-indulgent, materialistic, grasping, short-sighted, vindictive, calculating, ungrateful, grudging, brutal, profligate, disloyal, and on and on.) (b) A closely related objection has to do with whether virtue ethics can provide an adequate account of right action. This worry can take two forms. (i) One might think a virtue ethical account of right action is extensionally inadequate. It is possible to perform a right action without being virtuous and a virtuous person can occasionally perform the wrong action without that calling her virtue into question. If virtue is neither necessary nor sufficient for right action, one might wonder whether the relationship between rightness/wrongness and virtue/vice is close enough for the former to be identified in terms of the latter. (ii) Alternatively, even if one thought it possible to produce a virtue ethical account that picked out all (and only) right actions, one might still think that at least in some cases virtue is not what explains rightness (Adams 2006:6-8). Some virtue ethicists respond to the adequacy objection by rejecting the assumption that virtue ethics ought to be in the business of providing an account of right action in the first place. Following in the footsteps of Anscombe (1958) and MacIntyre (1985), Talbot Brewer (2009) argues that to work with the categories of rightness and wrongness is already to get off on the wrong foot. Contemporary conceptions of right and wrong action, built as they are around a notion of moral duty that presupposes a framework of divine (or moral) law or around a conception of obligation that is defined in contrast to self-interest, carry baggage the virtue ethicist is better off without. Virtue ethics can address the questions of how one should live, what kind of person one should become, and even what one should do without that committing it to providing an account of 'right action'. One might choose, instead, to work with aretaic concepts (defined in terms of virtues and vices) and axiological concepts (defined in terms of good and bad, better and worse) and leave out deontic notions (like right/wrong action, duty, and obligation) altogether. Other virtue ethicists wish to retain the concept of right action but note that in the current philosophical discussion a number of distinct qualities march under that banner. In some contexts, 'right action' identifies the best action an agent might perform in the circumstances. In others, it designates an action that is commendable (even if not the best possible). In still others, it picks out actions that are not blameworthy (even if not commendable). A virtue ethicist might choose to define one of these--for example, the best action--in terms of virtues and vices, but appeal to other normative concepts--such as legitimate expectations--when defining other conceptions of right action. As we observed in section 2, a virtue ethical account need not attempt to reduce all other normative concepts to virtues and vices. What is required is simply (i) that virtue is not reduced to some other normative concept that is taken to be more fundamental and (ii) that some other normative concepts are explained in terms of virtue and vice. This takes the sting out of the adequacy objection, which is most compelling against versions of virtue ethics that attempt to define all of the senses of 'right action' in terms of virtues. Appealing to virtues and vices makes it much easier to achieve extensional adequacy. Making room for normative concepts that are not taken to be reducible to virtue and vice concepts makes it even easier to generate a theory that is both extensionally and explanatorily adequate. Whether one needs other concepts and, if so, how many, is still a matter of debate among virtue ethicists, as is the question of whether virtue ethics even ought to be offering an account of right action. Either way virtue ethicists have resources available to them to address the adequacy objection. Insofar as the different versions of virtue ethics all retain an emphasis on the virtues, they are open to the familiar problem of (c) the charge of cultural relativity. Is it not the case that different cultures embody different virtues, (MacIntyre 1985) and hence that the v-rules will pick out actions as right or wrong only relative to a particular culture? Different replies have been made to this charge. One--the tu quoque, or "partners in crime" response--exhibits a quite familiar pattern in virtue ethicists' defensive strategy (Solomon 1988). They admit that, for them, cultural relativism is a challenge, but point out that it is just as much a problem for the other two approaches. The (putative) cultural variation in character traits regarded as virtues is no greater--indeed markedly less--than the cultural variation in rules of conduct, and different cultures have different ideas about what constitutes happiness or welfare. That cultural relativity should be a problem common to all three approaches is hardly surprising. It is related, after all, to the "justification problem" (see below) the quite general metaethical problem of justifying one's moral beliefs to those who disagree, whether they be moral sceptics, pluralists or from another culture. A bolder strategy involves claiming that virtue ethics has less difficulty with cultural relativity than the other two approaches. Much cultural disagreement arises, it may be claimed, from local understandings of the virtues, but the virtues themselves are not relative to culture (Nussbaum 1993). Another objection to which the tu quoque response is partially appropriate is (d) "the conflict problem." What does virtue ethics have to say about dilemmas--cases in which, apparently, the requirements of different virtues conflict because they point in opposed directions? Charity prompts me to kill the person who would be better off dead, but justice forbids it. Honesty points to telling the hurtful truth, kindness and compassion to remaining silent or even lying. What shall I do? Of course, the same sorts of dilemmas are generated by conflicts between deontological rules. Deontology and virtue ethics share the conflict problem (and are happy to take it on board rather than follow some of the utilitarians in their consequentialist resolutions of such dilemmas) and in fact their strategies for responding to it are parallel. Both aim to resolve a number of dilemmas by arguing that the conflict is merely apparent; a discriminating understanding of the virtues or rules in question, possessed only by those with practical wisdom, will perceive that, in this particular case, the virtues do not make opposing demands or that one rule outranks another, or has a certain exception clause built into it. Whether this is all there is to it depends on whether there are any irresolvable dilemmas. If there are, proponents of either normative approach may point out reasonably that it could only be a mistake to offer a resolution of what is, ex hypothesi, irresolvable. Another problem arguably shared by all three approaches is (e), that of being self-effacing. An ethical theory is self-effacing if, roughly, whatever it claims justifies a particular action, or makes it right, had better not be the agent's motive for doing it. Michael Stocker (1976) originally introduced it as a problem for deontology and consequentialism. He pointed out that the agent who, rightly, visits a friend in hospital will rather lessen the impact of his visit on her if he tells her either that he is doing it because it is his duty or because he thought it would maximize the general happiness. But as Simon Keller observes, she won't be any better pleased if he tells her that he is visiting her because it is what a virtuous agent would do, so virtue ethics would appear to have the problem too (Keller 2007). However, virtue ethics' defenders have argued that not all forms of virtue ethics are subject to this objection (Pettigrove 2011) and those that are are not seriously undermined by the problem (Martinez 2011). Another problem for virtue ethics, which is shared by both utilitarianism and deontology, is (f) "the justification problem." Abstractly conceived, this is the problem of how we justify or ground our ethical beliefs, an issue that is hotly debated at the level of metaethics. In its particular versions, for deontology there is the question of how to justify its claims that certain moral rules are the correct ones, and for utilitarianism of how to justify its claim that all that really matters morally are consequences for happiness or well-being. For virtue ethics, the problem concerns the question of which character traits are the virtues. In the metaethical debate, there is widespread disagreement about the possibility of providing an external foundation for ethics--"external" in the sense of being external to ethical beliefs--and the same disagreement is found amongst deontologists and utilitarians. Some believe that their normative ethics can be placed on a secure basis, resistant to any form of scepticism, such as what anyone rationally desires, or would accept or agree on, regardless of their ethical outlook; others that it cannot. Virtue ethicists have eschewed any attempt to ground virtue ethics in an external foundation while continuing to maintain that their claims can be validated. Some follow a form of Rawls's coherentist approach (Slote 2001; Swanton 2003); neo-Aristotelians a form of ethical naturalism. A misunderstanding of eudaimonia as an unmoralized concept leads some critics to suppose that the neo-Aristotelians are attempting to ground their claims in a scientific account of human nature and what counts, for a human being, as flourishing. Others assume that, if this is not what they are doing, they cannot be validating their claims that, for example, justice, charity, courage, and generosity are virtues. Either they are illegitimately helping themselves to Aristotle's discredited natural teleology (Williams 1985) or producing mere rationalizations of their own personal or culturally inculcated values. But McDowell, Foot, MacIntyre and Hursthouse have all outlined versions of a third way between these two extremes. Eudaimonia in virtue ethics, is indeed a moralized concept, but it is not only that. Claims about what constitutes flourishing for human beings no more float free of scientific facts about what human beings are like than ethological claims about what constitutes flourishing for elephants. In both cases, the truth of the claims depends in part on what kind of animal they are and what capacities, desires and interests the humans or elephants have. The best available science today (including evolutionary theory and psychology) supports rather than undermines the ancient Greek assumption that we are social animals, like elephants and wolves and unlike polar bears. No rationalizing explanation in terms of anything like a social contract is needed to explain why we choose to live together, subjugating our egoistic desires in order to secure the advantages of co-operation. Like other social animals, our natural impulses are not solely directed towards our own pleasures and preservation, but include altruistic and cooperative ones. This basic fact about us should make more comprehensible the claim that the virtues are at least partially constitutive of human flourishing and also undercut the objection that virtue ethics is, in some sense, egoistic. (g) The egoism objection has a number of sources. One is a simple confusion. Once it is understood that the fully virtuous agent characteristically does what she should without inner conflict, it is triumphantly asserted that "she is only doing what she wants to do and hence is being selfish." So when the generous person gives gladly, as the generous are wont to do, it turns out she is not generous and unselfish after all, or at least not as generous as the one who greedily wants to hang on to everything she has but forces herself to give because she thinks she should! A related version ascribes bizarre reasons to the virtuous agent, unjustifiably assuming that she acts as she does because she believes that acting thus on this occasion will help her to achieve eudaimonia. But "the virtuous agent" is just "the agent with the virtues" and it is part of our ordinary understanding of the virtue terms that each carries with it its own typical range of reasons for acting. The virtuous agent acts as she does because she believes that someone's suffering will be averted, or someone benefited, or the truth established, or a debt repaid, or ... thereby. It is the exercise of the virtues during one's life that is held to be at least partially constitutive of eudaimonia, and this is consistent with recognising that bad luck may land the virtuous agent in circumstances that require her to give up her life. Given the sorts of considerations that courageous, honest, loyal, charitable people wholeheartedly recognise as reasons for action, they may find themselves compelled to face danger for a worthwhile end, to speak out in someone's defence, or refuse to reveal the names of their comrades, even when they know that this will inevitably lead to their execution, to share their last crust and face starvation. On the view that the exercise of the virtues is necessary but not sufficient for eudaimonia, such cases are described as those in which the virtuous agent sees that, as things have unfortunately turned out, eudaimonia is not possible for them (Foot 2001, 95). On the Stoical view that it is both necessary and sufficient, a eudaimon life is a life that has been successfully lived (where "success" of course is not to be understood in a materialistic way) and such people die knowing not only that they have made a success of their lives but that they have also brought their lives to a markedly successful completion. Either way, such heroic acts can hardly be regarded as egoistic. A lingering suggestion of egoism may be found in the misconceived distinction between so-called "self-regarding" and "other-regarding" virtues. Those who have been insulated from the ancient tradition tend to regard justice and benevolence as real virtues, which benefit others but not their possessor, and prudence, fortitude and providence (the virtue whose opposite is "improvidence" or being a spendthrift) as not real virtues at all because they benefit only their possessor. This is a mistake on two counts. Firstly, justice and benevolence do, in general, benefit their possessors, since without them eudaimonia is not possible. Secondly, given that we live together, as social animals, the "self-regarding" virtues do benefit others--those who lack them are a great drain on, and sometimes grief to, those who are close to them (as parents with improvident or imprudent adult offspring know only too well). The most recent objection (h) to virtue ethics claims that work in "situationist" social psychology shows that there are no such things as character traits and thereby no such things as virtues for virtue ethics to be about (Doris 1998; Harman 1999). In reply, some virtue ethicists have argued that the social psychologists' studies are irrelevant to the multi-track disposition (see above) that a virtue is supposed to be (Sreenivasan 2002; Kamtekar 2004). Mindful of just how multi-track it is, they agree that it would be reckless in the extreme to ascribe a demanding virtue such as charity to people of whom they know no more than that they have exhibited conventional decency; this would indeed be "a fundamental attribution error." Others have worked to develop alternative, empirically grounded conceptions of character traits (Snow 2010; Miller 2013 and 2014; however see Upton 2016 for objections to Miller). There have been other responses as well (summarized helpfully in Prinz 2009 and Miller 2014). Notable among these is a response by Adams (2006, echoing Merritt 2000) who steers a middle road between "no character traits at all" and the exacting standard of the Aristotelian conception of virtue which, because of its emphasis on phronesis, requires a high level of character integration. On his conception, character traits may be "frail and fragmentary" but still virtues, and not uncommon. But giving up the idea that practical wisdom is the heart of all the virtues, as Adams has to do, is a substantial sacrifice, as Russell (2009) and Kamtekar (2010) argue. Even though the "situationist challenge" has left traditional virtue ethicists unmoved, it has generated a healthy engagement with empirical psychological literature, which has also been fuelled by the growing literature on Foot's Natural Goodness and, quite independently, an upsurge of interest in character education (see below). ## 4. Future Directions Over the past thirty-five years most of those contributing to the revival of virtue ethics have worked within a neo-Aristotelian, eudaimonist framework. However, as noted in section 2, other forms of virtue ethics have begun to emerge. Theorists have begun to turn to philosophers like Hutcheson, Hume, Nietzsche, Martineau, and Heidegger for resources they might use to develop alternatives (see Russell 2006; Swanton 2013 and 2015; Taylor 2015; and Harcourt 2015). Others have turned their attention eastward, exploring Confucian, Buddhist, and Hindu traditions (Yu 2007; Slingerland 2011; Finnigan and Tanaka 2011; McRae 2012; Angle and Slote 2013; Davis 2014; Flanagan 2015; Perrett and Pettigrove 2015; and Sim 2015). These explorations promise to open up new avenues for the development of virtue ethics. Although virtue ethics has grown remarkably in the last thirty-five years, it is still very much in the minority, particularly in the area of applied ethics. Many editors of big textbook collections on "moral problems" or "applied ethics" now try to include articles representative of each of the three normative approaches but are often unable to find a virtue ethics article addressing a particular issue. This is sometimes, no doubt, because "the" issue has been set up as a deontologicial/utilitarian debate, but it is often simply because no virtue ethicist has yet written on the topic. However, the last decade has seen an increase in the amount of attention applied virtue ethics has received (Walker and Ivanhoe 2007; Hartman 2013; Austin 2014; Van Hooft 2014; and Annas 2015). This area can certainly be expected to grow in the future, and it looks as though applying virtue ethics in the field of environmental ethics may prove particularly fruitful (Sandler 2007; Hursthouse 2007, 2011; Zwolinski and Schmidtz 2013; Cafaro 2015). Whether virtue ethics can be expected to grow into "virtue politics"--i.e. to extend from moral philosophy into political philosophy--is not so clear. Gisela Striker (2006) has argued that Aristotle's ethics cannot be understood adequately without attending to its place in his politics. That suggests that at least those virtue ethicists who take their inspiration from Aristotle should have resources to offer for the development of virtue politics. But, while Plato and Aristotle can be great inspirations as far as virtue ethics is concerned, neither, on the face of it, are attractive sources of insight where politics is concerned. However, recent work suggests that Aristotelian ideas can, after all, generate a satisfyingly liberal political philosophy (Nussbaum 2006; LeBar 2013a). Moreover, as noted above, virtue ethics does not have to be neo-Aristotelian. It may be that the virtue ethics of Hutcheson and Hume can be naturally extended into a modern political philosophy (Hursthouse 1990-91; Slote 1993). Following Plato and Aristotle, modern virtue ethics has always emphasised the importance of moral education, not as the inculcation of rules but as the training of character. There is now a growing movement towards virtues education, amongst both academics (Carr 1999; Athanassoulis 2014; Curren 2015) and teachers in the classroom. One exciting thing about research in this area is its engagement with other academic disciplines, including psychology, educational theory, and theology (see Cline 2015; and Snow 2015). Finally, one of the more productive developments of virtue ethics has come through the study of particular virtues and vices. There are now a number of careful studies of the cardinal virtues and capital vices (Pieper 1966; Taylor 2006; Curzer 2012; Timpe and Boyd 2014). Others have explored less widely discussed virtues or vices, such as civility, decency, truthfulness, ambition, and meekness (Calhoun 2000; Kekes 2002; Williams 2002; and Pettigrove 2007 and 2012). One of the questions these studies raise is "How many virtues are there?" A second is, "How are these virtues related to one another?" Some virtue ethicists have been happy to work on the assumption that there is no principled reason for limiting the number of virtues and plenty of reason for positing a plurality of them (Swanton 2003; Battaly 2015). Others have been concerned that such an open-handed approach to the virtues will make it difficult for virtue ethicists to come up with an adequate account of right action or deal with the conflict problem discussed above. Dan Russell has proposed cardinality and a version of the unity thesis as a solution to what he calls "the enumeration problem" (the problem of too many virtues). The apparent proliferation of virtues can be significantly reduced if we group virtues together with some being cardinal and others subordinate extensions of those cardinal virtues. Possible conflicts between the remaining virtues can then be managed if they are tied together in some way as part of a unified whole (Russell 2009). This highlights two important avenues for future research, one of which explores individual virtues and the other of which analyses how they might be related to one another.
epistemology-visual-thinking	## 1. Introduction Visual thinking is a feature of mathematical practice across many subject areas and at many levels. It is so pervasive that the question naturally arises: does visual thinking in mathematics have any epistemically significant roles? A positive answer begets further questions. Can we rationally arrive at a belief with the generality and necessity characteristic of mathematical theorems by attending to specific diagrams or images? If visual thinking contributes to warrant for believing a mathematical conclusion, must the outcome be an empirical belief? How, if at all can visual thinking contribute to understanding abstract mathematical subject matter? Visual thinking includes thinking with external visual representations (e.g., diagrams, symbol arrays, kinematic computer images) and thinking with internal visual imagery; often the two are used in combination, as when we are required to visually imagine a certain spatial transformation of an object represented by a diagram on paper or on screen. Almost always (and perhaps always) visual thinking in mathematics is used in conjunction with non-visual thinking. Possible epistemic roles include contributions to evidence, proof, discovery, understanding and grasp of concepts. The kinds and the uses of visual thinking in mathematics are numerous and diverse. This entry will deal with some of the topics in this area that have received attention and omit others. Among the omissions is the possible explanatory role of visual representations in mathematics. The topic of explanation within pure mathematics is tricky and best dealt with separately; for this an excellent starting place is the entry on explanation in mathematics (Mancosu 2011). Two other omissions are the development of logic diagrams (Euler, Venn, Pierce and Shin) and the nature and use of geometric diagrams in Euclid's Elements, both of which are well treated in the entry diagrams (Shin et al. 2013). The focus here is on visual thinking generally, which includes thinking with symbol arrays as well as with diagrams; there will be no attempt here to formulate a criterion for distinguishing between symbolic and diagrammatic thinking. However, the use of visual thinking in proving and in various kinds of discovery will be covered in what follows. Discussions of some related questions and some studies of historical cases not considered here are to be found in the collection Diagrams in Mathematics: History and Philosophy (Mumma and Panza 2012). ## 2. Historical Background "Mathematics can achieve nothing by concepts alone but hastens at once to intuition" wrote Kant (1781/9: A715/B743), before describing the geometrical construction in Euclid's proof of the angle sum theorem (Euclid, Book 1, proposition 32). In a review of 1816 Gauss echoes Kant: > > anybody who is acquainted with the essence of geometry knows that [the > logical principles of identity and contradiction] are able to > accomplish nothing by themselves, and that they put forth sterile > blossoms unless the fertile living intuition of the object itself > prevails everywhere. (Ewald 1996 [Vol. 1]: 300) > The word "intuition" here translates the German "Anschauung", a word which applies to visual imagination and perception, though it also has more general uses. By the late 19th century a different view had emerged, at least in foundational areas. In a celebrated text giving the first rigorous axiomatization of projective geometry, Pasch wrote: "the theorem is only truly demonstrated if the proof is completely independent of the figure" (Pasch 1882), a view expressed also by Hilbert in writing on the foundations of geometry (Hilbert 1894). A negative attitude to visual thinking was not confined to geometry. Dedekind, for example, wrote of an overpowering feeling of dissatisfaction with appeal to geometric intuitions in basic infinitesimal analysis (Dedekind 1872, Introduction). The grounds were felt to be uncertain, the concepts employed vague and unclear. When such concepts were replaced by precisely defined alternatives without allusions to space, time or motion, our intuitive expectations turned out to be unreliable (Hahn 1933). In some quarters this view turned into a general disdain for visual thinking in mathematics: "In the best books" Russell pronounced "there are no figures at all" (Russell 1901). Although this attitude was opposed by a few mathematicians, notably Klein (1893), others took it to heart. Landau, for example, wrote a calculus textbook without a single diagram (Landau 1934). But the predominant view was not so extreme: thinking in terms of figures was valued as a means of facilitating grasp of formulae and linguistic text, but only reasoning expressed by means of formulae and text could bear any epistemological weight. By the late 20th century the mood had swung back in favour of visualization: Mancosu (2005) provides an excellent survey. Some books advertise their defiance of anti-visual puritanism in their titles, for example Visual Geometry and Topology (Fomenko 1994) and Visual Complex Analysis (Needham 1997); mathematics educators turn their attention to pedagogical uses of visualization (Zimmerman and Cunningham 1991); the use of computer-generated imagery begins to bear fruit at research level (Hoffman 1987; Palais 1999), and diagrams find their way into research papers in abstract fields: see for example the papers on higher dimensional category theory by Joyal et al. (1996), Leinster (2004) and Lauda (2005, Other Internet Resources). But attitudes to the epistemology of visual thinking remain mixed. The discussion is mostly concerned with the role of diagrams in proofs. ## 3. Visual thinking and proof In some cases, it is claimed, a picture alone is a proof (Brown 1999: ch. 3). But that view is rare. Even the editor of Proofs without Words: Exercises in Visual Thinking, writes "Of course, 'proofs without words' are not really proofs" (Nelsen 1993: vi). Expressions of the other extreme are rare but can be found: > [the diagram] has no proper place in the proof as > such. For the proof is a syntactic object consisting only of sentences > arranged in a finite and inspectable array. (Tennant 1986) > > > Between the extremes we find the view that, even if no picture alone is a proof, visual representations can have a non-superfluous role in reasoning that constitutes a proof. (This is not to deny that there may be another proof of the same conclusion which does not involve any visual representation.) Geometric diagrams, graphs and maps, all carry information. Taking valid deductive reasoning to be the reliable extraction of information from information already obtained, Barwise and Etchemendy (1996:4) pose the following question: Why cannot the representations composing a proof be visual as well as linguistic? The sole reason for denying this role to visual representations is the thought that, with the possible exception of very restricted cases, visual thinking is unreliable, hence cannot contribute to proof. Is that right? Our concern here is thinking through the steps in a proof, either for the first time (a first successful attempt to construct a proof) or following a given proof. Clearly we want to distinguish between visual thinking which merely accompanies the process of thinking through the steps in a proof and visual thinking which is essential to the process. This is not always straightforward as a proof can be presented in different ways. How different can distinct presentations be and yet be presentations of the same proof? There is no context-invariant answer to this. Often mathematicians are happy to regard two presentations as presenting the same proof if the central idea is the same in both cases. But if one's main concern is with what is involved in thinking through a proof, its central idea is not enough to individuate it: the overall structure, the sequence of steps and perhaps some other factors affecting the cognitive processes involved will be relevant. Once individuation of proofs has been settled, we can distinguish between replaceable thinking and superfluous thinking, where these attributions are understood as relative to a given argument or proof. In the process of thinking through a proof, a given part of the thinking is replaceable if thinking of some other kind could stand in place of the given part in a process that would count as thinking through the same proof. A given part of the thinking is superfluous if its excision without replacement would be a process of thinking through the same proof. Superfluous thinking may be extremely valuable in facilitating grasp of the proof text and in enabling one to understand the idea underlying the proof steps; but it is not necessary for thinking through the proof. It is uncontentious that the visual thinking involved in symbol manipulations, for example in following the "algebraic" steps of proofs of basic lemmas about groups, can be essential, that is neither superfluous nor replaceable. The worry is about thinking visually with diagrams, where "diagram" is used widely to include all non-symbolic visual representations. Let us agree that there can be superfluous diagrammatic thinking in thinking through a proof. This leaves several possibilities. * (a) All diagrammatic thinking in a process of thinking through a proof is superfluous. * (b)Not all diagrammatic thinking in a process of thinking through a proof is superfluous; but if not superfluous it will be replaceable by non-diagrammatic thinking. * (c)Some diagrammatic thinking in a process of thinking through a proof is neither superfluous nor replaceable by non-diagrammatic thinking. ### 3.1 The reliability question The negative view stated earlier that diagrams can have no role in proof entails claim (a). The idea behind (a) is that, because diagrammatic reasoning is unreliable, if a process of thinking through an argument contains some non-superfluous diagrammatic thinking, that process lacks the epistemic security to be a case of thinking through a proof. This view, claim (a) in particular, is threatened by cases in which the reliability of the diagrammatic thinking is demonstrated non-visually. The clearest kind of example would be provided by a formal system which has diagrams in place of formulas among its syntactic objects, and types of inter-diagram transition for inference rules. Suppose you take in such a formal system and an interpretation of it, and then think through a proof of the system's soundness with respect to that interpretation; suppose you then inspect a sequence of diagrams, checking along the way that it constitutes a derivation in the system; suppose finally that you recover the interpretation to reach a conclusion. (The order is unimportant: one can go through the derivation first and then follow the soundness proof.) That entire process would constitute thinking through a proof of the conclusion; and the diagrammatic thinking involved would not be superfluous. Shin et al. (2013) report that formal diagrammatic systems of logic and geometry have been proven to be sound. People have indeed followed proofs in these systems. That is enough to refute claim (a), the claim that all diagrammatic thinking in thinking through a proof is superfluous. For a concrete example, Figure 1 presents a derivation of Euclid's first theorem, that on any straight line segment an equilateral triangle is constructible, in a formal diagrammatic system of a part of Euclidean geometry (Miller 2001). ![[a three by three array of rectangles each containing a diagram. Going left to right then top to bottom, the first has a line segment with each end having a dot. The second is a circle with a radius drawn and dots on each end of the radius line segment. The third is the same the second except another overlapping circle is drawn using the same radius line segment but with the first circle's center dot now on the perimeter and the first circle's perimeter dot now the center of the second circle, dots are added at the two points the circles intersect. The fourth diagram is identical to the third except a line segment is drawn from the top intersection dot to the first circle's center dot. The fifth diagram is like the fourth except a line segment is drawn from the top intersection dot to the center dot of the second circle. ...]](fig1.png) Figure 1 What about Tennant's claim that a proof is "a syntactic object consisting only of sentences" as opposed to diagrams? A proof is never a syntactic object. A formal derivation on its own is a syntactic object but not a proof. Without an interpretation of the language of the formal system the end-formula of the derivation says nothing; and so nothing is proved. Without a demonstration of the system's soundness with respect to the interpretation, one may lack sufficient reason to believe that all derivable conclusions are true. A formal derivation plus an interpretation and soundness proof can be a proof of the derived conclusion, but that whole package is not a syntactic object. Moreover, the part of the proof which really is a syntactic object, the formal derivation, need not consist solely of sentences; it can consist of diagrams. With claim (a) disposed of, consider again claim (b) that, while not all diagrammatic thinking in a process of thinking through a proof is superfluous, all non-superfluous diagrammatic thinking will be replaceable by non-diagrammatic thinking in a process of thinking through that same proof. The visual thinking in following the proof of Euclid's first theorem using Miller's formal system consists in going through a sequence of diagrams and at each step seeing that the next diagram results from a permitted alteration of the previous diagram. It is clear that in a process that counts as thinking through this proof, the diagrammatic thinking is neither superfluous nor replaceable by non-diagrammatic thinking. That knocks out (b), leaving only (c): some thinking that involves a diagram in thinking through a proof is neither superfluous nor replaceable by non-diagrammatic thinking (without changing the proof). ### 3.2 Visual means in non-formal proving Mathematical practice almost never proceeds by way of formal systems. Outside the context of formal diagrammatic systems, the use of diagrams is widely felt to be unreliable. A diagram can be unfaithful to the described construction: it may represent something with a property that is ruled out by the description, or without a property that is demanded by the description. This is exemplified by diagrams in the famous argument for the proposition that all triangles are isosceles: the meeting point of an angle bisector and the perpendicular bisector of the opposite side is represented as falling inside the triangle, when it has to be outside (Rouse Ball 1939; Maxwell 1959). Errors of this sort are comparatively rare, usually avoidable with a modicum of care, and not inherent in the nature of diagrams; so they do not warrant a general charge of unreliability. The major sort of error is unwarranted generalisation. Typically diagrams (and other non-verbal visual representations) do not represent their objects as having a property that is actually ruled out by the intention or specification of the object to be represented. But diagrams very frequently do represent their objects as having properties that, though not ruled out by the specification, are not demanded by it. Verbal descriptions can be discrete, in that they supply no more information than is needed. But visual representations are typically indiscrete, in that they supply too much detail. This is often unavoidable, because for many properties or kinds $F$, a visual representation cannot represent something as being $F$ without representing it as being $F$ in a particular way. Any diagram of a triangle, for instance, must represent it as having three acute angles or as having just two acute angles, even if neither property is required by the specification, as would be the case if the specification were "Let ABC be a triangle". As a result there is a danger that in using a diagram to reason about an arbitrary instance of class $K$, we will unwittingly rely on a feature represented in the diagram that is not common to all instances of the class $K$. Thus the risk of unwarranted generalisation is a danger inherent in the use of many diagrams. Indiscretion of diagrams is not confined to geometrical figures. The dot or pebble diagrams of ancient mathematics used to convince one of elementary truths of number theory necessarily display particular numbers of dots, though the truths are general. Here is an example, used to justify the formula for the $n$th triangular number, i.e., the sum of the first $n$ positive integers. ![[a grid of blue dots 5 wide and 7 deep, on the right side is a brace embracing the right column labeled n+1 and on the bottom a brace embracing the bottom row labeled n]](fig2.png) Figure 2 The conclusion drawn is that the sum of integers from 1 to $n$ is $(n \times n+1)/2$ for any positive integer $n$, but the diagram presents the case for $n = 6$. We can perhaps avoid representing a particular number of dots when we merely imagine a display of the relevant kind; or if a particular number is represented, our experience may not make us aware of the number--just as, when one imagines the sky on a starry night, for no particular number $k$ are we aware that exactly $k$ stars are represented. Even so, there is likely to be some extra specificity. For example, in imagining an array of dots of the form just illustrated, one is unlikely to imagine just two columns of three dots, the rectangular array for $n = 2$. Typically the subject will be aware of imagining an array with more than two columns. This entails that an image is likely to have unintended exclusions. In this case it would exclude the three-by-two array. An image of a triangle representing all angles as acute would exclude triangles with an obtuse angle or a right angle. The danger is that the visual reasoning will not be valid for the cases that are unintentionally excluded by the visual representation, with the result that the step to the conclusion is an unwarranted generalisation. What should we make of this? First, let us note that in a few cases the image or diagram will not be over-specific. When in geometry all instances of the relevant class are congruent to one another, for instance all circles or all squares, the image or diagram will not be over-specific for a generalisation about that class; so there will be no unintended exclusions and no danger of unwarranted generalisation. Here then are possibilities for reliable visual thinking in proving. To get clear about the other cases, where there is a danger of over generalizing, it helps to look at generalisation in ordinary non-visual reasoning. Schematically put, in reasoning about things of kind $K$, once we have shown that from certain premisses it follows that such-and-such a condition is true of arbitrary instance $c$, we can validly infer from those same premisses that that condition is true of all $K$s, with the proviso that neither the condition nor any premiss mentions $c$. The proviso is required, because if a premiss or the condition does mention $c$, the reasoning may depend on a property of $c$ that is not shared by all other $K$s and so the generalisation would be unsafe. For a trivial example consider a step from "$x = c$" to "$\forall x [x = c]$". A question we face is whether, in order to come to know the truth of a conclusion by following an argument involving generalisation on an arbitrary instance (a.k.a. universal generalisation, or universal quantifier introduction), the thinking must include a conscious, explicit check that the proviso is met. It is clearly not enough that the proviso is in fact met. For in that case it might just be the thinker's good luck that the proviso is met; hence the thinker would not know that the generalisation is valid and so would not have genuinely thought through the proof at that step. This leaves two options. The strict option is that without a conscious, explicit check one has not really thought through the proof. The relaxed option is that one can properly think through the proof without checking that the proviso is met, but only if one is sensitive to the potential error and would detect it in otherwise similar arguments. For then one is not just lucky that the proviso is met. Being sensitive in this context consists in being alert to dependence on features of the arbitrary instance not shared by all members of the class of generalisation, a state produced by a combination of past experience and current vigilance. Without compelling reason to prefer one of these options, decisions on what is to count as proving or following a proof must be conditional. How does all this apply to generalizing from visual thinking about an arbitrary instance? Take the example of the visual route to the formula for triangular numbers using the diagram of Figure 2. The diagram reveals that the formula holds for the 6th triangular number. The generalisation to all triangular numbers is justified only if the visuo-spatial method used is applicable to the $n$th triangular number for all positive integers $n$, that is, provided that the method used does not depend on a property not shared by all positive integers. A conscious, explicit check that this proviso is met requires making explicit the method exemplified for 6 and proving that the method is applicable for all positive integers in place of 6. (For a similar idea in the context of automating visual arguments, see Jamnik 2001). This is not done in practice when thinking visually, and so if we accept the strict option for thinking through a proof involving generalisation, we would have to accept that the visual route to the formula for triangular numbers does not amount to thinking through a proof of it; and the same would apply to the familiar visual routes to other general positive integer formulas, such as that $n^2 =$ the sum of the first $n$ odd numbers. But what if the strict option for proving by generalisation on an arbitrary instance is too strict, and the relaxed option is right? When arriving at the formula in the visual way indicated, one does not pay attention to the fact that the visual display represents the situation for the 6th triangular number; it is as if the mind had somehow extracted a general schema of visual reasoning from exposure to the particular case, and had then proceeded to reason schematically, converting a schematic result into a universal proposition. What is required, on the relaxed option, is sensitivity to the possibility that the schema is not applicable to all positive integers; one must be so alert to ways a schema of the given kind can fall short of universal applicability that if one had been presented with a schema that did fall short, one would have detected the failure. In the example at hand, the schema of visual reasoning involves at the start taking a number $k$ to be represented by a column of $k$ dots, thence taking the triangular array of $n$ columns to represent the sum of the first $n$ positive integers, thence taking that array combined with an inverted copy to make a rectangular array of $n$ columns of $n+1$ dots. For a schema starting this way to be universally applicable, it must be possible, given any positive integer $n$, for the sum of the first $n$ positive integers to be represented in the form of a triangular array, so that combined with an inverted copy one gets a rectangular array. This actually fails at the extreme case: $n = 1$. The formula $(n.(n + 1))/2$ holds for this case; but that is something we know by substituting "1" for the variable in the formula, not by the visual method indicated. That method cannot be applied to $n = 1$, because a single dot does not form a triangular array, and combined with a copy it does not form a rectangular array. But we can check that the method works for all positive integers after the first, using visual reasoning to assure ourselves that it works for 2 and that if the method works for $k$ it works for $k+1$. Together with this reflective thinking, the visual thinking sketched earlier constitutes following a proof of the formula for the $n$th triangular number for all integers $n > 1$, at least if the relaxed view of thinking through a proof is correct. Similar conclusions hold in the case of other "dot" arguments (Giaquinto 1993, 2007: ch. 8). So in some cases when the visual representation carries unwanted detail, the danger of over-generalisation in visual reasoning can be overcome. But the fact that this is frequently missed by commentators suggests that the required sensitivity is often absent. Missing an untypical case is a common hazard in attempts at visual proving. A well-known example is the proof of Euler's formula $V - E + F = 2$ for polyhedra by "removing triangles" of a triangulated planar projection of a polyhedron. One is easily convinced by the thinking, but only because the polyhedra we normally think of are convex, while the exceptions are not convex. But it is also easy to miss a case which is not untypical or extreme when thinking visually. An example is Cauchy's attempted proof (Cauchy 1813) of the claim that if a convex polygon is transformed into another polygon keeping all but one of the sides constant, then if some or all of the internal angles at the vertices increase, the remaining side increases, while if some or all of the internal angles at the vertices decrease, the remaining side decreases. The argument proceeds by considering what happens when one transforms a polygon by increasing (or decreasing) angles, angle by angle. But in a trapezoid, changing a single angle can turn a convex polygon into a concave polygon, and this invalidates the argument (Lyusternik 1963). The frequency of such mistakes indicates that visual arguments (other than symbol manipulations) often lack the transparency required for proof. Even when a visual argument is in fact sound, its soundness may not be clear, in which case the argument is not a way of proving the truth of the conclusion, though it may be a way of discovering it. But this is consistent with the claim that visual non-symbolic thinking can be (and often is) part of a way of proving something. An example from knot theory will substantiate the modal part of this claim. To present the example, we need some background information, which will be given with a minimum of technical detail. A knot is a tame closed non-self-intersecting curve in Euclidean 3-space. In other words, knots are just the tame curves in Euclidean 3-space which are homeomorphic to a circle. The word "tame" here stands for a property intended to rule out certain pathological cases, such as curves with infinitely nested knotting. There is more than one way of making this mathematically precise, but we have no need for these details. A knot has a specific geometric shape, size and axis-relative position. Now imagine it to be made of flexible yet unbreakable yarn that is stretchable and shrinkable, so that it can be smoothly transformed into other knots without cutting or gluing. Since our interest in a knot is the nature of its knottedness regardless of shape, size or axis-relative position, the real focus of interest is not just the knot but all its possible transforms. A way to think of this is to imagine a knot transforming continuously, so that every possible transform is realized at some time. Then the thing of central interest would be the object that persists over time in varying forms, with knots strictly so called being the things captured in each particular freeze frame. Mathematically, we represent the relevant entity as an equivalence class of knots. Two knots are equivalent iff one can be smoothly deformed into the other by stretching, shrinking, twisting, flipping, repositioning or in any other way that does not involve cutting, gluing or passing one strand through another. The relevant kind of deformation forbids eliminating a knotted part by shrinking it down to a point. Again there are mathematically precise definitions of knot-equivalence. Figure 3 gives diagrams of equivalent knots, instances of a trefoil. ![[a closed line which goes under, over, under, over, under, over itself forming a shape with three nodes]](fig3a.png) ![[a closed line which goes under, over, under, over, under, over itself but forming a shop closer to a figure 8 inside an oval]](fig3b.png) (a) (b) Figure 3 Diagrams like these are not merely illustrations; they also have an operational role in knot theory. But not any picture of a knot will do for this purpose. We need to specify: A knot diagram is a regular projection of a knot onto a plane which, when there is a crossing, tells us which strand passes over the other. Regularity here is a combination of conditions. In particular, regularity entails that not more than two points of the strict knot project to the same point on the plane, and that two points of the strict knot project to the same point on the plane only where there is a crossing. For more on diagrams in knot theory see (De Toffoli and Giardino 2014). A major task of knot theory is to find ways of telling whether two knot diagrams are diagrams of equivalent knots. In particular we will want to know if a given knot diagram represents a knot equivalent to an unknot, that is, a knot representable by a knot diagram without crossings. One way of showing that a knot diagram represents a knot equivalent to an unknot is to show that the diagram can be transformed into one without crossings by a sequence of atomic moves, known as Reidemeister moves. The relevant background fact is Reidemeister's theorem, which links the visualizable diagrammatic changes to the mathematically precise definition of knot equivalence: Two knots are equivalent if and only if there is a finite sequence of Reidemeister moves taking a knot diagram of one to a knot diagram of the other. Figure 4 illustrates. Each knot diagram is changed into the adjacent knot diagram by a Reidemeister move; hence the knot represented by the leftmost diagram is equivalent to the unknot. ![[a closed line that goes under, under, under, over, over, over but forming otherwise a shape much like figure 3a]](fig4a.png) ![[a closed line that goes over, under forming a shape much like a loop within a loop]]](fig4b.png) ![[a closed line that forms a distorted loop with no intersections]](fig4c.png) (a) (b) (c) Figure 4 In contrast to these, the knot presented by the left knot diagram of Figure 3, a trefoil, may seem impossible to deform into an unknot. And in fact it is. To prove it, we can use a knot invariant known as colourability. An arc in a knot diagram is a maximal part between crossings (or the whole thing if there are no crossings). Colourability is this: A knot diagram is colourable if and only if each of its arcs can be coloured one of three different colours so that (a) at least two colours are used and (b) at each crossing the three arcs are all coloured the same or all coloured differently. The reference to colours here is inessential. Colourability is in fact a specific case of a kind of combinatorial property known as mod $p$ labelling (for $p$ an odd prime). Colourability is a knot invariant in the sense that if one diagram of a knot is colourable every diagram of that knot and of any equivalent knot is colourable. (By Reidemeister's theorem this can be proved by showing that each Reidemeister move preserves colourability.) A standard diagram of an unknot, a diagram without crossings, is clearly not colourable because it has only one arc (the whole thing) and so two colours cannot be used. So in order to complete proving that the trefoil is not equivalent to an unknot, we only need prove that our trefoil diagram is colourable. This can be done visually. Colour each arc of the knot diagram one of the three colours red, green or blue so that no two arcs have the same colour (or visualize this). Then do a visual check of each crossing, to see that at each crossing the three meeting arcs are all coloured differently. That visual part of the proof is clearly non-superfluous and non-replaceable (without changing the proof). Moreover, the soundness of the argument is quite transparent. So here is a case of a non-formal, non-symbolic visual way of proving a mathematical truth. ### 3.3 A dispute: diagrams in proofs in analysis. Where notions involving the infinite are in play, such as many involving limits, the use of diagrams is famously risky. For this reason it has been widely thought that, beyond some very simple cases, arguments in real and complex analysis in which diagrams have a non-superfluous role are not genuine proofs. Bolzano [1817] expressed this attitude with regard to the intermediate value theorem for the real numbers (IVT) before giving a purely analytic proof, arguing that spatial thinking could not be used to help justify the IVT. James Robert Brown (1999) takes issue with Bolzano on this point. The IVT is this: If $f$ is a real-valued function of a real variable continuous on the closed interval $[a, b]$ and $f(a) < c < f(b)$, then for some $x$ in $(a, b), f(x) = c$. Brown focuses on the special case when $c = 0$. As the IVT can be deduced easily from this special case using the theorem that the difference of two continuous functions is continuous, there is no loss of generality here. Alluding to a diagram like Figure 5, Brown (1999) writes > We have a continuous line running from below to above > the $x$-axis. Clearly, it must cross that axis in doing > so. (1999: 26) > > Later he claims: > > Using the picture alone, we can be certain of this result--if > we can be certain of anything. (1999: 28) > > ![[a first quadrant graph, the x-axis labeled near the left with 'a' and near the right with 'b'; the y-axis labeled at the top with 'f(b)', in the middle with 'c' and near the bottom with 'f(a)'. A dotted horizontal line lines up with the 'c'. A solid curve starts the intersection of 'f(b)' and 'a', rambles horizontally for a short while before rising above the 'c' dotted line, dips below then rises again and ending at the intersection of 'f(b)' and 'b'. ]](fig5.png) Figure 5 Bolzano's diagram-free proof of the IVT is an argument from what later became known as the Dedekind completeness of the real numbers: every non-empty set of reals bounded above (below) has a least upper bound (greatest lower bound). The value of Bolzano's deduction of the IVT from the Dedekind completeness of the reals, according to Brown, is not that it proves the IVT but that it gives us confirmation of Dedekind completeness, just as an empirical hypothesis in empirical science gets confirmed by deducing some consequence of the hypothesis and observing those consequence to be true. This view assumes that we already know the IVT to be true by observing a diagram relevantly like Figure 5. That assumption is challenged by Giaquinto (2011). Once we distinguish graphical concepts from associated analytic concepts, the underlying argument from the diagram is essentially this. * 1. Any function $f$ which is $\varepsilon\textrm{-}\delta$ continuous on $[a, b]$ with $f (a) < 0 < f (b)$ has a visually continuous graphical curve from below the horizontal line representing the $x$-axis to above. * 2. Any visually continuous graphical curve from below a horizontal line to above it meets the line at a crossing point. * 3. Any function whose graphical curve meets the line representing the $x$-axis at a crossing point has a zero value. * 4. So, any $\varepsilon\textrm{-}\delta$ continuous function $f$ on $[a, b]$ with $f (a) < 0< f (b)$ has a zero value. What is inferred from the diagram is premiss 2. Premisses 1 and 3 are assumptions linking analytical with graphical conditions. These linking assumptions are disputed. With regard to premiss 1 Giaquinto (2011) argues that there are functions on the reals which meet the antecedent condition but do not have graphical curves, such as continuous but nowhere differentiable functions and functions which oscillate with unbounded frequency e.g., $f(x) = x \cdot\sin(1/x)$ for non-zero $x$ in $[-1, 1]$ and $f(0) = 0$. With regard to premiss 3 it is argued that, under the standard conventions of graphical representation of functions in a Cartesian co-ordinate frame, the graphical curve for $x^2 - 2$ in the rationals is the same as the graphical curve for $x^2- 2$ in the reals. This is because every real is a limit point of rationals; so for every point $P$ with one or both co-ordinates irrational, there are points arbitrarily close to $P$ with both co-ordinates rational; so no gaps would appear if irrational points were removed from the curve for $x^2- 2$ in the reals. But for $x$ in the rational interval [0, 2] the function $x^2- 2$ has no zero value, even though it has a graphical curve which visually crosses the line representing the $x$-axis. So one cannot read off the existence of a zero of $x^2- 2$ on the reals from the diagram; one needs to appeal to some property of the reals which the rationals lack, such as Dedekind completeness. This raises some obvious questions. Do any theorems of analysis have proofs in which diagrams have a non-superfluous role? Littlewood (1953: 54-5) thought so and gives an example which is examined in Giaquinto (1994). If so, can we demarcate this class of theorems by some mathematical feature of their content? Another question is whether there is a significantly broad class of functions on the reals for which we could prove an intermediate value theorem (i.e., restricted to that class). If there are theorems of analysis provable with diagrams we do not yet have a mathematical demarcation criterion for them. A natural place to look would be O-minimal structures on the reals--this was brought to the author's attention by Ethan Galebach. This is because of some remarkable theorems about such structures which exclude all the pathological (hence vision-defying) functions on the reals (Van den Dries 1998), such as continuous nowhere differentiable functions and "space-filling" curves i.e., continuous surjections $f:(0, 1)\rightarrow(0, 1)^2$. Is the IVT for functions in an O-minimal structure on the reals provable by visual means? Certainly one objection to the visual argument for the unrestricted IVT does not apply when the restriction is in place. This is the objection that continuous nowhere differentiable functions, having no graphical curve, provide counterexamples to the premiss that any $\varepsilon\textrm{-}\delta$ continuous function $f$ on $[a, b]$ with $f (a) < c < f (b)$ has a visually continuous graphical curve from below the horizontal line representing $y = c$ to above. But the existence of continuous functions with no graphical curve is not the only objection to the visual argument, contrary to a claim of Azzouni (2013: 327). There are also counterexamples to the premiss that any function that does have a graphical curve which visibly crosses the line representing $y = c$ takes $c$ as a value, e.g., the function $x^2 - 2$ on the rationals with $c = 0$. So the question of a visual proof of the IVT restricted to functions in an O-minimal structure on the reals is still open at the time of writing. ## 4. Visual thinking and discovery Though philosophical discussion of visual thinking in mathematics has concentrated on its role in proof, visual thinking may be more valuable for discovery than proof. Three kinds of discovery important in mathematical practice are these: * (1)propositional discovery (discovering, of a proposition, that it is true), * (2)discovering a proof strategy (or more loosely, getting the idea for a proof of a proposition), and * (3)discovering a property or kind of mathematical entity. In the following subsections visual discovery of these kinds will be discussed and illustrated. ### 4.1 Propositional discovery To discover a truth, as that expression is being used here, is to come to believe it by one's own lights (as opposed to reading it or being told) in a way that is reliable and involves no violation of epistemic rationality (given one's epistemic state). One can discover a truth without being the first to discover it (in this context); it is enough that one comes to believe it in an independent, reliable and rational way. The difference between merely discovering a truth and proving it is a matter of transparency: for proving or following a proof the subject must be aware of the way in which the conclusion is reached and the soundness of that way; this is not required for discovery. Sometimes one discovers something by means of visual thinking using background knowledge, resulting in a cogent argument from which one could construct a proof. A nice example is a visual argument that any knot diagram with a finite number of crossings can be turned into a diagram of an unknot by interchanging the over-strand and under-strand of some of its crossings (Adams 2001: 58-90). That argument is a bit too long to present accessibly here. For a short example, here is a way of discovering that the geometric mean of two positive numbers is less than or equal to their arithmetic mean (Eddy 1985) using Figure 6. ![[two circles of differing sizes next to each other and touching at one point, the larger left circle has a vertical diameter line drawn and adjacent, parallel on the left is a double arrow headed line labelled 'a'. The smaller circle has a similar vertical diameter line with a double arrow headed line labelled 'b' to the right. The bottom of the diameter lines are connected by a double headed arrow line labeled 'square root of (ab)'. Another line connects the centers of both circles and has a parallel double arrow headed line labeled '(a+b)/2'. A dashed horizontal line goes horizontally from the center of the smaller circle until it hits the diameter line of the larger circle. Between this intersection and the center of the larger circle is a double arrow headed line labeled '(a-b)/2'.]](fig6.png) Figure 6 Two circles (with diameters $a$ and $b$) meet at a single point. A line is drawn between their centres through their common point; its length is $(a + b)/2$, the sum of the two radii. This line is the hypotenuse of a right angled triangle with one other side of length $(a - b)/2$, the difference of the radii. Pythagoras's theorem is used to infer that the remaining side of the right-angled triangle has length $\sqrt{(ab)}$.Then visualizing what happens to the triangle when the diameter of the smaller circle varies between 0 and the diameter of the larger circle, one infers that $0 < \sqrt{(ab)} < (a + b)/2$; then verifying symbolically that $\sqrt{(ab)} = (a + b)/2$ when $a = b$, one concludes that for positive $a$ and $b$, $\sqrt{(ab)} \le (a + b)/2$. This thinking does not constitute a case of proving or following a proof of the conclusion, because it involves a step which we cannot clearly tell is valid. This is the step of attempting to visually imagine what would happen when the smaller circle varies in diameter between 0 and the diameter of the larger circle and inferring from the resulting experience that the line joining the centres of the circles will always be longer than the horizontal line from the centre of the smaller circle to the vertical diameter of the larger circle. This step seems sound (does not lead us into error) and may be sound; but its soundness is opaque. If in fact it is sound, the whole thinking process is a reliable way of reaching the conclusion; so in the absence of factors that would make it irrational to trust the thinking, it would be a way of discovering the conclusion to be true. ### 4.2 Discovering a proof strategy In some cases visual thinking inclines one to believe something on the basis of assumptions suggested by the visual representation that remain to be justified given the subject's current knowledge. In such cases there is always the danger that the subject takes the visual representation to show the correctness of the assumptions and ends up with an unwarranted belief. In such a case, even if the belief is true, the subject has not made a discovery, as the means of belief-acquisition is unreliable. Here is an example using Figure 7 (Montuchi and Page 1988). ![[A first quadrant graph, on the x-axis are marked (2 squareroot(k), 0) and further to the right (j,0). On the y-axis is marked (0,2(squareroot(k)) and further up, (0,j). Solid lines connect (0,2(squareroot(k)) to (2(squareroot(k),0) and (0,j) to (j,0). A dotted line goes from the origin in a roughly 45 degree angle the point where it intersects the (0,2(squareroot(k)) to (2(squareroot(k),0) line is labeled (squareroot(k),squareroot(k)). A curve tangent to that point with one end heading up and the other right is labeled 'xy=k'.]](fig7.png) Figure 7 Using this diagram one can come to think the following about the real numbers. When for a constant $k$ the positive values of $x$ and $y$ are constrained to satisfy the equation $x \cdot y = k$, the positive values of $x$ and $y$ for which $x + y$ is minimal are $x = \sqrt{k} = y$. (Let "#" denote this claim.) Suppose that one knows the conventions for representing functions by graphs in a Cartesian co-ordinate system, knows also that the diagonal represents the function $y = x$, and that a line segment with gradient -1 from $(0, b)$ to $(b, 0)$ represents the function $x + y = b$. Then looking at the diagram may incline one to think that for no positive value of $x$ does the value of $y$ in the function $x\cdot y = k$ fall below the value of $y$ in $x + y = 2\sqrt{k}$, and that these functions coincide just at the diagonal. From these beliefs the subject may (correctly) infer the conclusion #. But mere attention to the diagram cannot warrant believing that, for a given positive $x$-value, the $y$-value of $x\cdot y = k$ never falls below the $y$-value of $x + y = 2\sqrt{k}$ and that the functions coincide just at the diagonal; for the conventions of representation do not rule out that the curve of $x\cdot y = k$ meets the curve of $x + y = 2\sqrt{k}$ at two points extremely close to the diagonal, and that the former curve falls under the latter in between those two points. So the visual thinking is not in this case a means of discovering proposition #. But it is useful because it provides the idea for a proof of the conclusion--one of the major benefits of visual thinking in mathematics. In brief: for each equation $(x\cdot y = k$; $x + y = 2\sqrt{k})$ if $x = y$, their common value is $\sqrt{k}$. So the functions expressed by those equations meet at the diagonal. To show that, for a fixed positive $x$-value, the $y$-values of $x\cdot y = k$ never fall below the $y$-values of $x + y = 2\sqrt{k}$, it suffices to show that $2\sqrt{k} - x \le k/x$. As a geometric mean is less than or equal to the corresponding arithmetic mean, $\sqrt{[x \cdot (k/x)]} \le [x + (k/x)]/2$. So $2\sqrt{k} \le x + (k/x)$. So $2\sqrt{k} - x \le k/x$. In this example, visual attention to, and reasoning about, the diagram is not part of a way of discovering the conclusion. But if it gave one the idea for the argument just given, it would be part of what led to a way of discovering the conclusion, and that is important. Can visual thinking lead to discovery of an idea for a proof in more advanced contexts? Yes. Carter (2010) gives an example from free probability theory. The case is about certain permutations (those denoted by "$p$" with a circumflex in Carter 2010) on a finite set of natural numbers. Using specific kinds of diagram, easily seen properties of the diagrams lead one naturally to certain properties of the permutations (crossing and non-crossing, having neighbouring pairs), and to a certain operation (cancellation of neighbouring pairs). All of these have algebraic definitions, but the ideas defined were noticed by thinking in terms of the diagrams. For the relevant permutations $\sigma$, $\sigma(\sigma(n)) = n$; so a permutation can be represented by a set of lines joining dots. The permutations represented on the left and right in Figure 8 are non-crossing and crossing respectively, the former with neighbouring pairs $\{2, 3\}$ and $\{6, 7\}$. ![[a circle with 8 points on the circumference, a point at about 45 degrees is labeled '1', at 15 degrees, '2', at -15 degrees '3', at -45 degrees '4', at -135 degrees '5', at -165 degrees '6', at 165 degrees '7', and at 135 degrees '8'. Smooth curves in the interior of the circle connect point 1 to 4, 2 to 3, 5 to 8, and 6 to 7.]](fig8a.png) ![[a circle with 8 points on the circumference, a point at about 45 degrees is labeled '1', at 15 degrees, '2', at -15 degrees '3', at -45 degrees '4', at -135 degrees '5', at -165 degrees '6', at 165 degrees '7', and at 135 degrees '8'. Straight lines connect 1 to 6, 2 to 5, 3 to 8, and 4 to 7.]](fig8b.png) (a) (b) Figure 8 A permutation $\sigma$ of $\{1, 2, \ldots, 2p\}$ is defined to have a crossing just when there are $a$, $b$, $c$, $d$ in $\{1, 2, \ldots, 2p\}$ such that $a < b < c < d$ and $\sigma(a) = c$ and $\sigma(b) = d$. The focus is on the proof of a theorem which employs this notion. (The theorem is that when a permutation of $\{1, 2, \ldots, 2p\}$ of the relevant kind is non-crossing, there will be exactly $p+1$ R-equivalence classes, where $R$ is a certain equivalence relation on $\{1, 2, \ldots, 2p\}$ defined in terms of the permutation.) Carter says that the proofs of some lemmas "rely on a visualization of the setup", in that to grasp the correctness of one or more of the steps one needs to visualize the situation. There is also a nice example of some reasoning in terms of a diagram which gives the idea for a proof ("suggests a proof strategy") for the lemma that every non-crossing permutation has a neighbouring pair. Reflection on a diagram such as Figure 9 does the work. ![[A circle, a dashed interior curve connects an unmarked point at about 40 degrees to an unmarked point at -10 degrees (the second point is labeled 'j+1'). Another dashed interior curve connects this point to an unmarked point at about -100 degrees. A solid interior curve connects and unmarked point at about 10 degrees (labeled 'j') to another unmarked point at about -60 degrees (labeled 'j+a'). Between the labels 'j+1' and 'j+a' is another label 'j+2' and then a dotted line between 'j+2' and 'j+a'.]](fig9.png) Figure 9 The reasoning is this. Suppose that $\pi$ has no neighbouring pair. Choose $j$ such that $\pi(j) - j = a$ is minimal, that is, for all $k, \pi(j) - j \le \pi(k) - k$. As $\pi$ has no neighbouring pair, $\pi(j+1) \ne j$. So either $\pi(j+1)$ is less than $j$ and we have a crossing, or by minimality of $\pi(j) - j$, $\pi(j+1)$ is greater than $j+a$ and again we have a crossing. Carter reports that this disjunction was initially believed by thinking in term of the diagram, and the proof of the lemma given in the published paper is a non-diagrammatic "version" of that reasoning. In this case study, visual thinking is shown to contribute to discovery in several ways; in particular, by leading the mathematicians to notice crucial properties--the "definitions are based on the diagrams"--and in giving them the ideas for parts of the overall proof. ### 4.3 Discovering properties and kinds In this section I will illustrate and then discuss the use of visual thinking in discovering kinds of mathematical entity, by going through a few of the main steps leading to geometric group theory, a subject which really took off in the 1980s through the work of Mikhail Gromov. The material is set out nicely in greater depth in Starikova (2012). Sometimes it can be fruitful to think of non-spatial entities, such as algebraic structures, in terms of a spatial representation. An example is the representation of a finitely generated group by a Cayley graph. Let $(G, \cdot)$ be a group and $S$ a finite subset of $G$. Let $S^{-1}$ be the set of inverses of members of $S$. Then $(G, \cdot)$ is generated by $S$ if and only if every member of $G$ is the product (with respect to $\cdot$) of members of $S\cup S^{-1}$. In that case $(G, \cdot, S)$ is said to be a finitely generated group. Here are a couple of examples. First consider the group $S\_{3}$ of permutations of 3 elements under composition. Letting $\{a, b, c\}$ be the elements, all six permutations can be generated by $\rf$ and $\rr$ where $\rf$ (for "flip") fixes a and swaps $b$ with $c$, i.e., it takes to $\langle a, b, c\rangle$ to $\langle a, c, b\rangle$, and $\rr$ (for "rotate") takes $\langle a, b, c\rangle$ to $\langle c, a, b\rangle$. The Cayley graph for $(S\_{3}, \cdot, \{\rf, \rr\})$ is a graph whose vertices represent the members of $S\_{3}$ and two "colours" of directed edges, representing composition with $\rf$ and composition with $\rr$. Figure 10 illustrates: red directed edges represent composition with $\rr$ and black edges represent composition with $\rf$. So a red edge from a vertex $\rv$ representing $\rs$ in $S\_{3}$ ends at a vertex representing $\rs\rr$ and a black edge from $\rv$ ends at a vertex representing $\rs\rf$. (Notation: "$\rs\rr$" abbreviates "$\rs \cdot \rr$" which here denotes "$\rs$ followed by $\rr$"; same for "$\rf$" in place of "$\rr$".) A black edge has arrowheads both ways because $\rf$ is its own inverse, that is, flipping and flipping again takes you back to where you started. (Sometimes a pair of edges with arrows in opposite directions is used instead.) The symbol "$\re$" denotes the identity. ![[Two red equilateral triangles, one inside the other. The smaller triangle has arrows on each side pointing in a clockwise direction; the larger has arrows on each side in a counterclockwise direction. Black double arrow lines connect the respective vertices of each triangle. The top vertice of the outside triangle is labeled 'e', of the inside triangle 'f'; the bottom left vertice of the outside triangle is labeled 'r', of the inside triangle 'r'; the bottom right vertix of the outside triangle is labeled with 'rr',of the inside triangle with 'fr'.]](fig10.png) Figure 10 An example of a finitely generated group of infinite order is $(\mathbb{Z}, +, \{1\})$. We can get any integer by successively adding 1 or its additive inverse $-1$. Since 3 added to the inverse of 2 is 1, and 2 added to the inverse of 3 is $-1$, we can get any integer by adding members of $\{2, 3\}$ and their inverses. Thus both $\{1\}$ and $\{2, 3\}$ are generating sets for $(\mathbb{Z}, +)$. Figure 11 illustrates part of the Cayley graph for $(\mathbb{Z}, +, \{2, 3\})$. The horizontal directed edges represent +2. The directed edges ascending or descending obliquely represent $+3$. ![[Two horizontal parallel black lines with directional arrows pointing to the right. The top line has equidistant points marked '-2', '0', '2', '4' and the bottom line equidistant points marked '-1' (about half way between the upper line's '-2' and '0'), '1', '3', '5'. A red arrow goes from '-2' to '1', from somewhere to the left up to '0', from '0' to '3', from '-1' to '2', from '1' to '4, from '2' to '5', and from '3' to somewhere to the right up.]](fig11.png) Figure 11 Another example of a generated group of infinite order is $F\_2$, the free group generated by a pair of members. The first few iterations of its Cayley graph are shown in Figure 12, where $\{a, b\}$ is the set of generators and a right horizontal move between adjacent vertices represents composition with $a$, an upward vertical move represents composition with $b$, and leftward and downward moves represent composition with the inverse of $a$ and the inverse of $b$ respectively. The central vertex represents the identity. ![[A blue vertical line pointing up labeled 'b' crossed by a red horizontal line pointing right labeled 'b'. Each line is crossed by two smaller copies of the other line on either side of the main intersection. And, in turn, each of those smaller copies of the line are crossed by two smaller copies of the other line, again on either side of their main intersection.]](fig12.png) Figure 12 Thinking of generated groups in terms of their Cayley graphs makes it very natural to view them as metric spaces. A path is a sequence of consecutively adjacent edges, regardless of direction. For example in the Cayley graph for $(\mathbb{Z}, +, \{2, 3\})$ the edges from $-2$ to 1, from 1 to $-1$, from $-1$ to 2 (in that order) constitute a path, representing the action, starting from $-2$, of adding 3, then adding $-2$, then adding 3. Taking each edge to have unit length, the metric $d\_S$ for a group $G$ generated by a finite subset $S$ of $G$ is defined: for any $g$, $h \in G$, $d\_{S}(g, h) =$ the length of a shortest path from $g$ to $h$ in the Caley graph of $(G, \cdot, S)$. This is the word metric for this generated group. Viewing a finitely generated group as a metric space allows us to consider its growth function $\gamma(n)$ which is the cardinality of the "ball" of radius $\le n$ centred on the identity (the number of members of the group whose distance from the identity is not greater than $n$). A growth function for a given group depends on the set of generators chosen, but when the group is infinite the asymptotic behaviour as $n \rightarrow \infty$ of the growth functions is independent of the set of generators. Noticing the possibility of defining a metric on generated groups did not require first viewing diagrams of their Cayley graphs. This is because a word in the generators is just a finite sequence of symbols for the generators or their inverses (we omit the symbol for the group operation), and so has an obvious length visually suggested by the written form of the word, namely the number of symbols in the sequence; and then it is natural to define the distance between group members $g$ and $h$ to be the length of a shortest word that gets one from $g$ to $h$ by right multiplication, that is, $\textrm{min}\{\textrm{length}(w): w = g^{-1}h\}$. However, viewing generated groups by means of their Cayley graphs was the necessary starting point for geometric group theory, which enables us to view finitely generated groups of infinite order not merely as graphs or metric spaces but as geometric entities. The main steps on this route will be sketched briefly here; for more detail see Starikova (2012) and the references therein. The visual key is to start thinking in terms of the "coarse geometry" of the Cayley graph of the generated group, by zooming out in visual imagination so far that the discrete nature of the graph is transformed into a traditional geometrical object. For example, the Cayley graph of a generated group of finite order such as $(S\_{3}, \cdot, \{f, r\})$ illustrated in Figure 11 becomes a dot; the Cayley graph for $(\mathbb{Z}, +, \{2, 3\})$ illustrated in Figure 12 becomes an uninterrupted line infinite in both directions. The word metric of a generated group is discrete: the values are always in $N$. How is this visuo-spatial association of a discrete metric space with a continuous geometrical object achieved mathematically? By quasi-isometry. While an isometry from one metric space to another is a distance preserving map, a quasi-isometry is a map which preserves distances to within fixed linear bounds. Precisely put, a map $f$ from $(S, d)$ to $(S', d')$ is a quasi-isometry iff for some real constants $L > 0$ and $K \ge 0$ and all $x$, $y$ in $S$ \[ d(x, y)/L - K \le d'(f(x), f(y)) \le L \cdot d(x, y) + K. \] The spaces $(S, d)$ and $(S', d')$ are quasi-isometric spaces iff the quasi-isometry $f$ is also quasi-surjective, in the sense that there is a real constant $M \ge 0$ such that every point of $S'$ is no further than $M$ away from some point in the image of $f$. For example, $(\mathbb{Z}, d)$ is quasi-isometric to $(\mathbb{R}, d)$ where $d(x, y) = \|y - x\|$, because the inclusion map $\iota$ from $\mathbb{Z}$ to $\mathbb{R}$, $\iota(n) = n$, is an isometry hence a quasi-isometry with $L = 1$ and $K = 0$, and each point in $\mathbb{R}$ is no further than $1/2$ away from an integer (in $\mathbb{R}$). Also, it is easy to see that for any real number $x$, if $g(x) =$ the nearest integer to $x$ (or the greatest integer less than $x$ if it is midway between integers) then $g$ is a quasi-isometry from $\mathbb{R}$ to $\mathbb{Z}$ with $L = 1$ and $K =\frac{1}{2}$;. The relation between metric spaces of being quasi-isometric is an equivalence relation. Also, if $S$ and $T$ are generating sets of a group $(G, \cdot)$, the Cayley graphs of $(G, \cdot, S)$ and $(G, \cdot, T)$ with their word metrics are quasi-isometric spaces. This means that properties of a generated group which are quasi-isometric invariants will be independent of the choice of generating set, and therefore informative about the group itself. Moreover, it is easy to show that the Cayley graph of a generated group with word metric is quasi-isometric to a geodesic space.[1] A triangle with vertices $x$, $y$, $z$ in this space is the union of three geodesic segments, between $x$ and $y$, between $y$ and $z$, and between $z$ and $x$. This is the gateway for the application of Gromov's insights, some of which can be grasped with the help of visual geometric thinking. Here are some indications. Recall the Poincare open disc model of hyperbolic geometry: geodesics are diameters or arcs of circles orthogonal to the boundary, with unit distance represented by ever shorter Euclidean distances as one moves from the centre towards the boundary. (The boundary is not part of the model). All triangles have angle sum $< \pi$ (Figure 13, left), and there is a global constant d such that all triangles are d-thin in the following sense: A triangle $T$ is d-thin if and only if any point on one side of $T$ lies within d of some point on one of the other two sides. This condition is equivalent to the condition that each side of $T$ lies within the union of the d-neighbourhoods of the other two sides, as illustrated in Figure 13, right. There is no constant d such that all triangles in a Euclidean plane are d-thin, because for any d there are triangles large enough that the midpoint of a longest side lies further than d from all points on the other two sides. ![[a circle. In the interior are three arcs colored green, blue, and red. For all three smooth curves where each meets the circumference of the circle is marked as at a 90 degree angle. The green curve may actually be a straight line and goes from about 160 degrees to about -20 degrees. The blue curve goes from about 170 degrees to about 80 degrees. The red curve goes from about 90 degrees to about -25 degrees. Where the green and blue curves intersect is marked as an angle and labelled with the Greek letter alpha; where the blue and the red curves intersect is also marked as an angle and labelled with gamma; and with where the red and the green curves intersect and this labelled with beta.]](fig13a.png) ![[not sure how to describe this]](fig13b.png) (a) (b) Figure 13[2] The definition of thin triangles is sufficiently general to apply to any geodesic space and allows a generalisation of the concept of hyperbolicity beyond its original context: * A geodesic space is hyperbolic iff for some d all its triangles are d-thin. * A group is hyperbolic iff it has a Cayley graph quasi-isometric to a hyperbolic geodesic space. The class of hyperbolic groups is large and includes important subkinds, such as finite groups, free groups and the fundamental groups of surfaces of genus $\ge 2$. Some striking theorems have been proved for them. For example, for every hyperbolic group the word problem is solvable, and every hyperbolic group has a finite presentation. So we can reasonably conclude that the discovery of this mathematical kind, the hyperbolic groups, has been fruitful. How important was visual thinking to the discoveries leading to geometric group theory? Visual thinking was needed to discover Cayley graphs as a means of representing finitely generated groups. This is not the triviality it might seem: Cayley graphs must be distinguished from the diagrams we use to present them visually. A Cayley graph is a mathematical representation of a generated group, not a visual representation. It consists of the following components: a set $V$ ("vertices"), a set $E$ of ordered pairs of members of $V$ ("directed edges") and a partition of $E$ into distinguished subsets, ("colours", each one for representing right multiplication by a particular generator). The Cayley graph of a generated group of infinite order cannot be fully represented by a diagram given the usual conventions of representation for diagrams of graphs, and distinct diagrams may visually represent the same Cayley graph: both diagrams in Figure 14 can be labelled so that under the usual conventions they represent the Cayley graph of $(S\_{3}, \cdot, \{f, r\})$, already illustrated by Figure 10. So the Cayley graph cannot be a diagram. ![[two identical red triangles, one above the other and inverted. Both have arrows going clockwise around. Black lines with arrows pointing both ways link the respective vertices.]](fig14a.png) ![[two identical red triangles, one above the other. The bottom one has arrows going clockwise around and the top counterclockwise. Black lines with arrows pointing both ways link the respective vertices.]](fig14b.png) (a) (b) Figure 14 Diagrams of Cayley graphs were important in prompting mathematicians to think in terms of the coarse-grained geometry of the graphs, in that this idea arises just when one thinks in terms of "zooming out" visually. Gromov (1993) makes the point in a passage quoted in Starikova (2012:138) > This space [a Cayley graph with the word metric] may > appear boring and uneventful to a geometer's eye since it is > discrete and the traditional (e.g., topological and infinitesimal) > machinery does not run in [the group] G. To regain the geometric > perspective one has to change one's position and move the > observation point far away from G. Then the metric in G > seen from the distance $d$ becomes the original distance divided by > $d$ and for $d \rightarrow \infty$ the points in G coalesce > into a connected continuous solid unity which occupies the visual > horizon without any gaps and holes and fills our geometer's > heart with joy. > > In saying that one has to move the observation point far away from G so that the points coalesce into a unity which occupies the visual horizon, he makes clear that visual imagination is involved in a crucial step on the road to geometric group theory. Visual thinking is again involved in discovering hyperbolicity as a property of general geodesic spaces from thinking about the Poincare disk model of hyperbolic geometry. It is hard to see how this property would have been discovered without the use of visual resources. ## 5. Visual thinking and mental arithmetic While there is no reason to think that mental arithmetic (mental calculation in the integers and rational numbers) typically involves much visual thinking, there is strong evidence of substantial visual processing in the mental arithmetic of highly trained abacus users. In earlier times an abacus would be a rectangular board or table surface marked with lines or grooves along which pebbles or counters could be moved. The oldest surviving abacus, the Salamis abacus, dated around 300 BCE, is a white marble slab, with markings designed for monetary calculation (Fernandes 2015, Other Internet Resources). These were superseded by rectangular frames within which wires or rods parallel to the short sides are fixed, with moveable holed beads on them. There are several kinds of modern abacus -- the Chinese suanpan, the Russian schoty and the Japanese soroban for example -- each kind with variations. Evidence for visual processing in mental arithmetic comes from studies with well trained users of the soroban, an example of which is shown in Figure 15. ![[Picture of a soroban with 17 columns of beads, each column has 1 bead above the horizontal bar used to represent 5 and 4 beads below the bar each of which represents 1. Together the beads in each column can represent any digit from 0 to 9.]](fig15.png) Figure 15 Each column of beads represents a power of 10, increasing to the left. The horizontal bar, sometimes called the reckoning bar, separates the beads on each column into one bead of value 5 above and four beads of value 1 below. The number represented in a column is determined by the beads which are not separated from the reckoning bar. A column on which all beads are separated by a gap from the bar represents zero. For example, the number 6059 is represented on a portion of a schematic soroban in Figure 16. ![[A schematic soroban representing 6059. There are 8 places and the first four from the left are set to 0, then 6, then 0, then 5, then 9 ]](fig16.png) Figure 16 On some sorobans there is a mark on the reckoning bar at every third column; if a user chooses one of these as a unit column, the marks will help the user keep track of which columns represent which powers of ten. Calculations are made by using forefinger and thumb to move beads according to procedures for the standard four numerical operations and for extraction of square and cube roots (Bernazzani 2005,Other Internet Resources). Despite the fact that the soroban has a decimal place representation of numbers, the soroban procedures are not 'translations' of the procedures normally taught for the standard operations using arabic numerals. For example, multidigit addition on a soroban starts by adding highest powers of ten and proceeds rightwards to lower powers, instead of starting with units thence proceeding leftwards to tens, hundreds and so on. People trained to use a soroban often learn to do mental arithmetic by visualizing an abacus and imagining moving beads on it in accordance with the procedures learned for arithmetical calculations (Frank and Barner 2012). Mental abacus (MA), as this kind of mental arithmetic is known, compares favourably with other kinds of mental calculation for speed and accuracy (Kojima 1954) and MA users are often found among the medallists in the Mental Calculation World Cup. Although visual and manual motor imagery is likely to occur, cognitive scientists have probed the question whether the actual processes of MA calculation consist in or involve imagining performing operations on a physical abacus. Brain imaging studies provide one source of evidence bearing on this question. Comparing well-trained abacus calculators with matched controls, evidence has been found that MA involves neural resources of visuospatial working memory with a form of abacus which does not depend on the modality (visual or auditory) of the numerical inputs (Chen et al. 2006). Another imaging study found that, compared to controls without abacus training, subjects with long term MA training from a young age had enhanced brain white matter related to motor and visuospatial processes (Hu et al. 2011). Behavioural studies provide more evidence. Tests on expert and intermediate level abacus users strongly suggest that MA calculators mentally manipulate an abacus representation so that it passes through the same states that an actual abacus would pass through in solving an addition problem. Without using an actual abacus MA calculators were able to answer correctly questions about intermediates states unique to the abacus-based solution of a problem; moreover, their response times were a monotonic function of the position of the probed state in the sequence of states of the abacus process for solving the problem (Stigler 1984). On top of the 'intermediate states' evidence, there is 'error type' evidence. Mental addition tests comparing abacus users with American subjects revealed that abacus users made errors of a kind which the Americans did not make, but which were predictable from the distribution of errors in physical abacus addition (Stigler 1984). Another study found evidence that when a sequence of numbers is presented auditorily (as a verbal whole "three thousand five hundred and forty seven" or as a digit sequence "Three, five, four, seven") abacus experts encode it into an imaged abacus display, while non-experts encode it verbally (Hishitani 1990). Further evidence comes from behavioural interference studies. In these studies subjects have to perform mental calculations, with and without a task of some other kind to be performed during the calculation, with the aim of seeing which kinds of task interfere with calculation as measured by differences of reaction time and error rate. An early study found that a linguistic task interfered weakly with MA performance (unless the linguistic task was to answer a mathematical question), while motor and visual tasks interfered relatively strongly. These findings suggested to the paper's authors that MA representations are not linguistic in nature but rely on visual mechanisms and, for intermediate practitioners, on motor mechanisms as well (Hatano et al. 1977). These studies provide impressive evidence that MA does involve mental manipulation of a visualized abacus. However, limits of the known capacities for perceiving or representing pluralities of objects seem to pose a problem. We have a parallel individuation system for keeping track of up to four objects simultaneously and an approximate number system (ANS) which allows us to gauge roughly the cardinality of a set of things, with an error which increases with the size of the set. The parallel individuation system has a limit of three or four objects and the ANS represents cardinalities greater than four only approximately. Yet mental abacus users would need to hold in mind with precision abacus representations involving a much larger number of beads than four (and the way in which those beads are distributed on the abacus). For example, the number 439 requires a precise distribution of twelve beads. Frank and Barner (2012) address this problem. In some circumstances we can perceive a plurality of objects as a single entity, a set, and simultaneously perceive those objects as individuals. There is evidence that we can keep track of up to three such sets in parallel and simultaneously make reliable estimates of the cardinalities of the sets (if not more than four). If the sets themselves can be easily perceived as (a) divided into disjoint subsets, e.g. columns of beads on an abacus, and (b) structured in a familiar way, e.g. as a distribution of four beads below a reckoning bar and one above, we have the resources for recognising a three-digit number from its abacus representation. The findings of (Frank and Barner 2012) suggest that this is what happens in MA: a mental abacus is represented in visuospatial working memory by splitting it into a series of columns each of which is stored as a unit with its own detailed substructure. These cognitive investigations confirm the self-reports of mental abacus users that they calculate mentally by visualizing operating on an abacus as they would operate on a physical abacus. (See the 20-second movie Brief interview with mental abacus user, at the Stanford Language and Cognition Lab, for one such self-report.) There is good evidence that MA often involves processes linked to motor cognition in addition to active visual imagination. Intermediate abacus users often make hand movements, without necessarily attending to those movements during MA calculation, as shown in the second of the three short movies just mentioned. Experiments to test the possible role of motor processes in MA resulted in findings which led the authors to conclude that premotor processes involved in the planning of hand movements were involved in MA (Brooks et al. 2018). ## 6. A priori and a posteriori roles of visual experience In coming to know a mathematical truth visual experience can play a merely "enabling" role. For example, visual experience may have been a factor in a person's getting certain concepts involved in a mathematical proposition, thus enabling her to understand the proposition, without giving her reason to believe it. Or the visual experience of reading an argument in a text book may enable one to find out just what the argument is, without helping her tell that the argument is sound. In earlier sections visual experience has been presented as having roles in proof and propositional discovery that are not merely enabling. On the face of it this raises a puzzle: mathematics, as opposed to its application to natural phenomena, has traditionally been thought to be an a priori science; but if visual experience plays a role in acquiring mathematical knowledge which is not merely enabling, the result would surely be a posteriori knowledge, not a priori knowledge. Setting aside knowledge acquired by testimony (reading or hearing that such-&-such is the case), there remain plenty of cases where sensory experience seems to play an evidential role in coming to know some mathematical fact. ### 6.1 Evidential uses of visual experience A plausible example of the evidential use of sensory experience is the case of a child coming to know that $5 + 3 = 8$ by counting on her fingers. While there may be an important $a$ priori element in the child's appreciation that she can reliably generalise from the result of her counting experiment, getting that result by counting is an a posteriori route to it. For another example, consider the question: how many vertices does a cube have? With the background knowledge that cubes do not vary in shape and that material cubes do not differ from geometrical cubes in number of vertices (where a "vertex" of a material cube is a corner), one can find the answer by visually inspecting a material cube. Or if one does not have a material cube to hand, one can visually imagine a cube, and by attending to its top and bottom faces extract the information that the vertices of the cube are exactly the vertices of these two quadrangular faces. When one gets the answer by inspecting a material cube, the visual experience contributes to one's grounds for believing the answer and that contribution is part of what makes the belief state knowledge. So the role of the visual experience is evidential; hence the resulting knowledge is not a priori. When one gets the answer by visually imagining a cube, one is drawing on the accumulated cognitive effects of past experiences of seeing material cubes to bring to mind what a cube looks like; so the experience of visual imagining has an indirectly evidential role in this case. Do such examples show that mathematics is not an a priori science? Yes, if an a priori science is understood to be one whose knowable truths are all knowable only in an a priori way, without use of sense experience as evidence. No, if an a priori science is one whose knowable truths are all knowable in an a priori way, allowing that some may be knowable also in an a posteriori way. ### 6.2 An evidential use of visual experience in proving Many cases of proving something (or following a proof of it) involve making, or imagining making, changes in a symbol array. A standard presentation of the proof of left-cancellation in group theory provides an example. "Left-cancellation" is the claim that for any members $a$, $b$, $c$ of a group with operation $\cdot$ and identity element $\mathbf{e}$, if $a \cdot b = a \cdot c$, then $b = c$. Here is (the core of) a proof of it: \begin{align\} a \cdot b &= a \cdot c\\ a^{-1} \cdot (a \cdot b) &= a^{-1} \cdot (a \cdot c)\\ (a^{-1} \cdot a) \cdot b &= (a^{-1} \cdot a) \cdot c\\ \mathbf{e}\cdot b &= \mathbf{e} \cdot c\\ b &= c. \end{align\} Suppose that one comes to know left-cancellation by following this sequence of steps. Is this an a priori way of getting this knowledge? Although following a mathematical proof is thought to be a paradigmatically a priori way of getting knowledge, attention to the role of visual experience here throws this into doubt. The case for claiming that the visual experience has an evidential role is as follows. The visual experience reveals not only what the steps of the argument are but also that they are valid, thereby contributing to our grounds for accepting the argument and believing its conclusion. Consider, for example, the step from the second equation to the third. The relevant background knowledge, apart from the logic of identity, is that a group operation is associative. This fact is usually represented in the form of an equation that simply relocates brackets in an obvious way: \[ x \cdot (y \cdot z) = (x \cdot y) \cdot z \] We see that relocating the brackets in accord with this format, the left-hand term of the second equation is transformed into the left-hand term of the third equation, and the same for the right-hand terms. So the visual experience plays an evidential role in our recognising as valid the step from the second equation to the third. Hence this quite standard route to knowledge of left-cancellation turns out to be a posteriori, even though it is a clear case of following a proof. Against this, one may argue that the description just given of what is going on in following the proof is not strictly correct, as follows. Exactly the same proof can be expressed in natural language, using "the composition of $x$ with $y$" for "$x \cdot y$", but the result would be hard to take in. Or the proof can be presented using a different notational convention, one which forces a quite different expression of associativity. For example, we can use the Polish convention of putting the operation symbol before the operands: instead of "$x \cdot y$" we put "$\cdot x y$". In that case associativity would be expressed in the following way, without brackets: \[ \cdot x \cdot y z = \cdot \cdot x y z. \] The equations of the proof would then need to be re-symbolised; but what is expressed by each equation after re-symbolisation and the steps from one to the next would be exactly as before. So we would be following the very same proof, step by step. But we would not be using visual experiences involved to notice the relocation of brackets this time. This suggests that the role of the different visual experiences involved in following the argument in its different guises is merely to give us access to the common reasoning: the role of the experience is merely enabling. On this account the visual experience does not strictly and literally enable us to see that any of the steps are valid; rather, recognition of (or sensitivity to) the validity of the steps results from cognitive processing at a more abstract level. Which of these rival views is correct? Does our visual experience in following the argument presented with brackets (1) reveal to us the validity of some of the steps, given the relevant background knowledge ? Or (2) merely give us access to the argument? The core of the argument against view (1) is this: Seeing the relocation of brackets is not essential to following the argument. So seeing merely gives access to the argument; it does not reveal any step to be valid. The step to this conclusion is faulty. How one follows a proof may, and in this case does, depend on how it is presented, and different ways of following a proof may be different ways of coming to know its conclusion. While seeing the relocation of brackets is not essential to all ways of following this argument, it is essential to the normal way of following the argument when it is symbolically presented with brackets in the way given above. Associativity, expressed without symbols, is this: When the binary group operation is applied twice in succession on an ordered triple of operands $\langle a, b, c\rangle$, it makes no difference whether the first application is to the initial two operands or the final two operands. While this is the content of associativity, for ease of processing associativity is almost always expressed as a symbol-manipulation rule. Visual perception is used to tell in particular cases whether the rule thus expressed is correctly implemented, in the context of prior knowledge that the rule is correct. What is going on here is a familiar division of labour in mathematical thinking. We first establish the soundness of a rule of symbol-manipulation (in terms of the governing semantic conventions--in this case the matter is trivial); then we check visually that the rule is correctly implemented. Processing at a more abstract, semantic level is often harder than processing at a purely syntactic level; it is for this reason that we often resort to symbol-manipulation techniques as proxy for reasoning directly with meanings to solve a problem. (What is six eighths divided by three fifths, without using any symbolic technique?) When we do use symbol-manipulation in proving or following a proof, visual experience is required to discern that the moves conform to permitted patterns and thus contributes to our grounds for accepting the argument. Then the way of coming to know the conclusion has an a posteriori element. ### 6.3 A non-evidential use of visual experience Must a use of visual experience in knowledge acquisition be evidential, if the visual experience is not merely enabling? Here is an example which supports a negative answer. Imagine a square or look at a drawing of one. Each of its four sides has a midpoint. Now visualize the "inner" square whose sides run between the midpoints of adjacent sides of the original square (Figure 17, left). By visualizing this figure, it should be clear that the original square is composed precisely of the inner square plus four corner triangles, each side of the inner square being the base of a corner triangle. One can now visualize the corner triangles folding over, with creases along the sides of the inner square. The starting and end states of the imagery transformation can be represented by the left and right diagrams of Figure 17. ![[The first of identical squares in size. The first has lines connecting the midpoints of each adjacent pair of sides to form another square. The second has in addition lines connecting the midpoints of opposite pairs of sides. In addition the outer square of the second has dashed lines instead of solid.]](fig17a.png) ![[The second of identical squares in size. The first has lines connecting the midpoints of each adjacent pair of sides to form another square. The second has in addition lines connecting the midpoints of opposite pairs of sides. In addition the outer square of the second has dashed lines instead of solid.]](fig17b.png) (a) (b) Figure 17 Visualizing the folding-over within the remembered frame of the original square results in an image of the original square divided into square quarters, its quadrants, and the sides of the inner square seem to be diagonals of the quadrants. Many people conclude that the corner triangles can be arranged to cover the inner square exactly, without any gap or overlap. Thence they infer that the area of the original square is twice the size of the inner square. Let us assume that the propositions concerned are about Euclidean figures. Our concern is with the visual route to the following: > The parts of a square beyond its inner square (formed by > joining midpoints of adjacent sides of the original square) can be > arranged to fit the inner square exactly, without overlap or gap, > without change of size or shape. The experience of visualizing the corner triangles folding over can lead one to this belief. But it cannot provide good evidence for it. This is because visual experience (of sight or imagination) has limited acuity and so does not enable us to discriminate between a situation in which the outer triangles fit the inner square exactly and a situation in which they fit inexactly but well enough for the mismatch to escape visual detection. (This contrasts with the case of discovering the number of vertices of a cube by seeing or visualizing one.) Even though visualizing the square, the inner square and then visualizing the corner triangles folding over is constrained by the results of earlier perceptual experience of scenes with relevant similarities, we cannot draw from it reliable information about exact equality of areas, because perception itself is not reliable about exact equalities (or exact proportions) of continuous magnitudes. Though the visual experience could not provide good evidence for the belief, it is possible that we erroneously use the experience evidentially in reaching the belief. But it is also possible, when reaching the belief in the way described, that we do not take the experience to provide evidence. A non-evidential use is more likely, if when one arrives at the belief in this way one feels fairly certain of it, while aware that visual perception and imagination have limited acuity and so cannot provide evidence for a claim of exact fit. But what could the role of the visualizing experience possibly be, if it were neither merely enabling nor evidential? One suggestion is that we already have relevant beliefs and belief-forming dispositions, and the visualizing experience could serve to bring to mind the beliefs and to activate the belief-forming dispositions (Giaquinto 2007). These beliefs and dispositions will have resulted from prior possession of cognitive resources, some subject-specific such as concepts of geometrical figures, some subject-general such as symmetry perception about perceptually salient vertical and horizontal axes. A relevant prior belief in this case might be that a square is symmetric about a diagonal. A relevant disposition might be the disposition to believe that the quadrants of a square are congruent squares upon seeing or visualizing a square with a horizontal base plus the vertical and horizontal line segments joining midpoints of its opposite sides. (These dispositions differ from ordinary perceptual dispositions to believe what we see in that they are not cancelled when we mistrust the accuracy of the visual experience.) The question whether the resulting belief would be knowledge depends on whether the belief-forming dispositions are reliable (truth-conducive) and the pre-existing belief states are states of knowledge. As these conditions can be met without any violation of epistemic rationality, the visualizing route described incompletely here can be a route to knowledge. In that case we would have an example of a use of visual experience which is integral to a way of knowing a truth, which is not merely enabling and yet not evidential. A fuller account and discussion is given in chapters 3 and 4 of Giaquinto (2007). ## 7. Further uses of visual representations There are other significant uses of visual representations in mathematics. This final section briefly presents a couple of them. Although the use of diagrams in arguments in analysis faces special dangers (as noted in 3.3), the use of diagrams to illustrate symbolically presented operations can be very helpful. Consider, for example, this pair of operations $\{ f(x) + k, f(x + k) \}$. Grasping them and the difference between them can be aided by a visual illustration; similarly for the sets $\{ f(x + k), f(x - k) \}$, $\{ \|f(x)\|, f(\|x\|) \}$, $\{ f(x)^{-1}, f^{-1}(x), f(x^{-1}) \}$. While generalization on the basis of a visual illustration is unreliable, we can use them as checks against calculation errors and overgeneralization. The same holds for properties. Consider for example, functions for which $f(-x) = f(x)$, known as even functions, and functions for which $f(-x) = -f(x)$, the odd functions: it can be helpful to have in mind the images of graphs of $y = x^2$ and $y = x^{3}$ as instances of evenness and oddness, to remind one that even functions are symmetrical about the $y$-axis and odd functions have rotation symmetry by $\pi$ about the origin. They can serve as a reminder and check against over-generalisation: any general claim true of all odd functions, for example, must be true of $y = x^{3}$ in particular. The utility of visual representations in real and complex analysis is not confined to such simple cases. Visual representations can help us grasp what motivates certain definitions and arguments, and thereby deepen our understanding. Abundant confirmation of this claim can be gathered from working through the text Visual Complex Analysis (Needham 1997). Some mathematical subjects have natural visual representations, which then give rise to a domain of mathematical entities in their own right. This is true of geometry but is also true of subjects which become algebraic in nature very quickly, such as graph theory, knot theory and braid theory. Techniques of computer graphics now enable us to use moving images. For an example of the power of kinematic visual representations to provide and increase understanding of a subject, see the first two "chapters" of the online introduction to braid theory by Ester Dalvit (2012, Other Internet Resources). With regard to proofs, a minimal kind of understanding consists in understanding each line (proposition or formula) and grasping the validity of each step to a new line from earlier lines. But we can have that stepwise grasp of proof without any idea of why it proceeds by those steps. One has a more advanced (or deeper) kind of understanding when one has the minimal understanding and a grasp of the motivating idea(s) and strategy of the proof. The point is sharply expressed by Weyl (1995 [1932]: 453), quoted in (Tappenden 2005:150) > We are not very pleased when we are forced to accept > a mathematical truth by virtue of a complicated chain of formal > conclusions and computations, which we traverse blindly, link by link, > feeling our way by touch. We want first an overview of the aim and the > road; we want to understand the idea of the proof, the deeper > context. > > Occasionally the author of a proof gives readers the desired understanding by adding commentary. But this is not always needed, as the idea of a proof is sometimes revealed in the presentation of the proof itself. Often this is done by using visual representations. An example is Fisk's proof of Chvatal's "art gallery" theorem. This theorem is the answer to a combinatorial problem in geometry. Put concretely, the problem is this. Let the $n$ walls of a single-floored gallery make a polygon. What is the smallest number of stationary guards needed to ensure that every point of the gallery wall can be seen by a guard? If the polygon is convex (all interior angles < 180deg), one guard will suffice, as guards may rotate. But if the polygon is not convex, as in Figure 18, one guard may not be enough. ![[An irregular 9 sided polygon.]](fig18.png) Figure 18 Chvatal's theorem gives the answer: for a gallery with $n$ walls, $\llcorner n/3\lrcorner$ guards suffice, where $\llcorner n/3\lrcorner$ is the greatest integer $\le n/3$. (If this does not sound to you sufficiently like a mathematical theorem, it can be restated as follows: Let $S$ be a subset of the Euclidean plane. For a subset $B$ of $S$ let us say that $B$ supervises $S$ iff for each $x \in S$ there is a $y \in B$ such that the segment $xy$ lies within $S$. Then the smallest number $f(n)$ such that every set bounded by a simple $n$-gon is supervised by a set of $f(n)$ points is at most $\llcorner n/3.\lrcorner$ Here is Steve Fisk's proof. A short induction shows that every polygon can be triangulated, i.e., non-crossing edges between non-adjacent vertices ("diagonals") can be added so that the polygon is entirely composed of non-overlapping triangles. So take any $n$-sided polygon with a fixed triangulation. Think of it as a graph, a set of vertices and connected edges, as in Figure 19. ![[10 irregularly placed black dots with a solid black line connecting them to form an irregular 10 sided polygon. One black dot has dashed lines going to four other dots that are not adjacent to it and one of its adjacent dots has dashed lines going to three other non-adjacent dots (including one dot that was the endpoint for one of the first dots dashed lines), the dashed lines do not intersect.]](fig19.png) Figure 19 The first part of the proof shows that the graph is 3-colourable, i.e., every vertex can be coloured with one of just three colours (red, white and blue, say) so that no edge connects vertices of the same colour. The argument proceeds by induction on $n \ge 3$, the number of vertices. For $n = 3$ it is trivial. Assume it holds for all $k$, where $3 \le k < n$. Let triangulated polygon $G$ have $n$ vertices. Let $u$ and $v$ be any two vertices connected by diagonal edge $uv$. The diagonal $uv$ splits $G$ into two smaller graphs, both containing $uv$. Give $u$ and $v$ different colours, say red and white, as in Figure 20. ![[Same figure as before with one of the black dots split into two red dots side-by-side and another black dot split into two white dots side-by-side. This splits the previously joined figure into two smaller graphs.]](fig20.png) Figure 20 By the inductive assumption, we may colour each of the smaller graphs with the three colours so that no edge joins vertices of the same colour, keeping fixed the colours of $u$ and $v$. Pasting together the two smaller graphs as coloured gives us a 3-colouring of the whole graph. What remains is to show that $\llcorner n/3\lrcorner$ or fewer guards can be placed on vertices so that every triangle is in the view of a guard. Let $b$, $r$ and $w$ be the number of vertices coloured blue, red and white respectively. Let $b$ be minimal in $\{b, r, w\}$. Then $b \le r$ and $b \le w$. Then $2b \le r + w$. So $3b \le b + r + w = n$. So $b \le n/3$ and so $b \le \llcorner n/3\lrcorner$. Place a guard on each blue vertex. Done. The central idea of this proof, or the proof strategy, is clear. While the actual diagrams produced here are superfluous to the proof, some visualizing enables us to grasp the central idea. ## 8. Conclusion Thinking which involves the use of seen or visualized images, which may be static or moving, is widespread in mathematical practice. Such visual thinking may constitute a non-superfluous and non-replaceable part of thinking through a specific proof. But there is a real danger of over-generalisation when using images, which we need to guard against, and in some contexts, such as real and complex analysis, the apparent soundness of a diagrammatic inference is liable to be illusory. Even when visual thinking does not contribute to proving a mathematical truth, it may enable one to discover a truth, where to discover a truth is to come to believe it in an independent, reliable and rational way. Visual thinking can also play a large role in discovering a central idea for a proof or a proof-strategy; and in discovering a kind of mathematical entity or a mathematical property. The (non-superfluous) use of visual thinking in coming to know a mathematical truth does in some cases introduce an a posteriori element into the way one comes to know it, resulting in a posteriori mathematical knowledge. This is not as revolutionary as it may sound as a truth knowable a posteriori may also be knowable a priori. More interesting is the possibility that one can acquire some mathematical knowledge in a way in which visual thinking is essential but does not contribute evidence; in this case the role of the visual thinking may be to activate one's prior cognitive resources. This opens the possibility that non-superfluous visual thinking may result in a priori knowledge of a mathematical truth. Visual thinking may contribute to understanding in more than one way. Visual illustrations may be extremely useful in providing examples and non-examples of analytic concepts, thus helping to sharpen our grasp of those concepts. Also, visual thinking accompanying a proof may deepen our understanding of the proof, giving us an awareness of the direction of the proof so that, as Hermann Weyl put it, we are not forced to traverse the steps blindly, link by link, feeling our way by touch.
james-viterbo	## 1. Life and Writings James was born in Viterbo around 1255. Nothing is known about the early years of his life. It is presumed that he joined the order of the Hermits of St. Augustine somewhere around 1272. He is first mentioned in 1283 in the capitular acts of the Roman province of his order as a recently appointed lecturer (lector) in the Augustinian Convent of Viterbo. This means that he must have spent the previous five years (i.e., 1278-83) in Paris, as the Augustinian order required its lecturers to be trained in theology in that city for a period not exceeding five years. James returned to Paris not long after his appointment in Viterbo, for he is mentioned again in the capitular acts in 1288, this time as baccalaureus parisiensis, i.e., as holding a bachelor of theology degree from the University of Paris. After completing his course of studies, James was appointed Master of Theology in 1292 (Wippel 1974) or 1293 (Ypma 1974), teaching in Paris for the next seven years. His output during that time was considerable. All of his works in speculative theology and metaphysics date from the Parisian period. In 1299-1300, he was named definitor (i.e., member of the governing council) of the Roman province by the general chapter of his order, and in May of 1300 he became regent master of the order's studium generale in Naples. These would be the last two years of his academic career, however, for in September 1302 he was named Archbishop of Benevento by Pope Boniface VIII, possibly in gratitude for the support James had shown for the papal cause in his great treatise, De regimine christiano (On Christian Rulership). In December of the same year, at the request of the Angevin King, Charles II, he was transferred to the Archbishopric of Naples. He remained in Naples until his death, in 1307 or 1308. James wrote the bulk of his philosophically relevant work while in Paris. Nothing remains of his early production in that city. His commentary on Peter Lombard's Sentences has been lost, although still extant is a so-called Abbreviatio in I Sententiarum Aegidii Romani (Summary of the First Book of the Sentences of Giles of Rome), which recent scholarship suggests might be James' preparatory notes for lectures on the Sentences rather than a summary of Giles' commentary to the same (Tavolaro 2018), although it is unclear whether these notes were prepared during James' student years in Paris or whether they date from his later teaching in Naples. Also lost are a treatise on the animation of the heavens (De animatione caelorum), as well as a commentary on Aristotle's Metaphysics (Expositio primae philosophiae), two works to which James refers explicitly in his Quodlibets and which Ypma surmises must have been written during James' four-year residency as a "formed bachelor", i.e., between 1288 and 1292 (Ypma 1975). James' most significant works written while in Paris are his four quodlibetal questions and thirty-four disputed questions on the categories as they pertain to divine matters (Quaestiones de praedicamentis in divinis), although five of them have actually nothing to do with theology or the categories, as they deal with the human will and the habits (see Ypma 1975 for a possible explanation regarding their inclusion in the Quaestiones). These two works constitute the most important sources for our knowledge of James' philosophical thought. In Naples, James wrote the work for which he is arguably best known (Gutierrez 1939: 61-2), the De regimine christiano, an important work in the history of late medieval political thought, and the only philosophical work he wrote in the last eight or so years of his life. James is not the author of two works that have been regularly attributed to him by scholars, the De perfectione specierum (Ypma 1974, Tavolaro 2014), and the Quaestiones septem de verbo (Seven Disputed Questions on the Divine Word) (Guttierez 1939, Scanzillo 1972, Ypma 1975). The author of the De perfectione specierum is James of Saint Martin (Maier 1943), or James of Naples, as he is referred to in the manuscripts, an Augustinian Hermit who was active in the late fourteenth century; and the author of the Quaestiones septem de verbo, as Stephen Dumont has recently shown (Dumont 2018b), was James' confrere and near-successor at the head of the Augustinian school in Paris, Henry of Friemar. The following table provides the list of James' authentic works with approximate dates of composition. Single asterisks mark those which have been fully edited; double asterisks indicate those for which there exist only partial editions: 1. Lectura super IV libros Sententiarum (1286-1288) 2. Quaestiones Parisius disputatae De praedicamentis in divinis\\ (1293-1295) 3. Quaestio de animatione caelorum (1293-1296) 4. Expositio primae philosophiae (1293-1295) 5. Quodlibeta quattuor\* (1293-1298) 6. Abbreviatio In Sententiarum Aegidii Romani\\ (1286-1288 or 1300-1302) 7. De regimine christiano\* (1302) 8. Summa de peccatorum distinctione\\ (1300-1308) 9. Sermones diversarum rerum\\ (1303-1308) 10. Concordantia psalmorum David (1302-1308) 11. De confessione (1302-1308) 12. De episcopali officio\* (1302-1308) ## 2. Philosophical Theology ### 2.1 Theology as a Practical Science Like many of his contemporaries, James devotes serious attention to determining the status of theology as a science and to specifying its object, or rather, as the scholastics say, its subject. In Quodlibet III, q. 1, he asks whether theology is principally a practical or a speculative science. Unsurprisingly, perhaps, for an Augustinian, James responds that the end of theology resides principally not in the knowledge but in the love of God. The love of God, informed by grace, is what distinguishes the way in which Christians worship God from the way in which pagans worship their deities. For philosophers--James has Cicero in mind--religion is a species of justice; worship is owed to God as a sign of submission. For the Christian, by contrast, there can be no worship without an internal affection of the soul, i.e., without love. James allows that there is some recognition of this fact in Book X of the Nicomachean Ethics, for the happy man would not be "most beloved of God," as Aristotle claims he is, if he did not love God by making him the object of his theorizing. In this sense, it can be said that philosophy as well sees its end as the love of God as its principal subject. But there is a difference, James contends, in the manner in which a science based on natural reason aims for the love of God and the manner in which sacred science does so: sacred science tends to the love of God more perfectly. One way in which James illustrates the difference between both approaches is by contrasting the ways in which God is the "highest" object for metaphysics and for theology. The proper subject of metaphysics is being, not God, although God is the highest being. Theology, on the other hand, views God as its subject and considers being in relation to God. Thus, James concludes, "theology is called divine or of God in a much more excellent and principal way than metaphysics, for metaphysics considers God only in relation to common being, whereas theology considers common being in relation to God" (Quodl. III, q. 1, p. 20, 370-374). Another way in which James illustrates the difference between natural theology and sacred science is by means of the distinction between the love of desire (amor concupiscientiae) and the love of friendship (amor amicitiae) to which I will return in section 7.3. James defines the love of desire as "the love of some good which we want for ourselves or for others," and the love of friendship as "the love of someone for their own sake." The love of God philosophers have in mind, James contends, is the love of desire; it cannot, by the philosophers' own admission, be the love of friendship, for according to the Magna Moralia, friendship involves a form of community or sharing between the friends that cannot possibly obtain between mere mortals and the gods. Now although James concedes that a "community of life" between God and man cannot be achieved by natural means, it is attainable through the gift of grace. The particular friendship grace affords is called charity and it is to the conferring of charity that sacred scripture is principally ordered. ### 2.2 Divine Power Like all scholastics since the early thirteenth century, James subscribes to the distinction between God's ordained power, according to which "God can only do what he preordained he would do according to wisdom and will" (Quodl. I, q. 2, p. 17, 35-37) and his absolute power, according to which he can do whatever is "doable," i.e., whatever does not imply a contradiction. Problems concerning what God can or cannot do arise only in the latter case. James considers several questions: can God add an infinite number of created species to the species already in existence (Quodl. I, q. 2)? Can he make matter exist without form (Quodl. IV, q. 1)? Can he make an accident subsist without a substrate (Quodl. II, q. 1)? Can he create the seminal reason of a rational soul in matter (Quodl. III, q. 10)? In response to the first question, James explains, following Giles of Rome but against the opinion of Godfrey of Fontaines and Henry of Ghent, that God can by his absolute power add an infinite number of created species ad superius, in the ascending order of perfection, if not in actuality, then at least in potency. God cannot, however, add even one additional species of reality ad inferius, between prime matter and pure nothingness, not because this exceeds his power but because prime matter is contiguous to nothingness and leaves, so to speak, no room for God to exercise his power (Cote 2009). James is more hesitant about the second question. He is sympathetic both to the arguments of those who deny that God can make matter subsist independently of form and to the arguments of those who claim he can. Both positions can reasonably be held, because each argues from a different (and valid) perspective. Proponents of the first position argue from the point of view of reason: because they rightly believe that God cannot make what implies a contradiction, and because they believe (rightly or wrongly) that making matter exist without form does involve a contradiction, they conclude that God cannot make matter exist without form. Proponents of the second group argue from the perspective of God's omnipotence which transcends human reason: because they rightly assume that God's power exceeds human comprehension, they conclude (rightly or wrongly) that making matter exist without form is among those things exceeding human comprehension that God can make come to pass. Another question James considers is whether God can make an accident subsist without a subject or substrate. The question arises only with respect to what he calls "absolute accidents," namely quantity and quality, as opposed to relational accidents--the remaining categories of accident. God clearly cannot make relational accidents exist without a subject in which they inhere, for this would entail a contradiction. This is so because relations for James, as we will see in section 3.3 below, are modes, not things. What about absolute accidents? As a Catholic theologian, James is committed to the view that some quantities and qualities can subsist without a subject, for instance extension and color, a view for which he attempts to provide a philosophical justification. His position, in a nutshell, is that accidents are capable of existing independently if they are thing-like (dicunt rem). Numbers, place (locus), and time are not thing-like and are thus not capable of independent existence; extension, however, is and so can be made to exist without a subject. The same reasoning applies to quality. This is somewhat surprising, for according to the traditional account of the Eucharist, whereas extension may exist without a subject, the qualities, color, odor, texture, necessarily cannot; they inhere in the extension. James, however, holds that just as God can make thing-like quantities to exist without a subject, so too must he be able to make a thing-like quality exist without the subject in which it inheres. Just which qualities are capable of existing without a subject is determined by whether or not they are "modes of being," i.e., by whether or not they are relational. This seems to be the case with health and shape: health is a proportion of the humors, and so, relational; likewise, shape is related to parts of quantity, without which, therefore, it cannot exist. Colors and weight, by contrast, are non-relational, according to James, and are thus in principle capable of being made to exist without a subject. The fourth question James considers in relation to God's omnipotence raises the interesting problem of whether the rational soul can come from matter. James proceeds carefully, claiming not to provide a definitive solution but merely to investigate the issue (non determinando sed investigando). The upshot of the investigation is that although there are many good reasons (the soul's immortality, its spirituality and its per se existence) to say that God cannot produce the seminal reason of the rational soul in matter, in the end, James decides, with the help of Augustine, that such a possibility must be open to God. Thus, it is true in the order which God has de facto instituted, that the soul's incorruptibility is repugnant to matter, but this is not so in absolute terms: if God can miraculously cause something to come to existence through generation and confer immortality upon it (James is presumably thinking of Christ), then he can make it come to pass that souls are produced through generation without being subject to corruption. Likewise, although it appears inconceivable that something material could generate something endowed with per se existence, it is not impossible absolutely speaking: if God can confer separate existence upon an accident--despite the fact that accidents naturally inhere in their substrates--then, in like manner, he can confer separate existence upon a soul, although it has a seminal reason in matter. ### 2.3 Divine Ideas The scholastics held that because God is the creative cause of all natural beings, he must possess the ideas corresponding to each of his creatures. But because God is eternal and is not subject to change, the ideas must be eternally present in him, although creatures exist for only a finite period of time. This doctrine of course raised many difficulties, which each author addressed with varying degrees of success. One difficulty had to do with reconciling the multiplicity of ideas with God's unity: since there are many species of being, there must be a corresponding number of ideas; but God is one and, hence, cannot contain any multiplicity. Another, directly related, difficulty had to do with the ontological status of ideas: do ideas have any reality apart from God? If one denied them any kind of reality, it was hard to see how they could function as exemplar causes of things; but to assign too much reality to them was to run the risk of introducing multiplicity in God and undermining the ex nihilo character of creation. One influential solution to these difficulties was provided by Thomas Aquinas, who argued that divine ideas are nothing else but the diverse ways in which God's essence is capable of being imitated, so that God knows the ideas of things by knowing his essence. Ideas are not distinct from God's essence, though they are distinct from the essences of the things God creates (De veritate, q. 2, a. 3). One can discern two answers to the problem of divine ideas in the works of James of Viterbo. At an early stage of his career, in the Abbreviatio in Sententiarum Aegidii Romani--assuming one accepts, as seems reasonable, the early dating suggested by Ypma (1975)--James defends a position that is almost identical to that of Thomas Aquinas (Giustiniani 1979). In his Quodlibeta, however, he defends a position that is closer to that of Henry of Ghent. In the following I will sketch James' position in the Quodlibeta as it provides the most mature statement of his views. For detailed discussions, see Gossiaux (2007) and Cote (2018). Although James agreed with the notion that ideas are to be viewed as the differing ways in which God can be imitated, he did not think that one could make sense of the claim that God knows other things by cognizing his own essence unless one supposed that the essences of those things preexist in some way (aliquo modo) in God. James' solution is to distinguish two ways in which ideas are in God's intellect. They are in God's intellect, firstly, as identical with it, and, secondly, as distinct from it. The first mode of being is necessary as a means of acknowledging God's unity; but the second mode of being is just as necessary, for, as James puts it (Quodl. I, q. 5, p. 64, 65-67), "if God knows creatures before they exist, even insofar as they are other than him and distinct (from him), that which he knows is a cognized object, which must needs be something; for that which nowise exists and is absolutely nothing cannot be understood." But James also thinks that the necessity of positing distinct ideas in God follows from a consideration of God's essence. God enjoys the highest degree of nobility and goodness. His mode of knowledge must be commensurate with his nature. But according to Proclus, an author James is quite fond of quoting, the highest form of knowledge is knowledge through a thing's cause. That means that God knows things through his own essence. More precisely, he knows things by knowing his essence as the cause of those things, knowledge that is distinct in James's view from God's mere knowledge of himself. Although James' insistence on the distinctness of ideas with respect to God's essence is reminiscent of Henry of Ghent's teaching, it is important to note, as has been stressed by M. Gossiaux (2007), that James does not conceive of this distinctness as Henry does. For Henry, ideas possess "being of essence" (esse essentiae); James, by contrast, while referring to divine ideas as things (res), is careful to add that they are not things "in the absolute sense but only in a qualified sense," viz., as cognized objects (Quodl. I, q. 5, p. 63, 60). Thus, divine ideas for James possess a lesser degree of distinction from God's essence than do Henry of Ghent's. Nevertheless, because James did consider ideas to be distinct in some sense from God, his position would be viewed by some later authors--e.g., William of Alnwick--as compromising divine unity. (See Cote 2016) ## 3. Metaphysics ### 3.1 Analogy of Being The concept of being, all the medievals agreed, is common. What was debated was the nature of this commonness. According to James of Viterbo, all commonness is founded on some agreement, and this agreement can be either merely nominal or grounded in reality. Agreement is nominal when the same name is predicated of wholly different things, without there being any objective basis for the application of the common name; such is the case of equivocal names. Agreement is real in the following two cases: (1) if it is based on some essential resemblance between the many things to which a particular concept applies, in which case the concept applies to these many things by virtue of the self-same ratio and is said of them univocally; or (2) if that concept is truly common to the many things of which it is said, although it is not said of them in relation to the same nature (ratio), but rather it is said in a prior sense of one and in a posterior sense of others, insofar as they are related in a certain way to the first. A concept that is predicated of things in this way is said to be analogous, and the agreement displayed by the things to which it applies is said to be an agreement of attribution (convenientia attributionis). James believes that it is according to this sense of analogy that being is said of God and creatures, and of substance and accident (Quaestiones de divinis praedicamentis I, q. 1, p. 25, 674-80). For being is said in a prior sense of God and in a posterior sense of creatures by virtue of a certain relation between the two; likewise, being is said first of substance and secondarily of accidents, on account of the relation of posteriority accidents have to substance. The reason why being is said in a prior sense of God and in a secondary sense of creatures and, hence, the reason why the 'ratio' or nature of being is different in the two cases is that being, in God, is "the very thing which God is" (Quaestiones de divinis praedicamentis, q. 1, p. 16, 412), whereas created being is only being through something added to it. From this first difference follows a second, namely, that created being is being by virtue of being related to an agent, whereas uncreated being has no relation. These two differences can be summarized by saying that divine being is being through itself (per se), whereas created being is being through another (per aliud) (Quaestiones de divinis praedicamentis, q. 1, p. 16, 425-6). In sum, being is said of God and creature, but according to a different ratio: it is said of God according to the proper and perfect nature of being, but of creatures in a derivative or secondary way. ### 3.2 The Distinction of Being and Essence The question of how being and essence are related to each other, and in particular whether they are really identical or not, attracted considerable interest in the last quarter of the thirteenth century, with all major masters devoting some discussion to it. James of Viterbo is no exception. Drawing his inspiration from Anselm's semantics, he attempts to articulate a compromise solution (Gossiaux 2018) between the main competing solutions on offer in his day. James' most detailed discussion of the distinction between being and essence occurs in the context of a question that asks whether creation could be saved if being (esse) and essence were not different (Quodl. I, q. 4). His answer is that although he finds it difficult to see how one could account for creation if being and essence were not really different, he does not believe it is necessary to conceive of the real distinction in the way in which "certain Doctors" do. Which Doctors does he have in mind? In Quodl. I, q. 4, he summarizes the views of three authors: Godfrey of Fontaines, according to whom the distinction is only conceptual (secundum rationem); Henry of Ghent, for whom esse is only intentionally different from essence, a distinction that is less than a real distinction but greater than a verbal distinction; and finally, Giles of Rome, for whom esse is one thing (res), and essence another. Thus, James agrees with Giles, and disagrees with Henry and Godfrey, that the distinction between being and essence is real; however, he disagrees with Giles about the proper way of understanding the real distinction. The starting point of his analysis is Anselm's statement in the Monologion that the substantive lux (light), the infinitive lucere (to-emit-light), and the present participle lucens (emitting light) are related to each other in the same way as essentia (essence), esse (to be), and ens (being). The relation of lucere to lux, James tells us, is the relation of a concrete term to an abstract one; but this is just the relation that obtains between being and essence. Now, a concrete term signifies more things than the corresponding abstract term; for an abstract term signifies only a form, whereas the concrete term signifies the form and the "subject", i.e., the bearer of the form; it is said to signify the former principally, and the latter secondarily. So it is in the case of being and essence: esse signifies the form (principally) and the subject (secondarily), while essence signifies the form only. What is taken for granted in this analysis is that the distinction between a form and its bearer is a real one, at least in creatures. This is how James achieves his compromise solution: pace Godfrey of Fontaines and Henry of Ghent there is a real distinction (that between the subject and the form) but pace Giles, it is real only in qualified sense, since being principally signifies the same thing as essence. ### 3.3 Relations James devotes five of his Quaestiones de divinis praedicamentis (qq. 11-15), representing some 270 pages of edited text, to the question of relations. It is with a view to providing a proper account of divine relations, he explains, that it is "necessary to examine the nature of relation with such diligence" (Quaestiones de divinis praedicamentis, q. 11, p. 12, 300-301). But before turning to Trinitarian relations, James devotes the whole of q.11 to the status of relations in general. The following account focuses exclusively on q. 11. James in essence adopts Henry of Ghent's "modalist" solution, which was to exercise considerable influence among late thirteenth-century thinkers (Henninger 1989), although he disagrees with Henry about the proper way of understanding what a mode is. The question boils down to whether relations exist in some manner in extra-mental reality or solely through the operation of the intellect, like second intentions (species and genera). Many arguments can be adduced in support of each position, as Simplicius had already shown in his commentary on Aristotle's Categories--a work that would have a decisive influence on James' thought. For instance, in support of the view that relations are not real, one may point out that the intellect is able to apprehend relations between existents and non-existents, e.g., the relation between a father and his deceased son; yet, there cannot be anything real in the relation given that one of the two relata is a non-existent. But if so, then the same must be true of all relations, as the intellectual operation involved is the same in all cases. Another argument concerns the way in which relations come to be and cease to be. This appears to happen without any change taking place in the subject which the relation is said to affect. For instance, a child who has lost his mother is said to be an orphan until the age of eighteen, at which point it ceases to be one, although no change has occurred: "the relation recedes or ceases by reason of the mere passage of time." But good reasons can also be found in support of the opposing view. For one, Aristotle clearly considers relations to be real, as they constitute one of the ten categories that apply to things outside the soul. Furthermore, according to a view commonly held by the scholastics, the perfection of the universe cannot consist solely of the perfection of the individual things of which it is made; it is also determined by the relations those things have to each other; hence, those relations must be real. The correct solution to the question of whether relations are real or not, James contends, depends on assigning to a given relation no more but no less reality than is fitting to it. Those who rely on arguments such as the first two above to infer that relations are entirely devoid of reality are guilty of assigning relations too little reality; those who appeal to arguments such as the last two, showing that relations are distinct from their subjects in the way in which things are distinct from each other, assign too great a degree of reality to relations. The correct view must lie somewhere in between: relations are real, but are not distinct from their subjects in the way one thing is distinct from another. That they must be real is sufficiently shown by the first Simplician arguments mentioned above, to which James adds some others of his own. However, showing that they are not things is slightly more complicated. James' position, in fact, is that relations are not things "properly and absolutely speaking," but only "in a certain way according to a less proper way of speaking." A relation is not a thing in an absolute sense because of the "meekness" of its being, for which reason "it is like a middle point between being and non-being" (Quaestiones de divinis praedicamentis, q. 11, p. 30, 668-9). The reasoning behind this last statement is as follows: the more intrinsic some principle is to a thing, the more that thing is said to be through it; what is maximally intrinsic to a thing is its substance; a thing is therefore maximally said to be on account of its substance. Now a thing's being related to another is, in the constellation of accidents that qualify that thing, what is minimally intrinsic to it and thus farthest from its being, and so closest to non-being. But if relations are not things, at least in the absolute sense, what are they? James answers that they are modes of being of their foundations. "The mode of being of a thing does not differ from the thing in such a way as to constitute another essence or thing. The relation, therefore, is not different from its foundation" (Quaestiones de divinis praedicamentis, q. 11, p. 33, 745-7). Speaking of relations as modes allows us to acknowledge their reality, as attested by experience, without hypostasizing them. A certain number's being equal to another is clearly something distinct from the number itself. The number and its being equal are two "somethings" (aliqua), says James; they are not, however, two things; they are two in the sense that one is a thing (the number) and the other is a mode of being of the number. In making relations modes of being of the foundation, James was clearly taking his cue from Henry of Ghent, "the chief representative of the modalist theory of relation" (Henninger 1989). For Henry and James, relations are real in the sense that they are distinct from their foundations and belong to extra mental reality. However, James' understanding of the way in which a relation is a mode differs from Henry's. For Henry, a thing's mode is the same thing as its ratio or nature; it is the particular type of being that thing has, what "specifies" it. But according to James' understanding of the term, a mode lies beyond the ratio of a thing, like an accident of that thing (Quaestiones de divinis praedicamentis, p. 34, 767-8). In conclusion, one could say that in his discussion of relations, James was guided by the same motivation as many of his contemporaries, namely securing the objectivity of relations without conferring full-blooded existence upon them. Relations do exhibit some form of being, James believed, but it is a most faint one (debilissimum), the existence of a mode qua accident. ### 3.4 Individuation James discusses individuation in two places: Quodl. I, q. 21 and Quodl. II, q. 1. I will focus on the first treatment, because it is the lengthier of the two and because the tenor of James' brief remarks on individuation in Quodl. II, q. 1, despite certain similarities with his earlier discussion (Wippel 1994), make it hard to see how they fit into an overall theory of individuation. The question James faces in Quodl. I, q. 21 is a markedly theological one, namely whether, if the soul were to take on the ashes or the matter of another man at resurrection, the resulting individual would be the same as before resurrection. In order to answer that question, James tells us, it is first necessary to determine what the cause of numerical unity is in the case of composite beings. There have been numerous answers to that question and James provides a short account of each. Some philosophers have appealed to quantity as the principle of numerical unity; others to matter; others yet to matter as subtending indeterminate dimensions; finally, others have turned to form as the cause of individuation. According to James, each of these answers is part of the correct explanation though each is insufficient if taken on its own. The correct view, according to him, is that form and matter taken together are the principal causes of numerical identity in the composite, with quantity contributing something "in a certain manner." Form and matter, however, are principal causes in different ways; more precisely, each accounts for a different kind of numerical unity. For by 'singularity' we can really mean two distinct things: we can mean the mere fact of something's being this or that singular thing, or we can point to a thing qua "something complete and perfect within a certain species" (Quodl. I, 21, 227, 134-35). It is matter that accounts for the first kind of singularity, and form for the second. Put otherwise, the kind of unity that accrues to a thing on account of its being a mere singular results from the concurrence of the "substantial" unity provided by matter and the "accidental" unity provided by quantity. By contrast, the unity that characterizes a thing by virtue of the perfection or completeness it displays is conferred to it by the form, which is the principle of perfection and actuality in composites. Although James thinks he can quite legitimately enlist the support of such prestigious authorities as Aristotle and Averroes in favor of the view that matter and form together are constitutive of a thing's numerical unity, his solution has struck commentators as a somewhat contrived and ad hoc attempt to reach a compromise solution at all costs (Pickave 2007; Wippel 1994). James, it has been suggested, "seems to be driven by the desire to offer a compromise position with which everyone can to some extent agree" (Pickave 2007: 55). Such a suggestion does accord with James' oft-expressed preference for solutions that present a "middle way" (media via) among competing theories (Quaestiones de divinis praedicamentis, q. 11, p. 23, 513; Quodl. II, q. 7, p. 108, 118; De regimine christiano, 210; see also Quodl. II, q. 5, p. 65, 208-209), although these professions of moderation must sometimes be taken with a grain of salt, as we will see in Section 8 below. ## 4. Natural Philosophy (The Doctrine of Seminal Reasons) The belief that matter contains the 'seeds' of all the forms that can possibly accrue to it is one of the hallmarks of James of Viterbo's thought, as is the belief that the soul pre-contains, in the shape of "fitnesses" (idoneitates), all the sensitive, intellective, and volitional forms it is able to take on, though, as we will see, there is an important difference between both cases. I will present James' doctrine of "fitnesses" in the intellect in Section 6, and his doctrine of fitnesses in the will in Section 6. In this section, I review James' arguments in favor of seminal reasons (For a full account see Pickave and Cote 2018). James takes as the point of departure of his analysis of change the view commonly attributed to Aristotle by the medievals according to which substantial change involves a natural agent educing a form from the potency of matter. His contention is that for the form to be educed from prime matter it has to preexist in prime matter in an "incipient or inchoate" state. Otherwise the forms would have to be put into matter by an external cause. That cause could only be a natural agent or a supernatural one. It cannot be the latter, for then change would no longer be natural; but nor can it be the former because James holds that forms do not "migrate" from one substance to another. Hence the forms must preexist in matter. James holds that the inchoate form present in matter is the same as the full-fledged, actualized form, differing from it only modally (Quodl. II, q. 5, p. 70, 386-388). To the objection that if this were true nothing "new" would result from the process of change, he responded by pointing out that the assumption that natural change results in the emergence of new things is rejected by no less an authority than Averroes himself, who denies that natural agents "induce something extrinsic in matter" (Quodl. II, q. 5, p. 77, 621). What newness does result from natural change is accounted for by the modal difference between the potential and the actualized form. James holds that natural change requires two active principles: that of the potential form existing in potency in matter, i.e., the seminal reason, and that of the extrinsic natural agent acting on the matter. He explicitly denies that the potential form on its own is a sufficient cause of change (Quodl. II, q. 5, p. 89, 1012-1014). The first active principle is that of the inchoate form itself: it is active "by means of inclination", that is, it is active inasmuch as it naturally tends to its actualization. The second active principle is the extrinsic agent that educes the form from matter; it is active by means of transmutation, i.e., by efficiently causing the change by virtue of the power of acting (virtus agendi) conferred upon it by God. Both causes cooperate in the production of the effect. Although James teaches that there are preexisting ideas and volitions in the soul similarly to the way in which seminal reasons preexist in matter, there is an important difference between the two cases. For the "inclining principles" in matter, he explains in Quodl. III, q. 4 (p. 70, 416-424), stand farther from their actualization than do the inclining principles in the soul. They thus require more on the part of the extrinsic transmutative cause than do the soul's inclining principles. So much so, James explains, that it sometimes appears that all the work is being done by the extrinsic agent. This is not so, of course, as the "intrinsic inclining principle" also plays a role. Because James assigns such an important role to external agent causes in his account of natural change, one of the common complaints directed against theories of seminal reasons, namely that they deny the reality of secondary causes, clearly does not apply to his version of the theory. James' doctrine of seminal reasons would elicit considerable criticism in the early fourteenth century and beyond (Phelps 1980). The initial reaction came from Dominicans. Bernard of Auvergne wrote a series of Impugnationes (i.e., refutations) contra Jacobum de Viterbio (see Pattin 1962 and Cote 2016), attacking various aspects of James' metaphysics, including his theory of seminal reasons; and John of Naples later argued against James' distinction between the potency of matter and matter tout court. James' theory also encountered resistance from within the Augustinian Order, e.g., from Alphonsus Vargas of Toledo. Even arts masters entered the fray. The Milanese arts master Maino de Maineri devoted a lengthy section of one of his questions in his question-commentary on Averroes' De substantia orbis dating from 1315-1317 to a presentation and rebuttal of James' theory of natural change (see Cote 2019). ## 5. The Soul and Its Powers According to Aristotle in the De anima," the soul is in a sense all things." James wants to infer from this that all things must somehow preexist in the soul "by a certain conformity and resemblance." (Quodl. I, q. 7, p. 91, 403). James distinguished between three sorts of conformity: the conformity between the sense and the sensibles, that between the intellect and intelligibles, and that between the will and appetibles. He thus believed that all sensibles must preexist in the sense-power, all intelligibles in the intellective power, and all "appetibles" in the appetitive power, i.e., the will. They do not preexist in their fully actualized state, of course; but nor do they preexist as purely indeterminate capacities: James holds that they preexist as "incomplete actualities," innate and permanent qualities of the soul tending toward their actualization. Borrowing from Simplicius' commentary on Aristotle's Categories, James described a power of the soul as a "general (communis) aptitude (aptitudo) or a fitness (idoneitas) with respect to the complete act" (Quodl. I, q. 7, p. 92, 421) that is divided into further specific "fitnesses" (speciales idoneitates), "following the diversity of objects" of that power. For instance, intellect is a general idoneitas that is subdivided in specific idoneitates "following the diversity" of intelligibles. Despite what the phrase "following the diversity" would have us believe, James did not assert that the division of fitnesses in the soul exactly mapped the division of kinds of objects. Though he was clearly committed to some correspondence between the two, he believed it was not possible in this life to know how far the division into specific "fitnesses" proceeded, and hence how many "fitnesses" there are in the soul. In addition to explaining how the soul's powers relate to intelligibles, sensibles and appetibles, James also described their relation to occurrent acts of understanding, sensing, and willing. As with seminal reasons, an aptitude in potency in the soul and the corresponding actualized aptitude were not viewed as really but only as modally distinct (Quodl. III, q. 5, p. 84, 62-63). James also held that each power was both passive and active in relation to its actualization: passive inasmuch as a power qua power is not yet actualized, active insofar as it tends toward its actualization (Quodl. I, q. 12, p. 165, 281-285). To the argument that nothing could be both active and passive in the same respect, James responded that this was true only of transeunt actions, such as fire heating a pot, which require the active cause of change to be distinct from the passive recipient of change, not of immanent actions, as are the operations of the soul. Although James held that all the soul's powers were active, they were not so to the same degree: the will and its aptitudes were considered to be more active and thus "closer" to their actualization than those of the intellect; and the intellect and its aptitudes, in turn, were viewed as more active and closer to their actualization than those of the senses. Accordingly, James considered that the more active an aptitude or "inclining principle" was, the more causal power it had and the less causal input it required from other sources. From the foregoing it is easy to see what position James would take in what was a commonly discussed topic in the thirteenth century, namely the problem whether or not the "essence" of the soul was really different from its powers. The position espoused by the scholastics whose teachings James studies most carefully, namely Thomas Aquinas, Giles of Rome, and Godfrey of Fontaines was the soul was indeed really distinct from its powers. There was, however, a commonly discussed minority position, one that eschewed both real distinction and strict identity (which had few followers): that of Henry of Ghent. Henry believed that the powers of the soul were "intentionally," not really distinct from its essence. James, however, sided with Thomas, Giles, and Godfrey, against Henry (Quodl. II, q. 14, p. 160, 70-71; Quodl. III, q. 5, 56-84,63). His reasoning was as follows. Given that everyone agreed that there was a real distinction between the essence of the soul and one of its powers in act, that is, between the soul and, e.g., an occurrent act of willing, then if one denied that there was a real distinction between the soul and its powers, as Henry had, one would be committed to the existence of a real distinction between the power in act (the occurrent act of willing) and that same power in potency (the will qua capable of produce that act). But as we have already seen, James believed that something in potency is not really distinct from that same thing in act. Hence the soul's essence must be distinct from its powers. ## 6. Cognition James' longest discussion of cognition occurs in question 12 of his first Quodlibet, which asks whether it is necessary to posit an agent intellect in the human soul that is distinct from the possible intellect (for discussion see Cote 2013, and Solere 2018a and 2018b). Since the view that there is such a distinction was an important component of abstraction theories of knowledge, what was really at stake was whether abstraction theory provided the correct account of knowledge acquisition. According to abstraction theory, of which Aquinas was one of the most famous exponents, the mind derives its content from an engagement with sense-images (phantasms). The agent intellect, a "part" of the intellectual soul, was held to abstract intelligible species from the sense-images, and these species were then "received" by the possible intellect. Since the same power could not be both active and passive, the two intellects had to be distinct. As can be gathered from the preceding section, James would not be favorably disposed to such a theory. Since intelligibles, for him, preexisted in the soul in the form of "aptitudes" or "fitnesses," there was no need for them to be abstracted; and since the main reason for positing an agent intellect was to perform abstraction, there was no reason to suppose that such a faculty existed -- which is exactly the position he defended in question 12. James' theory of innate idoneitates in the soul thus entailed a wholesale rejection of abstraction theories of knowledge and of the model of the soul on which they rested (Cote 2013). James was well aware that by denying the distinction between the two intellects, he was opposing the consensus view of Aristotle commentators. Indeed, his views seem to run counter to the De anima itself, though, as he would mischievously point out, it was difficult to determine just what Aristotle's doctrine was, so obscure was its formulation (Quodl. I, q. 12, p. 169, 426--170, 439). He explained that what he was denying was not that there was a "difference" in the intellect--the soul's powers did exhibit such a difference since they were both active and passive--rather, what he was denying was that's the existence of a difference implied a real distinction of powers (Quodl. I, q. 12, p. 170, 440-45). As far as James was concerned the relevant question was not how intelligibles are abstracted so as to produce occurrent acts of cognition, but rather how innate intellectual aptitudes can develop into occurrent acts of cognition. The key to his solution lay in his view that aptitudes actively tend toward their completion. James described such active inclination as a kind of self-motion--formal self-motion-which he viewed as the main created causal contributor to a power's actualization. But of course "main causal contributor" does not mean "sole causal contributor.;" Although the soul's powers stand closer to their actualization and exercise greater formal self-motion than do the seminal reasons in matter, and although the intellect and its aptitudes likewise stand closer to their actualization and exercise greater self-motion than do the senses and their aptitudes, no "fitness" on its own is sufficient to bring about its actualization (Quodl. I, q. 7, p. 102, 777-778; something further is required. In the case of the senses, what is required is the "excitative" causality (excitatio) exercised by the physical change in the organ of sense; in the case of the intellect, it is the excitative causality supplied by the phantasm. In both cases, James held that the power's formal self-motion together with the "rousing" action stemming from the organ or the phantasm necessarily entailed their effect. James of Viterbo is thus part of a group of thinkers in the history of philosophy for whom the intellect itself, as opposed to extra-mental objects or their proxies in the soul, is the main causal factor not only of the production of acts of knowledge but of their conceptual content as well. Indeed, for James, that content is already present in the soul in the form of general and specific "fitnesses" or "aptitudes." The aptitudes are innate (naturaliter inditae, "always present in [the soul]" (Quodl. I, q. 7, p. 92, 422-423), ready to be actualized in the presence of the appropriate triggering factors. Few other scholastics were ready to espouse such an extreme form of innatism, which some scholars have likened to the theory innate ideas of the Early Moderns (Solere 2018a). ## 7. Ethics ### 7.1 Freedom of the Will The scholastics all agreed that the human will is free. They also agreed that the will and the intellect both played a role in the genesis of the voluntary act. What they disagreed about was which of the two faculties had the decisive role. For Henry of Ghent, the will was the sole cause of its free acts (Quodl. I, q. 17), so much so that he tended to relegate the intellect's role to that of a "sine qua non" cause. At the other end of the spectrum of opinions, Godfrey of Fontaines held that the will is always passive in regard to the intellect and that it is the intellect that exercises the decisive motion in producing the voluntary act (Quodl. III, q. 16). Although James of Viterbo used language apt to suggest that he wanted to steer a middle course between Henry and Godfrey (Quodl. II, q. 7), his preferences clearly lay with a position like that of Henry's, as can be gathered from his most detailed treatment of the question in Quodl. I, q. 7 (see Dumont 2018a for a complete treatment of the issue in James). James' thesis in Quodl. I, q. 7 was that the will both moves itself and is moved by another. An agent that is moved by another can be said to be free as long as it also moves itself in some way. The human will is just such an agent: it is moved by itself; but it is also moved by another, namely by the object of willing outside the soul, and it is moved by that same object insofar as it is apprehended by the intellect. To explain how this was so, James used Aquinas' distinction between the specification of an act of will (the will's willing this as opposed to that) and the exercise of the act (the will's actually willing this). The will formally moves itself in regard to the exercise of its act; it is moved by the object outside the soul in regard to the specification of its act; and it is moved by the intellect, or more precisely by the object as apprehended by the intellect, both in regard to the specification and the exercise of its act. James called the motion by which the will was specified by the object as apprehended by the intellect "metaphorical" motion, as distinct from "proper" motion. Final causes were metaphorical movers in this sense. James called the motion by which the will is moved to its exercise by the object as apprehended by the intellect efficient motion. This is prima facie a surprising move given what we know about James' theory of the soul as a self-mover. For even though James was keen to show that the will cannot, so to speak, "go it alone," did not making the object as understood by the intellect the efficient cause of the will's exercise to act risk tilting the causal balance too far in the direction of the intellect? But in fact if the intellect was an efficient cause, it wasn't so in the usual sense of the word "efficient." James claimed to follow Aristotle in Metaphysics in distinguishing two ways in which something could be moved efficiently. The first was through direct efficient causation; in this sense, only God was the efficient cause of the will's motion. The second sense was through "a certain connection" that obtains between two things that are rooted in the same subject. It is this second sense that is relevant for understanding how the intellect moves the will. As James understood the matter, "because the will and the intellect are rooted and connected in the same essence of the soul, it follows that when the soul is in actuality in relation to the intellect, there arises (fit) in the soul a certain inclination such that it becomes in actuality in relation to the will, with the result that (the will) moves itself." It is, he concluded, "on account of that inclination that the intellect moves the will;" and the motion by which it does so can be termed a "rousing up" (excitatio) ( Quodl. I, q. 7, p. 104, 849-855). Thus, despite what the language of "efficiency" might suggest, James' account of how the intellect moves the will is similar to his account of how the sense organ moves the power of sense and how the imagination moves the intellect: in all three cases the motion is of the "excitatory" variety; the power itself as the formal cause of its acts supplies the lion's share of the voluntary act's causality. If anything, the will itself can be considered to be an efficient cause (in the first sense of the word), if only per accidens. This is because the will, like heavy objects but unlike the intellect and the senses, belongs to the class of things that both move themselves formally and move others efficiently. Thus, just as a heavy object is formally self-moved as a result of its heaviness but moves another efficiently inasmuch as it divides the medium it traverses, so too the will is a self-mover that efficiently moves the intellect and the other faculties, and hence can be said to move itself per accidens (Quodl. I, q. 7, p. 98, 633-640). What is unclear is how any of the above makes the will's volitions free. After all, by James' own admission, the senses and the intellect are also formal self-movers, and yet they are not free. The difference is that whereas acts of sensation and intellection, given the self-motion of the senses and the intellect, will necessarily follow upon the motions of the organs of sense and the motion of the imagination, no act of willing according to James is necessitated by the efficient causality (in James' sense of the word) of the intellect. Thus, although any act of willing must be preceded by an act of the intellect, no act of the intellect necessarily entails a particular act of willing. This makes the will the only power of the soul that is essentially free, though some other powers may be called free "by participation," insofar as they are apt to be moved by the will. (Quodl. I, q. 7, p. 107, 935-944) ### 7.2 Connection of the Virtues Like Albert the Great and Thomas Aquinas, James of Viterbo holds that the moral virtues, considered as habits, i.e., virtuous dispositions or acts, are "connected." In other words, he believes that one cannot have one of the virtues without having the others as well. The virtues he has in mind are what he calls the "purely" moral virtues, that is, courage, justice, and temperance, which he distinguishes from prudence, which is a partly moral, partly intellectual virtue. In his discussion in Quodl. II, q. 17 James begins by granting that the question is difficult and proceeds to expound Aristotle's solution to it, which he will ultimately adopt. As James sees it, Aristotle proves in Nicomachean Ethics VI the connection of the purely moral virtues by showing their necessary relation to prudence, and this is to show that just as moral virtue cannot be had without prudence, prudence cannot be had without moral virtue. The connection of the purely moral virtues follows from this: they are necessarily connected because (1) each is connected to prudence and (2) prudence is connected to the virtues (Quodl. II, q. 17, p. 187, 436 - p. 188, 441). ### 7.3 Love of Self vs. Love of God Although there has never been any doubt among medieval theologians that man ought to love God more than he loves himself, medieval theologians did disagree about whether man can fulfill this obligation by purely natural means or whether he requires grace in order to do so. The consensus in the thirteenth century, shared, among other authors, by Thomas Aquinas, Giles of Rome, and Godfrey of Fontaines, was that although perfect love of God is possible only through grace, man does have the natural capacity to love God more than himself (See Osborne 2005 for discussion of the topic in the thirteenth century, including in James of Viterbo). Many of these authors believed that granting man such a capacity was necessary in order to safeguard the universally accepted principle that "grace perfects, not destroys, nature". For if man is naturally inclined to love himself more than God and is made to love God more than himself only through grace, then that natural inclination is not so much perfected as replaced by a different one. Against these authors, James of Viterbo in his Quodlibet II, question 20 famously defended the view that man naturally loves himself more than God. Before stating James' argument in support of this position, it is important to be attentive to the precise way in which he formulated the question, as well as to how he understands the term "love" (diligere). First, the question James raised in Quodl. II, q. 20, was not (a): "Does man naturally love God more than he loves himself?" or (b): "Does man naturally love himself more than God," but rather (c): "Does man naturally love God more than himself, or the converse (vel econverso)?" What (c), but not (a) or (b), rules out is that man can love himself and God equally. James did not think that this was so. He compared the case of self-love vs. love of God to the distinction between natural and supernatural knowledge: "Natural knowledge starts from creatures and proceeds thence to God; contrariwise, supernatural knowledge begins with God and proceeds thence to creatures." Since the distinction between natural and supernatural knowledge is exhaustive and mutually exclusive, there is no other kind of knowledge, and hence none that would proceed "equally" from creatures and God. But for the comparison between knowledge and love to hold, the conclusion that applies to the former must apply to the latter, and that conclusion is that if man does not love God more than himself, then he loves himself more than God; and conversely. Second, following the practice of many scholastics, James distinguished between two forms of love: the "love of desire" or "love of covetousness" (amor concupiscentiae), which he defined as the "love of some good which we want for ourselves or for others," and the love of friendship (amor amicitiae), or the love of someone for their own sake. Although James believed that rational creatures love God in both these ways, he was clear that the debate about whether or not man loves himself more than God or the converse concerned only the love of friendship; the debate, then, was about whether man naturally loves himself for his own sake more than he loves God for his own sake--which was the view James defended--or the converse. James offered two arguments to support his position. The first was based on the principle that the mode of natural love must be commensurate with the mode of being and, hence, of the mode of being one--since everything that is is one. Now a thing is one with itself by virtue of numerical identity, but it is one with something else by virtue of a certain conformity. For instance, Socrates is one with himself by virtue of his being Socrates, but he is one with Plato by virtue of the fact that both share the same form. But the being something has by virtue of numerical identity is "greater" than the being it has by reason of something it shares with another. And given that the species of natural love follows the mode of being, it follows that it is more perfect to love oneself than to love another (Quodl. II, q. 20, p. 206, 148 - p. 149, 165). The second argument attempted to derive the same conclusion from the principle that "God's love of charity elevates [human] nature's" love of God ( Quodl. II, q. 20, p. 207, 166-167). James reasoned that there was only one way in which charity could elevate nature, namely by making it love God above all else. But if this was so, it had to follow "that nature in itself cannot love God in this way (...). For if it could, it would not need to be elevated by charity." (Quodl. II, q. 20, p. 207, 181-184). And for James to say that nature itself cannot love God above all else was just to say that man naturally loves himself above all else. QED. But if charity elevated nature in this way, did it not "destroy" nature after all, by substituting for the inclination to love the self above all else the entirely different inclination to love God above all else? James' answer was that it did not, since "through charity man loves himself no less than before; he simply loves God more." (Quodl. II, q. 20, p. 210, 280-282). James' opposition to the consensus position on the issue of the love of self vs. love of God did not go unnoticed. It elicited considerable criticism in the years following his death, from such authors as Durand of Saint-Pourcain, Peter of Palude, and John of Naples (Jeschke 2009). ## 8. Political Thought Although James touches briefly on political issues in Quodl. I, q. 17 (see Kempshall, 1999, and Cote, 2012), his most extensive discussions occur in his celebrated De regimine christiano (On Christian Government), written in 1302 during the conflict pitting Boniface VIII against the king of France Philip IV (the Fair). De regimine christiano is often compared in aim and content with Giles of Rome's De ecclesiastica potestate (On Ecclesiastical Power), which offers one of the most extreme statements of pontifical supremacy in the thirteenth century; indeed, in the words of De regimine's editor, James' goal is "to formulate a theory of papal monarchy that is every bit as imposing and ambitious as that of [Giles]" (De regimine christiano: xxxiv). However, as scholars have also recognized, James shows a greater sensitivity to the distinction between nature and grace than Giles did (Arquilliere 1926). De regimine christiano is divided into two parts. The first, dealing with the theory of the Church, is of little philosophical interest, save for James' enlisting of Aristotle to show that all human communities, including the Church, are rooted in the "natural inclination of mankind." The second and longest part is devoted to defining the nature and extent of Christ's and the pope's power. One of James' most characteristic doctrines is found in Book II, chapter 7, where he turns to the question of whether temporal power must be "instituted" by spiritual power, in other words, whether it derives its legitimacy from the spiritual, or possesses a legitimacy of its own. James states outright that spiritual power does institute temporal power, but notes that there have been two views in this regard. Some, e. g., the proponents of the so-called "dualist" position such as John Quidort of Paris, hold that the temporal power derives directly from God and thus in no way needs to be instituted by the spiritual, while others, such as Giles of Rome in De ecclesiastica potestate, contend that the temporal derives wholly from the spiritual and is devoid of any legitimacy whatsoever "unless it is united with spiritual power in the same person or instituted by the spiritual power" (De regimine christiano: 211). James is dissatisfied with both positions and, as he so often does, endeavors to find a "middle way" between them. His solution is to say that the "being" of the temporal power's institution comes both from God--by way of man's natural inclination--in "a material and incomplete sense," and from the spiritual power by which it is "perfected and formed." This is a very clever solution. On the one hand, by rooting the temporal power in man's natural inclination, albeit in the imperfect sense just mentioned, James was acknowledging the legitimacy of temporal rule independently of its connection to the spiritual, thus "avoid[ing] the extreme and implausible view of [Giles of Rome]" (Dyson 2009: xxix). On the other hand, making the natural origins of temporal power merely the incomplete matter of its being was a way of stressing its subordination and inferiority to the spiritual order, in keeping with his papalist convictions. Still, James' very choice of analogies to illustrate the relationship between the spiritual and temporal realms showed that his solution lay much closer to the theocratic position espoused by Giles of Rome than his efforts to find a "middle way" would have us believe. Thus, comparing the spiritual power's relation to the temporal in terms of the relation of light to color, he explains that although "color has something of the nature of light, (...) it has such a feeble light that, unless there is present a more excellent light by which it may be formed, not in its own nature but in its power, it cannot move the vision" (De regimine christiano: 211). In other words, James is telling us that although temporal power does originate in man's natural inclinations, it is ineffectual qua power unless it is informed by the spiritual.
vives	## 1. Life and main works Juan Luis Vives was born in Valencia, Spain on March 6, 1493 (not 1492, as is often found in the literature on him). His parents were Jewish cloth merchants who had converted to Catholicism and who strove to live with the insecurities of their precarious situation. His father, Luis Vives Valeriola (1453-1524), had been prosecuted in 1477 for secretly practicing Judaism. A second trial took place in 1522 and ended two years later when he was burned at the stake. His mother, Blanquina March (1473-1508), became a Christian in 1491, one year before the decree expelling Jews from Spain. She died in 1508 of the plague. Twenty years after her death, she was charged with having visited a clandestine synagogue. Her remains were exhumed and publicly burned. In his youth, Vives attended the Estudio General of his hometown. In 1509, he moved to Paris and enrolled as a freshman in the faculty of arts. He was never to return to Spain. Vives began his studies at the College de Lisieux, where Juan Dolz had just started a triennial course, but soon moved to the College de Beauvais, where he attended the lectures of Jan Dullaert (d.1513). From the fall of 1512, Vives started to attend the course of the Aragonese Gaspar Lax (1487-1560) at the College de Montaigu. Through Nicolas Berault (c.1470-c.1545), who was an associate of Guillaume Bude (1467-1540) and taught at various colleges in Paris, Vives also came into contact with the Parisian humanist circle. In 1514, Vives left Paris without having taken any formal academic degree and moved to the Low Countries. He settled in Bruges, where he would spend most of his life. About this time, he was introduced to Erasmus and appointed as tutor to the Flemish nobleman William of Croy. From 1517 until Croy's premature death in 1521, Vives lived in Louvain and taught at the Collegium Trilingue, a humanist foundation based on Erasmian educational principles. In this period he wrote 'Fabula de homine' ('A Fable about Man,' 1518), an early version of his views on the nature and purpose of mankind; De initiis, sectis et laudibus philosophiae (On the Origins, Schools and Merits of Philosophy, 1518), a short essay on the history of philosophy; In pseudodialecticos (Against the Pseudo-Dialecticians, 1519), a lively and trenchant attack on scholastic logic; as well as a critical edition, with an extensive commentary, of Augustine's De civitate Dei (City of God, 1522), which was commissioned by Erasmus. From 1523 to 1528, Vives divided his time between England, which he visited on six occasions, and Bruges, where he married Margarita Valldaura in 1524. In England he attended the court of Henry VIII and Catherine of Aragon, and was tutor to their daughter, Mary. He also held a lectureship at Corpus Christi College, Oxford, and associated with English humanists such as Thomas More and Thomas Linacre. During these years he published De institutione feminae Christianae (The Education of a Christian Woman, 1524), in which he set out pedagogical principles for the instruction of women; the extremely popular Introductio ad sapientiam (Introduction to Wisdom, 1524), a short handbook of ethics, blending Stoicism and Christianity; and De subventione pauperum (On Assistance to the Poor, 1526), a program for the organization of public relief, which he dedicated to the magistrates of Bruges. In 1528 he lost the favor of Henry VIII by siding with his fellow countrywoman Catherine of Aragon in the matter of the divorce. He was placed under house arrest for a time, before being allowed to return to Bruges. The last twelve years of Vives' life were his most productive, and it was in this period that he published several of the works for which he is best known today. These include De concordia et discordia in humano genere (On Concord and Discord in Humankind, 1529), a piece of social criticism emphasizing the value of peace and the absurdity of war; De disciplinis (On the Disciplines, 1531), an encyclopedic treatise providing an extensive critique of the foundations of contemporary education, as well as a program for its renewal; and De anima et vita (On the Soul and Life, 1538), a study of the soul and its interaction with the body, which also contains a penetrating analysis of the emotions. De veritate fidei Christianae (On the Truth of the Christian Faith), the most thorough discussion of his religious views, was published posthumously in 1543. He died in Bruges on May 6, 1540. ## 2. Dialectic and language Vives' career as a leading northern European humanist starts with the publication in 1519 of In pseudodialecticos, a satirical diatribe in which he voices his opposition to scholastic logic on several counts. He follows in the footsteps of earlier humanists such as Lorenzo Valla (1406-57) and Rudolph Agricola (1443-85), who set about to replace the scholastic curriculum, based on syllogistic and disputation, with a treatment of logic oriented toward the use of the topics, a technique of verbal association aiming at the invention and organization of material for arguments, and persuasion. Vives' severe censure of scholastic logic derived from his own unhappy experience with the scholastic curriculum at Paris. Therefore, as he himself emphasized, no one could accuse him of "condemning it because he did not understand it" (Opera omnia, 1964, III, 38; all quotations below are from this edition). Erasmus wrote to More that no one was better suited for the battle against the dialecticians, in whose ranks he had served for many years. The main targets of Vives' criticism are Peter of Spain's Summule logicales, a work dating from the thirteenth century but which still held an important place in the university curriculum, and the theory of the property of terms, a semantic theory dealing with properties of linguistic expressions such as signification, i.e., the meaning of a word regardless of context, and supposition, i.e., the meaning of a word within the context of its use in a proposition. He repudiates the use of technical jargon, accessible only to a narrow group of professionals, and maintains that if scholastic logicians made an effort to speak plainly and according to common usage, many of their conundrums would disappear. Instead, they choose to fritter away their ingenuity on logically ambiguous propositions known as sophismata. Vives provides many examples of such propositions, which in his view make no sense whatever and are certainly of no use. Many of these, such as "Some animal is not man, therefore some man is not animal" (III, 54), were standard scholastic examples. Others, such as "Only any non-donkey c of any man except Socrates and another c belonging to this same man begins contingently to be black" (III, 40), are intended as a mockery of the futile quibbling he associated with scholastic method. Since dialectic, like rhetoric and grammar, deals with language, its rules should be adapted to the rules of ordinary language; but with what language, he asks, have these propositions to do? Moreover, dialectic should not be learned for its own sake, but as a support for the other arts; therefore, no more effort should be spent on it than is absolutely necessary. Vives' criticism is also informed by ethical concerns and the demand for a method that would be of use in everyday life rather than in academic disputations. In contrast to the standard interpretation, it has been argued that In pseudodialecticos is not a serious refutation of scholasticism, but rather a sophistical exercise replete with fallacious arguments, whose purpose is to teach a valuable lesson in scholastic dialectic challenging the reader to detect the many logical fallacies that it contains. According to this interpretation, In pseudodialecticos offers no serious arguments against scholastic logical theory or principles (see Perreiah, 2014, ch.4). A more detailed criticism can be found in De disciplinis of 1531. This encyclopedic treatise is divided into three parts: De causis corruptarum artium (On the Causes of the Corruption of the Arts), seven books devoted to a thoroughgoing critique of the foundations of contemporary education; De tradendis disciplinis (On Handing Down the Disciplines), five books in which Vives outlines his program for educational reform; and five shorter treatises De artibus (On the Arts), dealing mainly with logic and metaphysics. These five treatises include De prima philosophia (On First Philosophy), a compendium of Aristotelian physics and metaphysics from a Christian point of view; De censura veri (On the Assessment of Truth), a discussion of the proposition and the forms of argumentation; De explanatione cuiusque essentiae (On the Explanation of Each Essence); De instrumento probabilitatis (On the Instrument of Probability), which contains a theory of knowledge, as well as a detailed account of dialectical invention; and De disputatione (On Disputation), in which he discusses non-formal proofs. In these treatises, Vives not only continues the trends in humanist dialectic initiated by Valla and Agricola, but also displays a familiarity with philosophical technicalities that was unusual among humanists and that reveals the more traditionally Aristotelian aspects of his thought. His appraisal of the Aristotelian corpus is summarised in Censura de Aristotelis operibus (Assessment of Aristotle's Works, 1538). A posthumously published treatise entitled Dialectices libri quatuor (Four Books of Dialectic, 1550) appears to be a youthful work that Vives evidently did not consider suitable for publication. Vives' criticism of scholastic logic hinges on a profound analysis of the arts of discourse. For him, the supremacy of the sermo communis (ordinary discourse) over the abstract language of metaphysics is indisputable. Philosophy ought not to invent the language and subject of its own specific investigation (VI, 140). In De causis corruptarum artium, he writes: "enraged against nature, about which they know nothing, the dialecticians have constructed another one for themselves--that is to say, the nature of formalities, individual natures (ecceitates), realities, relations, Platonic ideas and other monstrosities which cannot be understood even by those who have invented them" (VI, 190-1). Instead of the formal language of the dialecticians, which he found completely unsuited to interpret reality, he proposes the less rigorous but more concrete universe of everyday communication, which answers all our practical needs and aims to provide a knowledge that is useful. ## 3. Epistemology and history Vives was pessimistic about the possibility of attaining knowledge as understood in Aristotelian terms; and his thought anticipates the moderate skepticism of early modern philosophers such as Francisco Sanches (1551-1623) and Pierre Gassendi (1592-1655). Vives belongs, like Francis Bacon (1561-1626), to the so-called "maker's knowledge" tradition, which regards knowledge as a kind of making or as a capacity to make (see A. Perez-Ramos, Francis Bacon's Idea of Science and the Maker's Knowledge Tradition, Oxford and New York: Clarendon Press and Oxford University Press, 1988). He often insists on the practical nature of knowledge; for instance, in De causis corruptarum artium, he maintains that peasants and artisans know nature far better than many philosophers (VI, 190). In Satellitium animi (The Soul's Escort, 1524), a collection of aphorisms dedicated to Princess Mary, he points out that "man knows as far as he can make" (IV, 63). A central tenet of the maker's knowledge tradition is that man cannot gain access to nature's intimate works, since these, as opera divina (divine works), are only known to God, their maker. According to Vives, things have two different layers: one external, consisting in the sensible accidents of the thing, and another, internal, and therefore hidden, which is the essence of the thing (III, 197). "The true and genuine essences of all things", he writes, "are not known by us in themselves. They hide concealed in the innermost part of each thing where our mind, enclosed by the bulk of the body and the darkness of life, cannot penetrate" (III, 406-7). Vives subscribes to the Aristotelian principle that all of our knowledge has its origin in perception. We cannot learn anything, he maintains, except through the senses (III, 193 and 378). But he also maintains that the human mind "must realize that, since it is locked up in a dark prison and surrounded by obscurity, it is prevented from understanding many things and cannot clearly observe or know what it wants: neither the concealed essence of material things, nor the quality and character of immaterial things; nor can it, on account of the gloom of the body, use its acuity and swiftness" (III, 329). In other words, since the senses cannot grasp what is incorporeal or hidden, sense perception does not yield any knowledge of the essence of things but only of their accidents. Vives' view, however, is that sensory knowledge must nonetheless be transcended by means of reasoning. Yet, according to him, the best that human reason can accomplish in this process is to provide a judgment grounded in all the available evidence, thereby increasing the probability of the conclusion. In his view, our knowledge of the essence of a thing is only an approximate guess based on the sensible operations of the thing in question (III, 122). The most reliable guide for human inquiry, he argues, is mankind's natural propensity toward what is good and true. This light of our mind, as he also calls it, is always, directly or indirectly, inclined toward what is good and true, and can be regarded as the beginning and origin of prudence and of all sciences and arts. This natural propensity can be perfected if it is subjected to teaching and exercise, just as the seeds of plants grow better if they are cultivated by the industrious hands of a farmer. He found philosophical grounds for this idea in Cicero's report (see, e.g., De natura deorum (On the Nature of the Gods), I.43-5) of the Hellenistic theory of anticipationes (anticipations) and of naturales informationes (natural notions), which we have not learned from teachers or custom, but are instead derived and received from nature (III, 356-7). The topics, which Vives conceives as a reflection of the ontological order, represent another valuable instrument for human inquiry. In his view, the topics are a set of universal aspects of things that help to bring order to the great variety of nature. As such, they play an important role as organizing principles of knowledge. They are like a grid through which knowledge can be acquired and arguments formulated (see Nauta, 2015). Nevertheless, human knowledge can be nothing other than a finite participation in creation. Because of the limitations that characterize man's fallen state, investigations into the realm of nature can only lead to conjectures, and not to firm and indubitable knowledge, which we neither deserve nor need. In De prima philosophia, Vives writes: "human inquiry comes to conjectural conclusions, for we do not deserve certain knowledge (scientia), stained by sin as we are and hence burdened with the great weight of the body; nor do we need it, for we see that man is ordained lord and master of everything in the sublunary world" (III, 188). In his opinion, certainty is not a prerequisite for advances in science and philosophy; and as a criterion for scientific progress and for the rational conduct of life, he advocates a method consisting in sound judgment based on experience. History, seen as the sum of all human experience, is therefore of great importance for all branches of learning. In De tradendis disciplinis, Vives maintains that "history appears to surpass all disciplines, since it either gives birth to or nourishes, develops [and] cultivates all arts" (VI, 291). In this sense, history is not primarily regarded as the memory of great deeds or the source of useful examples, but as a process of development. In principle, every new generation is better equipped than the preceding one, since it can derive advantage from all earlier experience: "It is therefore clear - he maintains - that if only we apply our minds sufficiently, we can formulate better opinions about matters of life and nature than could Aristotle, Plato, or any of the ancients" (VI, 6-7). According to Vives, the saying "we are like dwarfs on the shoulders of giants" is plainly false. We are not dwarfs, nor were they giants. All humans are composed of the same structure (VI, 39). The idea of progress plays an essential role in Vives' conception of intellectual history, and several of the cultural problems he deals with, such as the causes of the corruption of the arts, are approached from an historical perspective. ## 4. Moral and social philosophy Vives' moral philosophy stems mainly from his Christian humanism and is aimed at the reform of both individuals and society. He often proclaims the superiority of Christian ethics over pagan wisdom (I, 23; VI, 209-10). In De causis corruptarum artium, he argues at length that Aristotle's ethics, on account of its worldly conception of happiness and virtue, is completely incompatible with the Christianity: "we cannot serve both Christ and Aristotle, whose teaching are diametrically opposed to each other" (VI, 218). He has more sympathy for Platonism and Stoicism, which he believes are broadly in line with Christian morality. In De initiis, sectis et laudibus philosophiae, he even asserts: "I do not think, in fact, that there is any truer Christian than the Stoic sage" (III, 17). In Introductio ad sapientiam, inspired by the teaching of the Stoics, he recommends self-knowledge as the first step toward virtue, which he regards as the culmination of human perfection. We should not, in his judgment, call anything our own except our soul, in which learning and virtue, or their opposites, are to be found. The body is "a bondslave of the soul" (mancipium animi), while such things as riches, power, nobility, honor, dignity, and their contraries, are completely external to man (I, 2; VI, 401). Vice follows from a wrong judgment about the value of things: "Nothing - he writes - is more destructive in human life than a corrupt judgment which gives no object its proper value" (I, 1). To be wise, however, is not only to have true opinions about things, but also to translate this knowledge into action by desiring honorable things and avoiding evil (I, 2). Wisdom therefore requires the subordination of the passions to the control of the intellect. Vives holds that the best means to secure the reform of society is through the moral and practical training of the individual. In his view, there are two related paths which are necessary to develop our humanity: education and action. Education is fundamental in order for us to rise above our animal instincts and realize our full potential as human beings. However, learning needs to be applied in every day life, especially for the public good (see Verbeke, 2014). Man, by his own nature, is a social being: "Experience proves every day that man was created by God for society, both in this and in the eternal life. For this reason God inspired in man an admirable disposition of benevolence and good will toward other men" (VI, 222-3). In the first book of De subventione pauperum, which consists of a theoretical discussion of the human condition, he stresses not only our need for and dependency on others, but also our natural inclination to love and help one another. He regards the development of society as a distinctly human achievement, based on the ability to profit from experience and turn knowledge to useful ends. Social problems, such as poverty and war, are the result of emotional disorders. During Vives' lifetime, Europe experienced dissension and war between princes and within the church, as well as the increased threat posed by Muslim expansion into Western Europe. He addressed the problem of political and religious disturbances in several works, which also deal with the psychological origins of discord, the proper conduct of all the offices of the commonweal, and the theme of Christian harmony. His mayor political text on European war and peace is De concordia et discordia in humano genere where he sets out the case for the origins of discord in society and he aims to show how peace and concord can be fostered through knowledge of human nature, especially the emotions. According to him, the virtue of the people can only be maintained and promoted in peace. "In war, as in a sick body, no member exercises its office properly" (V, 180-1). Vives deplored the Italian war between France and Spain (1521-6), which, he felt, completely ignored the rights of the suffering population, and accused Francis I and Charles V of irresponsibility and criminal ambition (VI, 480). In these circumstances, he often referred to the notion of natural law, which, as he explains in his preface to Cicero's De legibus (On the Laws), has the same validity everywhere because it was impressed on the heart of every human being before birth (V, 494). ## 5. Psychology Vives' philosophical reflections on the human soul are mainly concentrated in De anima et vita, published in 1538, which provides the psychological underpinning for many of his educational ideas and can be characterized as a prolegomenon to moral philosophy. He attempts to reconcile the Aristotelian view of the soul as an organizing and animating principle with the Platonic conception of the soul as an immaterial and immortal substance. He also pays close attention to physiology and, following the Galenic tradition, maintains that our mental capacities depend on the temperament of our body. The structure of the treatise is indebted to the traditional approach of faculty psychology, in which the soul is said to be composed of a number of different faculties or powers, each directed toward a different object and responsible for a distinct operation. The first book covers the functions of the vegetative soul (nutrition, growth and reproduction), of the sensitive soul (the five external senses), and of the cogitative soul (the internal senses, i.e., a variety of cognitive faculties, including imagination, fantasy and the estimative power, which are located in the three ventricles of the brain, and whose acts follows from the acts of the external senses). The second book deals with the functions of the rational soul and its three faculties (mind, will, and memory), as well as with topics stemming from Aristotle's Parva naturalia, such as sleep, dreams, and longevity. The third and final book explores the emotions, which Vives, rejecting the Stoic view, regards as natural responses to the way things appear to us and as essential constituents of human life. He defines the soul as "the principal agent inhabiting a body adapted to life (agens praecipuum, habitans in corpore apto ad vitam)." The soul is called an agent in the sense that it acts through instruments--e.g., heat, humors and spirits--by means of its own power. That it is the principal agent means that, even though its instruments act on the body, they do not operate by means of their own power but only through the power that they receive from the soul (III, 335-6). The organs governing our rational functions consist of fine and bright spirits exhaled from the pericardial blood. Although the heart is the source and origin of all the rational soul's operations, the head is their workshop. In fact, the mind does not apprehend, nor is it affected, unless the spirits reach the brain (III, 365-6). The enormous importance Vives attaches to the exploration of the emotions is reflected in the fact that he describes the branch of philosophy that provides a remedy for the severe diseases of the soul not only as "the foundation of all morality, private as well as public" (III, 299-300), but also as "the supreme form of learning and knowledge" (I, 17). Emotions (affectus sive affectiones) are defined as "the acts of those faculties which nature gave to our souls for the pursuit of good and the avoidance of evil, by means of which we are led toward the good and we move away from or against evil". He stresses that the terms "good" and "evil" in this definition mean, not what is in reality good or evil, but rather what each person judges to be good or evil (III, 422). Therefore, the more pure and elevated the judgment, the more it takes account of what is genuinely good and true, admitting fewer and less intense emotions and becoming disturbed more rarely. Immoderate and confused movements, on the other hand, are the result of ignorance, thoughtlessness, and false judgments, since we judge the good or evil to be greater than it really is (III, 425). One of the most distinctive features of Vives' study of the human soul is the fundamental role that psychological inquiry came to play in his reform program. His use of psychological principles in his writings often surpasses that of previous authors in both scope and detail. He applies these principles, for instance, not only to individual conduct and education, but also to professional practice, social reform, and practical affairs in general. According to Vives, psychology is relevant to all disciplines. "The study of man's soul", he writes in De tradendis disciplinis, "exercises a most helpful influence on all kinds of knowledge, because our knowledge is determined by the intelligence and comprehension of our minds, not by the things themselves" (VI, 375). ## 6. Influence Vives' works, which went through hundreds of editions and were translated into several vernacular languages, continued to be widely read and extremely influential during the century after their publication. His critical attitude toward the Aristotelian orthodoxy of his day left a mark on several authors. Mario Nizolio (1488-1567) cites Vives numerous times in De veris principiis et vera ratione philosophandi contra pseudophilosophos (On the True Principles and True Method of Philosophizing against the Pseudo-Philosophers, 1553), an attack on Aristotelian dialectic and metaphysics, which G. W. Leibniz (1646-1716) considered to be worth editing more than a hundred years later. In Quod nihil scitur (That Nothing is Known, 1581), one of the best systematic expositions of philosophical skepticism produced during the sixteenth century, the Portuguese philosopher and medical writer Francisco Sanches displays a familiarity with De disciplinis, and there are indications that he might also have been acquainted with In pseudodialecticos. In Exercitationes paradoxicae adversus Aristoteleos (Paradoxical Exercises against the Aristotelians, 1624), a skeptical attack on Aristotelianism, Pierre Gassendi says that reading Vives gave him courage and helped him to free himself from the dogmatism of Peripatetic philosophy. Psychology was another area in which Vives enjoyed considerable success. Philip Melanchthon (1497-1560) recommended De anima et vita in the prefatory letter to his Commentarius de anima (Commentary on the Soul, 1540). Vives' influence on the naturalistic pedagogy of the Spaniard Juan Huarte de San Juan (c.1529-1588), in his celebrated El examen de ingenios para las ciencias (The Examination of Men's Wits, 1575), is undeniable. In his discussion of the passions of the soul, the Jesuit Francisco Suarez (1548-1617) counted Vives among his authorities, pointing out that the study of the emotions belongs to natural philosophy and medicine as well as to moral philosophy. Vives' treatise was also an important source of inspiration for Robert Burton (1577-1640), who, in The Anatomy of Melancholy (1621), repeatedly quotes from De anima et vita. The reference to Vives by Rene Descartes (1596-1650) in Les Passions de l'ame (1649) suggests that he had read the book. Although Vives is rarely mentioned in the scholarly literature on the Scottish philosophy of "Common Sense," the impact of his thought on leading exponents of the school was significant. William Hamilton (1788-1856) praised Vives' insights on memory and the laws of association. In his "Contributions Towards a History of the Doctrine of Mental Suggestion or Association", he quotes extensive portions of Vives' account of memory and maintains that the observations of "the Spanish Aristotelian" comprise "nearly all of principal moment that has been said upon this subject, either before or since". Moreover, Vives' Dialectices libri quatuor (1550) was one of the primary sources of Thomas Reid (1710-1796) in his "A Brief Account of Aristotle's Logic" (1774). During the second half of the nineteenth century and the fist decades of the twentieth, Vives was read and studied by philosophers such as Ernest Renan (1823-1892), Friedrich Albert Lange (1828-1875), Wilhelm Dilthey (1833-1911), Pierre Duhem (1861-1916), Ernst Cassirer (1874-1945), and Jose Ortega y Gasset (1883-1955). Lange regards him as one of the most important reformers of philosophy of his time and a precursor of both Bacon and Descartes. According to Ortega y Gasset, Vives' method, firmly based on experience, and his emphasis on the need to found a new culture grounded, not in barren speculation, but in the usefulness of knowledge, anticipate some elements of the modern Zeitgeist.
freewill	## 1. Major Historical Contributions ### 1.1 Ancient and Medieval Period One finds scholarly debate on the 'origin' of the notion of free will in Western philosophy. (See, e.g., Dihle (1982) and, in response Frede (2011), with Dihle finding it in St. Augustine (354-430 CE) and Frede in the Stoic Epictetus (c. 55-c. 135 CE).) But this debate presupposes a fairly particular and highly conceptualized concept of free will, with Dihle's later 'origin' reflecting his having a yet more particular concept in view than Frede. If, instead, we look more generally for philosophical reflection on choice-directed control over one's own actions, then we find significant discussion in Plato and Aristotle (cf. Irwin 1992). Indeed, on this matter, as with so many other major philosophical issues, Plato and Aristotle give importantly different emphases that inform much subsequent thought. In Book IV of The Republic, Plato posits rational, spirited, and appetitive aspects to the human soul. The wise person strives for inner 'justice', a condition in which each part of the soul plays its proper role--reason as the guide, the spirited nature as the ally of reason, exhorting oneself to do what reason deems proper, and the passions as subjugated to the determinations of reason. In the absence of justice, the individual is enslaved to the passions. Hence, freedom for Plato is a kind of self-mastery, attained by developing the virtues of wisdom, courage, and temperance, resulting in one's liberation from the tyranny of base desires and acquisition of a more accurate understanding and resolute pursuit of the Good (Hecht 2014). While Aristotle shares with Plato a concern for cultivating virtues, he gives greater theoretical attention to the role of choice in initiating individual actions which, over time, result in habits, for good or ill. In Book III of the Nicomachean Ethics, Aristotle says that, unlike nonrational agents, we have the power to do or not to do, and much of what we do is voluntary, such that its origin is 'in us' and we are 'aware of the particular circumstances of the action'. Furthermore, mature humans make choices after deliberating about different available means to our ends, drawing on rational principles of action. Choose consistently well (poorly), and a virtuous (vicious) character will form over time, and it is in our power to be either virtuous or vicious. A question that Aristotle seems to recognize, while not satisfactorily answering, is whether the choice an individual makes on any given occasion is wholly determined by his internal state--perception of his circumstances and his relevant beliefs, desires, and general character dispositions (wherever on the continuum between virtue and vice he may be)--and external circumstances. He says that "the man is the father of his actions as of children"--that is, a person's character shapes how she acts. One might worry that this seems to entail that the person could not have done otherwise--at the moment of choice, she has no control over what her present character is--and so she is not responsible for choosing as she does. Aristotle responds by contending that her present character is partly a result of previous choices she made. While this claim is plausible enough, it seems to 'pass the buck', since 'the man is the father' of those earlier choices and actions, too. We note just a few contributions of the subsequent centuries of the Hellenistic era. (See Bobzien 1998.) This period was dominated by debates between Epicureans, Stoics, and the Academic Skeptics, and as it concerned freedom of the will, the debate centered on the place of determinism or of fate in governing human actions and lives. The Stoics and the Epicureans believed that all ordinary things, human souls included, are corporeal and governed by natural laws or principles. Stoics believed that all human choice and behavior was causally determined, but held that this was compatible with our actions being 'up to us'. Chrysippus ably defended this position by contending that your actions are 'up to you' when they come about 'through you'--when the determining factors of your action are not external circumstances compelling you to act as you do but are instead your own choices grounded in your perception of the options before you. Hence, for moral responsibility, the issue is not whether one's choices are determined (they are) but in what manner they are determined. Epicurus and his followers had a more mechanistic conception of bodily action than the Stoics. They held that all things (human soul included) are constituted by atoms, whose law-governed behavior fixes the behavior of everything made of such atoms. But they rejected determinism by supposing that atoms, though law-governed, are susceptible to slight 'swerves' or departures from the usual paths. Epicurus has often been understood as seeking to ground the freedom of human willings in such indeterministic swerves, but this is a matter of controversy. If this understanding of his aim is correct, how he thought that this scheme might work in detail is not known. (What little we know about his views in this matter stem chiefly from the account given in his follower Lucretius's six-book poem, On the Nature of Things. See Bobzien 2000 for discussion.) A final notable figure of this period was Alexander of Aphrodisias, the most important Peripatetic commentator on Aristotle. In his On Fate, Alexander sharply criticizes the positions of the Stoics. He goes on to resolve the ambiguity in Aristotle on the question of the determining nature of character on individual choices by maintaining that, given all such shaping factors, it remains open to the person when she acts freely to do or not to do what she in fact does. Many scholars see Alexander as the first unambiguously 'libertarian' theorist of the will (for more information about such theories see section 2 below). Augustine (354-430) is the central bridge between the ancient and medieval eras of philosophy. His mature thinking about the will was influenced by his early encounter with late classical Neoplatonist thought, which is then transformed by the theological views he embraces in his adult Christian conversion, famously recounted in his Confessions. In that work and in the earlier On the Free Choice of the Will, Augustine struggles to draw together into a coherent whole the doctrines that creaturely misuse of freedom, not God, is the source of evil in the world and that the human will has been corrupted through the 'fall' from grace of the earliest human beings, necessitating a salvation that is attained entirely through the actions of God, even as it requires, constitutively, an individual's willed response of faith. The details of Augustine's positive account remain a matter of controversy. He clearly affirms that the will is by its nature a self-determining power--no powers external to it determine its choice--and that this feature is the basis of its freedom. But he does not explicitly rule out the will's being internally determined by psychological factors, as Chrysippus held, and Augustine had theological reasons that might favor (as well as others that would oppose) the thesis that all things are determined in some manner by God. Scholars divide on whether Augustine was a libertarian or instead a kind of compatibilist with respect to metaphysical freedom. (Macdonald 1999 and Stump 2006 argue the former, Baker 2003 and Couenhoven 2007 the latter.) It is clear, however, that Augustine thought that we are powerfully shaped by wrongly-ordered desires that can make it impossible for us to wholeheartedly will ends contrary to those desires, for a sustained period of time. This condition entails an absence of something more valuable, 'true freedom', in which our wills are aligned with the Good, a freedom that can be attained only by a transformative operation of divine grace. This latter, psychological conception of freedom of will clearly echoes Plato's notion of the soul's (possible) inner justice. Thomas Aquinas (1225-1274) attempted to synthesize major strands of Aristotle's systematic philosophy with Christian theology, and so Aquinas begins his complex discussion of human action and choice by agreeing with Aristotle that creatures such as ourselves who are endowed with both intellect and will are hardwired to will certain general ends ordered to the most general goal of goodness. Will is rational desire: we cannot move towards that which does not appear to us at the time to be good. Freedom enters the picture when we consider various means to these ends and move ourselves to activity in pursuit of certain of them. Our will is free in that it is not fixed by nature on any particular means, and they generally do not appear to us either as unqualifiedly good or as uniquely satisfying the end we wish to fulfill. Furthermore, what appears to us to be good can vary widely--even, over time, intra-personally. So much is consistent with saying that in a given total circumstance (including one's present beliefs and desires), one is necessitated to will as one does. For this reason, some commentators have taken Aquinas to be a kind of compatibilist concerning freedom and causal or theological determinism. In his most extended defense of the thesis that the will is not 'compelled' (DM 6), Aquinas notes three ways that the will might reject an option it sees as attractive: (i) it finds another option more attractive, (ii) it comes to think of some circumstance rendering an alternative more favorable "by some chance circumstance, external or internal", and (iii) the person is momentarily disposed to find an alternative attractive by virtue of a non-innate state that is subject to the will (e.g., being angry vs being at peace). The first consideration is clearly consistent with compatibilism. The second at best points to a kind of contingency that is not grounded in the activity of the will itself. And one wanting to read Aquinas as a libertarian might worry that his third consideration just passes the buck: even if we do sometimes have an ability to directly modify perception-coloring states such as moods, Aquinas's account of will as rational desire seems to indicate that we will do so only if it seems to us on balance to be good to do so. Those who read Aquinas as a libertarian point to the following further remark in this text: "Will itself can interfere with the process [of some cause's moving the will] either by refusing to consider what attracts it to will or by considering its opposite: namely, that there is a bad side to what is being proposed..." (Reply to 15; see also DV 24.2). For discussion, see MacDonald (1998), Stump (2003, ch. 9) and especially Hoffman & Michon (2017), which offers the most comprehensive analysis of relevant texts to date. John Duns Scotus (1265/66-1308) was the stoutest defender in the medieval era of a strongly libertarian conception of the will, maintaining on introspective grounds that will by its very nature is such that "nothing other than the will is the total cause" of its activity (QAM). Indeed, he held the unusual view that not only up to but at the very instant that one is willing X, it is possible for one to will Y or at least not to will X. (He articulates this view through the puzzling claim that a single instant of time comprises two 'instants of nature', at the first but not the second of which alternative possibilities are preserved.) In opposition to Aquinas and other medieval Aristotelians, Scotus maintained that a precondition of our freedom is that there are two fundamentally distinct ways things can seem good to us: as practically advantageous to us or as according with justice. Contrary to some popular accounts, however, Scotus allowed that the scope of available alternatives for a person will be more or less constricted. He grants that we are not capable of willing something in which we see no good whatsoever, nor of positively repudiating something which appears to us as unqualifiedly good. However, in accordance with his uncompromising position that nothing can be the total cause of the will other than itself, he held that where something does appear to us as unqualifiedly good (perfectly suited both to our advantage and justice)--viz., in the 'beatific vision' of God in the afterlife--we still can refrain from willing it. For discussion, see John Duns Scotus, SS5.2. ### 1.2 Modern Period and Twentieth Century The problem of free will was an important topic in the modern period, with all the major figures wading into it (Descartes 1641 [1988], 1644 [1988]; Hobbes 1654 [1999], 1656 [1999]; Spinoza 1677 [1992]; Malebranche 1684 [1993]; Leibniz 1686 [1991]; Locke 1690 [1975]; Hume 1740 [1978], 1748 [1975]; Edwards 1754 [1957]; Kant 1781 [1998], 1785 [1998], 1788 [2015]; Reid 1788 [1969]). After less sustained attention in the 19th Century (most notable were Schopenhauer 1841 [1999] and Nietzsche 1886 [1966]), it was widely discussed again among early twentieth century philosophers (Moore 1912; Hobart 1934; Schlick 1939; Nowell-Smith 1948, 1954; Campbell 1951; Ayer 1954; Smart 1961). The centrality of the problem of free will to the various projects of early modern philosophers can be traced to two widely, though not universally, shared assumptions. The first is that without belief in free will, there would be little reason for us to act morally. More carefully, it was widely assumed that belief in an afterlife in which a just God rewards and punishes us according to our right or wrong use of free will was key to motivating us to be moral (Russell 2008, chs. 16-17). Life before death affords us many examples in which vice is better rewarded than virtue and so without knowledge of a final judgment in the afterlife, we would have little reason to pursue virtue and justice when they depart from self-interest. And without free will there can be no final judgement. The second widely shared assumption is that free will seems difficult to reconcile with what we know about the world. While this assumption is shared by the majority of early modern philosophers, what specifically it is about the world that seems to conflict with freedom differs from philosopher to philosopher. For some, the worry is primarily theological. How can we make sense of contingency and freedom in a world determined by a God who must choose the best possible world to create? For some, the worry was primarily metaphysical. The principle of sufficient reason--roughly, the idea that every event must have a reason or cause--was a cornerstone of Leibniz's and Spinoza's metaphysics. How does contingency and freedom fit into such a world? For some, the worry was primarily scientific (Descartes). Given that a proper understanding of the physical world is one in which all physical objects are governed by deterministic laws of nature, how does contingency and freedom fit into such a world? Of course, for some, all three worries were in play in their work (this is true especially of Leibniz). Despite many disagreements about how best to solve these worries, there were three claims that were widely, although not universally, agreed upon. The first was that free will has two aspects: the freedom to do otherwise and the power of self-determination. The second is that an adequate account of free will must entail that free agents are morally responsible agents and/or fit subjects for punishment. Ideas about moral responsibility were often a yard stick by which analyses of free will were measured, with critics objecting to an analysis of free will by arguing that agents who satisfied the analysis would not, intuitively, be morally responsible for their actions. The third is that compatibilism--the thesis that free will is compatible with determinism--is true. (Spinoza, Reid, and Kant are the clear exceptions to this, though some also see Descartes as an incompatibilist [Ragland 2006].) Since a detailed discussion of these philosophers' accounts of free will would take us too far afield, we want instead to focus on isolating a two-step strategy for defending compatibilism that emerges in the early modern period and continued to exert considerable force into the early twentieth century (and perhaps is still at work today). Advocates of this two-step strategy have come to be known as "classical compatibilists". The first step was to argue that the contrary of freedom is not determinism but external constraint on doing what one wants to do. For example, Hobbes contends that liberty is "the absence of all the impediments to action that are not contained in the nature and intrinsical quality of the agent" (Hobbes 1654 [1999], 38; cf. Hume 1748 [1975] VIII.1; Edwards 1754 [1957]; Ayer 1954). This idea led many compatibilists, especially the more empiricist-inclined, to develop desire- or preference-based analyses of both the freedom to do otherwise and self-determination. An agent has the freedom to do otherwise than $\phi$ just in case if she preferred or willed to do otherwise, she would have done otherwise (Hobbes 1654 [1999], 16; Locke 1690 [1975]) II.xx.8; Hume 1748 [1975] VIII.1; Moore 1912; Ayer 1954). The freedom to do otherwise does not require that you are able to act contrary to your strongest motivation but simply that your action be dependent on your strongest motivation in the sense that had you desired something else more strongly, then you would have pursued that alternative end. (We will discuss this analysis in more detail below in section 2.2.) Similarly, an agent self-determines her $\phi$-ing just in case $\phi$ is caused by her strongest desires or preferences at the time of action (Hobbes 1654 [1999]; Locke 1690 [1975]; Edwards 1754 [1957]). (We will discuss this analysis in more detail below in section 2.4.) Given these analyses, determinism seems innocuous to freedom. The second step was to argue that any attempt to analyze free will in a way that putatively captures a deeper or more robust sense of freedom leads to intractable conundrums. The most important examples of this attempt to capture a deeper sense of freedom in the modern period are Immanuel Kant (1781 [1998], 1785 [1998], 1788 [2015]) and Thomas Reid (1788 [1969]) and in the early twentieth century C. A. Campbell (1951). These philosophers argued that the above compatibilist analyses of the freedom to do otherwise and self-determination are, at best, insufficient for free will, and, at worst, incompatible with it. With respect to the classical compatibilist analysis of the freedom to do otherwise, these critics argued that the freedom to do otherwise requires not just that an agent could have acted differently if he had willed differently, but also that he could have willed differently. Free will requires more than free action. With respect to classical compatibilists' analysis of self-determination, they argued that self-determination requires that the agent--rather than his desires, preferences, or any other mental state--cause his free choices and actions. Reid explains: > > > I consider the determination of the will as an effect. This effect > must have a cause which had the power to produce it; and the cause > must be either the person himself, whose will it is, or some other > being.... If the person was the cause of that determination of > his own will, he was free in that action, and it is justly imputed to > him, whether it be good or bad. But, if another being was the cause of > this determination, either producing it immediately, or by means and > instruments under his direction, then the determination is the act and > deed of that being, and is solely imputed to him. (1788 [1969] IV.i, > 265) > > > Classical compatibilists argued that both claims are incoherent. While it is intelligible to ask whether a man willed to do what he did, it is incoherent to ask whether a man willed to will what he did: > > > For to ask whether a man is at liberty to will either motion or rest, > speaking or silence, which he pleases, is to ask whether a man can > will what he wills, or be pleased with what he is > pleased with? A question which, I think, needs no answer; and they who > make a question of it must suppose one will to determine the acts of > another, and another to determine that, and so on in > infinitum. (Locke 1690 [1975] II.xx.25; cf. Hobbes 1656 [1999], > 72) > > > In response to libertarians' claim that self-determination requires that the agent, rather than his motives, cause his actions, it was objected that this removes the agent from the natural causal order, which is clearly unintelligible for human animals (Hobbes 1654 [1999], 38). It is important to recognize that an implication of the second step of the strategy is that free will is not only compatible with determinism but actually requires determinism (cf. Hume 1748 [1975] VIII). This was a widely shared assumption among compatibilists up through the mid-twentieth century. Spinoza's Ethics (1677 [1992]) is an important departure from the above dialectic. He endorses a strong form of necessitarianism in which everything is categorically necessary as opposed to the conditional necessity embraced by most compatibilists, and he contends that there is no room in such a world for divine or creaturely free will. Thus, Spinoza is a free will skeptic. Interestingly, Spinoza is also keen to deny that the nonexistence of free will has the dire implications often assumed. As noted above, many in the modern period saw belief in free will and an afterlife in which God rewards the just and punishes the wicked as necessary to motivate us to act morally. According to Spinoza, so far from this being necessary to motivate us to be moral, it actually distorts our pursuit of morality. True moral living, Spinoza thinks, sees virtue as its own reward (Part V, Prop. 42). Moreover, while free will is a chimera, humans are still capable of freedom or self-determination. Such self-determination, which admits of degrees on Spinoza's view, arises when our emotions are determined by true ideas about the nature of reality. The emotional lives of the free persons are ones in which "we desire nothing but that which must be, nor, in an absolute sense, can we find contentment in anything but truth. And so in so far as we rightly understand these matters, the endeavor of the better part of us is in harmony with the order of the whole of Nature" (Part IV, Appendix). Spinoza is an important forerunner to the many free will skeptics in the twentieth century, a position that continues to attract strong support (see Strawson 1986; Double 1992; Smilansky 2000; Pereboom 2001, 2014; Levy 2011; Waller 2011; Caruso 2012; Vilhauer 2012. For further discussion see the entry skepticism about moral responsibility). It is worth observing that in many of these disputes about the nature of free will there is an underlying dispute about the nature of moral responsibility. This is seen clearly in Hobbes (1654 [1999]) and early twentieth century philosophers' defenses of compatibilism. Underlying the belief that free will is incompatible with determinism is the thought that no one would be morally responsible for any actions in a deterministic world in the sense that no one would deserve blame or punishment. Hobbes responded to this charge in part by endorsing broadly consequentialist justifications of blame and punishment: we are justified in blaming or punishing because these practices deter future harmful actions and/or contribute to reforming the offender (1654 [1999], 24-25; cf. Schlick 1939; Nowell-Smith 1948; Smart 1961). While many, perhaps even most, compatibilists have come to reject this consequentialist approach to moral responsibility in the wake of P. F. Strawson's 1962 landmark essay 'Freedom and Resentment' (though see Vargas (2013) and McGeer (2014) for contemporary defenses of compatibilism that appeal to forward-looking considerations) there is still a general lesson to be learned: disputes about free will are often a function of underlying disputes about the nature and value of moral responsibility. ## 2. The Nature of Free Will ### 2.1 Free Will and Moral Responsibility As should be clear from this short discussion of the history of the idea of free will, free will has traditionally been conceived of as a kind of power to control one's choices and actions. When an agent exercises free will over her choices and actions, her choices and actions are up to her. But up to her in what sense? As should be clear from our historical survey, two common (and compatible) answers are: (i) up to her in the sense that she is able to choose otherwise, or at minimum that she is able not to choose or act as she does, and (ii) up to her in the sense that she is the source of her action. However, there is widespread controversy both over whether each of these conditions is required for free will and if so, how to understand the kind or sense of freedom to do otherwise or sourcehood that is required. While some seek to resolve these controversies in part by careful articulation of our experiences of deliberation, choice, and action (Nozick 1981, ch. 4; van Inwagen 1983, ch. 1), many seek to resolve these controversies by appealing to the nature of moral responsibility. The idea is that the kind of control or sense of up-to-meness involved in free will is the kind of control or sense of up-to-meness relevant to moral responsibility (Double 1992, 12; Ekstrom 2000, 7-8; Smilansky 2000, 16; Widerker and McKenna 2003, 2; Vargas 2007, 128; Nelkin 2011, 151-52; Levy 2011, 1; Pereboom 2014, 1-2). Indeed, some go so far as to define 'free will' as 'the strongest control condition--whatever that turns out to be--necessary for moral responsibility' (Wolf 1990, 3-4; Fischer 1994, 3; Mele 2006, 17). Given this connection, we can determine whether the freedom to do otherwise and the power of self-determination are constitutive of free will and, if so, in what sense, by considering what it takes to be a morally responsible agent. On these latter characterizations of free will, understanding free will is inextricably linked to, and perhaps even derivative from, understanding moral responsibility. And even those who demur from this claim regarding conceptual priority typically see a close link between these two ideas. Consequently, to appreciate the current debates surrounding the nature of free will, we need to say something about the nature of moral responsibility. It is now widely accepted that there are different species of moral responsibility. It is common (though not uncontroversial) to distinguish moral responsibility as answerability from moral responsibility as attributability from moral responsibility as accountability (Watson 1996; Fischer and Tognazzini 2011; Shoemaker 2011. See Smith (2012) for a critique of this taxonomy). These different species of moral responsibility differ along three dimensions: (i) the kind of responses licensed toward the responsible agent, (ii) the nature of the licensing relation, and (iii) the necessary and sufficient conditions for licensing the relevant kind of responses toward the agent. For example, some argue that when an agent is morally responsible in the attributability sense, certain judgments about the agent--such as judgments concerning the virtues and vices of the agent--are fitting, and that the fittingness of such judgments does not depend on whether the agent in question possessed the freedom to do otherwise (cf. Watson 1996). While keeping this controversy about the nature of moral responsibility firmly in mind (see the entry on moral responsibility for a more detailed discussion of these issues), we think it is fair to say that the most commonly assumed understanding of moral responsibility in the historical and contemporary discussion of the problem of free will is moral responsibility as accountability in something like the following sense: An agent $S$ is morally accountable for performing an action $\phi$ $=\_{df.}$ $S$ deserves praise if $\phi$ goes beyond what can be reasonably expected of $S$ and $S$ deserves blame if $\phi$ is morally wrong. The central notions in this definition are praise, blame, and desert. The majority of contemporary philosophers have followed Strawson (1962) in contending that praising and blaming an agent consist in experiencing (or at least being disposed to experience (cf. Wallace 1994, 70-71)) reactive attitudes or emotions directed toward the agent, such as gratitude, approbation, and pride in the case of praise, and resentment, indignation, and guilt in the case of blame. (See Sher (2006) and Scanlon (2008) for important dissents from this trend. See the entry on blame for a more detailed discussion.) These emotions, in turn, dispose us to act in a variety of ways. For example, blame disposes us to respond with some kind of hostility toward the blameworthy agent, such as verbal rebuke or partial withdrawal of good will. But while these kinds of dispositions are essential to our blaming someone, their manifestation is not: it is possible to blame someone with very little change in attitudes or actions toward the agent. Blaming someone might be immediately followed by forgiveness as an end of the matter. By 'desert', we have in mind what Derk Pereboom has called basic desert: > > > The desert at issue here is basic in the sense that the agent would > deserve to be blamed or praised just because she has performed the > action, given an understanding of its moral status, and not, for > example, merely by virtue of consequentialist or contractualist > considerations. (2014, 2) > > > As we understand desert, if an agent deserves blame, then we have a strong pro tanto reason to blame him simply in virtue of his being accountable for doing wrong. Importantly, these reasons can be outweighed by other considerations. While an agent may deserve blame, it might, all things considered, be best to forgive him unconditionally instead. When an agent is morally responsible for doing something wrong, he is blameworthy: he deserves hard treatment marked by resentment and indignation and the actions these emotions dispose us toward, such as censure, rebuke, and ostracism. However, it would seem unfair to treat agents in these ways unless their actions were up to them. Thus, we arrive at the core connection between free will and moral responsibility: agents deserve praise or blame only if their actions are up to them--only if they have free will. Consequently, we can assess analyses of free will by their implications for judgments of moral responsibility. We note that some might reject the claim that free will is necessary for moral responsibility (e.g., Frankfurt 1971; Stump 1988), but even for these theorists an adequate analysis of free will must specify a sufficient condition for the kind of control at play in moral responsibility. In what follows, we focus our attention on the two most commonly cited features of free will: the freedom to do otherwise and sourcehood. While some seem to think that free will consists exclusively in either the freedom to do otherwise (van Inwagen 2008) or in sourcehood (Zagzebski 2000), many philosophers hold that free will involves both conditions--though philosophers often emphasize one condition over the other depending on their dialectical situation or argumentative purposes (cf. Watson 1987). In what follows, we will describe the most common characterizations of these two conditions. ### 2.2 The Freedom to Do Otherwise For most newcomers to the problem of free will, it will seem obvious that an action is up to an agent only if she had the freedom to do otherwise. But what does this freedom come to? The freedom to do otherwise is clearly a modal property of agents, but it is controversial just what species of modality is at stake. It must be more than mere possibility: to have the freedom to do otherwise consists in more than the mere possibility of something else's happening. A more plausible and widely endorsed understanding claims the relevant modality is ability or power (Locke 1690 [1975], II.xx; Reid 1788 [1969], II.i-ii; D. Locke 1973; Clarke 2009; Vihvelin 2013). But abilities themselves seem to come in different varieties (Lewis 1976; Horgan 1979; van Inwagen 1983, ch. 1; Mele 2003; Clarke 2009; Vihvelin 2013, ch. 1; Franklin 2015; Cyr and Swenson 2019; Hofmann 2022; Whittle 2022), so a claim that an agent has 'the ability to do otherwise' is potentially ambiguous or indeterminate; in philosophical discussion, the sense of ability appealed to needs to be spelled out. A satisfactory account of the freedom to do otherwise owes us both an account of the kind of ability in terms of which the freedom to do otherwise is analyzed, and an argument for why this kind of ability (as opposed to some other species) is the one constitutive of the freedom to do otherwise. As we will see, philosophers sometimes leave this second debt unpaid. The contemporary literature takes its cue from classical compatibilism's recognized failure to deliver a satisfactory analysis of the freedom to do otherwise. As we saw above, classical compatibilists (Hobbes 1654 [1999], 1656 [1999]; Locke 1690 [1975]; Hume 1740 [1978], 1748 [1975]; Edwards 1754 [1957]; Moore 1912; Schlick 1939; Ayer 1954) sought to analyze the freedom to do otherwise in terms of a simple conditional analysis of ability: Simple Conditional Analysis: An agent $S$ has the ability to do otherwise if and only if, were $S$ to choose to do otherwise, then $S$ would do otherwise. Part of the attraction of this analysis is that it obviously reconciles the freedom to do otherwise with determinism. While the truth of determinism entails that one's action is inevitable given the past and laws of nature, there is nothing about determinism that implies that if one had chosen otherwise, then one would not do otherwise. There are two problems with the Simple Conditional Analysis. The first is that it is, at best, an analysis of free action, not free will (cf. Reid 1788 [1969]; Chisholm 1966; 1976, ch. 2; Lehrer 1968, 1976). It only tells us when an agent has the ability to do otherwise, not when an agent has the ability to choose to do otherwise. One might be tempted to think that there is an easy fix along the following lines: *Simple Conditional Analysis\: An agent $S$ has the ability to choose otherwise if and only if, were $S$ to desire or prefer to choose otherwise, then $S$ would choose otherwise. The problem is that we often fail to choose to do things we want to choose, even when it appears that we had the ability to choose otherwise (one might think the same problem attends the original analysis). Suppose that, in deciding how to spend my evening, I have a desire to choose to read and a desire to choose to watch a movie. Suppose that I choose to read. By all appearances, I had the ability to choose to watch a movie. And yet, according to the Simple Conditional Analysis\**, I lack this freedom, since the conditional 'if I were to desire to choose to watch a movie, then I would choose to watch a movie' is false. I do desire to choose to watch a movie and yet I do not choose to watch a movie. It is unclear how to remedy this problem. On the one hand, we might refine the antecedent by replacing 'desire' with 'strongest desire' (cf. Hobbes 1654 [1999], 1656 [1999]; Edwards 1754 [1957]). The problem is that this assumes, implausibly, that we always choose what we most strongly desire (for criticisms of this view see Reid 1788 [1969]; Campbell 1951; Wallace 1999; Holton 2009). On the other hand, we might refine the consequent by replacing 'would choose to do otherwise' with either 'would probably choose to do otherwise' or 'might choose to do otherwise'. But each of these proposals is also problematic. If 'probably' means 'more likely than not', then this revised conditional still seems too strong: it seems possible to have the ability to choose otherwise even when one's so choosing is unlikely. If we opt for 'might', then the relevant sense of modality needs to be spelled out. Even if there are fixes to these problems, there is a yet deeper problem with these analyses. There are some agents who clearly lack the freedom to do otherwise and yet satisfy the conditional at the heart of these analyses. That is, although these agents lack the freedom to do otherwise, it is, for example, true of them that if* they chose otherwise, they would do otherwise. Picking up on an argument developed by Keith Lehrer (1968; cf. Campbell 1951; Broad 1952; Chisholm 1966), consider an agoraphobic, Luke, who, when faced with the prospect of entering an open space, is subject not merely to an irresistible desire to refrain from intentionally going outside, but an irresistible desire to refrain from even choosing to go outside. Given Luke's psychology, there is no possible world in which he suffers from his agoraphobia and chooses to go outside. It may well nevertheless be true that if Luke chose to go outside, then he would have gone outside. After all, any possible world in which he chooses to go outside will be a world in which he no longer suffers (to the same degree) from his agoraphobia, and thus we have no reason to doubt that in those worlds he would go outside as a result of his choosing to go outside. The same kind of counterexample applies with equal force to the conditional 'if $S$ desired to choose otherwise, then $S$ would choose otherwise'. While simple conditional analyses admirably make clear the species of ability to which they appeal, they fail to show that this species of ability is constitutive of the freedom to do otherwise. Agents need a stronger ability to do otherwise than characterized by such simple conditionals. Some argue that the fundamental source of the above problems is the conditional nature of these analyses (Campbell 1951; Austin 1961; Chisholm 1966; Lehrer 1976; van Inwagen 1983, ch. 4). The sense of ability relevant to the freedom to do otherwise is the 'all-in sense'--that is, holding everything fixed up to the time of the decision or action--and this sense, so it is argued, can only be captured by a categorical analysis of the ability to do otherwise: Categorical Analysis: An agent $S$ has the ability to choose or do otherwise than $\phi$ at time $t$ if and only if it was possible, holding fixed everything up to $t$, that $S$ choose or do otherwise than $\phi$ at $t$. This analysis gets the right verdict in Luke's case. He lacks the ability to do otherwise than refrain from choosing to go outside, according to this analysis, because there is no possible world in which he suffers from his agoraphobia and yet chooses to go outside. Unlike the above conditional analyses, the Categorical Analysis requires that we hold fixed Luke's agoraphobia when considering alternative possibilities. If the Categorical Analysis is correct, then free will is incompatible with determinism. According to the thesis of determinism, all deterministic possible worlds with the same pasts and laws of nature have the same futures (Lewis 1979; van Inwagen 1983, 3). Suppose John is in deterministic world $W$ and refrains from raising his hand at time $t$. Since $W$ is deterministic, it follows that any possible world $W^\$ that has the same past and laws up to $t$ must have the same future, including John's refraining from raising his hand at $t$. Therefore, John lacked the ability, and thus freedom, to raise his hand. This argument, carefully articulated in the late 1960s and early 1970s by Carl Ginet (1966, 1990) and Peter van Inwagen (1975, 1983) and refined in important ways by John Martin Fischer (1994), has come to be known as the Consequence Argument. van Inwagen offers the following informal statement of the argument: > > > If determinism is true, then our acts are the consequences of the laws > of nature and events in the remote past. But it is not up to us what > went on before we were born [i.e., we do not have the ability to > change the past], and neither is it up to us what the laws of nature > are [i.e., we do not have the ability to break the laws of nature]. > Therefore, the consequences of these things (including our present > acts) are not up to us. (van Inwagen 1983, 16; cf. Fischer 1994, ch. > 1) > > > Like the Simple Conditional Analysis, a virtue of the Categorical Analysis* is that it spells out clearly the kind of ability appealed to in its analysis of the freedom to do otherwise, but like the Simple Conditional Analysis, critics have argued that the sense of ability it captures is not the sense at the heart of free will. The objection here, though, is not that the analysis is too permissive or weak, but rather that it is too restrictive or strong. While there have been numerous different replies along these lines (e.g., Lehrer 1980; Slote 1982; Watson 1986. See the entry on arguments for incompatibilism for a more extensive discussion of and bibliography for the Consequence Argument), the most influential of these objections is due to David Lewis (1981). Lewis contended that van Inwagen's argument equivocated on 'is able to break a law of nature'. We can distinguish two senses of 'is able to break a law of nature': (Weak Thesis) I am able to do something such that, if I did it, a law of nature would be broken. (Strong Thesis) I am able to do something such that, if I did it, it would constitute a law of nature's being broken or would cause a law of nature to be broken. If we are committed to the Categorical Analysis, then those desiring to defend compatibilism seem to be committed to the sense of ability in 'is able to break a law of nature' along the lines of the strong thesis. Lewis agrees with van Inwagen that it is "incredible" to think humans have such an ability (Lewis 1981, 113), but maintains that compatibilists need only appeal to the ability to break a law of nature in the weak sense. While it is absurd to think that humans are able to do something that is a violation of a law of nature or causes a law of nature to be broken, there is nothing incredible, so Lewis claimed, in thinking that humans are able to do something such that if they did it, a law of nature would be broken. In essence, Lewis is arguing that incompatibilists like van Inwagen have failed to adequately motivate the restrictiveness of the Categorical Analysis. Some incompatibilists have responded to Lewis by contending that even the weak ability is incredible (van Inwagen 2004). But there is a different and often overlooked problem for Lewis: the weak ability seems to be too weak. Returning to the case of John's refraining from raising his hand, Lewis maintains that the following three propositions are consistent: (i) John is able to raise his hand. (ii) A necessary condition for John's raising his hand fails to obtain (i.e., that the laws of nature or past are different than they actually are). (iii) John is not able to do anything that would constitute this necessary condition's obtaining or cause this necessary condition to obtain (i.e., he is unable to do anything that would constitute or cause a law of nature to be broken or the past to be different). One might think that (ii) and (iii) are incompatible with (i). Consider again Luke, our agoraphobic. Suppose that his agoraphobia affects him in such a way that he will only intentionally go outside if he chooses to go outside, and yet his agoraphobia makes it impossible for him to make this choice. In this case, a necessary condition for Luke's intentionally going outside is his choosing to go outside. Moreover, Luke is not able to choose or cause himself to choose to go outside. Intuitively, this would seem to imply that Luke lacks the freedom to go outside. But this implication does not follow for Lewis. From the fact that Luke is able to go outside only if he chooses to go outside and the fact that Luke is not able to choose to go outside, it does not follow, on Lewis's account, that Luke lacks the ability to go outside. Consequently, Lewis's account fails to explain why Luke lacks the ability to go outside (cf. Speak 2011). (For other important criticisms of Lewis, see Ginet [1990, ch. 5] and Fischer [1994, ch. 4].) While Lewis may be right that the Categorical Analysis is too restrictive, his argument, all by itself, doesn't seem to establish this. His argument is successful only if (a) he can provide an alternative analysis of ability that entails that Luke's agoraphobia robs him of the ability to go outside and (b) does not entail that determinism robs John of the ability to raise his hand (cf. Pendergraft 2010). Lewis must point out a principled difference between these two cases. As should be clear from the above, the Simple Conditional Analysis is of no help. However, some recent work by Michael Smith (2003), Kadri Vihvelin (2004; 2013), and Michael Fara (2008) have attempted to fill this gap. What unites these theorists--whom Clarke (2009) has called the 'new dispositionalists'--is their attempt to appeal to recent advances in the metaphysics of dispositions to arrive at a revised conditional analysis of the freedom to do otherwise. The most perspicuous of these accounts is offered by Vihvelin (2004), who argues that an agent's having the ability to do otherwise is solely a function of the agent's intrinsic properties. (It is important to note that Vihvelin [2013] has come to reject the view that free will consists exclusively in the kind of ability analyzed below.) Building on Lewis's work on the metaphysics of dispositions, she arrives at the following analysis of ability: Revised Conditional Analysis of Ability: $S$ has the ability at time $t$ to do $X$ iff, for some intrinsic property or set of properties $B$ that $S$ has at $t$, for some time $t'$ after $t$, if $S$ chose (decided, intended, or tried) at $t$ to do $X$, and $S$ were to retain $B$ until $t'$, $S$'s choosing (deciding, intending, or trying) to do $X$ and $S$'s having $B$ would jointly be an $S$-complete cause of $S$'s doing $X$. (Vihvelin 2004, 438) Lewis defines an '$S$-complete cause' as "a cause complete insofar as havings of properties intrinsic to [$S$] are concerned, though perhaps omitting some events extrinsic to [$S$]" (cf. Lewis 1997, 156). In other words, an $S$-complete cause of $S$'s doing $\phi$ requires that $S$ possess all the intrinsic properties relevant to $S$'s causing $S$'s doing $\phi$. This analysis appears to afford Vihvelin the basis for a principled difference between agoraphobics and merely determined agents. We must hold fixed an agent's phobias since they are intrinsic properties of agents, but we need not hold fixed the laws of nature because these are not intrinsic properties of agents. (It should be noted that the assumption that intrinsic properties are wholly separable from the laws of nature is disputed by 'dispositional essentialists.' See the entry on metaphysics of causation.) Vihvelin's analysis appears to be restrictive enough to exclude phobics from having the freedom to do otherwise, but permissive enough to allow that some agents in deterministic worlds have the freedom to do otherwise. But appearances can be deceiving. The new dispositionalist claims have received some serious criticism, with the majority of the criticisms maintaining that these analyses are still too permissive (Clarke 2009; Whittle 2010; Franklin 2011b). For example, Randolph Clarke argues that Vihvelin's analysis fails to overcome the original problem with the Simple Conditional Analysis. He writes, "A phobic agent might, on some occasion, be unable to choose to A and unable to A without so choosing, while retaining all that she would need to implement such a choice, should she make it. Despite lacking the ability to choose to A, the agent might have some set of intrinsic properties B such that, if she chose to A and retained B, then her choosing to A and her having B would jointly be an agent-complete cause of her A-ing" (Clarke 2009, p. 329). The Categorical Analysis, and thus incompatibilism about free will and determinism, remains an attractive option for many philosophers precisely because it seems that compatibilists have yet to furnish an analysis of the freedom to do otherwise that implies that phobics clearly lack the ability to choose or do otherwise that is relevant to moral responsibility and yet some merely determined agents have this ability. ### 2.3 Freedom to Do Otherwise vs. Sourcehood Accounts Some have tried to avoid these lingering problems for compatibilists by arguing that the freedom to do otherwise is not required for free will or moral responsibility. What matters for an agent's freedom and responsibility, so it is argued, is the source of her action--how her action was brought about. The most prominent strategy for defending this move appeals to 'Frankfurt-style cases'. In a ground-breaking article, Harry Frankfurt (1969) presented a series of thought experiments intended to show that it is possible that agents are morally responsible for their actions and yet they lack the ability to do otherwise. While Frankfurt (1971) took this to show that moral responsibility and free will come apart--free will requires the ability to do otherwise but moral responsibility does not--if we define 'free will' as 'the strongest control condition required for moral responsibility' (cf. Wolf 1990, 3-4; Fischer 1994, 3; Mele 2006, 17), then if Frankfurt-style cases show that moral responsibility does not require the ability to do otherwise, then they also show that free will does not require the ability to do otherwise. Let us consider this challenge in more detail. Here is a representative Frankfurt-style case: > > Imagine, if you will, that Black is a quite nifty (and even generally > nice) neurosurgeon. But in performing an operation on Jones to remove > a brain tumor, Black inserts a mechanism into Jones's brain > which enables Black to monitor and control Jones's activities. > Jones, meanwhile, knows nothing of this. Black exercises this control > through a sophisticated computer which he has programmed so that, > among other things, it monitors Jones's voting behavior. If > Jones were to show any inclination to vote for Bush, then the > computer, through the mechanism in Jones's brain, intervenes to > ensure that he actually decides to vote for Clinton and does so vote. > But if Jones decides on his own to vote for Clinton, the computer does > nothing but continue to monitor--without affecting--the > goings-on in Jones's head. (Fischer 2006, 38) > Fischer goes on to suppose that Jones "decides to vote for Clinton on his own", without any interference from Black, and maintains that in such a case Jones is morally responsible for his decision. Fischer draws two interrelated conclusions from this case. The first, negative conclusion, is that the ability to do otherwise is not necessary for moral responsibility. Jones is unable to refrain from deciding to vote for Clinton, and yet, so long as Jones decides to vote for Clinton on his own, his decision is free and one for which he is morally responsible. The second, positive conclusion, is that freedom and responsibility are functions of the actual sequence. What matters for an agent's freedom and moral responsibility is not what might have happened, but how his action was actually brought about. What matters is not whether the agent had the ability to do otherwise, but whether he was the source of his actions. The success of Frankfurt-style cases is hotly contested. An early and far-reaching criticism is due to David Widerker (1995), Carl Ginet (1996), and Robert Kane (1996, 142-43). According to this criticism, proponents of Frankfurt-style cases face a dilemma: either these cases assume that the connection between the indicator (in our case, the absence of Jones's showing any inclination to decide to vote for Bush) and the agent's decision (here, Jones's deciding to vote for Clinton) is deterministic or not. If the connection is deterministic, then Frankfurt-style cases cannot be expected to convince incompatibilists that the ability to do otherwise is not necessary for moral responsibility and/or free will, since Jones's action will be deterministically brought about by factors beyond his control, leading incompatibilists to conclude that Jones is not morally responsible for his decision. But if the connection is nondeterministic, then it is possible even in the absence of showing any inclination to decide to vote for Bush, that Jones decides to vote for Bush, and so he retains the ability to do otherwise. Either way Frankfurt-style cases fail to show that Jones is both morally responsible for his decision and yet is unable to do otherwise. While some have argued that even Frankfurt-style cases that assume determinism are effective (see, e.g., Fischer 1999, 2010, 2013 and Haji and McKenna 2004 and for criticisms of this approach, see Goetz 2005, Palmer 2005, 2014, Widerker and Goetz 2013, and Cohen 2017), the majority of proponents of Frankfurt-style cases have attempted to revise these cases so that they are explicitly nondeterministic and yet still show that the agent was morally responsible even though he lacked the ability to do otherwise--or, at least that he lacked any ability to do otherwise that could be relevant to grounding the agent's moral responsibility (see, e.g., Mele and Robb 1998, 2003, Pereboom 2001, 2014, McKenna 2003, Hunt 2005, and for criticisms of these cases see Ginet 2002, Timpe 2006, Widerker 2006, Franklin 2011c, Moya 2011, Palmer 2011, 2013, Robinson 2014, Capes 2016, Capes and Swenson 2017, and Elzein 2017). Supposing that Frankfurt-style cases are successful, what exactly do they show? In our view, they show neither that free will and moral responsibility do not require an ability to do otherwise in any sense nor that compatibilism is true. Frankfurt-style cases are of clear help to the compatibilists' position (though see Speak 2007 for a dissenting opinion). The Consequence Argument raises a powerful challenge to the cogency of compatibilism. But if Frankfurt-style cases are successful, agents can act freely in the sense relevant to moral responsibility while lacking the ability to do otherwise in the all-in sense. This allows compatibilists to concede that the all-in ability to do otherwise is incompatible with determinism, and yet insist that it is irrelevant to the question of the compatibility of determinism with moral responsibility (and perhaps even free will, depending on how we define this) (cf. Fischer 1987, 1994. For a challenge to the move from not strictly necessary to irrelevant, see O'Connor [2000, 20-22] and in reply, Fischer [2006, 152-56].). But, of course, showing that an argument for the falsity of compatibilism is irrelevant does not show that compatibilism is true. Indeed, many incompatibilists maintain that Frankfurt-style cases are successful and defend incompatibilism not via the Consequence Argument, but by way of arguments that attempt to show that agents in deterministic worlds cannot be the 'source' of their actions in the way that moral responsibility requires (Stump 1999; Zagzebski 2000; Pereboom 2001, 2014). Thus, if successful, Frankfurt-style cases would be at best the first step in defending compatibilism. The second step must offer an analysis of the kind of sourcehood constitutive of free will that entails that free will is compatible with determinism (cf. Fischer 1982). Furthermore, while proponents of Frankfurt-style cases often maintain that these cases show that no ability to do otherwise is necessary for moral responsibility ("I have employed the Frankfurt-type example to argue that this sense of control [i.e. the one required for moral responsibility] need not involve any alternative possibilities" [Fischer 2006, p. 40; emphasis ours]), we believe that this conclusion overreaches. At best, Frankfurt-style cases show that the ability to do otherwise in the all-in sense--in the sense defined by the Categorical Analysis--is not necessary for free will or moral responsibility (cf. Franklin 2015). To appreciate this, let us assume that in the above Frankfurt-style case Jones lacks the ability to do otherwise in the all-in sense: there is no possible world in which we hold fixed the past and laws and yet Jones does otherwise, since all such worlds include Black and his preparations for preventing Jones from doing otherwise should Jones show any inclination. Even if this is all true, it should take only a little reflection to recognize that in this case Jones is able to do otherwise in certain weaker senses we might attach to that phrase, and compatibilists in fact still think that the ability to do otherwise in some such senses is necessary for free will and moral responsibility. Consequently, even though Frankfurt-style cases have, as a matter of fact, moved many compatibilists away from emphasizing ability to do otherwise to emphasizing sourcehood, we suggest that this move is best seen as a weakening of the ability-to-do-otherwise condition on moral responsibility (but see Cyr 2017 and Kittle 2019 for criticisms of this claim). (A potentially important exception to this claim is Sartorio [2016], who appealing to some controversial ideas in the metaphysics of causation appears to argue that no sense of the ability to do otherwise is necessary for control in the sense at stake for moral responsibility, but instead what matters is whether the agent is the cause of the action. We simply note that Sartorio's account of causation is a modal one [see especially Sartorio (2016, 94-95, 132-37)] and thus it is far from clear that her account of freedom and responsibility is really an exception.) ### 2.4 Compatibilist Accounts of Sourcehood In this section, we will assume that Frankfurt-style cases are successful in order to consider two prominent compatibilist attempts to construct analyses of the sourcehood condition (though see the entry on compatibilism for a more systematic survey of compatibilist theories of free will). The first, and perhaps most popular, compatibilist model is a reasons-responsiveness model. According to this model, an agent's action $\phi$ is free just in case the agent or manner in which the action is brought about is responsive to the reasons available to the agent at the time of action. While compatibilists develop this kind of account in different ways, the most detailed proposal is due to John Martin Fischer (1994, 2006, 2010, 2012; Fischer and Ravizza 1998. For similar compatibilist treatments of reasons-responsiveness, see Wolf 1990, Wallace 1994, Haji 1998, Nelkin 2011, McKenna 2013, Vargas 2013, Sartorio 2016). Fischer and Ravizza argue that an agent's action is free and one for which he is morally responsible only if the mechanism that issued in the action is moderately reasons-responsive (Fischer and Ravizza 1998, ch. 3). By 'mechanism', Fischer and Ravizza simply mean "the way the action was brought about" (38). One mechanism they often discuss is practical deliberation. For example, in the case of Jones discussed above, his decision to vote for Clinton on his own was brought about by the process of practical deliberation. What must be true of this process, this mechanism, for it to be moderately reasons-responsive? Fischer and Ravizza maintain that moderate reasons-responsiveness consists in two conditions: reasons-receptivity and reasons-reactivity. A mechanism's reasons-receptivity depends on the agent's cognitive capacities, such as being capable of understanding moral reasons and the implications of their actions (69-73). The second condition is more important for us in the present context. A mechanism's reasons-reactivity depends on how the mechanism would react given different reasons for action. Fischer and Ravizza argue that the kind of reasons-reactivity at stake is weak reasons-reactivity, where this merely requires that there is some possible world in which the laws of nature remain the same, the same mechanism operates, there is a sufficient reason to do otherwise, and the mechanism brings about the alternative action in response to this sufficient reason (73-76). On this analysis, while Jones, due to the activity of Black, lacks the 'all-in' sense of the ability to do otherwise, he is nevertheless morally responsible for deciding to vote for Clinton because his action finds its source in Jones's practical deliberation that is moderately reasons-responsive. Fischer and Ravizza's theory of freedom and responsibility has shifted the focus of much recent debate to questions of sourcehood. Moreover, one might argue that this theory is a clear improvement over classical compatibilism with respect to handling cases of phobia. By focusing on mechanisms, Fischer and Ravizza can argue that our agoraphobic Luke is not morally responsible for deciding to refrain from going outside because the mechanism that issues in this action--namely his agoraphobia--is not moderately reasons-responsive. There is no world with the same laws of nature as our own, this mechanism operates, and yet it reacts to a sufficient reason to go outside. No matter what reasons there are for Luke to go outside, when acting on this mechanism, he will always refrain from going outside (cf. Fischer 1987, 74). Before turning to our second compatibilist model, it is worth noting that it would be a mistake to think that Fischer and Ravizza's account is a sourcehood account to the exclusion of the ability to do otherwise in any sense. As we have just seen, Fischer and Ravizza place clear modal requirements on mechanisms that issue in actions with respect to which agents are free and morally responsible. Indeed, this should be clear from the very idea of reasons-responsiveness. Whether one is responsive depends not merely on how one does respond, but also on how one would respond. Thus, any account that makes reasons-responsiveness an essential condition of free will is an account that makes the ability to do otherwise, in some sense, necessary for free will (Fischer [2018] concedes this point, though, as noted above, the reader should consider Sartorio [2016] as a potential counterexample to this claim). The second main compatibilist model of sourcehood is an identification model. Accounts of sourcehood of this kind lay stress on self-determination or autonomy: to be the source of her action the agent must self-determine her action. Like the contemporary discussion of the ability to do otherwise, the contemporary discussion of the power of self-determination begins with the failure of classical compatibilism to produce an acceptable definition. According to classical compatibilists, self-determination simply consists in the agent's action being determined by her strongest motive. On the assumption that some compulsive agents' compulsions operate by generating irresistible desires to act in certain ways, the classical compatibilist analysis of self-determination implies that these compulsive actions are self-determined. While Hobbes seems willing to accept this implication (1656 [1999], 78), most contemporary compatibilists concede that this result is unacceptable. Beginning with the work of Harry Frankfurt (1971) and Gary Watson (1975), many compatibilists have developed identification accounts of self-determination that attempt to draw a distinction between an agent's desires or motives that are internal to the agent and those that are external. The idea is that while agents are not (or at least may not be) identical to any motivations (or bundle of motivations), they are identified with a subset of their motivations, rendering these motivations internal to the agent in such a way that any actions brought about by these motivations are self-determined. The identification relation is not an identity relation, but something weaker (cf. Bratman 2000, 39n12). What the precise nature of the identification relation is and to which attitudes an agent stands in this relation is hotly disputed. Lippert-Rasmussen (2003) helpfully divides identification accounts into two main types. The first are "authority" accounts, according to which agents are identified with attitudes that are authorized to speak for them (368). The second are authenticity accounts, according to which agents are identified with attitudes that reveal who they truly are (368). (But see Shoemaker 2015 for an ecumenical account of identification that blends these two accounts.) Proposed attitudes to which agents are said to stand in the identification relation include higher-order desires (Frankfurt 1971), cares or loves (Frankfurt 1993, 1994; Shoemaker 2003; Jaworska 2007; Sripada 2016), self-governing policies (Bratman 2000), the desire to make sense of oneself (Velleman 1992, 2009), and perceptions (or judgments) of the good (or best) (Watson 1975; Stump 1988; Ekstrom 1993; Mitchell-Yellin 2015). The distinction between internal and external motivations allows identification theorists to enrich classical compatibilists' understanding of constraint, while remaining compatibilists about free will and determinism. According to classical compatibilists, the only kind of constraint is external (e.g., broken cars and broken legs), but addictions and phobias seem just as threatening to free will. Identification theorists have the resources to concede that some constraints are internal. For example, they can argue that our agoraphobic Luke is not free in refraining from going outside even though this decision was caused by his strongest desires because he is not identified with his strongest desires. On compatibilist identification accounts, what matters for self-determination is not whether our actions are determined or undetermined, but whether they are brought about by motives with which the agent is identified: exercises of the power of self-determination consists in an agent's actions being brought about, in part, by an agent's motives with which she is identified. (It is important to note that while we have distinguished reasons-responsive accounts from identification accounts, there is nothing preventing one from combing both elements in a complete analysis of free will.) Even if these reasons-responsive and identification compatibilist accounts of sourcehood might successfully side-step the Consequence Argument, they must come to grips with a second incompatibilist argument: the Manipulation Argument. The general problem raised by this line of argument is that whatever proposed compatibilist conditions for an agent $S$'s being free with respect to, and morally responsible for, some action $\phi$, it will seem that agents can be manipulated into satisfying these conditions with respect to $\phi$ and, yet, precisely because they are manipulated into satisfying these conditions, their freedom and responsibility seem undermined. The two most influential forms of the Manipulation Argument are Pereboom's Four-case Argument (2001, ch. 4; 2014, ch. 4) and Mele's Zygote Argument (2006, ch. 7. See Todd 2010, 2012 for developments of Mele's argument). As the structure of Mele's version is simpler, we will focus on it. Imagine a goddess Diana who creates a zygote $Z$ in Mary in some deterministic world. Suppose that Diana creates $Z$ as she does because she wants Jones to be murdered thirty years later. From her knowledge of the laws of nature in her world and her knowledge of the state of the world just prior to her creating $Z$, she knows that a zygote with precisely $Z$'s constitution located in Mary will develop into an agent Ernie who, thirty years later, will murder Jones as a result of his moderately reasons-responsive mechanism and on the basis of motivations with which he is identified (whatever those might be). Suppose Diana succeeds in her plan and Ernie murders Jones as a result of her manipulation. Many judge that Ernie is not morally responsible for murdering Jones even though he satisfies both the reasons-responsive and identification criteria. There are two possible lines of reply open to compatibilists. On the soft-line reply, compatibilists attempt to show that there is a relevant difference between manipulated agents such as Ernie and agents who satisfy their account (McKenna 2008, 470). For example, Fischer and Ravizza propose a second condition on sourcehood: in addition to a mechanism's being moderately reasons-responsive, an agent is morally responsible for the output of such a mechanism only if the agent has come to take responsibility for the mechanism, where an agent has taken responsibility for a mechanism $M$ just in case (i) she believes that she is an agent when acting from $M$, (ii) she believes that she is an apt target for blame and praise for acting from $M$, and (iii) her beliefs specified in (i) and (ii) are "based, in an appropriate way, on [her] evidence" (Fischer and Ravizza 1998, 238). The problem with this reply is that we can easily imagine Diana creating Ernie so that his murdering Jones is a result not only of a moderately reasons-responsive mechanism, but also a mechanism for which he has taken responsibility. On the hard-line reply, compatibilists concede that, despite initial appearances, the manipulated agent is free and morally responsible and attempt to ameliorate the seeming counterintuitiveness of this concession (McKenna 2008, 470-71). Here compatibilists might point out that the idea of being manipulated is worrisome only so long as the manipulators are interfering with an agent's development. But if the manipulators simply create a person, and then allow that person's life to unfold without any further inference, the manipulators' activity is no threat to freedom (McKenna 2008; Fischer 2011; Sartorio 2016, ch. 5). (For other responses to the Manipulation Argument, see Kearns 2012; Sripada 2012; McKenna 2014.) ### 2.5 Libertarian Accounts of Sourcehood Despite these compatibilist replies, to some the idea that the entirety of a free agent's life can be determined, and in this way controlled, by another agent will seem incredible. Some take the lesson of the Manipulation Argument to be that no compatibilist account of sourcehood or self-determination is satisfactory. True sourcehood--the kind of sourcehood that can actually ground an agent's freedom and responsibility--requires, so it is argued, that one's action not be causally determined by factors beyond one's control. Libertarians, while united in endorsing this negative condition on sourcehood, are deeply divided concerning which further positive conditions may be required. It is important to note that while libertarians are united in insisting that compatibilist accounts of sourcehood are insufficient, they are not committed to thinking that the conditions of freedom spelled out in terms either of reasons-responsiveness or of identification are not necessary. For example, Stump (1988, 1996, 2010) builds a sophisticated libertarian model of free will out of resources originally developed within Frankfurt's identification model (see also Ekstrom 1993, 2000; Franklin 2014) and nearly all libertarians agree that exercises of free will require agents to be reasons-responsive (e.g., Kane 1996; Clarke 2003, chs. 8-9; Franklin 2018, ch. 2). Moreover, while this section focuses on libertarian accounts of sourcehood, we remind readers that most (if not all) libertarians think that the freedom to do otherwise is also necessary for free will and moral responsibility. There are three main libertarian options for understanding sourcehood or self-determination: non-causal libertarianism (Ginet 1990, 2008; McCann 1998; Lowe 2008; Goetz 2009; Pink 2017; Palmer 2021), event-causal libertarianism (Wiggins 1973; Kane 1996, 1999, 2011, 2016; Mele 1995, chs. 11-12; 2006, chs. 4-5; 2017; Ekstrom 2000, 2019; Clarke 2003, chs. 2-6; Franklin 2018), and agent-causal libertarianism (Reid 1788 [1969]; Chisholm 1966, 1976; Taylor 1966; O'Connor 2000; Clarke 1993; 1996; 2003, chs. 8-10; Griffith 2010; Steward 2012). Non-causal libertarians contend that exercises of the power of self-determination need not (or perhaps even cannot) be caused or causally structured. According to this view, we control our volition or choice simply in virtue of its being ours--its occurring in us. We do not exert a special kind of causality in bringing it about; instead, it is an intrinsically active event, intrinsically something we do. While there may be causal influences upon our choice, there need not be, and any such causal influence is wholly irrelevant to understanding why it occurs. Reasons provide an autonomous, non-causal form of explanation. Provided our choice is not wholly determined by prior factors, it is free and under our control simply in virtue of being ours. Non-causal views have failed to garner wide support among libertarians since, for many, self-determination seems to be an essentially causal notion (cf. Mele 2000 and Clarke 2003, ch. 2). This dispute hinges on the necessary conditions on the concept of causal power, and relatedly on whether power simpliciter admits causal and non-causal variants. For discussion, see O'Connor (2021). Most libertarians endorse an event-causal or agent-causal account of sourcehood. Both these accounts maintain that exercises of the power of self-determination consist partly in the agent's bringing about her choice or action, but they disagree on how to analyze an agent's bringing about her choice. While event-causal libertarianism admits of different species, at the heart of this view is the idea that self-determining an action requires, at minimum, that the agent cause the action and that an agent's causing his action is wholly reducible to mental states and other events involving the agent nondeviantly causing his action. Consider an agent's raising his hand. According to the event-causal model at its most basic level, an agent's raising his hand consists in the agent's causing his hand to rise and his causing his hand to rise consists in apt mental states and events involving the agent--such as the agent's desire to ask a question and his belief that he can ask a question by raising his hand--nondeviantly causing his hand to rise. (The nondeviance clause is required since it seems possible that an event be brought about by one's desires and beliefs and yet not be self-determined, or even an action for that matter, due to the unusual causal path leading from the desires and beliefs to action. Imagine a would-be accomplice of an assassin believes that his dropping his cigarette is the signal for the assassin to shoot his intended victim and he desires to drop his cigarette and yet this belief and desire so unnerve him that he accidentally drops his cigarette. While the event of dropping the cigarette is caused by a relevant desire and belief it does not seem to be self-determined and perhaps is not even an action [cf. Davidson 1973].) To fully spell out this account, event-causal libertarians must specify which mental states and events are apt (cf. Brand 1979)--which mental states and events are the springs of self-determined actions--and what nondeviance consists in (cf. Bishop 1989). (We note that this has proven very difficult, enough so that some take the problem to spell doom for event-causal theories of action. Such philosophers [e.g., Taylor 1966 and Sehon 2005] take agential power to be conceptually and/or ontologically primitive and understand reasons explanations of action in irreducibly teleological terms. See Stout 2010 for a brisk survey of discussions of this topic.) For ease, in what follows we will assume that apt mental states are an agent's reasons that favor the action. Event-causal libertarians, of course, contend that self-determination requires more than nondeviant causation by agents' reasons: for it is possible that agents' actions in deterministic worlds are nondeviantly caused by apt mental states and events. Self-determination requires nondeterministic causation, in a nondeviant way, by an agent's reasons. While historically many have thought that nondeterministic causation is impossible (Hobbes 1654 [1999], 1656 [1999]; Hume 1740 [1978], 1748 [1975]), with the advent of quantum physics and, from a very different direction, an influential essay by G.E.M. Anscombe (1971), it is now widely assumed that nondeterministic (or probabilistic) causation is possible. There are two importantly different ways to understand nondeterministic causation: as the causation of probability or as the probability of causation. Under the causation of probability model, a nondeterministic cause $C$ causes (or causally contributes to) the objective probability of the outcome's occurring rather than the outcome itself. On this account, $S$'s reasons do not cause his decision but there being a certain antecedent objective probability of its occurring, and the decision itself is uncaused. On the competing probability of causation model, a nondeterministic cause $C$ causes the outcome of a nondeterministic process. Given that $C$ is a nondeterministic cause of the outcome, it was possible given the exact same past and laws of nature that $C$ not cause the outcome (perhaps because it was possible that some other event cause some other outcome)--the probability of this causal transaction's occurring was less than $1$. Given that event-causal libertarians maintain that self-determined actions, and thus free actions, must be caused, they are committed to the probability of causation model of nondeterministic causation (cf. Franklin 2018, 25-26). (We note that Balaguer [2010] is skeptical of the above distinction, and it is thus unclear whether he should best be classified as a non-causal or event-causal libertarian, though see Balaguer [2014] for evidence that it is best to treat him as a non-causalist.) Consequently, according to event-causal libertarians, when an agent $S$ self-determines his choice $\phi$, then $S$'s reasons $r\_1$ nondeterministically cause (in a nondeviant way) $\phi$, and it was possible, given the past and laws, that $r\_1$ not have caused $\phi$, but rather some of $S$'s other reasons $r\_2$ nondeterministically caused (in a nondeviant way) a different action $\psi$. Agent-causal libertarians contend that the event-causal picture fails to capture self-determination, for it fails to accord the agent with a power to settle what she does. Pereboom offers a forceful statement of this worry: > > > On an event-causal libertarian picture, the relevant causal conditions > antecedent to the decision, i.e., the occurrence of certain > agent-involving events, do not settle whether the decision will occur, > but only render the occurrence of the decision about $50\%$ > probable. In fact, because no occurrence of antecedent events settles > whether the decision will occur, and only antecedent events are > causally relevant, nothing settles whether the decision will > occur. (Pereboom 2014, 32; cf. Watson 1987, 1996; Clarke 2003 [ch. 8], 2011; Griffith 2010; Shabo 2011, 2013; > Steward 2012 [ch. 3]; and Schlosser 2014); and for critical > assessment, see Clarke 2019. > > > On the event-causal picture, the agent's causal contribution to her actions is exhausted by the causal contribution of her reasons, and yet her reasons leave open which decisions she will make, and this seems insufficient for self-determination. But what more must be added? Agent-causal libertarians maintain that self-determination requires that the agent herself play a causal role over and above the causal role played by her reasons. Some agent-causal libertarians deny that an agent's reasons play any direct causal role in bringing about an agent's self-determined actions (Chisholm 1966; O'Connor 2000, ch. 5), whereas others allow or even require that self-determined actions be caused in part by the agent's reasons (Clarke 2003, ch. 9; Steward 2012, ch. 3). But all agent-causal libertarians insist that exercises of the power of self-determination do not reduce to nondeterministic causation by apt mental states: agent-causation does not reduce to event-causation. Agent-causal libertarianism seems to capture an aspect of self-determination that neither the above compatibilists accounts nor event-causal libertarian accounts capture. (Some compatibilists even accept this and try to incorporate agent-causation into a compatibilist understanding of free will. See Markosian 1999, 2012; Nelkin 2011.) These accounts reduce the causal role of the self to states and events to which the agent is not identical (even if he is identified with them). But how can self-determination of my actions wholly reduce to determination of my actions by things other than the self? Richard Taylor nicely expresses this intuition: "If I believe that something not identical to myself was the cause of my behavior--some event wholly external to myself, for instance, or even one internal to myself, such as a nerve impulse, volition, or whatnot--then I cannot regard the behavior as being an act of mine, unless I further believed that I was the cause of that external or internal event" (1974, 55; cf. Franklin 2016). Despite its powerful intuitive pull for some, many have argued that agent-causal libertarianism is obscure or even incoherent. The stock objection used to be that the very idea of agent-causation--causation by agents that is not reducible to causation by mental states and events involving the agent--is incoherent, but this objection has become less common due to pioneering work by Chisholm (1966, 1976), Taylor (1974), O'Connor (2000, 2011), Clarke (2003), and Steward 2012, ch. 8). More common objections now concern, first, how to understand the relationship between agent-causation and an agent's reasons (or motivations in general), and, second, the empirical adequacy of agent-causal libertarianism. With respect to the first worry, it is widely assumed that the only (or at least best) way to understand reasons-explanation and motivational influence is within a causal account of reasons, where reasons cause our actions (Davidson 1963; Mele 1992). If agent-causal libertarians accept that self-determined actions, in addition to being agent-caused, must also be caused by agents' reasons that favored those actions, then agent-causal libertarians need to explain how to integrate these causes (for a detailed attempt to do just this, see Clarke 2003, ch. 8). Given that these two causes seem distinct, is it not possible that the agent cause his decision to $\phi$ and yet the agent's reasons simultaneously cause an incompatible decision to $\psi$? If agent-causal libertarians side-step this difficult question by denying that reasons cause action, then they must explain how reasons can explain and motivate action without causing it; and this has turned out to be no easy task. (For more general attempts to understand reasons-explanation and motivation within a non-causal framework see Schueler 1995, 2003; Sehon 2005). For further discussion see the entry on incompatibilist (nondeterministic) theories of free will. Finally, we note that some recent philosophers have questioned the presumed difference between event- and agent-causation by arguing that all causation is object or substance causation. They argue that the dominant tendency to understand 'garden variety' causal transactions in the world as relations between events is an unfortunate legacy of David Hume's rejection of substance and causation as basic metaphysical categories. On the competing metaphysical picture of the world, the event or state of an object's having some property such as mass is its having a causal power, which in suitable circumstances it exercises to bring about a characteristic effect. Applied to human agents in an account of free will, the account suggests a picture on which an agent's having desires, beliefs, and intentions are rational powers to will particular courses of action, and where the agent's willing is not determined in any one direction, she wills freely. An advantage for the agent-causalist who embraces this broader metaphysics is 'ideological' parsimony. For different developments and defenses of this approach, see Lowe (2008), Swinburne (2013), and O'Connor (2021); and for reason to doubt that a substance-causal metaphysics helps to allay skepticism concerning free will, see Clarke and Reed (2015). ## 3. Do We Have Free Will? Most philosophers theorizing about free will take themselves to be attempting to analyze a near-universal power of mature human beings. But as we've noted above, there have been free will skeptics in both ancient and (especially) modern times. (Israel 2001 highlights a number of such skeptics in the early modern period.) In this section, we summarize the main lines of argument both for and against the reality of human freedom of will. ### 3.1 Arguments Against the Reality of Free Will There are both a priori and empirical arguments against free will (See the entry on skepticism about moral responsibility). Several of these start with an argument that free will is incompatible with causal determinism, which we will not rehearse here. Instead, we focus on arguments that human beings lack free will, against the background assumption that freedom and causal determinism are incompatible. The most radical a priori argument is that free will is not merely contingently absent but is impossible. Nietzsche 1886 [1966] argues to this effect, and more recently it has been argued by Galen Strawson (1986, ch. 2; 1994, 2002). Strawson associates free will with being 'ultimately morally responsible' for one's actions. He argues that, because how one acts is a result of, or explained by, "how one is, mentally speaking" ($M$), for one to be responsible for that choice one must be responsible for $M$. To be responsible for $M$, one must have chosen to be $M$ itself--and that not blindly, but deliberately, in accordance with some reasons $r\_1$. But for that choice to be a responsible one, one must have chosen to be such as to be moved by $r\_1$, requiring some further reasons $r\_2$ for such a choice. And so on, ad infinitum. Free choice requires an impossible infinite regress of choices to be the way one is in making choices. There have been numerous replies to Strawson's argument. Mele (1995, 221ff.) argues that Strawson misconstrues the locus of freedom and responsibility. Freedom is principally a feature of our actions, and only derivatively of our characters from which such actions spring. The task of the theorist is to show how one is in rational, reflective control of the choices one makes, consistent with there being no freedom-negating conditions. While this seems right, when considering those theories that make one's free control to reside directly in the causal efficacy of one's reasons (such as compatibilist reasons-responsive accounts or event-causal libertarianism), it is not beside the point to reflect on how one came to be that way in the first place and to worry that such reflection should lead one to conclude that true responsibility (and hence freedom) is undermined, since a complete distal source of any action may be found external to the agent. Clarke (2003, 170-76) argues that an effective reply may be made by indeterminists, and, in particular, by nondeterministic agent-causal theorists. Such theorists contend that (i) aspects of 'how one is, mentally speaking', fully explain an agent's choice without causally determining it and (ii) the agent himself causes the choice that is made (so that the agent's antecedent state, while grounding an explanation of the action, is not the complete causal source of it). Since the agent's exercise of this power is causally undetermined, it is not true that there is a sufficient 'ultimate' source of it external to the agent. Finally, Mele (2006, 129-34, and 2017, 212-16) and O'Connor (2009b) suggest that freedom and moral responsibility come in degrees and grow over time, reflecting the fact that 'how one is, mentally speaking' is increasingly shaped by one's own past choices. Furthermore, some choices for a given individual may reflect more freedom and responsibility than others, which may be the kernel of truth behind Strawson's sweeping argument. (For discussion of the ways that nature, nurture, and contingent circumstances shape our behavior and raise deep issues concerning the extent of our freedom and responsibility, see Levy 2011 and Russell 2017, chs. 10-12.) A second family of arguments against free will contend that, in one way or another, nondeterministic theories of freedom entail either that agents lack control over their choices or that the choices cannot be adequately explained. These arguments are variously called the 'Mind', 'Rollback', or 'Luck' argument, with the latter admitting of several versions. (For statements of such arguments, see van Inwagen 1983, ch. 4; 2000; Haji 2001; Mele 2006; Shabo 2011, 2013, 2020; Coffman 2015). We note that some philosophers advance such arguments not as parts of a general case against free will, but merely as showing the inadequacy of specific accounts of free will [see, e.g., Griffith 2010].) They each describe imagined cases--individual cases, or comparison of intra- or inter-world duplicate antecedent conditions followed by diverging outcomes--designed to elicit the judgment that the occurrence of a choice that had remained unsettled given all prior causal factors can only be a 'matter of chance', 'random', or 'a matter of luck'. Such terms have been imported from other contexts and have come to function as quasi-technical, unanalyzed concepts in these debates, and it is perhaps more helpful to avoid such proxies and to conduct the debates directly in terms of the metaphysical notion of control and epistemic notion of explanation. Where the arguments question whether an undetermined agent can exercise appropriate control over the choice he makes, proponents of nondeterministic theories often reply that control is not exercised prior to, but at the time of the choice--in the very act of bringing it about (see, e.g., Clarke 2005 and O'Connor 2007). Where the arguments question whether undetermined choices can be adequately explained, the reply often consists in identifying a form of explanation other than the form demanded by the critic--a 'noncontrastive' explanation, perhaps, rather than a 'contrastive' explanation, or a species of contrastive explanation consistent with indeterminism (see, e.g., Kane 1999; Clarke, 2003, ch. 8; and Franklin 2011a; 2018, ch. 5). We now consider empirical arguments against human freedom. Some of these stem from the physical sciences (while making assumptions concerning the way physical phenomena fix psychological phenomena) and others from neuroscience and psychology. It used to be common for philosophers to argue that there is empirical reason to believe that the world in general is causally determined, and since human beings are parts of the world, they are too. Many took this to be strongly confirmed by the spectacular success of Isaac Newton's framework for understanding the universe as governed everywhere by fairly simple, exceptionless laws of motion. But the quantum revolution of the early twentieth century has made that 'clockwork universe' image at least doubtful at the level of basic physics. While quantum mechanics has proven spectacularly successful as a framework for making precise and accurate predictions of certain observable phenomena, its implications for the causal structure of reality is still not well understood, and there are competing indeterministic and deterministic interpretations. See the entry on quantum mechanics for detailed discussion.) It is possible that indeterminacy on the small-scale, supposing it to be genuine, 'cancels out' at the macroscopic scale of birds and buildings and people, so that behavior at this scale is virtually deterministic. But this idea, once common, is now being challenged empirically, even at the level of basic biology. Furthermore, the social, biological, and medical sciences, too, are rife with merely statistical generalizations. Plainly, the jury is out on all these inter-theoretic questions. But that is just a way to say that current science does not decisively support the idea that everything we do is pre-determined by the past, and ultimately by the distant past, wholly out of our control. For discussion, see Balaguer (2009), Koch (2009), Roskies (2014), Ellis (2016). Maybe, then, we are subject to myriad causal influences, but the sum total of these influences doesn't determine what we do, they only make it more or less likely that we'll do this or that. Now some of the a priori no-free-will arguments above center on nondeterministic theories according to which there are objective antecedent probabilities associated with each possible choice outcome. Why objective probabilities of this kind might present special problems beyond those posed by the absence of determinism has been insufficiently explored to date. (For brief discussion, see Vicens 2016 and O'Connor 2016.) But one philosopher who argues that there is reason to hold that our actions, if undetermined, are governed by objective probabilities and that this fact calls into question whether we act freely is Derk Pereboom (2001, ch. 3; 2014, ch. 3). Pereboom notes that our best physical theories indicate that statistical laws govern isolated, small-scale physical events, and he infers from the thesis that human beings are wholly physically composed that such statistical laws will also govern all the physical components of human actions. Finally, Pereboom maintains that agent-causal libertarianism offers the correct analysis of free will. He then invites us to imagine that the antecedent probability of some physical component of an action occurring is $0.32$. If the action is free while not violating the statistical law, then, in a scenario with a large enough number of instances, this action would have to be freely chosen close to $32$ percent of the time. This leads to the problem of "wild coincidences": > > > if the occurrence of these physical components were settled by the > choices of agent-causes, then their actually being chosen close to 32 > percent of the time would amount to a coincidence no less wild than > the coincidence of possible actions whose physical components have an > antecedent probability of about 0.99 being chosen, over large enough > number of instances, close to 99 percent of the time. The proposal > that agent-caused free choices do not diverge from what the > statistical laws predict for the physical components of our actions > would run so sharply counter to what we would expect as to make it > incredible. (2014, 67) > > > Clarke (2010) questions the implicit assumption that free agent-causal choices should be expected not to conform to physical statistical laws, while O'Connor (2009a) challenges the more general assumption that freedom requires that agent-causal choices not be governed by statistical laws of any kind, as they plausibly would be if the relevant psychological states/powers are strongly emergent from physical states of the human brain. Finally, Runyan 2018 argues that Pereboom's case rests on an implausible empirical assumption concerning the evolution of objective probabilities concerning types of behavior over time. Pereboom's empirical basis for free will skepticism is very general. Others see support for free will skepticism from specific findings and theories in the human sciences. They point to evidence that we can be unconsciously influenced in the choices we make by a range of factors, including ones that are not motivationally relevant; that we can come to believe that we chose to initiate a behavior that in fact was artificially induced; that people subject to certain neurological disorders will sometimes engage in purposive behavior while sincerely believing that they are not directing them. Finally, a great deal of attention has been given to the work of neuroscientist Benjamin Libet (2002). Libet conducted some simple experiments that seemed to reveal the existence of 'preparatory' brain activity (the 'readiness potential') shortly before a subject engages in an ostensibly spontaneous action. (Libet interpreted this activity as the brain's 'deciding' what to do before we are consciously settled on a course of action.) Wegner (2002) surveys all of these findings (some of which are due to his own work as a social psychologist) and argues on their basis that the experience of conscious willing is 'an illusion'. For criticism of such arguments, see Mele (2009); Nahmias (2014); Mudrik et al. (2022); and several contributions to Maoz and Sinnott-Armstrong (2022). Libet's interpretation of the readiness potential has come in for severe criticism. After extensive subsequent study, neuroscientists are uncertain what it signifies. For thorough review of the evidence, see Schurger et al. (2021). While Pereboom and others point to these empirical considerations in defense of free will skepticism, other philosophers see them as reasons to favor a more modest free will agnosticism (Kearns 2015) or to promote revisionism about the 'folk idea of free will' (Vargas 2013; Nichols 2015). ### 3.2 Arguments for the Reality of Free Will If one is a compatibilist, then a case for the reality of free will requires evidence for our being effective agents who for the most part are aware of what we do and why we are doing it. If one is an incompatibilist, then the case requires in addition evidence for causal indeterminism, occurring in the right locations in the process leading from deliberation to action. Many think that we already have third-personal 'neutral' scientific evidence for much of human behavior's satisfying modest compatibilist requirements, such as Fischer and Ravizza's reasons-responsiveness account. However, given the immaturity of social science and the controversy over whether psychological states 'reduce' in some sense to underlying physical states (and what this might entail for the reality of mental causation), this claim is doubtful. A more promising case for our satisfying (at least) compatibilist requirements on freedom is that effective agency is presupposed by all scientific inquiry and so cannot rationally be doubted (which fact is overlooked by some of the more extreme 'willusionists' such as Wegner). However, effective intervention in the world (in scientific practice and elsewhere) does not (obviously) require that our behavior be causally undetermined, so the 'freedom is rationally presupposed' argument cannot be launched for such an understanding of freedom. Instead, incompatibilists usually give one of the following two bases for rational belief in freedom (both of which can be given by compatibilists, too). First, philosophers have long claimed that we have introspective evidence of freedom in our experience of action, or perhaps of consciously attended or deliberated action. Augustine and Scotus, discussed earlier, are two examples among many. In recent years, philosophers have been more carefully scrutinizing the experience of agency and a debate has emerged concerning its contents, and in particular whether it supports an indeterministic theory of human free action. For discussion, see Deery et al. (2013), Guillon (2014), Horgan (2015), and Bayne (2017). Second, philosophers (e.g., Reid 1788 [1969], Swinburne 2013) sometimes claim that our belief in the reality of free will is epistemically basic, or reasonable without requiring independent evidential support. Most philosophers hold that some beliefs have that status, on pain of our having no justified beliefs whatever. It is controversial, however, just which beliefs do because it is controversial which criteria a belief must satisfy to qualify for that privileged status. It is perhaps necessary that a basic belief be 'instinctive' (unreflectively held) for all or most human beings; that it be embedded in regular experience; and that it be central to our understanding of an important aspect of the world. Our belief in free will seems to meet these criteria, but whether they are sufficient is debated. (O'Connor 2019 proposes that free will belief is epistemically basic but defeasible.) Other philosophers defend a variation on this stance, maintaining instead that belief in the reality of moral responsibility is epistemically basic, and that since moral responsibility entails free will, or so it is claimed, we may infer the reality of free will (see, e.g., van Inwagen 1983, 206-13). ## 4. Theological Wrinkles ### 4.1 Free Will and God's Power, Knowledge, and Goodness A large portion of Western philosophical work on free will has been written within an overarching theological framework, according to which God is the ultimate source, sustainer, and end of all else. Some of these thinkers draw the conclusion that God must be a sufficient, wholly determining cause for everything that happens; all of them suppose that every creaturely act necessarily depends on the explanatorily prior, cooperative activity of God. It is also commonly presumed by philosophical theists that human beings are free and responsible (on pain of attributing evil in the world to God alone, and so impugning His perfect goodness). Hence, those who believe that God is omni-determining typically are compatibilists with respect to freedom and (in this case) theological determinism. Edwards (1754 [1957]) is a good example. But those who suppose that God's sustaining activity (and special activity of conferring grace) is only a necessary condition on the outcome of human free choices need to tell a more subtle story, on which omnipotent God's cooperative activity can be (explanatorily) prior to a human choice and yet the outcome of that choice be settled only by the choice itself. For important medieval discussions--the apex of philosophical reflection on theological concerns--see the relevant portions of Al-Ghazali IP, Aquinas BW and Scotus QAM. Three positions (given in order of logical strength) on God's activity vis-a-vis creaturely activity were variously defended by thinkers of this area: mere conservationism, concurrentism, and occasionalism. These positions turn on subtle distinctions, which have recently been explored by Freddoso (1988), Kvanvig and McCann (1991), Koons (2002), Grant (2016 and 2019), and Judisch (2016). Many suppose that there is a challenge to human freedom stemming not only from God's perfect power but also from his perfect knowledge. A standard argument for the incompatibility of free will and causal determinism has a close theological analogue. Recall van Inwagen's influential formulation of the 'Consequence Argument': > > > If determinism is true, then our acts are the consequences of the laws > of nature and events in the remote past. But it is not up to us what > went on before we were born, and neither is it up to us what the laws > of nature are. Therefore, the consequences of these things (including > our present acts) are not up to us. (van Inwagen 1983, 16) > > > And now consider an argument that turns on God's comprehensive and infallible knowledge of the future: > > > If infallible divine foreknowledge is true, then our acts are the > (logical) consequences of God's beliefs in the remote past. > (Since God cannot get things wrong, his believing that > something will be so entails that it will be so.) But it is not up to > us what beliefs God had before we were born, and neither is it up to > us that God's beliefs are necessarily true. Therefore, the > consequences of these things (including our present acts) are not up > to us. > > > An excellent discussion of these arguments in tandem and attempts to point to relevant disanalogies between causal determinism and infallible foreknowledge may be found in the introduction to Fischer (1989). See also the entry on foreknowledge and free will. Another issue concerns how knowledge of God, the ultimate Good, would impact human freedom. Many philosophical theologians, especially the medieval Aristotelians, were drawn to the idea that human beings cannot but will that which they take to be an unqualified good. (As noted above, Duns Scotus is an exception to this consensus, as were Ockham and Suarez subsequently, but their dissent is limited.) Hence, if there is an afterlife, in which humans 'see God face to face,' they will inevitably be drawn to Him. Following Pascal, Murray (1993, 2002) argues that a good God would choose to make His existence and character less than certain for human beings, for the sake of preserving their freedom. (He will do so, the argument goes, at least for a period of time in which human beings participate in their own character formation.) If it is a good for human beings that they freely choose to respond in love to God and to act in obedience to His will, then God must maintain an 'epistemic distance' from them lest they be overwhelmed by His goodness or power and respond out of necessity, rather than freedom. (See also the other essays in Howard-Snyder and Moser 2002.) If it is true that God withholds our ability to be certain of his existence for the sake of our freedom, then it is natural to conclude that humans will lack freedom in heaven. And it is anyways common to traditional Jewish, Christian, and Muslim theologies to maintain that humans cannot sin in heaven. Even so, traditional Christian theology at least maintains that human persons in heaven are free. What sort of freedom is in view here, and how does it relate to mundane freedom? Two good recent discussions of these questions are Pawl and Timpe (2009) and Tamburro (2017). ### 4.2 God's Freedom Finally, there is the question of the freedom of God himself. Perfect goodness is an essential, not acquired, attribute of God. God cannot lie or be in any way immoral in His dealings with His creatures (appearances notwithstanding). Unless we take the minority position on which this is a trivial claim, since whatever God does definitionally counts as good, this appears to be a significant, inner constraint on God's freedom. Did we not contemplate immediately above that human freedom would be curtailed by our having an unmistakable awareness of what is in fact the Good? And yet is it not passing strange to suppose that God should be less than perfectly free? One suggested solution to this puzzle takes as its point of departure the distinction noted in section 2.3 between the ability to do otherwise and sourcehood, proposing that the core metaphysical feature of freedom is being the ultimate source, or originator, of one's choices. For human beings or any created persons who owe their existence to factors outside themselves, the only way their acts of will could find their ultimate origin in themselves is for such acts not to be determined by their character and circumstances. For if all my willings were wholly determined, then if we were to trace my causal history back far enough, we would ultimately arrive at external factors that gave rise to me, with my particular genetic dispositions. My motives at the time would not be the ultimate source of my willings, only the most proximate ones. Only by there being less than deterministic connections between external influences and choices, then, is it be possible for me to be an ultimate source of my activity, concerning which I may truly say, "the buck stops here." As is generally the case, things are different on this point in the case of God. As Anselm observed, even if God's character absolutely precludes His performing certain actions in certain contexts, this will not imply that some external factor is in any way a partial origin of His willings and refrainings from willing. Indeed, this would not be so even if he were determined by character to will everything which He wills. God's nature owes its existence to nothing. Thus, God would be the sole and ultimate source of His will even if He couldn't will otherwise. Well, then, might God have willed otherwise in any respect? The majority view in the history of philosophical theology is that He indeed could have. He might have chosen not to create anything at all. And given that He did create, He might have created any number of alternatives to what we observe. But there have been noteworthy thinkers who argued the contrary position, along with others who clearly felt the pull of the contrary position even while resisting it. The most famous such thinker is Leibniz (1710 [1985]), who argued that God, being both perfectly good and perfectly powerful, cannot fail to will the best possible world. Leibniz insisted that this is consistent with saying that God is able to will otherwise, although his defense of this last claim is notoriously difficult to make out satisfactorily. Many read Leibniz, malgre lui, as one whose basic commitments imply that God could not have willed other than He does in any respect. One might challenge Leibniz's reasoning on this point by questioning the assumption that there is a uniquely best possible Creation (an option noted by Adams 1987, though he challenges instead Leibniz's conclusion based on it). One way this could be is if there is no well-ordering of worlds: some pairs of worlds are sufficiently different in kind that they are incommensurate with each other (neither is better than the other, nor are they equal) and no world is better than either of them. Another way this could be is if there is no upper limit on goodness of worlds: for every possible world God might have created, there are others (infinitely many, in fact) which are better. If such is the case, one might argue, it is reasonable for God to arbitrarily choose which world to create from among those worlds exceeding some threshold value of overall goodness. However, William Rowe (2004) has countered that the thesis that there is no upper limit on goodness of worlds has a very different consequence: it shows that there could not be a morally perfect Creator! For suppose our world has an on-balance moral value of $n$ and that God chose to create it despite being aware of possibilities having values higher than $n$ that He was able to create. It seems we can now imagine a morally better Creator: one having the same options who chooses to create a better world. For critical replies to Rowe, see Almeida (2008, ch. 1), Kray (2010), and Zimmerman (2018). Finally, Norman Kretzmann (1997, 220-25) has argued in the context of Aquinas's theological system that there is strong pressure to say that God must have created something or other, though it may well have been open to Him to create any of a number of contingent orders. The reason is that there is no plausible account of how an absolutely perfect God might have a resistible motivation--one consideration among other, competing considerations--for creating something rather than nothing. (It obviously cannot have to do with any sort of utility, for example.) The best general understanding of God's being motivated to create at all--one which in places Aquinas himself comes very close to endorsing--is to see it as reflecting the fact that God's very being, which is goodness, necessarily diffuses itself. Perfect goodness will naturally communicate itself outwardly; God who is perfect goodness will naturally create, generating a dependent reality that imperfectly reflects that goodness. Wainwright (1996) discusses a somewhat similar line of thought in the Puritan thinker Jonathan Edwards. Alexander Pruss (2016), however, raises substantial grounds for doubt concerning this line of thought; O'Connor (2022) offers a rejoinder.
voltaire	## 1. Voltaire's Life: The Philosopher as Critic and Public Activist Voltaire only began to identify himself with philosophy and the philosophe identity during middle age. His work Lettres philosophiques, published in 1734 when he was forty years old, was the key turning point in this transformation. Before this date, Voltaire's life in no way pointed him toward the philosophical destiny that he was later to assume. His early orientation toward literature and libertine sociability, however, shaped his philosophical identity in crucial ways. ### 1.1 Voltaire's Early Years (1694-1726) Francois-Marie d'Arouet was born in 1694, the fourth of five children, to a well-to-do public official and his well bred aristocratic wife. In its fusion of traditional French aristocratic pedigree with the new wealth and power of royal bureaucratic administration, the d'Arouet family was representative of elite society in France during the reign of Louis XIV. The young Francois-Marie acquired from his parents the benefits of prosperity and political favor, and from the Jesuits at the prestigious College Louis-le-Grand in Paris he also acquired a first-class education. Francois-Marie also acquired an introduction to modern letters from his father who was active in the literary culture of the period both in Paris and at the royal court of Versailles. Francois senior appears to have enjoyed the company of men of letters, yet his frustration with his son's ambition to become a writer is notorious. From early in his youth, Voltaire aspired to emulate his idols Moliere, Racine, and Corneille and become a playwright, yet Voltaire's father strenuously opposed the idea, hoping to install his son instead in a position of public authority. First as a law student, then as a lawyer's apprentice, and finally as a secretary to a French diplomat, Voltaire attempted to fulfill his father's wishes. But in each case, he ended up abandoning his posts, sometimes amidst scandal. Escaping from the burdens of these public obligations, Voltaire would retreat into the libertine sociability of Paris. It was here in the 1720s, during the culturally vibrant period of the Regency government between the reigns of Louis XIV and XV (1715-1723), that Voltaire established one dimension of his identity. His wit and congeniality were legendary even as a youth, so he had few difficulties establishing himself as a popular figure in Regency literary circles. He also learned how to play the patronage game so important to those with writerly ambitions. Thanks, therefore, to some artfully composed writings, a couple of well-made contacts, more than a few bon mots, and a little successful investing, especially during John Law's Mississippi Bubble fiasco, Voltaire was able to establish himself as an independent man of letters in Paris. His literary debut occurred in 1718 with the publication of his Oedipe, a reworking of the ancient tragedy that evoked the French classicism of Racine and Corneille. The play was first performed at the home of the Duchesse du Maine at Sceaux, a sign of Voltaire's quick ascent to the very pinnacle of elite literary society. Its published title page also announced the new pen name that Voltaire would ever after deploy. During the Regency, Voltaire circulated widely in elite circles such as those that congregated at Sceaux, but he also cultivated more illicit and libertine sociability as well. This pairing was not at all uncommon during this time, and Voltaire's intellectual work in the 1720s--a mix of poems and plays that shifted between playful libertinism and serious classicism seemingly without pause--illustrated perfectly the values of pleasure, honnetete, and good taste that were the watchwords of this cultural milieu. Philosophy was also a part of this mix, and during the Regency the young Voltaire was especially shaped by his contacts with the English aristocrat, freethinker,and Jacobite Lord Bolingbroke. Bolingbroke lived in exile in France during the Regency period, and Voltaire was a frequent visitor to La Source, the Englishman's estate near Orleans. The chateau served as a reunion point for a wide range of intellectuals, and many believe that Voltaire was first introduced to natural philosophy generally, and to the work of Locke and the English Newtonians specifically, at Bolingbroke's estate. It was certainly true that these ideas, especially in their more deistic and libertine configurations, were at the heart of Bolingbroke's identity. ### 1.2 The English Period (1726-1729) Yet even if Voltaire was introduced to English philosophy in this way, its influence on his thought was most shaped by his brief exile in England between 1726-29. The occasion for his departure was an affair of honor. A very powerful aristocrat, the Duc de Rohan, accused Voltaire of defamation, and in the face of this charge the untitled writer chose to save face and avoid more serious prosecution by leaving the country indefinitely. In the spring of 1726, therefore, Voltaire left Paris for England. It was during his English period that Voltaire's transition into his mature philosophe identity began. Bolingbroke, whose address Voltaire left in Paris as his own forwarding address, was one conduit of influence. In particular, Voltaire met through Bolingbroke Jonathan Swift, Alexander Pope, and John Gay, writers who were at that moment beginning to experiment with the use of literary forms such as the novel and theater in the creation of a new kind of critical public politics. Swift's Gulliver's Travels, which appeared only months before Voltaire's arrival, is the most famous exemplar of this new fusion of writing with political criticism. Later the same year Bolingbroke also brought out the first issue of the Craftsman, a political journal that served as the public platform for his circle's Tory opposition to the Whig oligarchy in England. The Craftsman helped to create English political journalism in the grand style, and for the next three years Voltaire moved in Bolingbroke's circle, absorbing the culture and sharing in the public political contestation that was percolating all around him. Voltaire did not restrict himself to Bolingbroke's circle alone, however. After Bolingbroke, his primary contact in England was a merchant by the name of Everard Fawkener. Fawkener introduced Voltaire to a side of London life entirely different from that offered by Bolingbroke's circle of Tory intellectuals. This included the Whig circles that Bolingbroke's group opposed. It also included figures such as Samuel Clarke and other self-proclaimed Newtonians. Voltaire did not meet Newton himself before Sir Isaac's death in March, 1727, but he did meet his sister--learning from her the famous myth of Newton's apple, which Voltaire would play a major role in making famous. Voltaire also came to know the other Newtonians in Clarke's circle, and since he became proficient enough with English to write letters and even fiction in the language, it is very likely that he immersed himself in their writings as well. Voltaire also visited Holland during these years, forming important contacts with Dutch journalists and publishers and meeting Willem's Gravesande and other Dutch Newtonian savants. Given his other activities, it is also likely that Voltaire frequented the coffeehouses of London even if no firm evidence survives confirming that he did. It would not be surprising, therefore, to learn that Voltaire attended the Newtonian public lectures of John Theophilus Desaguliers or those of one of his rivals. Whatever the precise conduits, all of his encounters in England made Voltaire into a very knowledgeable student of English natural philosophy. ### 1.3 Becoming a Philosophe When French officials granted Voltaire permission to re-enter Paris in 1729, he was devoid of pensions and banned from the royal court at Versailles. But he was also a different kind of writer and thinker. It is no doubt overly grandiose to say with Lord Morley that, "Voltaire left France a poet and returned to it a sage." It is also an exaggeration to say that he was transformed from a poet into a philosophe while in England. For one, these two sides of Voltaire's intellectual identity were forever intertwined, and he never experienced an absolute transformation from one into the other at any point in his life. But the English years did trigger a transformation in him. After his return to France, Voltaire worked hard to restore his sources of financial and political support. The financial problems were the easiest to solve. In 1729, the French government staged a sort of lottery to help amortize some of the royal debt. A friend perceived an opportunity for investors in the structure of the government's offering, and at a dinner attended by Voltaire he formed a society to purchase shares. Voltaire participated, and in the fall of that year when the returns were posted he had made a fortune. Voltaire's inheritance from his father also became available to him at the same time, and from this date forward Voltaire never again struggled financially. This result was no insignificant development since Voltaire's financial independence effectively freed him from one dimension of the patronage system so necessary to aspiring writers and intellectuals in the period. In particular, while other writers were required to appeal to powerful financial patrons in order to secure the livelihood that made possible their intellectual careers, Voltaire was never again beholden to these imperatives. The patronage structures of Old Regime France provided more than economic support to writers, however, and restoring the credit upon which his reputation as a writer and thinker depended was far less simple. Gradually, however, through a combination of artfully written plays, poems, and essays and careful self-presentation in Parisian society, Voltaire began to regain his public stature. In the fall of 1732, when the next stage in his career began to unfold, Voltaire was residing at the royal court of Versailles, a sign that his re-establishment in French society was all but complete. During this rehabilitation, Voltaire also formed a new relationship that was to prove profoundly influential in the subsequent decades. He became reacquainted with Emilie Le Tonnier de Breteuil,the daughter of one of his earliest patrons, who married in 1722 to become the Marquise du Chatelet. Emilie du Chatelet was twenty-nine years old in the spring of 1733 when Voltaire began his relationship with her. She was also a uniquely accomplished woman. Du Chatelet's father, the Baron de Breteuil, hosted a regular gathering of men of letters that included Voltaire, and his daughter, ten years younger than Voltaire, shared in these associations. Her father also ensured that Emilie received an education that was exceptional for girls at the time. She studied Greek and Latin and trained in mathematics, and when Voltaire reconnected with her in 1733 she was a very knowledgeable thinker in her own right even if her own intellectual career, which would include an original treatise in natural philosophy and a complete French translation of Newton's Principia Mathematica--still the only complete French translation ever published--had not yet begun. Her intellectual talents combined with her vivacious personality drew Voltaire to her, and although Du Chatelet was a titled aristocrat married to an important military officer, the couple was able to form a lasting partnership that did not interfere with Du Chatelet's marriage. This arrangement proved especially beneficial to Voltaire when scandal forced him to flee Paris and to establish himself permanently at the Du Chatelet family estate at Cirey. From 1734, when this arrangement began, to 1749, when Du Chatelet died during childbirth, Cirey was the home to each along with the site of an intense intellectual collaboration. It was during this period that both Voltaire and Du Chatelet became widely known philosophical figures, and the intellectual history of each before 1749 is most accurately described as the history of the couple's joint intellectual endeavors. ### 1.4 The Newton Wars (1732-1745) For Voltaire, the events that sent him fleeing to Cirey were also the impetus for much of his work while there. While in England, Voltaire had begun to compose a set of letters framed according to the well-established genre of a traveler reporting to friends back home about foreign lands. Montesquieu's 1721 Lettres Persanes, which offered a set of fictionalized letters by Persians allegedly traveling in France, and Swift's 1726 Gulliver's Travels were clear influences when Voltaire conceived his work. But unlike the authors of these overtly fictionalized accounts, Voltaire innovated by adopting a journalistic stance instead, one that offered readers an empirically recognizable account of several aspects of English society. Originally titled Letters on England, Voltaire left a draft of the text with a London publisher before returning home in 1729. Once in France, he began to expand the work, adding to the letters drafted while in England, which focused largely on the different religious sects of England and the English Parliament, several new letters including some on English philosophy. The new text, which included letters on Bacon, Locke, Newton and the details of Newtonian natural philosophy along with an account of the English practice of inoculation for smallpox, also acquired a new title when it was first published in France in 1734: Lettres philosophiques. Before it appeared, Voltaire attempted to get official permission for the book from the royal censors, a requirement in France at the time. His publisher, however, ultimately released the book without these approvals and without Voltaire's permission. This made the first edition of the Lettres philosophiques illicit, a fact that contributed to the scandal that it triggered, but one that in no way explains the furor the book caused. Historians in fact still scratch their heads when trying to understand why Voltaire's Lettres philosophiques proved to be so controversial. The only thing that is clear is that the work did cause a sensation that subsequently triggered a rapid and overwhelming response on the part of the French authorities. The book was publicly burned by the royal hangman several months after its release, and this act turned Voltaire into a widely known intellectual outlaw. Had it been executed, a royal lettre de cachet would have sent Voltaire to the royal prison of the Bastille as a result of his authorship of Lettres philosophiques; instead, he was able to flee with Du Chatelet to Cirey where the couple used the sovereignty granted by her aristocratic title to create a safe haven and base for Voltaire's new position as a philosophical rebel and writer in exile. Had Voltaire been able to avoid the scandal triggered by the Lettres philosophiques, it is highly likely that he would have chosen to do so. Yet once it was thrust upon him, he adopted the identity of the philosophical exile and outlaw writer with conviction, using it to create a new identity for himself, one that was to have far reaching consequences for the history of Western philosophy. At first, Newtonian science served as the vehicle for this transformation. In the decades before 1734, a series of controversies had erupted, especially in France, about the character and legitimacy of Newtonian science, especially the theory of universal gravitation and the physics of gravitational attraction through empty space. Voltaire positioned his Lettres philosophiques as an intervention into these controversies, drafting a famous and widely cited letter that used an opposition between Newton and Descartes to frame a set of fundamental differences between English and French philosophy at the time. He also included other letters about Newtonian science in the work while linking (or so he claimed) the philosophies of Bacon, Locke, and Newton into an English philosophical complex that he championed as a remedy for the perceived errors and illusions perpetuated on the French by Rene Descartes and Nicolas Malebranche. Voltaire did not invent this framework, but he did use it to enflame a set of debates that were then raging, debates that placed him and a small group of young members of the Royal Academy of Sciences in Paris into apparent opposition to the older and more established members of this bastion of official French science. Once installed at Cirey, both Voltaire and Du Chatelet further exploited this apparent division by engaging in a campaign on behalf of Newtonianism, one that continually targeted an imagined monolith called French Academic Cartesianism as the enemy against which they in the name of Newtonianism were fighting. The centerpiece of this campaign was Voltaire's Elements de la Philosophie de Newton, which was first published in 1738 and then again in 1745 in a new and definitive edition that included a new section, first published in 1740, devoted to Newton's metaphysics. Voltaire offered this book as a clear, accurate, and accessible account of Newton's philosophy suitable for ignorant Frenchman (a group that he imagined to be large). But he also conceived of it as a machine de guerre directed against the Cartesian establishment, which he believed was holding France back from the modern light of scientific truth. Vociferous criticism of Voltaire and his work quickly erupted, with some critics emphasizing his rebellious and immoral proclivities while others focused on his precise scientific views. Voltaire collapsed both challenges into a singular vision of his enemy as "backward Cartesianism". As he fought fiercely to defend his positions, an unprecedented culture war erupted in France centered on the character and value of Newtonian natural philosophy. Du Chatelet contributed to this campaign by writing a celebratory review of Voltaire's Elements in the Journal des savants, the most authoritative French learned periodical of the day. The couple also added to their scientific credibility by receiving separate honorable mentions in the 1738 Paris Academy prize contest on the nature of fire. Voltaire likewise worked tirelessly rebutting critics and advancing his positions in pamphlets and contributions to learned periodicals. By 1745, when the definitive edition of Voltaire's Elements was published, the tides of thought were turning his way, and by 1750 the perception had become widespread that France had been converted from backward, erroneous Cartesianism to modern, Enlightened Newtonianism thanks to the heroic intellectual efforts of figures like Voltaire. ### 1.5 From French Newtonian to Enlightenment Philosophe (1745-1755) This apparent victory in the Newton Wars of the 1730s and 1740s allowed Voltaire's new philosophical identity to solidify. Especially crucial was the way that it allowed Voltaire's outlaw status, which he had never fully repudiated, to be rehabilitated in the public mind as a necessary and heroic defense of philosophical truth against the enemies of error and prejudice. From this perspective, Voltaire's critical stance could be reintegrated into traditional Old Regime society as a new kind of legitimate intellectual martyrdom. Since Voltaire also coupled his explicitly philosophical writings and polemics during the 1730s and 1740s with an equally extensive stream of plays, poems, stories, and narrative histories, many of which were orthogonal in both tone and content to the explicit campaigns of the Newton Wars, Voltaire was further able to reestablish his old identity as an Old Regime man of letters despite the scandals of these years. In 1745, Voltaire was named the Royal Historiographer of France, a title bestowed upon him as a result of his histories of Louis XIV and the Swedish King Charles II. This royal office also triggered the writing of arguably Voltaire's most widely read and influential book, at least in the eighteenth century, Essais sur les moeurs et l'esprit des nations (1751), a pioneering work of universal history. The position also legitimated him as an officially sanctioned savant. In 1749, after the death of du Chatelet, Voltaire reinforced this impression by accepting an invitation to join the court of the young Frederick the Great in Prussia, a move that further assimilated him into the power structures of Old Regime society. Had this assimilationist trajectory continued during the remainder of Voltaire's life, his legacy in the history of Western philosophy might not have been so great. Yet during the 1750s, a set of new developments pulled Voltaire back toward his more radical and controversial identity and allowed him to rekindle the critical philosophe persona that he had innovated during the Newton Wars. The first step in this direction involved a dispute with his onetime colleague and ally, Pierre-Louis Moreau de Maupertuis. Maupertuis had preceded Voltaire as the first aggressive advocate for Newtonian science in France. When Voltaire was preparing his own Newtonian intervention in the Lettres philosophiques in 1732, he consulted with Maupertuis, who was by this date a pensioner in the French Royal Academy of Sciences. It was largely around Maupertuis that the young cohort of French academic Newtonians gathered during the Newton wars of 1730s and 40s, and with Voltaire fighting his own public campaigns on behalf of this same cause during the same period, the two men became the most visible faces of French Newtonianism even if they never really worked as a team in this effort. Like Voltaire, Maupertuis also shared a relationship with Emilie du Chatelet, one that included mathematical collaborations that far exceeded Voltaire's capacities. Maupertuis was also an occasional guest at Cirey, and a correspondent with both du Chatelet and Voltaire throughout these years. But in 1745 Maupertuis surprised all of French society by moving to Berlin to accept the directorship of Frederick the Great's newly reformed Berlin Academy of Sciences. Maupertuis's thought at the time of his departure for Prussia was turning toward the metaphysics and rationalist epistemology of Leibniz as a solution to certain questions in natural philosophy. Du Chatelet also shared this tendency, producing in 1740 her Institutions de physiques, a systematic attempt to wed Newtonian mechanics with Leibnizian rationalism and metaphysics. Voltaire found this Leibnizian turn dyspeptic, and he began to craft an anti-Leibnizian discourse in the 1740s that became a bulwark of his brand of Newtonianism. This placed him in opposition to Du Chatelet, even if this intellectual rift in no way soured their relationship. Yet after she died in 1749, and Voltaire joined Maupertuis at Frederick the Great's court in Berlin, this anti-Leibnizianism became the centerpiece of a rift with Maupertuis. Voltaire's public satire of the President of the Royal Academy of Sciences of Berlin published in late 1752, which presented Maupertuis as a despotic philosophical buffoon, forced Frederick to make a choice. He sided with Maupertuis, ordering Voltaire to either retract his libelous text or leave Berlin. Voltaire chose the latter, falling once again into the role of scandalous rebel and exile as a result of his writings. ### 1.6 Fighting for Philosophie (1755-1778) This event proved to be Voltaire's last official rupture with establishment authority. Rather than returning home to Paris and restoring his reputation, Voltaire instead settled in Geneva. When this austere Calvinist enclave proved completely unwelcoming, he took further steps toward independence by using his personal fortune to buy a chateau of his own in the hinterlands between France and Switzerland. Voltaire installed himself permanently at Ferney in early 1759, and from this date until his death in 1778 he made the chateau his permanent home and capital, at least in the minds of his intellectual allies, of the emerging French Enlightenment. During this period, Voltaire also adopted what would become his most famous and influential intellectual stance, announcing himself as a member of the "party of humanity" and devoting himself toward waging war against the twin hydras of fanaticism and superstition. While the singular defense of Newtonian science had focused Voltaire's polemical energies in the 1730s and 1740s, after 1750 the program became the defense of philosophie tout court and the defeat of its perceived enemies within the ecclesiastical and aristo-monarchical establishment. In this way, Enlightenment philosophie became associated through Voltaire with the cultural and political program encapsulated in his famous motto, "Ecrasez l'infame!" ("Crush the infamy!"). This entanglement of philosophy with social criticism and reformist political action, a contingent historical outcome of Voltaire's particular intellectual career, would become his most lasting contribution to the history of philosophy. The first cause to galvanize this new program was Diderot and d'Alembert's Encyclopedie. The first volume of this compendium of definitions appeared in 1751, and almost instantly the work became buried in the kind of scandal to which Voltaire had grown accustomed. Voltaire saw in the controversy a new call to action, and he joined forces with the project soon after its appearance, penning numerous articles that began to appear with volume 5 in 1755. Scandal continued to chase the Encyclopedie, however, and in 1759 the work's publication privilege was revoked in France, an act that did not kill the project but forced it into illicit production in Switzerland. During these scandals, Voltaire fought vigorously alongside the project's editors to defend the work, fusing the Encyclopedie's enemies, particularly the Parisian Jesuits who edited the monthly periodical the Journal de Trevoux, into a monolithic "infamy" devoted to eradicating truth and light from the world. This framing was recapitulated by the opponents of the Encyclopedie, who began to speak of the loose assemblage of authors who contributed articles to the work as a subversive coterie of philosophes devoted to undermining legitimate social and moral order. As this polemic crystallized and grew in both energy and influence, Voltaire embraced its terms and made them his cause. He formed particularly close ties with d'Alembert, and with him began to generalize a broad program for Enlightenment centered on rallying the newly self-conscious philosophes (a term often used synonymously with the Encyclopedistes) toward political and intellectual change. In this program, the philosophes were not unified by any shared philosophy but through a commitment to the program of defending philosophie itself against its perceived enemies. They were also imagined as activists fighting to eradicate error and superstition from the world. The ongoing defense of the Encyclopedie was one rallying point, and soon the removal of the Jesuits--the great enemies of Enlightenment, the philosophes proclaimed--became a second unifying cause. This effort achieved victory in 1763, and soon the philosophes were attempting to infiltrate the academies and other institutions of knowledge in France. One climax in this effort was reached in 1774 when the Encyclopediste and friend of Voltaire and the philosophes, Anne-Robert Jacques Turgot, was named Controller-General of France, the most powerful ministerial position in the kingdom, by the newly crowned King Louis XVI. Voltaire and his allies had paved the way for this victory through a barrage of writings throughout the 1760s and 1770s that presented philosophie like that espoused by Turgot as an agent of enlightened reform and its critics as prejudicial defenders of an ossified tradition. Voltaire did bring out one explicitly philosophical book in support this campaign, his Dictionnaire philosophique of 1764-1770. This book republished his articles from the original Encyclopedie while adding new entries conceived in the spirit of the original work. Yet to fully understand the brand of philosophie that Voltaire made foundational to the Enlightenment, one needs to recognize that it just as often circulated in fictional stories, satires, poems, pamphlets, and other less obviously philosophical genres. Voltaire's most widely known text, for instance, Candide, ou l'Optimisme, first published in 1759, is a fictional story of a wandering traveler engaged in a set of farcical adventures. Yet contained in the text is a serious attack on Leibnizian philosophy, one that in many ways marks the culmination of Voltaire's decades long attack on this philosophy started during the Newton wars. Philosophie a la Voltaire also came in the form of political activism, such as his public defense of Jean Calas who, Voltaire argued, was a victim of a despotic state and an irrational and brutal judicial system. Voltaire often attached philosophical reflection to this political advocacy, such as when he facilitated a French translation of Cesare Beccaria's treatise on humanitarian justice and penal reform and then prefaced the work with his own essay on justice and religious toleration (Calas was a French protestant persecuted by a Catholic monarchy). Public philosophic campaigns such as these that channeled critical reason in a direct, oppositionalist way against the perceived injustices and absurdities of Old Regime life were the hallmark of philosophie as Voltaire understood the term. ### 1.7 Voltaire, Philosophe Icon of Enlightenment Philosophie (1778-Present) Voltaire lived long enough to see some of his long-term legacies start to concretize. With the ascension of Louis XVI in 1774 and the appointment of Turgot as Controller-General, the French establishment began to embrace the philosophes and their agenda in a new way. Critics of Voltaire and his program for philosophie remained powerful, however, and they would continue to survive as the necessary backdrop to the positive image of the Enlightenment philosophe as a modernizer, progressive reformer, and courageous scourge against traditional authority that Voltaire bequeathed to later generations. During Voltaire's lifetime, this new acceptance translated into a final return to Paris in early 1778. Here, as a frail and sickly octogenarian, Voltaire was welcomed by the city as the hero of the Enlightenment that he now personified. A statue was commissioned as a permanent shrine to his legacy, and a public performance of his play Irene was performed in a way that allowed its author to be celebrated as a national hero. Voltaire died several weeks after these events, but the canonization that they initiated has continued right up until the present. Western philosophy was profoundly shaped by the conception of the philosophe and the program for Enlightenment philosophie that Voltaire came to personify. The model he offered of the philosophe as critical public citizen and advocate first and foremost, and as abstruse and systematic thinker only when absolutely necessary, was especially influential in the subsequent development of the European philosophy. Also influential was the example he offered of the philosopher measuring the value of any philosophy according by its ability to effect social change. In this respect, Karl Marx's famous thesis that philosophy should aspire to change the world, not merely interpret it, owes more than a little debt Voltaire. The link between Voltaire and Marx was also established through the French revolutionary tradition, which similarly adopted Voltaire as one of its founding heroes. Voltaire was the first person to be honored with re-burial in the newly created Pantheon of the Great Men of France that the new revolutionary government created in 1791. This act served as a tribute to the connections that the revolutionaries saw between Voltaire's philosophical program and the cause of revolutionary modernization as a whole. In a similar way, Voltaire remains today an iconic hero for everyone who sees a positive linkage between critical reason and political resistance in projects of progressive, modernizing reform. ## 2. Voltaire's Enlightenment Philosophy Voltaire's philosophical legacy ultimately resides as much in how he practiced philosophy, and in the ends toward which he directed his philosophical activity, as in any specific doctrine or original idea. Yet the particular philosophical positions he took, and the way that he used his wider philosophical campaigns to champion certain understandings while disparaging others, did create a constellation appropriately called Voltaire's Enlightenment philosophy. True to Voltaire's character, this constellation is best described as a set of intellectual stances and orientations rather than as a set of doctrines or systematically defended positions. Nevertheless, others found in Voltaire both a model of the well-oriented philosophe and a set of particular philosophical positions appropriate to this stance. Each side of this equation played a key role in defining the Enlightenment philosophie that Voltaire came to personify. ### 2.1 Liberty Central to this complex is Voltaire's conception of liberty. Around this category, Voltaire's social activism and his relatively rare excursions into systematic philosophy also converged. In 1734, in the wake of the scandals triggered by the Lettres philosophiques, Voltaire wrote, but left unfinished at Cirey, a Traite de metaphysique that explored the question of human freedom in philosophical terms. The question was particularly central to European philosophical discussions at the time, and Voltaire's work explicitly referenced thinkers like Hobbes and Leibniz while wrestling with the questions of materialism, determinism, and providential purpose that were then central to the writings of the so-called deists, figures such as John Toland and Anthony Collins. The great debate between Samuel Clarke and Leibniz over the principles of Newtonian natural philosophy was also influential as Voltaire struggled to understand the nature of human existence and ethics within a cosmos governed by rational principles and impersonal laws. Voltaire adopted a stance in this text somewhere between the strict determinism of rationalist materialists and the transcendent spiritualism and voluntarism of contemporary Christian natural theologians. For Voltaire, humans are not deterministic machines of matter and motion, and free will thus exists. But humans are also natural beings governed by inexorable natural laws, and his ethics anchored right action in a self that possessed the natural light of reason immanently. This stance distanced him from more radical deists like Toland, and he reinforced this position by also adopting an elitist understanding of the role of religion in society. For Voltaire, those equipped to understand their own reason could find the proper course of free action themselves. But since many were incapable of such self-knowledge and self-control, religion, he claimed, was a necessary guarantor of social order. This stance distanced Voltaire from the republican politics of Toland and other materialists, and Voltaire echoed these ideas in his political musings, where he remained throughout his life a liberal, reform-minded monarchist and a skeptic with respect to republican and democratic ideas. In the Lettres philosophiques, Voltaire had suggested a more radical position with respect to human determinism, especially in his letter on Locke, which emphasized the materialist reading of the Lockean soul that was then a popular figure in radical philosophical discourse. Some readers singled out this part of the book as the major source of its controversy, and in a similar vein the very materialist account of "Ame," or the soul, which appeared in volume 1 of Diderot and d'Alembert's Encyclopedie, was also a flashpoint of controversy. Voltaire also defined his own understanding of the soul in similar terms in his own Dictionnaire philosophique. What these examples point to is Voltaire's willingness, even eagerness, to publicly defend controversial views even when his own, more private and more considered writings often complicated the understanding that his more public and polemical writings insisted upon. In these cases, one often sees Voltaire defending less a carefully reasoned position on a complex philosophical problem than adopting a political position designed to assert his conviction that liberty of speech, no matter what the topic, is sacred and cannot be violated. Voltaire never actually said "I disagree with what you say, but I will defend to the death your right to say it." Yet the myth that associates this dictum with his name remains very powerful, and one still hears his legacy invoked through the redeclaration of this pronouncement that he never actually declared. Part of the deep cultural tie that joins Voltaire to this dictum is the fact that even while he did not write these precise words, they do capture, however imprecisely, the spirit of his philosophy of liberty. In his voluminous correspondence especially, and in the details of many of his more polemical public texts, one does find Voltaire articulating a view of intellectual and civil liberty that makes him an unquestioned forerunner of modern civil libertarianism. He never authored any single philosophical treatise on this topic, however, yet the memory of his life and philosophical campaigns was influential in advancing these ideas nevertheless. Voltaire's influence is palpably present, for example, in Kant's famous argument in his essay "What is Enlightenment?" that Enlightenment stems from the free and public use of critical reason, and from the liberty that allows such critical debate to proceed untrammeled. The absence of a singular text that anchors this linkage in Voltaire's collected works in no way removes the unmistakable presence of Voltaire's influence upon Kant's formulation. ### 2.2 Hedonism Voltaire's notion of liberty also anchored his hedonistic morality, another key feature of Voltaire's Enlightenment philosophy. One vehicle for this philosophy was Voltaire's salacious poetry, a genre that both reflected in its eroticism and sexual innuendo the lived culture of libertinism that was an important feature of Voltaire's biography. But Voltaire also contributed to philosophical libertinism and hedonism through his celebration of moral freedom through sexual liberty. Voltaire's avowed hedonism became a central feature of his wider philosophical identity since his libertine writings and conduct were always invoked by those who wanted to indict him for being a reckless subversive devoted to undermining legitimate social order. Voltaire's refusal to defer to such charges, and his vigor in opposing them through a defense of the very libertinism that was used against him, also injected a positive philosophical program into these public struggles that was very influential. In particular, through his cultivation of a happily libertine persona, and his application of philosophical reason toward the moral defense of this identity, often through the widely accessible vehicles of poetry and witty prose, Voltaire became a leading force in the wider Enlightenment articulation of a morality grounded in the positive valuation of personal, and especially bodily, pleasure, and an ethics rooted in a hedonistic calculus of maximizing pleasure and minimizing pain. He also advanced this cause by sustaining an unending attack upon the repressive and, to his mind, anti-human demands of traditional Christian asceticism, especially priestly celibacy, and the moral codes of sexual restraint and bodily self-abnegation that were still central to the traditional moral teachings of the day. This same hedonistic ethics was also crucial to the development of liberal political economy during the Enlightenment, and Voltaire applied his own libertinism toward this project as well. In the wake of the scandals triggered by Mandeville's famous argument in The Fable of the Bees (a poem, it should be remembered) that the pursuit of private vice, namely greed, leads to public benefits, namely economic prosperity, a French debate about the value of luxury as a moral good erupted that drew Voltaire's pen. In the 1730s, he drafted a poem called Le Mondain that celebrated hedonistic worldly living as a positive force for society, and not as the corrupting element that traditional Christian morality held it to be. In his Essay sur les moeurs he also joined with other Enlightenment historians in celebrating the role of material acquisition and commerce in advancing the progress of civilization. Adam Smith would famously make similar arguments in his founding tract of Enlightenment liberalism, On the Wealth of Nations, published in 1776. Voltaire was certainly no great contributor to the political economic science that Smith practiced, but he did contribute to the wider philosophical campaigns that made the concepts of liberty and hedonistic morality central to their work both widely known and more generally accepted. The ineradicable good of personal and philosophical liberty is arguably the master theme in Voltaire's philosophy, and if it is, then two other themes are closely related to it. One is the importance of skepticism, and the second is the importance of empirical science as a solvent to dogmatism and the pernicious authority it engenders. ### 2.3 Skepticism Voltaire's skepticism descended directly from the neo-Pyrrhonian revival of the Renaissance, and owes a debt in particular to Montaigne, whose essays wedded the stance of doubt with the positive construction of a self grounded in philosophical skepticism. Pierre Bayle's skepticism was equally influential, and what Voltaire shared with these forerunners, and what separated him from other strands of skepticism, such as the one manifest in Descartes, is the insistence upon the value of the skeptical position in its own right as a final and complete philosophical stance. Among the philosophical tendencies that Voltaire most deplored, in fact, were those that he associated most powerfully with Descartes who, he believed, began in skepticism but then left it behind in the name of some positive philosophical project designed to eradicate or resolve it. Such urges usually led to the production of what Voltaire liked to call "philosophical romances," which is to say systematic accounts that overcome doubt by appealing to the imagination and its need for coherent explanations. Such explanations, Voltaire argued, are fictions, not philosophy, and the philosopher needs to recognize that very often the most philosophical explanation of all is to offer no explanation at all. Such skepticism often acted as bulwark for Voltaire's defense of liberty since he argued that no authority, no matter how sacred, should be immune to challenge by critical reason. Voltaire's views on religion as manifest in his private writings are complex, and based on the evidence of these texts it would be wrong to call Voltaire an atheist, or even an anti-Christian so long as one accepts a broad understanding of what Christianity can entail. But even if his personal religious views were subtle, Voltaire was unwavering in his hostility to church authority and the power of the clergy. For similar reasons, he also grew as he matured ever more hostile toward the sacred mysteries upon which monarchs and Old Regime aristocratic society based their authority. In these cases, Voltaire's skepticism was harnessed to his libertarian convictions through his continual effort to use critical reason as a solvent for these "superstitions" and the authority they anchored. The philosophical authority of romanciers such as Descartes, Malebranche, and Leibniz was similarly subjected to the same critique, and here one sees how the defense of skepticism and liberty, more than any deeply held opposition to religiosity per se, was often the most powerful motivator for Voltaire. From this perspective, Voltaire might fruitfully be compared with Socrates, another founding figure in Western philosophy who made a refusal to declaim systematic philosophical positions a central feature of his philosophical identity. Socrates's repeated assertion that he knew nothing was echoed in Voltaire's insistence that the true philosopher is the one who dares not to know and then has the courage to admit his ignorance publicly. Voltaire was also, like Socrates, a public critic and controversialist who defined philosophy primarily in terms of its power to liberate individuals from domination at the hands of authoritarian dogmatism and irrational prejudice. Yet while Socrates championed rigorous philosophical dialectic as the agent of this emancipation, Voltaire saw this same dialectical rationalism at the heart of the dogmatism that he sought to overcome. Voltaire often used satire, mockery and wit to undermine the alleged rigor of philosophical dialectic, and while Socrates saw this kind of rhetorical word play as the very essence of the erroneous sophism that he sought to alleviate, Voltaire cultivated linguistic cleverness as a solvent to the false and deceptive dialectic that anchored traditional philosophy. ### 2.4 Newtonian Empirical Science Against the acceptance of ignorance that rigorous skepticism often demanded, and against the false escape from it found in sophistical knowledge--or what Voltaire called imaginative philosophical romances--Voltaire offered a different solution than the rigorous dialectical reasoning of Socrates: namely, the power and value of careful empirical science. Here one sees the debt that Voltaire owed to the currents of Newtonianism that played such a strong role in launching his career. Voltaire's own critical discourse against imaginative philosophical romances originated, in fact, with English and Dutch Newtonians, many of whom were expatriate French Huguenots, who developed these tropes as rhetorical weapons in their battles with Leibniz and European Cartesians who challenged the innovations of Newtonian natural philosophy. In his Principia Mathematica (1687; 2nd rev. edition 1713), Newton had offered a complete mathematical and empirical description of how celestial and terrestrial bodies behaved. Yet when asked to explain how bodies were able to act in the way that he mathematically and empirically demonstrated that they did, Newton famously replied "I feign no hypotheses." From the perspective of traditional natural philosophy, this was tantamount to hand waving since offering rigorous causal accounts of the nature of bodies in motion was the very essence of this branch of the sciences. Newton's major philosophical innovation rested, however, in challenging this very epistemological foundation, and the assertion and defense of Newton's position against its many critics, not least by Voltaire, became arguably the central dynamic of philosophical change in the first half of the eighteenth century. While Newtonian epistemology admitted of many variations, at its core rested a new skepticism about the validity of apriori rationalist accounts of nature and a new assertion of brute empirical fact as a valid philosophical understanding in its own right. European Natural philosophers in the second half of the seventeenth century had thrown out the metaphysics and physics of Aristotle with its four part causality and teleological understanding of bodies, motion and the cosmic order. In its place, however, a new mechanical causality was introduced that attempted to explain the world in equally comprehensive terms through the mechanisms of an inert matter acting by direct contact and action alone. This approach lead to the vortical account of celestial mechanics, a view that held material bodies to be swimming in an ethereal sea whose action pushed and pulled objects in the manner we observe. What could not be observed, however, was the ethereal sea itself, or the other agents of this supposedly comprehensive mechanical cosmos. Yet rationality nevertheless dictated that such mechanisms must exist since without them philosophy would be returned to the occult causes of the Aristotelian natural tendencies and teleological principles. Figuring out what these point-contact mechanisms were and how they worked was, therefore, the charge of the new mechanical natural philosophy of the late seventeenth century. Figures such as Descartes, Huygens, and Leibniz established their scientific reputations through efforts to realize this goal. Newton pointed natural philosophy in a new direction. He offered mathematical analysis anchored in inescapable empirical fact as the new foundation for a rigorous account of the cosmos. From this perspective, the great error of both Aristotelian and the new mechanical natural philosophy was its failure to adhere strictly enough to empirical facts. Vortical mechanics, for example, claimed that matter was moved by the action of an invisible agent, yet this, the Newtonians began to argue, was not to explain what is really happening but to imagine a fiction that gives us a speciously satisfactory rational explanation of it. Natural philosophy needs to resist the allure of such rational imaginings and to instead deal only with the empirically provable. Moreover, the Newtonians argued, if a set of irrefutable facts cannot be explained other then by accepting the brute facticity of their truth, this is not a failure of philosophical explanation so much as a devotion to appropriate rigor. Such epistemological battles became especially intense around Newton's theory of universal gravitation. Few questioned that Newton had demonstrated an irrefutable mathematical law whereby bodies appear to attract one another in relation to their masses and in inverse relation to the square of the distance between them. But was this rigorous mathematical and empirical description a philosophical account of bodies in motion? Critics such as Leibniz said no, since mathematical description was not the same thing as philosophical explanation, and Newton refused to offer an explanation of how and why gravity operated the way that it did. The Newtonians countered that phenomenal descriptions were scientifically adequate so long as they were grounded in empirical facts, and since no facts had yet been discerned that explained what gravity is or how it works, no scientific account of it was yet possible. They further insisted that it was enough that gravity did operate the way that Newton said it did, and that this was its own justification for accepting his theory. They further mocked those who insisted on dreaming up chimeras like the celestial vortices as explanations for phenomena when no empirical evidence existed to support of such theories. The previous summary describes the general core of the Newtonian position in the intense philosophical contests of the first decades of the eighteenth century. It also describes Voltaire's own stance in these same battles. His contribution, therefore, was not centered on any innovation within these very familiar Newtonian themes; rather, it was his accomplishment to become a leading evangelist for this new Newtonian epistemology, and by consequence a major reason for its widespread dissemination and acceptance in France and throughout Europe. A comparison with David Hume's role in this same development might help to illuminate the distinct contributions of each. Both Hume and Voltaire began with the same skepticism about rationalist philosophy, and each embraced the Newtonian criterion that made empirical fact the only guarantor of truth in philosophy. Yet Hume's target remained traditional philosophy, and his contribution was to extend skepticism all the way to the point of denying the feasibility of transcendental philosophy itself. This argument would famously awake Kant's dogmatic slumbers and lead to the reconstitution of transcendental philosophy in new terms, but Voltaire had different fish to fry. His attachment was to the new Newtonian empirical scientists, and while he was never more than a dilettante scientist himself, his devotion to this form of natural inquiry made him in some respects the leading philosophical advocate and ideologist for the new empirico-scientific conception of philosophy that Newton initiated. For Voltaire (and many other eighteenth-century Newtonians) the most important project was defending empirical science as an alternative to traditional natural philosophy. This involved sharing in Hume's critique of abstract rationalist systems, but it also involved the very different project of defending empirical induction and experimental reasoning as the new epistemology appropriate for a modern Enlightened philosophy. In particular, Voltaire fought vigorously against the rationalist epistemology that critics used to challenge Newtonian reasoning. His famous conclusion in Candide, for example, that optimism was a philosophical chimera produced when dialectical reason remains detached from brute empirical facts owed a great debt to his Newtonian convictions. His alternative offered in the same text of a life devoted to simple tasks with clear, tangible, and most importantly useful ends was also derived from the utilitarian discourse that Newtonians also used to justify their science. Voltaire's campaign on behalf of smallpox inoculation, which began with his letter on the topic in the Lettres philosophiques, was similarly grounded in an appeal to the facts of the case as an antidote to the fears generated by logical deductions from seemingly sound axiomatic principles. All of Voltaire's public campaigns, in fact, deployed empirical fact as the ultimate solvent for irrational prejudice and blind adherence to preexisting understandings. In this respect, his philosophy as manifest in each was deeply indebted to the epistemological convictions he gleaned from Newtonianism. ### 2.5 Toward Science without Metaphysics Voltaire also contributed directly to the new relationship between science and philosophy that the Newtonian revolution made central to Enlightenment modernity. Especially important was his critique of metaphysics and his argument that it be eliminated from any well-ordered science. At the center of the Newtonian innovations in natural philosophy was the argument that questions of body per se were either irrelevant to, or distracting from, a well focused natural science. Against Leibniz, for example, who insisted that all physics begin with an accurate and comprehensive conception of the nature of bodies as such, Newton argued that the character of bodies was irrelevant to physics since this science should restrict itself to a quantified description of empirical effects only and resist the urge to speculate about that which cannot be seen or measured. This removal of metaphysics from physics was central to the overall Newtonian stance toward science, but no one fought more vigorously for it, or did more to clarify the distinction and give it a public audience than Voltaire. The battles with Leibnizianism in the 1740s were the great theater for Voltaire's work in this regard. In 1740, responding to Du Chatelet's efforts in her Institutions de physiques to reconnect metaphysics and physics through a synthesis of Leibniz with Newton, Voltaire made his opposition to such a project explicit in reviews and other essays he published. He did the same during the brief revival of the so-called "vis viva controversy" triggered by du Chatelet's treatise, defending the empirical and mechanical conception of body and force against those who defended Leibniz's more metaphysical conception of the same thing. In the same period, Voltaire also composed a short book entitled La Metaphysique de Newton, publishing it in 1740 as an implicit counterpoint to Chatelet's Institutions. This tract did not so much articulate Newton's metaphysics as celebrate the fact that he avoided practicing such speculations altogether. It also accused Leibniz of becoming deluded by his zeal to make metaphysics the foundation of physics. In the definitive 1745 edition of his Elements de la philosophie de Newton, Voltaire also appended his tract on Newton's metaphysics as the book's introduction, thus framing his own understanding of the relationship between metaphysics and empirical science in direct opposition to Chatelet's Leibnizian understanding of the same. He also added personal invective and satire to this same position in his indictment of Maupertuis in the 1750s, linking Maupertuis's turn toward metaphysical approaches to physics in the 1740s with his increasingly deluded philosophical understanding and his authoritarian manner of dealing with his colleagues and critics. While Voltaire's attacks on Maupertuis crossed the line into ad hominem, at their core was a fierce defense of the way that metaphysical reasoning both occludes and deludes the work of the physical scientist. Moreover, to the extent that eighteenth-century Newtonianism provoked two major trends in later philosophy, first the reconstitution of transcendental philosophy a la Kant through his "Copernican Revolution" that relocated the remains of metaphysics in the a priori categories of reason, and second, the marginalization of metaphysics altogether through the celebration of philosophical positivism and the anti-speculative scientific method that anchored it, Voltaire should be seen as a major progenitor of the latter. By also attaching what many in the nineteenth century saw as Voltaire's proto-positivism to his celebrated campaigns to eradicate priestly and aristo-monarchical authority through the debunking of the "irrational superstitions" that appeared to anchor such authority, Voltaire's legacy also cemented the alleged linkage that joined positivist science on the one hand with secularizing disenchantment and dechristianization on the other. In this way, Voltaire should be seen as the initiator of a philosophical tradition that runs from him to Auguste Comte and Charles Darwin, and then on to Karl Popper and Richard Dawkins in the twentieth century.
voluntarism-theological	## 1. Metaethical and normative theological voluntarism To be a theological voluntarist with respect to some moral status is to hold that entities have that status in virtue of some act of divine will. But some instances of this view are metaethical theses; some instances of it are normative theses. Consider, for example, theological voluntarism about the status of acts as obligatory or non-obligatory. One might hold that there is a single supreme obligation, the obligation to obey God. Every particular type of act that one might perform thus has its moral status as obligatory or non-obligatory in virtue of God's having commanded the performance of acts of that type or God's not having commanded acts of that type. This is a common version of divine command theory, according to which all of the more workaday obligations that we are under (not to steal from each other, not to murder each other, to help each other out when it would not be inconvenient, etc.) bind us as a result of the exercise of God's supreme practical authority. The view just described is a version of normative theological voluntarism. It is a normative view because it asserts that some normative state of affairs obtains--namely, the normative state of affairs its being obligatory to obey God. And it is a version of theological voluntarism because it holds that all other normative states of affairs, at least those involving obligation, obtain in virtue of God's commanding activity. Metaethical theological voluntarism, by contrast, does not assert the obtaining of any normative state of affairs. (For a contrary view, see Heathwood 2012, p. 7.) It is possible for one to be a metaethical theological voluntarist and to hold that no normative states of affairs obtain. Rather, metaethical theological voluntarists aim to say something interesting and informative about moral concepts, properties, or states of affairs; and they want to say something interesting and informative about them by connecting them to acts of the divine will. Metaethical theological voluntarists might claim that (e.g.) obligation is a theological concept, or that the property of being obligatory is a theological property, or that obligations are caused immediately by the divine will. But note that none of these views asserts that there are any obligations. ### 1.1 Theological voluntarism and theism One does not have to be a theist in order to be a theological voluntarist. One can affirm normative theological voluntarism or metaethical theological voluntarism while failing to affirm theism; atheists and agnostics can be theological voluntarists of either stripe. With respect to normative theological voluntarism: one might claim that while it is true that any being that merits the title of 'God' merits obedience, we should not believe that there is such a being. (Compare: if, through some glitch in promotions, there happened to be no lieutenants in the army at some time, it would not cease to be true that privates ought to obey lieutenants. One could believe that that there are no lieutenants while believing that privates ought to obey their lieutenants.) With respect to metaethical theological voluntarism: one might claim that, for example, the concept of obligation is ineliminably theistic, though there is no God; that God does not exist counts not against metaethical theological voluntarism but rather against the claim that the concept of obligation has application. (Compare: one might believe that 'sin' is properly defined as 'offense against God.' One can clearly affirm this definition while rejecting God's existence; all that one is committed to thereby is that there really are no sins.) ### 1.2 Theological voluntarism and moral skepticism Call a 'moral skeptic' one who disbelieves or withholds judgment on the claim that any normative state of affairs obtains. One can affirm metaethical theological voluntarism while being a moral skeptic; one cannot affirm normative theological voluntarism while being a moral skeptic. A metaethical theological voluntarist might claim that no normative state of affairs could be made to obtain without certain acts of divine will, but because there is no God, or because there is a God that has not performed the requisite acts of will, no normative states of affairs obtain. A normative theological voluntarist cannot, however, be a moral skeptic. Because the normative theological voluntarist is committed to the obtaining of at least one normative state of affairs--for example, its being obligatory to obey God--the conjunction of moral skepticism and normative theological voluntarism is not a coherent combination of views. My concern in the rest of this article will be with the metaethical version of theological voluntarism; any further references to theological voluntarism are, unless otherwise noted, to the metaethical version of the position. Theological voluntarism thus understood is consistent either with the affirmation or with the denial of theism and moral skepticism. Taking a negative stand on theism or a positive stand on moral skepticism should not prevent one from taking seriously theological voluntarism as a philosophical position. This is an important point, because it is often thought that theological voluntarism is only for theists, or only for moral nonskeptics. While it is true that some of the arguments for theological voluntarism take theism, or the existence of moral obligations, as premises, not all of them do. ## 2. Metaethical theological voluntarism Metaethics is concerned with the formulation of interesting and informative accounts of normative concepts, properties, and states of affairs; and a metaethics that is a version of theological voluntarism will formulate such accounts in terms of some acts of divine will. This statement of the position is highly abstract, but it cannot be made less abstract without making difficult choices among rival formulations of the view. ### 2.1 Considerations in Favor The considerations to be offered in favor of theological voluntarism are, at this level, similarly abstract. I will discuss three types of consideration: historical, theological, and metaethical. #### Historical considerations in favor of theological voluntarism Some of the considerations in favor of metaethical theological voluntarism are historical. Both theists and nontheists have been impressed by the extent to which at least some moral concepts developed in tandem with theological concepts, and it may therefore be the case that there could be no adequate explication of some moral concepts without appeal to theological ones. On this view, it is not merely historical accident that at least some moral concepts had their origin in contexts of theistic belief and practice; rather, these concepts have their origin essentially in such contexts, and become distorted and unintelligible when exported from those contexts (see, for example, Anscombe 1958). #### Theological considerations in favor of theological voluntarism Some of the considerations in favor of theological voluntarism have their source in matters regarding the divine nature. Several such arguments are summarized in Idziak 1979 (pp. 8-10). Some appeal to omnipotence: since God is both omnipotent and impeccable, theological voluntarism must be true: for if God cannot act in a way that is morally wrong, then God's power would be limited by other normative states of affairs were theological voluntarism not the case. Some appeal to God's freedom: since God is free and impeccable, theological voluntarism must be true: for if moral requirements existed prior to God's willing them, requirements that an impeccable God could not violate, God's liberty would be compromised. Some appeal to God's status as supremely lovable and deserving of allegiance: if theism is true, then the world of value must be a theocentric one, and so any moral view that does not place God at its center is bound to be inadequate. Even if individually insufficient as justifications for adopting theological voluntarism, collectively they may suggest some desiderata for a moral view: that God must be at the center of a moral theory, and, in particular, that the realm of the moral must be dependent on God's free choices. It seems that any moral theory that met these desiderata would count as a version of theological voluntarism. #### Metaethical considerations in favor of theological voluntarism A third set of considerations in favor of theological voluntarism has its source in metaethics proper, in the attempt to provide adequate philosophical accounts of the various formal features exhibited by moral concepts, properties, and states of affairs. One might claim, that is, that theological voluntarism makes the best sense of the formal features of morality that both theists and nontheists acknowledge. Consider first the normativity of morals. Both theists and nontheists have been impressed by the weirdness of normativity, with its very otherness, and have thought that whatever we say about normativity, it will have to be a story not about natural properties but nonnatural ones (cf. Moore 1903, section 13). John Mackie, an atheist, and George Mavrodes, a theist, have both drawn from this the same moral: if there is a God, then the normativity of morality can be understood in theistic terms; otherwise, the normativity of morality is unintelligible (Mavrodes 1986; Mackie 1977, p. 48). As Robert Adams has suggested, given the serious difficulties present in understanding moral properties as natural properties, it is worthwhile taking seriously the hypothesis that morality is not just a nonnatural matter but a supernatural one (Adams 1973, p. 105). For the standard objections against understanding normativity as a nonnatural property concern our inability to say anything further about that nonnatural property itself and about our ability to grasp that property (see, e.g., M. Smith 1994, pp. 21-25). But if morality is to be understood in terms of God's commands, we can give an informative account of what these unusual properties are; and if it is understood in terms of God's commands, then we can give an informative account of how God, being the creator and sustainer of us rational beings, can ensure that we can have an adequate epistemic grasp of the moral domain (Adams 1979a, pp. 137-138). Consider next the impartiality of morals. The domain of the moral, unlike the domain of value generally, is governed by the requirements of impartiality. To use Sidgwick's phrase, the point of view of morality is not one's personal point of view but rather "the point of view ... of the Universe" (Sidgwick 1907, p. 382). But, to remark on the perfectly obvious, the Universe does not have a point of view. Various writers have employed fictions to try to provide some sense to this idea: Adam Smith's impartial and benevolent spectator, Firth's ideal observer, and Rawls' contractors who see the world sub specie aeternitatis come to mind most immediately (Smith 1759, Pt III, Ch 8; Firth 1958; and Rawls 1971, p. 587). But theological voluntarism can provide a straightforward understanding of the impartiality of morals by appealing to the claim that the demands of morality arise from the demands of someone who in fact has an impartial and supremely deep love for all of the beings that are morality's proper objects. Consider next the overridingness of morals. The domain of the moral, it is commonly thought, consists in a range of values that can demand absolute allegiance, in the sense that it is never reasonable to act contrary to what those values finally require. One deep difficulty with this view, formulated in a number of ways but perhaps most memorably by Sidgwick (1907, pp. 497-509), is that it is hard to see how moral value automatically trumps other kinds of value (e.g. prudential value) when they conflict. But if the domain of the moral is to be understood in terms of the will of a being who can make it possible that, or even ensure that, the balance of reasons is always in favor of acting in accordance with the moral demand, then the overridingness of morals becomes far easier to explain (Layman 2006; Evans 2013, pp. 29-30). Consider next the content of morals. There is a strong case to be made that moral judgments cannot have just any content: they must be concerned, somehow, with what exhibits respect for certain beings, or with what promotes their interests (cf. Foot 1958, pp. 510-512; M. Smith 1994, p. 40; Cuneo & Shafer-Landau 2014, pp. 404-407). Theological voluntarism has a ready explanation for the content of morals being what it is: it is that moral demands arise from a being that loves that being's creation. So there are some general reasons to think theological voluntarism promising. The reasons are stronger yet when one is proceeding from theistic starting points. (This is not trivial, since a number of theistic philosophers reject theological voluntarism.) But these reasons, while suggestive, are rather generic: they point to the promise possessed by theological voluntarism, though they do not fix for us on a particular formulation of the view. The general schema for a particular theological voluntarist position is 'evaluative status M stands in dependence relation D to divine act A' (cf. Quinn 1999, p. 53, which I follow here except to substitute the more general 'evaluative' for Quinn's more specific 'moral'). So there are at least three choices that have to be made. We need to say something about what sorts of evaluative statuses depend on God's will. We need to say something about what are the relevant acts of divine will. And we need to say something about what the dependence relation is supposed to be. (These are not independent questions, of course.) ### 2.2 Which evaluative statuses? A metaethical view can be more or less comprehensive, aiming to cover more or fewer evaluative statuses. A metaethical view might claim to provide an account of all evaluative notions, or of all normative notions, or of all moral notions, or or some set of moral notions. (Roughly, and taking the notion of an evaluative property as fundamental: for a notion to be normative is for it to be a certain sort of evaluative notion, one that is essentially action-guiding; for a notion to be moral is for it to be a certain sort of normative notion, one that exhibits impartiality.) No one claims that theological voluntarism provides an account of all evaluative notions. The real contenders are the latter three. There are good reasons to reject the claim that all normative notions are to be understood in relation to God's will. The main reason is that, as we will see below, it is important that there be items with normative statuses independent of God's will in order to explain how God's will, even if free, is not arbitrary. And it is not as if the view that some normative statuses are not to be explained in terms of God's will must be repugnant to a theocentric metaethics: for, after all, one might understand such statuses in theological, even if not voluntaristic, terms. Adams, for example, understands some notions of goodness in terms of likeness to God, an understanding that is unquestionably theocentric though not voluntaristic (Adams 1999, pp. 28-38; also Murphy 2011, pp. 148-160). Most of the current debate over the evaluative statuses to be explained by theological voluntarism, then, concerns whether the entire set or only some proper subset of moral statuses is to be understood in both theological and voluntaristic terms. Quinn's 1978 work offers a theological voluntarist view on which all moral statuses are to be understood in terms of God's will. But Adams rejects this view, and Quinn, following Adams and Alston (1990), eventually rejected it as well. These writers hold that only moral properties in the "obligation family," properties like those of being obligatory, being permissible, being required, and being right (where being right involves a constraint on conduct, rather than being merely fitting), are to be understood in theological voluntarist terms. Call their view the restricted moral view; call the view that all moral statuses are to be understood in voluntarist terms the unrestricted moral view. The restricted moral view has been defended with more and less impressive arguments. (See Murphy 2012.) The less impressive arguments are those that appeal to the idea that there must be moral properties that are not explained in terms of God's will in order to deal with some of the classic objections to theological voluntarism. To preserve the notion that God is good, one might say, we need to restrict the aspirations of theological voluntarism to those of explaining a proper subset of moral notions, leaving the remainder for an account of God's goodness; or, to make intelligible the commands that God chooses to give, we need to set aside some group of moral notions to be explained in other than theological voluntarist terms and that can therefore enter into our account of the intelligibility of God's choices to give certain commands rather than others. But these considerations are not, after all, entirely persuasive. For one might well explain the notion that God is good and account for the intelligibility of divine commands by appeal to normative notions that are nonmoral. (See Section 3 below for further discussion of these arguments.) More plausible are arguments that suggest that there is something in particular about obligation that makes it fit for a theological voluntarist explanation, some feature that is not shared with notions like moral virtue and moral good. Adams suggests, with some plausibility, that the notion of obligation is ineliminably social, that it must involve a relationship between persons, a relationship in which a demand is made (Adams 1987b, p. 264; also Adams 1999, pp. 245-246; for a nontheistic view defending a similar position, see Darwall 2006). This feature of obligation makes it different from notions of goodness and virtue, which do not seem to have this essentially social element. That obligation is special in this way does not, of course, show that notions of moral virtue and moral goodness do not also need to be treated in a theological voluntarist way. It could be that even if obligation most obviously requires this treatment, the points made earlier about the promise of theological voluntarism also extend to other moral notions, even if in a less pressing way. (See Section 3.3 for further discussion, and evaluation, of this point.) There are no obviously decisive reasons for the theological voluntarist to adhere to either the restricted or the unrestricted moral view. (For a set of arguments that the restricted theological voluntarists have failed to avoid objections to theological voluntarism while leaving the positive arguments for the view intact, see Murphy 2012.) But theological voluntarists have wanted to say at least that the properties in the obligation family are to be accounted for in terms of this view. In the remainder of this article, it will be assumed that theological voluntarism is about properties in the obligation family, though we will occasionally consider how the view could be extended to other moral properties as well. ### 2.3 Which act of divine will? Assume, then, that theological voluntarism is an account of obligation-type properties. A second issue concerning the proper formulation of the view concerns the relevant act of divine will. Is the requisite act of divine will to be understood as an act of commanding, or instead as some mental act like choosing, intending, preferring, or wishing? And if one holds that the act of the divine will is a mental act, should the mental act to which the theological voluntarist appeals in order to account for obligation be one whose object is the action that is made obligatory, or one whose object is the state of affairs that the action is obligatory? We have, to simplify matters, three options: * (1) That it is obligatory for A to ph depends on God's commanding A to ph. * (2) That it is obligatory for A to ph depends on God's willing that A ph. * (3) That it is obligatory for A to ph depends on God's willing that it be obligatory for A to ph. One might think that the central issue here would be to decide between the speech-act view (1) and the mental acts views (2) and (3); it might be thought to be less important, an issue of intramural interest only, to decide between (2) and (3). But this is not right. The important debate is between (1) and (2). For (3) is, understood in one way, no competitor with either (1) or (2); and understood differently, it has little argumentative support. #### The dispute between (1) and (2) There is an ongoing debate whether (1) or (2) is the better formulation of theological voluntarism about obligation. There are initially plausible points on both sides of the issue. In favor of (1), one might appeal to the centrality of the image of God as commander in the Abrahamic faiths. In favor of (2), one might appeal to the centrality to theistic belief and practice of the idea that doing God's will is the standard for the moral life. With only these initial points, there can be no resolution, and so defenders of these two formulations of theological voluntarism have sought other argumentative routes. One might try to reduce (2) to absurdity. One who is rational does not intend what one knows will not happen; and, on the orthodox conception of God, God is both rational and omniscient. This entails that God will never intend something that will not happen. If obligation arises from divine intentions, though, then no obligations will ever be violated. Since this is absurd, one should prefer (1) over (2). But defenders of (2) have a plausible response. First, defenders of (1) are in no better a position than defenders of (2). For it is a sincerity condition on the giving of commands that the commander intend that the commanded perform the action; and so if this objection reduces (2) to absurdity, the only way that the defender of (1) can avoid having his or her position reduced to absurdity is by holding that God is not necessarily sincere. Second, the notion of intention admits of various readings, and there is a reading of intention suitable for theological voluntarism that does not have the untoward result that no created rational beings could ever act contrary to a divine intention. It is standard to distinguish between God's antecedent and God's consequent will: God's consequent will is God's will absolutely considered, as bearing on all actual circumstances; God's antecedent will is God's will considered with respect to some proper subset of actual circumstances. (To use an example of Aquinas's, one drawn from the discussion in which the distinction between antecedent and consequent will is made [Summa Theologiae, Ia, Q. 19, A. 6]: while in one way God wills that all persons be saved, in another way God does not will that all be saved; indeed, God wills that some persons be damned. What makes this coherent is that the sense of willing in which God wills that all be saved is antecedent: prior to a consideration of all of the particulars of persons' situations, God wills their salvation; but in light of all of the particulars of persons' situations--including the circumstance that some persons have willingly rejected friendship with God--God wills their damnation.) So while there is a sense in which it is true that all that God intends must come to pass, this sense is that of consequent intending rather than antecedent intending. On (2), then, one can say that the theological voluntarist holds that obligation depends on certain of God's antecedent intentions. (See also Murphy 1998, pp. 17-21.) We might ask, in order to bring the differences between these views better out into the light, what our considered opinion is in cases in which a divine antecedent intention that A ph and a divine command that A ph pull apart. It is far from clear that it is a real option for God to command that A ph while not intending that A ph. Though this possibility is endorsed by Wierenga 1983 (p. 390) and is at least entertained in Murphy 1998 (p. 9), for God to issue such a command would be for God to command insincerely--something that many would be loath to allow. (See also Adams 1999, p. 260 and Murphy 2002, 2.5). But the other option appears unproblematic enough. God might intend for humans to act a certain way while not commanding them to do so. In such a scenario, one might ask, is obligation engendered? If yes, then it seems to count in favor of the divine will view (2); if no, then it seems to count in favor of the divine command view (1). Adams claims that obligations are not engendered in such cases; actually made demands are necessary. He offers three reasons for preferring the command conception in these cases. The first is that holding that obligation is a matter of divine command rather than divine will makes possible a distinction between the obligatory and the supererogatory: we can say that while God's commands somehow makes certain acts obligatory, if God does will that we perform some act but does not command it, performing that act is supererogatory. The second is an appeal to the idea that theological voluntarism is a social conception of obligation: obligation arises in the context of the social relationship between God and created rational beings. (Not all versions of theological voluntarism affirm this; see the discussion of (3) below.) But, Adams says, in social relationships obligations arise only when demands are actually made. And the third reason is that there is something unsavory about obligations allegedly resulting from an act of divine will that is not expressed as a command: "Games in which one party incurs guilt for failing to guess the unexpressed wishes of the other party are not nice games. They are no nicer if God is thought of as a party to them" (Adams 1999, p. 261). But the defender of the divine will view (2) has some responses available. The defender of this view can say, with respect to the first point, that a divine will view can capture the difference between the obligatory and the supererogatory not by appeal to the difference between acts of divine will that are expressed as commands and those that are not but rather as a difference between distinct types of act of divine will: the difference, say, between what God intends that we do and what God merely prefers that we do (cf. Quinn 1999, p. 56). Or, in the alternative, the distinction can be made within the divine will: one might characterize as obligatory those actions that God wills that one perform, and as supererogatory those actions that God wills that one perform if one is willing to do so, so that if one performs the action, one is doing what God wills, but if one does not, one is not doing what God wills that one not do. With respect to the second and third points, the defender of the divine will view can directly challenge Adams' view that obligations generated within social relationships must always be expressed as demands. Spouses, for example, often take themselves to be obligated by what their spouses intend with respect to their behavior; indeed, it would be unseemly to hold oneself to be bound by one's spouse's will only if the spouse has actually made a demand on one. ("How can you blame me for not helping you empty the dishwasher? You didn't tell me to!" does not often go over well.) One often wants another to perform some action without being told to; many actions have their value only through being performed without being prompted by a command. But, on (1), no act of the form 'ph-ing, though God has not told me to ph' could ever be obligatory. Here is a thought experiment that may help to decide the dispute between these two camps. For it to be possible for one to give another a command to ph, there must be a linguistic practice available to the addressee in terms of which the speaker can formulate a command. This is not just for the sake of having the means to communicate a command: rather, commands are essentially linguistic items, and cannot be defined except in such terms. Imagine, though, that a certain created rational being, Mary, inhabits a linguistic community in which there is no practice of commanding. One can successfully make assertions to Mary, and among these assertions can be assertions about one's own psychological states, but one cannot successfully command Mary to do anything. Here is the question: so long as Mary's linguistic resources are confined to those afforded by this practice, can God impose obligations on her? The defender of (1) will have to say No: for Mary cannot be commanded to do anything. The defender of (2) will have to say Yes: God could have an antecedent intention that Mary perform some action and (to sidestep worries about being under an obligation that one cannot know about) could inform Mary that God has that intention with respect to her conduct. The debate between defenders of (1) and defenders of (2) is ongoing, and at present far from conclusive. (For some recent interventions into this debate, see Mann 2005b, Miller 2009a, and Jordan 2012.) #### Option (3) The other formulation of theological voluntarism that we noted is that in which the act of divine will is that of willing that the state of affairs that it is obligatory for A to ph obtain. Unlike the formulations (1) and (2), which admit of various sorts of dependence relationship between the act of divine will and the obligation, (3) is limited to something like a causal picture. (It obviously could not be that its being obligatory for A to ph is identical to God's willing that it be obligatory for A to ph, on pain of a vicious regress.) The idea expressed here is that ultimately all obligations are present because of efficacious acts of the divine will, in particular, acts of willing that those obligations be in force. This account is compatible with (1) and (2), because it could be that the way that God makes it the case that an act is obligatory is necessarily through the giving of commands (as in (1)) or through antecedently intending (as in (2)) the performance of that action. So it is not really, in its most general form, a competitor to (1) and (2). It could be made a competitor by adding claims about the way that the divine will brings it about that these obligations obtain. A defender of (3) might add that on his or her view the divine intention that A be obligated to ph is the immediate, total, and exclusive cause of its being obligatory for A to ph (cf. Quinn 1999, p. 55). If so, then a divine command that A ph or a divine antecedent intention that A ph could not be partial or mediate causes of its being obligatory for A to ph. But note that even thus strengthened (3) is compatible with (1) or (2) understood as an identity claim: if the claim is that obligations just are divine commands or divine intentions, then the compatibility with (1) and (2) is reestablished. Even if it turns out that (3) is not an obvious competitor with (1) and (2), it is still worth asking whether it is true. Though Quinn eventually rejected (3), at one point he argued for it by appeal to divine sovereignty: because every state of affairs that obtains that does not involve God's existing depends on God's will, a fortiori every normative state of affairs that does not involve God's existing depends on God's will. (Quinn thought that its being obligatory to obey God is a state of affairs that involves God's existing [Quinn 1990, pp. 298-299], but for a reason to reject that claim see Murphy 1998, pp. 12-13.) It does not seem, though, that this argument would support (3) in the strengthened version that holds that the dependence must be immediate, total, and exclusive. After all, very few folks want to say that every state of affairs that is brought about by God's will is brought about by God's will exclusively, totally, and immediately. While it is plausibly part of theism that every state of affairs that obtains, apart from those that involve God, is somehow dependent on God's will, this does not show that deontic states of affairs are more interestingly connected to the divine will than states of affairs involving mathematics, or physics, or accounting (Murphy 1998, pp. 14-16). We may put (3) to the side, then. While some formulations of it may very well be true, those formulations for which there is argumentative support do not establish much in the way of interesting metaethical conclusions. The debate concerning whether (1) and (2) is the more adequate formulation of theological voluntarism is ongoing, and we should thus proceed in a way that is as far as possible neutral between the two (though admittedly unwieldy) by saying that As moral obligations to ph depend on God's commands/intentions that A ph. (By 'intentions' I will mean antecedent intentions.) Allowing for both of these possibilities, what can be said about the relationship of dependence holding between divine commands/intentions and moral obligations? ### 2.4 What sort of dependence? The third issue that must be dealt with in providing a formulation of theological voluntarism is that of the specification of the dependence relationship that holds between divine commands/intentions that A ph and the moral obligation of A to ph. There have been several options considered whose nature and merits are worth discussing here. On an analysis view, it is part of the meaning of 'it is morally obligatory for A to ph' that God commands/intends that A ph. On a reduction view, the state of affairs its being obligatory for A to ph is the state of affairs God's commanding/intending that A ph. On a supervenience view, its being obligatory for A to ph supervenes on God's commanding/intending that A ph. On a causal view, necessarily, its being obligatory for A to ph is caused by God's commanding/intending that A ph, and necessarily, God's commanding/intending that A ph causes it to be obligatory for A to ph. #### Causation The causal view is defended by Quinn (1979, 1990, 1999), and in a particularly strong form: on Quinn's view, the causal connection between God's antecedently intending that A ph and its being obligatory for A to ph exhibits totality, exclusivity, activity, immediacy, and necessity. There are at least three serious difficulties for the causation formulation. The first we may call the 'Humean worry'. Once we allow that its being morally obligatory to ph is distinct from God's commanding/intending ph-ing, there is the question of what reason we would have for thinking that its being morally obligatory to ph necessarily obtains if God's commanding/intending ph-ing obtains. And whatever answer the defender of this view offers, it must be consistent with the causation formulation of theological voluntarism. But it is unclear what would do the trick. One way to try to make this necessary connection is by holding that there is a prior moral obligation to obey God; and so, whenever God gives a command/has an intention that one perform some action, it follows that one is morally obligated to perform the action commanded/intended. But we cannot take this route, because if the causation formulation is correct, then all moral obligations are caused entirely by God's commanding/intending activity; there cannot be, then, this prior moral obligation to obey God that would serve as part of the explanans for the necessary connection between divine commands/intentions and moral obligations. The causal view that is a version of (2) (that is, that its being obligatory for A to ph depends causally on God's willing that A ph) should be distinguished carefully from the causal view that is a version of (3) (that is, that its being obligatory for A* to ph* depends causally on God's willing that it be obligatory for A* to ph). The causal formulation of (3) has at least some plausibility as a result of God's sovereignty and omnipotence--though it is in the end unclear why we should move from the claim that God is the ultimate source of all being to the claim that, for all deontic states of affairs, God's willing that that deontic state of affairs obtain is the immediate, total, and exclusive* cause of its obtaining. Most of us would not, after all, find intuitively compelling a move from the claim that God is ultimate source of all being to the claim that, for all physical states of affairs, God's willing that that physical state of affairs obtain is the immediate, total, and exclusive cause of its obtaining. (That view is 'occasionalism', and occasionalism is a distinctly minority view even among theists; see Murphy 2011 (pp. 140-142) for a comparison between theological voluntarism and occasionalism.) The causal view as an instance of (2), though, seems to have even less in the way of argumentative support. Why would one think that God's intending that A ph is an immediate, total, and exclusive cause of a deontic state of affairs' obtaining? In the absence of some evidence for such a connection, it is hard to see why one would be attracted to this formulation of theological voluntarism. The second worry about the causal formulation I will call the 'lack of precedent worry'. Moral properties and states of affairs supervene on nonmoral properties and states of affairs. The intuitive idea is that there can be no differences in moral status without some difference in nonmoral status. The causal formulation satisfies the supervenience constraint--the differences in nonmoral status concern God's commands/intentions--but it does so in a way that is unprecedented and mysterious. When we look at the specific ways in which changes in nonmoral facts can make a difference to the moral facts that hold, there is a pretty limited number of intelligible relationships that can hold between these nonmoral facts and the moral facts that supervene on them. A nonmoral fact can be part of what constitutes a reason to perform an action. (That you promised to ph can be cited in explaining why you have a reason to ph your promising to ph constitutes, at least in part, the reason that you have to ph.) It can be part of an enabling condition for that reason. (The existence of a social practice of promising can be cited in explaining why you have a reason to ph; the existence of that practice might explain why your promise has the reason-giving force that it has.) It can be cited as a defeater-defeater for a reason. (While the fact the promisee told you that you need not fulfill your promise to ph typically releases you from your promise to ph, the fact that you threatened to beat up the promisee if he or she did not tell you that you need not fulfill your promise invalidates that release, and can be cited in explaining why you have a reason to ph.) But while theological voluntarism holds that a fact--the fact that God commands/intends that one ph--explains why one has a reason (in this case, an obligation) to ph, the causal view holds that this fact falls into none of the familiar explanatory categories: it is not constitutive of the reason, it is not an enabling condition for the reason, it is not a defeater-defeater for the reason. The way that the fact is supposed to explain the reason is merely causal: it just brings the reason about, exclusively, totally, immediately. This is an entirely unfamiliar phenomenon: nowhere else do we encounter a merely causal connection between a nonmoral fact and a moral one. (The appeal to the very strangeness of divine causation itself is not sufficient to answer the objection. For there is an extra strangeness here: that the relationship between nonmoral and moral facts is in every case with which we are familiar a rational relationship, whereas on the causal formulation of theological voluntarism the relationship is merely causal. Creation ex nihilo does not constitute carte blanche to multiply strangenesses. Interestingly, though, Wielenberg 2018 (p. 18), which rejects the project of theistic ethics, appeals to just this sort of causation to explain the supervenience of the moral on the non-moral.) The third worry is the 'no authority worry'. Theological voluntarism can be defended on the basis of considerations proper to metaethics--that, for example, theological voluntarism provides the best explanation for the impartiality of morals, or for its overridingness, or for its normativity, or for its content. But theological voluntarists have tended to argue that theological voluntarism has something specific to offer to theists. One of these benefits on offer is that theological voluntarism fits well with the centrality of the virtue of obedience in theistic thought and practice (Quinn 1992, p. 510; Adams 1973, pp. 99-103). God is a being who is to be obeyed, is someone who is a practical authority over us. For one to be a practical authority over another is, at least, for one to have some sort of control over others' reasons for action. Whatever else practical authority is, it is the ability to make a difference with respect to someone's reasons to act. The control involved in practical authority is, however, of a specific sort: it is constitutive control. When a party is an authority over another, his or her dictates constitute, at least in part, reasons for action for that other. (One piece of evidence for this is that when we take A to be an authority over us, we will cite 'A told us to' as a reason for action.) But if God's commands to ph have merely causal power to bring about obligations to ph, then the resultant state of affairs that is the reason for action is its being obligatory to ph--a state of affairs that need not be in any way constituted by God's issuing any commands. No version of theological voluntarism that is built simply around God's causal role in actualizing moral obligations implies that God is a practical authority. (See Murphy 2002, 4.3.) #### Supervenience The supervenience account is defended by Hare (2001). Suppose that we continue to interpret supervenience intuitively as the no-difference-in-moral-properties-without-some-difference-in-nonmoral-properties thesis. We can see very quickly that the theological voluntarist has to say something more about the sort of supervenience he or she has in mind in order to present what is genuinely a theological voluntarist account of moral obligation. For suppose that one puts forward a view on which both of the following claims are true: the moral law does not depend on, nor is it identical with, God's commands; but God necessarily commands us to follow the moral law. While it is obvious is that this not a version of theological voluntarism at all--moral obligation in no way depends on divine command--it satisfies the intuitive description of what is involved in the supervenience of the moral on the nonmoral: for there could be, on this view, no differences in moral status without some difference in divine commands. So if one is to put forward a supervenience formulation of theological voluntarism, then one will have to either be a little bit more doctrinaire about the supervenience relationship, so that it will exclude the nonvoluntarist view just described, or one will have to say more than that moral obligations supervene on divine commands. For our purposes here, they come to the same thing: that there is something more to the supervenience formulation of theological voluntarism than the claim that there are no differences in agents' moral obligations without some differences in the divine commands that have been imposed on that agent. What is called for here is, pretty obviously, just some particular relationship of ontological dependence. It will not be that of causation, for reasons we have already examined. But neither does the defender of the supervenience view want it to be the extreme dependence of moral obligations on divine commands affirmed by the reduction formulation, on which moral obligations just are divine commands. To avoid collapse into the reduction formulation, it has to hold that moral obligations are distinct from divine commands. It can make this distinction in one of two ways. It could say that moral obligation is wholly distinct from divine command--that is, that the state of affairs its being morally obligatory to ph is not constituted even in part by God's commanding ph-ing. Or it could say that moral obligation is only partially constituted by divine command--that is, that the state of affairs its being obligatory to ph, while not identical with God's commanding ph-ing, includes the state of affairs God's commanding ph-ing (and some other state of affairs besides). Let us consider each of these possibilities in turn. Suppose first that the defender of the supervenience view affirms that moral obligation is wholly distinct from divine command. If so, then all of the arguments that were raised against the causation formulation can be leveled against the supervenience view. The no authority issue will arise. Because the states of affairs its being obligatory to ph and God's commanding ph-ing will be distinct, the supervenience account lacks the resources to underwrite divine authority. For God is authoritative only if God's commands are themselves reasons for action, but if the states of affairs its being obligatory to ph and God's commanding ph-ing are distinct, then God's commands will not be themselves reasons for action on the adequately strengthened supervenience view. And if these commands are not themselves reasons for action, then God does not constitutively actualize reasons for action by His commands; and if God does not constitutively actualize reasons for action by His commands, then God is not authoritative. The lack of precedent issue will arise. For the adequately strengthened supervenience view cannot view obligations as constituted by divine commands, and no theological voluntarist worthy of the name will see God's commands as merely enablers or defeater-defeaters for obligations; and so the relationship between divine commands and moral obligations is bound to be unprecedented and mysterious. And the Humean issue will arise. For the causation view is, after all, just the adequately strengthened supervenience view plus the claim that the dependence relationship involved in a particular sort of causal dependence. So, understood as affirming a dependence relationship between wholly distinct moral obligations and divine commands, the supervenience view has all of the problems of the causation view. So the only hope for the supervenience formulation is to hold that God's commands are proper parts of moral obligations: for if those commands are identical with moral obligations, then the supervenience view collapses into the reduction view, and if moral obligations are wholly distinct from divine commands, then the supervenience view fails for the reasons that the causation view fails. There are, however, serious difficulties for this partial constitution version: in particular, if one is committed to saying that God's commanding ph-ing partly constitutes its being morally obligatory to ph, it is hard to see what state of affairs the theological voluntarist would be tempted to say is also necessary for moral obligation to be fully constituted. Obviously this other state of affairs cannot be one that involves moral obligation, on pain of circularity. (So, the theological voluntarist cannot say that its being morally obligatory to ph just is the complex state of affairs consisting both of God's commanding ph-ing and its being morally obligatory to do what God commands.) Further, in order to remain faithful to the basic idea of the supervenience version, we would have to say that any state of affairs that is held to constitute its being morally obligatory to ph along with God's commanding ph-ing must be a state of affairs that is certain to obtain if God's commanding ph-ing obtains. Otherwise, it might be the case that its being morally obligatory to ph does not supervene on God's commanding ph-ing, for there would be two possible worlds, in both of which God's commanding ph-ing obtains, but in only one of which does its being morally obligatory to ph obtain. This runs contrary to even the basic idea of the supervenience view, on which there are no differences in moral obligations without a difference in divine commands. These limitations make it hard to imagine what a motivated version of this form of the supervenience view would look like. We have to imagine a view of the following form. It is nonnegotiable that the state of affairs its being morally obligatory to ph is partially constituted by God's commanding ph-ing. It is nonnegotiable that there is, apart from God's commanding ph-ing, at least one state of affairs S that partially constituted its being morally obligatory to ph. It is nonnegotiable that S either obtains necessarily or at the very least necessarily obtains if God's commanding ph-ing obtains (otherwise moral obligation would not supervene on divine command). And it is nonnegotiable that S not involve moral obligation. The only remotely plausible candidates for S that come to mind are normative states of affairs that fall short of the obligatory, for example, ph-ing's being good (or virtuous, or praiseworthy). One might say, for example, that its being morally obligatory to ph is constituted jointly by God's commanding ph-ing and ph-ing's being virtuous. But it is unclear what motivation one would have for affirming such a position. It cannot be for the sake of making sure that God cannot impose a moral obligation to do something that is not virtuous: for, ex hypothesi, we know already that ph-ing's being virtuous obtains whenever God's commanding ph-ing obtains, for otherwise its being morally obligatory to ph would not supervene on God's commanding ph-ing. The difficulty that faces the defender of the supervenience view can be framed as a dilemma. If the defender of that view holds that the state of affairs its being morally obligatory to ph is wholly distinct from God's commanding ph-ing, then he or she is refuted by the considerations that refute the causation view. If, on the other hand, the defender of the supervenience view holds that the state of affairs its being morally obligatory to ph is partially but not wholly constituted by God's commanding ph-ing, then there is pressure to explain why he or she does not simply affirm the reduction view, on which its being morally obligatory to ph just is God's commanding ph-ing. Unless the defender of the supervenience view identifies the state of affairs that, in addition to God's commanding ph-ing, makes for a moral obligation to ph, then his or her unwillingness to adopt the reduction view will look unmotivated and arbitrary. #### Analysis According to the analysis view, defended in Adams 1973, the concept of the morally obligatory is to be analyzed as that of being commanded by a loving God. Adams did not put this view forward as an account of the meaning of 'obligation' generally, but only of its meaning as employed in Judeo-Christian moral discourse. As evidence for this analysis, Adams appealed to the freedom with which users of that discourse moved between claims of the form 'x is obligatory' and 'x is God's will' or 'x is God's command.' There are a couple of central difficulties for this position. The first is that it seems to imply that those inside and those outside the Judeo-Christian practice of moral discourse have never disagreed when one has affirmed a claim of the form ' ph-ing is obligatory' and the other denied a claim of that form. For they do not, on Adams' account, mean the same thing when they use these terms. In atheistic moral discourse, a masterful user of the language can say 'it is not true that God has commanded ph-ing, but ph-ing is nonetheless obligatory'; in Judeo-Christian moral discourse, on Adams' view, one shows oneself to be either unintelligible or not a masterful user of moral language if one were to speak thus. Adams was aware of this difficulty, and attempted to mitigate it: he argued that the agreement over which items the term 'obligatory' applied to, and the appropriate attitudinal and volitional responses to those things correctly described as 'obligatory,' made possible substantive moral discourse (Adams 1973, pp. 116-120). But all this seems to do is to explain how a simulacrum of genuine moral discourse is preserved; it does not show that what we get is the real thing. The second difficulty is that of dealing with those within the Judeo-Christian tradition of moral discourse who employed or continue to employ moral language in a way that is out of step with Adams' analysis. Now, it is not sufficient to refute a suggested analysis of some term that users of that term have questioned or even rejected that analysis. But if we take the task of analyzing terms to be that of making explicit and systematizing the platitudes employing that term affirmed by masterful users of that term (Smith 1994, pp. 29-32), and we note that many thoughtful Jews and Christians who otherwise appear to be masterful users of the language of moral obligation have rejected, either explicitly or implicitly, the notion that an act is obligatory if and only if it has been commanded by God, then we would have some reason to doubt whether the analysis formulation of theological voluntarism is defensible. #### Reduction Adams' maneuver in the face of these difficulties was to move from the analysis to the reduction version of theological voluntarism. He decided that the meaning of the term 'morally obligatory' was common to theists and nontheists. There is a common concept of the morally obligatory, a common concept that makes possible substantive agreement and disagreement between theists and nontheists. This common concept is neutral between theism and nontheism. But, following the now standard Kripke-Putnam line, Adams affirms that there are necessary a posteriori truths, among which are included property identifications. He argued that the property being wrong is identical to the property being contrary to the commands of (a loving) God because the property being contrary to the commands of (a loving) God best fills the role assigned by the concept of wrongness (Adams 1979a, pp. 133-142; see also Adams 1999, pp. 252-258). By conceptual analysis alone we can know only that wrongness is a property of actions (and perhaps intentions and attitudes); that people are generally opposed to what they regard as wrong; that wrongness is a reason, perhaps a conclusive reason, for opposing an act; and that there are certain acts (e.g. torture for fun) that are wrong. But given traditional theistic beliefs, the best candidate property to fill the role set by the concept of wrongness is that of being contrary to (a loving) God's commands. For that property is an objective property of actions. Further, given Christian views about the content of God's commands, this identification fits well with widespread pre-theoretical intuitions about wrongness; and given Christian views about human receptivity to divine communication and God's willingness to communicate both naturally and supernaturally, God's commands have a causal role in our acquisition of moral knowledge (Adams 1979, p. 139; see also Adams 1999, pp. 257). The reduction formulation avoids the most troublesome implications of the analysis formulation, for it allows that there is a common concept of obligation, so that those within the Judeo-Christian tradition and those outside it can engage in moral debate and can have substantive agreements and disagreements with each other, and so that those within the Judeo-Christian tradition can raise substantive questions about the relationship between God and obligation without ipso facto excluding themselves from the class of masterful users of the moral concepts of that community. The reduction formulation allows that the concept of obligation may be nontheistic while the property that best fills the role assigned to it by that concept is a theistic one. #### Analysis vs. reduction Nevertheless, it remains an open question whether the reduction view is superior to the analysis view. One might argue that Adams' analogy to 'H2O is water' is inappropriate, as the identification with water with H2O is clearly a posteriori, whereas the identification of the morally obligatory with the commanded by God is a priori. For if Adams is right in his characterization of the concept of obligation, it is not as if those who do not have the ability to infer from 'this is morally obligatory' to 'this is commanded by a loving God' (and vice versa) are just missing out on an interesting extra fact, the way that those without rudimentary chemistry are missing out on an interesting extra fact if they do not know that water is H2O. The term 'water' can play its role in our practical lives perfectly well without our knowing that it is H2O. The term 'morally obligatory' cannot play its role in our practical lives without our knowing that the morally obligatory is the commanded by God. No unintelligibility creeps into the life of agents that do not grasp that water is H2O; unintelligibility creeps into the life of agents that do not grasp that the morally obligatory is what is commanded by God. Why might one think that the masterful use of 'morally obligatory' requires recognition that the morally obligatory is what is commanded by God? If Adams is right, it is part of the meaning of obligation that obligations are social in character (Adams 1999, p. 233) and involve actually made demands by one party in the social relationship on another (Adams 1999, pp. 245-246). It is the fact that a demand is actually made that gives sense to the notion that one has to perform an action, rather than merely that it would be good, even the best, to do it (Adams 1999, p. 246). But if it is part of the meaning of 'morally obligatory' that one is part of a certain social relationship in which demands are actually made, then it is no longer just an interesting further fact that the property that best answers to the concept 'morally obligatory' is the property being commanded by God. Rather, one who denies that there is a God or that God actually makes demands on human beings must fail to use the term 'morally obligatory' masterfully. For think of the other marks of the moral, especially those of impartiality and overridingness. For one to think of an act as obligatory is for one to think of it as being actually imposed on one as a demand; for every obligation, on Adams' view, there is someone who imposes that obligation by commanding. It is clear a priori that the only being that could impose the sort of obligation that could plausibly be classified as moral would be God. How, then, could one be a masterful user of 'moral obligation' without grasping that moral obligations are demands imposed by God? This analysis view would not, unlike Adams' earlier formulation, require the subdivision of linguistic communities. One could say that the meaning of 'morally obligatory' includes 'being commanded by God,' for both theists and nontheists. Those who do not grasp that it is of the essence of obligations to be divinely commanded--whether theists or nontheists--fail to be masterful users of the language of moral obligation. To embrace this view is to return to the position of Anscombe 1958, according to which we should hold that the concept of obligation is inherently theological. On this view, we should not allow that Judeo-Christian moral practice has a different concept of obligation. Rather, the theological understanding of obligation is the authentic one, and nontheological concepts of obligation are unintelligible truncations. ## 3. Perennial difficulties for metaethical theological voluntarism Apart from the difficulties that must be handled by particular formulations of theological voluntarism, there are a number of objections that have been levelled against theological voluntarist views as such. A wide variety of these objections are helpfully discussed in Quinn 1999 (pp. 65-71) and Evans 2013 (pp. 88-117). Here I will consider only two objections, but they are the two that are characteristically taken to be the most powerful perennial objections to theological voluntarism: first, that theological voluntarism is incompatible with any substantive sense in which God is good; second, that theological voluntarism entails the arbitrariness of morality. While these objections have been answered plausibly in recent formulations of theological voluntarism, the way that these objections have been answered leave theological voluntarists open to a different objection: that theological voluntarism is not adequately motivated as a philosophical position, even for theists. I conclude with a brief discussion of this worry. ### 3.1 Theological voluntarism and God's goodness God is, by definition, good. This is both a fixed point concerning God's nature and a plausibility-making feature of theological voluntarism. If one were to deny that God is good (understood de dicto--that is, 'if there is a being that qualifies as God, then that being is good'), one would call one's own competence in use of the term 'God' into question. And even if it were allowed that one can employ the term 'God' masterfully while denying that God is good, if one were to deny that God is good, then one would undercut one's capacity to defend theological voluntarism. For theological voluntarism is plausible only if God is an exalted being; but a being that is not good is not an exalted being. That God is good is a fixed point for theistic discourse in general and for theological voluntarism in particular provides the basis for a common objection to theological voluntarism: that theological voluntarism makes it impossible to say that, in any substantive sense, God is good. The most straightforward formulation of the objection is as follows. For God to be good is for God to be morally good. But if moral goodness is to be understood in theological voluntarist terms, then God's goodness consists only in God's measuring up to a standard that God has set for Godself. While this is perhaps an admirable resoluteness--it is, other things being equal, a good thing to live up to your own standards--it is hardly the sort of thing that provokes in us the admiration that God's goodness is supposed to provoke. Now, one might dispute the claim that if God's goodness consists simply in God's living up to a standard that God has set for Himself, then that goodness is far less admirable than we would have supposed. (See, for a nice discussion of this issue, Clark 1982, esp. pp. 341-343.) Suppose, though, that we grant this part of the argument. How powerful is the objection from God's goodness against theological voluntarism? As we noted earlier, theological voluntarism comes in a variety of strengths. One dimension along which a theological voluntarist view might be assessed as stronger or weaker is in terms of the range of normative properties that it attempts to account for in theological voluntarist terms. The strength of the objection from God's goodness is directly proportional to the size of the range of normative properties that one wishes to explain in theological voluntarist terms (see also Alston 1990). If one wishes only to account for a proper subset of moral notions, such as obligation, with one's theological voluntarism, then the objection from God's goodness is very weak; if one wishes to provide a sweeping account of normativity in theological voluntarist terms, then the objection is much stronger. Suppose, for example, that one defends a version of theological voluntarism that accounts only for obligation. If moral obligation only is dependent on acts of the divine will, one can appeal to moral notions other than deontic ones in order to provide a substantive sense in which God is good. Granting to some extent the force of the objection, we can say, on this view, that God's moral goodness cannot consist in God's adhering to what is morally obligatory. But there are other ways to assess God morally other than in terms of the morally obligatory. Adams, for example, holds that God should be understood as benevolent and as just, and indeed concedes that his theological voluntarist account of obligation as the divinely commanded is implausible unless God is thus understood (Adams 1999, pp. 253-255). The ascription to God of these moral virtues is entirely consistent with his theological voluntarism, for his theological voluntarism is not meant to provide any account of the moral virtues. One can hold that God's moral goodness involves supereminent possession of the virtues, at least insofar as those virtues do not presuppose weakness and vulnerability. God is good because God is supremely just, loyal, faithful, benevolent, and so forth. It seems that ascribing to God supereminent possession of these virtues would be enough to account for God's supreme moral goodness: it is, after all, in such terms that God is praised in the Psalms. It has been argued that this appeal to God's justice is illegitimate within a theological voluntarist account, because what is just is a matter of moral requirement, and so to suppose that God's acting justly is metaphysically prior to God's imposing all moral requirements by way of commanding is incoherent (Hooker 2001, p. 334). But the theological voluntarist may deny that acting justly is morally required prior to God's commanding it, any more than acting courageously, temperately, or prudently are morally required prior to God's commanding us to act courageously, temperately, or prudently. Just as one can coherently acknowledge the excellence of temperance while wondering whether one is morally obligated to act temperately, one can coherently acknowledge the excellence of justice while wondering whether one is morally obligated to act justly. It thus seems an available strategy for the theological voluntarist who holds a restricted view of the range of moral properties explained by God's commands/intentions to appeal to justice in accounting for God's goodness. Matters become more difficult for theological voluntarist views that aim to provide accounts of all moral notions in terms of God's will. If one held to such an ambitious version of theological voluntarism--if one were to hold, say, that a state of affairs is morally good because it is a state of affairs that God wishes to obtain for its own sake, and that a character trait is a moral virtue because it is a property that God wants one to have for its own sake, and that an action is morally obligatory because it is antecedently intended by God, and so forth--then obviously the gambit employed by the less ambitious theological voluntarist is unavailable. The more ambitious theological voluntarist should hold, instead, that God's goodness is not to be understood in moral terms. God's being good might be understood in terms of God's being good to us, where us includes all created rational beings, or all created sentient beings, or whatever class of created beings to which one thinks that God has a special relationship. What it is for God to be good to us would be for God to be loving--to will each of our goods, and to do so in a way that plays no favorites. This understanding of 'loving' does not run afoul of theological voluntarism construed as an account of all moral goods, because 'our goods' is to be interpreted in terms of prudential goodness, what makes each of us well-off (but see Chandler 1985). Suppose, though, that one were to go all the way, holding that theological voluntarism is the correct account of all normative notions: on this extremely ambitious view, anything that is intrinsically action-guiding depends on God's will. I think that the 'God is good to us' understanding of God's goodness is ruled out on this approach: for the notion of 'good to us' is a normative notion. Perhaps one could hold, on this view, that 'God is good' affirms of God some sort of metaphysical goodness, fullness of being. This surely makes God exalted, but it is not clear whether the will of such a being is plausibly understood as the source of all normative statuses. It is also less than clear that 'God is good,' on this reading, is the claim that God possesses a particular perfection, rather than is merely a reminder that God has a variety of perfections. To sum up, then: for each of the various formulations of theological voluntarism, there seems to be some way of answering the charge that the view undercuts the notion that God is good. But the strain needed to answer the charge becomes greater the wider the range of normative properties that the formulation of theological voluntarism aims to explain. ### 3.2 Theological voluntarism and arbitrariness It is also an extraordinarily popular charge against theological voluntarism that it entails, objectionably, that morality is arbitrary. There is, however, more than one objection here, and the different objections need to be distinguished and answered individually. One claim is that theological voluntarism implies that God's commands/intentions, on which moral statuses depend, must be arbitrary. A distinct claim is that theological voluntarism implies that the content of morality is itself arbitrary, that it is of the essence of morality to exhibit a certain rational structure, and that theological voluntarism precludes its having that structure. I will consider each of these objections in turn. One arbitrariness objection against theological voluntarism is that if theological voluntarism is true, then God's commands/intentions must be arbitrary; and it cannot be that morality could wholly depend on something arbitrary; and so theological voluntarism must be false. In favor of the claim that if theological voluntarism were true, then morality would be arbitrary: morality would be arbitrary, on theological voluntarism, if God lacks reasons for the commands/intentions that God gives/has; but because theological voluntarism holds that reasons depend on God's commands/intentions, it is clear that there could ultimately be no reason for God's commanding/intending one thing rather than another. In favor of the claim that morality could not wholly depend on something arbitrary: when we say that some moral state of affairs obtains, we take it that there is a reason for that moral state of affairs obtaining rather than another. Moral states of affairs do not just happen to obtain. Just as in the case of the objection from God's goodness, the strength of this version of the objection from arbitrariness depends on the formulation of theological voluntarism that is being attacked. The arbitrariness objection becomes more difficult to answer the stronger the relationship between God's intentions/commands and moral properties is held to be; and it becomes more difficult to answer the more normative properties one attempts to account for by appeal to God's intentions/commands. The arbitrariness objection has less force if one holds that, say, only moral obligations are to be accounted for by theological voluntarism. The claim made by the objector is that morality is arbitrary on theological voluntarism, because God has no reason for having one set of commands/intentions rather than another. But this is so only if one appeals to the very strong form of theological voluntarism on which all normative states of affairs depend on God's will. If one holds that only moral obligations are determined by God's will, then God might have moral reasons for selecting one set of commands/intentions rather than another: that, for example, one set of commands/intentions is more benevolent, or just, or loyal, than another. If one holds that all moral properties are determined by God's will, then God might have nonmoral reasons for selecting one set of commands/intentions rather than another: that one set of commands/intentions is more loving than another. The fewer normative properties that a version of theological voluntarism attempts to account for, the less susceptible it is to the claim that theological voluntarism implies the arbitrariness of God's commands/intentions. (For further discussion of the role that restriction of theological voluntarism to a proper subset of normative statuses has had in answering these perennial objections, see Murphy 2012.) Now, one might respond on behalf of this version of the arbitrariness objection that even if it is true that there can be reasons for God to choose the commands/intentions that God chooses, it is unlikely that these reasons would wholly determine God's choice of commands/intentions, and so there would be some latitude for arbitrariness in God's choices/intentions. But of itself this is not much of a worry. The initial claim pressed against theological voluntarism was that it made all of God's commands/intentions ultimately arbitrary, and morality could not depend on something so thoroughly arbitrary. But the chastened claim--that there is some arbitrariness in God's commands--is far less troubling on its own. We are already familiar with morality depending to some extent on arbitrary facts about the world: if one thinks about the particular requirements that he or she is under, one will note straightaway the extent to which these requirements have resulted from contingent and indeed fluky facts about oneself, one's relationships, and one's circumstances. It does not seem that allowing that God has some choices to make concerning what to command/intend with respect to the conduct of created rational beings that are undetermined by reasons must introduce an intolerable arbitrariness into the total set of divine commands/intentions. (See also Carson 2012.) Allowing for such pockets of divine discretion does not provide backing for this version of the objection from arbitrariness, but rather offers a premise for the other version of the objection from arbitrariness. This other version of the objection from arbitrariness holds that moral states of affairs exhibit a certain rational structure that they would not have if theological voluntarism were true. Here is the idea, roughly formulated. Suppose that some moral state of affairs obtains--that it is the case that murder is wrong, or that lying is objectionable, or that courage is a virtue, or that Sharon's snubbing me in that way was unforgivable. The idea is that for any such moral state of affairs, the following is true: either we can provide a justification for the obtaining of that moral state of affairs, or that moral state of affairs is necessary. A justification of an obtaining moral state of affairs A is some obtaining moral state of affairs B (where A is not identical with B), which in conjunction with the other non-moral facts entails that A obtains. So, for example: it may be the justification for murder's being prima facie wrong that murder is an intentional harm (non-moral fact) and intentionally harming is prima facie wrong (obtaining moral state of affairs). Now, presumably not all moral states of affairs can be justified: eventually there will be basic moral states of affairs, for which no justification can be given. But it would be very unsatisfactory to say that these basic moral states of affairs just happen to obtain. So any basic moral states of affairs must obtain necessarily. Perhaps unrelieved suffering's being bad is a state of affairs of this sort, or perhaps rational beings' being worthy of respect. The claim that the structure of morality is not arbitrary is, put positively, the claim that every obtaining moral state of affairs either has a justification or is necessary. And, thus, what those who claim that theological voluntarism entails that morality is objectionably arbitrary mean is that if theological voluntarism is true, then there are some moral states of affairs that both lack a justification and are not necessary. The view that God's commands/intentions are not wholly determined by reasons offer our basis for holding that there are some moral states of affairs that both lack a justification and are not necessary. For consider some act of ph-ing that is not subsumed under any other issued divine command and which is such that God lacks decisive reasons to command or not to command its performance. In the possible world in which God issues a command to ph, there is a moral state of affairs--its being obligatory to ph--which lacks a justification (for the action is subsumed under no other divine command) and is not necessary (for God might have failed to command the action). One might respond to this sort of worry by proposing that what God commands/intends with respect to human action, God commands/intends necessarily. But this seems either to understate the divine freedom or to overstate the determination of God's commands by reasons (Murphy 2002, pp. 83-85). More plausible is the denial of the claim that morality must exhibit the particular structure presupposed in the objection. While I think that in general the subsumption model of justification is innocuous enough--even particularists can affirm it, if they affirm even the most minimal doctrine of moral supervenience--its appeal to necessary moral states of affairs as the only proper starting point is dubious. It is not clear why the starting points for justification have to be necessary moral states of affairs, for two reasons. First, if these moral states of affairs are basic, then of course their moral status must not be explained by appeal to other moral states of affairs, but that does not mean they must be necessary; they might be contingent, and have their moral status explained in some way other than an appeal to another moral state of affairs. It could be, for example, that the explanation of them appeals to a contingent nonmoral state of affairs plus some necessary state of affairs concerning a connection between that contingent nonmoral state of affairs and the moral state of affairs. Theological voluntarism would be an instance of this latter model. (For a discussion of a similar objection raised against theological voluntarism by Ralph Cudworth (1731), and a similar response on behalf of theological voluntarism, see Schroeder 2005.) Second, the appeal to necessary moral states of affairs as the stopping point for explanation seems to assume that necessary moral states of affairs somehow are not in need of explanation. But the very fact that some state of affairs obtains necessarily does not entail that its obtaining does not require explanation, and there is no reason why moral states of affairs would be special on this score. So one might think that not only is it possible for justifications to bottom out in contingent moral states of affairs, there is no reason to think that such justifications are as such any less adequate than justifications that bottom out in necessary moral states of affairs. (See Murphy 2011, pp. 47-49.) ### 3.3 Is theological voluntarism adequately motivated? Both with respect to the objection from God's goodness and with respect to the objection from arbitrariness, the now-standard theological voluntarist response is not to bite the bullet but rather to restrict the range of normative properties of which theological voluntarism is supposed to provide an account. So Adams, Quinn, and Alston all recommend theological voluntarism only as a theory about properties like being morally obligatory, and not about any other normative properties. The worry is that allowing that there is adequate motivation to refuse to understand these other normative properties in theological voluntarist terms might commit one to holding that there is adequate motivation to refuse to understand obligation-type properties in theological voluntarist terms. Look, one might say: if you are willing to hold that all moral properties other than those in the obligation family are to be understood in non-theological voluntarist terms, what is to stop us from holding that obligation is to be understood in non-theological voluntarist terms as well? If we are willing to give up theological voluntarism in some moral domains, why not in all of them? The most well-developed account of why we should treat obligation as special is Adams', on which obligation is apt for theological voluntarist treatment because of its intrinsic link to demands made within social relationships (Adams 1987b, and Adams 1999, pp. 231-258; see also Evans 2014, p. 27). But it is also unclear whether this is persuasive. We may grant that obligations result from demands, but only if we emphasize (as Adams does) that it is demands from authorities that result in obligations. But what makes someone an authority is that by his or her dictates he or she can give reasons for action of a certain kind. There is some dispute over what kind of reasons they must be, but for our purposes we can just follow Joseph Raz, who holds that genuine authorities give "protected reasons" by their dictates, where a protected reason to ph is a reason to ph and a reason to disregard some reasons against ph-ing. If one is an authority over another with respect to ph-ing, then one's dictate that the other ph is a protected reason for the other to ph (Raz 1979, p. 18). But now here is the question. We agree that obligations arise from authoritative dictates. And we agree that for a dictate to be authoritative is for it to constitute a certain sort of reason, let us say a protected one. But why, then, would we identify obligations with protected reasons that result from demands, rather than (as Raz does) with protected reasons themselves, whatever their source? If, after all, there were other ways of producing protected reasons other than through the giving of commands, what would be the point of saying 'oh, but though there is a protected reason to ph, it isn't really obligatory to ph.' Surely if there were any point to this remark it would be purely verbal, and of no philosophical / normative interest. It turns out, then, that whether Adams' move is enough to motivate theological voluntarism about obligation is dependent on whether there are in fact any protected reasons (or reasons of whatever structure that one thinks that authoritative dictates must give) that are not dependent on demands being made. If there are such reasons--which natural law theorists, for example, would hold--then Adams' gambit will not work, and the theological voluntarist will have to look elsewhere for motivation to understand obligation in those terms. Now, one might say that by appealing to the idea that obligations arise from demands made within social relationships as the basis for the theological voluntarist's account of moral obligation, we are misconstruing the key claim that they make about the social nature of obligation. The social nature of obligation, voluntarists might retort, first and foremost concerns the fact that when one is under an obligation, there is someone else who is entitled to hold one accountable for failures to adhere to the content of that obligation (Adams 1999, p. 233; Evans 2013, p 14). In the case of the moral law, the party who has standing to hold us accountable is God. Even if one accepts this claim about the nature of obligation -- it has some plausibility, as there is some basis for holding that not everything that one has protected reason to do counts as obligatory -- and the claim that only God has adequate standing to hold us accountable for adhering to morality, it is still not at all clear how to provide a satisfactory rationale for a theological voluntarist view of moral obligation. For theological voluntarism with respect to moral obligations holds that the existence of moral obligations depends on God's bringing them about via God's command or some other act of divine will; but there is no obvious argument for the view that God's being essential to holding us accountable for following the norms of morality entails God's commanding or willing those moral norms into existence. Even if we grant, then, to the theological voluntarists their desired premises regarding the social character of obligation and God's ideal status as someone to hold us accountable for adhering to the norms of morality, theological voluntarists still have a worrisome gap to overcome in offering positive reasons to affirm that view.
voting	## 1. The Rationality of Voting The act of voting has an opportunity cost. It takes time and effort that could be used for other valuable things, such as working for pay, volunteering at a soup kitchen, or playing video games. Further, identifying issues, gathering political information, thinking or deliberating about that information, and so on, also take time and effort which could be spent doing other valuable things. Economics, in its simplest form, predicts that rational people will perform an activity only if doing so maximizes expected utility. However, economists have long worried that, that for nearly every individual citizen, voting does not maximize expected utility. This leads to the "paradox of voting"(Downs 1957): Since the expected costs (including opportunity costs) of voting appear to exceed the expected benefits, and since voters could always instead perform some action with positive overall utility, it's surprising that anyone votes. However, whether voting is rational or not depends on just what voters are trying to do. Instrumental theories of the rationality of voting hold that it can be rational to vote when the voter's goal is to influence or change the outcome of an election, including the "mandate" the winning candidate receives. (The mandate theory of elections holds that a candidate's effectiveness in office, i.e., her ability to get things done, is in part a function of how large or small a lead she had over her competing candidates during the election.) In contrast, the expressive theory of voting holds that voters vote in order to express themselves and their fidelity to certain groups or ideas. Alternatively, one might hold that voting is rational because it is has consumption value; many people enjoy political participation for its own sake or for being able to show others that they voted. Finally, if one believes, as most democratic citizens say they do (Mackie 2010), that voting is a substantial moral obligation, then voting could be rational because it is necessary to discharge one's obligation. ### 1.1 Voting to Change the Outcome One reason a person might vote is to influence, or attempt to change, the outcome of an election. Suppose there are two candidates, D and R. Suppose Sally prefers D to R. Suppose she correctly believes that D would do a trillion dollars more overall good than R would do. If her beliefs were correct, then by hypothesis, it would be best if D won. Here, casting the expected value difference between the two candidates in monetary terms is a simplifying assumption. Whether political outcomes can be described in monetary terms as such is not without controversy. To illustrate, suppose the difference between two candidates came down entirely to how many lives would be lost in the way they would conduct a current war. Whether we can translate "lives lost" into dollar terms is controversial. Further, whether we can commensurate all the distinct goods and harms a candidate might cause onto a common scale is also controversial. Even if the expected value difference between two candidates could be expressed on some common value scale, such as in monetary terms, this leaves open whether the typical voter is aware of or can generally estimate that difference. Empirical work generally finds that most voters are badly informed, and further, that many of them are not voting for the purpose of promoting certain policies or platforms over others (Achen and Bartels 2016; Kinder and Kalmoe 2017; Mason 2017). Beyond that, estimating the value difference between candidates requires evaluating complex counterfactuals, estimating what various candidates are likely to achieve, and determining what the outcomes of these actions would be (Freiman 2020). These worries aside, even if Sally is correct that D will do a trillion dollars more good than R, this does not yet show it is rational for Sally to vote for D. Instead, this depends on how likely it is that her vote will make a difference. In much the same way, it might be worth $200 million to win the lottery, but that does not imply it is rational to buy a lottery ticket. Suppose Sally's only goal, in voting, is to change the outcome of the election between two major candidates. In that case, the expected value of her vote ($U\_v$) is: \[ U\_v = p[V(D) - V(R)] - C \] where p represents the probability that Sally's vote is decisive, $[V(D) - V(R)]$ represents (in monetary terms) the difference in the expected value of the two candidates, and C represents the opportunity cost of voting. In short, the value of her vote is the value of the difference between the two candidates discounted by her chance of being decisive, minus the opportunity cost of voting. In this way, voting is indeed like buying a lottery ticket. Unless $p[V(D) - V(R)] > C$, then it is (given Sally's stated goals) irrational for her to vote. The equation above models the rationality of Sally's choice to vote under the assumption that she is simply trying to change the outcome of the election, and gets no further benefit from voting. Further, the equation assumes her vote confers no other benefit to others than having some chance of changing which candidate wins. However, these are controversial simplifying assumptions. It is possible that the choice to cast a vote may induce others to vote, might improve the quality of the ground decision by adding cognitive diversity, might have some marginal influence on which candidates or platforms parties run, or might have some other effect not modeled in the equation above. Again, it is controversial among some philosophers whether the difference in value between two candidates can be expressed, in principle, in monetary terms. Nevertheless, the point generalizes in some way. If we are discussing the instrumental value of a vote, then the general point is that the vote depends on the expected difference in value between the chosen candidate and the next best alternative, discounted by the probability of the vote breaking a tie, and we must then take into account the opportunity cost of voting. For instance, if two candidates were identical except that one would save one more life than another, but one had a 1 in 1 billion chance of being decisive, and instead of voting one could save a drowning toddler, then it seems voting is not worthwhile, even if we cannot assign an exact monetary value to thee consequences. There is some debate among economists, political scientists, and philosophers over the precise way to calculate the probability that a vote will be decisive. Nevertheless, they generally agree that the probability that the modal individual voter in a typical election will break a tie is small. Binomial models of voting estimate the probability of a vote being decisive by modeling voters as if they were weighted coins and then asking what the probability is that a weighted coin will come up heads exactly 50% of the time. These models generally imply that the probability of being decisive, if any candidate has a lead, is vanishingly small, which in turn implies that the expected benefit of voting (i.e., $p[V(D) - V(R)]$) for a good candidate is worth far less than a millionth of a penny (G. Brennan and Lomasky 1993: 56-7, 119). A more optimistic estimate in the literature, which uses statistical estimate techniques based on past elections, claims that in a typical presidential election, American voters have widely varying chances of being decisive depending on which state they vote in. This model still predicts that a typical vote in a "safe" states, like California, has a vanishingly small chance of making a difference, but suggests that that a vote in very close states could have on the order of a 1 in 10 million chance of breaking a tie (Edlin, Gelman, and Kaplan 2007). Thus, on both of these popular models, whether voting for the purpose of changing the outcome is rational depends upon the facts on the ground, including how close the election is and how significant the value difference is between the candidates. The binomial model suggests it will almost never be rational to vote, while the statistical model suggests it will be rational for voters to vote in sufficiently close elections or in swing states. However, some claim even these assessments are optimistic. One worry is that if a major election in most places came down to a single vote, the issue might be decided in the courts after extensive lawsuits (Somin 2013). Further, in making such estimates, we have assumed that voters can reliably identify which candidate is better and reliably estimate how much better that candidate is. But perhaps voters cannot. After all, showing that individual votes matter is a double-edged sword; the more expected good an individual vote can do, the more expected harm it can do (J. Brennan 2011a; Freiman 2020). ### 1.2 Voting to Change the "Mandate" One popular response to the paradox of voting is to posit that voters are not trying to determine who wins, but instead trying to change the "mandate" the elected candidate receives. The assumption here is that an elected official's efficacy--i.e., her ability to get things done in office--depends in part on how large of a majority vote she received. If that were true, I might vote for what I expect to be the winning candidate in order to increase her mandate, or vote against the expected winner to reduce her mandate. The virtue of the mandate hypothesis, if it were true, is that it could explain why it would be rational to vote even in elections where one candidate enjoys a massive lead coming into the election. However, the mandate argument faces two major problems. First, even if we assume that such mandates exist, to know whether voting is rational, we would need to know how much the nth voter's vote increases the marginal effectiveness of her preferred candidate, or reduces the marginal effectiveness of her dispreferred candidate. Suppose voting for the expected winning candidate costs me $15 worth of my time. It would be rational for me to vote only if I believed my individual vote would give the winning candidate at least $15 worth of electoral efficacy (and I care about the increased efficiency as much or more than my opportunity costs). In principle, whether individual votes change the "mandate" this much is something that political scientists could measure, and indeed, they have tried to do so. But this brings us to the second, deeper problem: Political scientists have done extensive empirical work trying to test whether electoral mandates exist, and they now roundly reject the mandate hypothesis (Dahl 1990b; Noel 2010). A winning candidate's ability to get things done is generally not affected by how small or large of a margin she wins by. Perhaps voting is rational not as a way of trying to change how effective the elected politician will be, but instead as a way of trying to change the kind of mandate the winning politician enjoys (Guerrero 2010). Perhaps a vote could transform a candidate from a delegate to a trustee. A delegate tries to do what she believes her constituents want, but a trustee has the normative legitimacy to do what she believes is best. Suppose for the sake of argument that trustee representatives are significantly more valuable than delegates, and that what makes a representative a trustee rather than a delegate is her large margin of victory. Unfortunately, this does not yet show that the expected benefits of voting exceed the expected costs. Suppose (as in Guerrero 2010: 289) that the distinction between a delegate and trustee lies on a continuum, like difference between bald and hairy. To show voting is rational, one would need to show that the marginal impact of an individual vote, as it moves a candidate a marginal degree from delegate to trustee, is higher than the opportunity cost of voting. If voting costs me $15 worth of time, then, on this theory, it would be rational to vote only if my vote is expected to move my favorite candidate from delegate to trustee by an increment worth at least $15 (Guerrero 2010: 295-297). Alternatively, suppose that there were a determinate threshold (either known or unknown) of votes at which a winning candidate is suddenly transformed from being a delegate to a trustee. By casting a vote, the voter has some chance of decisively pushing her favored candidate over this threshold. However, just as the probability that her vote will decide the election is vanishingly small, so the probability that her vote will decisively transform a representative from a delegate into a trustee would be vanishingly small. Indeed, the formula for determining decisiveness in transforming a candidate into a trustee would be roughly the same as determining whether the voter would break a tie. Thus, suppose it's a billion or even a trillion dollars better for a representative to be a trustee rather than a candidate. Even if so, the expected benefit of an individual vote is still less than a penny, which is lower than the opportunity cost of voting. Again, it's wonderful to win the lottery, but that doesn't mean it's rational to buy a ticket. ### 1.3 Other Reasons to Vote Other philosophers have attempted to shift the focus on other ways individual votes might be said to "make a difference". Perhaps by voting, a voter has a significant chance of being among the "causally efficacious set" of votes, or is in some way causally responsible for the outcome (Tuck 2008; Goldman 1999). On these theories, what voters value is not changing the outcome, but being agents who have participated in causing various outcomes. These causal theories of voting claim that voting is rational provided the voter sufficiently cares about being a cause or among the joint causes of the outcome. Voters vote because they wish to bear the right kind of causal responsibility for outcomes, even if their individual influence is small. What these alternative theories make clear is that whether voting is rational depends in part upon what the voters' goals are. If their goal is to in some way change the outcome of the election, or to change which policies are implemented, then voting is indeed irrational, or rational only in unusual circumstances or for a small subset of voters. However, perhaps voters have other goals. The expressive theory of voting (G. Brennan and Lomasky 1993) holds that voters vote in order to express themselves. On the expressive theory, voting is a consumption activity rather than a productive activity; it is more like reading a book for pleasure than it is like reading a book to develop a new skill. On this theory, though the act of voting is private, voters regard voting as an apt way to demonstrate and express their commitment to their political team. Voting is like wearing a Metallica T-shirt at a concert or doing the wave at a sports game. Sports fans who paint their faces the team colors do not generally believe that they, as individuals, will change the outcome of the game, but instead wish to demonstrate their commitment to their team. Even when watching games alone, sports fans cheer and clap for their teams. Perhaps voting is like this. This "expressive theory of voting" is untroubled by and indeed partly supported by the empirical findings that most voters are ignorant about basic political facts (Somin 2013; Delli Carpini and Keeter, 1996). The expressive theory is also untroubled by and indeed partly supported by work in political psychology showing that most citizens suffer from significant "intergroup bias": we tend to automatically form groups, and to be irrationally loyal to and forgiving of our own group while irrationally hateful of other groups (Lodge and Taber 2013; Haidt 2012; Westen, Blagov, Harenski, Kilts, and Hamann 2006; Westen 2008). Voters might adopt ideologies in order to signal to themselves and others that they are certain kinds of people. For example, suppose Bob wants to express that he is a patriot and a tough guy. He thus endorses hawkish military actions, e.g., that the United States nuke Russia for interfering with Ukraine. It would be disastrous for Bob were the US to do what he wants. However, since Bob's individual vote for a militaristic candidate has little hope of being decisive, Bob can afford to indulge irrational and misinformed beliefs about public policy and express those beliefs at the polls. Another simple and plausible argument is that it can be rational to vote in order to discharge a perceived duty to vote (Mackie 2010). Surveys indicate that most citizens in fact believe there is a duty to vote or to "do their share" (Mackie 2010: 8-9). If there are such duties, and these duties are sufficiently weighty, then it would be rational for most voters to vote. ## 2. The Moral Obligation to Vote Surveys show that most citizens in contemporary democracies believe there is some sort of moral obligation to vote (Mackie 2010: 8-9). Other surveys show most moral and political philosophers agree (Schwitzgebel and Rust 2010). They tend to believe that citizens have a duty to vote even when these citizens rightly believe their favored party or candidate has no serious chance of winning (Campbell, Gurin, and Mill 1954: 195). Further, most people seem to think that the duty to vote specifically means a duty to turn out to vote (perhaps only to cast a blank ballot), rather than a duty to vote a particular way. On this view, citizens have a duty simply to cast a vote, but nearly any good-faith vote is morally acceptable. Many popular arguments for a duty to vote rely upon the idea that individual votes make a significant difference. For instance, one might argue that that there is a duty to vote because there is a duty to protect oneself, a duty to help others, or to produce good government, or the like. However, these arguments face the problem, as discussed in section 1, that individual votes have vanishingly small instrumental value (or disvalue) For instance, one early hypothesis was that voting might be a form of insurance, meant to to prevent democracy from collapsing (Downs 1957: 257). Following this suggestion, suppose one hypothesizes that citizens have a duty to vote in order to help prevent democracy from collapsing. Suppose there is some determinate threshold of votes under which a democracy becomes unstable and collapses. The problem here is that just as there is a vanishingly small probability that any individual's vote would decide the election, so there is a vanishingly small chance that any vote would decisively put the number of votes above that threshold. Alternatively, suppose that as fewer and fewer citizens vote, the probability of democracy collapsing becomes incrementally higher. If so, to show there is a duty to vote, one would first need to show that the marginal expected benefits of the nth vote, in reducing the chance of democratic collapse, exceed the expected costs (including opportunity costs). A plausible argument for a duty to vote would thus not depend on individual votes having significant expected value or impact on government or civic culture. Instead, a plausible argument for a duty to vote should presume that individual votes make little difference in changing the outcome of election, but then identify a reason why citizens should vote anyway. One suggestion (Beerbohm 2012) is that citizens have a duty to vote to avoid complicity with injustice. On this view, representatives act in the name of the citizens. Citizens count as partial authors of the law, even when the citizens do not vote or participate in government. Citizens who refuse to vote are thus complicit in allowing their representatives to commit injustice. Perhaps failure to resist injustice counts as kind of sponsorship. (This theory thus implies that citizens do not merely have a duty to vote rather than abstain, but specifically have a duty to vote for candidates and policies that will reduce injustice.) Another popular argument, which does not turn on the efficacy of individual votes, is the "Generalization Argument": > > What if everyone were to stay home and not vote? The results would be > disastrous! Therefore, I (you/she) should vote. (Lomasky and G. > Brennan 2000: 75) > This popular argument can be parodied in a way that exposes its weakness. Consider: > > What if everyone were to stay home and not farm? Then we would all > starve to death! Therefore, I (you/she) should each become farmers. > (Lomasky and G. Brennan 2000: 76) > The problem with this argument, as stated, is that even if it would be disastrous if no one or too few performed some activity, it does not follow that everyone ought to perform it. Instead, one conclude that it matters that sufficient number of people perform the activity. In the case of farming, we think it's permissible for people to decide for themselves whether to farm or not, because market incentives suffice to ensure that enough people farm. However, even if the Generalization Argument, as stated, is unsound, perhaps it is on to something. There are certain classes of actions in which we tend to presume everyone ought to participate (or ought not to participate). For instance, suppose a university places a sign saying, "Keep off the newly planted grass." It's not as though the grass will die if one person walks on it once. If I were allowed to walk on it at will while the rest of you refrained from doing so, the grass would probably be fine. Still, it would seem unfair if the university allowed me to walk on the grass at will but forbade everyone else from doing so. It seems more appropriate to impose the duty to keep off the lawn equally on everyone. Similarly, if the government wants to raise money to provide a public good, it could just tax a randomly chosen minority of the citizens. However, it seems more fair or just for everyone (at least above a certain income threshold) to pay some taxes, to share in the burden of providing police protection. We should thus ask: is voting more like the first kind of activity, in which it is only imperative that enough people do it, or the second kind, in which it's imperative that everyone do it? One difference between the two kinds of activities is what abstention does to others. If I abstain from farming, I don't thereby take advantage of or free ride on farmers' efforts. Rather, I compensate them for whatever food I eat by buying that food on the market. In the second set of cases, if I freely walk across the lawn while everyone else walks around it, or if I enjoy police protection but don't pay taxes, it appears I free ride on others' efforts. They bear an uncompensated differential burden in maintaining the grass or providing police protection, and I seem to be taking advantage of them. A defender of a duty to vote might thus argue that non-voters free ride on voters. Non-voters benefit from the government that voters provide, but do not themselves help to provide government. There are at least a few arguments for a duty to vote that do not depend on the controversial assumption that individual votes make a difference: 1. The Generalization/Public Goods/Debt to Society Argument: Claims that citizens who abstain from voting thereby free ride on the provision of good government, or fail to pay their "debts to society". 2. The Civic Virtue Argument: Claims that citizens have a duty to exercise civic virtue, and thus to vote. 3. The Complicity Argument: Claims that citizens have a duty to vote (for just outcomes) in order to avoid being complicit in the injustice their governments commit. However, there is a general challenge to these arguments in support of a duty to vote. Call this the particularity problem: To show that there is a duty to vote, it is not enough to appeal to some goal G that citizens plausibly have a duty to support, and then to argue that voting is one way they can support or help achieve G. Instead, proponents of a duty to vote need to show specifically that voting is the only way, or the required way, to support G (J. Brennan 2011a). The worry is that the three arguments above might only show that voting is one way among many to discharge the duty in question. Indeed, it might not be even be an especially good way, let alone the only or obligatory way to discharge the duty. For instance, suppose one argues that citizens should vote because they ought to exercise civic virtue. One must explain why a duty to exercise civic virtue specifically implies a duty to vote, rather than a duty just to perform one of thousands of possible acts of civic virtue. Or, if a citizen has a duty to to be an agent who helps promote other citizens' well-being, it seems this duty could be discharged by volunteering, making art, or working at a productive job that adds to the social surplus. If a citizen has a duty to to avoid complicity in injustice, it seems that rather than voting, she could engage in civil disobedience; write letters to newspaper editors, pamphlets, or political theory books; donate money; engage in conscientious abstention; protest; assassinate criminal political leaders; or do any number of other activities. It's unclear why voting is special or required. Note that the particularity problem need not be framed in consequentialist terms, i.e., both defenders and critics of the duty to vote need not say that what determines whether voting is morally required depends on whether voting has the highest expected consequences. Rather, the issue is whether voting is simply one of many ways to discharge an underlying duty or respond to underlying reasons, or whether voting is in some way special and unique, such that these reasons select voting in particular as an obligatory means of responding to these underlying reasons. Maskivker (2019) responds partly to this objection by saying, in effect, "Why not both?" J. Brennan (2011a) and Freiman (2020) say that the underlying grounds for any duty to vote can be discharged (and discharged better) through actions other than voting. Maskivker takes this to suggest not that voting is optional, but that one should vote (if one is already sufficiently well-informed and publicly-spirited) and also performs these other actions. Maskivker grounds her argument on a deontological duty of easy aid: if one can provide aid to others at very low cost to oneself, then one should do so. For already well-informed citizens, voting is an instance of easy aid. ### 2.1 A General Moral Obligation Not to Vote? While many hold that it is obligatory to vote, a few have argued that many people have an obligation not to vote under special circumstances. For instance, Sheehy (2002) argues that voting when one is indifferent to the election is unfair. He argues that if one's vote makes a difference, it could be to disappoint what otherwise would be have been the majority coalition, whose position is now thwarted by those who, by hypothesis, have no preference. Another argument holds that voting might be wrong because it is an ineffective form of altruism. Freiman (2020) argues that when people discharge their obligations to help and aid others, they are obligated to pursue effective rather than ineffective forms of altruism (see also MacAskill 2015). For instance, suppose one has an obligation to give a certain amount to charity each year. This obligation is not fundamentally about spending a certain percentage of one's money. If a person gave 10% of their income to a charity that did no good at all, or which made the world worse, one would not have discharged the obligation to act beneficently. Similarly, Freiman argues, if a person is voting for the purpose of aiding and helping others, then they would at the very least need to be sufficiently well-informed to vote for the better candidate, a condition few voters meet (see section 3.2 below). In part because most voters are in no position to judge whether they are voting for the better or worse candidates, and in part simply because individual votes make little difference, votes and most other forms of political action (such as donating to political campaigns, canvassing, volunteering, and the like) are highly ineffective forms of altruism. Freiman claims that we are instead obligated to pursue effective forms of altruism, such as collecting and making donations to the Against Malaria Foundation. ## 3. Moral Obligations Regarding How One Votes Most people appear to believe that there is a duty to cast a vote (perhaps including a blank ballot) rather than abstain (Mackie 2010: 8-9), but this leaves open whether they believe there is a duty to vote in any particular way. Some philosophers and political theorists have argued there are ethical obligations attached to how one chooses to vote. For instance, many deliberative democrats (see Christiano 2006) believe not only that every citizen has a duty to vote, but also that they must vote in publicly-spirited ways, after engaging in various forms of democratic deliberation. In contrast, some (G. Brennan and Lomasky 1993; J. Brennan 2009, 2011a) argue that while there is no general duty to vote (abstention is permissible), those citizens who do choose to vote have duties affecting how they vote. They argue that while it is not wrong to abstain, it is wrong to vote badly, in some theory-specified sense of "badly". Note that the question of how one ought to vote is distinct from the question of whether one ought to have the right to vote. The right to vote licenses a citizen to cast a vote. It requires the state to permit the citizen to vote and then requires the state to count that vote. This leaves open whether some ways a voter could vote could be morally wrong, or whether other ways of voting might be morally obligatory. In parallel, my right of free association arguably includes the right to join the Ku Klux Klan, while my right of free speech arguably includes the right to advocate an unjust war. Still, it would be morally wrong for me to do either of these things, though doing so is within my rights. Thus, just as someone can, without contradiction, say, "You have the right to have racist attitudes, but you should not," so a person can, without contradiction, say, "You have the right to vote for that candidate, but you should not." A theory of voting ethics might include answers to any of the following questions: 1. The Intended Beneficiary of the Vote: Whose interests should the voter take into account when casting a vote? May the voter vote selfishly, or should she vote sociotropically? If the latter, on behalf of which group ought she vote: her demographic group(s), her local jurisdiction, the nation, or the entire world? Is it permissible to vote when one has no stake in the election, or is otherwise indifferent to the outcome? 2. The Substance of the Vote: Are there particular candidates or policies that the voter is obligated to support, or not to support? For instance, is a voter obligated to vote for whatever would best produce the most just outcomes, according to the correct theory of justice? Must the voter vote for candidates with good character? May the voter vote strategically, or must she vote in accordance with her sincere preferences? 3. Epistemic Duties Regarding Voting: Are voters required to have a particular degree of knowledge, or exhibit a particular kind of epistemic rationality, in forming their voting preferences? Is it permissible to vote in ignorance, on the basis of beliefs about social scientific matters that are formed without sufficient evidence? ### 3.1 The Expressivist Ethics of Voting Recall that one important theory of voting behavior holds that most citizens vote not in order to influence the outcome of the election or influence government policies, but in order to express themselves (G. Brennan and Lomasky 1993). They vote to signal to themselves and to others that they are loyal to certain ideas, ideals, or groups. For instance, I might vote Democrat to signal that I'm compassionate and fair, or Republican to signal I'm responsible, moral, and tough. If voting is primarily an expressive act, then perhaps the ethics of voting is an ethics of expression (G. Brennan and Lomasky 1993: 167-198). We can assess the morality of voting by asking what it says about a voter that she voted like that: > > To cast a Klan ballot is to identify oneself in a morally > significant way with the racist policies that the organization > espouses. One thereby lays oneself open to associated moral liability > whether the candidate has a small, large, or zero probability of > gaining victory, and whether or not one's own vote has an > appreciable likelihood of affecting the election result. (G. Brennan > and Lomasky 1993: 186) > The idea here is that if it's wrong (even if it's within my rights) in general for me to express sincere racist attitudes, and so it's wrong for me to express sincere racist commitments at the polls. Similar remarks apply to other wrongful attitudes. To the extent it is wrong for me to express sincere support for illiberal, reckless, or bad ideas, it would also be wrong for me to vote for candidates who support those ideas. Of course, the question of just what counts as wrongful and permissible expression is complicated. There is also a complicated question of just what voting expresses. What I think my vote expresses might be different from what it expresses to others, or it might be that it expresses different things to different people. The expressivist theory of voting ethics acknowledges these difficulties, and replies that whatever we would say about the ethics of expression in general should presumably apply to expressive voting. ### 3.2 The Epistemic Ethics of Voting Consider the question: What do doctors owe patients, parents owe children, or jurors owe defendants (or, perhaps, society)? Doctors owe patients proper care, and to discharge their duties, they must 1) aim to promote their patients' interests, and 2) reason about how to do so in a sufficiently informed and rational way. Parents similarly owe such duties to their children. Jurors similarly owe society at large, or perhaps more specifically the defendant, duties to 1) try to determine the truth, and 2) do so in an informed and rational way. The doctors, parents, and jurors are fiduciaries of others. They owe a duty of care, and this duty of care brings with it certain epistemic responsibilities. One might try to argue that voters owe similar duties of care to the governed. Perhaps voters should vote 1) for what they perceive to be the best outcomes (consistent with strategic voting) and 2) make such decisions in a sufficiently informed and rational way. How voters vote has significant impact on political outcomes, and can help determine matters of peace and war, life and death, prosperity and poverty. Majority voters do not just choose for themselves, but for everyone, including dissenting minorities, children, non-voters, resident aliens, and people in other countries affected by their decisions. For this reason, voting seems to be a morally charged activity (Christiano 2006; J. Brennan 2011a; Beerbohm 2012). That said, one clear disanalogy between the relationship doctors have with patients and voters have with the governed is that individual voters have only a vanishingly small chance of making a difference. The expected harm of an incompetent individual vote is vanishingly small, while the expected harm of incompetent individual medical decisions is high. However, perhaps the point holds anyway. Define a "collectively harmful activity" as an activity in which a group is imposing or threatening to impose harm, or unjust risk of harm, upon other innocent people, but the harm will be imposed regardless of whether individual members of that group drop out. It's plausible that one might have an obligation to refrain from participating in such activities, i.e., a duty to keep one's hands clean. To illustrate, suppose a 100-member firing squad is about to shoot an innocent child. Each bullet will hit the child at the same time, and each shot would, on its own, be sufficient to kill her. You cannot stop them, so the child will die regardless of what you do. Now, suppose they offer you the opportunity to join in and shoot the child with them. You can make the 101st shot. Again, the child will die regardless of what you do. Is it permissible for you join the firing squad? Most people have a strong intuition that it is wrong to join the squad and shoot the child. One plausible explanation of why it is wrong is that there may be a general moral prohibition against participating in these kinds of activities. In these kinds of cases, we should try to keep our hands clean. Perhaps this "clean-hands principle" can be generalized to explain why individual acts of ignorant, irrational, or malicious voting are wrong. The firing-squad example is somewhat analogous to voting in an election. Adding or subtracting a shooter to the firing squad makes no difference--the girl will die anyway. Similarly, with elections, individual votes make no difference. In both cases, the outcome is causally overdetermined. Still, the irresponsible voter is much like a person who volunteers to shoot in the firing squad. Her individual bad vote is of no consequence--just as an individual shot is of no consequence--but she is participating in a collectively harmful activity when she could easily keep her hands clean (J. Brennan 2011a, 68-94). ## 4. The Justice of Compulsory Voting Voting rates in many contemporary democracies are (according to many observers) low, and seem in general to be falling. The United States, for instance, barely manages about 60% in presidential elections and 45% in other elections (Brennan and Hill 2014: 3). Many other countries have similarly low rates. Some democratic theorists, politicians, and others think this is problematic, and advocate compulsory voting as a solution. In a compulsory voting regime, citizens are required to vote by law; if they fail to vote without a valid excuse, they incur some sort of penalty. One major argument for compulsory voting is what we might call the Demographic or Representativeness Argument (Lijphart 1997; Engelen 2007; Galston 2011; Hill in J. Brennan and Hill 2014: 154-173; Singh 2015). The argument begins by noting that in voluntary voting regimes, citizens who choose to vote are systematically different from those who choose to abstain. The rich are more likely to vote than the poor. The old are more likely to vote than the young. Men are more likely to vote than women. In many countries, ethnic minorities are less likely to vote than ethnic majorities. More highly educated people are more likely to vote than less highly educated people. Married people are more likely to vote than non-married people. Political partisans are more likely to vote than true independents (Leighley and Nagler 1992; Evans 2003: 152-6). In short, under voluntary voting, the electorate--the citizens who actually choose to vote--are not fully representative of the public at large. The Demographic Argument holds that since politicians tend to give voters what they want, in a voluntary voting regime, politicians will tend to advance the interests of advantaged citizens (who vote disproportionately) over the disadvantaged (who tend not to vote). Compulsory voting would tend to ensure that the disadvantaged vote in higher numbers, and would thus tend to ensure that everyone's interests are properly represented. Relatedly, one might argue compulsory voting helps citizens overcome an "assurance problem" (Hill 2006). The thought here is that an individual voter realizes her individual vote has little significance. What's important is that enough other voters like her vote. However, she cannot easily coordinate with other voters and ensure they will vote with her. Compulsory voting solves this problem. For this reason, Lisa Hill (2006: 214-15) concludes, "Rather than perceiving the compulsion as yet another unwelcome form of state coercion, compulsory voting may be better understood as a coordination necessity in mass societies of individual strangers unable to communicate and coordinate their preferences." Whether the Demographic Argument succeeds or not depends on a few assumptions about voter and politician behavior. First, political scientists overwhelmingly find that voters do not vote their self-interest, but instead vote for what they perceive to be the national interest. (See the dozens of papers cited at Brennan and Hill 2014: 38-9n28.) Second, it might turn out that disadvantaged citizens are not informed enough to vote in ways that promote their interests--they might not have sufficient social scientific knowledge to know which candidates or political parties will help them (Delli Carpini and Keeter 1996; Caplan 2007; Somin 2013). Third, it may be that even in a compulsory voting regime, politicians can get away with ignoring the policy preferences of most voters (Gilens 2012; Bartels 2010). In fact, contrary to many theorists' expectations, it appears that compulsory voting has no significant effect on individual political knowledge (that is, it does not induce ignorant voters to become better informed), individual political conversation and persuasion, individual propensity to contact politicians, the propensity to work with others to address concerns, participation in campaign activities, the likelihood of being contacted by a party or politician, the quality of representation, electoral integrity, the proportion of female members of parliament, support for small or third parties, support for the left, or support for the far right (Birch 2009; Highton and Wolfinger 2001). Political scientists have also been unable to demonstrate that compulsory voting leads to more egalitarian or left-leaning policy outcomes. The empirical literature so far shows that compulsory voting gets citizens to vote, but it's not clear it does much else. ## 5. The Ethics of Vote Buying Many citizens of modern democracies believe that vote buying and selling are immoral (Tetlock 2000). Many philosophers agree; they argue it is wrong to buy, trade, or sell votes (Satz 2010: 102; Sandel 2012: 104-5). Richard Hasen reviews the literature on vote buying and concludes that people have offered three main arguments against it. He says, > > Despite the almost universal condemnation of core vote buying, > commentators disagree on the underlying rationales for its > prohibition. Some offer an equality argument against vote buying: the > poor are more likely to sell their votes than are the wealthy, leading > to political outcomes favoring the wealthy. Others offer an efficiency > argument against vote buying: vote buying allows buyers to engage in > rent-seeking that diminishes overall social wealth. Finally, some > commentators offer an inalienability argument against vote buying: > votes belong to the community as a whole and should not be alienable > by individual voters. This alienability argument may support an > anti-commodification norm that causes voters to make public-regarding > voting decisions. (Hasen 2000: 1325) > Two of the concerns here are consequentialist: the worry is that in a regime where vote-buying is legal, votes will be bought and sold in socially destructive ways. However, whether vote buying is destructive is a subject of serious social scientific debate; some economists think markets in votes would in fact produce greater efficiency (Buchanan and Tullock 1962; Haefele 1971; Mueller 1973; Philipson and Snyder 1996; Hasen 2000: 1332). The third concern is deontological: it holds that votes are just not the kind of thing that ought be for sale, even if it turned out that vote-buying and selling did not lead to bad consequences. Many people think vote selling is wrong because it would lead to bad or corrupt voting. But, if that is the problem, then perhaps the permissibility of vote buying and selling should be assessed on a case-by-case basis. Perhaps the rightness or wrongness of individual acts of vote buying and selling depends entirely on how the vote seller votes (J. Brennan 2011a: 135-160; Brennan and Jaworski 2015: 183-194). Suppose I pay a person to vote in a good way. For instance, suppose I pay indifferent people to vote on behalf of women's rights, or for the Correct Theory of Justice, whatever that might be. Or, suppose I think turnout is too low, and so I pay a well-informed person to vote her conscience. It is unclear why we should conclude in either case that I have done something wrong, rather than conclude that I have done everyone a small public service. Certain objections to vote buying and selling appear to prove too much; these objections lead to conclusions that the objectors are not willing to support. For instance, one common argument against voting selling is that paying a person to vote imposes an externality on third parties. However, so does persuading others to vote or to vote in certain ways (Freiman 2014: 762). If paying you to vote for X is wrong because it imposes a third party cost, then for the sake of consistency, I should also conclude that persuading you to vote for X, say, on the basis of a good argument, is equally problematic. As another example, some object to voting markets on the grounds that votes should be for the common good, rather than for narrow self-interest (Satz 2010: 103; Sandel 2012: 10). Others say that voting should "be an act undertaken only after collectively deliberating about what it is in the common good" (Satz 2010: 103). Some claim that vote markets should be illegal for this reason. Perhaps it's permissible to forbid vote selling because commodified votes are likely to be cast against the common good. However, if that is sufficient reason to forbid markets in votes, then it is unclear why we should not, e.g., forbid highly ignorant, irrational, or selfish voters from voting, as their votes are also unusually likely to undermine the common good (Freiman 2014: 771-772). Further these arguments appear to leave open that a person could permissibly sell her vote, provided she does so after deliberating and provided she votes for the common good. It might be that if vote selling were legal, most or even all vote sellers would vote in destructive ways, but that does not show that vote selling is inherently wrong. One pressing issue, though, is whether vote buying is compatible with the secret ballot (Maloberti 2018). Regardless of whether vote buying is enforced through legal means (such as through enforceable contracts) or social means (such as through the reputation mechanism in eBay or through simply social disapproval), to enforce vote buying seems to require that voters in some way actively prove they voted in various ways. But, if so, then this will partly eliminate the secret ballot and possibly lead to increased clientelism, in which politicians make targeted promises to particular bands of voters rather than serve the common good (Maloberti 2018). Not all objections to vote-buying have this consequentialist flavor. Some argue that vote buying is wrong for deontological grounds, for instance, on the grounds that vote buying in some way is incompatible with the social meaning of voting (e.g. Walzer 1984). Some view voting is an expressive act, and the meaning of that expression is socially-determined. To buy and sell votes may signal disrespect to others in light of this social meaning. ## 6. Who Should Be Allowed to Vote? Should Everyone Receive Equal Voting Rights? The dominant view among political philosophers is that we ought to have some sort of representative democracy, and that each adult ought to have one vote, of equal weight to every other adult's, in any election in her jurisdiction. This view has recently come under criticism, though, both from friends and foes of democracy. Before one even asks whether "one person, one vote" is the right policy, one needs to determine just who counts as part of the demos. Call this the boundary problem or the problem of constituting the demos (Goodin 2007: 40; Ron 2017). Democracy is the rule of the people. But one fundamental question is just who constitutes "the people". This is no small problem. Before one can judge that a democracy is fair, or adequately responds to citizens' interests, one needs to know who "counts" and who does not. One might be inclined to say that everyone living under a particular government's jurisdiction is part of the demos and is thus entitled to a vote. However, in fact, most democracies exclude children and teenagers, felons, the mentally infirm, and non-citizens living in a government's territory from being able to vote, but at the same time allow their citizens living in foreign countries to vote (Lopez-Guerra 2014: 1). There are a number of competing theories here. The "all affected interests" theory (Dahl 1990a: 64) holds that anyone who is affected by a political decision or a political institution is part of the demos. The basic argument is that anyone who is affected by a political decision-making process should have some say over that process. However, this principle suffers from multiple problems. It may be incoherent or useless, as we might not know or be able to know who is affected by a decision until after the decision is made (Goodin 2007: 52). For example (taken from Goodin 2007: 53), suppose the UK votes on whether to transfer 5% of its GDP to its former African colonies. We cannot assess whether the members of the former African colonies are among the affected interests until we know what the outcome of the vote is. If the vote is yay, then they are affected; if the vote is nay, then they are not. (See Owen 2012 for a response.) Further, the "all affected interests" theory would often include non-citizens and exclude citizens. Sometimes political decisions made in one country have a significant effect on citizens of another country; sometimes political decisions made in one country have little or no effect on some of the citizens of that country. One solution (Goodin 2007: 55) to this problem (of who counts as an affected party) is to hold that all people with possibly or potentially affected interests constitute part of the polity. This principle implies, however, that for many decisions, the demos is smaller than the nation-state, and for others, it is larger. For instance, when the United States decides whether to elect a warmongering or pacifist candidate, this affects not only Americans, but a large percentage of people worldwide. Other major theories offered as solutions to the boundary problem face similar problems. For example, the coercion theory holds that anyone subject to coercion from a political body ought to have a say (Lopez-Guerra 2005). But this principle might be also be seen as over-inclusive (Song 2009), as it would require that resident aliens, tourists, or even enemy combatants be granted a right to vote, as they are also subject to a state's coercive power. Further, who will be coerced depends on the outcome of a decision. If a state decides to impose some laws, it will coerce certain people, and if the state declines to impose those laws, then it will not. If we try to overcome this by saying anyone potentially subject to a given state's coercive power ought to have a say, then this seems to imply that almost everyone worldwide should have a say in most states' major decisions. The commonsense view of the demos, i.e., that the demos includes all and only adult members of a nation-state, may be hard to defend. Goodin (2007: 49) proposes that what makes citizens special is that their interests are interlinked. This may be an accidental feature of arbitrarily-decided national borders, but once these borders are in place, citizens will find that their interests tend to more linked together than with citizens of other polities. But whether this is true is also highly contingent. ### 6.1 Democratic Challenges to One Person, One Vote The idea of "One person, one vote" is supposedly grounded on a commitment to egalitarianism. Some philosophers believe that democracy with equal voting rights is necessary to ensure that government gives equal consideration to everyone's interests (Christiano 1996, 2008). However, it is not clear that giving every citizen an equal right to vote reliably results in decisions that give equal consideration to everyone's interests. In many decisions, many citizens have little to nothing at stake, while other citizens have a great deal at stake. Thus, one alternative proposal is that citizens' votes should be weighted by how much they have a stake in the decision. This preserves equality not by giving everyone an equal chance of being decisive in every decision, but by giving everyone's interests equal weight. Otherwise, in a system of one person, one vote, issues that are deeply important to the few might continually lose out to issues of only minor interest to the many (Brighouse and Fleurbaey 2010). There are a number of other independent arguments for this conclusion. Perhaps proportional voting enhances citizens' autonomy, by giving them greater control over those issues in which they have greater stakes, while few would regard it as significant loss of autonomy were they to have reduced control over issues that do not concern them. Further, though the argument for this conclusion is too technical to cover here in depth (Brighouse and Fleurbaey 2010; List 2013), it may be that apportioning political power according to one's stake in the outcome can overcome some of the well-known paradoxes of democracy, such as the Condorcet Paradox (which show that democracies might have intransitive preferences, i.e., the majority might prefer A to B, B to C, and yet also prefer C to A). However, even if this proposal seems plausible in theory, it is unclear how a democracy might reliably instantiate this in practice. Before allowing a vote, a democratic polity would need to determine to what extent different citizens have a stake in the decision, and then somehow weight their votes accordingly. In real life, special-interests groups and others would likely try to use vote weighting for their own ends. Citizens might regard unequal voting rights as evidence of corruption or electoral manipulation (Christiano 2008: 34-45). ### 6.2 Non-Democratic Challenges to One Person, One Vote Early defenders of democracy were concerned to show democracy is superior to aristocracy, monarchy, or oligarchy. However, in recent years, epistocracy has emerged as a major contender to democracy (Estlund 2003, 2007; Landemore 2012). A system is said to be epistocratic to the extent that the system formally allocates political power on the basis of knowledge or political competence. For instance, an epistocracy might give university-educated citizens additional votes (Mill 1861), exclude citizens from voting unless they can pass a voter qualification exam, weigh votes by each voter's degree of political knowledge while correcting for the influence of demographic factors, or create panels of experts who have the right to veto democratic legislation (Caplan 2007; J. Brennan 2011b; Lopez-Guerra 2014; Mulligan 2015). Arguments for epistocracy generally center on concerns about democratic incompetence. Epistocrats hold that democracy imbues citizens with the right to vote in a promiscuous way. Ample empirical research has shown that the mean, median, and modal levels of basic political knowledge (let alone social scientific knowledge) among citizens is extremely low (Somin 2013; Caplan 2007; Delli Carpini and Keeter 1996). Further, political knowledge makes a significant difference in how citizens vote and what policies they support (Althaus 1998, 2003; Caplan 2007; Gilens 2012). Epistocrats believe that restricting or weighting votes would protect against some of the downsides of democratic incompetence. One argument for epistocracy is that the legitimacy of political decisions depends upon them being made competently and in good faith. Consider, as an analogy: In a criminal trial, the jury's decision is high stakes; their decision can remove a person's rights or greatly harm their life, liberty, welfare, or property. If a jury made its decision out of ignorance, malice, whimsy, or on the basis of irrational and biased thought processes, we arguably should not and probably would not regard the jury's decision as authoritative or legitimate. Instead, we think the criminal has a right to a trial conducted by competent people in good faith. In many respects, electoral decisions are similar to jury decisions: they also are high stakes, and can result in innocent people losing their lives, liberty, welfare, or property. If the legitimacy and authority of a jury decision depends upon the jury making a competent decision in good faith, then perhaps so should the legitimacy and authority of most other governmental decisions, including the decisions that electorates and their representatives make. Now, suppose, in light of widespread voter ignorance and irrationality, it turns out that democratic electorates tend to make incompetent decisions. If so, then this seems to provide at least presumptive grounds for favoring epistocracy over democracy (J. Brennan 2011b). Some dispute whether epistocracy would in fact perform better than democracy, even in principle. Epistocracy generally attempts to generate better political outcomes by in some way raising the average reliability of political decision-makers. Political scientists Lu Hong and Scott Page (2004) adduced a mathematical theorem showing that under the right conditions, cognitive diversity among the participants in a collective decision more strongly contributes to the group making a smart decision than does increasing the individual participants' reliability. On the Hong-Page theorem, it is possible that having a large number of diverse but unreliable decision-makers in a collective decision will outperform having a smaller number of less diverse but more reliable decision-makers. There is some debate over whether the Hong-Page theorem has any mathematical substance (Thompson 2014 claims it does not), whether real-world political decisions meet the conditions of the theorem, and if so, to what extent that justifies universal suffrage, or merely shows that having widespread but restricted suffrage is superior to having highly restricted suffrage (Landemore 2012; Somin 2013: 113-5). Relatedly, Condorcet's Jury Theorem holds that under the right conditions, provided the average voter is reliable, as more and more voters are added to a collective decision, the probability that the democracy will make the right choice approaches 1 (List and Goodin 2001). However, assuming the theorem applies to real-life democratic decisions, whether the theorem supports or condemns democracy depends on how reliable voters are. If voters do systematically worse than chance (e.g., Althaus 2003; Caplan 2007), then the theorem instead implies that large democracies almost always make the wrong choice. One worry about certain forms of epistocracy, such as a system in which voters must earn the right to vote by passing an examination, is that such systems might make decisions that are biased toward members of certain demographic groups. After all, political knowledge is not evenly dispersed among all demographic groups. (At the very least, the kinds of knowledge political scientists have been studying are not evenly distributed. Whether other kinds of knowledge are better distributed is an open question.) On average, in the United States, on measures of basic political knowledge, whites know more than blacks, people in the Northeast know more than people in the South, men know more than women, middle-aged people know more than the young or old, and high-income people know more than the poor (Delli Carpini and Keeter 1996: 137-177). If such a voter examination system were implemented, the resulting electorate would be whiter, maler, richer, more middle-aged, and better employed than the population at large. Democrats might reasonably worry that for this very reason an epistocracy would not take the interests of non-whites, women, the poor, or the unemployed into proper consideration. However, at least one form of epistocracy may be able to avoid this objection. Consider, for instance, the "enfranchisement lottery": > > The enfranchisement lottery consists of two devices. First, there > would be a sortition to disenfranchise the vast majority of the > population. Prior to every election, all but a random sample of the > public would be excluded. I call this device the exclusionary > sortition because it merely tells us who will not be entitled to vote > in a given contest. Indeed, those who survive the sortition (the > pre-voters) would not be automatically enfranchised. Like everyone in > the larger group from which they are drawn, pre-voters would be > assumed to be insufficiently competent to vote. This is where the > second device comes in. To finally become enfranchised and vote, > pre-voters would gather in relatively small groups to participate in a > competence-building process carefully designed to optimize their > knowledge about the alternatives on the ballot. (Lopez-Guerra > 2014: 4; cf. Ackerman and Fishkin 2005) > Under this scheme, no one has any presumptive right to vote. Instead, everyone has, by default, equal eligibility to be selected to become a voter. Before the enfranchisement lottery takes place, candidates would proceed with their campaigns as they do in democracy. However, they campaign without knowing which citizens in particular will eventually acquire the right to vote. Immediately before the election, a random but representative subset of citizens is then selected by lottery. These citizens are not automatically granted the right to vote. Instead, the chosen citizens merely acquire permission to earn the right to vote. To earn this right, they must then participate in some sort of competence-building exercise, such as studying party platforms or meeting in a deliberative forum with one another. In practice this system might suffer corruption or abuse, but, epistocrats respond, so does democracy in practice. For epistocrats, the question is which system works better, i.e., produces the best or most substantively just outcomes, all things considered. One important deontological objection to epistocracy is that it may be incompatible with public reason liberalism (Estlund 2007). Public reason liberals hold that distribution of coercive political power is legitimate and authoritative only if all reasonable people subject to that power have strong enough grounds to endorse a justification for that power (Vallier and D'Agostino 2013). By definition, epistocracy imbues some citizens with greater power than others on the grounds that these citizens have greater social scientific knowledge. However, the objection goes, reasonable people could disagree about just what counts as expertise and just who the experts are. If reasonable people disagree about what counts as expertise and who the experts are, then epistocracy distributes political power on terms not all reasonable people have conclusive grounds to endorse. Epistocracy thus distributes political power on terms not all reasonable people have conclusive grounds to endorse. (See, however, Mulligan 2015.)
voting-methods	## 1. The Problem: Who Should be Elected? Suppose that there is a group of 21 voters who need to make a decision about which of four candidates should be elected. Let the names of the candidates be $A$, $B$, $C$ and $D$. Your job, as a social planner, is to determine which of these 4 candidates should win the election given the opinions of all the voters. The first step is to elicit the voters' opinions about the candidates. Suppose that you ask each voter to rank the 4 candidates from best to worst (not allowing ties). The following table summarizes the voters' rankings of the candidates in this hypothetical election scenario. \| \| \| \| --- \| --- \| \| # Voters \| Ranking \| \| 3 \| $A\s B\s C\s D$ \| \| 5 \| $A\s C\s B\s D$ \| \| 7 \| $B\s D\s C\s A$ \| \| 6 \| $C\s B\s D\s A$ \| Read the table as follows: Each row represents a ranking for a group of voters in which candidates to the left are ranked higher. The numbers in the first column indicate the number of voters with that particular ranking. So, for example, the third row in the table indicates that 7 voters have the ranking $B\s D\s C\s A$ which means that each of the 7 voters rank $B$ first, $D$ second, $C$ third and $A$ last. Suppose that, as the social planner, you do not have any personal interest in the outcome of this election. Given the voters' expressed opinions, which candidate should win the election? Since the voters disagree about the ranking of the candidates, there is no obvious candidate that best represents the group's opinion. If there were only two candidates to choose from, there is a very straightforward answer: The winner should be the candidate or alternative that is supported by more than 50 percent of the voters (cf. the discussion below about May's Theorem in Section 4.2). However, if there are more than two candidates, as in the above example, the statement "the candidate that is supported by more than 50 percent of the voters" can be interpreted in different ways, leading to different ideas about who should win the election. One candidate who, at first sight, seems to be a good choice to win the election is $A$. Candidate $A$ is ranked first by more voters than any other candidate. ($A$ is ranked first by 8 voters, $B$ is ranked first by 7; $C$ is ranked first by 6; and $D$ is not ranked first by any of the voters.) Of course, 13 people rank $A$ last. So, while more voters rank $A$ first than any other candidate, more than half of the voters rank $A$ last. This suggests that $A$ should not be elected. None of the voters rank $D$ first. This fact alone does not rule out $D$ as a possible winner of the election. However, note that every voter ranks candidate $B$ above candidate $D$. While this does not mean that $B$ should necessarily win the election, it does suggest that $D$ should not win the election. The choice, then, boils down to $B$ and $C$. It turns out that there are good arguments for each of $B$ and $C$ to be elected. The debate about which of $B$ or $C$ should be elected started in the 18th-century as an argument between the two founding fathers of voting theory, Jean-Charles de Borda (1733-1799) and M.J.A.N. de Caritat, Marquis de Condorcet (1743-1794). For a history of voting theory as an academic discipline, including Condorcet's and Borda's writings, see McLean and Urken (1995). I sketch the intuitive arguments for the election of $B$ and $C$ below. Candidate $C$ should win. Initially, this might seem like an odd choice since both $A$ and $B$ receive more first place votes than $C$ (only 6 voters rank $C$ first while 8 voters rank $A$ first and 7 voters rank $B$ first). However, note how the population would vote in the various two-way elections comparing $C$ with each of the other candidates: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| # Voters \| $C$ versus $A$ \| $C$ versus $B$ \| $C$ versus $D$ \| \| 3 \| $\bA\s \gB\s \bC\s \gD$ \| $\gA\s \bB\s \bC\s \gD$ \| $\gA\s \gB\s \bC\s \bD$ \| \| 5 \| $\bA\s \bC\s \gB\s \gD$ \| $\gA\s \bC\s \bB\s \gD$ \| $\gA\s \bC\s \gB\s \bD$ \| \| 7 \| $\gB\s \gD\s \bC\s \bA$ \| $\bB\s \gD\s \bC\s \gA$ \| $\gB\s \bD\s \bC\s \gA$ \| \| 6 \| $ \bC\s \gB\s \gD\s \bA$ \| $ \bC\s \bB\s \gD\s \gA$ \| $ \bC\s \gB\s \bD\s \gA$ \| \| Totals: \| 13 rank $C$ above $A$8 rank $A$ above $C$ \| 11 rank $C$ above $B$10 rank $B$ above $C$ \| 14 rank $C$ above $D$7 rank $D$ above $C$ \| Condorcet's idea is that $C$ should be declared the winner since she beats every other candidate in a one-on-one election. A candidate with this property is called a Condorcet winner. We can similarly define a Condorcet loser. In fact, in the above example, candidate $A$ is the Condorcet loser since she loses to every other candidate in a one-on-one election. Candidate $B$ should win. Consider $B$'s performance in the one-on-one elections. \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| # Voters \| $B$ versus $A$ \| $B$ versus $C$ \| $B$ versus $D$ \| \| 3 \| $\bA\s \bB\s \gC\s \gD$ \| $\gA\s \bB\s \bC\s \gD$ \| $\gA\s \bB\s \gC\s \bD$ \| \| 5 \| $\bA\s \gC\s \bB\s \gD$ \| $\gA\s \bC\s \bB\s \gD$ \| $\gA\s \gC\s \bB\s \bD$ \| \| 7 \| $\bB\s \gD\s \gC\s \bA$ \| $\bB\s \gD\s \bC\s \gA$ \| $\bB\s \bD\s \gC\s \gA$ \| \| 6 \| $ \gC\s \bB\s \gD\s \bA$ \| $ \bC\s \bB\s \gD\s \gA$ \| $ \gC\s \bB\s \bD\s \gA$ \| \| Totals: \| 13 rank $B$ above $A$8 rank $A$ above $B$ \| 10 rank $B$ above $C$11 rank $C$ above $B$ \| 21 rank $B$ above $D$0 rank $D$ above $B$ \| Candidate $B$ performs the same as $C$ in a head-to-head election with $A$, loses to $C$ by only one vote and beats $D$ in a landslide (everyone prefers $B$ over $D$). Borda suggests that we should take into account all of these facts when determining which candidate best represents the overall group opinion. To do this, Borda assigns a score to each candidate that reflects how much support he or she has among the electorate. Then, the candidate with the largest score is declared the winner. One way to calculate the score for each candidate is as follows (I will give an alternative method, which is easier to use, in the next section): * $A$ receives 24 points (8 votes in each of the three head-to-head elections) * $B$ receives 44 points (13 points in the competition against $A$, plus 10 in the competition against $C$ plus 21 in the competition against $D$) * $C$ receives 38 points (13 points in the competition against $A$, plus 11 in the competition against $B$ plus 14 in the competition against $D$) * $D$ receives 20 points (13 points in the competition against $A$, plus 0 in the competition against $B$ plus 7 in the competition against $C$) The candidate with the highest score (in this case, $B$) is the one who should be elected. Both Condorcet and Borda suggest comparing candidates in one-on-one elections in order to determine the winner. While Condorcet tallies how many of the head-to-head races each candidate wins, Borda suggests that one should look at the margin of victory or loss. The debate about whether to elect the Condorcet winner or the Borda winner is not settled. Proponents of electing the Condorcet winner include Mathias Risse (2001, 2004, 2005) and Steven Brams (2008); Proponents of electing the Borda winner include Donald Saari (2003, 2006) and Michael Dummett (1984). See Section 3.1.1 for further issues comparing the Condorcet and Borda winners. The take-away message from this discussion is that in many election scenarios with more than two candidates, there may not always be one obvious candidate that best reflects the overall group opinion. The remainder of this entry will discuss different methods, or procedures, that can be used to determine the winner(s) given the a group of voters' opinions. Each of these methods is intended to be an answer to the following question: Given a group of people faced with some decision, how should a central authority combine the individual opinions so as to best reflect the "overall group opinion"? A complete analysis of this question would incorporate a number of different issues ranging from central topics in political philosophy about the nature of democracy and the "will of the people" to the psychology of decision making. In this article, I focus on one aspect of this question: the formal analysis of algorithms that aggregate the opinions of a group of voters (i.e., voting methods). Consult, for example, Riker 1982, Mackie 2003, and Christiano 2008 for a more comprehensive analysis of the above question, incorporating many of the issues raised in this article. ### 1.1 Notation In this article, I will keep the formal details to a minimum; however, it is useful at this point to settle on some terminology. Let $V$ and $X$ be finite sets. The elements of $V$ are called voters and I will use lowercase letters $i, j, k, \ldots$ or integers $1, 2, 3, \ldots$ to denote them. The elements of $X$ are called candidates, or alternatives, and I will use uppercase letters $A, B, C, \ldots $ to denote them. Different voting methods require different types of information from the voters as input. The input requested from the voters are called ballots. One standard example of a ballot is a ranking of the set of candidates. Formally, a ranking of $X$ is a relation $P$ on $X$, where $Y\mathrel{P} Z$ means that "$Y$ is ranked above $Z$," satisfying three constraints: (1) $P$ is complete: any two distinct candidates are ranked (for all candidates $Y$ and $Z$, if $Y\ne Z$, then either $Y\mathrel{P} Z$ or $Z\mathrel{P} Y$); (2) $P$ is transitive: if a candidate $Y$ is ranked above a candidate $W$ and $W$ is ranked above a candidate $Z$, then $Y$ is ranked above $Z$ (for all candidates $Y, Z$, and $W$, if $Y\mathrel{P} W$ and $W\mathrel{P} Z$, then $Y\mathrel{P} Z$); and (3) $P$ is irreflexive: no candidate is ranked above itself (there is no candidate $Y$ such that $Y\mathrel{P} Y$). For example, suppose that there are three candidates $X =\{A, B, C\}$. Then, the six possible rankings of $X$ are listed in the following table: \| \| \| \| --- \| --- \| \| # Voters \| Ranking \| \| $n\_1$ \| $A\s B\s C$ \| \| $n\_2$ \| $A\s C\s B$ \| \| $n\_3$ \| $B\s A\s C$ \| \| $n\_4$ \| $B\s C\s A$ \| \| $n\_5$ \| $C\s A\s B$ \| \| $n\_6$ \| $C\s B\s A$ \| I can now be more precise about the definition of a Condorcet winner (loser). Given a ranking from each voter, the majority relation orders the candidates in terms of how they perform in one-on-one elections. More precisely, for candidates $Y$ and $Z$, write $Y \mathrel{>\_M} Z$, provided that more voters rank candidate $Y$ above candidate $Z$ than the other way around. So, if the distribution of rankings is given in the above table, we have: \[\begin{align} A\mathrel{>\_M} B\ &\text{ just in case } n\_1 + n\_2 + n\_5 > n\_3 + n\_4 + n\_6 \\ A\mathrel{>\_M} C\ &\text{ just in case } n\_1 + n\_2 + n\_3 > n\_4 + n\_5 + n\_6 \\ B \mathrel{>\_M} C\ &\text{ just in case } n\_1 + n\_3 + n\_4 > n\_2 + n\_5 + n\_6 \end{align}\] A candidate $Y$ is called the Condorcet winner in an election scenario if $Y$ is the maximum of the majority ordering $>\_M$ for that election scenario (that is, $Y$ is the Condorcet winner if $Y\mathrel{>\_M} Z$ for all other candidates $Z$). The Condorcet loser is the candidate that is the minimum of the majority ordering. Rankings are one type of ballot. In this article, we will see examples of other types of ballots, such as selecting a single candidate, selecting a subset of candidates or assigning grades to candidates. Given a set of ballots $\mathcal{B}$, a profile for a set of voters specifies the ballot selected by each voter. Formally, a profile for set of voters $V=\{1,\ldots, n\}$ and a set of ballots $\mathcal{B}$ is a sequence $\bb=(b\_1,\ldots, b\_n)$, where for each voter $i$, $b\_i$ is the ballot from $\mathcal{B}$ submitted by voter $i$. A voting method is a function that assigns to each possible profile a group decision. The group decision may be a single candidate (the winning candidate), a set of candidates (when ties are allowed), or an ordering of the candidates (possibly allowing ties). Note that since a profile identifies the voter associated with each ballot, a voting method may take this information into account. This means that voting methods can be designed that select a winner (or winners) based only on the ballots of some subset of voters while ignoring all the other voters' ballots. An extreme example of this is the so-called Arrovian dictatorship for voter $d$ that assigns to each profile the candidate ranked first by $d$. A natural way to rule out these types of voting methods is to require that a voting method is anonymous: the group decision should depend only on the number of voters that chose each ballot. This means that if two profiles are permutations of each other, then a voting method that is anonymous must assign the same group decision to both profiles. When studying voting methods that are anonymous, it is convenient to assume the inputs are anonymized profiles. An anonymous profile for a set of ballots $\mathcal{B}$ is a function from $\mathcal{B}$ to the set of integers $\mathbb{N}$. The election scenario discussed in the previous section is an example of an anonymized profile (assuming that each ranking not displayed in the table is assigned the number 0). In the remainder of this article (unless otherwise specified), I will restrict attention to anonymized profiles. I conclude this section with a few comments on the relationship between the ballots in a profile and the voters' opinions about the candidates. Two issues are important to keep in mind. First, the ballots used by a voting method are intended to reflect some aspect of the voters' opinions about the candidates. Voters may choose a ballot that best expresses their personal preference about the set of candidates or their judgements about the relative strengths of the candidates. A common assumption in the voting theory literature is that a ranking of the set of candidates expresses a voter's ordinal preference ordering over the set of candidates (see the entry on preferences, Hansson and Grune-Yanoff 2009, for an extended discussion of issues surrounding the formal modeling of preferences). Other types of ballots represent information that cannot be inferred directly from a voter's ordinal preference ordering, for example, by describing the intensity of a preference for a particular candidate (see Section 2.3). Second, it is important to be precise about the type of considerations voters take into account when selecting a ballot. One approach is to assume that voters choose sincerely by selecting the ballot that best reflects their opinion about the the different candidates. A second approach assumes that the voters choose strategically. In this case, a voter selects a ballot that she expects to lead to her most desired outcome given the information she has about how the other members of the group will vote. Strategic voting is an important topic in voting theory and social choice theory (see Taylor 2005 and Section 3.3 of List 2013 for a discussion and pointers to the literature), but in this article, unless otherwise stated, I assume that voters choose sincerely (cf. Section 4.1). ## 2. Examples of Voting Methods A quick survey of elections held in different democratic societies throughout the world reveals a wide variety of voting methods. In this section, I discuss some of the key methods that have been analyzed in the voting theory literature. These methods may be of interest because they are widely used (e.g., Plurality Rule or Plurality Rule with Runoff) or because they are of theoretical interest (e.g., Dodgson's method). I start with the most widely used method: Plurality Rule: Each voter selects one candidate (or none if voters can abstain), and the candidate(s) with the most votes win. Plurality rule (also called First Past the Post) is a very simple method that is widely used despite its many problems. The most pervasive problem is the fact that plurality rule can elect a Condorcet loser. Borda (1784) observed this phenomenon in the 18th century (see also the example from Section 1). \| \| \| \| --- \| --- \| \| # Voters \| Ranking \| \| 1 \| $A\s B\s C$ \| \| 7 \| $A\s C\s B$ \| \| 7 \| $B\s C\s A$ \| \| 6 \| $C\s B\s A$ \| Candidate $A$ is the Condorcet loser (both $B$ and $C$ beat candidate $A$, 13 - 8); however, $A$ is the plurality rule winner (assuming the voters vote for the candidate that they rank first). In fact, the plurality ranking ($A$ is first with 8 votes, $B$ is second with 7 votes and $C$ is third with 6 votes) reverses the majority ordering $C\mathrel{>\_M} B\mathrel{>\_M} A$. See Laslier 2012 for further criticisms of Plurality Rule and comparisons with other voting methods discussed in this article. One response to the above phenomenon is to require that candidates pass a certain threshold to be declared the winner. Quota Rule: Suppose that $q$, called the quota, is any number between 0 and 1. Each voter selects one candidate (or none if voters can abstain), and the winners are the candidates that receive at least $q \times \# V$ votes, where $\# V$ is the number of voters. Majority Rule is a quota rule with $q=0.5$ (a candidate is the strict or absolute majority winner if that candidate receives strictly more than $0.5 \times \# V$ votes). Unanimity Rule is a quota rule with $q=1$. An important problem with quota rules is that they do not identify a winner in every election scenario. For instance, in the above election scenario, there are no majority winners since none of the candidates are ranked first by more than 50% of the voters. A criticism of both plurality and quota rules is that they severely limit what voters can express about their opinions of the candidates. In the remainder of this section, I discuss voting methods that use ballots that are more expressive than simply selecting a single candidate. Section 2.1 discusses voting methods that require voters to rank the alternatives. Section 2.2 discusses voting methods that require voters to assign grades to the alternatives (from some fixed set of grades). Finally, Section 2.3 discusses two voting methods in which the voters may have different levels of influence on the group decision. In this article, I focus on voting methods that either are familiar or help illustrate important ideas. Consult Brams and Fishburn 2002, Felsenthal 2012, and Nurmi 1987 for discussions of voting methods not covered in this article. ### 2.1 Ranking Methods: Scoring Rules and Multi-Stage Methods The voting methods discussed in this section require the voters to rank the candidates (see section 1.1 for the definition of a ranking). Providing a ranking of the candidates is much more expressive than simply selecting a single candidate. However, ranking all of the candidates can be very demanding, especially when there is a large number of them, since it can be difficult for voters to make distinctions between all the candidates. The most well-known example of a voting method that uses the voters' rankings is Borda Count: Borda Count: Each voter provides a ranking of the candidates. Then, a score (the Borda score) is assigned to each candidate by a voter as follows: If there are $n$ candidates, give $n-1$ points to the candidate ranked first, $n-2$ points to the candidate ranked second,..., 1 point to the candidate ranked second to last and 0 points to candidate ranked last. So, the Borda score of candidate $A$, denoted $\BS(A)$, is calculated as follows (where $\#U$ denotes the number elements in the set $U)$: \[\begin{align} \BS(A) =\ &(n-1)\times \# \{i\ \|\ i \text{ ranks $A$ first}\}\\ &+ (n-2)\times \# \{i\ \|\ i \text{ ranks $A$ second}\} \\ &+ \cdots \\ &+ 1\times \# \{i\ \|\ i \text{ ranks $A$ second to last}\}\\ &+ 0\times \# \{i\ \|\ i \text{ ranks $A$ last}\} \end{align}\] The candidate with the highest Borda score wins. Recall the example discussed in the introduction to Section 1. For each alternative, the Borda scores can be calculated using the above method: \[\begin{align} \BS(A) &= 3 \times 8 + 2 \times 0 + 1 \times 0 + 0 \times 13 = 24 \\ \BS(B) &= 3 \times 7 + 2 \times 9 + 1 \times 5 + 0 \times 0 = 44 \\ \BS(C) &= 3 \times 6 + 2 \times 5 + 1 \times 10 + 0 \times 0 = 38 \\ \BS(D) &= 3 \times 0 + 2 \times 7 + 1 \times 6 + 0 \times 8 = 20 \end{align}\] Borda Count is an example of a scoring rule. A scoring rule is any method that calculates a score based on weights assigned to candidates according to where they fall in the voters' rankings. That is, a scoring rule for $n$ candidates is defined as follows: Fix a sequence of numbers $(s\_1, s\_2, \ldots, s\_n)$ where $s\_k\ge s\_{k+1}$ for all $k=1,\ldots, n-1$. For each $k$, $s\_k $ is the score assigned to a alternatives ranked in position $k$. Then, the score for alternative $A$, denoted $Score(A)$, is calculated as follows: \[\begin{align} \textit{Score}(A)=\ &s\_1\times \# \{i\ \|\ i \text{ ranks $A$ first}\}\\ &+ s\_2\times \# \{i\ \|\ i \text{ ranks $A$ second}\}\\ &+ \cdots \\ &+ s\_n\times \# \{i\ \|\ i \text{ ranks $A$ last}\}. \end{align}\] Borda count for $n$ alternatives uses scores $(n-1, n-2, \ldots, 0)$ (call $\BS(X)$ the Borda score for candidate $X$). Note that Plurality Rule can be viewed as a scoring rule that assigns 1 point to the first ranked candidate and 0 points to the other candidates. So, the plurality score of a candidate $X$ is the number of voters that rank $X$ first. Building on this idea, $k$-Approval Voting is a scoring method that gives 1 point to each candidate that is ranked in position $k$ or higher, and 0 points to all other candidates. To illustrate $k$-Approval Voting, consider the following election scenario: \| \| \| \| --- \| --- \| \| # Voters \| Ranking \| \| 2 \| $A\s D\s B\s C$ \| \| 2 \| $B\s D\s A\s C$ \| \| 1 \| $C\s A\s B\s D$ \| * The winners according to 1-Approval Voting (which is the same as Plurality Rule) are $A$ and $B.$ * The winner according 2-Approval Voting is $D.$ * The winners according to 3-Approval Voting are $A$ and $B.$ Note that the Condorcet winner is $A$, so none of the above methods guarantee that the Condorcet winner is elected (whether $A$ is elected using 1-Approval or 3-Approval depends on the tie-breaking mechanism that is used). A second way to make a voting method sensitive to more than the voters' top choice is to hold "multi-stage" elections. The idea is to successively remove candidates that perform poorly in the election until there is one candidate that is ranked first by more than 50% of the voters (i.e., there is a strict majority winner). The different stages can be actual "runoff" elections in which voters are asked to evaluate a reduced set of candidates; or they can be built in to the way the winner is calculated by asking voters to submit rankings over the set of all candidates. The first example of a multi-stage method is used to elect the French president. Plurality with Runoff: Start with a plurality vote to determine the top two candidates (the candidates ranked first and second according to their plurality scores). If a candidate is ranked first by more than 50% of the voters, then that candidate is declared the winner. If there is no candidate with a strict majority of first place votes, then there is a runoff between the top two candidates (or more if there are ties). The candidate(s) with the most votes in the runoff elections is(are) declared the winner(s). Rather than focusing on the top two candidates, one can also iteratively remove the candidate(s) with the fewest first-place votes: The Hare Rule: The ballots are rankings of the candidates. If a candidate is ranked first by more than 50% of the voters, then that candidate is declared the winner. If there is no candidate with a strict majority of first place votes, repeatedly delete the candidate or candidates that receive the fewest first-place votes (i.e., the candidate(s) with the lowest plurality score(s)). The first candidate to be ranked first by strict majority of voters is declared the winner (if there is no such candidate, then the remaining candidate(s) are declared the winners). The Hare Rule is also called Ranked-Choice Voting, Alternative Vote, and Instant Runoff. If there are only three candidates, then the above two voting methods are the same (removing the candidate with the lowest plurality score is the same as keeping the two candidates with highest and second-highest plurality score). The following example shows that they can select different winners when there are more than three candidates: \| \| \| \| --- \| --- \| \| # Voters \| Ranking \| \| 7 \| $A\s B\s C\s D$ \| \| 5 \| $B\s C\s D\s A$ \| \| 4 \| $D\s B\s C\s A$ \| \| 3 \| $C\s D\s A\s B$ \| \| Candidate $A$ is the Plurality with Runoff winner Candidate $D$ is the Hare Rule winner \| Candidate $A$ is the Plurality with Runoff winner: Candidates $A$ and $B$ are the top two candidates, being ranked first by 7 and 5 voters, respectively. In the runoff election (using the rankings from the above table), the groups voting for candidates $C$ and $D$ transfer their support to candidates $B$ and $A,$ respectively, with $A$ winning 10 - 9. Candidate $D$ is the Hare Rule winner: In the first round, candidate $C$ is eliminated since she is only ranked first by 3 voters. This group's votes are transferred to $D$, giving him 7 votes. This means that in the second round, candidate $B$ is ranked first by the fewest voters (5 voters rank $B$ first in the profile with candidate $C$ removed), and so is eliminated. After the elimination of candidate $B$, candidate $D$ has a strict majority of the first-place votes: 12 voters ranking him first (note that in this round the group in the second column transfers all their votes to $D$ since $C$ was eliminated in an earlier round). The core idea of multi-stage methods is to successively remove candidates that perform "poorly" in an election. For the Hare Rule, performing poorly is interpreted as receiving the fewest first place votes. There are other ways to identify "poorly performing" candidates in an election scenario. For instance, the Coombs Rule successively removes candidates that are ranked last by the most voters (see Grofman and Feld 2004 for an overview of Coombs Rule). Coombs Rule: The ballots are rankings of the candidates. If a candidate is ranked first by more than 50% of the voters, then that candidate is declared the winner. If there is no candidate with a strict majority of first place votes, repeatedly delete the candidate or candidates that receive the most last-place votes. The first candidate to be ranked first by a strict majority of voters is declared the winner (if there is no such candidate, then the remaining candidate(s) are declared the winners). In the above example, candidate $B$ wins the election using Coombs Rule. In the first round, $A$, with 9 last-place votes, is eliminated. Then, candidate $B$ receives 12 first-place votes, which is a strict majority, and so is declared the winner. There is a technical issue that is important to keep in mind regarding the above definitions of the multi-stage voting methods. When identifying the poorly performing candidates in each round, there may be ties (i.e., there may be more than one candidate with the lowest plurality score or more than one candidate ranked last by the most voters). In the above definitions, I assume that all of the poorly performing candidates will be removed in each round. An alternative approach would use a tie-breaking rule to select one of the poorly performing candidates to be removed at each round. ### 2.2 Voting by Grading The voting methods discussed in this section can be viewed as generalizations of scoring methods, such as Borda Count. In a scoring method, a voter's ranking is an assignment of grades (e.g., "1st place", "2nd place", "3rd place", ... , "last place") to the candidates. Requiring voters to rank all the candidates means that (1) every candidate is assigned a grade, (2) there are the same number of possible grades as the number of candidates, and (3) different candidates must be assigned different grades. In this section, we drop assumptions (2) and (3), assuming a fixed number of grades for every set of candidates and allowing different candidates to be assigned the same grade. The first example gives voters the option to either select a candidate that they want to vote for (as in plurality rule) or to select a candidate that they want to vote against. Negative Voting: Each voter is allowed to choose one candidate to either vote for (giving the candidate one point) or to vote against (giving the candidate -1 points). The winner(s) is(are) the candidate(s) with the highest total number of points (i.e., the candidate with the greatest score, where the score is the total number of positive votes minus the total number of negative votes). Negative voting is tantamount to allowing the voters to support either a single candidate or all but one candidate (taking a point away from a candidate $C$ is equivalent to giving one point to all candidates except $C$). That is, the voters are asked to choose a set of candidates that they support, where the choice is between sets consisting of a single candidate or sets consisting of all except one candidate. The next voting method generalizes this idea by allowing voters to choose any subset of candidates: Approval Voting: Each voter selects a subset of the candidates (where the empty set means the voter abstains) and the candidate(s) with selected by the most voters wins. If a candidate $X$ is in the set of candidates selected by a voter, we say that the voter approves of candidate $X$. Then, the approval winner is the candidate with the most approvals. Approval voting has been extensively discussed by Steven Brams and Peter Fishburn (Brams and Fishburn 2007; Brams 2008). See, also, the recent collection of articles devoted to approval voting (Laslier and Sanver 2010). Approval voting forces voters to think about the decision problem differently: They are asked to determine which candidates they approve of rather than selecting a single candidate to voter for or determining the relative ranking of the candidates. That is, the voters are asked which candidates are above a certain "threshold of acceptance". Ranking a set of candidates and selecting the candidates that are approved are two different aspects of a voters overall opinion about the candidates. They are related but cannot be derived from each other. See Brams and Sanver 2009, for examples of voting methods that ask voters to both select a set of candidates that they approve and to (linearly) rank the candidates. Approval voting is a very flexible method. Recall the election scenario illustrating the $k$-Approval Voting methods: \| \| \| \| --- \| --- \| \| # Voters \| Ranking \| \| 2 \| $\underline{A}\s D\s B\s C$ \| \| 2 \| $\underline{B}\s D\s A\s C$ \| \| 1 \| $\underline{C}\s \underline{A}\s B\s D$ \| In this election scenario, $k$-Approval for $k=1,2,3$ cannot guarantee that the Condorcet winner $A$ is elected. The Approval ballot $(\{A\},\{B\}, \{A, C\})$ does elect the Condorcet winner. In fact, Brams (2008, Chapter 2) proves that if there is a unique Condorcet winner, then that candidate may be elected under approval voting (assuming that all voters vote sincerely: see Brams 2008, Chapter 2, for a discussion). Note that approval voting may also elect other candidates (perhaps even the Condorcet loser). Whether this flexibility of Approval Voting should be seen as a virtue or a vice is debated in Brams, Fishburn and Merrill 1988a, 1988b and Saari and van Newenhizen 1988a, 1988b. Approval Voting asks voters to express something about their intensity of preference for the candidates by assigning one of two grades: "Approve" or "Don't Approve". Expanding on this idea, some voting methods assume that there is a fixed set of grades, or a grading language, that voters can assign to each candidate. See Chapters 7 and 8 from Balinksi and Laraki 2010 for examples and a discussion of grading languages (cf. Morreau 2016). There are different ways to determine the winner(s) given a profile of ballots that assign grades to each candidate. The main approach is to calculate a "group" grade for each candidate, then select the candidate with the best overall group grade. In order to calculate a group grade for each candidate, it is convenient to use numbers for the grading language. Then, there are two natural ways to determine the group grade for a candidate: calculating the mean, or average, of the grades or calculating the median of the grades. Cumulative Voting: Each voter is asked to distribute a fixed number of points, say ten, among the candidates in any way they please. The candidate(s) with the most total points wins the election. Score Voting (also called Range Voting): The grades are a finite set of numbers. The ballots are an assignment of grades to the candidates. The candidate(s) with the largest average grade is declared the winner(s). Cumulative Voting and Score Voting are similar. The important difference is that Cumulative Voting requires that the sum of the grades assigned to the candidates by each voter is the same. The next procedure, proposed by Balinski and Laraki 2010 (cf. Bassett and Persky 1999 and the discussion of this method at rangevoting.org), selects the candidate(s) with the largest median grade rather than the largest mean grade. Majority Judgement: The grades are a finite set of numbers (cf. discussion of common grading languages). The ballots are an assignment of grades to the candidates. The candidate(s) with the largest median grade is(are) declared the winner(s). See Balinski and Laraki 2007 and 2010 for further refinements of this voting method that use different methods for breaking ties when there are multiple candidates with the largest median grade. I conclude this section with an example that illustrates Score Voting and Majority Judgement. Suppose that there are 3 candidates $\{A, B, C\}$, 5 grades $\{0,1,2,3,4\}$ (with the assumption that the larger the number, the higher the grade), and 5 voters. The table below describes an election scenario. The candidates are listed in the first row. Each row describes an assignment of grades to a candidate by a set of voters. \| \| \| \| --- \| --- \| \| \| Grade (0-4) for: \| \| # Voters \| $A$ \| $B$ \| $C$ \| \| 1 \| 4 \| 3 \| 1 \| \| 1 \| 4 \| 3 \| 2 \| \| 1 \| 2 \| 0 \| 3 \| \| 1 \| 2 \| 3 \| 4 \| \| 1 \| 1 \| 0 \| 2 \| \| Mean: \| 2.6 \| 1.8 \| 2.4 \| \| Median: \| 2 \| 3 \| 2 \| The bottom two rows give the mean and median grade for each candidate. Candidate $A$ is the score voting winner with the greatest mean grade, and candidate $B$ is the majority judgement winner with the greatest median grade. There are two types of debates about the voting methods introduced in this section. The first concerns the choice of the grading language that voters use to evaluate the candidates. Consult Balinski and Laraki 2010 amd Morreau 2016 for an extensive discussion of the types of considerations that influence the choice of a grading language. Brams and Potthoff 2015 argue that two grades, as in Approval Voting, is best to avoid certain paradoxical outcomes. To illustrate, note that, in the above example, if the candidates are ranked by the voters according to the grades that are assigned, then candidate $C$ is the Condorcet winner (since 3 voters assign higher grades to $C$ than to $A$ or $B$). However, neither Score Voting nor Majority Judgement selects candidate $C$. The second type of debate concerns the method used to calculate the group grade for each candidate (i.e., whether to use the mean as in Score Voting or the median as in Majority Judgement). One important issue is whether voters have an incentive to misrepresent their evaluations of the candidates. Consider the voter in the middle column that assigns the grade of 2 to $A$, 0 to $B$, and 3 to $C$. Suppose that these grades represents the voter's true evaluations of the candidates. If this voter increases the grade for $C$ to 4 and decreases the grade for $A$ to 1 (and the other voters do not change their grades), then the average grade for $A$ becomes 2.4 and the average grade for $C$ becomes 2.6, which better reflects the voter's true evaluations of the candidates (and results in $C$ being elected according to Score Voting). Thus, this voter has an incentive to misrepresent her grades. Note that the median grades for the candidates do not change after this voter changes her grades. Indeed, Balinski and Laraki 2010, chapter 10, argue that using the median to assign group grades to candidates encourages voters to submit grades that reflect their true evaluations of the candidates. The key idea of their argument is as follows: If a voter's true grade matches the median grade for a candidate, then the voter does not have an incentive to assign a different grade. If a voter's true grade is greater than the median grade for a candidate, then raising the grade will not change the candidate's grade and lowering the voter's grade may result in the candidate receiving a grade that is lowering than the voter's true evaluation. Similarly, if a voter's true grade is lower than the median grade for a candidate, then lowering the grade will not change the candidate's grade and raising the voter's grade may result in the candidate receiving a grade that is higher than the voter's true evaluation. Thus, if voters are focused on ensuring that the group grades for the candidates best reflects their true evaluations of the candidates, then voters do not have an incentive to misrepresent their grades. However, as pointed out in Felsenthal and Machover 2008 (Example 3.3), voters can manipulate the outcome of an election using Majority Judgement to ensure a preferred candidate is elected (cf. the discussion of strategic voting in Section 4.1 and Section 3.3 of List 2013). Suppose that the voter in the middle column assigns the grade of 4 to candidate $A$, 0 to candidate $B$ and 3 to candidate $C$. Assuming the other voters do not change their grades, the majority judgement winner is now $A$, which the voter ranks higher than the original majority judgement winner $B.$ Consult Balinski and Laraki 2010, 2014 and Edelman 2012b for arguments in favor of electing candidates with the greatest median grade; and Felsenthal and Machover 2008, Gehrlein and Lepelley 2003, and Laslier 2011 for arguments against electing candidates with the greatest median grade. ### 2.3 Quadratic Voting and Liquid Democracy In this section, I briefly discuss two new approaches to voting that do not fit nicely into the categories of voting methods introduced in the previous sections. While both of these methods can be used to select representatives, such as a president, the primary application is a group of people voting directly on propositions, or referendums. Quadratic Voting: When more than 50% of the voters support an alternative, most voting methods will select that alternative. Indeed, when there are only two alternatives, such as when voting for or against a proposition, there are many arguments that identify majority rule as the best and most stable group decision method (May 1952; Maskin 1995). One well-known problem with always selecting the majority winner is the so-called tyranny of the majority. A complete discussion of this issue is beyond the scope of this article. The main problem from the point of view of the analysis of voting methods is that there may be situations in which a majority of the voters weakly support a proposition while there is a sizable minority of voters that have a strong preference against the proposition. One way of dealing with this problem is to increase the quota required to accept a proposition. However, this gives too much power to a small group of voters. For instance, with Unanimity Rule a single voter can block a proposal from being accepted. Arguably, a better solution is to use ballots that allow voters to express something about their intensity of preference for the alternatives. Setting aside issues about interpersonal comparisons of utility (see, for instance, Hausman 1995), this is the benefit of using the voting methods discussed in Section 2.2, such as Score Voting or Majority Judgement. These voting methods assume that there is a fixed set of grades that the voters use to express their intensity of preference. One challenge is finding an appropriate set of grades for a population of voters. Too few grades makes it harder for a sizable minority with strong preferences to override the majority opinion, but too many grades makes it easy for a vocal minority to overrule the majority opinion. Using ideas from mechanism design (Groves and Ledyard 1977; Hylland and Zeckhauser 1980), the economist E. Glen Weyl developed a voting method called Quadratic Voting that mitigates some of the above issues (Lalley and Weyl 2018a). The idea is to think of an election as a market (Posner and Weyl, 2018, Chapter 2). Each voter can purchase votes at a costs that is quadratic in the number of votes. For instance, a voter must pay $25 for 5 votes (either in favor or against a proposition). After the election, the money collected is distributed on a pro rata basis to the voters. There are a variety of economic arguments that justify why voters should pay $v^2$ to purchase $v$ votes (Lalley and Weyl 2018b; Goeree and Zhang 2017). See Posner and Weyl 2015 and 2017 for further discussion and a vigorous defense of the use of Quadratic Voting in national elections. Consult Laurence and Sher 2017 for two arguments against the use of Quadratic Voting. Both arguments are derived from the presence of wealth inequality. The first argument is that it is ambiguous whether the Quadratic Voting decision really outperforms a decision using majority rule from the perspective of utilitarianism (see Driver 2014 and Sinnott-Armstrong 2019 for overviews of utilitarianism). The second argument is that any vote-buying mechanism will have a hard time meeting a legitimacy requirement, familiar from the theory of democratic institutions (cf. Fabienne 2017). Liquid Democracy: Using Quadratic Voting, the voters' opinions may end up being weighted differently: Voters that purchase more of a voice have more influence over the election. There are other reasons why some voters' opinions may have more weight than others when making a decision about some issue. For instance, a voter may have been elected to represent a constituency, or a voter may be recognized as an expert on the issue under consideration. An alternative approach to group decision making is direct democracy in which every citizen is asked to vote on every political issue. Asking the citizens to vote on every issue faces a number of challenges, nicely explained by Green-Armytage (2015, pg. 191): > > Direct democracy without any option for representation is problematic. > Even if it were possible for every citizen to learn everything they > could possibly know about every political issue, people who did this > would be able to do little else, and massive amounts of time would be > wasted in duplicated effort. Or, if every citizen voted but most > people did not take the time to learn about the issues, the results > would be highly random and/or highly sensitive to overly simplistic > public relations campaigns. Or, if only a few citizens voted, > particular demographic and ideological groups would likely be > under-represented > One way to deal with some of the problems raised in the above quote is to use proxy voting, in which voters can delegate their vote on some issues (Miller 1969). Liquid Democracy is a form of proxy voting in which voters can delegate their votes to other voters (ideally, to voters that are well-informed about the issue under consideration). What distinguishes Liquid Democracy from proxy voting is that proxies may further delegate the votes entrusted to them. For example, suppose that there is a vote to accept or reject a proposition. Each voter is given the option to delegate their vote to another voter, called a proxy. The proxies, in turn, are given the option to delegate their votes to yet another voter. The voters that decide to not transfer their votes cast a vote weighted by the number of voters who entrusted them as a proxy, either directly or indirectly. While there has been some discussion of proxy voting in the political science literature (Miller 1969; Alger 2006; Green-Armytage 2015), most studies of Liquid Democracy can be found in the computer science literature. A notable exception is Blum and Zuber 2016 that justifies Liquid Democracy, understood as a procedure for democratic decision-making, within normative democratic theory. An overview of the origins of Liquid Democracy and pointers to other online discussions can be found in Behrens 2017. Formal studies of Liquid Democracy have focused on: the possibility of delegation cycles and the relationship with the theory of judgement aggregation (Christoff and Grossi 2017); the rationality of delegating votes (Bloembergen, Grossi and Lackner 2018); the potential problems that arise when many voters delegate votes to only a few voters (Kang et al. 2018; Golz et al. 2018); and generalizations of Liquid Democracy beyond binary choices (Brill and Talmon 2018; Zhang and Zhou 2017). ### 2.4 Criteria for Comparing Voting Methods This section introduced different methods for making a group decision. One striking fact about the voting methods discussed in this section is that they can identify different winners given the same collection of ballots. This raises an important question: How should we compare the different voting methods? Can we argue that some voting methods are better than others? There are a number of different criteria that can be used to compare and contrast different voting methods: 1. Pragmatic concerns: Is the procedure easy to use? Is it legal to use a particular voting method for a national or local election? The importance of "ease of use" should not be underestimated: Despite its many flaws, plurality rule (arguably the simplest voting procedure to use and understand) is, by far, the most commonly used method (cf. the discussion by Levin and Nalebuff 1995, p. 19). Furthermore, there are a variety of consideration that go into selecting an appropriate voting method for an institution (Edelman 2012a). 2. Behavioral considerations: Do the different procedures really lead to different outcomes in practice? An interesting strand of research, behavorial social choice, incorporates empirical data about actual elections into the general theory of voting (This is discussed briefly in Section 5. See Regenwetter et al. 2006, for an extensive discussion). 3. Information required from the voters: What type of information do the ballots convey? While ranking methods (e.g., Borda Count) require the voter to compare all of the candidates, it is often useful to ask the voters to report something about the "intensities" of their preferences over the candidates. Of course, there is a trade-off: Limiting what voters can express about their opinions of the candidates often makes a procedure much easier to use and understand. Also related to these issues is the work of Brennan and Lomasky 1993 (among others) on expressive voting (cf. Wodak 2019 and Aragones et al. 2011 for analyses along these lines touching on issues raised in this article). 4. Axiomatic characterization results and voting paradoxes: Much of the work in voting theory has focused on comparing and contrasting voting procedures in terms of abstract principles that they satisfy. The goal is to characterize the different voting procedures in terms of normative principles of group decision making. See Sections 3 and 4.2 for discussions. ## 3. Voting Paradoxes In this section, I introduce and discuss a number of voting paradoxes -- i.e., anomalies that highlight problems with different voting methods. Consult Saari 1995 and Nurmi 1999 for penetrating analyses that explain the underlying mathematics behind the different voting paradoxes. ### 3.1 Condorcet's Paradox A very common assumption is that a rational preference ordering must be transitive (i.e., if $A$ is preferred to $B$, and $B$ is preferred to $C$, then $A$ must be preferred to $C$). See the entry on preferences (Hansson and Grune-Yanoff 2009) for an extended discussion of the rationale behind this assumption. Indeed, if a voter's preference ordering is not transitive, for instance, allowing for cycles (e.g., an ordering of $A, B, C$ with $A \succ B \succ C \succ A$, where $X\succ Y$ means $X$ is strictly preferred to $Y$), then there is no alternative that the voter can be said to actually support (for each alternative, there is another alternative that the voter strictly prefers). Many authors argue that voters with cyclic preference orderings have inconsistent opinions about the candidates and should be ignored by a voting method (in particular, Condorcet forcefully argued this point). A key observation of Condorcet (which has become known as the Condorcet Paradox) is that the majority ordering may have cycles (even when all the voters submit rankings of the alternatives). Condorcet's original example was more complicated, but the following situation with three voters and three candidates illustrates the phenomenon: \| \| \| \| --- \| --- \| \| # Voters \| Ranking \| \| 1 \| $A\s B\s C$ \| \| 1 \| $B\s C\s A$ \| \| 1 \| $C\s A\s B$ \| Note that we have: * Candidate $A$ beats candidate $B$ in a one-on-one election: 2 voters rank $A$ above $B$ compared to 1 voter ranking $B$ above $A$. * Candidate $B$ beats candidate $C$ in a one-on-one election: 2 voters rank $B$ above $C$ compared to 1 voter ranking $C$ above $B$. * Candidate $C$ beats candidate $A$ in a one-on-one election: 2 voters rank $C$ above $A$ compared to 1 voter ranking $A$ above $C$. That is, there is a majority cycle $A>\_M B >\_M C >\_M A$. This means that there is no Condorcet winner. This simple, but fundamental observation has been extensively studied (Gehrlein 2006; Schwartz 2018). #### 3.1.1 Electing the Condorcet Winner The Condorcet Paradox shows that there may not always be a Condorcet winner in an election. However, one natural requirement for a voting method is that if there is a Condorcet winner, then that candidate should be elected. Voting methods that satisfy this property are called Condorcet consistent. Many of the methods introduced above are not Condorcet consistent. I already presented an example showing that plurality rule is not Condorcet consistent (in fact, plurality rule may even elect the Condorcet loser). The example from Section 1 shows that Borda Count is not Condorcet consistent. In fact, this is an instance of a general phenomenon that Fishburn (1974) called Condorcet's other paradox. Consider the following voting situation with 81 voters and three candidates from Condorcet 1785. \| \| \| \| --- \| --- \| \| # Voters \| Ranking \| \| 30 \| $A\s B\s C$ \| \| 1 \| $A\s C\s B$ \| \| 29 \| $B\s A\s C$ \| \| 10 \| $B\s C\s A$ \| \| 10 \| $C\s A\s B$ \| \| 1 \| $C\s B\s A$ \| The majority ordering is $A >\_M B >\_M C$, so $A$ is the Condorcet winner. Using the Borda rule, we have: \[\begin{align} \BS(A) &= 2\times 31 + 1 \times 39 + 0 \times 11 = 101 \\ \BS(B) &= 2 \times 39 + 1 \times 31 + 0 \times 11 = 109 \\ \BS(C) &= 2 \times 11 + 1 \times 11 + 0 \times 59 = 33 \end{align}\] So, candidate $B$ is the Borda winner. Condorcet pointed out something more: The only way to elect candidate $A$ using any scoring method is to assign more points to candidates ranked second than to candidates ranked first. Recall that a scoring method for 3 candidates fixes weights $s\_1\ge s\_2\ge s\_3$, where $s\_1$ points are assigned to candidates ranked 1st, $s\_2$ points are assigned to candidates ranked 2nd, and $s\_3$ points are assigned to candidates ranked last. To simplify the calculation, assume that candidates ranked last receive 0 points (i.e., $s\_3=0$). Then, the scores assigned to candidates $A$ and $B$ are: \[\begin{align} Score(A) &= s\_1 \times 31 + s\_2 \times 39 + 0 \times 11 \\ Score(B) &= s\_1 \times 39 + s\_2 \times 31 + 0 \times 11 \end{align}\] So, in order for $Score(A) > Score(B)$, we must have $(s\_1 \times 31 + s\_2 \times 39) > (s\_1 \times 39 + s\_2 \times 31)$, which implies that $s\_2 > s\_1$. But, of course, it is counterintuitive to give more points for being ranked second than for being ranked first. Peter Fishburn generalized this example as follows: Theorem (Fishburn 1974). For all $m\ge 3$, there is some voting situation with a Condorcet winner such that every scoring rule will have at least $m-2$ candidates with a greater score than the Condorcet winner. So, no scoring rule is Condorcet consistent, but what about other methods? A number of voting methods were devised specifically to guarantee that a Condorcet winner will be elected, if one exists. The examples below give a flavor of different types of Condorcet consistent methods. (See Brams and Fishburn, 2002, and Fishburn, 1977, for more examples and a discussion of Condorcet consistent methods.) Condorcet's Rule: Each voter submits a ranking of the candidates. If there is a Condorcet winner, then that candidate wins the election. Otherwise, all candidates tie for the win. Black's Procedure: Each voter submits a ranking of the candidates. If there is a Condorcet winner, then that candidate is the winner. Otherwise, use Borda Count to determine the winners. Nanson's Method: Each voter submits a ranking of the candidates. Calculate the Borda score for each candidate. The candidates with a Borda score below the average of the Borda scores are eliminated. The Borda scores of the candidates are re-calculated and the process continues until there is only one candidate remaining. (See Niou, 1987, for a discussion of this voting method.) Copeland's Rule: Each voter submits a ranking of the candidates. A win-loss record for candidate $B$ is calculated as follows: \[ WL(B) = \#\{C\ \mid\ B >\_M C\} - \#\{C\ \mid\ C >\_M B\} \] The Copeland winner is the candidate that maximizes the win-loss record. Schwartz's Set Method: Each voter submits a ranking of the candidates. The winners are the smallest set of candidates that are not beaten in a one-on-one election by any candidate outside the set (Schwartz 1986). Dodgson's Method: Each voter submits a ranking of the candidates. For each candidate, determine the fewest number of pairwise swaps in the voters' rankings needed to make that candidate the Condorcet winner. The candidate(s) with the fewest swaps is(are) declared the winner(s). The last method was proposed by Charles Dodgson (better known by the pseudonym Lewis Carroll). Interestingly, this is an example of a procedure in which it is computationally difficult to compute the winner (that is, the problem of calculating the winner is NP-complete). See Bartholdi et al. 1989 for a discussion. These voting methods (and the other Condorcet consistent methods) guarantee that a Condorcet winner, if one exists, will be elected. But, should a Condorcet winner be elected? Many people argue that there is something amiss with a voting method that does not always elect a Condorcet winner (if one exists). The idea is that a Condorcet winner best reflects the overall group opinion and is stable in the sense that it will defeat any challenger in a one-on-one contest using Majority Rule. The most persuasive argument that the Condorcet winner should not always be elected comes from the work of Donald Saari (1995, 2001). Consider again Condorcet's example of 81 voters. \| \| \| \| --- \| --- \| \| # Voters \| Ranking \| \| 30 \| $A\s B\s C$ \| \| 1 \| $A\s C\s B$ \| \| 29 \| $B\s A\s C$ \| \| 10 \| $B\s C\s A$ \| \| 10 \| $C\s A\s B$ \| \| 1 \| $C\s B\s A$ \| This is another example that shows that Borda's method need not elect the Condorcet winner. The majority ordering is \[ A >\_M B >\_M C, \] while the ranking given by the Borda score is \[ B >\_{\Borda} A >\_{\Borda} C. \] However, there is an argument that candidate $B$ is the best choice for this electorate. Saari's central observation is to note that the 81 voters can be divided into three groups: \| \| \| \| \| --- \| --- \| --- \| \| \| # Voters \| Ranking \| \| \| 10 \| $A\s B\s C$ \| \| Group 1 \| 10 \| $B\s C\s A$ \| \| \| 10 \| $C\s A\s B$ \| \| \| 1 \| $A\s C\s B$ \| \| Group 2 \| 1 \| $C\s B\s A$ \| \| \| 1 \| $B\s A\s C$ \| \| Group 3 \| 20 \| $A\s B\s C$ \| \| 28 \| $B\s A\s C$ \| Groups 1 and 2 constitute majority cycles with the voters evenly distributed among the three possible rankings. Such profiles are called Condorcet components. These profiles form a perfect symmetry among the rankings. So, within each of these groups, it is natural to assume that the voters' opinions cancel each other out; therefore, the decision should depend only on the voters in group 3. In group 3, candidate $B$ is the clear winner. Balinski and Laraki (2010, pgs. 74-83) have an interesting spin on Saari's argument. Let $V$ be a ranking voting method (i.e., a voting method that requires voters to rank the alternatives). Say that $V$ cancels properly if for all profiles $\bR$, if $V$ selects $A$ as a winner in $\bP$, then $V$ selects $A$ as a winner in any profile $\bP+\bC$, where $\bC$ is a Condorcet component and $\bP+\bC$ is the profile that contains all the rankings from $\bP$ and $\bC$. Balinski and Laraki (2010, pg. 77) prove that there is no Condorcet consistent voting method that cancels properly. (See the discussion of the multiple districts paradox in Section 3.3 for a proof of a closely related result.) ### 3.2 Failures of Monotonicity A voting method is monotonic provided that receiving more support from the voters is always better for a candidate. There are different ways to make this idea precise (see Fishburn, 1982, Sanver and Zwicker, 2012, and Felsenthal and Tideman, 2013). For instance, moving up in the rankings should not adversely affect a candidate's chances to win an election. It is easy to see that Plurality Rule is monotonic in this sense: The more voters that rank a candidate first, the better chance the candidate has to win. Surprisingly, there are voting methods that do not satisfy this natural property. The most well-known example is Plurality with Runoff. Consider the two scenarios below. Note that the only difference between the them is the ranking of the fourth group of voters. This group of two voters ranks $B$ above $A$ above $C$ in scenario 1 and swaps $B$ and $A$ in scenario 2 (so, $A$ is now their top-ranked candidate; $B$ is ranked second; and $C$ is still ranked third). \| \| \| \| \| --- \| --- \| --- \| \| # Voters \| Scenario 1Ranking \| Scenario 2Ranking \| \| 6 \| $A\s B\s C$ \| $A\s B\s C$ \| \| 5 \| $C\s A\s B$ \| $C\s A\s B$ \| \| 4 \| $B\s C\s A$ \| $B\s C\s A$ \| \| 2 \| $\bB\s \bA\s C$ \| $\bA\s \bB\s C$ \| \| Scenario 1: Candidate $A$ is the Plurality with Runoff winner \| \| Scenario 2: Candidate $C$ is the Plurality with Runoff winner \| In scenario 1, candidates $A$ and $B$ both have a plurality score of 6 while candidate $C$ has a plurality score of 5. So, $A$ and $B$ move on to the runoff election. Assuming the voters do not change their rankings, the 5 voters that rank $C$ transfer their support to candidate $A$, giving her a total of 11 to win the runoff election. However, in scenario 2, even after moving up in the rankings of the fourth group ($A$ is now ranked first by this group), candidate $A$ does not win this election. In fact, by trying to give more support to the winner of the election in scenario 1, rather than solidifying $A$'s win, the last group's least-preferred candidate ended up winning the election! The problem arises because in scenario 2, candidates $A$ and $B$ are swapped in the last group's ranking. This means that $A$'s plurality score increases by 2 and $B$'s plurality score decreases by 2. As a consequence, $A$ and $C$ move on to the runoff election rather than $A$ and $B$. Candidate $C$ wins the runoff election with 9 voters that rank $C$ above $A$ compared to 8 voters that rank $A$ above $C$. The above example is surprising since it shows that, when using Plurality with Runoff, it may not always be beneficial for a candidate to move up in some of the voter's rankings. The other voting methods that violate monotonicity include Coombs Rule, Hare Rule, Dodgson's Method and Nanson's Method. See Felsenthal and Nurmi 2017 for further discussion of voting methods that are not monotonic. ### 3.3 Variable Population Paradoxes In this section, I discuss two related paradoxes that involve changes to the population of voters. No-Show Paradox: One way that a candidate may receive "more support" is to have more voters show up to an election that support them. Voting methods that do not satisfy this version of monotonicity are said to be susceptible to the no-show paradox (Fishburn and Brams 1983). Suppose that there are 3 candidates and 11 voters with the following rankings: \| \| \| \| --- \| --- \| \| # Voters \| Ranking \| \| 4 \| $A\s B\s C$ \| \| 3 \| $B\s C\s A$ \| \| 1 \| $C\s A\s B$ \| \| 3 \| $C\s B\s A$ \| \| Candidate $C$ is the Plurality with Runoff winner \| In the first round, candidates $A$ and $C$ are both ranked first by 4 voters while $B$ is ranked first by only 3 voters. So, $A$ and $C$ move to the runoff round. In this round, the voters in the second column transfer their votes to candidate $C$, so candidate $C$ is the winner beating $A$ 7-4. Suppose that 2 voters in the first group do not show up to the election: \| \| \| \| --- \| --- \| \| # Voters \| Ranking \| \| $\mathbf{2}$ \| $A\s B\s C$ \| \| 3 \| $B\s C\s A$ \| \| 1 \| $C\s A\s B$ \| \| 3 \| $C\s B\s A$ \| \| Candidate $B$ is the Plurality with Runoff winner \| In this election, candidate $A$ has the lowest plurality score in the first round, so candidates $B$ and $C$ move to the runoff round. The first group's votes are transferred to $B$, so $B$ is the winner beating $C$ 5-4. Since the 2 voters that did not show up to this election rank $B$ above $C$, they prefer the outcome of the second election in which they did not participate! Plurality with Runoff is not the only voting method that is susceptible to the no-show paradox. The Coombs Rule, Hare Rule and Majority Judgement (using the tie-breaking mechanism from Balinski and Laraki 2010) are all susceptible to the no-show paradox. It turns out that always electing a Condorcet winner, if one exists, makes a voting method susceptible to the above failure of monotonicity. Theorem (Moulin 1988). If there are four or more candidates, then every Condorcet consistent voting method is susceptible to the no-show paradox. See Perez 2001, Campbell and Kelly 2002, Jimeno et al. 2009, Duddy 2014, Brandt et al. 2017, 2019, and Nunez and Sanver 2017 for further discussions and generalizations of this result. Multiple Districts Paradox: Suppose that a population is divided into districts. If a candidate wins each of the districts, one would expect that candidate to win the election over the entire population of voters (assuming that the two districts divide the set of voters into disjoint sets). This is certainly true for Plurality Rule: If a candidate is ranked first by the most voters in each of the districts, then that candidate will also be ranked first by a the most voters over the entire population. Interestingly, this is not true for all voting methods (Fishburn and Brams 1983). The example below illustrates the paradox for Coombs Rule. \| \| \| \| \| --- \| --- \| --- \| \| \| # Voters \| Ranking \| \| District 1 \| 3 \| $A\s B\s C$ \| \| 3 \| $B\s C\s A$ \| \| 3 \| $C\s A\s B$ \| \| 1 \| $C\s B\s A$ \| \| District 2 \| 2 \| $A\s B\s C$ \| \| 3 \| $B\s A\s C$ \| \| District 1: Candidate $B$ is the Coombs winner \| \| District 2: Candidate $B$ is the Coombs winner \| Candidate $B$ wins both districts: District 1: There are a total of 10 voters in this district. None of the candidates are ranked first by 6 or more voters, so candidate $A$, who is ranked last by 4 voters (compared to 3 voters ranking each of $C$ and $B$ last), is eliminated. In the second round, candidate $B$ wins the election since 6 voters rank $B$ above $C$ and 4 voters rank $C$ above $B$. District 2: There are a total of 5 voters in this district. Candidate $B$ is ranked first by a strict majority of voters, so $B$ wins the election. Combining the two districts gives the following table: \| \| \| \| \| --- \| --- \| --- \| \| \| # Voters \| Ranking \| \| Districts 1 + 2 \| 5 \| $A\s B\s C$ \| \| 3 \| $B\s C\s A$ \| \| 3 \| $C\s A\s B$ \| \| 1 \| $C\s B\s A$ \| \| 3 \| $B\s A\s C$ \| \| Candidate $A$ is the Coombs winner \| There are 15 total voters in the combined districts. None of the candidates are ranked first by 8 or more of the voters. Candidate $C$ receives the most last-place votes, so is eliminated in the first round. In the second round, candidate $A$ is beats candidate $B$ by 1 vote (8 voters rank $A$ above $B$ and 7 voters rank $B$ above $A$), and so is declared the winner. Thus, even though $B$ wins both districts, candidate $A$ wins the election when the districts are combined. The other voting methods that are susceptible to the multiple-districts paradox include Plurality with Runoff, The Hare Rule, and Majority Judgement. Note that these methods are also susceptible to the no-show paradox. As is the case with the no-show paradox, every Condorcet consistent voting method is susceptible to the multiple districts paradox (see Zwicker, 2016, Proposition 2.5). I sketch the proof of this from Zwicker 2016 (pg. 40) since it adds to the discussion at the end of Section 3.1 about whether the Condorcet winner should be elected. Suppose that $V$ is a voting method that always selects the Condorcet winner (if one exists) and that $V$ is not susceptible to the multiple-districts paradox. This means that if a candidate $X$ is among the winners according to $V$ in each of two districts, then $X$ must be among the winners according to $V$ in the combined districts. Consider the following two districts. \| \| \| \| \| --- \| --- \| --- \| \| \| # Voters \| Ranking \| \| District 1 \| 2 \| $A\s B\s C$ \| \| 2 \| $B\s C\s A$ \| \| 2 \| $C\s A\s B$ \| \| District 2 \| 1 \| $A\s B\s C$ \| \| 2 \| $B\s A\s C$ \| Note that in district 2 candidate $B$ is the Condorcet winner, so must be the only winner according to $V$. In district 1, there are no Condorcet winners. If candidate $B$ is among the winners according to $V$, then, in order to not be susceptible to the multiple districts paradox, $B$ must be among the winners in the combined districts. In fact, since $B$ is the only winner in district 2, $B$ must be the only winner in the combined districts. However, in the combined districts, candidate $A$ is the Condorcet winner, so must be the (unique) winner according to $V$. This is a contradiction, so $B$ cannot be among the winners according to $V$ in district 1. A similar argument shows that neither $A$ nor $C$ can be among the winners according to $V$ in district 1 by swapping $A$ and $B$ in the first case and $B$ with $C$ in the second case in the rankings of the voters in district 2. Since $V$ must assign at least one winner to every profile, this is a contradiction; and so, $V$ is susceptible to the multiple districts paradox. One last comment about this paradox: It is an example of a more general phenomenon known as Simpson's Paradox (Malinas and Bigelow 2009). See Saari (2001, Section 4.2) for a discussion of Simpson's Paradox in the context of voting theory. ### 3.4 The Multiple Elections Paradox The paradox discussed in this section, first introduced by Brams, Kilgour and Zwicker (1998), has a somewhat different structure from the paradoxes discussed above. Voters are taking part in a referendum, where they are asked their opinion directly about various propositions (cf. the discussion of Quadratic Voting and Liquid Democracy in Section 2.3). So, voters must select either "yes" (Y) or "no" (N) for each proposition. Suppose that there are 13 voters who cast the following votes for the three propositions (so voters can cast one of eight possible votes): \| \| \| \| --- \| --- \| \| # Voters \| Propositions \| \| 1 \| YYY \| \| 1 \| YYN \| \| 1 \| YNY \| \| 3 \| YNN \| \| 1 \| NYY \| \| 3 \| NYN \| \| 3 \| NNY \| \| 0 \| NNN \| When the votes are tallied for each proposition separately, the outcome is N for each proposition (N wins 7-6 for all three propositions). Putting this information together, this means that NNN is the outcome of this election. However, there is no support for this outcome in this population of voters. This raises an important question about what outcome reflects the group opinion: Viewing each proposition separately, there is clear support for N on each proposition; however, there is no support for the entire package of N for all propositions. Brams et al. (1998, pg. 234) nicely summarise the issue as follows: > > The paradox does not just highlight problems of aggregation and > packaging, however, but strikes at the core of social > choice--both what it means and how to uncover it. In our view, > the paradox shows there may be a clash between two different meanings > of social choice, leaving unsettled the best way to uncover what this > elusive quantity is. > See Scarsini 1998, Lacy and Niou 2000, Xia et al. 2007, and Lang and Xia 2009 for further discussion of this paradox. A similar issue is raised by Anscombe's paradox (Anscombe 1976), in which: > > It is possible for a majority of voters to be on the losing side of a > majority of issues. > This phenomenon is illustrated by the following example with five voters voting on three different issues (the voters either vote 'yes' or 'no' on the different issues). \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| Issue 1 \| Issue 2 \| Issue 3 \| \| Voter 1 \| yes \| yes \| no \| \| Voter 2 \| no \| no \| no \| \| Voter 3 \| no \| yes \| yes \| \| Voter 4 \| yes \| no \| yes \| \| Voter 5 \| yes \| no \| yes \| \| Majority: \| yes \| no \| yes \| However, a majority of the voters (voters 1, 2 and 3) do not support the majority outcome on a majority of the issues (note that voter 1 does not support the majority outcome on issues 2 and 3; voter 2 does not support the majority outcome on issues 1 and 3; and voter 3 does not support the majority outcome on issues 1 and 2)! The issue is more interesting when the voters do not vote directly on the issues, but on candidates that take positions on the different issues. Suppose there are two candidates $A$ and $B$ who take the following positions on the three issues: \| \| \| \| \| \| --- \| --- \| --- \| --- \| \| \| Issue 1 \| Issue 2 \| Issue 3 \| \| Candidate $A$ \| yes \| no \| yes \| \| Candidate $B$ \| no \| yes \| no \| Candidate $A$ takes the majority position, agreeing with a majority of the voters on each issue, and candidate $B$ takes the opposite, minority position. Under the natural assumption that voters will vote for the candidate who agrees with their position on a majority of the issues, candidate $B$ will win the election (each of the voters 1, 2 and 3 agree with $B$ on two of the three issues, so $B$ wins the election 3-2)! This version of the paradox is known as Ostrogorski's Paradox (Ostrogorski 1902). See Kelly 1989; Rae and Daudt 1976; Wagner 1983, 1984; and Saari 2001, Section 4.6, for analyses of this paradox, and Pigozzi 2005 for the relationship with the judgement aggregation literature (List 2013, Section 5). ## 4. Topics in Voting Theory ### 4.1 Strategizing In the discussion above, I have assumed that voters select ballots sincerely. That is, the voters are simply trying to communicate their opinions about the candidates under the constraints of the chosen voting method. However, in many contexts, it makes sense to assume that voters choose strategically. One need only look to recent U.S. elections to see concrete examples of strategic voting. The most often cited example is the 2000 U.S. election: Many voters who ranked third-party candidate Ralph Nader first voted for their second choice (typically Al Gore). A detailed overview of the literature on strategic voting is beyond the scope of this article (see Taylor 2005 and Section 3.3 of List 2013 for discussions and pointers to the relevant literature; also see Poundstone 2008 for an entertaining and informative discussion of the occurrence of this phenomenon in many actual elections). I will explain the main issues, focusing on specific voting rules. There are two general types of manipulation that can be studied in the context of voting. The first is manipulation by a moderator or outside party that has the authority to set the agenda or select the voting method that will be used. So, the outcome of an election is not manipulated from within by unhappy voters, but, rather, it is controlled by an outside authority figure. To illustrate this type of control, consider a population with three voters whose rankings of four candidates are given in the table below: \| \| \| \| --- \| --- \| \| # Voters \| Ranking \| \| 1 \| $B\s D\s C\s A$ \| \| 1 \| $A\s B\s D\s C$ \| \| 1 \| $C\s A\s B\s D$ \| Note that everyone prefers candidate $B$ over candidate $D$. Nonetheless, a moderator can ask the right questions so that candidate $D$ ends up being elected. The moderator proceeds as follows: First, ask the voters if they prefer candidate $A$ or candidate $B$. Since the voters prefer $A$ to $B$ by a margin of 2 to 1, the moderator declares that candidate $B$ is no longer in the running. The moderator then asks voters to choose between candidate $A$ and candidate $C$. Candidate $C$ wins this election 2-1, so candidate $A$ is removed. Finally, in the last round the chairman asks voters to choose between candidates $C$ and $D$. Candidate $D$ wins this election 2-1 and is declared the winner. A second type of manipulation focuses on how the voters themselves can manipulate the outcome of an election by misrepresenting their preferences. Consider the following two election scenarios with 7 voters and 3 candidates: \| \| \| \| \| --- \| --- \| --- \| \| # Voters \| Scenario 1Ranking \| Scenario 2Ranking \| \| 1 \| $C\s D\s B\s A$ \| $C\s D\s B\s A$ \| \| 1 \| $B\s A\s C\s D$ \| $B\s A\s C\s D$ \| \| 1 \| $A\s \bC\s \bB\s \bD$ \| $A\s \bB\s \bD\s \bC$ \| \| 1 \| $A\s C\s D\s B$ \| $A\s C\s D\s B$ \| \| 1 \| $D\s C\s A\s B$ \| $D\s C\s A\s B$ \| \| Scenario 1: Candidate $C$ is the Borda winner ($\BS(A)=9, \BS(B)=5, \BS(C)=10$, and $\BS(D)=6$) \| \| Scenario 2: Candidate $A$ is the Borda winner ($\BS(A)=9, \BS(B)=6, \BS(C)=8$, and $\BS(D)=7$) \| The only difference between the two election scenarios is that the third voter changed the ranking of the bottom three candidates. In election scenario 1, the third voter has candidate $A$ ranked first, then $C$ ranked second, $B$ ranked third and $D$ ranked last. In election scenario 2, this voter still has $A$ ranked first, but ranks $B$ second, $D$ third and $C$ last. In election scenario 1, candidate $C$ is the Borda Count winner (the Borda scores are $\BS(A)=9, \BS(B)=5, \BS(C)=10$, and $\BS(D)=6$). In the election scenario 2, candidate $A$ is the Borda Count winner (the Borda scores are $\BS(A)=9, \BS(B)=6, \BS(C)=8$, and $\BS(D)=7$). According to her ranking in election scenario 1, this voter prefers the outcome in election scenario 2 (candidate $A$, the Borda winner in election scenario 2, is ranked above candidate $C$, the Borda winner in election scenario 1). So, if we assume that election scenario 1 represents the "true" preferences of the electorate, it is in the interest of the third voter to misrepresent her preferences as in election scenario 2. This is an instance of a general result known as the Gibbard-Satterthwaite Theorem (Gibbard 1973; Satterthwaite 1975): Under natural assumptions, there is no voting method that guarantees that voters will choose their ballots sincerely (for a precise statement of this theorem see Theorem 3.1.2 from Taylor 2005 or Section 3.3 of List 2013). ### 4.2 Characterization Results Much of the literature on voting theory (and, more generally, social choice theory) is focused on so-called axiomatic characterization results. The main goal is to characterize different voting methods in terms of abstract principles of collective decision making. See Pauly 2008 and Endriss 2011 for interesting discussions of axiomatic characterization results from a logician's point-of-view. Consult List 2013 and Gaertner 2006 for introductions to the vast literature on axiomatic characterizations in social choice theory. In this article, I focus on a few key axioms and results and how they relate to the voting methods and paradoxes discussed above. I start with three core principles. Anonymity: The names of the voters do not matter: If two voters swap their ballots, then the outcome of the election is unaffected. Neutrality: The names of the candidates, or alternatives, do not matter: If two candidates are exchanged in every ballot, then the outcome of the election changes accordingly. Universal Domain: There are no restrictions on the voter's choice of ballots. In other words, no profile of ballots can be ignored by a voting method. One way to make this precise is to require that voting methods are total functions on the set of all profiles (recall that a profile is a sequence of ballots, one from each voter). These properties ensure that the outcome of an election depends only on the voters' ballots, with all the voters and candidates being treated equally. Other properties are intended to rule out some of the paradoxes and anomalies discussed above. In section 4.1, there is an example of a situation in which a candidate is elected, even though all the voters prefer a different candidate. The next principle rules out such situations: Unanimity (also called the Pareto Principle): If candidate $A$ is ranked above candidate $B$ by all voters, then candidate $B$ should not win the election. These are natural properties to impose on any voting method. A surprising consequence of these properties is that they rule out another natural property that one may want to impose: Say that a voting method is resolute if the method always selects one winner (i.e., there are no ties). Suppose that $V$ is a voting method that requires voters to rank the candidates and that there are at least 3 candidates and enough voters to form a Condorcet component (a profile generating a majority cycle with voters evenly distributed among the different rankings). First, consider the situation when there are exactly 3 candidates (in this case, we do not need to assume Unanimity). Divide the set of voters into three groups of size $n$ and consider the Condorcet component: \| \| \| \| --- \| --- \| \| # Voters \| Ranking \| \| $n$ \| $A\s B\s C$ \| \| $n$ \| $B\s C\s A$ \| \| $n$ \| $C\s A\s B$ \| By Universal Domain and resoluteness, $V$ must select exactly one of $A$, $B$, or $C$ as the winner. Assume that $V$ select $A$ as the winner (the argument when $V$ selects the other candidates is similar). Now, consider the profile in which every voter swaps candidate $A$ and $B$ in their rankings: \| \| \| \| --- \| --- \| \| # Voters \| Ranking \| \| $n$ \| $B\s A\s C$ \| \| $n$ \| $A\s C\s B$ \| \| $n$ \| $C\s B\s A$ \| By Neutrality and Universal Domain, $V$ must elect candidate $B$ in this election scenario. Now, consider the profile in which every voter in the above election scenario swaps candidates $B$ and $C$: \| \| \| \| --- \| --- \| \| # Voters \| Ranking \| \| $n$ \| $C\s A\s B$ \| \| $n$ \| $A\s B\s C$ \| \| $n$ \| $B\s C\s A$ \| By Neutrality and Universal Domain, $V$ must elect candidate $C$ in this election scenario. Notice that this last election scenario can be generated by permuting the voters in the first election scenario (to generate the last election scenario from the first election scenario, move the first group of voters to the 2nd position, the 2nd group of voters to the 3rd position and the 3rd group of voters to the first position). But this contradicts Anonymity since this requires $V$ to elect the same candidate in the first and third election scenario. To extend this result to more than 3 candidates, consider a profile in which candidates $A$, $B$, and $C$ are all ranked above any other candidate and the restriction to these three candidates forms a Condorcet component. If $V$ satisfies Unanimity, then no candidate except $A$, $B$ or $C$ can be elected. Then, the above argument shows that $V$ cannot satisfy Resoluteness, Universal Domain, Neutrality, and Anonymity. That is, there are no Resolute voting methods that satisfy Universal Domain, Anonymity, Neutrality, and Unanimity for 3 or more candidates (note that I have assumed that the number of voters is a multiple of 3, see Moulin 1983 for the full proof). Section 3.2 discussed examples in which candidates end up losing an election as a result of more support from some of the voters. There are many ways to state properties that require a voting method to be monotonic. The following strong version (called Positive Responsiveness in the literature) is used to characterize majority rule when there are only two candidates: Positive Responsiveness: If candidate $A$ is a winner or tied for the win and moves up in some of the voter's rankings, then candidate $A$ is the unique winner. I can now state our first characterization result. Note that in all of the example discussed above, it is crucial that there are three or more candidates (for example, stating Condorcet's paradox requires there to be three or more candidates). When there are only two candidates, or alternatives, Majority Rule (choose the alternative ranked first by more than 50% of the voters) can be singled out as "best": Theorem (May 1952). A voting method for choosing between two candidates satisfies Neutrality, Anonymity, Unanimity and Positive Responsiveness if and only if the method is majority rule. See May 1952 for a precise statement of this theorem and Asan and Sanver 2002, Maskin 1995, and Woeginger 2003 for alternative characterizations of majority rule. A key assumption in the proof May's theorem and subsequent results is the restriction to voting on two alternatives. When there are only two alternatives, the definition of a ballot can be simplified since a ranking of two alternatives boils down to selecting the alternative that is ranked first. The above characterizations of Majority Rule work in a more general setting since they also allow voters to abstain (which is ambiguous between not voting and being indifferent between the alternatives). So, if the alternatives are $\{A,B\}$, then there are three possible ballots: selecting $A$, selecting $B$, or abstaining (which is treated as selecting both $A$ and $B$). A natural question is whether there are May-style characterization theorems for more than two alternatives. A crucial issue is that rankings of more than two alternatives are much more informative than selecting an alternative or abstaining. By restricting the information required from a voter to selecting one of the alternatives or abstaining, Goodin and List 2006 prove that the axioms used in May's Theorem characterize Plurality Rule when there are more than two alternatives. They also show that a minor modification of the axioms characterize Approval Voting when voters are allowed to select more than one alternative. Note that focusing on voting methods that limit the information required from the voters to selecting one or more of the alternatives hides all the interesting phenomena discussed in the previous sections, such as the existence of a Condorcet paradox. Returning to the study of voting methods that require voters to rank the alternatives, the most important characterization result is Ken Arrow's celebrated impossibility theorem (1963). Arrow showed that there is no social welfare function (a social welfare function maps the voters' rankings (possibly allowing ties) to a single social ranking) satisfying universal domain, unanimity, non-dictatorship (there is no voter $d$ such that for all profiles, if $d$ ranks $A$ above $B$ in the profile, then the social ordering ranks $A$ above $B$) and the following key property: Independence of Irrelevant Alternatives: The social ranking (higher, lower, or indifferent) of two candidates $A$ and $B$ depends only on the relative rankings of $A$ and $B$ for each voter. This means that if the voters' rankings of two candidates $A$ and $B$ are the same in two different election scenarios, then the social rankings of $A$ and $B$ must be the same. This is a very strong property that has been extensively criticized (see Gaertner, 2006, for pointers to the relevant literature, and Cato, 2014, for a discussion of generalizations of this property). It is beyond the scope of this article to go into detail about the proof and the ramifications of Arrow's theorem (see Morreau, 2014, for this discussion), but I note that many of the voting methods we have discussed do not satisfy the above property. A striking example of a voting method that does not satisfy Independence of Irrelevant Alternatives is Borda Count. Consider the following two election scenarios: \| \| \| \| \| --- \| --- \| --- \| \| # Voters \| Scenario 1Ranking \| Scenario 2Ranking \| \| 3 \| $A\s B\s C\s \bX$ \| $A\s B\s C\s \bX$ \| \| 2 \| $B\s C\s A\s \bX$ \| $B\s C\s \bX\s A$ \| \| 2 \| $C\s A\s B\s \bX$ \| $C\s \bX\s A\s B$ \| \| Scenario 1: The Borda ranking is $A >\_{\Borda} B >\_{\Borda} C >\_{\Borda} X$ ($\BS(A)=15$, $\BS(B)=14$, $\BS(C)=13$, and $\BS(X)=0$) \| \| Scenario 2: The Borda ranking is $C >\_{\Borda} B >\_{\Borda} A >\_{\Borda} X$ ($\BS(A)=11$, $\BS(B)=12$, $\BS(C)=13$, and $\BS(X)=6$) \| Notice that the relative rankings of candidates $A$, $B$ and $C$ are the same in both election scenarios. In the election scenario 2, the ranking of candidate $X$, that is uniformly ranked in last place in election scenario 1, is changed. The ranking according to the Borda score of the candidates in election scenario 1 puts $A$ first with 15 points, $B$ second with 14 points, $C$ third with 13 points, and $X$ last with 0 points. In election scenario 2, the ranking of $A$, $B$ and $C$ is reversed: Candidate $C$ is first with 13 voters; candidate $B$ is second with 12 points; candidate $A$ is third with 11 points; and candidate $X$ is last with 6 points. So, even though the relative rankings of candidates $A$, $B$ and $C$ do not differ in the two election scenarios, the position of candidate $X$ in the voters' rankings reverses the Borda rankings of these candidates. In Section 3.3, it was noted that a number of methods (including all Condorcet consistent methods) are susceptible to the multiple districts paradox. An example of a method that is not susceptible to the multiple districts paradox is Plurality Rule: If a candidate receives the most first place votes in two different districts, then that candidate must receive the most first place votes in the combined the districts. More generally, no scoring rule is susceptible to the multiple districts paradox. This property is called reinforcement: Reinforcement: Suppose that $N\_1$ and $N\_2$ are disjoint sets of voters facing the same set of candidates. Further, suppose that $W\_1$ is the set of winners for the population $N\_1$, and $W\_2$ is the set of winners for the population $N\_2$. If there is at least one candidate that wins both elections, then the winner(s) for the entire population (including voters from both $N\_1$ and $N\_2$) is the set of candidates that are in both $W\_1$ and $W\_2$ (i.e., the winners for the entire population is $W\_1\cap W\_2$). The reinforcement property explicitly rules out the multiple-districts paradox (so, candidates that win all sub-elections are guaranteed to win the full election). In order to characterize all scoring rules, one additional technical property is needed: Continuity: Suppose that a group of voters $N\_1$ elects a candidate $A$ and a disjoint group of voters $N\_2$ elects a different candidate $B$. Then there must be some number $m$ such that the population consisting of the subgroup $N\_2$ together with $m$ copies of $N\_1$ will elect $A$. We then have: Theorem (Young 1975). Suppose that $V$ is a voting method that requires voters to rank the candidates. Then, $V$ satisfies Anonymity, Neutrality, Reinforcement and Continuity if and only if the method is a scoring rule. See Merlin 2003 and Chebotarev and Smais 1998 for surveys of other characterizations of scoring rules. Additional axioms single out Borda Count among all scoring methods (Young 1974; Gardenfors 1973; Nitzan and Rubinstein 1981). In fact, Saari has argued that "any fault or paradox admitted by Borda's method also must be admitted by all other positional voting methods" (Saari 1989, pg. 454). For example, it is often remarked that Borda Count (and all scoring rules) can be easily manipulated by the voters. Saari (1995, Section 5.3.1) shows that among all scores rules Borda Count is the least susceptible to manipulation (in the sense that it has the fewest profiles where a small percentage of voters can manipulate the outcome). I have glossed over an important detail of Young's characterization of scoring rules. Note that the reinforcement property refers to the behavior of a voting method on different populations of voters. To make this precise, the formal definition of a voting method must allow for domains that include profiles (i.e., sequences of ballots) of different lengths. To do this, it is convenient to assume that the domain of a voting method is an anonymized profile: Given a set of ballots $\mathcal{B}$, an anonymous profile is a function $\pi:\mathcal{B}\rightarrow\mathbb{N}$. Let $\Pi$ be the set of all anonymous profiles. A variable domain voting method assigns a non-empty set of voters to each anonymous profile--i.e., it is a function $V:\Pi\rightarrow \wp(X)-\emptyset$). Of course, this builds in the property of Anonymity into the definition of a voting method. For this reason, Young (1975) does not need to state Anonymity as a characterizing property of scoring rules. Young's axioms identify scoring rules out of the set of all functions defined from ballots that are rankings of candidates. In order to characterize the voting methods from Section 2.2, we need to change the set of ballots. For example, in order to characterize Approval Voting, the set of ballots $\mathcal{B}$ is the set of non-empty subsets of the set of candidates--i.e., $\mathcal{B}=\wp(X)-\emptyset$ (selecting the ballot $X$ consisting of all candidates means that the voter abstains). Two additional axioms are needed to characterize Approval Voting: Faithfulness: If there is exactly one voter in the population, then the winners are the set of voters chosen by that voter. Cancellation: If all candidates receive the same number of votes (i.e., they are elements of the same number of ballots) from the participating voters, then all candidates are winning. We then have: Theorem (Fishburn 1978b; Alos-Ferrer 2006 ). A variable domain voting method where the ballots are non-empty sets of candidates is Approval Voting if and only if it satisfies Faithfulness, Cancellation, and Reinforcement. Note that Approval Voting satisfies Neutrality even though it is not listed as one of the characterizing properties in the above theorem. This is because Alos-Ferrer (2006) showed that Neutrality is a consequence of Faithfulness, Cancellation and Reinforcement. See Fishburn 1978a and Baigent and Xu 1991 for alternative characterizations of Approval Voting, and Xu 2010 for a survey of the characterizations of Approval Voting (cf. the characterization of Approval Voting from Goodin and List 2006). Myerson (1995) introduced a general framework for characterizing abstract scoring rules that include Borda Count and Approval Voting as examples. The key idea is to think of a ballot, called a signal or a vote, as a function from candidates to a set $\mathcal{V}$, where $\mathcal{V}$ is a set of numbers. That is, the set of ballots is a subset of $\mathcal{V}^X$ (the set of functions from $X$ to $\mathcal{V}$). Then, an anonymous profile of signals assigns a score to each candidate $X$ by summing the numbers assigned to $X$ by each voter. This allows us to define voting methods by specifying the set of ballots: * Plurality Rule: The ballots are functions assigning 0 or 1 to the candidates such that exactly one candidate is assigned 1: $\{v\ \|\ v\in \{0,1\}^X$ and there is an $A\in X$ such that $v(A)=1$ and for all $B$, if $B\ne A$, then $v(B)=0\}$ * Approval Voting: The ballots are functions assigning 0 or 1 to the candidates: $\{v\ \|\ v\in \{0,1\}^X \}$ * Borda Count: The ballots are functions assigning numbers from the set $\{\#X, \#X-1,\ldots,0\}$ such that each candidate is assigned exactly one of the numbers: $\{v\ \|\ v\in\{\# X, \# X - 1, \ldots, 0\}^X$ such that $v$ is a bijection$\}$ * Range Voting: The ballots are assignments of real numbers between 0 and 1 to candidates: $[0,1]^X = \{v \ \|\ v:X\rightarrow [0,1] \}$ * Cumulative Voting: The ballots are assignments of real numbers between 0 and 1 to candidates such that the assignments sum to 1: $ \{v \ \|\ v\in [0,1]^X$ and $\sum\_{A\in X} v(A)=1\}$ * Formal Utilitarian: The ballots are assignments of real numbers to candidates: $\mathbb{R}^X = \{v \ \|\ v:X\rightarrow\mathbb{R}\}$. Myerson (1995) showed that an abstract voting rule is an abstract scoring rule if and only if it satisfies Reinforcement, Universal Domain (i.e. it is defined for all anonymous profiles), a version of the Neutrality property (adapted to the more abstract setting), and the Continuity property, which is called Overwhelming Majority. Pivato (2013) generalizes this result, and Gaertner and Xu (2012) provide a related characterization result (using different properties). Pivato (2014) characterizes Formal Utilitarian and Range Voting within the class of abstract scoring rules, and Mace (2018) extends this approach to cover a wider class of grading voting methods (including Majority Judgement). ### 4.3 Voting to Track the Truth The voting methods discussed above have been judged on procedural grounds. This "proceduralist approach to collective decision making" is defined by Coleman and Ferejohn (1986, p. 7) as one that "identifies a set of ideals with which any collective decision-making procedure ought to comply. ... [A] process of collective decision making would be more or less justifiable depending on the extent to which it satisfies them." The authors add that a distinguishing feature of proceduralism is that "what justifies a [collective] decision-making procedure is strictly a necessary property of the procedure -- one entailed by the definition of the procedure alone." Indeed, the characterization theorems discussed in the previous section can be viewed as an implementation of this idea (cf. Riker 1982). The general view is to analyze voting methods in terms of "fairness criteria" that ensure that a given method is sensitive to all of the voters' opinions in the right way. However, one may not be interested only in whether a collective decision was arrived at "in the right way," but in whether or not the collective decision is correct. This epistemic approach to voting is nicely explained by Joshua Cohen (1986, p. 34): > > An epistemic interpretation of voting has three main elements: (1) an > independent standard of correct decisions -- that is, an account > of justice or of the common good that is independent of current > consensus and the outcome of votes; (2) a cognitive account of voting > -- that is, the view that voting expresses beliefs about what the > correct policies are according to the independent standard, not > personal preferences for policies; and (3) an account of decision > making as a process of the adjustment of beliefs, adjustments that are > undertaken in part in light of the evidence about the correct answer > that is provided by the beliefs of others. > > Under this interpretation of voting, a given method is judged on how well it "tracks the truth" of some objective fact (the truth of which is independent of the method being used). A comprehensive comparison of these two approaches to voting touches on a number of issues surrounding the justification of democracy (cf. Christiano 2008); however, I will not focus on these broader issues here. Instead, I briefly discuss an analysis of Majority Rule that takes this epistemic approach. The most well-known analysis comes from the writings of Condorcet (1785). The following theorem, which is attributed to Condorcet and was first proved formally by Laplace, shows that if there are only two options, then majority rule is, in fact, the best procedure from an epistemic point of view. This is interesting because it also shows that a proceduralist analysis and an epistemic analysis both single out Majority Rule as the "best" voting method when there are only two candidates. Assume that there are $n$ voters that have to decide between two alternatives. Exactly one of these alternatives is (objectively) "correct" or "better." The typical example here is a jury deciding whether or not a defendant is guilty. The two assumptions of the Condorcet jury theorem are: Independence: The voters' opinions are probabilistically independent (so, the probability that two or more voters are correct is the product of the probability that each individual voter is correct). Voter Competence: The probability that a voter makes the correct decision is greater than 1/2 (and this probability is the same for all voters, though this is not crucial). See Dietrich 2008 for a critical discussion of these two assumptions. The classic theorem is: Condorcet Jury Theorem. Suppose that Independence and Voter Competence are both satisfied. Then, as the group size increases, the probability that the majority chooses the correct option increases and converges to certainty. See Nitzan 2010 (part III) and Dietrich and Spiekermann 2013 for modern expositions of this theorem, and Goodin and Spiekermann 2018 for implications for the theory of democracy. Condorcet envisioned that the above argument could be adapted to voting situations with more than two alternatives. Young (1975, 1988, 1995) was the first to fully work out this idea (cf. List and Goodin 2001 who generalize the Condorcet Jury Theorem to more than two alternatives in a different framework). He showed (among other things) that the Borda Count can be viewed as the maximum likelihood estimator for identifying the best candidate. Conitzer and Sandholm (2005), Conitzer et al. (2009), Xia et al. (2010), and Xia (2016) take these ideas further by classifying different voting methods according to whether or not the methods can be viewed as a maximum likelihood estimator (for a noise model). The most general results along these lines can be found in Pivato 2013 which contains a series of results showing when voting methods can be interpreted as different kinds of statistical 'estimators'. ### 4.4 Computational Social Choice One of the most active and exciting areas of research that is focused, in part, on the study of voting methods and voting paradoxes is computational social choice. This is an interdisciplinary research area that uses ideas and techniques from theoretical computer science and artificial intelligence to provide new perspectives and to ask new questions about methods for making group decisions; and to use voting methods in computational domains, such as recommendation systems, information retrieval, and crowdsourcing. It is beyond the scope of this article to survey this entire research area. Readers are encouraged to consult the Handbook of Computational Social Choice (Brandt et al. 2016) for an overview of this field (cf. also Endriss 2017). In the remainder of this section, I briefly highlight some work from this research area related to issues discussed in this article. Section 4.1 discussed election scenarios in which voters choose their ballots strategically and briefly introduced the Gibbard-Satterthwaite Theorem. This theorem shows that every voting method satisfying natural properties has profiles in which there is some voter, called a manipulator, that can achieve a better outcome by selecting a ballot that misrepresents her preferences. Importantly, in order to successfully manipulate an election, the manipulator must not only know which voting method is being used but also how the other members of society are voting. Although there is some debate about whether manipulation in this sense is in fact a problem (Dowding and van Hees 2008; Conitzer and Walsh, 2016, Section 6.2), there is interest in mechanisms that incentivize voters to report their "truthful" preferences. In a seminal paper, Bartholdi et al. (1989) argue that the complexity of computing which ballot will lead to a preferred outcome for the manipulator may provide a barrier to voting insincerely. See Faliszewski and Procaccia 2010, Faliszewski et al. 2010, Walsh 2011, Brandt et al. 2013, and Conitzer and Walsh 2016 for surveys of the literature on this and related questions, such as the the complexity of determining the winner given a voting method and the complexity of determining which voter or voters should be bribed to change their vote to achieve a given outcome. One of the most interesting lines of research in computational social choice is to use techniques and ideas from AI and theoretical computer science to design new voting methods. The main idea is to think of voting methods as solutions to an optimization problem. Consider the space of all rankings of the alternatives $X$. Given a profile of rankings, the voting problem is to find an "optimal" group ranking (cf. the discussion or distance-based rationalizations of voting methods from Elkind et al. 2015). What counts as an "optimal" group ranking depends on assumptions about the type of the decision that the group is making. One assumption is that the voters have real-valued utilities for each candidate, but are only able to report rankings of the alternatives (it is assumed that the rankings represent the utility functions). The voting problem is to identify the candidates that maximizes the (expected) social welfare (the average of the voters' utilities), given the partial information about the voters' utilities--i.e., the profile of rankings of the candidates. See Pivato 2015 for a discussion of this approach to voting and Boutilier et al. 2015 for algorithms that solve different versions of this problem. A second assumption is that there is an objectively correct ranking of the alternatives and the voters' rankings are noisy estimates of this ground truth. This way of thinking about the voting problem was introduced by Condorcet and discussed in Section 4.3. Procaccia et al. (2016) import ideas from the theory of error-correcting codes to develop an interesting new approach to aggregate rankings viewed as noisy estimates of some ground truth. ## 5. Concluding Remarks ### 5.1 From Theory to Practice As with any mathematical analysis of social phenomena, questions abound about the "real-life" implications of the theoretical analysis of the voting methods given above. The main question is whether the voting paradoxes are simply features of the formal framework used to represent an election scenario or formalizations of real-life phenomena. This raises a number of subtle issues about the scope of mathematical modeling in the social sciences, many of which fall outside the scope of this article. I conclude with a brief discussion of two questions that shed some light on how one should interpret the above analysis. *How likely* is a Condorcet Paradox or any of the other voting paradoxes?** There are two ways to approach this question. The first is to calculate the probability that a majority cycle will occur in an election scenario. There is a sizable literature devoted to analytically deriving the probability of a majority cycle occurring in election scenarios of varying sizes (see Gehrlein 2006, and Regenwetter et al. 2006, for overviews of this literature). The calculations depend on assumptions about the distribution of rankings among the voters. One distribution that is typically used is the so-called impartial culture, where each ranking is possible and occurs with equal probability. For example, if there are three candidates, and it is assumed that the voters' ballots are rankings of the candidates, then each possible ranking can occur with probability 1/6. Under this assumption, the probability of a majority cycle occurring has been calculated (see Gehrlein 2006, for details). Riker (1982, p. 122) has a table of the relevant calculations. Two observations about this data: First, as the number of candidates and voters increases, the probability of a majority cycles increases to certainty. Second, for a fixed number of candidates, the probability of a majority cycle still increases, though not necessarily to certainty (the number of voters is the independent variable here). For example, if there are five candidates and seven voters, then the probability of a majority cycle is 21.5 percent. This probability increases to 25.1 percent as the number of voters increases to infinity (keeping the number of candidates fixed) and to 100 percent as the number of candidates increases to infinity (keeping the number of voters fixed). Prima facie, this result suggests that we should expect to see instances of the Condorcet and related paradoxes in large elections. Of course, this interpretation takes it for granted that the impartial culture is a realistic assumption. Many authors have noted that the impartial culture is a significant idealization that almost certainly does not occur in real-life elections. Tsetlin et al. (2003) go even further arguing that the impartial culture is a worst-case scenario in the sense that any deviation results in lower probabilities of a majority cycle (see Regenwetter et al. 2006, for a complete discussion of this issue, and List and Goodin 2001, Appendix 3, for a related result). A second way to argue that the above theoretical observations are robust is to find supporting empirical evidence. For instance, is there evidence that majority cycles have occurred in actual elections? While Riker (1982) offers a number of intriguing examples, the most comprehensive analysis of the empirical evidence for majority cycles is provided by Mackie (2003, especially Chapters 14 and 15). The conclusion is that, in striking contrast to the probabilistic analysis referenced above, majority cycles typically have not occurred in actual elections. However, this literature has not reached a consensus about this issue (cf. Riker 1982): The problem is that the available data typically does not include voters' opinions about all pairwise comparison of candidates, which is needed to determine if there is a majority cycle. So, this information must be inferred (for example, by using statistical methods) from the given data. A related line of research focuses on the influence of factors, such as polls (Reijngoud and Endriss 2012), social networks (Santoro and Beck 2017, Stirling 2016) and deliberation among the voters (List 2018), on the profiles of ballots that are actually realized in an election. For instance, List et al. 2013 has evidence suggesting that deliberation reduces the probability of a Condorcet cycle occurring. How do the different voting methods compare in actual elections? In this article, I have analyzed voting methods under highly idealized assumptions. But, in the end, we are interested in a very practical question: Which method should a group adopt? Of course, any answer to this question will depend on many factors that go beyond the abstract analysis given above (cf. Edelman 2012a). An interesting line of research focuses on incorporating empirical evidence into the general theory of voting. Evidence can come in the form of a computer simulation, a detailed analysis of a particular voting method in real-life elections (for example, see Brams 2008, Chapter 1, which analyzes Approval voting in practice), or as in situ experiments in which voters are asked to fill in additional ballots during an actual election (Laslier 2010, 2011). The most striking results can be found in the work of Michael Regenwetter and his colleagues. They have analyzed datasets from a variety of elections, showing that many of the usual voting methods that are considered irreconcilable (e.g., Plurality Rule, Borda Count and the Condorcet consistent methods from Section 3.1.1) are, in fact, in perfect agreement. This suggests that the "theoretical literature may promote overly pessimistic views about the likelihood of consensus among consensus methods" (Regenwetter et al. 2009, p. 840). See Regenwetter et al. 2006 for an introduction to the methods used in these analyses and Regenwetter et al. 2009 for the current state-of-the-art. ### 5.2 Further Reading My objective in this article has been to introduce different voting methods and to highlight key results and issues that facilitate comparisons between the voting methods. To dive more into the details of the topics introduced in this article, see Saari 2001, 2008, Nurmi 1998, Brams and Fishburn 2002, Zwicker 2012, and the collection of articles in Felsenthal and Machover 2012. Some important topics related to the study of voting methods not discussed in this article include: * Saari's influential geometric approach to study of voting methods and paradoxes (Saari 1995); * The study of measures of voting power: the probability that a single voter is decisive in an election (Gelman et al. 2002; Felsenthal and Machover 1998); * The study of probabilistic voting methods: a voting method in which the output is a lottery over the set of candidates rather than a ranking or set of candidates (Brandt 2017); * Questions about whether it is rational to vote in an election (Brennan 2016); and * The study of methods for ensuring fair and proportional representation (Balinksi and Young 1982; Pukelsheim 2017). Finally, consult List 2013 and Morreau 2014 for a discussion of broader issues in theory of social choice.
wang-yangming	## 1. Life This philosopher's family name was "Wang," his personal name was "Shouren," and his "courtesy name" was "Bo-an."[1] However, he is normally known today as "Wang Yangming," based on a nickname he adopted when he was living in the Yangming Grotto of Kuaiji Mountain. Born in 1472 near Hangzhou in what is now Zhejiang Province, Wang was the son of a successful official. As such, he would have received a fairly conventional education, with a focus on the Four Books of the Confucian tradition: the Analects (the sayings of Confucius and his immediate disciples), the Great Learning (believed to consist of an opening statement by Confucius with a commentary on it by his leading disciple, Zengzi), the Mean (attributed to Zisi, the grandson of Confucius, who was also a student of Zengzi), and the Mengzi (the sayings and dialogues of Mencius, a student of Zisi). The young Wang would have literally committed these classics to memory, along with the commentaries on them by the master of orthodox Confucianism, Zhu Xi (1130-1200). The study of these classics-cum-commentary was thought to be morally edifying; however, people also studied them in order to pass the civil service examinations, which were the primary route to government power, and with it wealth and prestige. At the age of seventeen (1489), Wang had a conversation with a Daoist priest that left him deeply intrigued with this alternative philosophical system and way of life. Wang was also attracted to Buddhism, and remained torn between Daoism, Buddhism and Confucianism for much of his early life. Whereas Confucianism emphasizes our ethical obligations to others, especially family members, and public service in government, the Daoism and Buddhism of Wang's era encouraged people to overcome their attachment to the physical world. Wang continued the serious study of Zhu Xi's interpretation of Confucianism, but was disillusioned by an experience in which he and a friend made a determined effort to apply what they took to be Zhu Xi's method for achieving sagehood: > > ...my friend Qian and I discussed the idea that to become a sage > or a worthy one must investigate all the things in the world. But how > can a person have such tremendous energy? I therefore pointed to the > bamboos in front of the pavilion and told him to investigate them and > see. Day and night Qian went ahead trying to investigate to the > utmost the Pattern of the bamboos. He exhausted his mind and > thoughts, and on the third day he was tired out and took sick. At > first I said that it was because his energy and strength were > insufficient. Therefore I myself went to try to investigate to the > utmost. From morning till night, I was unable to find the Pattern of > the bamboos. On the seventh day I also became sick because I thought > too hard. In consequence we sighed to each other and said that it was > impossible to be a sage or a worthy, for we do not have the tremendous > energy to investigate things that they have. (Translation modified > from Chan 1963, 249) > As we shall see (Section 2, below), it is unclear whether Wang and his friend were correctly applying what Zhu Xi meant by "the investigation of things." However, Wang's experience of finding it impractical to seek for the Pattern of the universe in external things left a deep impression on him, and influenced the later course of his philosophy. Wang continued to study Daoism as well as Buddhism, but also showed a keen interest in military techniques and the craft of writing elegant compositions. Meanwhile, he progressed through the various levels of the civil service examinations, finally passing the highest level in 1499. After this, Wang had a meteoric rise in the government, including distinguished service in offices overseeing public works, criminal prosecution, and the examination system. During this period, Wang began to express disdain for overly refined literary compositions like those he had produced in his earlier years. Wang would later criticize those who "waste their time competing with one another writing flowery compositions in order to win acclaim in their age, and...no longer comprehend conduct that honors what is fundamental, esteems what is real, reverts to simplicity, and returns to purity" (Tiwald and Van Norden 2014, 275). In addition, Wang started to turn his back on Daoism and Buddhism, which he came to regard as socially irresponsible: "...simply because they did not understand what it is to rest in the ultimate good and instead exerted their selfish minds toward the achievement of excessively lofty goals, [Buddhists and Daoists] were lost in vagaries, illusions, emptiness, and stillness and had nothing to do with the family, state or world" (Tiwald and Van Norden 2014, 244, gloss mine). Nonetheless, Wang continued to show an intimate familiarity with Daoist and Buddhist literature and concepts throughout his career. A life-changing event for Wang occurred in 1506. A eunuch who had assumed illegitimate influence at court had several able officials imprisoned for opposing him. Wang wrote a "memorial" to the emperor in protest. The eunuch responded by having Wang publicly beaten and exiled to an insignificant position in a semi-civilized part of what is now Guizhou Province. Wang had to face considerable physical and psychological hardship in this post, but through these challenges he achieved a deep philosophical awakening (1508), which he later expressed in a poem he wrote for his students: > > Everyone has within an unerring compass; > The root and source of > the myriad transformations lies in the mind. > I laugh when I > think that, earlier, I saw things the other way around; > > Following branches and leaves, I searched outside! (Ivanhoe 2009, 181) > In other words, looking outside oneself for moral truth, as he and his friend Qian had tried to do when studying the bamboos, was ignoring the root of moral insight, which is one's own innate understanding. Fortunately for Wang, in the year when his term of service in Guizhou was over (1510), the eunuch who had Wang beaten and banished was himself executed. Wang's official career quickly returned to its stratospheric level of achievement, with high-ranking posts and exceptional achievements in both civil and military positions. This inevitably led to opposition from the faction-ridden court. Wang was even accused of conspiring with the leader of a rebellion that Wang had himself put down. Wang had begun to attract devoted disciples even before his exile to Guizhou, and they gradually compiled the Record for Practice (the anthology of his sayings, correspondence, and dialogues that is one of our primary sources for Wang's philosophy). It is reflective of Wang's philosophy that the discussions recorded in this work occurred in the midst of his active life in public affairs. Near the end of his life, Wang was called upon to suppress yet another rebellion (1527). The night before he left, one of his disciples recorded the "Inquiry on the Great Learning," which was intended as a primer of Wang's philosophy for new disciples. Wang put down the rebellion, but his health had been declining for several years, and he died soon afterward (1529). On his deathbed, Wang said, " 'This mind' is luminous and bright. What more is there to say?"[2] ## 2. Intellectual Context During Wang's lifetime, the dominant intellectual movement was Neo-Confucianism (in Chinese, Daoxue, or the Learning of the Way). Neo-Confucianism traces its origins to Han Yu and Li Ao in the late Tang dynasty (618-906), but it only came to intellectual maturity in the Song and Southern Song dynasties (960-1279), with the theorizing of Zhou Dunyi, Zhang Zai, Cheng Yi, his brother Cheng Hao, and Zhu Xi. Neo-Confucianism originally developed as a Confucian reaction against Buddhism. Ironically, though, the Neo-Confucians were deeply influenced by Buddhism and adopted many key Buddhists concepts, including the notions that the diverse phenomena of the universe are manifestations of some underlying unity, and that selfishness is the fundamental vice. Zhu Xi synthesized the thought of earlier Neo-Confucians, and his interpretation of the Four Books became the basis of the civil service examinations (see Section 1, above). Zhu held that everything in existence has two aspects, Pattern and qi. Pattern (li, also translated "principle") is the underlying structure of the universe. Qi is the spatio-temporal stuff that constitutes concrete objects. The complete Pattern is fully present in each entity in the universe; however, entities are individuated by having distinct allotments of qi (variously translated as "ether," "psychophysical stuff," "vital fluid," "material force," and others). In addition to having spatio-temporal location, qi has qualitative distinctions, described metaphorically in terms of degrees of "clarity" and "turbidity." The more "clear" the qi, the greater the extent to which the Pattern manifests. This explains how speciation is possible. For example, plants have more "clear" qi than do rocks, which is reflected in the more complex repertoire of interactions plants have with their environments. Neo-Confucian metaphysics, ethics, and philosophical psychology are systematically connected. Humans have the most clear qi of any being. However, some humans have especially clear qi, and as a result manifest the virtues, while others have comparatively turbid qi, and therefore are prone to vice. The state of one's qi is not fixed, though. Through ethical cultivation, any human can clarify his qi, and thereby become more virtuous. Similarly, through laxness, one can allow one's qi to morally deteriorate. Because the Pattern is fully present in each human, everyone has complete innate ethical knowledge. However, due to the obscurations of qi, the manifestations of this knowledge are sporadic and inconsistent. Thus, a king who shows benevolence when he takes pity on an ox being led to slaughter and spares it may fail to show benevolence in his treatment of his own subjects (Mengzi 1A7). Similarly, a person might manifest righteousness when he disdains to even consider an illicit sexual relationship; however, the same person might see nothing shameful about flattering the worst traits and most foolish policies of a ruler whom he serves (Mengzi 3B3). Because humans all share the same Pattern, we are parts of a potentially harmonious whole. In some sense, we form "one body" with other humans and the rest of the universe. Consequently, the fundamental human virtue is benevolence, which involves recognition of our shared nature with others.[3] Correspondingly, the fundamental vice is selfishness. Cheng Hao uses a medical metaphor to explain the relationship between ethics and personal identity: > > A medical text describes numbness of the hands and feet as being > "unfeeling." This expression describes it perfectly. > Benevolent people regard Heaven, Earth, and the myriad things as one > Substance. Nothing is not oneself. If you recognize something as > yourself, there are no limits to how far [your compassion] will go. > But if you do not identify something as part of yourself, you will > have nothing to do with it. This is like how, when hands and feet are > "unfeeling," the qi does not flow and one does > not identify them as part of oneself. (Tiwald and Van Norden 2014, > 201) > Just as those whose hands and feet are "unfeeling" are not bothered by injuries to their own limbs, so do those who are ethically "unfeeling" fail to show concern for other humans. In both cases, one fails to act appropriately because one fails to recognize that something is a part of oneself. Neo-Confucians regarded the ancient philosopher Mencius (4th century BCE) as having presented an especially incisive interpretation of Confucianism. They referred to him as the "Second Sage," second in importance only to Confucius himself, and often appealed to his vocabulary, arguments, and examples.[4] In particular, Neo-Confucians adopted Mencius's view that human nature is good, and that wrongdoing is the result of humans ignoring the promptings of their innate virtues. Physical desires are one of the primary causes for the failure to heed one's innate moral sense. There is nothing intrinsically immoral about desires for fine food, sex, wealth, etc., but we will be led to wrongdoing if we pursue them without giving thought to the well-being of others (Mengzi 6A15). Neo-Confucians also adopted Mencius's list of four cardinal virtues: benevolence, righteousness, wisdom, and propriety. Benevolence is manifested in the emotion of compassion, which involves sympathizing with the suffering of others and acting appropriately on that sympathy. For instance, a benevolent ruler could never be indifferent to the well-being of his subjects, and would work tirelessly to alleviate their suffering. Righteousness is manifested in the emotion of disdain or shame at the thought of doing what is dishonorable, especially in the face of temptations of wealth or physical desire. For example, a righteous person would not cheat at a game, or accept a bribe. Wisdom is manifested in approval or disapproval that reflects sound judgment about means-end deliberation and about the character of others. Thus, a wise official would know which policies to support and which to oppose, and would be inventive in devising solutions to complex problems. Propriety is manifested in respect or deference toward elders and legitimate authorities, particularly as expressed in ceremonies or etiquette. For instance, a person with propriety would willingly defer to elder family members most of the time, but would be motivated to serve a guest before his elder brother. The preceding views are shared, in broad outline, by all Neo-Confucian philosophers. The primary issue they debated was the proper method of ethical cultivation. In other words, how is it possible for us to consistently and reliably access and act on our innate ethical knowledge? Other disagreements among them (on seemingly recondite issues of metaphysics and the interpretation of the Confucian classics) can often be traced to more fundamental disagreements over ethical cultivation. For Zhu Xi, almost all humans are born with qi that is so turbid that it is impossible for them to consistently know what is virtuous and act accordingly without external assistance. The remedy is carefully studying the classics written by the ancient sages, ideally under the guidance of a wise teacher. Although the classics were written in response to concrete historical situations, their words allow us to grasp the Pattern at a certain level of abstraction from its instantiations. However, Zhu Xi states that merely understanding the Pattern is insufficient for the exercise of virtue. One must achieve Sincerity (cheng), a continual awareness of moral knowledge in the face of temptations.[5] In his own lifetime, Zhu Xi's leading critic was Lu Xiangshan (1139-1193). Lu argued that, because the Pattern is fully present in the mind of each human, it is not necessary to engage in an intellectually demanding process of study in order to recover one's moral knowledge: > > Righteousness and Pattern are in the minds of human beings. As a > matter of fact, these are what Heaven has endowed us with, and they > can never be effaced or eliminated [from our minds]. If one becomes > obsessed with [desires for] things and reaches the point where one > violates Pattern and transgresses righteousness, usually this is > simply because one fails to reflect upon these things [i.e., the > righteousness and Pattern that lie within one]. If one truly is able > to turn back and reflect upon these things, then what is right and > wrong and what one should cleave to and what one should subtly reject > will begin to stir, separate, become clear, and leave one resolute and > without doubts. (Tiwald and Van Norden 2014, 251-52, glosses in > original translation) > In emphasizing the innate human capacity for moral judgment, Lu adopted the phrase "pure knowing" from Mengzi 7A15. In its original context, Mencius writes that "Among babes in arms there are none that do not know to love their parents," and describes this as an instance of "pure knowing," which humans "know without pondering" (Tiwald and Van Norden 2014, 218). Lu explains: > > Pure knowing lies within human beings; although some people become > mired in dissolution, pure knowing still remains undiminished and > enduring [within them]. ... Truly, if they can turn back and seek > after it, then, without needing to make a concerted effort, what is > right and wrong, what is fine and foul, will become exceedingly clear, > and they will decide for themselves what to like and dislike, what to > pursue and what to abandon. (Tiwald and Van Norden 2014, 252) > Although there were always Confucians like Lu who disagreed with Zhu Xi, the latter's interpretation became dominant after the government sponsored it as the official interpretation for the civil service examinations. By the time of Wang Yangming, Zhu Xi's approach had ossified into a stale orthodoxy that careerists mindlessly parroted in an effort to pass the examinations. ## 3. Unity of Knowing and Acting Some aspects of Wang's philosophy can be understood as refining or drawing out the full implications of Lu Xiangshan's critique of Zhu Xi. Like Lu, Wang stressed that the Pattern is fully present in the mind of every person: "The mind is Pattern. Is there any affair outside the mind? Is there any Pattern outside the mind?" (Tiwald and Van Norden 2014, 264)[6] As a result, the theoretical study of ethics is unnecessary. All a moral agent needs to do is exercise "pure knowing." This view may seem naive at first glance. However, this is partially due to the different foci of Chinese and Western ethics. Metaethics and normative ethical theory are substantial parts of Western ethics, but less central to Chinese ethics. In addition, recent Western ethics manifests an interest in abstract ethical dilemmas that is perhaps disproportionate to the relevance of these quandaries in practice. In contrast, the focus of much Chinese ethics, particularly Confucianism, is applied ethical cultivation. Consequently, Lu and Wang were primarily interested in having a genuine positive effect on the ethical lives of their followers. There is some plausibility to the claim that (even in our own complex, multicultural intellectual context) most of us know what our basic ethical obligations are. For example, as a teacher, I have an obligation to grade assignments promptly and fairly. As a colleague, I have an obligation to take my turn serving as chair. As a parent, I have an obligation to make time to share my children's interests. Due to selfish desires, it is tempting to procrastinate in grading essays, or try to evade departmental service, or ignore the emotional needs of one's children. However, it is not as if it is genuinely difficult to figure out that these are our obligations. Wang's most distinctive and well-known doctrine is the unity of knowing and acting (zhi xing he yi). In order to grasp the significance of this doctrine, consider a student who is being questioned by a college honor code panel after being caught plagiarizing an essay. If the honor code panel asked the student whether he knew that plagiarism was wrong, it would not be surprising for the student to reply, "Yes, I knew it was wrong to plagiarize my essay, but I wanted so much to get an easy A that I gave in to temptation." Western philosophers would describe this as a case of akrasia, or weakness of will. Zhu Xi, along with many Western philosophers like Aristotle, would acknowledge that the student had correctly described his psychological state. However, Wang Yangming would deny that the student actually knew that plagiarism was wrong: "There never have been people who know but do not act. Those who 'know' but do not act simply do not yet know" (Tiwald and Van Norden 2014, 267). One of Wang's primary claims is that merely verbal assent is inadequate to demonstrate actual knowledge: "One cannot say he knows filial piety or brotherly respect simply because he knows how to say something filial or brotherly. Knowing pain offers another good example. One must have experienced pain oneself in order to know pain." Wang also adduces cold and hunger as things that one must experience in order to know. We might object that Wang's example shows, at most, that someone must have had an experience of goodness or evil at some point in his life in order to know what it means for something to be good or evil. This falls short of the conclusion that Wang needs, though, for just as I can know what you mean when you say "I am hungry" even if I am not currently motivated to eat, so might I know that "plagiarism is wrong," even if I am not motivated to avoid plagiarism myself in this circumstance. A more promising line of argument for Wang is his appropriation of an example from the Great Learning (Commentary 6), which suggests that loving the good is "like loving a lovely sight," while hating evil is "like hating a hateful odor" (Tiwald and Van Norden 2014, 191). To recognize an odor as disgusting (a kind of knowing) is to be repulsed by it (a motivation), which leads to avoiding the odor or eliminating its source (actions). The Chinese phrase rendered "loving a lovely sight" (hao hao se) has connotations of finding someone sexually attractive.[7] To regard someone as sexually attractive (a kind of cognition) is to be drawn toward them (a kind of motivation that can lead to action). To this, one might object that Wang's examples show, at most, that knowing something is good or bad requires that one have some level of appropriate motivation. Even if we grant the relevance of Wang's examples, it is possible to have an intrinsic motivation that does not result in action. Recognizing that someone is sexually attractive certainly does not always result in pursuing an assignation, for example. Perhaps Wang's most compelling line of argument is a pragmatic one about the aim or purpose behind discussing knowing and acting. He allows that it may be useful and legitimate to discuss knowing separately from acting or acting separately from knowing, but only in order to address the failings of specific kinds of individuals. On the one hand, "there is a type of person in the world who foolishly acts upon impulse without engaging in the slightest thought or reflection. Because they always act blindly and recklessly, it is necessary to talk to them about knowing...," without emphasizing acting. On the other hand, "[t]here is also a type of person who is vague and irresolute; they engage in speculation while suspended in a vacuum and are unwilling to apply themselves to any concrete actions" (268). These latter people benefit from advice that emphasizes action, without necessarily discussing knowledge. However, those in Wang's era who distinguish knowledge and action (and here he has in mind those who follow the orthodox philosophy of Zhu Xi) "separate knowing and acting into two distinct tasks to perform and think that one must first know and only then can one act." As a result, they become nothing more than pedantic bookworms, who study ethics without ever living up to its ideals or trying to achieve positive change in the world around them. Wang concludes, "My current teaching regarding the unity of knowing and acting is a medicine directed precisely at this disease." ## 4. Interpretation of the Great Learning In the standard Confucian curriculum of Wang's era, the Great Learning was the first of the Four Books that students were assigned, and Zhu Xi's commentary on it often made a lasting impression on them. In the opening of the Great Learning, Confucius describes the steps in self-cultivation: > > The ancients who desired to enlighten the enlightened Virtue of the > world would first put their states in order. Those who desired to put > their states in order would first regulate their families. Those who > desired to regulate their families would first cultivate their > selves. Those who desired to cultivate their selves would first > correct their minds. Those who desired to correct their minds would > first make their thoughts have Sincerity. Those who desired to make > their thoughts have Sincerity would first extend their knowledge. > Extending knowledge lies in ge wu. (Translation > slightly modified from Tiwald and Van Norden 2014, 188-189) > Ge wu is left unexplained in the Great Learning. However, Zhu Xi, following the interpretation of Cheng Yi, claims that > > ...what is meant by "extending knowledge lies in ge > wu" is that desiring to extend my knowledge lies in > encountering things and exhaustively investigating their Pattern. In > general, the human mind is sentient and never fails to have knowledge, > while the things of the world never fail to have the Pattern. It is > only because the Pattern is not yet exhaustively investigated that > knowledge is not fully fathomed. Consequently, at the beginning of > education in the Great Learning, the learner must be made to encounter > the things of the world, and never fail to follow the Pattern that one > already knows and further exhaust it, seeking to arrive at the > farthest points. When one has exerted effort for a long time, one > day, like something suddenly cracking open, one will know in a manner > that binds it all together. (Translation slightly modified from Tiwald > and Van Norden 2014, 191) > Following Zhu Xi's interpretations, most translators today render ge wu as "investigating things." Wang and his friend Qian were trying to "investigate things" in this manner when they stared attentively at the bamboos, struggling to grasp their underlying Pattern. It is questionable, though, whether this is what Zhu Xi had in mind. Zhu certainly thought the Pattern was present in everything, and could potentially be investigated in even the most mundane of objects. However, he emphasized that the best method to learn about the Pattern was through studying the classics texts of Confucianism, particularly the Four Books. As important as learning was for Zhu Xi, he stressed that it was possible to know what is right and wrong yet not act on it: "Knowledge and action always need each other. It's like how eyes cannot walk without feet, but feet cannot see without eyes. If we discuss them in terms of their sequence, knowledge comes first. But if we discuss them in terms of importance, action is what is important" (Tiwald and Van Norden 2014, 180-81). What connects knowledge and action, on Zhu Xi's account, is Sincerity. Sincerity (cheng) is a subtle and multifaceted notion in Neo-Confucian thought. However, the Great Learning focuses on one key aspect of it: "What is meant by 'making thoughts have Sincerity' is to let there be no self-deception. It is like hating a hateful odor, or loving a lovely sight. The is called not being conflicted" (Tiwald and Van Norden 2014, 191). In other words, humans engage in wrongdoing through a sort of self-deception in which they allow themselves to ignore the promptings of their moral sense, and become motivated solely by their physical desires for sex, food, wealth, etc. Once one knows what is right and wrong, one must then make an effort to keep that knowledge present to one's consciousness, so that it becomes motivationally efficacious. As Zhu Xi explains, "...those who desire to cultivate themselves, when they know to do good in order to avoid the bad, must then genuinely make an effort and forbid self-deception, making their hatred of the [ethically] hateful be like their hating a hateful odor, and their loving what is good like their loving a lovely sight" (Tiwald and Van Norden 2014, 192). Wang challenged almost every aspect of Zhu Xi's interpretation of the Great Learning. He regarded Zhu Xi's account as not just theoretically mistaken but dangerously misleading for those seeking to improve themselves ethically. However, Wang recognized that students who had memorized Zhu Xi's interpretation to prepare for the civil service examinations would have difficulty understanding the text any other way. Consequently, when he accepted a new disciple, Wang often began by explaining his alternative interpretation of the Great Learning, and invited students to ask questions. Wang's approach is preserved in a brief but densely argued work, "Questions on the Great Learning." Let's begin at the apparent foundation of the Great Learning's program: ge wu. Whereas Zhu Xi argues that ge wu is literally "reaching things," meaning to intellectually grasp the Pattern in things and situations, Wang argues that ge wu means "rectifying things," including both one's own thoughts and the objects of those thoughts. For Zhu Xi, ge wu is primarily about gaining knowledge, while for Wang it is about motivation and action.[8] The fact that the Great Learning explicates ge wu in terms of "extending knowledge" might seem to support Zhu Xi's interpretation. However, Wang offers a plausible alternative explanation: "To fully extend one's knowledge is not like the so-called 'filling out' of what one knows that later scholars [like Zhu Xi] talk about. It is simply to extend fully the 'pure knowing' of my own mind. Pure knowing is what [Mencius] was talking about when he said that all humans have 'the mind of approval and disapproval.' The mind of approval and disapproval 'knows without pondering' and is 'capable without learning' " (Tiwald and Van Norden 2014, 248, glosses mine).[9] Wang is suggesting that "extending knowledge" refers simply to exercising our innate faculty of moral awareness, which Mencius refers to as "pure knowing." As David S. Nivison explains, "We might say that Wang's 'extending' of knowledge is more like extending one's arm than, say, extending one's vocabulary" (Nivison 1996a, 225). According to Zhu Xi, the opening of the Great Learning lists a series of steps that are, at least to some extent, temporally distinct. Wang demurs: > > While one can say that there is an ordering of first and last in this > sequence of spiritual training, the training itself is a unified whole > that cannot be divided into any ordering of first and last. While > this sequence of spiritual training cannot be divided into any > ordering of first and last, only when every aspect of its practice is > highly refined can one be sure that it will not be deficient in the > slightest degree. (Tiwald and Van Norden 2014, 250) > Wang's argument here is similar to the one he made about the verbal distinction between knowing and acting (Section 3, above). He claimed that the ancient sages recognized that knowing and acting were ultimately one thing, but sometimes discussed them separately for pedagogic purposes, to help those who underemphasized one aspect of this unity. Similarly, Wang suggests, the Great Learning uses multiple terms to describe various aspects of the unified exercise of moral agency, but does not mean to suggest that these are actually distinct temporal stages: > > ...we can say that "self," "mind," > "thoughts," "knowledge," and > "things" describe the sequence of spiritual training. > While each has its own place, in reality they are but a single thing. > We can say that "ge wu," "extending," > "making Sincere," "correcting," and > "cultivating" describe the spiritual training used in the > course of this sequence. While each has its own name, in reality they > are but a single affair. What do we mean by "self"? It > is the way we refer to the physical operations of the mind. What do > we mean by "mind"? It is the way we refer to the luminous > and intelligent master of the person. (Translation slightly modified > from Tiwald and Van Norden 2014, 247) > Wang even interprets the key metaphor of the Great Learning in a very different way from Zhu Xi. For Zhu Xi, hating evil "like hating a hateful odor," and loving good "like loving a lovely sight" are goals that we must aspire to, but can only achieve after an arduous process of ethical cultivation. For Wang Yangming, these phrases describe what our attitudes toward good and evil can and should be at the very inception of ethical cultivation: > > ...the Great Learning gives us examples of true knowing > and acting, saying it is "like loving a lovely sight" or > "hating a hateful odor." Seeing a lovely sight is a case > of knowing, while loving a lovely sight is a case of acting. As soon > as one sees that lovely sight, one naturally loves it. It is not as > if you first see it and only then, intentionally, you decide to love > it. Smelling a hateful odor is a case of knowing, while hating a > hateful odor is a case of acting. As soon as one smells that hateful > odor, one naturally hates it. It is not as if you first smell it and > only then, intentionally, you decide to hate it. Consider the case of > a person with a stuffed-up nose. Even if he sees a malodorous > object right in front of him, the smell does not reach him, and so he > does not hate it. This is simply not to know the hateful > odor. (Tiwald and Van Norden 2014, 267) > In summary, Zhu Xi and Wang agree that the Great Learning is an authoritative statement on ethical cultivation, expressing the wisdom of the ancient sages. However, for Zhu Xi, it is analogous to a recipe, with distinct steps that must be performed in order. For Wang, the Great Learning is analogous to a description of a painting, in which shading, coloring, composition, perspective and other factors are aspects of a unified effect. ## 5. Metaphysics Wang was not primarily interested in theoretical issues. However, some of his comments suggest a subtle metaphysical view that supports his conception of ethics. This metaphysics is phrased in terms of the Pattern/qi framework (see Section 2, above), and also makes use of the Substance/Function distinction. Substance (ti), literally body, is an entity in itself, while Function is its characteristic or appropriate activity or manifestation: a lamp is Substance, its light is function; an eye is Substance, seeing is its Function; water is Substance, waves are its Function. The Substance/Function distinction goes back to the Daoists of the Han dynasty (202 BCE-220 CE) and became central among Chinese Buddhists before being picked up by Neo-Confucians. Part of the attraction of this vocabulary is that it gives philosophers drawn to a sort of monism the ability to distinguish between two aspects of what they regard as ultimately a unity. Wang argues that the human mind is identical with the Pattern of the universe, and as such it forms "one body" (yi ti, one Substance) with "Heaven, Earth, and the myriad creatures" of the world. The difference between the virtuous and the vicious is that the former recognize that their minds form one body with everything else, while the latter, "because of the space between their own physical form and those of others, regard themselves as separate" (Tiwald and Van Norden 2014, 241). As evidence for the claim that the minds of all humans form one body with the rest of the universe, Wang appeals to a thought experiment first formulated by Mencius: "when they see a child [about to] fall into a well, [humans] cannot avoid having a mind of alarm and compassion for the child" (Tiwald and Van Norden 2014, 241-242; Mengzi 2A6). Wang then anticipates a series of objections, and offers further thought experiments to motivate the conclusion that only some underlying metaphysical identity between humans and other things can account for the broad range of our reactions to them: > > Someone might object that this response is because the child belongs > to the same species. But when they hear the anguished cries or see > the frightened appearance of birds or beasts, they cannot avoid a > sense of being unable to bear it. ... Someone might object that > this response is because birds and beasts are sentient creatures. But > when they see grass or trees uprooted and torn apart, they cannot > avoid feeling a sense of sympathy and distress. ... Someone might > object that this response is because grass and trees have life and > vitality. But when they see tiles and stones broken and destroyed, > they cannot avoid feeling a sense of concern and regret. ... This > shows that the benevolence that forms one body [with Heaven, Earth, > and the myriad creatures] is something that even the minds of petty > people possess. (Tiwald and Van Norden 2014, 242) > Wang's first three thought experiments seem fairly compelling at first glance. Almost all humans would, at least as a first instinct, have "alarm and compassion" if suddenly confronted with the sight of a child about to fall into a well. In addition, humans often do show pity for the suffering of non-human animals. Finally, the fact that humans maintain public parks and personal gardens shows some kind of concern for plants. It is not a decisive objection to Wang's view that humans often fail to manifest benevolence to other humans, non-human animals, or plants. Wang is arguing that all humans manifest these responses sometimes (and that this is best explained by his favored metaphysics). Like all Neo-Confucians, Wang readily acknowledges that selfish desires frequently block the manifestation of our shared nature. However, there are three lines of objection to Wang's view that are harder to dismiss. (1) There is considerable empirical evidence that some humans never manifest any compassion for the suffering of others. In the technical literature of psychology, these people are a subset of those identified as having "Antisocial Personality Disorder" as defined in DSM-V.[10] (2) Wang asserts that when humans "see tiles and stones broken and destroyed, they cannot avoid feeling a sense of concern and regret." This claim is important for Wang's argument, because he takes this reaction to be evidence for the conclusion that our minds are ultimately "one Substance" with everything in the universe, not merely with members of our species, or other sentient creatures, or other living things. We can perhaps motivate Wang's intuition by considering how we might react if we saw that someone had spray painted graffiti on Half Dome in Yosemite National Park. The defacement of this scenic beauty would probably provoke sadness in those of us with an eye for natural beauty. However, it is certainly not obvious that everyone manifests even sporadic concern for "tiles and stones," which is what he needs for his conclusion. (3) Even if we grant Wang his intuitions, it is not clear that the particular metaphysics he appeals to is the best explanation of those intuitions. As Darwin himself suggested, our compassion for other humans can be explained in evolutionary terms. In addition, the "biophilia hypothesis" (Wilson [1984]) provides an evolutionary explanation for the human fondness for other animals and plants. There is not an obvious evolutionary explanation for why humans seem engaged by non-living natural beauty, like mountain peaks, but as we have seen it seems questionable how common this trait is. ## 6. Influence Wang is regarded, along with Lu Xiangshan, as one of the founders of the Lu-Wang School of Neo-Confucianism, or the School of Mind. This is one of the two major wings of Neo-Confucianism, along with the Cheng-Zhu School (named after Cheng Yi and Zhu Xi), or School of Pattern. He has frequently been an inspiration for critics of the orthodox Cheng-Zhu School, not just in China but also in Japan. In Japan, his philosophy is referred to as Oyomeigaku, and its major adherents included Nakae Toju (1608-1648) and Kumazawa Banzan (1619-1691). Wang's thought was also an inspiration for some of the leaders of the Meiji Restoration (1868), which began Japan's rapid modernization.[11] In China, Confucianism underwent a significant shift during the Qing dynasty (1644-1911) with the development of the Evidential Research movement. Evidential Research emphasized carefully documented and tightly argued work on concrete issues of philology, history, and even mathematics and civil engineering. Such scholars generally looked with disdain on "the Song-Ming Confucians" (including both Zhu Xi and Wang Yangming), whom they accused of producing "empty words" that could not be substantiated.[12] One of the few Evidential Research scholars with a serious interest in ethics was Dai Zhen (1724-1777). However, he too was critical of both the Cheng-Zhu and the Lu-Wang schools. The three major prongs of Dai's critique of Neo-Confucians like Wang were that they (1) encouraged people to treat their subjective opinions as the deliverances of some infallible moral sense, (2) projected Buddhist-inspired concepts back onto the ancient Confucian classics, and (3) ignored the ethical value of ordinary physical desires. One of the major trends in contemporary Chinese philosophy is "New Confucianism." New Confucianism is a movement to adapt Confucianism to modern thought, showing how it is consistent with democracy and modern science. New Confucianism is distinct from what we in the West call Neo-Confucianism, but it adopts many Neo-Confucian concepts, in particular the view that humans share a trans-personal nature which is constituted by the universal Pattern. Many New Confucians agree with Mou Zongsan (1909-1995) that Wang had a deeper and more orthodox understanding of Confucianism than did Zhu Xi.[13] Wang's philosophy is of considerable intrinsic interest, because of the ingenuity of his arguments, the systematicity of his views, and the precision of his textual exegesis. Beyond that, Wang's work has the potential to inform contemporary ethics. Although his particular metaphysics may not be appealing, many of his ideas can be naturalized. It may be hard to believe that everything is unified by a shared, underlying Pattern, but it does seem plausible that we are deeply dependent upon one another and upon our natural environment for our survival and our identities. I am a husband, a father, a teacher, and a researcher, but only because I have a wife, children, students, and colleagues. In some sense, we do form "one body" with others, and Wang provides provocative ideas about how we should respond to this insight. In addition, Wang's fundamental criticism of Zhu Xi's approach, that it produces pedants who only study and talk about ethics, rather than people who strive to actually be ethical, has considerable contemporary relevance, particularly given the empirical evidence that our current practices of ethical education have little positive effect on ethical behavior (Schwitzgebel and Rust 2014; and cf. Schwitzgebel 2013 (Other Internet Resources)).
war	## 1. Traditionalists and Revisionists Contemporary just war theory is dominated by two camps: traditionalist and revisionist.[3] The traditionalists might as readily be called legalists. Their views on the morality of war are substantially led by international law, especially the law of armed conflict. They aim to provide those laws with morally defensible foundations. States (and only states) are permitted to go to war only for national defence, defence of other states, or to intervene to avert "crimes that shock the moral conscience of mankind" (Walzer 2006: 107). Civilians may not be targeted in war, but all combatants, whatever they are fighting for, are morally permitted to target one another, even when doing so foreseeably harms some civilians (so long as it does not do so excessively).[4] Revisionists question the moral standing of states and the permissibility of national defence, argue for expanded permissions for humanitarian intervention, problematise civilian immunity, and contend that combatants fighting for wrongful aims cannot do anything right, besides lay down their weapons. Most revisionists are moral revisionists only: they deny that the contemporary law of armed conflict is intrinsically morally justified, but believe, mostly for pragmatic reasons, that it need not be substantially changed. Some, however, are both morally and legally revisionist. And even moral revisionists' disagreement with the traditionalists is hardly ersatz: most believe that, faced with a clash between what is morally and what is legally permitted or prohibited, individuals should follow their conscience rather than the law.[5] The traditionalist view received its most prominent exposition the same year as it was decisively codified in international law, in the first additional protocol to the Geneva Conventions. Michael Walzer's Just and Unjust Wars, first published in 1977, has been extraordinarily influential among philosophers, political scientists, international lawyers, and military practitioners. Among its key contributions were its defence of central traditionalist positions on national defence, humanitarian intervention, discrimination, and combatant equality. Early revisionists challenged Walzer's views on national defence (Luban 1980a) and humanitarian intervention (Luban 1980b). Revisionist criticism of combatant equality and discrimination followed (Holmes 1989; McMahan 1994; Norman 1995). Since then there has been an explosion of revisionist rebuttals of Walzer (for example Rodin 2002; McMahan 2004b; McPherson 2004; Arneson 2006; Fabre 2009; McMahan 2009; Fabre 2012). Concurrently, many philosophers welcomed Walzer's conclusions, but rejected his arguments. They have accordingly sought firmer foundations for broadly traditionalist positions on national defence (Benbaji 2014; Moore 2014), humanitarian intervention (Coady 2002), discrimination (Rodin 2008b; Dill and Shue 2012; Lazar 2015c), and especially combatant equality (Zohar 1993; Kutz 2005; Benbaji 2008; Shue 2008; Steinhoff 2008; Emerton and Handfield 2009; Benbaji 2011). We will delve deeper into these debates in what follows. First, though, some methodological groundwork. Traditionalists and revisionists alike often rely on methodological or second-order premises, to the extent that one might think that the first-order questions are really just proxy battles through which they work out their deeper disagreements (Lazar and Valentini forthcoming). ## 2. How Should We Think about the Morality of War? ### 2.1 Historical vs Contemporary Just War Theory For the sake of concision this entry discusses only contemporary analytical philosophers working on war. Readers are directed to the excellent work of philosophers and intellectual historians such as Greg Reichberg, Pablo Kalmanovitz, Daniel Schwartz, and Rory Cox to gain further insights about historical just war theory (see, in particular, Cox 2016; Kalmanovitz 2016; Reichberg 2016; Schwartz 2016). ### 2.2 Institutions and Actions Within contemporary analytical philosophy, there are two different ways in which moral and political philosophers think about war (Lazar and Valentini forthcoming). On the first, institutionalist, approach, philosophers' primary goal is to establish what the institutions regulating war should be. In particular, we should prescribe morally justified laws of war. We then tell individuals and groups that they ought to follow those laws. On the second approach, we should focus first on the moral reasons that apply directly to individual and group actions, without the mediating factor of institutions. We tell individuals and groups to act as their moral reasons dictate. Since this approach focuses not on the institutions that govern our interactions, but on those interactions themselves, we will call it the "interactional" approach.[6] In general, the institutionalist approach is favoured by indirect consequentialists and contractualists. Indirect consequentialists believe these institutions are justified just in case they will in fact have better long-run results than any feasible alternative institutions (see Mavrodes 1975; Dill and Shue 2012; Shue 2013; Waldron 2016). Contractualists believe these institutions ground or reflect either an actual or a hypothetical contract among states and/or their citizens, which specifies the terms of their interaction in war (see Benbaji 2008, 2011, 2014; Statman 2014). Non-contractualist deontologists and direct- or act-consequentialists tend to prefer the interactional approach. Their central question is: what moral reasons bear directly on the permissibility of killing in war? This focus on killing might seem myopic--war involves much more violence and destruction than the killing alone. However, typically this is just a heuristic device; since we typically think of killing as the most presumptively wrongful kind of harm, whatever arguments one identifies that justify killing are likely also to justify lesser wrongs. And if the killing that war involves cannot be justified, then we should endorse pacifism. Any normative theory of war should pay attention both to what the laws of war should be, and to what we morally ought to do. These are two distinct but equally important questions. And they entail the importance of a third: what ought we to do all things considered, for example when law and morality conflict? Too much recent just war theory has focused on arguing that philosophical attention should be reserved to one of the first two of these questions (Buchanan 2006; Shue 2008, 2010; Rodin 2011b). Not enough has concentrated on the third (though see McMahan 2008; Lazar 2012a). Although this entry touches on the first question, it focuses on the second. Addressing the first requires detailed empirical research and pragmatic political speculation, both of which are beyond my remit here. Addressing the third takes us too deep into the minutiae of contemporary just war theory for an encyclopaedia entry. What's more, even institutionalists need some answer to the second question--and so some account of the interactional morality of war. Rule-consequentialists need an account of the good (bad) that they are hoping that the ideal laws of war will maximise (minimise) in the long run. This means, for example, deciding whether to aim to minimise all harm, or only to minimise wrongful harm. The latter course is much more plausible--we wouldn't want laws of war that, for example, licensed genocide just in case doing so leads to fewer deaths overall. But to follow this course, we need to know which harms are (extra-institutionally) wrongful. Similarly, contractualists typically acknowledge various constraints on the kinds of rules that could form the basis of a legitimate contract, which, again, we cannot work out without thinking about the extra-institutional morality of war (Benbaji 2011). ### 2.3 Overarching Disputes in Contemporary Analytical Just War Theory Even within interactional just war theory, several second-order disagreements underscore first-order disputes. First: when thinking about the ethics of war, what kinds of cases should we use to test our intuitions and our principles? We can start by thinking about actual wars and realistic wartime scenarios, paying attention to international affairs and military history. Or, more clinically, we can construct hypothetical cases to isolate variables and test their impact on our intuitions. Some early revisionists relied heavily on highly artificial cases (e.g., McMahan 1994; Rodin 2002). They were criticized for this by traditionalists, who generally use more empirically-informed examples (Walzer 2006). But one's standpoint on the substantive questions at issue between traditionalists and revisionists need not be predetermined by one's methodology. Revisionists can pay close attention to actual conflicts (e.g., Fabre 2012). Traditionalists can use artificial hypotheticals (e.g., Emerton and Handfield 2009; Lazar 2013). Abstraction forestalls unhelpful disputes over historical details. It also reduces bias--we are inclined to view actual conflicts through the lens of our own political allegiances. But it also has costs. We should be proportionately less confident of our intuitions the more removed the test case is from our lived experience. Philosophers' scenarios involving mind-control, armed pedestrians, trolleys, meteorites, and incredibly complicated causal sequences are pure exercises in imagination. How can we trust our judgements about such cases more than we trust our views on actual, realistic scenarios? What's more, abandoning the harrowing experience of war for sanitized hypothetical cases might be not merely epistemically unsound, but also disrespectful of the victims of war. Lastly, cleaned-up examples often omit morally relevant details--for instance, assuming that everyone has all the information relevant to their choice, rather than acknowledging the "fog of war", and making no allowances for fear or trauma. Artificial hypotheticals have their place, but any conclusions they support must be tested against the messy reality of war. What's more, our intuitive judgements should be the starting-point for investigation, rather than its end. The second divide is related to the first. Reductivists think that killing in war must be justified by the same properties that justify killing outside of war. Non-reductivists, sometimes called exceptionalists, think that some properties justify killing in war that do not justify killing outside of war.[7] Most exceptionalists think that specific features of killing in war make it morally different from killing in ordinary life--for example, the scale of the conflict, widespread and egregious non-compliance with fundamental moral norms, the political interests at stake, the acute uncertainty, the existence of the law of armed conflict, or the fact that the parties to the conflict are organized groups. A paradigm reductivist, by contrast, might argue that justified wars are mere aggregates of justified acts of individual self- and other-defence (see Rodin 2002; McMahan 2004a). Reductivists are much more likely to use far-fetched hypothetical cases, since they think there is nothing special about warfare. The opposite is true for exceptionalists. Walzer's first critics relied on reductivist premises to undermine the principles of national defence (Luban 1980a; Rodin 2002), discrimination (Holmes 1989; McMahan 1994), and combatant equality (Holmes 1989; McMahan 1994). Many traditionalists replied by rejecting reductivism, arguing that there is something special about war that justifies a divergence from the kinds of judgements that are appropriate to other kinds of conflict (Zohar 1993; Kutz 2005; Benbaji 2008; Dill and Shue 2012). Again, some philosophers buck these overarching trends (for reductivist traditionalist arguments, see e.g., Emerton and Handfield 2009; Lazar 2015c; Haque 2017; for non-reductivist revisionist arguments, see e.g., Ryan 2016). The debate between reductivism and exceptionalism is overblown--the concept of "war" is vague, and while typical wars involve properties that are not instantiated in typical conflicts outside of war, we can always come up with far-fetched hypotheticals that don't involve those properties, which we wouldn't call "wars". But this masks a deeper methodological disagreement: when thinking about the morality of war, should we start by thinking about war, or by thinking about the permissible use of force outside of war? Should we model justified killing in war on justified killing outside of war? Or, in focusing on the justification of killing in war, might we then discover that there are some non-canonical cases of permissible killing outside of war? My own view is that thinking about justified killing outside of war has its place, but must be complemented by thinking about war directly. Next, we can distinguish between individualists and collectivists; and we can subdivide them further into evaluative and descriptive categories. Evaluative individualists think that a collective's moral significance is wholly reducible to its contribution to the well-being of the individuals who compose it. Evaluative collectivists think that collectives can matter independently of how they contribute to individual well-being. Descriptive individualists think that any act that might appear to be collective is reducible to component acts by individuals. Descriptive collectivists deny this, thinking that some acts are irreducibly collective.[8] Again, the dialectic of contemporary just war theory involves revisionists first arguing that we cannot vindicate traditionalist positions on descriptively and evaluatively individualist grounds, with some traditionalists then responding by rejecting descriptive (Kutz 2005; Walzer 2006; Lazar 2012b) and evaluative individualism (Zohar 1993). And again there are outliers--individualist traditionalists (e.g., Emerton and Handfield 2009) and collectivist revisionists (e.g., Bazargan 2013). Unlike the reductivist/exceptionalist divide, the individualist/collectivist split cannot be resolved by thinking about the morality of war on its own. War is a useful test case for theories of collective action and the value of collectives, but no more than that. Intuitions about war are no substitute for a theory of collective action. Perhaps some collectives have value beyond their contribution to the well-being of their members. For example, they might instantiate justice, or solidarity, which can be impersonally valuable (Temkin 1993). It is doubtful, however, that groups have interests independent from the well-being of their members. On the descriptive side, even if we can reduce collective actions to the actions of individual members, this probably involves such complicated contortions that we should seriously question whether it is worth doing (Lazar 2012b). ### 2.4 Dividing up the Subject Matter Traditionally, just war theorists divide their enquiry into reflection on the resort to war--jus ad bellum--and conduct in war--jus in bello. More recently, they have added an account of permissible action post-war, or jus post bellum. Others suggest an independent focus on war exit, which they have variously called jus ex bello and jus terminatio (Moellendorf 2008; Rodin 2008a). These Latin labels, though unfortunately obscurantist, serve as a useful shorthand. When we refer to ad bellum justice, we mean to evaluate the permissibility of the war as a whole. This is particularly salient when deciding to launch the war. But it is also crucial for the decision to continue fighting. Jus ex bello, then, fits within jus ad bellum. The jus in bello denotes the permissibility of particular actions that compose the war, short of the war as a whole. ### 2.5 The Decisive Role of Necessity and Proportionality Traditional just war theory construes jus ad bellum and jus in bello as sets of principles, satisfying which is necessary and sufficient for a war's being permissible. Jus ad bellum typically comprises the following six principles: 1. Just Cause: the war is an attempt to avert the right kind of injury. 2. Legitimate Authority: the war is fought by an entity that has the authority to fight such wars. 3. Right Intention: that entity intends to achieve the just cause, rather than using it as an excuse to achieve some wrongful end. 4. Reasonable Prospects of Success: the war is sufficiently likely to achieve its aims. 5. Proportionality: the morally weighted goods achieved by the war outweigh the morally weighted bads that it will cause. 6. Last Resort (Necessity): there is no other less harmful way to achieve the just cause. Typically the jus in bello list comprises: 1. Discrimination: belligerents must always distinguish between military objectives and civilians, and intentionally attack only military objectives. 2. Proportionality: foreseen but unintended harms must be proportionate to the military advantage achieved. 3. Necessity: the least harmful means feasible must be used. These all matter to the ethics of war, and will be addressed below. However, it is unhelpful to view them as a checklist of necessary and sufficient conditions. When they are properly understood, only proportionality and necessity (in the guise of last resort in the jus ad bellum) are necessary conditions for a war, or an act in a war, to be permissible, since no matter how badly going to war fails the other ad bellum criteria (for example) it might still be permissible because it is the least awful of one's alternatives, and so satisfies the necessity and proportionality constraints. To get an intuitive grasp on necessity and proportionality, note that if someone threatens my life, then killing her would be proportionate; but if I could stop her by knocking her out, then killing her would be unnecessary, and so impermissible. The necessity and proportionality constraints have the same root: with few exceptions (perhaps when it is deserved), harm is intrinsically bad. Harms (and indeed all bads) that we cause must therefore be justified by some positive reason that counts in their favour--such as good achieved or evil averted (Lazar 2012a). Both the necessity and proportionality constraints involve comparing the bads caused by an action with the goods that it achieves. They differ only in the kinds of options they compare. The use of force is proportionate when the harm done is counterbalanced by the good achieved in averting a threat. To determine this, we typically compare the candidate course of action with what would happen if we allowed the threat to eventuate. Of course, in most cases we will have more than one means of averting or mitigating the threat. And a harmful option can be permissible only if all the harm that it involves is justified by a corresponding good achieved. If some alternative would as successfully avert the threat, but cause less harm, then the more harmful option is impermissible, because it involves unnecessary harm. Where an option O aims to avert a threat T, we determine O's necessity by comparing it with all the other options that will either mitigate or avert T. We determine its proportionality by comparing it with the harm suffered if T should come about. The only difference between the proportionality and necessity constraints is that the former involves comparing one's action with a very specific counterfactual scenario--in which we don't act to avert the threat--while the latter involves comparing it with all your available options that have some prospect of averting or mitigating the threat. In my view, we should simply expand this so that the necessity constraint compares all your available options bar none. Then proportionality would essentially involve comparing each option with the alternative of doing nothing, while necessity would involve comparing all options (including doing nothing) in terms of their respective balances of goods and bads. On this approach, necessity would subsume proportionality. But this is a technical point with little substantive payoff. More substantively, necessity and proportionality judgements concern consequences, and yet they are typically made ex ante, before we know what the results of our actions will be. They must therefore be modified to take this uncertainty into account. The most obvious solution is simply to refer to expected threats and expected harms, where the expected harm of an option O is the probability-weighted average of the harms that might result if I take O, and the expected threat is the probability-weighted average of the consequences of doing nothing to prevent the threat--allowing for the possibility that the threat might not eventuate at all (Lazar 2012b). We would also have to factor in the options' probability of averting the threat. This simple move obscures a number of important and undertheorised issues that we cannot discuss in detail here. For now, we must simply note that proportionality and necessity must be appropriately indexed to the agent's uncertainty. Necessity and proportionality judgements involve weighing harms inflicted and threats averted, indeed all relevant goods and bads. The simplest way to proceed would be to aggregate the harms to individual people on each side, and call the act proportionate just in case it averts more harm than it causes, and necessary just in case no alternative involves less harm. But few deontologists, and indeed few non-philosophers, think in this naively aggregative way. Instead we should weight harms (etc.) according to factors such as whether the agent is directly responsible for them, and whether they are intended or merely foreseen.[9] Many also think that we can, perhaps even must, give greater importance in our deliberations to our loved ones (for example) than to those tied to us only by the common bonds of humanity (Hurka 2007; Lazar 2013; for criticism, see Lefkowitz 2009). Similarly, we might justifiably prioritise defending our own state's sovereignty and territorial integrity, even when doing so would not be impartially best.[10] Only when all these and other factors (many of which are discussed below) are taken into consideration can we form defensible conclusions about which options are necessary and proportionate. The other elements of the ethics of war contribute to the evaluation of proportionality and necessity, in one (or more) of three ways: identifying positive reasons in favour of fighting; delineating the negative reasons against fighting; or as staging-posts on the way to judgements of necessity and proportionality. Given the gravity of the decision to go to war, only very serious threats can give us just cause to fight. So if just cause is satisfied, then you have weighty positive reasons to fight. Lacking just cause does not in itself aggravate the badness of fighting, but does make it less likely that the people killed in pursuing one's war aims will be liable to be killed (more on this below, and see McMahan 2005a), which makes killing very hard to justify. Even if having a just cause is not strictly speaking a necessary condition for warfare to be permissible, the absence of a just cause makes it very difficult for a war to satisfy proportionality. If legitimate authority is satisfied then additional positive reasons count in favour of fighting (see below). If it is not satisfied, then this adds an additional reason against fighting, which must be overcome for fighting to be proportionate. The "reasonable prospects of success" criterion is a surmountable hurdle on the way to proportionality. Typically, if a war lacks reasonable prospects of success, then it will be disproportionate, since wars always involve causing significant harms, and if those harms are likely to be pointless then they are unlikely to be justified. But of course sometimes one's likelihood of victory is very low, and yet fighting is still the best available option, and so necessary and proportionate. Having reasonable prospects of success matters only for the same reasons that necessity and proportionality matter. If necessity and proportionality are satisfied, then the reasonable prospects of success standard is irrelevant. Right intention may also be irrelevant, but insofar as it matters its absence would be a reason against fighting; having the right intention does not give a positive reason to fight. Lastly, discrimination is crucial to establishing proportionality and necessity, because it tells us how to weigh the lives taken in war. ## 3. Jus ad Bellum ### 3.1 Just Cause Wars destroy lives and environments. In the eight years following the Iraq invasion in 2003, half a million deaths were either directly or indirectly caused by the war (Hagopian et al. 2013). Direct casualties from the Second World War numbered over 60 million, about 3 per cent of the world's population. War's environmental costs are less commonly researched, but are obviously also extraordinary (Austin and Bruch 2000). Armed forces use fuels in Olympian quantities: in the years from 2000-2013, the US Department of Defense accounted for around 80% of US federal government energy usage, between 0.75 and 1 quadrillion BTUs per year--a little less than all the energy use that year in Denmark and Bulgaria, a little more than Slovakia and Serbia (Energy Information Administration 2015a,b). They also directly and indirectly destroy habitats and natural resources--consider, for example, the Gulf war oil spill (El-Baz and Makharita 1994). For both our planet and its inhabitants, wars are truly among the very worst things we can do. War can be necessary and proportionate only if it serves an end worth all this death and destruction. Hence the importance of having a just cause. And hence too the widespread belief that just causes are few and far between. Indeed, traditional just war theory recognizes only two kinds of justification for war: national defence (of one's own state or of an ally) and humanitarian intervention. What's more, humanitarian intervention is permissible only to avert the very gravest of tragedies--"crimes that shock the moral conscience of mankind" (Walzer 2006: 107). Walzer argued that states' claims to sovereignty and territorial integrity are grounded in the human rights of their citizens, in three ways. First, states ensure individual security. Rights to life and liberty have value "only if they also have dimension" (Walzer 2006: 58), which they derive from states' borders--"within that world, men and women... are safe from attack; once the lines are crossed, safety is gone" (Walzer 2006: 57). Second, states protect a common life, made by their citizens over centuries of interaction. If the common life of a political community is valued by its citizens, then it is worth fighting for. Third, they have also formed a political association, an organic social contract, whereby individuals have, over time and in informal ways, conceded aspects of their liberty to the community, to secure greater freedom for all. These arguments for national defence are double-edged. They helped explain why wars of national defence are permissible, but also make justifying humanitarian intervention harder. One can in principle successfully conclude a war in defence of oneself or one's allies without any lasting damage to the political sovereignty or territorial integrity of any of the contending parties. In Walzer's view, humanitarian interventions, in which one typically defends people against their own state, necessarily undermine political sovereignty and territorial integrity. So they must meet a higher burden of justification. Walzer's traditionalist stances on national defence and humanitarian intervention met heavy criticism. Early sceptics (Doppelt 1978; Beitz 1980; Luban 1980a) challenged Walzer's appeal to the value of collective freedom, noting that in diverse political communities freedom for the majority can mean oppression for the minority (see also Caney 2006). In modern states, can we even speak of a single common life? Even if we can, do wars really threaten it, besides in extreme cases? And even if our common life and culture were threatened, would their defence really justify killing innocent people? Critics also excoriated Walzer's appeal to individual rights (see especially Wasserstrom 1978; Luban 1980b). They questioned the normative purchase of his metaphor of the organic social contract (if hypothetical contracts aren't worth the paper they're not written on, then what are metaphorical contracts worth?). They challenged his claim that states guarantee individual security: most obviously, when humanitarian intervention seems warranted, the state is typically the greatest threat to its members. David Rodin (2002) advanced the quintessentially reductivist critique of Walzer, showing that his attempt to ground state defence in individual defensive rights could not succeed. He popularized the "bloodless invasion objection" to this argument for national defensive rights. Suppose an unjustly aggressing army would secure its objectives without loss of life if only the victim state offers no resistance (the 2001 invasion of Afghanistan, and the 2003 invasion of Iraq, arguably meet this description, as might some of Russia's territorial expansions). If the right of national defence is grounded in states' members' rights to security, then in these cases there would be no right of national defence, because their lives are at risk only if the victim state fights back. And yet we typically do think it permissible to fight against annexation and regime change. By undermining the value of sovereignty, revisionists lowered the bar against intervening militarily in other states. Often these arguments were directly linked: some think that if states cannot protect the security of their members, then they lack any rights to sovereignty that a military intervention could undermine (Shue 1997).[11] Caney (2005) argues that military intervention could be permissible were it to serve individual human rights better than non-intervention. Others countenance so-called "redistributive wars", fought on behalf of the global poor to force rich states to address the widespread violations of fundamental human rights caused by their economic policies (Luban 1980b; Fabre 2012; Lippert-Rasmussen 2013; Overland 2013). Other philosophers, equally unpersuaded by Walzer's arguments, nonetheless reject a substantively revisionist take on just cause. If the individual self-defence-based view of jus ad bellum cannot justify lethal defence against "lesser aggression", then we could follow Rodin (2014), and argue for radically revisionist conclusions about just cause; or we could instead reject the individual self-defence-based approach to justifying killing in war (Emerton and Handfield 2014; Lazar 2014). Some think we can solve the "problem of lesser aggression" by invoking the importance of deterrence, as well as the impossibility of knowing for sure that aggression will be bloodless (Fabre 2014). Others think that we must take proper account of people's interest in having a democratically elected, or at least home-grown, government, to justify national defence. On one popular account, although no individual could permissibly kill to protect her own "political interests", when enough people are threatened, their aggregated interests justify going to war (Hurka 2007; Frowe 2014). Counterintuitively, this means that more populous states have, other things equal, more expansive rights of national defence. However, perhaps states have a group right to national defence, which requires only that a sufficient number of individuals have the relevant political interests--any excess over the threshold is morally irrelevant. Many already think about national self-determination in this way: the population of the group seeking independence has to be sufficiently large before we take their claim seriously, but differences above that threshold matter much less (Margalit and Raz 1990). The revisionist take on humanitarian intervention might also have some troubling results. If sovereignty and territorial integrity matter little, then shouldn't we use military force more often? As Kutz (2014) has argued, revisionist views on national defence might license the kind of military adventurism that went so badly wrong in Iraq, where states have so little regard for sovereignty that they go to war to improve the domestic political institutions of their adversaries. We can resolve this worry in one of two ways. First, recall just how infrequently military intervention succeeds. Since it so often not only fails, but actually makes things worse, we should use it only when the ongoing crimes are so severe that we would take any risk to try to stop them. Second, perhaps the political interests underpinning the state's right to national defence are not simply interests in being part of an ideal liberal democracy, but in being governed by, very broadly, members of one's own nation, or perhaps even an interest in collective self-determination. This may take us back to Walzer's "romance of the nation-state", but people clearly do care about something like this. Unless we want to restrict rights of national defence to liberal democracies alone (bearing in mind how few of them there are in the world), we have to recognize that our political interests are not all exclusively liberal-democratic. What of redistributive wars? Too often arguments on this topic artfully distinguish between just cause and other conditions of jus ad bellum (Fabre 2012). Even when used by powerful states against weak adversaries, military force is rarely a moral triumph. It tends to cause more problems than it solves. Redistributive wars, as fought on behalf of the "global poor" against the "global rich", would obviously fail to achieve their objectives, indeed they would radically exacerbate the suffering of those they aim to help. So they would be disproportionate, and cannot satisfy the necessity constraint. The theoretical point that, in principle, not only national defence and humanitarian intervention could give just causes for war is sound. But this example is in practice irrelevant (for a robust critique of redistributive wars, see Benbaji 2014). And yet, given the likely path of climate change, the future might see resource wars grow in salience. As powerful states find themselves lacking crucial resources, held by other states, we might find that military attack is the best available means to secure these resources, and save lives. Perhaps in some such circumstances resource wars could be a realistic option. ### 3.2 Just Peace The goods and bads relevant to ad bellum proportionality and necessity extend far beyond the armistice. This is obvious, but has recently received much-needed emphasis, both among philosophers and in the broader public debate sparked by the conflicts in Iraq and Afghanistan (Bass 2004; Coady 2008; May 2012). Achieving your just cause is not enough. The aftermath of the war must also be sufficiently tolerable if the war is to be proportionate, all things considered. It is an open question how far into the future we have to look to assess the morally relevant consequences of conflict. ### 3.3 Legitimate Authority Historically, just war theory has been dominated by statists. Most branches of the tradition have had some version of a "legitimate", "proper" or "right" authority constraint, construed as a necessary condition for a war to be ad bellum just.[12] In practice, this means that sovereigns and states have rights that non-state actors lack. International law gives only states rights of national defence and bestows "combatant rights" primarily on the soldiers of states. Although Walzer said little about legitimate authority, his arguments all assume that states have a special moral standing that non-state actors lack. The traditionalist, then, says it matters that the body fighting the war have the appropriate authority to do so. Some think that authority is grounded in the overall legitimacy of the state. Others think that overall legitimacy is irrelevant--what matters is whether the body fighting the war is authorized to do so by the polity that it represents (Lazar forthcoming-b). Either way, states are much more likely to satisfy the legitimate authority condition than non-state actors. Revisionists push back: relying on reductivist premises, they argue that killing in war is justified by the protection of individual rights, and our licence to defend our rights need not be mediated through state institutions. Either we should disregard the legitimate authority condition or we should see it as something that non-state actors can, in fact, fulfil (Fabre 2008; Finlay 2010; Schwenkenbecher 2013). Overall, state legitimacy definitely seems relevant for some questions in war (Estlund 2007; Renzo 2013). But authorization is more fundamental. Ideally, the body fighting the war should be authorized to do so by the institutions of a constitutional democracy. Looser forms of authorization are clearly possible; even a state that is not overall legitimate might nonetheless be authorized by its polity to fight wars of national defence. Authorization of this kind matters to jus ad bellum in two ways. First, fighting a war without authorization constitutes an additional wrong, which has to be weighed against the goods that fighting will bring about, and must pass the proportionality and necessity tests. When a government involves its polity in a war, it uses the resources of the community at large, as well as its name, and exposes it to both moral and prudential risks (Lazar forthcoming-b). Doing this unauthorized is obviously deeply morally problematic. Any form of undemocratic decision-making by governments is objectionable; taking decisions of this magnitude without the population's granting you the right to do so is especially wrong. Second, authorization can allow the government to act on positive reasons for fighting that would otherwise be unavailable. Consider the claim that wars of national defence are in part justified by the political interests of the citizens of the defending state--interests, for example, in democratic participation or in collective self-determination. A government may defend these aggregated political interests only if it is authorized to do so. Otherwise fighting would contravene the very interests in self-determination that it is supposed to protect. But if it is authorized, then that additional set of reasons supports fighting. As a result, democratic states enjoy somewhat more expansive war rights than non-democratic states and non-state movements. The latter two groups cannot often claim the same degree of authorization as democratic states. Although this might not vindicate the current bias in international law towards states, it does suggest that it corresponds to something more than the naked self-interest of the framers of international law--which were, of course, states. This obviously has significant implications for civil wars (see Parry 2016). ### 3.4 Proportionality The central task of the proportionality constraint, recall, is to identify reasons that tell in favour of fighting and those that tell against it. Much of the latter task is reserved for the discussion of jus in bello below, since it concerns weighing lives in war. Among the goods that help make a war proportionate, we have already considered those in the just cause and others connected to just peace and legitimate authority. Additionally, many also think that proportionality can be swayed by reasonable partiality towards one's own state and co-citizens. Think back to the political interests that help justify national defence. If we were wholly impartial, then we should choose the course that will best realise people's political interests overall. So if fighting the defensive war would undermine the political interests of the adversary state's citizens more than it would undermine our own, then we should refuse to fight. But this is not how we typically think about the permission to resort to war: we are typically entitled to be somewhat partial towards the political interests of our co-citizens. Some propose further constraints on what goods can count towards the proportionality of a war. McMahan and McKim (1993) argued that benefits like economic progress cannot make an otherwise disproportionate war proportionate. This is probably true in practice, but perhaps not in principle--that would require a kind of lexical priority between lives taken and economic benefits, and lexical priorities are notoriously hard to defend. After all, economic progress saves lives. Some goods lack weight in ad bellum proportionality, not because they are lexically inferior to other values at stake, but because they are conditional in particular ways. Soldiers have conditional obligations to fulfil their roles, grounded in their contracts, oaths, and their co-citizens' legitimate expectations. That carrying out an operation fulfils my oath gives me a reason to perform that operation, which has to be weighed in the proportionality calculation (Lazar 2015b). But these reasons cannot contribute to ad bellum proportionality in the same way, because they are conditional on the war as a whole being fought. Political leaders cannot plausibly say: "were it not for all the oaths that would be fulfilled by fighting, this war would be disproportionate". This is because fighting counts as fulfilling those oaths only if the political leader decides to take her armed forces to war. Another reason to differentiate between proportionality ad bellum and in bello is that the relevant comparators change for the two kinds of assessment. In a loose sense, we determine proportionality by asking whether some option is better than doing nothing. The comparator for assessing the war as a whole, then, is not fighting at all, ending the war as a whole. That option is not available when considering particular actions within the war--one can only decide whether or not to perform this particular action. ### 3.5 Last Resort (Necessity) Are pre-emptive wars, fought in anticipation of an imminent enemy attack, permissible? What of preventive wars, in which the assault occurs prior to the enemy having any realistic plan of attack (see, in general, Shue and Rodin 2007)? Neoconservatives have recently argued, superficially plausibly, that the criterion of last resort can be satisfied long before the enemy finally launches an attack (see President 2002). The right answer here is boringly familiar. In principle, of course this is possible. But, in practice, we almost always overestimate the likelihood of success from military means and overlook the unintended consequences of our actions. International law must therefore retain its restrictions, to deter the kind of overzealous implementation of the last-resort principle that we saw in the 2003 invasion of Iraq (Buchanan and Keohane 2004; Luban 2004). Another frequently discussed question: what does the "last" in last resort really mean? The idea is simple, and is identical to in bello necessity. Going to war must be compared with the alternative available strategies for dealing with the enemy (which also includes the various ways in which we could submit). Going to war is literally a last resort when no other available means has any prospect of averting the threat. But our circumstances are not often this straitened. Other options always have some chance of success. So if you have a diplomatic alternative to war, which is less harmful than going to war, and is at least as likely to avert the threat, then going to war is not a last resort. If the diplomatic alternative is less harmful, as well as less likely to avert the threat, then the question is whether the reduction in expected harm is great enough for us to be required to accept the reduction in likelihood of averting the threat. If not, then war is your last resort.[13] ## 4. Jus in Bello ### 4.1 Walzer and his Critics The traditionalist jus in bello, as reflected in international law, holds that conduct in war must satisfy three principles: 1. Discrimination: Targeting noncombatants is impermissible.[14] 2. Proportionality: Collaterally harming noncombatants (that is, harming them foreseeably, but unintendedly) is permissible only if the harms are proportionate to the goals the attack is intended to achieve.[15] 3. Necessity: Collaterally harming noncombatants is permissible only if, in the pursuit of one's military objectives, the least harmful means feasible are chosen.[16] These principles divide the possible victims of war into two classes: combatants and noncombatants. They place no constraints on killing combatants.[17] But--outside of "supreme emergencies", rare circumstances in which intentionally killing noncombatants is necessary to avert an unconscionable threat--noncombatants may be killed only unintendedly and, even then, only if the harm they suffer is necessary and proportionate to the intended goals of the attack.[18] Obviously, then, much hangs on what makes one a combatant. This entry adopts a conservative definition. Combatants are (most) members of the organized armed forces of a group that is at war, as well as others who directly participate in hostilities or have a continuous combat function (for discussion, see Haque 2017). Noncombatants are not combatants. There are, of course, many hard cases, especially in asymmetric wars, but they are not considered here. "Soldier" is used interchangeably with "combatant" and "civilian" interchangeably with "noncombatant". Both traditionalist just war theory and international law explicitly license fighting in accordance with these constraints, regardless of one's objectives. In other words, they endorse: Combatant Equality: Soldiers who satisfy Discrimination, Proportionality, and Necessity fight permissibly, regardless of what they are fighting for. [19] We discuss Proportionality and Necessity below; for now let us concentrate on Michael Walzer's influential argument for Discrimination and Combatant Equality, which has proved very controversial. Individual human beings enjoy fundamental rights to life and liberty, which prohibit others from harming them in certain ways. Since fighting wars obviously involves depriving others of life and liberty, according to Walzer, it can be permissible only if each of the victims has, "through some act of his own ... surrendered or lost his rights" (Walzer 2006: 135). He then claims that, "simply by fighting", all combatants "have lost their title to life and liberty" (Walzer 2006: 136). First, merely by posing a threat to me, a person alienates himself from me, and from our common humanity, and so himself becomes a legitimate target of lethal force (Walzer 2006: 142). Second, by participating in the armed forces, a combatant has "allowed himself to be made into a dangerous man" (Walzer 2006: 145), and thus surrendered his rights. By contrast, noncombatants are "men and women with rights, and... they cannot be used for some military purpose, even if it is a legitimate purpose" (Walzer 2006: 137). This introduces the concept of liability into the debate, which we need to define carefully. On most accounts, that a person is liable to be killed means that she is not wronged by being killed. Often this is understood, as it was in Walzer, in terms of rights: everyone starts out with a right to life, but that right can be forfeited or lost, such that one can be killed without that right being violated or infringed. Walzer and his critics all agreed that killing a person intentionally is permissible only if either she has lost the protection of her right to life, or if the good achieved thereby is very great indeed, enough that, though she is wronged, it is not all things considered wrong to kill her. Her right is permissibly infringed. Walzer and his critics believe that such cases are very rare in war, arising only when the alternative to intentionally violating people's right to life is an imminent catastrophe on the order of Nazi victory in Europe (this is an example of a supreme emergency). These simple building blocks give us both Discrimination and Combatant Equality--the former, because noncombatants, in virtue of retaining their rights, are not legitimate objects of attack; the latter, because all combatants lose their rights, regardless of what they are fighting for: hence, as long as they attack only enemy combatants, they fight legitimately, because they do not violate anyone's rights. These arguments have faced withering criticism. The simplest objection against Combatant Equality brings it into conflict with Proportionality (McMahan 1994; Rodin 2002; Hurka 2005). Unintended noncombatant deaths are permissible only if proportionate to the military objective sought. This means the objective is worth that much innocent suffering. But military objectives are merely means to an end. Their worth depends on how valuable the end is. How many innocent deaths would be proportionate to Al Shabab's successfully gaining control of Mogadishu now or to Iraq's capturing Kuwaiti territory and oil reserves in 1991? In each case the answer is obvious: none. Proportionality is about weighing the evil inflicted against the evil averted (Lee 2012). But the military success of unjust combatants does not avert evil, it is itself evil. Evil intentionally inflicted can only add to, not counterbalance, unintended evils. Combatant Equality cannot be true. Other arguments against Combatant Equality focus on Walzer's account of how one loses the right to life. They typically start by accepting his premise that permissible killing in war does not violate the rights of the victims against being killed, at least for intentional killing.[20] This contrasts with the view that sometimes people's rights to life can be overridden, so war can be permissible despite infringing people's rights. Walzer's critics then show that his account of how we lose our right to life is simply not plausible. Merely posing a threat to others--even a lethal threat--is not sufficient to warrant the loss of one's fundamental rights, because sometimes one threatens others' lives for very good reasons (McMahan 1994). The soldiers of the Kurdish Peshmerga, heroically fighting to rescue Yazidis from ISIL's genocidal attacks, do not thereby lose their rights not to be killed by their adversaries. Posing threats to others in the pursuit of a just aim, where those others are actively trying to thwart that just aim, cannot void or vitiate one's fundamental natural rights against being harmed by those very people. The consent-based argument is equally implausible as a general defence for Combatant Equality. Unjust combatants have something to gain from waiving their rights against lethal attack, if doing so causes just combatants to effect the same waiver. And on most views, many unjust combatants have nothing to lose, since by participating in an unjust war they have at least weakened if not lost those rights already. Just combatants, by contrast, have something to lose, and nothing to gain. So why would combatants fighting for a just cause consent to be harmed by their adversaries, in the pursuit of an unjust end? Walzer's case for Combatant Equality rests on showing that just combatants lose their rights to life. His critics have shown that his arguments to this end fail. So Combatant Equality is false. But they have shown more than this. Inspired by Walzer to look at the conditions under which we lose our rights to life, his critics have made theoretical advances that place other central tenets of jus in bello in jeopardy. They argued, contra Walzer, that posing a threat is not sufficient for liability to be killed (McMahan 1994, 2009). But they also showed that posing the threat oneself is not necessary for liability either. This is more controversial, but revisionists have long argued that liability is grounded, in war as elsewhere, in one's responsibility for contributing to a wrongful threat. The US president, for example, is responsible for a drone strike she orders, even though she does not fire the weapon herself. As many have noted, this argument undermines Discrimination (McMahan 1994; Arneson 2006; Fabre 2012; Frowe 2014). In many states, noncombatants play an important role in the resort to military force. In modern industrialized countries, as much as 25 per cent of the population works in war-related industries (Downes 2006: 157-8; see also Gross 2010: 159; Valentino et al. 2010: 351); we provide the belligerents with crucial financial and other services; we support and sustain the soldiers who do the fighting; we pay our taxes and in democracies we vote. Our contributions to the state's capacity over time give it the strength and support to concentrate on war.[21] If the state's war is unjust, then many noncombatants are responsible for contributing to wrongful threats. If that is enough for them to lose their rights to life, then they are permissible targets. McMahan (2011a) has sought to avert this troubling implication of his arguments by contending that almost all noncombatants on the unjust side (unjust noncombatants) are less responsible than all unjust combatants. But this involves applying a double standard, talking up the responsibility of combatants, while talking down that of noncombatants, and mistakes a central element in his account of liability to be killed. On his view, a person is liable to be killed in self- or other-defence in virtue of being, of those able to bear an unavoidable and indivisible harm, the one who is most responsible for this situation coming about (McMahan 2002, 2005b). Even if noncombatants are only minimally responsible for their states' unjust wars--that is, they are not blameworthy, they merely voluntarily acted in a way that foreseeably contributed to this result--on McMahan's view this is enough to make them liable to be killed, if doing so is necessary to save the lives of wholly innocent combatants and noncombatants on the just side (see especially McMahan 2009: 225). One response is to reject this comparative account of how responsibility determines liability, and argue for a non-comparative approach, according to which one's degree of responsibility must be great enough to warrant such a severe derogation from one's fundamental rights. But if we do this, we must surely concede that many combatants on the unjust side are not sufficiently responsible for unjustified threats to be liable to be killed. Whether through fear, disgust, principle or ineptitude, many combatants are wholly ineffective in war, and contribute little or nothing to threats posed by their side. The much-cited research of S. L. A. Marshall claimed that only 15-25 per cent of Allied soldiers in the Second World War who could have fired their weapons did so (Marshall 1978). Most soldiers have a natural aversion to killing, which even intensive psychological training may not overcome (Grossman 1995). Many contribute no more to unjustified threats than do noncombatants. They also lack the "mens rea" that might make liability appropriate in the absence of a significant causal contribution. They are not often blameworthy. The loss of their right to life is not a fitting response to their conduct. If Walzer is right that in war, outside of supreme emergencies, we may intentionally kill only people who are liable to be killed, and if a significant proportion of unjust combatants and noncombatants are responsible to the same degree as one another for unjustified threats, and if liability is determined by responsibility, then we must decide between two unpalatable alternatives. If we set a high threshold of responsibility for liability, to ensure that noncombatants are not liable to be killed, then we will also exempt many combatants from liability. In ordinary wars, which do not involve supreme emergencies, intentionally killing such non-liable combatants would be impermissible. This moves us towards a kind of pacifism--though warfare can in principle be justified, it is so hard to fight without intentionally killing the non-liable that in practice we must be pacifists (May 2015). But if we set the threshold of responsibility low, ensuring that all unjust combatants are liable, then many noncombatants will be liable too, thus rendering them permissible targets and seriously undermining Discrimination. We are torn between pacifism on the one hand, and realism on the other. This is the "responsibility dilemma" for just war theory (Lazar 2010). ### 4.2 Killing Combatants Just war theory has meaning only if we can explain why killing some combatants in war is allowed, but we are not thereby licensed to kill everyone in the enemy state. Here the competing forces of realism and pacifism are at their most compelling. It is unsurprising, therefore, that so much recent work has focused on this topic. We cannot do justice to all the arguments here, but will instead consider three kinds of response: all-out revisionist; moderate traditionalist; and all-out traditionalist. The first camp faces two challenges: to justify intentionally killing apparently non-liable unjust combatants; but to do this without reopening the door to Combatant Equality, or indeed further undermining Discrimination. Their main move is to argue that, despite appearances, all and only unjust combatants are in fact liable to be killed. McMahan argues that liability to be killed need not, in fact, presuppose responsibility for an unjustified threat. Instead, unjust combatants' responsibility for just combatants' reasonable beliefs that they are liable may be enough to ground forfeiture of their rights (McMahan 2011a). Some argue that combatants' responsibility for being in the wrong place at the wrong time is enough (likening them to voluntary human shields).[22] More radically still, some philosophers abandon the insistence on individual responsibility, arguing that unjust combatants are collectively responsible for contributing to unjustified threats, even if they are individually ineffective (or even counterproductive) (Kamm 2004; Bazargan 2013). Lazar (forthcoming-a) suggests these arguments are unpersuasive. Complicity might be relevant to the costs one is required to bear in war, but most liberals will baulk at the idea of losing one's right to life in virtue of things that other people did. And if combatants can be complicitously liable for what their comrades-in-arms did, then why shouldn't noncombatants be complicitously liable also? Blameworthy responsibility for other people's false beliefs does seem relevant to the ethics of self- and other-defence. That said, consider an idiot who pretends to be a suicide bomber as a prank, and is shot by a police officer (Ferzan 2005; McMahan 2005c). Is killing him objectively permissible? It seems doubtful. The officer's justified belief that the prankster posed a threat clearly diminishes the wrongfulness of killing him (Lazar 2015a). And certainly the prankster's fault excuses the officer of any guilt. But killing the prankster still seems objectively wrong. Even if someone's blameworthy responsibility for false beliefs could make killing him objectively permissible, most philosophers agree that many unjust combatants are not to blame for the injustice of their wars (McMahan 1994; Lazar 2010). And it is much less plausible that blameless responsibility for beliefs can make one a permissible target. Even if it did, this would count in favour of moderate Combatant Equality, since most just combatants are also blamelessly responsible for unjust combatants' reasonable beliefs that they are liable to be killed. Moderate traditionalists think we can avoid the realist and pacifist horns of the responsibility dilemma only by conceding a moderate form of Combatant Equality. The argument proceeds in three stages. First, endorse a non-comparative, high threshold of responsibility for liability, such that most noncombatants in most conflicts are not responsible enough to be liable to be killed. This helps explain why killing civilians in war is so hard to justify. Of course, it also entails that many combatants will be innocent too. The second step, then, is to defend the principle of Moral Distinction, according to which killing civilians is worse than killing soldiers. This is obviously true if the soldiers are liable and the civilians are not. But the challenge is to show that killing non-liable civilians is worse than killing non-liable soldiers. If we can do that, then the permissibility of intentionally killing non-liable soldiers does not entail that intentionally killing non-liable noncombatants is permissible. Of course, one might still argue that, even if Moral Distinction is true, we should endorse pacifism. But, and this is the third stage, the less seriously wrongful some act is, the lesser the good that must be realised by performing that act, for it to be all things considered permissible. If intentionally killing innocent combatants is not the worst kind of killing one can do, then the good that must be realised for it to be all things considered permissible is less than is the case for, for example, intentionally killing innocent civilians, which philosophers tend to think can be permissible only in a supreme emergency. This could mean that intentionally killing innocent soldiers is permissible even in the ordinary circumstances of war. Warfare can be justified, then, by a combination of liability and lesser evil grounds. Some unjust combatants lose their rights not to be killed. Others' rights can be overridden without that implying that unjust noncombatants' rights may be overridden too. We can reject the pacifist horn of the responsibility dilemma. But a moderate Combatant Equality is likely to be true: since killing innocent combatants is not the worst kind of killing, it is correspondingly easier for unjust combatants to justify using lethal force (at least against just combatants). This increases the range of cases in which they can satisfy Discrimination, Proportionality, and Necessity, and so fight permissibly. Much hangs, then, on the arguments for Moral Distinction. Some focus on why killing innocent noncombatants is especially wrongful; others on why killing innocent combatants is not so bad. This section considers the second kind of argument, returning to the first in the next section. The revisionists' arguments mentioned above might not ground liability, but do perhaps justify some reason to prefer harming combatants. Combatants can better avoid harm than noncombatants. Combatants surely do have somewhat greater responsibilities to bear costs to avert the wrongful actions of their comrades-in-arms than do noncombatants. And the readiness of most combatants to fight--regardless of whether their cause is just--likely means that even just combatants have somewhat muddied status relative to noncombatants. They conform to their opponents' rights only by accident. They have weaker grounds for complaint when they are wrongfully killed than do noncombatants, who more robustly respect the rights of others (on robustness and respect, see Pettit 2015). Additionally, when combatants kill other combatants, they typically believe that they are doing so permissibly. Most often they believe that their cause is just, and that this is a legitimate means to bring it about. But, insofar as they are lawful combatants, they will also believe that international law constrains their actions, so that by fighting in accordance with it they are acting permissibly. Lazar (2015c) argues that killing people when you know that doing so is objectively wrong is more seriously objectionable than doing so when you reasonably believe that you are acting permissibly. The consent-based argument for Combatant Equality fails because of its empirical, not its normative premise. If combatants in fact waived their rights not to be killed by their adversaries, even when fighting a just war, then that would clearly affect their adversaries' reasons for action, reducing the wrongfulness of killing anyone who had waived that right. The problem is that they have not waived their rights not to be killed. However, they often do offer a more limited implicit waiver of their rights. The purpose of having armed forces, and the intention of many who serve in them, is to protect civilians from the predations of war. This means both countering threats to and drawing fire away from them. Combatants interpose themselves between the enemy and their civilian compatriots, and fight on their compatriots' behalf. If they abide by the laws of war, they clearly distinguish themselves from the civilian population, wearing a uniform and carrying their weapons openly. They implicitly say to their adversaries: "you ought to put down your weapons. But if you are going to fight, then fight us". This constitutes a limited waiver of their rights against harm. Like a full waiver, it alters the reasons confronting their adversaries--under these circumstances, other things equal it is worse to kill the noncombatants. Of course, in most cases unjust combatants ought simply to stop fighting. But this conditional waiver of their opponents' rights means that, if they are not going to put down arms, they do better to target combatants than noncombatants. Of course, one might think that in virtue of their altruistic self-sacrifice, just combatants are actually the least deserving of the harms of war (Tadros 2014). But, first, warfare is not a means for ensuring that people get their just deserts. More importantly, given that their altruism is specifically intended to draw fire away from their compatriot noncombatants, it would be perverse to treat this as a reason to do precisely what they are trying to prevent. These arguments and others suggest that killing innocent combatants is not the worst kind of killing one can do. It might therefore be all things considered permissible in the ordinary circumstances of war, provided enough good is achieved thereby. If unjust combatants attack only just combatants, and if they achieve some valuable objective by doing so--defence of their comrades, their co-citizens, or their territory--they therefore might fight permissibly, even though they violate the just combatants' rights (Kamm 2004; Hurka 2005; Kamm 2005; Steinhoff 2008; Lazar 2013). At least, it is more plausible that they can fight permissibly than if we regarded every just combatant's death as equivalent to the worst kind of murder. This does not vindicate Combatant Equality--it simply shows that, more often than one might think, unjust combatants can fight permissibly. Add to that the fact that all wars are morally heterogeneous, involving just and unjust phases (Bazargan 2013), and we quickly see that even if Combatant Equality in the laws of war lacks fundamental moral foundations, it is a sensible approximation of the truth. Some philosophers, however, seek a more robust defence of Combatant Equality. The three most prominent lines are institutionalist. A contractualist argument (Benbaji 2008, 2011) starts by observing that states (and their populations) need disciplined armies for the purposes of national defence. If soldiers always had to decide for themselves whether a particular war was just, many states could not raise armies when they need to. They would be unable to deter aggression. All states, and all people, benefit from an arrangement whereby individual combatants waive their rights not to be killed by one another--allowing them to obey their state's commands without second-guessing every deployment. Combatants tacitly consent to waive their rights in this way, given common knowledge that fighting in accordance with the laws of war involves such a waiver. Moreover, their assent is "morally effective" because it is consistent with a fair and optimal contract among states. International law does appear to change the moral standing of combatants. If you join the armed forces of a state, you know that, at international law, you thereby become a legitimate target in armed conflict. This has to be relevant to the wrongfulness of harming you, even if you are fighting for a just cause. But Benbaji's argument is more ambitious than this. He thinks that soldiers waive their rights not to be killed by one another--not the limited, conditional waiver described above, but an outright waiver, that absolves their adversaries of any wrongdoing (though it does not so absolve their military and political leaders). The first problem with this proposal is that it rests on contentious empirical speculation about whether soldiers in fact consent in this way. But setting that aside, second, it is radically statist, implying that international law simply doesn't apply to asymmetric conflicts between states and non-state actors, since the latter are not part of the appropriate conventions. This gives international law shallow foundations, which fail to support the visceral outrage that breaches of international law typically evoke. It also suggests that states that either don't ratify major articles of international law, or that withdraw from agreements, can escape its strictures. This seems mistaken. Third, we typically regard waivers of fundamental rights as reversible when new information comes to light. Why shouldn't just combatants be allowed to withdraw their rights-waiver when they are fighting a just war? Many regard the right to life as inalienable; even if we deny this, we must surely doubt whether you can alienate it once and for all, under conditions of inadequate information. Additionally, suppose that you want to join the armed forces only to fight a specific just war (McMahan 2011b). Why should you waive your rights against harm in this case, given that you plan only to fight now? Fourth, and most seriously, even if Benbaji's argument explained why killing combatants in war is permissible regardless of the cause you are serving, it cannot explain why unintentionally killing noncombatants as a side-effect of one's actions is permissible. By joining the armed forces of their state, soldiers at least do something that implies their consent to the regime of international law that structures that role. But noncombatants do not consent to this regime. Soldiers fighting for unjust causes will inevitably kill many innocent civilians. If those deaths cannot be rendered proportionate, then Combatant Equality does not hold. The second institutionalist argument starts from the belief that we have a duty to obey the law of our legitimate state. This gives unjust combatants, ordered to fight an unjust war, some reason to obey those orders. We can ground this in different ways. Estlund (2007) argues that the duty to obey orders derives from the epistemic authority of the state--it is more likely than an individual soldier to know whether this war is just (see Renzo 2013 for criticism); Cheyney Ryan (2011) emphasizes the democratic source of the state's authority, as well as the crucial importance of maintaining civilian control of the military. These are genuine moral reasons that should weigh in soldiers' deliberations. But are they really weighty enough to ground Combatant Equality? It seems doubtful. They cannot systematically override unjust combatants' obligations not to kill innocent people. This point stands regardless of whether these reasons weigh in the balance, or are exclusionary reasons that block others from being considered (Raz 1985). The rights of innocent people not to be killed are the weightiest, most fundamental rights around. For some other reason to outweigh them, or exclude them from deliberation, it would have to be extremely powerful. Combatants' obligations to obey orders simply are not weighty enough--as everyone recognises with respect to obedience to unlawful in bello commands (McMahan 2009: 66ff). Like the first argument, the third institutionalist argument grounds Combatant Equality in its long-term results. But instead of focusing on states' ability to defend themselves, it emphasizes the importance of limiting the horrors of war, given that we know that people deceive themselves about the justice of their cause (Shue 2008, 2010; Dill and Shue 2012; Shue 2013; Waldron 2016). Since combatants and their leaders almost always believe themselves to be in the right, any injunction to unjust combatants to lay down their arms would simply be ignored, while any additional permissions to harm noncombatants would be abused by both sides. In almost all wars, it is sufficient to achieve military victory that you target only combatants. If doing this will minimize wrongful deaths in the long run, we should enjoin that all sides, regardless of their aims, respect Discrimination. Additionally, while it is extremely difficult to secure international agreement even about what in fact constitutes a just cause for war (witness the controversy over the Rome statute on crimes of aggression, which took many years of negotiation before diplomats agreed an uneasy compromise), the traditionalist principles of jus in bello already have broad international support. They are hard-won concessions that we should abandon only if we are sure that the new regime will be an improvement (Roberts 2008). Although this argument is plausible, it doesn't address the same question as the act-focused arguments that preceded it. One thing we can ask is: given a particular situation, what ought we to do? How ought soldiers to act in Afghanistan, or Mali, or Syria, or Somalia? And when we ask this question, we shouldn't start by assuming that we or they will obviously fail to comply with any exacting moral standards that we might propose (Lazar 2012a; Lazar and Valentini forthcoming). When considering our own actions, and those of people over whom we have influence, we should select from all the available options, not rule some out because we know ourselves to be too immoral to take them. When designing institutions and laws, on the other hand, of course we should think about how people are likely to respond to them. We need to answer both kinds of questions: what really ought I to do? And what should the laws be, given my and others' predictable frailty? A moderate Combatant Equality, then, is the likely consequence of avoiding the pacifist horn of the responsibility dilemma. To show that killing in war is permissible, we need to show that intentionally killing innocent combatants is not as seriously wrongful as intentionally killing innocent noncombatants. And if killing innocent combatants is not the worst kind of killing, it can more plausibly be justified by the goods achieved in ordinary wars, outside of supreme emergencies. On this view, contrary to the views of both Walzer and his critics, much of the intended killing in justified wars is permissible not because the targets are liable to be killed, but because infringing their rights is a permissible lesser evil. But this principle applies regardless of whether you are on the just or the unjust side. This in turn increases the range of cases in which combatants fighting on the unjust side will be able to fight permissibly: instead of needing to achieve some good comparable to averting a supreme emergency in order to justify infringing the rights of just combatants, they need only achieve more prosaic kinds of goods, since these are not the worst kinds of rights infringements. So unjust combatants' associative duties to protect one another and their compatriots, their duties to obey their legitimate governments, and other such considerations, can sometimes make intentionally killing just combatants a permissible lesser evil, and unintentionally killing noncombatants proportionate. This means that the existing laws of war are a closer approximation of combatants' true moral obligations than many revisionists think. Nonetheless, much of the killing done by unjust combatants in war is still objectively wrong. ### 4.3 Sparing Civilians The middle path in just war theory depends on showing that killing civilians is worse than killing soldiers. This section discusses arguments to explain why killing civilians is distinctly objectionable. We discuss the significance of intentional killing when considering proportionality, below. These arguments are discussed at great length in Lazar (2015c), and are presented only briefly here. They rest on a key point: Moral Distinction says that killing civilians is worse than killing soldiers. It does not say that killing civilians is worse than killing soldiers, other things equal. Lazar holds that stronger principle but does not think that the intrinsic differences between killing civilians and killing soldiers--the properties that are necessarily instantiated in those two kinds of killings--are weighty enough to provide Moral Distinction with the kind of normative force needed to protect noncombatants in war. That protection depends on mobilising multiple foundations for Moral Distinction, which include many properties that are contingently but consistently instantiated in acts that kill civilians and kill soldiers, which make killing civilians worse. We cannot ground Moral Distinction in any one of these properties alone, since each is susceptible to counterexamples. But when they are all taken together, they justify a relatively sharp line between harming noncombatants and harming combatants. There are, of course, hard cases, but these must be decided by appealing to the salient underlying properties rather than to the mere fact of membership in one group or the other. First, at least deliberately killing civilians in war usually fails even the most relaxed interpretation of the necessity constraint. This is not always true--killing is necessary if it is effective at achieving your objective, and no other effective options are available. Killing civilians sometimes meets this description. It is often effective: the blockade of Germany helped end the first world war, though it may have caused as many as half a million civilian deaths; Russian targeting of civilians in Chechnya reduced Russian combatant casualties (Lyall 2009); Taliban anti-civilian tactics have been effective in Afghanistan. And these attacks are often the last recourse of groups at war (Valentino 2004); when all other options have failed or become too costly, targeting civilians is relatively easy to do. Indeed, as recent terrorist attacks have shown (Mumbai and Paris, for example), fewer than ten motivated gunmen with basic weaponry can bring the world's most vibrant cities screeching to a halt. So, killing civilians can satisfy the necessity constraint. Nonetheless, attacks on civilians are often wholly wanton, and there is a special contempt expressed in killing innocent people either wantonly or for its own sake. At least if you have some strategic goal in sight, you might believe that something is at stake that outweighs the innocent lives taken. Those who kill civilians pointlessly express their total disregard for their victims in doing so. Second, even when killing civilians is effective, it is usually so opportunistically (Quinn 1989; Frowe 2008; Quong 2009; Tadros 2011). That is, the civilians' suffering is used as a means to compel their compatriots and their leaders to end their war. Sieges and aerial bombardments of civilian population centres seek to break the will of the population and of their government. Combatants, by contrast, are almost always killed eliminatively--their deaths are not used to derive a benefit that could not be had without using them in this way; instead they are killed to solve a problem that they themselves pose. This too seems relevant to the relative wrongfulness of these kinds of attacks. Of course, at the strategic level every death is intended as a message to the enemy leadership, that the costs of continuing to fight outweigh the benefits. But at the tactical level, where the actual killing takes place, soldiers typically kill soldiers eliminatively, while they kill civilians opportunistically. If this difference is morally important, as many think, and if acts that kill civilians are opportunistic much more often than are acts that kill soldiers, then acts that kill civilians are, in general, worse than acts that kill soldiers. This lends further support to Moral Distinction. Third, as already noted above, the agent's beliefs can affect the objective seriousness of her act of killing. Killing someone when you have solid grounds to think that doing so is objectively permissible wrongs that person less seriously than when your epistemic basis for harming them is weaker. More precisely, killing an innocent person is more seriously wrongful the more reason the killer had to believe that she was not liable to be killed (Lazar 2015a). Last, in ordinary thinking about the morality of war, the two properties most commonly cited to explain the distinctive wrongfulness of harming civilians, after their innocence, are their vulnerability and their defencelessness. Lazar (2015c) suspects that the duties to protect the vulnerable and not to harm the defenceless are almost as basic as the duty not to harm the innocent. (Note that these duties apply only when their object is morally innocent.) Obviously, on any plausible analysis, civilians are more vulnerable and defenceless than soldiers, so if killing innocent people who are more vulnerable and defenceless is worse than killing those who are less so, then killing civilians is worse than killing soldiers. Undoubtedly soldiers are also often vulnerable too--one thinks of the "Highway of Death", in Iraq 1991, when American forces destroyed multiple armoured divisions of the Iraqi army, which were completely unprotected (many of the personnel in those divisions escaped into the desert). But this example just shows that killing soldiers, when they are vulnerable and defenceless, is harder to justify than when they are not. Provided the empirical claim that soldiers are less vulnerable and defenceless than civilians is true, this simply supports the case for Moral Distinction. ### 4.4 Proportionality Holding the principle of Moral Distinction allows one to escape the realist and pacifist horns of the responsibility dilemma, while still giving responsibility its due. Even revisionists who deny moderate Combatant Equality could endorse Moral Distinction, and thereby retain the very plausible insight that it is worse to kill just noncombatants than to kill just combatants. And, if they are to account for most people's considered judgements about war, even pacifists need some account of why killing civilians is worse than killing soldiers. However, Moral Distinction is not Discrimination. It is a comparative claim, and it says nothing about intentions. Discrimination, by contrast, prohibits intentionally attacking noncombatants, except in supreme emergencies. It is the counterpart of Proportionality, which places a much weaker bar on unintentionally killing noncombatants. Only a terrible crisis could make it permissible to intentionally attack noncombatants. But the ordinary goods achieved in individual battles can justify unintentional killing. What justifies this radical distinction? This is one of the oldest questions in normative ethics (though for the recent debate, see Quinn 1989; Rickless 1997; McIntyre 2001; Delaney 2006; Thomson 2008; Tadros 2015). On most accounts, those who intend harm to their victims show them a more objectionable kind of disrespect than those who unavoidably harm them as a side-effect. Perhaps the best case for the significance of intentions is, first, in a general argument that mental states are relevant to objective permissibility (Christopher 1998; see also Tadros 2011). And second, we need a rich and unified theoretical account of the specific mental states that matter in this way, into which intentions fit. It may be that the special prohibition of intentional attacks on civilians overstates the moral truth. Intentions do matter. Other things equal, intentional killings are worse than unintended killings (though some unintended killings that are wholly negligent or indifferent to the victim are nearly as bad as intentional killings). But the difference between them is not categorical. It cannot sustain the contrast between a near-absolute prohibition on one hand, and a sweeping permission on the other. Of course, this is precisely the kind of nuance that would be disastrous if implemented in international law or if internalized as a norm by combatants. Weighing lives in war is informationally incredibly demanding. Soldiers need a principle they can apply. Discrimination is that principle. It is not merely a rule of thumb, since it entails something that is morally grounded--killing civilians is worse than killing soldiers. But it is also a rule of thumb, because it draws a starker contrast between intended and unintended killing than is intrinsically morally justified. As already noted, proportionality and necessity contain within them almost every other question in the ethics of war; we now consider two further points. First, proportionality in international law is markedly different from the version of the principle that first-order moral theory supports. At law, an act of war is proportionate insofar as the harm to civilians is not excessive in relation to the concrete and direct military advantage realized thereby. As noted above, in first-order moral terms, this is unintelligible. But there might be a better institutional argument for this neutral conception of proportionality. Proportionality calculations involve many substantive value judgements--for example, about the significance of moral status, intentions, risk, vulnerability, defencelessness, and so on. These are all highly controversial topics. Reasonable disagreement abounds. Many liberals think that coercive laws should be justified in terms that others can reasonably accept, rather than depending on controversial elements of one's overarching moral theory (Rawls 1996: 217). The law of armed conflict is coercive; violation constitutes a war crime, for which one can be punished. Of course, a more complex law would not be justiciable, but we also have principled grounds for not basing international law on controversial contemporary disputes in just war theory. Perhaps the current standard can be endorsed from within a wider range of overarching moral theories than could anything closer to the truth. Second, setting aside the law and focusing again on morality, many think that responsibility is crucial to thinking about proportionality, in the following way. Suppose the Free Syrian Army (FSA) launches an assault on Raqqa, stronghold of ISIL. They predict that they will cause a number of civilian casualties in their assault, but that this is only because ISIL has chosen to operate from within a civilian area, forcing people to be "involuntary human shields". Some think that ISIL's responsibility for putting those civilians at risk allows the FSA to give those civilians' lives less weight in their deliberations than would be appropriate if ISIL had not used them as human shields (Walzer 2009; Keinon 2014). But one could also consider the following: Even if ISIL is primarily at fault for using civilians as cover, why should this mean that those civilians enjoy weaker protections against being harmed? We typically think that one should only lose or forfeit one's rights through one's own actions. But on this argument, civilians enjoy weaker protections against being killed through no fault or choice of their own. Some might think that more permissive standards apply for involuntary human shields because of the additional value of deterring people from taking advantage of morality in this kind of way (Smilansky 2010; Keinon 2014). But that argument seems oddly circular: we punish people for taking advantage of our moral restraint by not showing moral restraint. What's more, this changes the act from one that foreseeably kills civilians as an unavoidable side-effect of countering the military threat to one that kills those civilians as a means to deter future abuses. This instrumentalizes them in a way that makes harming them still harder to justify. ### 4.5 Necessity The foregoing considerations are all also relevant to necessity. They allow us to weigh the harms at stake, so that we can determine whether the morally weighted harm inflicted can be reduced at a reasonable cost to the agents. The basic structure of necessity is the same in bello as it is ad bellum, though obviously the same differences in substance arise as for proportionality. Some reasons apply only to in bello necessity judgements, not to ad bellum ones, because they are conditional on the background assumption that the war as a whole will continue. This means that we cannot reach judgements of the necessity of the war as a whole by simply aggregating our judgements about the individual actions that together constitute the war. For example, in bello one of the central questions when applying the necessity principle is: how much risk to our own troops are we required to bear in order to minimize harms to the innocent? Some option can be necessary simply in virtue of the fact that it saves some of our combatants' lives. Ad bellum, evaluating the war as a whole, we must of course consider the risk to our own combatants. But we do so in a different way--we ask whether the goods achieved by the war as a whole will justify putting our combatants at risk. We don't then count among the goods achieved by the war the fact that multiple actions within the war will save the lives of individual combatants. We cannot count averting threats that will arise only if we decide to go to war among the goods that justify the decision to go to war. This relates directly to the largely ignored requirement in international law that combatants must > > take all feasible precautions in the choice of means and methods of > attack with a view to avoiding, and in any event to minimizing, > incidental loss of civilian life, injury to civilians and damage to > civilian objects. (Geneva Convention, Article 57, 2(a)(ii)) > This has deep moral foundations: combatants in war are morally required to reduce the risk to innocents until doing so further would involve an unreasonably high cost to them, which they cannot be required to bear. Working out when that point is reached involves thinking through: soldiers' role-obligations to assume risks; the difference between doing harm to civilians and allowing it to happen to oneself or one's comrades-in-arms; the importance of associative duties to protect one's comrades; and all the considerations already adduced in favour of Moral Distinction. This calculus is very hard to perform. My own view is that combatants ought to give significant priority to the lives of civilians (Walzer and Margalit 2009; McMahan 2010b). This is in stark contrast to existing practice (Luban 2014). ## 5. The Future of Just War Theory Much recent work has used either traditionalist or revisionist just war theory to consider new developments in the practice of warfare, especially the use of drones, and the possible development of autonomous weapons systems. Others have focused on the ethics of non-state conflicts, and asymmetric wars. Very few contemporary wars fit the nation-state model of the mid-twentieth century, and conflicts involving non-state actors raise interesting questions for legitimate authority and the principle of Discrimination in particular (Parry 2016). A third development, provoked by the terrible failure to plan ahead in Iraq and Afghanistan, is the wave of reflection on the aftermath of war. This topic, jus post bellum, is addressed separately. As to the philosophical foundations of just war theory: the traditionalist and revisionist positions are now well staked out. But the really interesting questions that remain to be answered should be approached without thinking in terms of that split. Most notably, political philosophers may have something more to contribute to the just war theory debate. It would be interesting, too, to think with a more open mind about the institutions of international law (nobody has yet vindicated the claim that the law of armed conflict has authority, for example), and also about the role of the military within nation-states, outside of wartime (Ryan 2016). The collective dimensions of warfare could be more fully explored. Several philosophers have considered how soldiers "act together" when they fight (Zohar 1993; Kutz 2005; Bazargan 2013). But few have reflected on whether group agency is present and morally relevant in war. And yet it is superficially very natural to discuss wars in these terms, especially in evaluating the war as a whole. When the British parliament debated in late 2015 whether to join the war against ISIL in Syria and Iraq, undoubtedly each MP was thinking also about what she ought to do. But most of them were asking themselves what the United Kingdom ought to do. This group action might be wholly reducible to the individual actions of which it is composed. But this still raises interesting questions: in particular, how should I justify my actions, as an individual who is acting on behalf of the group? Must I appeal only to reasons that apply to me? Or can I act on reasons that apply to the group's other members or to the group as a whole? And can I assess the permissibility of my actions without assessing the group action of which they are part? Despite the prominence of collectivist thinking in war, discussion of war's group morality is very much in its infancy.
james-ward	## 1. Life James Ward was born in Hull, Yorkshire, on 27th January 1843, as a son of James Ward, senior, a merchant with some scientific and philosophical ambition (in a book entitled God, Man and the Bible, he tried to give a scientific explanation of the Flood). His father's enterprises were disastrous, however, so that lack of resources was to be a constant source of anxiety for the family, which included an uneducated, hard-working, deeply religious mother, six sisters and a brother (another son died). After one of his father's bankruptcies, Ward remained for two years without schooling and occupation. At the age of thirteen and a half, the boy was let free to occupy his time as he wished. In the small country town where his family had moved, Waterloo near Liverpool, he developed a spirit of solitude and a love of nature. A 'Memoir' written by the philosopher's daughter, Olwen Ward Campbell, enables us to get a glimpse of his life in these early years: "Waterloo was only a village in those days, and wild stretches of sandhills lay beyond it for many miles. Here he used to wander for hours on end, alone with his thoughts and the sea birds" (1927: 7). In addition to the family's economic difficulties, Ward's youth was shaped by the narrow Calvinistic faith in which he was raised. In 1863 he entered Spring Hill College (later incorporated in Mansfield College, Oxford) with the aim of becoming a minister, but became increasingly skeptical of the doctrines he was expected to preach. In the middle of his religious crisis, Ward moved to Germany for a temporary period of study. In Gottingen, he attended lectures by one of the most influential philosophers of the time, Rudolf Hermann Lotze. It is probably from Lotze that Ward gained a first substantial appreciation of the philosophy of Leibniz, whose theory of monads was later to provide the basis of his own metaphysics. Upon returning to England, Ward made a sincere yet vain attempt to retain his faith and accepted a pastorate in Cambridge. His spiritual crisis reached its climax and the point of no return in 1872, when he definitively abandoned the project of becoming a minister and applied as a non-collegiate student at Cambridge. One year later, he won a scholarship in Moral Science at Trinity College. Important lessons were learned in the anguish of these vicissitudes and deliberations. In a letter to a friend, Ward writes: "I am coming to see more clearly every day that man is only half free, or rather that his freedom is not what I once thought it. It is not the power to choose anything, but only the power to choose between alternatives offered, and what these shall be circumstances determine quite as will" (59). Surely, this insight owes more to the struggle of existence than to abstract speculation. Interestingly, Ward's mature philosophy comprises a defense of human freedom and sustains the idea that--independently of circumstances outside of our control--we always retain a capacity for responsible action. Ward made the best out of the difficult challenge of beginning a new life at the age of thirty. In 1874 he obtained a first in his Tripos; one year later, he won a fellowship with a dissertation entitled 'The Relation of Physiology to Psychology,' part of which was published in the first issue of Mind in 1876. These successes marked the beginning of a distinguished career. He first became known for his article on 'Psychology' (1886) in the Encyclopaedia Britannica, in which he criticised Mill and Bain's associationism, and in the years 1896-98 he delivered the prestigious Gifford Lectures, later published under the title Naturalism and Agnosticism (1899). In this book, Ward attacked the various forms of scientific materialism that were current in the second half of the nineteenth century. The lectures were well received by his contemporaries and there was a widespread agreement that Ward had been effective in his refutation. According to Alfred Edward Taylor, who reviewed the book in Mind in 1900, "one may assert without much fear of contradiction that Prof. Ward's Gifford Lectures are the philosophical book of the last year" (Taylor 1900: 244). G. E. Moore and Bertrand Russell were among Ward's students at Cambridge, where he had been elected to the new Chair of Mental Philosophy and Logic in 1897. Moore has left a vivid testimony of his personality and teaching style; as he recalls, > > > We sat around a table in his rooms at Trinity, and Ward had a large > notebook open on the table before him. But he did not read his > lectures; he talked; and, while he talked, he was obviously thinking > hard about the subject he was talking of and searching for the best way > of putting what he wanted to convey. (Moore 1942: 17) Moore describes Ward as a melancholy personality, overwhelmed by the difficulties of philosophical thinking--'Das Denken ist schwer' was one of his favorite sayings. (16) And while Moore was also impressed by Ward's 'extreme sincerity and conscientiousness' (18), Russell singles him out as one for whom he had a 'great personal affection' (Russell 1946: 10). Ward's reputation as a philosopher must have been very high, for he was invited to deliver a second series of Gifford Lectures. These were held in the years 1907-1910 and published in 1911 as The Realm of Ends or Pluralism and Theism; it is in this book that he provides the most complete exposition of his idealistic metaphysics. ## 2. Metaphysics Ward reached his metaphysics--a theory of interacting monads--by way of criticism of the two main philosophical tendencies of his day, scientific materialism and 'absolute' (or 'monistic') idealism. The critique of materialism is developed at considerable length in the first series of Gifford Lectures, Naturalism and Agnosticism. Ward observes here that the view of reality as a system of inert material particles subject to strict deterministic laws cannot explain the world's contingency, a fundamental aspect of experienced reality that needs to be taken at face value. Moreover, materialism fails to provide an adequate ontology for biology, for it cannot explain the emergence of life from lifeless matter. The doctrine is also inconsistent with the latest development of physics, which is moving away from the seventeenth-century, Democritean conception of the atom as an inert particle--a microscopic, indivisible 'thing.' Ward's overall conclusion is that materialism is so beset with difficulties that the real question is not whether it is true, but why it has come to be believed in the first place. Ward traces the origin of this doctrine in a tendency to confound abstractions with concrete realities. At the beginning of inquiry, the scientist is faced with a concrete whole of experience, but takes notice only of some aspects of it. This is wholly justified, but errors are likely to be committed if he overlooks the richness of the empirical basis from which his notions are derived. He is then apt to "confound his descriptive apparatus with the actual phenomena it is devised to describe" (1899: Vol. 1, 81). Materialism is the metaphysics of the scientist lacking understanding of his own procedure. These reflections led Ward to the idealistic conclusion that reality must be interpreted 'in terms of Mind.' Absolute idealism--the then widespread view that reality is a cosmic consciousness or single 'Experience'--will not do, for Ward regards it as explanatorily vacuous. The theory that the world of finite things is but the 'appearance' of the One does nothing to make the nature of the world more intelligible. Why does the One appear in the way it does? And why does it have to appear at all? These questions find no answer in the works of great monistic thinkers such as Spinoza and Hegel, nor do their British followers do any better in this respect. Ward substantiates this claim with a quotation from Bradley's Appearance and Reality (1893), where he admits that the 'fact of fragmentariness,' that is, why the One appears in the form of a multiplicity, cannot be explained. The upshot of these two parallel lines of arguments--the critique of materialism and Absolute idealism--is that some form of pluralistic idealism must be true. And it is only natural at this point to turn to Leibniz, and especially so for a thinker so well acquainted with German philosophy. According to Ward, however, Leibniz's metaphysics needs to be amended in one fundamental respect: > > > ...the well-known Monadology of Leibniz may be taken > as the type, to which all modern attempts to construct a pluralistic > philosophy more or less conform. But the theology on which Leibniz from > the outset strove to found his Monadology, is, in the first > instance at all events, set aside; and in particular his famous > doctrine of pre-established harmony is rejected altogether. (1911: > 53-54) Ward's main crux is that pre-established harmony fails to leave any room open for contingency and genuine novelties. In Leibniz's system, evolution can only be interpreted as 'preformation,' the gradual unfolding of what in a compressed form is already present at the beginning. This is inconsistent with a correct understanding of evolution, which involves 'epigenesis'--the coming into being of unexpected facts. Interestingly, the same criticism is advanced against philosophies such as Hegel's that conceives of natural history as the 'externalization' of a primordial 'Idea.' Ward notices, however, that Hegel's description of nature--as 'a bacchantic God' (148), and as providing 'the spectacle of a contingency that runs out into endless detail' (139)--is insightful and wholly in line with his own; indeed, he goes so far as to say that 'there is a strong undercurrent of [Leibnizian] pluralism running through the whole of Hegel's philosophy.' (159) Part of Leibniz's motivation for holding to pre-established harmony was the alleged impossibility of making sense of transeunt causation--that is, causation understood as a direct relation between distinct monads (as opposed to immanent causation, which holds between successive stages of a monad's life). If direct interaction is impossible, how to account for the correlations between things except by assuming that it had been established by a supreme Architect? In the hands of Leibniz, this surprising view as to the nature of causation becomes the foundation of an original proof of God's existence. Ward attacks Leibniz's reasoning at its very basis. Causal interaction was understood by Leibniz in terms of the doctrine of physical influx. On that model, direct causal interaction is rejected because properties cannot exist unless as inhering in substances; in passing from one substance to another, however, the communicated property would have to exist in a detached condition, which seems absurd. Could this simple piece of reasoning suffice to reject what we seem to experience all the time, namely our power to act directly upon external realities and to be immediately affected by them? In his answer to Leibniz, Ward remarks that the theory of physical influx categorizes the terms of a causal relation as substances to which qualities inhere; on this theory, even the monads are therefore conceived as if they were things. This involves a special sort of category-mistake: 'Monads'--Ward says--"are conative, that is are feeling and striving subjects or persons in the widest sense, not inert particles or things" (260). In other words, since monads are not 'things' but 'subjects,' the argument against direct monadic interaction is blatantly invalid. As Ward puts it: "If the Leibnizian assumption, that there are no beings devoid of perception and spontaneity... is otherwise sound, then the objections to transeunt action between things become irrelevant" (219). Hence, in a polemical reversion of Leibniz's famous metaphor, he concludes that 'all monads have windows' (260). ## 3. Natural Philosophy If pre-established harmony fails to account for the world's contingency, it must be asked whether Ward's theory of interacting monads can explain nature's order and regularity. The element of contingency that pervades the world is immediately accounted for by the assumption that the ultimate constituents of reality are 'living' subjects, but how do orderly processes emerge? Ward formulates this question as follows: "Can we conceive such an interaction of spontaneous agents..., taking on the appearance of mechanism?" (1927: 239). The solution consists in interpreting nature after a social analogy. The following thought-experiment illustrates the idea: "Let us imagine a great multitude of human beings, varying in tastes and endowments as widely as human beings are known to do, and let us suppose this multitude suddenly to find themselves, as Adam and Eve did, in an ample Paradise enriched sufficiently with diverse natural resources to make the attainment of high civilization possible" (1911: 54). It is obvious to Ward that such individuals would gradually achieve a hierarchical, specialized form of social organization. To a hypothetical external observer, the initial phases of the new life of the selected individuals will appear utterly chaotic, but in due course order and regularity would emerge: "...in place of an incoherent multitude, all seemingly acting at random, we should have a social and economic organization, every member of which had his appropriate place and function" (55-56). It is after this fashion that nature should be understood. Consider an apparently inert object such as a piece of rock. This is not an aggregate of material atoms, but a society of active subjects. Starting out from a state of mutual isolation, Ward speculates, a plurality of monads might end up finding a satisfactory modus vivendi. These monads might then continue to stick together--either because they are unwilling to break out of the social structures in which they are embedded, or because they have become unable to do so. To clarify this concept, Ward appeals to an ingenious simile: > > > ...as there are some individuals who are restless, enterprising > and inventing to the end of their days, so there are others who early > become supine and contented, the slaves of custom and let well alone. > The simpler their standard of well-being and the less differentiated > their environment the more monotonous their behavior will be and the > more inert they will appear. (60) Thus, monads might get imprisoned within a form of life in the same way in which persons might get imprisoned in their habits. While the more energetic monads will try to escape routine, other will acquiesce in it. It is the passivity and repetitiveness of the monad's behavior that generates 'the appearance of mechanism.' In spite of the rejection of pre-established harmony, this conception still owes much to Leibniz's teaching. On Ward's view, the physical objects apprehended in ordinary experience are the appearance to us of a structured conglomerate, or societal whole, of living subjects (1927: 239). Such appearances are not illusions. Our immediate sensations fail to reproduce the object's real internal constitution, yet they do point toward realities actually existing independently of the perceiving subject. Ward's metaphysical idealism thus comes with a form of epistemological realism, while his conception of every-day physical objects is virtually identical with Leibniz's notion of phenomena bene fundata. Stated in Kantian terminology, physical objects are the appearances to us of things whose noumenal being or intrinsic nature is mental or experiential. According to Ward (1911: 392), Kant too was forcefully drawn to this position, even in his critical phase and in spite of his notorious claim that noumena lie outside the categories of the understanding and sensible intuition. Ward's social account of the natural world provides the foundation for an original account of natural laws that he conceived of as statistical, evolved and transient. In the first place, in a universe that is ultimately social, natural laws must be of the like of economic or anthropological laws; as such, they hold for groups of monads rather than for individual ones and are statistical generalizations rather than eternally fixed decrees. Secondly, natural laws are to be regarded as the products of evolution because they record the habitual behavior of groups of socialized monads, and habits have to be acquired or 'learned' over time. Thirdly, it is a historical platitude that societies emerge and decay. Since the natural laws dominating a given epoch depend upon that epoch's social order, natural laws cannot be regarded as immutable; like all finite realities, they too are likely to change in the course of cosmic history. Another notable Cambridge thinker and Leibnizian scholar, Charles Dunbar Broad, nicely summarized Ward's conception with the following words: > > > ...it seems quite impossible to explain the higher types of > mental fact materialistically, whilst it does not seem impossible to > regard physical and chemical laws as statistical uniformities about > very large collections of very stupid minds. (Broad 1975: 169) Ward's account might be profitably compared with a similar one provided by Charles Sanders Peirce in his important essay of 1891, 'The Architecture of Theories.' Here, Peirce advocates the view that "the only intelligible theory of the universe is that of objective idealism, that matter is effete mind, inveterate habits becoming physical laws" (153). Furthermore, he contends that "the only possible way of accounting for the laws of nature and for uniformity in general is to suppose them results of evolution. This supposes them not to be absolute, not to be obeyed precisely. It makes an element of indeterminacy, spontaneity, or absolute chance in nature" (148). Despite the analogies, a major point of controversy is represented by Peirce's tychism--the doctrine hinted at in the just quoted passage that 'absolute chance' (tyche) is of the essence of the universe. The disagreement is not as to whether there is a level of reality that escapes external determination--a striking anticipation of Heisenberg's indeterminacy principle that both Peirce and Ward endorse. The question is as to how that most fundamental level is to be understood. In a passage that can only be interpreted as a reply to Peirce, Ward writes: > > > Some pluralists, very ill advisedly as I think, have identified this > element [contingency, what is unpredictable in the world] with pure > chance and even proposed to elevate it to the place of a guiding > principle under the title of 'tychism' [...] But every > act of a conative agent is determined by--what may, in a wide > sense, be called--a motive, and motivation is incompatible with > chance, though in the concrete it be not reducible to law. (1911: > 76) This critique rests upon the argument that--since monads are 'subjects' in the full sense of the term--they cannot act at random; specifically, monads act either in order to secure self-preservation or to achieve self-realization. This doctrine is phenomenologically well-grounded. Our own psychic life provides us with an immediate grasp of the inner life of a monadic creature, and it is clear to Ward that we do experience ourselves as motivated in this way. In sum, for Ward it is the 'teleological' that grounds the 'mechanical.' A monad always aims at self-preservation and self-realization; this is the simplest way in which the monad's conatus--or will to live--manifests itself. At the same time, self-preservation and self-realization are two fundamental metaphysical parameters, the defining features of each monad's 'character.' If the need for self-preservation predominates, the monad will acquiesce in its acquired status. These are Broad's 'stupid minds' and represents the conservative elements in nature. Per contra, 'intelligent' monads actively seek self-realization. Their importance in the cosmic scheme cannot be underestimated; eventually, it is their craving for novelties that powers evolution and prevents history from coming to a halt. ## 4. Philosophy of Mind Ward's theory of monads is a radical form of panpsychism; on this view, experience is not solely ubiquitous--it is all there truly his. In the last decades of the nineteenth century, the debate on panpsychism was launched by the publication in 1878 of William Kingdon Clifford's 'On the Nature of Things-in-Themselves.' This essay was perfectly attuned to the Zeitgeist, since it linked panpsychism with evolutionary theory. Clifford appeals to the non-emerge argument: the transition from inert matter to conscious beings would be unintelligible, a kind of unfathomable creatio ex nihilo; hence, we are forced to conclude that 'experience' is an intrinsic, aboriginal feature of matter. Ward apparently agrees with the general drift of this argument. In an essay significantly entitled 'Mechanism and Morals,' he explains that one need not fear the theory of evolution; this does not degrade either consciousness or human life, but leads to a spiritualized view of matter: > > > It is interesting... to notice that in the support which it > lends to pampsychist [sic] views the theory of evolution seems > likely to have an effect on science the precise opposite of that which > it exercised at first. That was a leveling down, this will be a > leveling up. At first it appeared as if man were only to be linked with > the ape, now it would seem that the atom, if reality at all, may be > linked with man. (1927: 247) Ward's endorsement of Clifford's evolutionary argument, however, points towards a tension in his philosophy. If evolution means 'epigenesis'--the coming into being of what was not potentially present at the beginning--it is not clear why Ward should be worried by the generation of mind out of matter. Conscious existence might well be one of the novelties evolution is capable of producing. Ward does, at any rate, vehemently reject Clifford's specific version of panpsychism. Clifford speculated that each atom of matter was associated with a quantum of experience, a small piece of 'mind-stuff.' This piece of 'mind-stuff' was conceived as an atom of experience, amounting to something less than a complete thought or feeling. Clifford also held that thoughts and feelings could be constituted simply by way of combination. "When molecules are so combined as to form the brain and nervous system of a vertebrate," he wrote, "the corresponding elements of mind-stuff are so combined as to form some kind of consciousness." Analogously, "when matter takes the complex form of a living human brain, the corresponding mind-stuff takes the form of a human consciousness, having intelligence and volition" (65). In this way, Clifford believed, it would be possible to explain why the genesis of complex material structures in the course of evolution is accompanied by the parallel emergence of higher forms of sentience and mental activity. The theory is best regarded as a form of psycho-physical parallelism; this is, at all events, how Ward interprets it in Naturalism and Agnosticism, just before subjecting 'Clifford's wild speculations concerning mind-stuff' (1899: Vol. 2, 12) to a devastating critique. Ward's criticizes Clifford's theory on three main accounts. (1) In the first place, the whole theory presupposes a point-to-point correspondence between matter and mind. Clifford's conception of an atom, however, is obsolete: 'if the speculations of Lord Kelvin and others are to be accepted, and the prime atom itself is a state of motion in a primitive homogeneous medium [the ether], what is the mental equivalent of this primordial medium?' (114) In other words, Clifford's theory couples material atoms with simple ideas. But if the atom is not a simple particle but a state of an underlying substance, it is not so easy to see what precisely in our consciousness could correspond to it. (2) Secondly, Clifford's views--'a maze of psychological barbarism' (15)--totally misrepresent the nature of human consciousness. It is inconceivable how there could be experiences apart from subjects of experiences or larger wholes of consciousness, such as the smallest pieces of mind-dust will have to be: "Nobody bent on psychological precision would speak of ideas as either conscious or intelligent, but still less would he speak of ideas existing in isolation apart from, and prior to, a consciousness and intelligence" (15-16). Clifford fails to recognize the fundamental psychological fact of the unity of consciousness; the psychical field is not made up of simple sensations in the way in which a mosaic-picture is made up of little stones. (3) Lastly, the mind-dust theory faces its own problem of emergence. Why should the transition from inert matter to consciousness be regarded as more mysterious than the transition from little pieces of consciousness to the single, unified consciousness of a human being? "Allowing that it [mind-dust] is not mind, he [Clifford] makes no attempt to show how from such dust a living mind could ever spring" (15). Thus, Clifford's theory has no explanatory advantage over competitor accounts of the mind-body relation; there is still a 'gap' that needs to be closed. One easily sees that Ward's theory of monads does not face any of these difficulties. (1) The correlation between the psychical and the physical is not a one-to-one correspondence between discrete physical atoms and discrete psychical atoms, but is explained in terms of a Leibnizian theory of 'confused perception.' (2) There are no 'subjectless' experiences, but all experienced contents are 'owned' by the monads, each of which is a genuine unity containing distinct perceptual contents--Leibniz's 'petites perceptions' (1911: 256), but is not literally 'made up' by them. The relationship between the mind and the monads in the body, Ward also remarks in this connection (196), is not that of a whole to its parts but that of a dominant to its subordinate monads. (3) And lastly, there are no mysterious transitions in Ward's theory. The growth of the mind--from the inchoate experiences of infants to the relatively sophisticated experiences of adult human beings--shows that 'lower' states are superseded by 'higher' ones thanks to intersubjective, social intercourse. Hence, Ward sums up his discussion by claiming the superiority of a Leibnizian approach over Clifford's mind-stuff theory: > > > Had he [Clifford] followed Leibniz instead, he could have speculated > as to simple minds to his heart's content, but would never have > imagined that absurdity, 'a piece of mind-stuff,' to which > his fearless and logical interpretation of atomistic psychology had led > him; he would never have imagined that ... mind ... could be > described in terms that have a meaning only when applied to the > complexity of material structure. (1899: Vol. 2, 16) Nevertheless, Ward's theory does leave several questions open. In the first place, if our conscious apprehension of the external world is mediated by our direct apprehension of the monads constituting the body and more specifically the brain--as Ward contends (1911: 257-258)--why are we not aware of our neurons? Secondly, Ward's theory escapes the composition-problem, but this is now replaced by the problem of monadic interaction. How does the dominant monad influence the lower ones and rule over them? Thirdly, there is a question as to the nature of space. Granted that direct monadic interaction is real, where does it occur? Interaction would seem to require a common dimension or medium in order to be possible at all; what kind of dimension could host spiritual beings such as the monads are supposed to be? Disappointingly enough, on all these crucial issues Ward has nothing to say. In the absence of a satisfactory treatment of these topics, however, his theory becomes intrinsically unstable, as further reflection might show that the positions just reached will have to be abandoned. ## 5. Natural Theology As in the rest of his philosophy, evolutionary theory also looms large in Ward's natural theology. The existence of God, Ward frankly recognizes, cannot be proved. However, a theory of interacting monads cannot guarantee that the universe will keep moving towards a state of increasing harmony and cohesion: "without such a spiritual continuity as theism alone seems able to ensure," Ward remarks, "it looks as if a pluralistic world were condemned to a Sisyphean task" (215). At the same time, the worldly monads need the leadership of a supreme agent if harmony has to be widespread across the entire universe. Accordingly, Ward's main problem is to understand how God can be related to the world of monads and guide them towards the realization of societies of ever increasing perfections. It is clear to Ward that there must be an ontological gap between God and the monads. God cannot be a primus inter pares--only a monad among other monads (as Leibniz's language sometimes suggests). Since God is the world's ratio essendi--its ontological foundation, rather than its creator in time--He must be regarded as altogether transcendent. At the same time, since God has to lure the evolutionary process, He cannot merely maintain the world in existence, but must somehow actively enter into it; His relation to the world must therefore involve an aspect of immanence. This gives rise to a dilemma: how could God be both immanent and transcendent? Ward can do no better than to provide an analogy in which God is compared with a Cosmic Artist: > > > We may discern perhaps a faint and distant analogy... in what > we are wont to style the creations of genius... the immortal works > of art, the things of beauty that are a joy forever, we regard > as... the spontaneous output of productive imagination, of a free > spirit that embodies itself in its work, lives in it and loves it. > (238-239) Needless to say, this analogy is too 'faint and distant' to be of any real help. Ward strikes an original note in his attempt to accommodate the reality of God with epigenesis. Since he believes that novelties could only be generated by the interplay of genuinely free agents, he conceives of God as limiting himself in the exercise of his power, so as to let the monads free: "Unless creators are created," he says, "nothing is really created" (437). Having renounced omnipotence, God cannot have knowledge of a future he does not control anymore. This means that the notion of divine omniscience--which in the case of Leibniz takes the form of the doctrine of complete concepts--must be abandoned. God lacks knowledge of future contingents; nevertheless, he is aware of all the possibilities open to a monad at any moment, so that the future will never come to him as unexpected: "God, who knows both tendencies and possibilities completely, is beyond surprise and his purpose beyond frustration" (479). Clearly, such a God cannot be the perfect, immutable God of Leibniz and of classical theism, but is a developing God whose knowledge of the world increases with the unfolding of cosmic history. How does such a God lure the world towards greater perfection? This question too is left unanswered; always candid, Ward acknowledges that God's "modus operandi...is to us inscrutable" (479). ## 6. Legacy Ward's metaphysics has had very little impact upon later Anglo-American philosophy, but it was not without influence in the Cambridge philosophical world in the first two or three decades of the twentieth century. Bertrand Russell explicitly recognizes his indebtedness to his teacher in his scholarly study of 1900, A Critical Examination of the Philosophy of Leibniz. Although he complained that Ward's philosophy was 'dull and antiquate' in a letter to Ottoline Morrell (Griffin 1991: 39), it is arguable that the influence run deeper, as Russell cultivated an interest in those questions at the verge between natural philosophy, philosophy of mind and metaphysics that so much mattered to Ward. This becomes clear as soon as one considers that he is not solely the celebrated author of 'On Denoting' (1905) and The Principles of Mathematics (1903), but also of The Analysis of Mind (1921) and The Analysis of Matter (1927). Other Cambridge thinkers that might have been influenced by Ward are John Ellis McTaggart, C. D. Broad, and Alfred North Whitehead. In The Nature of Existence (1921/1927) McTaggart developed a system of idealistic metaphysics that bears some resemblance with Leibniz's theory of monads. Broad says in his 'Autobiography' (1959) that his idealist professor had 'little influence' on his thought (50), yet he favorably (and acutely) reviewed Ward's The Realm of Ends and devoted an important study to Leibniz's philosophy. Arguably, the greatest ascertained influence was exerted on Alfred North Whitehead; the system of metaphysics that he exposed in Process and Reality (1929) incorporates all the crucial ideas of Ward's philosophy of nature, while his conception of the God-World relationship is a way of bringing to completion Ward's fragmentary account. (Basile 2009: 61-62, 142-43) One might reasonably expect that, outside of Cambridge, his philosophy should have been taken into serious consideration by thinkers with an interest in Naturphilosophie, and especially so by a philosopher usually referred to as an 'idealist' such as Robin G. Collingwood, but there seems to be no documentary evidence of this. American process philosopher and theologian Charles Hartshorne, whose speculative system approximates to Whitehead's, displays in print knowledge of Ward's works. (1963: 19) Ward's philosophy is little known today and scholarly discussions of his thought are scanty. In the widely read A Hundred Years of Philosophy, John Passmore compares it unfavourably with McTaggart's: "In McTaggart's case, the difficulty is to give a summary account of a highly intricate pattern of argument; in Ward's case, to decide what he really meant to say on the issues of central philosophical importance" (82). In an age in love with logical technicalities, this is like putting a heavy stone on the philosopher's grave. Passmore's judgment is not entirely justified; the essentials of Ward's philosophical credo are clear enough. The reasons for the oblivion in which his thought has fallen are historical and have much to do with a general shift of philosophical concerns, as he belonged to a philosophical world that was swept away as a whole by the mounting tide of linguistic philosophy--a world in which other towering figures were, just to mention a few names, forgotten but truly exceptional personalities like Francis Herbert Bradley, Bernard Bosanquet, Samuel Alexander, and George Santayana. It is not a sheer accident that Ward's chair was inherited by Moore, and then by Wittgenstein. Nevertheless, it would be a mistake to regard his philosophy as little more than a relic from the past. The question concerning the mind's place in nature is still at very centre of philosophical debate; panpsychism has been recently recognized as a hypothesis worth contemplating by acclaimed philosophers such as Nagel (1979) and Chalmers (1996); more surprisingly perhaps, the idea that the riddle of the universe might be solved by a theory of interacting monads has resurfaced in recent 'revisionary' work by Galen Strawson (2006: 274). At a deeper level, Ward's thought is admirable for his sincere attachment to the ideal of philosophical speculation as a non-dogmatic source of orientation. His belief in the creative powers and 'freedom' of the monad--which he defended from Peircian tychism, scientistic determinism and traditional theism--implicitly grounds an ethics of responsibility, while his evolutionary theism (a form of meliorism, rather than of optimism) inspires a confident yet not naive attitude towards life and its challenges. Surely, Ward's own vicissitudes--from debilitating poverty to public recognition as Cambridge Professor of Mental Philosophy--are reflected in his grand vision of reality as comprising an indefinite number of conative subjects obstinately striving to achieve self-realization and social harmonization; in this only sustained, as he was wont to say, by 'faith in light to come' (1927: 66).
transmission-justification-warrant	## 1. Introduction Arthur has the measles and stays in a closed environment with his sister Mafalda. If Mafalda ends up contracting the measles herself because of her staying in close contact with Arthur, there must be something--the measles virus--which is transmitted from Arthur to Mafalda in virtue of a relation--the relation of staying in close contact with one another--instantiated by the two siblings. Epistemologists have been devoting their attention to the fact that epistemic properties--like being justified or known--are often transmitted in a similar way. An example (not discussed in this entry) is the transmission of knowledge via testimony. A different but equally important phenomenon (which occupies center stage in this entry) is the transmission of epistemic justification across an inference or argument from one proposition to another. Consider proposition $q$ constituting the conclusion of an argument having proposition $p$ as premise (where $p$ can be a set or conjunction of propositions). If $p$ is justified for a subject $s$ by evidence $e$, this justification may transmit to $q$ if $s$ is aware of the entailment between $p$ and $q$. When this happens, $q$ is justified for $s$ in virtue of her justification for $p$ based on $e$ and her knowledge of the inferential relation from $p$ to $q$. Consider this example from Wright (2002): Toadstool $\rE\_1$. Three hours ago, Jones inadvertently consumed a large risotto of Boletus Satana. $\rP\_1$. Jones has absorbed a lethal quantity of the toxins that toadstool contains. Therefore: $\rQ\_1$. Jones will shortly die. Here $s$ can deductively infer $\rQ\_1$ from $\rP\_1$ given background information.[1] Suppose $s$ acquires justification for $\rP\_1$ by learning $\rE\_1$. $ s$ will also acquire justification for $\rQ\_1$ in virtue of her knowledge of the inferential relation from $\rP\_1$ to $\rQ\_1$ and her justification for $\rP\_1$.[2] Thus, $s$'s justification for $\rP\_1$ will transmit to $\rQ\_1$. It is widely recognized that epistemic justification sometimes fails to transmit across an inference or argument. In interesting cases of transmission failure, $s$ has justification for believing $p$ and knows the inferential link between $p$ and $q$, but $s$ has no justification for believing $q$ in virtue of her justification for $p$ and her knowledge of the inferential link. Here is an example from Wright (2003). Suppose $s$'s background information entails that Jessica and Jocelyn are indistinguishable twins. Consider this possible reasoning: Twins $\rE\_2$. This girl looks just like Jessica. $\rP\_2$. This girl is actually Jessica. Therefore: $\rQ\_2$. This girl is not Jocelyn. In Twins $\rE\_2$ can give $s$ justification for believing $\rP\_2$ only if $s$ has independent justification for believing $\rQ\_2$ in the first instance. Suppose that $s$ does have independent justification for believing $\rQ\_2$, and imagine that $s$ learns $\rE\_2$. In this case $s$ will acquire justification for believing $\rP\_2$ from $\rE\_2$. But it is intuitive that $s$ will acquire no justification for $\rQ\_2$ in virtue of her justification for believing $\rP\_2$ based on $\rE\_2$ and her knowledge of the inferential link between $\rP\_2$ and $\rQ\_2$. So, Twins instantiates transmission failure when $\rQ\_2$ is independently justified. An argument incapable of transmitting to its conclusion a specific justification for its premise(s)--e.g., a justification based on evidence $e$--may be able to transmit to the same conclusion a different justification for its premise(s)--e.g., one based on different evidence $e^\$. Replace for instance $\rE\_2$ with $\rE\_2^\.$ This girl's passport certifies she is Jessica. in Twins, $\rE\_2^\$ can provide $s$ with justification for believing $\rP\_2$ even if $s$ has no independent justification for $\rQ\_2$. Suppose then that $s$ has no independent justification for $\rQ\_2$, and that she acquires $\rE\_2^\$. It is intuitive that $s$ will acquire justification from $\rE\_2^\$ for $\rP\_2$ that does transmit to $\rQ\_2$. Now the inference from $\rP\_2$ to $\rQ\_2$ instantiates epistemic transmission. Although many of the epistemologists participating in the debate on epistemic transmission and transmission failure speak of transmission of warrant, rather than justification, almost all of them use the term 'warrant' to refer to some kind of epistemic justification (in Sect. 3.3 we will however consider an account of warrant transmission failure in which 'warrant' is interpreted differently.) Most epistemologists investigating epistemic transmission and transmission failure--e.g., Wright (2011, 2007, 2004, 2003, 2002 and 1985), Davies (2003, 2000 and 1998), Dretske (2005), Pryor (2004), Moretti (2012) and Moretti & Piazza (2013)--broadly identify the epistemic property capable of being transmitted with propositional* justification.[3] Only a few authors explicitly focus on transmission of doxastic justification--e.g., Silins (2005), Davies (2009) and Tucker (2010a and 2010b [OIR]). In this entry we follow the majority in discussing transmission and transmission failure of justification as phenomena primarily pertaining to propositional justification. (See however the supplement on Transmission of Propositional Justification versus Transmission of Doxastic Justification.) Epistemologists typically concentrate on transmission of (propositional or doxastic) justification across deductively valid (given background information) arguments. The fact that justification can transmit across deduction is crucial for our cognitive processes because it makes the advancement of knowledge--or justified belief--through deductive reasoning possible. Suppose evidence $e$ gives you justification for believing hypothesis $p$, and you know that $p$ entails another proposition $q$ that you haven't directly checked. If the justification you have for believing $p$ transmits to its unchecked prediction $q$ through the entailment, you acquire justification for believing $q$ too. Epistemologists may analyze epistemic transmission across inductive (or ampliative) inferences too. Yet this topic has received much less attention in the literature on epistemic transmission. (See however interesting remarks in Tucker 2010a.) In the remaining part of this entry we will focus on transmission and transmission failure of propositional justification across deductive inference. Unless differently specified, by 'epistemic justification' or 'justification' we always mean 'propositional justification'. ## 2. Epistemic Transmission As said, $s$'s justification for $p$ based on evidence $e$ transmits across entailment from $p$ to $p$'s consequence $q$ whenever $q$ is justified for $s$ in virtue of $s$'s justification for $p$ based on $e$ and her knowledge of $q$'s deducibility from $p$. This initial characterization can be distilled into three conditions individually necessary and jointly sufficient for epistemic transmission: > > > $s$'s justification for $p$ based on $e$ transmits to > $p$'s logical consequence $q$ if and only if: > > > > (C1) > $s$ has justification for believing $p$ based on $e$, > (C2) > $s$ knows that $q$ is deducible from $p$, > (C3) > $s$ has justification for believing $q$ in virtue of > the satisfaction of > (C1) > and > (C2). > > Condition (C3) is crucial for distinguishing transmission of justification across (known) entailment from closure of justification across (known) entailment. Saying that $s$'s justification for believing $p$ is closed under $p$'s (known) entailment to $q$ is saying that: > > > If > > > > (C1) > $s$ has justification for believing $p$ and > (C2) > $s$ knows that $p$ entails $q$, then > (C3$^{\textrm{c}}$) > $s$ has justification for believing $q$. > > One can coherently accept the above principle--known as the principle of epistemic closure--but deny a corresponding principle of epistemic transmission, cashed out in terms of the conditions outlined above: > > > If > > > > (C1) > $s$ has justification for believing $p$ and > (C2) > $s$ knows that $p$ entails $q$, then > (C3) > $s$ has justification for believing $q$ in virtue of > the satisfaction of (C1) and (C2). > > The principle of epistemic closure just requires that when $s$ has justification for believing $p$ and knows that $q$ is deducible from $p, s$ have justification for believing $q$. In addition, the principle of epistemic transmission requires this justification to be had in virtue of her having justification for $p$ and knowing that $p$ entails $q$ (for further discussion see Tucker 2010b [OIR]). Another important distinction is the one between the principle of epistemic transmission and a different principle of transmission discussed by Pritchard (2012a) under the label of evidential transmission principle. According to it, > > > If $s$ perceptually knows that $p$ in virtue of evidence set > $e$, and $s$ competently deduces $q$ from $p$ (thereby coming > to believe that $q$ while retaining her knowledge that $p)$, then > $s$ knows that $q$, where that knowledge is sufficiently supported > by $e$. (Pritchard 2012a: 75) > > > Pritchard's principle, to begin with, concerns (perceptual) knowledge and not propositional justification. Moreover, it deserves emphasis that there are inferences that apparently satisfy Pritchard's principle but fail to satisfy the principle of epistemic transmission. Consider for instance the following triad: Zebra $\rE\_3$. The animals in the pen look like zebras. $\rP\_3$. The animals in the pen are zebras. Therefore: $\rQ\_3$. The animals in the pen are not mules cleverly disguised to look like zebras. According to Pritchard, the evidence set of a normal zoo-goer standing before the zebra enclosure typically includes, above and beyond $\rE\_3$, the background proposition that $(\rE\_B)$ to disguise mules like zebras would be very costly and time-consuming, would bring no comparable benefit and would be relatively easy to unmask. Suppose $s$ knows $\rP\_3$ on the basis of her evidence set, and competently deduces $\rQ\_3$ from $\rP\_3$ thereby coming to believe $\rQ\_3$. Pritchard's evidential transmission principle apparently accommodates the fact that $s$ thereby comes to know $\rQ\_3$. For her evidence set--because of its inclusion of $\rE\_B$--sufficiently supports $\rQ\_3$. But the principle of epistemic transmission is not satisfied. Although (C1$\_{\textit{Zebra}}$) $s$ has justification for $\rP\_3$ and (C2$\_{\textit{Zebra}}$) she knows that $\rP\_3$ entails $\rQ\_3$, she has justification for believing $\rQ\_3$ in virtue of, not (C1$\_{\textit{Zebra}}$) and (C2$\_{\textit{Zebra}}$), but the independent support lent to it by $\rE\_B$. Condition (C3) suffices to distinguish the notion of epistemic transmission from different notions in the neighborhood. However, as it stands, it is still unsuitable for the purpose to completely characterize epistemic transmission. The problem is that there are cases in which it is intuitive that the justification for $p$ based on $e$ transmits to $q$ even if (C3), strictly speaking, is not satisfied. These cases can be described as situations in which only part of the justification that $s$ has for $q$ is based on her justification for $p$ and her knowledge of the entailment from $p$ to $q$. Consider this example. Suppose you are traveling on a train heading to Edinburgh. At 16:00, as you enter Newcastle upon Tyne, you spot the train station sign. Then, at 16:05, the ticket controller tells you that you are not yet in Scotland. Now consider the following reasoning: Journey $\rE\_4$. At 16:05 the ticket controller tells you that you are not yet in Scotland. $\rP\_4$. You are not yet in Scotland. Therefore: $\rQ\_4$. You are not yet in Edinburgh. As you learn $\rE\_4$, given suitable background information, you get justification for $\rP\_4$; moreover, to the extent to which you know that not being in Scotland is sufficient for not being in Edinburgh, you also acquire via transmission justification for $\rQ\_4$. This additional justification is transmitted irrespective of the fact that you already have justification for $\rQ\_4$, acquired by spotting the train station sign 'Newcastle upon Tyne'. If (C3) were read as requiring that the whole of the justification available for a proposition $q$ were had in virtue of the satisfaction of (C1) and (C2), cases like these would become invisible. A way to deal with this complication is to amend the tripartite analysis of epistemic transmission by turning (C3) into (C3$^{+}$), saying that at least part of the justification that $s$ has for $q$ has been achieved by her in virtue of the satisfaction of (C1) and (C2). Let us say that a justification for $q$ is an additional justification for $q$ whenever it is not a first-time justification for it. Condition (C3$^{+}$) can be reformulated as this disjunction: (C3$^{+}$) $s$ has first-time justification for $q$ in virtue of the satisfaction of (C1) and (C2), or $s$ has an additional justification for $q$ in virtue of the satisfaction of (C1) and (C2). Much of the extant literature on epistemic transmission concentrates on examples of transmission of first-time justification. These examples include Toadstool. We have seen, however, that what intuitively transmits in other cases is simply additional justification. Epistemologists have identified at least two--possibly overlapping--kinds of additional justification (cf. Moretti & Piazza 2013). One is what can be called independent justification because it appears--intuitively--independent of the original justification for $q$. This notion of justification can probably be sharpened by appealing to counterfactual analysis. Suppose $s$'s justification for $p$ based on $e$ transmits to $p$'s logical consequence $q$. This justification transmitted to $q$ is an additional independent justification just in case these three conditions are met: (IN1) $s$ was already justified in believing $q$ before acquiring $e$, (IN2) as $s$ acquires $e, s$ is still justified in believing $q$, and (IN3) if $s$ had not been antecedently justified in believing $q$, upon learning $e, s$ would have acquired via transmission a first-time justification for believing $q$. Journey instantiates transmission of justification by meeting (IN1), (IN2), and (IN3). Thus, Journey exemplifies a case of transmission of additional independent justification. Consider now that justification for a proposition or belief normally comes in degrees of strength. The second kind of additional justification can be characterized as quantitatively strengthening justification. Suppose again that $s$'s justification for $p$ based on $e$ transmits to $p$'s consequence $q$. This justification transmitted to $q$ is an additional quantitatively strengthening justification just in case these two conditions are satisfied: (STR1) $s$ was already justified in believing $q$ before acquiring $e$, and (STR2) as $s$ acquires $e$, the strength of $s$'s justification for believing $q$ increases. Here is an example from Moretti (2012). Your background information says that only one ticket out of 5,000 of a fair lottery has been bought by a person born in 1970, and that all other tickets have been bought by older or younger people. Consider now this reasoning: Lottery $\rE\_5$. The lottery winner's passport certifies she was born in 1980. $\rP\_5$. The lottery's winner was born in 1980. Therefore: $\rQ\_5$. The lottery's winner was not born in 1970. Given its high chance, $\rQ\_5$ is already justified on your background information only. It is intuitive that as you learn $\rE\_5$, you acquire an additional quantitatively strengthening justification for $\rQ\_5$ via transmission. For your justification for $\rP\_5$ transmitted to $\rQ\_5$ is intuitively quantitatively stronger than your initial justification for $\rQ\_5$. In many cases when $q$ receives via transmission from $p$ an additional independent justification, $q$ also receives a quantitatively strengthening justification. This is not always true though. For there seem to be cases in which an additional independent justification transmitted from $p$ to $q$ intuitively lessens an antecedent justification for $q$ (cf. Wright 2011). An interesting question is whether it is true that as $q$ gets via transmission from $p$ an additional quantitatively strengthening justification, $q$ also gets an independent justification. This seems true in some cases--for instance in Lottery. It is controversial, however, if it is the case that whenever $q$ gets via transmission a quantitatively strengthening justification, $q$ also gets an independent justification. Wright (2011) and Moretti & Piazza (2013) describe two examples in which a subject allegedly receives via transmission a quantitatively strengthening but not independent additional justification. To summarize, additional justification comes in two species at least: independent justification and quantitatively strengthening justification. This enables us to lay down three specifications of the general condition (C3$^{+}$) necessary for justification transmission, each of which represents a condition necessary for the transmission of one particular type of justification. Let's call these specifications (C3$^{\textrm{ft}}$), (C3$^{\textrm{ai}}$), and (C3$^{\textrm{aqs}}$). (C3$^{\textrm{ft}}$) $s$ has first time justification for $q$ in virtue of the satisfaction of (C1) and (C2). (C3$^{\textrm{ai}}$) $s$ has additional independent justification for $q$ in virtue of the satisfaction of (C1) and (C2). (C3$^{\textrm{aqs}}$) $s$ has additional quantitatively strengthening justification for $q$ in virtue of the satisfaction of (C1) and (C2). Transmission of first-time justification makes the advancement of justified belief through deductive reasoning possible. Yet the acquisition of first-time justification for $q$ isn't the sole possible improvement of one's epistemic position relative to $q$ that one could expect from transmission of justification. Moretti & Piazza (2013), for instance, describe a variety of different ways in which s's epistemic standing toward a proposition can be improved upon acquiring an additional justification for it via transmission. We conclude this section with a note about transmissivity as resolving doubts. Let us say that $s$ doubts $q$ just in case $s$ either disbelieves or withholds belief about $q$, namely, refrains from both believing and disbelieving $q$ after deciding about $q$. $s$'s doubting $q$ should be distinguished from $s$'s being open minded about $q$ and from $s$'s having no doxastic attitude whatsoever towards $q$ (cf. Tucker 2010a). Let us say that a deductively valid argument from $p$ to $q$ is able to resolve doubt about its conclusion just in case it is possible for $s$ to be rationally moved from doubting $q$ to believing $q$ solely in virtue of grasping the argument from $p$ to $q$ and the evidence offered for $p$. Some authors (e.g., Davies 1998, 2003, 2004, 2009; McLaughlin 2000; Wright 2002, 2003, 2007) have proposed that we should conceive of an argument's epistemic transmissivity in a way that is very closely related or even identical to the argument's ability to resolve doubt about its conclusion. Whereas some of these authors have eventually conceded that epistemic transmissivity cannot be defined as ability to resolve doubt (e.g., Wright 2011), others have attempted to articulate this view in full (see mainly Davies 2003, 2004, 2009). However, there is nowadays wide agreement in the literature that the property of being a transmissive argument doesn't coincide with the one of being an argument able to resolve doubt about its conclusion (see for example, Beebee 2001; Coliva 2010; Markie 2005; Pryor 2004; Bergmann 2004, 2006; White 2006; Silins 2005; Tucker 2010a). A good reason to think so is that whereas the property of being transmissive appears to be a genuinely epistemic property of an argument, the one of resolving doubt seems to be only a dialectical feature of it, which varies with the audience whose doubt the argument is used to address. ## 3. Failure of Epistemic Transmission ### 3.1 Trivial Transmission Failure versus Non-Transmissivity It is acknowledged that justification sometimes fails to transmit across known entailment (the acknowledgment dates back at least to Wright 1985). Moreover, it is no overstatement to say that the literature has investigated transmission failure more extensively than transmission of justification. As we have seen, justification based on $e$ transmits from $p$ to $q$ across the entailment if and only if (C1) $s$ has justification for $p$ based on $e$, (C2) $s$ knows that $q$ is deducible from $p$, and (C3$^{+}$) at least part of $s$'s justification for $q$ is based on the satisfaction of (C1) and (C2). It follows from this characterization that no justification based on e transmits from $p$ to $q$ across the entailment if (C1), (C2), or (C3$^{+}$) are not satisfied. These are cases of transmission failure. Some philosophers have argued that knowledge and justification are not always closed under competent (single-premise or multiple-premise) deduction. In the recent literature, the explanation of closure failure has often been essayed in terms of agglomeration of epistemic risk. This type of explanation is less controversial when applied to multi-premise deduction. An example of it concerning justification is the deduction from a high number $n$ of premises, each of which specifies that a different ticket in a fair lottery won't win, which are individually justified even though each premise is somewhat risky, to the conclusion that none of these $n$ tickets will win, which is too risky to be justified. (For more controversial cases in which knowledge or justification closure would fail across single-premise deduction because of risk accumulation, see for example Lasonen-Aarnio 2008 and Schechter 2013; for responses, see for instance Smith 2013 and 2016.) These cases of failure of epistemic closure can be taken to also involve failure of justification transmission. For instance, it can be argued that even if (C1$^{\}$) $s$ has justification for each of the $n$ premises stating that a ticket won't win, and (C2$^{\}$) $s$ knows that it follows from these premises that none of the $n$ tickets will win, $s$ has no justification--and, therefore, no justification in virtue of the satisfaction of (C1$^{\}$) and (C2$^{\}$)--for believing that none of the $n$ tickets will win. In these cases, transmission failure appears to be a consequence of closure failure, and it is therefore natural to treat the former simply as a symptomatic manifestation of the latter. For this reason, we will follow the current literature on transmission failure in not discussing these cases. (For an analysis of closure failure, see the entry on epistemic closure, especially section 6.) The current literature treats transmission failure as a self-standing phenomenon, in the sense that it focuses on cases in which transmission failure is not imputed--and thus not considered reducible--to an underlying failure of closure. For the rest of this entry, we shall follow suit and investigate transmission failure in this pure or genuine form. The most trivial cases of transmission failure are such that (C3$^{+}$) is unsatisfied because (C1) or (C2) are unsatisfied, but it is also true that (C3$^{+}$) would have been satisfied if both (C1) and (C2) have been satisfied (cf. Tucker 2010b [OIR]). It deserves emphasis that these cases involve arguments that aren't unsuitable for the purpose to transmit justification depending on evidence $e$: had the epistemic circumstances been congenial to the satisfaction of (C1) and (C2), these arguments would have transmitted the justification based on $e$ from $p$ to $q$. As we could put the point, these arguments are transmissive of the justification depending on $e$.[4] Cases of transmission failure of a more interesting variety are those in which, regardless of the validity of closure of justification, (C3$^{+}$) isn't satisfied because it could not have been satisfied, no matter whether or not (C1) and (C2) have been satisfied. These cases concern arguments non-transmissive of justification depending on a given evidence, i.e., arguments incapable of transmitting justification depending on that evidence under any epistemic circumstance. An example in point is Twins. Suppose first that $s$ has independent justification for $\rQ\_2$. In those circumstances (C1$\_{\textit{Twins}}$) $s$ has justification from $\rE\_2$ for $\rP\_2$. Furthermore (C2$\_{\textit{Twins}}$) $s$ does know that $\rP\_2$ entails $\rQ\_2$. However, $s$ hasn't justification for $\rQ\_2$ in virtue of the satisfaction of (C1$\_{\textit{Twins}}$) and (C2$\_{\textit{Twins}}$). So (C3$^{+}$) is not met. Suppose now that $s$ has no independent justification for $\rQ\_2$. Then, it isn't the case that (C1$\_{\textit{Twins}}$) $s$ has justification from $\rE\_2$ for $\rP\_2$. So, (C3$^{+}$) is not met either. Since there is no other possibility--either $s$ has independent justification for $\rQ\_2$ or she doesn't--it cannot be the case that $s$'s belief that $\rQ\_2$ is justified in virtue of her justification for $\rP\_2$ from $\rE\_2$ and her knowledge that $\rP\_2$ entails $\rQ\_2$. Note that none of the cases of transmission failure of justification considered above entails failure of epistemic closure. For in none of these cases $s$ has justification for believing $p, s$ knows that $p$ entails $q$, and $s$ fails to have justification for believing $q$. Unsurprisingly, the epistemologists contributing to the literature on transmission failure have principally devoted their attention to cases involving non-transmissive arguments. Epistemologists have endeavored to identify conditions whose satisfaction suffices to make an argument non-transmissive of justification based on a given evidence. The next section reviews the most influential of these proposals. ### 3.2 Varieties of Non-Transmissive Arguments Some non-transmissive arguments explicitly feature their conclusion among their premises. Suppose $p$ is justified for $s$ by $e$, and consider the premise-circular argument that deduces $p$ from itself. This argument cannot satisfy (C3$^{+}$) even if (C1) and (C2) are satisfied. The reason is that no part of $s$'s justification for $p$ can be acquired in virtue of, among other things, the satisfaction of (C2). For if (C1) is satisfied, $p$ is justified for $s$ by $e$ independently of $s$'s knowledge of $p$'s self-entailment, thus not in virtue of (C2). Non-transmissive arguments are not necessarily premise-circular. A different source of non-transmissivity instantiating a subtler form of circularity is the dependence of evidential relations on background or collateral information. This type of dependence is a rather familiar phenomenon: the boiling of a kettle gives one justification for believing that the temperature of the liquid inside is approximately 100degC only if one knows that the liquid is water and that atmospheric pressure is the one of the sea-level. It doesn't if one knows that the kettle is on the top of a high mountain, or the liquid is, say, sulfuric acid. Wright argues, for instance, that the following epistemic set-up, which he calls the information-dependence template, suffices for an argument's inability to transmit justification. > > > A body of evidence, $e$, is an information-dependent justification > for a particular proposition $p$ if whether $e$ justifies $p$ > depends on what one has by way of collateral information, $i$. > [...] Such a relationship is always liable to generate examples > of transmission failure: it will do so just when the particular $e, > p$, and $i$ have the feature that needed elements of the relevant > $i$ are themselves entailed by $p$ (together perhaps with other > warranted premises). In that case, any warrant supplied by $e$ for > $p$ will not be transmissible to those elements of $i$. (Wright > 2003: 59, edited.) > > > The claim that $s$'s justification from $e$ for $p$ requires $s$ to have background information $i$ is customarily understood as equivalent (in this context) to the claim that $s$'s justification from $e$ for $p$ depends on some type of independent justification for believing or accepting $i$.[5] Instantiating the information-dependence template appears sufficient for an argument's inability to transmit first-time justification. Consider again this triad: Twins $\rE\_2$. This girl looks just like Jessica. $\rP\_2$. This girl is actually Jessica. Therefore: $\rQ\_2$. This girl is not Jocelyn. Suppose $s$'s background information entails that Jessica and Jocelyn are indistinguishable twins. Imagine that $s$ acquires $\rE\_2$. It is intuitive that $\rE\_2$ could justify $\rP\_2$ for $s$ only if $s$ had independent justification for believing $\rQ\_2$ in the first instance. Thus, Twins instantiates the information-dependence template. Note that $s$ acquires first-time justification for $\rQ\_2$ in Twins only if (C1$\_{\textit{Twins}}$) $\rE\_2$ gives her justification for $\rP\_2$, (C2$\_{\textit{Twins}}$) $s$ knows that $\rP\_2$ entails $\rQ\_2$ and (C3$^{\textrm{ft}}\_{\textit{Twins}}$) $s$ acquires first-time justification for believing $\rQ\_2$ in virtue of (C1$\_{\textit{Twins}}$) and (C2$\_{\textit{Twins}}$). The satisfaction of (C3$^{\textrm{ft}}\_{\textit{Twins}}$) requires $s$'s justification for believing $\rQ\_2$ not to be independent of $s$'s justification from $\rE\_2$ for $\rP\_2$. However, if (C1$\_{\textit{Twins}}$) is true, the informational-dependence template requires $s$ to have justification for believing $\rQ\_2$ independently of $s$'s justification from $\rE\_2$ for $\rP\_2$. Thus, when the information-dependence template is instantiated, (C1$\_{\textit{Twins}}$) and (C3$^{\textrm{ft}}\_{\textit{Twins}}$) cannot be satisfied at once. In general, no argument satisfying this template together with a given evidence will be transmissive of first-time justification based on that evidence. One may wonder whether a deductive argument from $p$ to $q$ instantiating the information-dependence template is unable to transmit additional justification for $q$. The answer seems affirmative when the additional justification is independent justification. Suppose the information-dependence template is instantiated such that $s$'s justification for $p$ from $e$ depends on $s$'s independent justification for $q$. Note that $s$ acquires additional independent justification for $q$ only if (C1) $e$ gives her justification for $p$, (C2) $s$ knows that $p$ entails $q$, and (C3$^{\textrm{ai}}$) $s$ acquires an additional independent justification in virtue of (C1) and (C2). This additional justification is independent of $s$'s antecedent justification for $q$ only if, in particular, condition (IN3) of the characterization of additional independent justification is satisfied. (IN3) says that if $s$ had not been antecedently justified in believing $q$, upon learning $e, s$ would have acquired via transmission a first-time justification for believing $q$. (IN3) entails that if $q$ were not antecedently justified for $s, e$ would still justify $p$ for $s$. Hence, the satisfaction of (IN3) is incompatible with the instantiation of the information-dependence template, which entails that if $s$ had not antecedent justification for $q, e$ would not justify $p$ for $s$. The instantiation of the information-dependence template then precludes transmission of independent justification. As suggested by Wright (2007), the instantiation of the information-dependence template might also appear sufficient for an argument's inability to transmit additional quantitatively strengthening justification. This claim might appear intuitively plausible: perhaps it is reasonable that if the justification from $e$ for $p$ depends on independent justification for another proposition $q$, the strength of the justification available for $q$ sets an upper bound to the strength of the justification possibly supplied by $e$ for $p$. However, the examples by Wright (2011) and Moretti & Piazza (2013) mentioned in Sect. 2 appear to undermine this intuition. For they involve arguments that instantiate the information-dependent template yet seem to transmit quantitatively strengthening justification to their conclusions. (Alspector-Kelly 2015 suggests that these arguments are a symptom that Wright's explanation of non-transmissivity is overall inadequate.) Some authors have attempted Bayesian formalizations of the information dependence template (see the supplement on Bayesian Formalizations of the Information-Dependence Template.) Furthermore, Coliva (2012) has proposed a variant of the same template. In accordance with the information-dependence template, $s$'s justification from $e$ for $p$ fails to transmit to $p$'s consequence $q$ whenever $s$'s possessing that justification for $p$ requires $s$'s independent justification for $q$. According to Coliva's variant, $s$'s justification from $e$ for $p$ fails to transmit to $q$ whenever $s$'s possessing the latter justification for $p$ requires $s$'s independent assumption of $q$, whether this assumption is justified or not. Pryor (2012: SS VII) can be read as pressing objections against Coliva's template, which are addressed in Coliva (2012). We have seen that non-transmissivity may depend on premise-circularity or reliance on collateral information. There is at least a third possibility: an argument can be non-transmissive of the justification for its premise(s) based on given evidence because the evidence justifies directly the conclusion--i.e., independently of the argument itself (cf. Davies 2009). In this case the argument instantiates indirectness, for $s$'s going through the argument would result in nothing but an indirect (and unneeded) detour for justifying its conclusion. If $e$ directly justifies $q$, no part of the justification for $q$ is based on, among other things, $s$'s knowledge of the inferential relation between $p$ and $q$. So (C3$^{+}$) is unfulfilled whether or not (C1) and (C2) are fulfilled. Here is an example from Wright (2002): Soccer $\rE\_6$. Jones has just kicked the ball between the white posts. $\rP\_6$. Jones has just scored a goal. Therefore: $\rQ\_6$. A game of soccer is taking place. Suppose $s$ learns evidence $\rE\_6$. On ordinary background information, (C1$\_{\textit{Socc}}$) $\rE\_6$ justifies $\rP\_6$ for $s$, (C2$\_{\textit{Socc}}$) $s$ knows that $\rP\_6$ entails $\rQ\_6$, and $\rE\_6$ also justifies $\rQ\_6$ for $s$. It seems false, however, that $\rQ\_6$ is justified for $s$ in virtue of the satisfaction of (C1$\_{\textit{Socc}}$) and (C2$\_{\textit{Socc}}$). Quite the reverse, $\rQ\_6$ seems justified for $s$ by $\rE\_6$ independently the satisfaction of (C1$\_{\textit{Socc}}$) and (C2$\_{\textit{Socc}}$). This is so because in the (imagined) actual scenario it seems true that $s$ would still possess a justification for $\rQ\_6$ based on $\rE\_6$ even if $\rE\_6$ did not justify $\rP\_6$ for $s$. In fact suppose $s$ had noticed that the referee's assistant raised her flag to signal Jones's off-side. Against this altered background information, $\rE\_6$ would no longer justify $\rP\_6$ for $s$ but it would still justify $\rQ\_6$ for $s$. Thus, Soccer is non-transmissive of the justification depending on $\rE\_6$. In general, no argument instantiating indirectness in relation to some evidence is transmissive of justification based on that evidence. The information-dependence template and indirectness are diagnostics for a deductive argument's inability to transmit the justification for its premise(s) $p$ based on evidence $e$ to its conclusion $q$, where $e$ is conceived of as a believed proposition capable of supplying inferential and (typically) fallible justification for $p$. (Note that even though $e$ is conceived of as a belief, the collateral information $i$, which is part of the template, doesn't need to be believed on some views.) The justification for a proposition $p$ might come in other forms. For instance, it has been proposed that a proposition $p$ about the perceivable environment around the subject $s$ can be justified by the fact that $s$ sees that $p$ (where 'sees' is taken to be factive). In this case, $s$ is claimed to attain a kind of non-inferential and infallible justification for believing $p$. This view has been explored by epistemological disjunctivists (see for instance McDowell 1982, 1994 and 2008, and Pritchard 2007, 2008, 2009, 2011 and 2012a). One might find it intuitively plausible that when $s$ sees that $p, s$ attains non-inferential and infallible justification for believing $p$ that doesn't rely on $s$'s background information. Since this justification for believing $p$ would be a fortiori unconstrained by $s$'s independent justification for believing any consequence $q$ of $p$, in these cases the information-dependence template could not possibly be instantiated. Therefore, one might be tempted to conclude that when $s$ sees that $p, s$ acquires a justification that typically transmits to the propositions that $s$ knows to be deducible from $p$ (cf. Wright 2002). Pritchard (2009, 2012a) comes very close to endorsing this view explicitly. The latter contention has not stayed unchallenged. Suppose one accepts a notion of epistemic justification with internalist resonances saying that a factor $J$ is relevant to $s$'s justification only if $s$ is able to determine, by reflection alone, whether $J$ is or is not realized. On this notion, $s$'s seeing that $p$ cannot provide $s$ with justification for believing $p$ unless $s$ can rationally claim that she is seeing that $p$ upon reflection alone. Seeing that $p$, however, is subjectively indistinguishable from hallucinating that $p$, or from being in some other delusional state in which it merely seems to $s$ that she is seeing that $p$. Hence, one may find it compelling that $s$ can claim by reflection alone that she's seeing that $p$ only if $s$ has an independent reason for ruling out that it merely seems to her that she does (cf. Wright 2011). If this is true, for many deductive arguments from $p$ to $q, s$ won't be able to acquire non-inferential and infallible justification for believing $p$ of the type described by the disjunctivist and transmit it to $q$. This will happen whenever $q$ is the logical negation of a proposition ascribing to $s$ some delusionary mental state in which it merely seems to her that she directly perceives that $p$. Wright's disjunctive template is meant to be a diagnostic of transmission failure of non-inferential justification when epistemic justification is conceived of in the internalist fashion suggested above.[6] According to Wright (2000), for any propositions $p, q$ and $r$ and subject $s$, the disjunctive template is instantiated whenever: (D1) $p$ entails $q$; (D2) $s$'s justification for $p$ consists in $s$'s being in a state subjectively indistinguishable from a state in which $r$ would be true; (D3) $r$ is incompatible with $p$; (D4) $r$ would be true if $q$ were false. To see how this template works, consider again the following triad: Zebra $\rE\_3$. The animals in the pen look like zebras. $\rP\_3$. The animals in the pen are zebras. Therefore: $\rQ\_3$. The animals in the pen are not mules cleverly disguised to look like zebras. The justification from $\rE\_3$ for $\rP\_3$ arguably fails to transmit across the inference from $\rP\_3$ to $\rQ\_3$ because of the satisfaction of the information-dependence template. For it seems true that $s$ could acquire a justification for believing $\rP\_3$ on the basis of $\rE\_3$ only if $s$ had an independent justification for believing $\rQ\_3$. Now suppose that $s$'s justification for $\rP\_3$ is based on, not $\rE\_3$ but $s$'s seeing that $\rP\_3$. Let's call Zebra\* the corresponding variant of Zebra. Given the non-inferential nature of the justification considered for $\rP\_3$, Zebra\* could not instantiate the information-dependence template. However, it is easy to check that Zebra\* instantiates the disjunctive template. To begin with, (D1$\_{\textit{Zebra}}$) $\rP\_3$ entails $\rQ\_3$; (D2$\_{\textit{Zebra}}$) $s$'s justification for believing $\rP\_3$ is constituted by $s$'s seeing that $\rP\_3$, which is subjectively indistinguishable from the state that $s$ would be in if it were true that $\rR\_{\textit{Zebra}}$. the animals in the pen are mules cleverly disguised to look like zebras; (D3$\_{\textit{Zebra}}$) $\rR\_{\textit{Zebra}}$ is incompatible with $\rP\_3$; and, trivially, (D4$\_{\textit{Zebra}}$) if $\rQ\_3$ were false $\rR\_{\textit{Zebra}}$ would be true. Since Zebra\* instantiates the disjunctive template, it is non-transmissive of at least first-time justification. In fact note that $s$ acquires first-time justification for $\rQ\_3$ in this case if and only if (C1$\_{\textit{Zebra}}$) $s$ has justification for $\rP\_3$ based on seeing that $\rP\_3$, (C2$\_{\textit{Zebra}}$) $s$ knows that $\rP\_3$ entails $\rQ\_3$, and (C3$^{\textrm{ft}}\_{\textit{Zebra}}$) $s$ acquires first-time justification for believing $\rQ\_3$ in virtue of (C1$\_{\textit{Zebra}}$) and (C2$\_{\textit{Zebra}}$). Also note that (C3$^{\textrm{ft}}\_{\textit{Zebra}}$) requires $s$'s justification for believing $\rQ\_3$ not to be independent of $s$'s justification for $\rP\_3$ based on seeing that $\rP\_3$. However, if (C1$\_{\textit{Zebra}}$) is true, the disjunctive template requires $s$ to have justification for believing $\rQ\_3$ independent of $s$'s justification for $\rP\_3$ based on her seeing that $\rP\_3$. (For if $s$ could not independently exclude that $\rQ\_3$ is false, given (D4), (D3) and (D2), $s$ could not exclude that the incompatible alternative $\rR\_{\textit{Zebra}}$ to $\rP\_3$, which $s$ cannot subjectively distinguish from $\rP\_3$ on the ground of her seeing that $\rP\_3$, is true.) Thus, when the disjunctive template is instantiated, (C1$\_{\textit{Zebra}}$) and (C3$^{\textrm{ft}}\_{\textit{Zebra}}$) cannot be satisfied at once. The disjunctive template has been criticized by McLaughlin (2003) on the ground that the template is instantiated whenever the justification for $p$ is fallible, i.e. compatible with $p$'s falsity. Here is an example from Brown (2004). Take this deductive argument: Fox $\rP\_7$. The animal in the garbage is a fox. Therefore: $\rQ\_7$. The animal in the garbage is not a cat. Suppose $s$ has a fallible justification for believing $\rP\_7$ based on $s$'s experience as if the animal in the garbage is a fox. Take now $\rR\_{\textit{fox}}$ to be $\rP\_7$'s logical negation. Since the justification that $s$ has for $\rP\_7$ is fallible, condition (D2) above is met by default. As one can easily check, conditions (D1), (D3), and (D4) are also met. So, Fox instantiates the disjunctive template. Yet it is intuitive that Fox does transmit justification to its conclusion. One could respond to McLaughlin that his objection is misplaced because the disjunctive template is meant to apply to infallible, and not fallible, justification. A more interesting response to McLaughlin is to refine some condition listed in the disjunctive template to block McLaughlin's argument while letting this template account for transmission failure of both fallible and infallible justification. Wright (2011) for instance suggests replacing (D3) with the following condition:[7] (D3$^{\}$) $r$ is incompatible with some presupposition of the cognitive project of obtaining justification for $p$ in the relevant fashion. According to Wright's (2011) characterization, a presupposition of a cognitive project is any condition such that doubting it before carrying out the project would rationally commit one to doubting the significance or competence of the project irrespective of its outcome.[8] For a wide class of cognitive projects, examples of these presuppositions include: the normal and proper functioning of the relevant cognitive faculties, the reliability of utilized instruments, the obtaining of the circumstances congenial to the proposed method of investigation, the soundness of relevant principles of inference utilized in developing and collating one's results, and so on. With (D3$^{\}$) in the place of (D3), Fox no longer instantiates the disjunctive template. For the truth of $\rR\_{\textit{fox}}$, stating that the animal in the garbage is not a fox, appears to jeopardize no presupposition of the cognitive project of obtaining perceptual justification for $\rP\_7$. Thus (D3$^{\}$) is not fulfilled. On the other hand, arguments that intuitively don't transmit do satisfy (D3$^{\}$). Take Zebra\. In this case $\rR\_{\textit{fox}}$ states that the animals in the pen are mules cleverly disguised to look like zebras. Since $\rR\_{\textit{fox}}$ entails that conditions are unsuitable for attaining perceptual justification for believing $\rP\_3$, $\rR\_{\textit{fox}}$ looks incompatible with a presupposition of the cognitive project of obtaining perceptual justification for $\rP\_3$. Thus, $\rR\_{\textit{fox}}$ does satisfy (D3$^{\}$) in this case. ### 3.3 Non-Standard Accounts In this section we outline two interesting branches in the literature on transmission failure of justification and warrant across valid inference. We start with Smith's (2009) non-standard account of non-transmissivity of justification. Then, we present Alspector-Kelly's (2015) account of non-transmissivity of Plantinga's warrant. According to Smith (2009), epistemic justification requires reliability. $s$'s belief that $p$, held on the basis of $s$'s belief that $e$, is reliable in Smith's sense just in case in all possible worlds including $e$ as true and that are as normal (from the perspective of the actual world) as the truth of $e$ permits, $p$ is also true. Consider Zebra again. Zebra $\rE\_3$. The animals in the pen look like zebras. $\rP\_3$. The animals in the pen are zebras. Therefore: $\rQ\_3$. The animals in the pen are not mules cleverly disguised to look like zebras. $s$'s belief that $\rP\_3$ is reliable in Smith's sense when it is based on $s$'s belief that $\rE\_3$. (Disguising mules to make them look like zebras is certainly an abnormal practice.) Thus, all possible $\rE\_3$-worlds that are as normal as the truth of $\rE\_3$ permits aren't worlds in which the animals in the pen are cleverly disguised mules. Rather, they are $\rP\_3$-worlds--i.e., worlds in which the animals in the pen are zebras. Smith describes two ways in which a belief can possess this property of being reliable. One is that a belief that $p$ has it in virtue of the modal relationship with its basis $e$. In this case, $e$ is a contributing reliable basis. Another possibility is when it is the content of the belief that $p$, rather than the belief's modal relationship with its basis $e$, that guarantees by itself the belief's reliability. In this case, $e$ is a non-contributing reliable basis. An example of the first kind is $s$'s belief that $\rP\_3$, which is reliable because of its modal relationship with $\rE\_3$. There are obviously many sufficiently normal worlds in which $\rP\_3$ is false, but no sufficiently normal world in which $\rP\_3$ is false and $\rE\_3$ true. An example of the second kind is $s$'s belief that $\rQ\_3$ as based on $\rE\_3$. It is this belief's content, and not its modal relationship with $\rE\_3$, that guarantees its reliability. As Smith puts it, there are no sufficiently normal worlds in which $\rE\_3$ is true and $\rQ\_3$ is false, but this is simply because there are no sufficiently normal worlds in which $\rQ\_3$ is false. According to Smith, a deductive inference from $p$ to $q$ fails to transmit to $q$ justification relative to $p$'s basis $e$, if $e$ is a contributing reliable basis for believing $p$ but is a non-contributing reliable basis for believing $q$. In this case the inference fails to explain $q$'s reliability: if $s$ deduced one proposition from another, she would reliably believe $q$, but not--not even in part--in virtue of having inferred $q$ from $p$ (as held on the basis of $e)$. Zebra fails to transmit to $\rQ\_3$ justification relative to $\rP\_3$'s basis $\rE\_3$ in this sense.[9] Let's turn to Alspector-Kelly (2015). As we have seen, Wright's analysis of failure of warrant transmission interprets the epistemic good whose transmission is in question as, roughly, the same as epistemic justification.[10] By so doing, Wright departs from Plantinga's (1993a, 1993b) influential understanding of warrant as the epistemic quality that (whatever it is) suffices to turn true belief into knowledge. One might wonder, however, if there are deductive arguments incapable of transmitting warrant in Plantinga's sense. (Hereafter we use 'warrant' only to refer to Plantinga's warrant.) Alspector-Kelly answers this question affirmatively contending that certain arguments cannot transmit warrant because the cognitive project of establishing their conclusion via inferring it from their premise is procedurally self-defeating. Alspector-Kelly follows Wright (2012) in characterizing a cognitive project as a pair of a question and a procedure that one executes to answer the question. An enabling condition of a cognitive project is, for Alspector-Kelly, any proposition such that, unless it is true, one cannot learn the answer to its defining question by executing the procedure associated with it. That a given object $o$ is illuminated, for example, is an enabling condition of the cognitive project of determining its color by looking at it. For one cannot learn by sight that $o$ is of a given color unless $o$ is illuminated. Enabling conditions can be opaque. An enabling condition of a cognitive project is opaque, relative to some actual or possible result, if it is the case that whenever the execution of the associated procedure yields this result, it would have produced the same result had the condition been unfulfilled. The enabling condition that $o$ be illuminated of the cognitive project just considered is not opaque, since looking at $o$ never produces the same response about $o$'s color when $o$ is illuminated and when it isn't. (In the second case, it produces no response.) Now consider the cognitive project of establishing by sight whether ($\rP\_3)$ the animals enclosed in the pen are zebras. An enabling condition of this project states that ($\rQ\_3)$ the animals in the pen are not mules disguised to look like zebras. For one couldn't learn whether $\rP\_3$ is true by looking, if $\rQ\_3$were true. $\rQ\_3$ is opaque with respect to the possible outcome that $\rP\_3$. In fact, suppose $\rQ\_3$ is satisfied because the pen contains (undisguised) zebras. In this case, looking at them will attest they are zebras, and the execution of the procedure associated with this project will yield the outcome that $\rP\_3$. But this is exactly the outcome that would be generated by the execution of the same procedure if $\rQ\_3$ were not satisfied. In this case too, looking at the animals would produce the response that $\rP\_3$. We can now elucidate the notion of a procedurally self-defeating cognitive project. The distinguishing feature of any project of this type is that it seeks to answer the question whether an opaque enabling condition--call it '$q$'--of the project itself is fulfilled. A project of this type has the defect that it necessarily produces the response that $q$ is fulfilled when $q$ is unfulfilled. Given this, it is intuitive that it cannot yield warrant to believe $q$. As an example of a procedurally self-defeating project, consider trying to answer the question whether your informant is sincere by asking her this very question. Your informant's being sincere is an enabling condition of the very project you are carrying out. If your informant were not sincere, you couldn't learn anything from her. This condition is also opaque with respect to the possible response that she is sincere. For your informant would respond that she is sincere both in case she is so and in case she isn't. Since the execution of this project is guaranteed to yield the result that your informant is sincere when she isn't, it is intuitive that it cannot yield warrant to believe that the informant is sincere. Alspector-Kelly contends that arguments like Zebra don't transmit warrant because the cognitive project of determining inferentially whether their conclusion is true is procedurally self-defeating in the sense advertised. Let's apply the explanation to Zebra. Imagine you carry out Zebra. This initially commits you to executing the project of establishing whether $\rP\_3$ by looking. Suppose you get $\rE\_3$ and hence $\rP\_3$ as an outcome. As we have seen, $\rQ\_3$ is an opaque enabling condition relative to the outcome $\rP\_3$ of the cognitive project you have executed. Imagine that now, having received $\rP\_3$ as a response to your first question, you embark in the second project which carrying out Zebra commits you to: establishing whether $\rQ\_3$ is true by inference from $\rP\_3$. This project appears doomed. $\rQ\_3$ is an enabling condition of the initial project of determining whether $\rP\_3$, which is opaque with respect to the very premise $\rP\_3$. It follows from this that $\rQ\_3$ is also an enabling condition of the second project. For if $\rQ\_3$ were unfulfilled, you couldn't learn $\rP\_3$ and, a fortiori, you couldn't learn anything else--$\rQ\_3$ included--by inference from $\rP\_3$. Another consequence is that $\rQ\_3$ is an enabling condition of the second project that is opaque relative to the very outcome that $\rQ\_3$. Suppose first that $\rQ\_3$ is true and that, having visually inspected the pen, you have verified $\rP\_3$. You then execute the inferential procedure associated with the second project and produce the outcome that $\rQ\_3$. Now consider the case in which $\rQ\_3$ is false. Looking into the pen still generates the outcome that $\rP\_3$ is true. Thus, when you execute the procedure associated with the second project, you still infer that $\rQ\_3$ is true. Since you would get the result that $\rQ\_3$ is true even if $\rQ\_3$ were false, the second project is procedurally self-defeating. In conclusion, carrying out Zebra commits you to executing a project that cannot generate warrant for believing its conclusion $\rQ\_3$. This explanation of non-transmissivity generalizes to all structurally similar arguments. ## 4. Applications The notions of transmissive and non-transmissive argument, above and beyond being investigated for their own sake, have been put to work in relation to specific philosophical problems and issues. An important problem is whether Moore's infamous proof of an external world is successful, and whether so are structurally similar Moorean arguments directed against the perceptual skeptic and--more recently--the non-believer. Another issue pertains to the solution of McKinsey paradox. A third issue concerns Boghossian's (2001, 2003) explanation of our logical knowledge via implicit definitions, criticized as resting on a non-transmissive argument schema (see for instance Ebert 2005 and Jenkins 2008). As the debate focusing on the last topic is only at an early stage of its development, it is preferable to concentrate on the first two, which will be reviewed in respectively Sect. 4.1 and Sect. 4.2 below. ### 4.1 Moorean Proofs, Perceptual Justification and Religious Justification Much of the contemporary debate on Moore's proof of an external world (see Moore 1939) is interwoven with the topic of epistemic transmission and its failure. Moore's proof can be reconstructed as follows: Moore $\rE\_8$. My experience is in all respects as of a hand held up in front of my face. $\rP\_8$. Here is a hand. Therefore: $\rQ\_8$. There is a material world (since any hand is a material object existing in space). Evidence $\rE\_8$ in Moore is constituted by a proposition believed by $s$. One might suggest, however, that this is a misinterpretation of Moore's proof (and variants of it that we shall consider shortly). One might argue that what is meant to give $s$ justification for believing $\rP\_8$ is $s$'s experience of a hand. Nevertheless, most of the epistemologists participating in this debate implicitly or explicitly assume that one's experience as if $p$ and one's belief that one has an experience as if $p$ have the same justifying power (cf. White 2006 and Silins 2007). Many philosophers find Moore's proof unsuccessful. Philosophers have proposed explanations of this impression according to which Moore is non-transmissive in some of the senses described in Sect. 3 (see mainly Wright 1985, 2002, 2007 and 2011).[11] A different explanation is that Moore's proof does transmit justification but is dialectically ineffective (see mainly Pryor 2004). According to Wright, there exist cornerstone propositions (or simply cornerstones), where $c$ is a cornerstone for an area of discourse $d$ just in case for any proposition $p$ belonging to $d, p$ could not be justified for any subject $s$ if $s$ had no independent justification for accepting $c$ (see mainly Wright 2004).[12] Cornerstones for the area of discourse about perceivable things are for instance the logical negations of skeptical conjectures, such as the proposition that one's experiences are nothing but one's hallucinations caused by a Cartesian demon or the Matrix. Wright contends that the conclusion $\rQ\_8$ of Moore is also a cornerstone for the area of discourse about perceivable things. Adapting terminology introduced by Pryor (2004), Wright's conception of the architecture of perceptual justification thus treats $\rQ\_8$ conservatively with respect to any perceptual hypothesis $p$: if $s$ had no independent justification for $\rQ\_8$, no proposition $e$ describing an apparent perception could supply $s$ with prima facie justification for any perceptual hypothesis $p$. It follows from this that Moore instantiates the information-dependence template considered in Sect. 3.2. For $\rQ\_8$ is part of the collateral information which $s$ needs independent justification for if $s$ is to receive some justification for $\rP\_8$ from $\rE\_8$. Hence Moore is non-transmissive (see mainly Wright 2002). Note that the thesis that Moore's proof is epistemologically useless because non-transmissive in this sense is compatible with the claim that by learning $\rE\_8, s$ does acquire a justification for believing $\rP\_8$. For instance, Wright (2004) contends that we all have a special kind of non-evidential justification, which he calls entitlement, for accepting $\rQ\_8$ as well as other cornerstones in general.[13] So, by learning $\rE\_8$ we do acquire justification for $\rP\_8$. Wright's analysis of Moore's proof and Wright's conservatism have mostly been criticized in conjunction with his theory of entitlement. A presentation of these objections is beyond the scope of this entry. (See however Davies 2004; Pritchard 2005; Jenkins 2007. For a defense of Wright's views see for instance Neta 2007 and Wright 2014.) As anticipated, other philosophers contend that Moore's proof does transmit justification and that its ineffectiveness has a different explanation. An important conception of the architecture of perceptual justification, called dogmatism in Pryor (2000, 2004), embraces a generalized form of liberalism about perceptual justification. This form of liberalism is opposed to Wright's conservatism, and claims that to have prima facie perceptual justification for believing $p$ from an apparent perception that $p, s$ doesn't need independent justification for believing the negation of skeptical conjectures or non-perceiving hypotheses like $\notQ\_8$. This is so, for the dogmatist, because our experiences give us immediate and prima facie justification for believing their contents. Saying that perceptual justification is immediate is saying that it doesn't presuppose--not even in part--justification for anything else. Saying that justification is prima facie is saying that it can be defeated by additional evidence. Our perceptual justification would be defeated, for example, by evidence that a relevant non-perceiving hypothesis is true or just as probable as its negation. For instance, $s$'s perceptual justification for $\rP\_8$ would be defeated by evidence that $\notQ\_8$ is true, or that $\rQ\_8$ and $\notQ\_8$ are equally probable. On this point the dogmatist and Wright do agree. They disagree on whether $s$'s perceptual justification for $\rP\_8$ requires independent justification for believing or accepting $\rQ\_8$. The dogmatist denies that $s$ needs that independent justification. Thus, according to the dogmatist, Moore's proof transmits the perceptual justification available for its premise to the conclusion.[14] The dogmatist argues (or may argue), however, that Moore's proof is dialectically flawed (cf. Pryor 2004). The contention is that Moore's is unsuccessful because it is useless for the purpose to convince the idealist or the external world skeptic that there is an external (material) world. In short, neither the idealist nor the global skeptic believes that there is an external world. Since they don't believe $\rQ\_8$, they are rationally required to distrust any perceptual evidence offered in favor of $\rP\_8$ in the first instance. For this reason they both will reject Moore's proof as one based on an unjustified premise.[15] Moretti (2014) suggests that the dogmatist could alternatively contend that Moore's proof is non-transmissive because $s$'s experience of a hand gives $s$ immediate justification for believing both the premise $\rP\_8$ and the conclusion $\rQ\_8$ of the proof at once. According to this diagnosis, Moore's proof is epistemically flawed because it instantiates a variant of indirectness (in which the evidence is an experience of a hand). Pryor's analysis of Moore's proof has principally been criticized in conjunction with his liberalism in epistemology of perception. (See Cohen 2002, 2005; Schiffer 2004; Wright 2007; White 2006; Siegel & Silins 2015.) Some authors (e.g., Wright 2003; Pryor 2004; White 2006; Silins 2007; Neta 2010) have investigated whether certain variants of Moore are transmissive of justification. These arguments start from a premise like $\rP\_8$, describing a (supposed) perceivable state of affairs of the external world, and deduce from it the logical negation of a relevant skeptical conjecture. Consider for example the variant of Moore that starts from $\rP\_8$ and replaces $\rQ\_8$ with: $\rQ\_8^\$. It is not the case that I am a handless brain in a vat fed with the hallucination of a hand held up in front of my face. Let's call Moore\** this variant of Moore. While dogmatists a la Pryor argue that Moore\* is transmissive but dialectically flawed (cf. Pryor 2004), conservatives a la Wright contend that it is non-transmissive (cf. Wright 2007). Although it remains very controversial whether or not Moore is transmissive, epistemologists have found some prima facie reason to think that arguments like Moore\* are non-transmissive. An important difference between Moore and Moore\* is this: whereas the logical negation of $\rQ\_8$ does not explain the evidential statement $\rE\_8$ ("My experience is in all respects as of a hand held up in front of my face") adduced in support of $\rP\_8$, the logical negation of $\rQ\_8^\$--$\notQ\_8^\$--somewhat explains $\rE\_8$. Since $\notQ\_8^\$ provides a potential explanation of $\rE\_8$, is it intuitive that $\rE\_8$ is evidence* (perhaps very week) for believing $\notQ\_8^\$. It is easy to conclude from this that $s$ cannot acquire justification for believing $\rQ\_8^\$ via transmission through Moore\* upon learning $\rE\_8$. For it is intuitive that if this were the case, $\rE\_8$ should count as evidence for $\rQ\_8^\$. But this is impossible: one and the same proposition cannot simultaneously be evidence for a hypothesis and its logical negation. By formalizing intuitions of this type, White (2006) has put forward a simple Bayesian argument to the effect that Moore\** and similar variants of Moore are not transmissive of justification.[16] (See Silins 2007 for discussion. For responses to White, see Weatherson 2007; Kung 2010; Moretti 2015.) From the above analysis it is easy to conclude that the information-dependence template is satisfied by Moore\* and akin proofs. In fact note that if $\rE\_8$ is evidence for both $\rP\_8$ and $\notQ\_8^\$, it seems correct to say that $s$ can acquire a justification for believing $\rP\_8$ only if $s$ has independent justification for disbelieving* $\notQ\_8^\$ and thus believing* $\rQ\_8^\$. Since $\notQ\_8^\$ counts as a non-perceiving hypothesis for Pryor, this gives us a reason to doubt dogmatism (cf. White 2006).[17] Coliva (2012, 2015) defends a view--baptized by her moderatism--that aims to be a middle way between Wright's conservatism and Pryor's dogmatism. The moderate contends--against the conservative--that cornerstones cannot be justified and that to possess perceptual justification $s$ needs no justification for accepting any cornerstones. On the other hand, the moderate claims--against the dogmatist--that to possess perceptual justification $s$ needs to assume (without justification) relevant cornerstones.[18] By relying on her variant of the information-dependent template described in Sect. 3.2, Coliva concludes that neither Moore nor any proof like Moore\* is transmissive. (For a critical discussion of moderatism see Avnur 2017; Baghramian 2017; Millar 2017; Volpe 2017; Coliva 2017.) Epistemological disjunctivists like McDowell and Pritchard have argued that in paradigmatic cases of perceptual knowledge, what is meant to give $s$ justification for believing $\rP\_8$ is, not $\rE\_8$, but $s$'s factive state of seeing that $\rP\_8$. This seems to have consequences for the question whether Moore\* transmits propositional justification. Pritchard explicitly defends the claim that when $s$ sees that $\rP\_8$, thereby learning that $\rP\_8, s$ can come to know by inference from $\rP\_8$ the negation of any skeptical hypothesis inconsistent with $\rP\_8$, like $\rQ\_8^\$ (cf. Pritchard 2012a: 129-30). This may encourage the belief that, for Pritchard, Moore\** can transmit the justification for $\rP\_8$, based on s's seeing that $\rP\_8$, to $\rQ\_8^\$ (see Lockhart 2018 for an explicit argument to this effect). This claim must however be handled with some care. As we have seen in Sect. 2, Pritchard contends that when one knows $p$ on the basis of evidence $e$, one can know $p$'s consequence $q$ by inference from $p$ only if $e$ sufficiently supports $q$. For Pritchard this condition is met when $s$'s support for believing $\rP\_8$ is constituted by $s$'s seeing that $\rP\_8, s$'s epistemic situation is objectively good (i.e., $s$'s cognitive faculties are working properly in a cooperative environment) and the skeptical hypothesis ruled out by $\rQ\_8^\$ has not been epistemically motivated. For, in this case, $s$ has a reflectively accessible factive support for believing $\rP\_8$ that entails--and so sufficiently supports--$\rQ\_8^\$. Thus, in this case, nothing stands in the way of $s$ competently deducing $\rQ\_8^\$ from $\rP\_8$, thereby coming to know $\rQ\_8^\$ on the basis of $\rP\_8$. If upon deducing* one proposition from another, $s$ comes to justifiably believe $\rQ\_8^\$ for the first-time, the inference from $\rP\_8$ to $\rQ\_8^\$ presumably transmits doxastic justification. (See the supplement on Transmission of Propositional Justification versus Transmission of Doxastic Justification.) It is more dubious, however, that when $s$'s support for $\rP\_8$ is constituted by $s$'s seeing that $\rP\_8$, Moore\* is also transmissive of propositional justification. For instance, one might contend that Moore\* is non-transmissive because it instantiates the disjunctive template described in Sect. 3.2 (cf. Wright 2002). To start with, $\rP\_8$ entails $\rQ\_8^\$, so (D1) is satisfied. Let $\rR^{\}\_{\textit{Moore}}$ be the proposition that this is no hand but $s$ is victim of a hallucination of a hand held up before her face. Since $\rR^{\}\_{\textit{Moore}}$ would be true if $\rQ\_8^\$ were false, (D4) is also satisfied. Furthermore, take the grounds of $s$'s justification for $\rP\_8$ to be $s$'s seeing that $\rP\_8$. Since this experience is a state for $s$ indistinguishable from one in which $\rR^{\}\_{\textit{Moore}}$ is true, (D2) is also satisfied. Finally, it might be argued that the proposition that one is not hallucinating is a presupposition of the cognitive project of learning about one's environment through perception. It follows that $\rR^{\}\_{\textit{Moore}}$ is incompatible with a presupposition of $s$'s cognitive project of learning about one's environment through perception. Thus (D3$^{\}$) appears fulfilled too. So, Moore\** won't transmit the justification that $s$ has for $\rP\_8$ to $\rQ\_8^\$. To resist this conclusion, a disjunctivist might insist that Moore\*, relative to $s$'s support for $\rP\_8$ supplied by $s$'s seeing that $\rP\_8$, doesn't always instantiate the disjunctive template* because (D2) isn't necessarily fulfilled (cf. Lockhart 2018.) By invoking a distinction drawn by Pritchard (2012a), one might contend that (D2) isn't necessarily fulfilled because $s$, though unable to introspectively discriminate seeing that $\rP\_8$ from hallucinating it, may have evidence that favors the hypothesis that she is in the first state, which makes the state reflectively accessible. For Pritchard, this happens--as we have seen--when $s$ sees that $\rP\_8$ in good epistemic conditions and the skeptical conjecture that $s$ is having a hallucination of a hand hasn't been epistemically motivated. In this case, $s$ can come to know $\rQ\_8^\$ by inference from $\rP\_8$ even if she is unable to introspectively discriminate one situation from the other. Pritchard's thesis that, in good epistemic conditions, $s$'s factive support for believing $\rP\_8$ coinciding with $s$'s seeing that $\rP\_8$ is reflectively accessible is controversial (cf. Piazza 2016 and Lockhart 2018). Since this thesis is essential to the contention that Moore\** may not instantiate the disjunctive template and may thus be transmissive of propositional justification, the latter contention is also controversial. So far we have focused on perceptual justification. Now let's switch to religious justification. Shaw (2019) aims to motivate a view in religious epistemology--he contends that the 'theist in the street', though unaware of the traditional arguments for theism, can find independent rational support for believing that God exists through proofs for His existence. These proofs are based on religious experiences and parallel in structure Moore's proof of an external world interpreted as having the premise supported by an experience. A Moorean proof for the existence of God is one that deduces a belief that God exists from what Alston (1991) calls a 'manifestation belief' (M-belief, for short). An M-belief is a belief based non-inferentially on a corresponding mystical experience. Typical M-beliefs are about God's acts toward a given subject at a given time: for example, beliefs about God's bringing comfort to me, reproving me for some wrongdoing, or demonstrating His love for me, and so on. Here is Moorean proof for the existence of God: God $\rP\_9$. God is comforting me just now. Therefore: $\rQ\_9$. God exists. Note that the 'good' case in which my experience that $\rP\_9$ is veridical has a corresponding, subjectively indistinguishable, 'bad' case, in which it only seems to me that $\rP\_9$ because I'm suffering from a delusional religious experience. A concern is, therefore, that God may instantiate the disjunctive template. In this case, my experience as if $\rP\_9$ could actually justify $\rP\_9$ only if I had independent justification for $\rQ\_9$ (cf. Pritchard 2012b). If so, God isn't transmissive. Shaw (2019) concedes that God is dialectically ineffective. To defend its transmissivity, he appeals to religious epistemological disjunctivism, which says that an M-belief that $p$ can enjoy infallible rational support in virtue of one's pneuming that $p$, where this mental state is both factive and accessible on reflection. Shaw intends 'pneuming that $p$' to stand as a kind of religious-perceptual analogue to 'seeing that $p$' (cf. Shaw 2016, 2019). When it comes to God, my belief that $\rP\_9$ is supposed to be justified by my pneuming that God is comforting me just now. As we have seen in the case of Moore\, a worry is that conceiving of the justification for $\rP\_9$ along epistemological disjunctivist lines may not suffice to exempt God* from the charge of instantiating the disjunctive template and being thus non-transmissive. ### 4.2 McKinsey's Paradox McKinsey (1991, 2002, 2003, 2007) has offered a reductio argument for the incompatibility of first-person privileged access to mental content and externalism about mental content. The privileged access thesis roughly says that it is necessarily true that if $s$ is thinking that $x$, then $s$ can in principle know a priori (or in a non-empirical way) that she is thinking that $x$. Externalism about mental content roughly says that predicates of the form 'is thinking that $x$'--e.g., 'is thinking that water is wet'--express properties that are wide, in the sense that possession of these properties by $s$ logically or conceptually implies the existence of relevant contingent objects external to $s$'s mind--e.g., water. McKinsey argues that $s$ may reason along these lines: Water $\rP\_{10}$. I am thinking that water is wet. $\rP\_{11}$. If I am thinking that water is wet, then I have (or my linguistic community has) been embedded in an environment that contains water. Therefore: $\rQ\_{10}$. I have (or my linguistic community has) been embedded in an environment that contains water. Water produces an absurdity. If the privileged access thesis is true, $s$ knows $\rP\_{10}$ non-empirically. If semantic externalism is true, $s$ knows $\rP\_{11}$ a priori by conceptual analysis. Since $\rP\_{10}$ and $\rP\_{11}$ do entail $\rQ\_{10}$ and knowledge is presumably closed under known entailment, $s$ knows $\rQ\_{10}$--which is an empirical proposition--by simply competently deducing it from $\rP\_{10}$ and $\rP\_{11}$ and without conducting any empirical investigation. Since this is absurd, McKinsey concludes that the privileged access thesis or semantic externalism must be false. One way to resist McKinsey's incompatibilist conclusion that the privileged access thesis and externalism about mental content cannot be true together is to argue that Water is non-transmissive. Since knowledge is presumably closed under known entailment, it remains true that $s$ cannot know $\rP\_{10}$ and $\rP\_{11}$ while failing to know $\rQ\_{10}$. However, McKinsey's paradox originates from the stronger conclusion--motivated by the claim that Water is a deductively valid argument featuring premises knowable non-empirically--that $s$, by running Water, could know non-empirically the empirical proposition $\rQ\_{10}$ that she or members of her community have had contact with water. This is precisely what could not happen if Water is non-transmissive: in this case $s$ couldn't learn $\rQ\_{10}$ on the basis of her non-empirical justification for $\rP\_{10}$ and $\rP\_{11}$, and her knowledge of the entailment between $\rP\_{10}$, $\rP\_{11}$, and $\rQ\_{10}$ (see mainly Wright 2000, 2003, 2011).[19] A first possibility to defend the thesis that Water is non-transmissive is to argue that it instantiates the disjunctive template (considered in Sect. 3.2) (cf. Wright 2000, 2003, 2011). If Water is non-transmissive, $s$ could acquire a justification for, or knowledge of, $\rP\_{10}$ and $\rP\_{11}$ only if $s$ had an independent justification for, or knowledge of, $\rQ\_{10}$. (And to avoid the absurd result that McKinsey recoils from, this independent justification should be empirical.) If this diagnosis is correct, one need not deny $\rP\_{10}$ or $\rP\_{11}$ to reject the intuitively false claim that $s$ could know the empirical proposition $\rQ\_{10}$ in virtue of only non-empirical knowledge. To substantiate the thesis that Water instantiates the disjunctive template one should first emphasize that the kind of externalism about mental content underlying $\rP\_{10}$ is compatible with the possibility that $s$ suffers from illusion of content. Were this to happen with $\rP\_{10}, s$ would seem to introspect that she believes that water is wet whereas there is nothing like that content to be believed by $s$ in the first instance. Consider: $\rR\_{\textit{water}}$. 'water' refers to no natural kind so that there is no content expressed by the sentence 'water is wet'. $s$'s state of having introspective justification for believing $\rP\_{10}$ is arguably subjectively indistinguishable from a situation in which $\rR\_{\textit{water}}$ is true. Thus condition (D2) of the disjunctive template is met. Moreover, $\rR\_{\textit{water}}$ appears incompatible with an obvious presupposition of $s$'s cognitive project of attaining introspective justification for believing $\rP\_{10}$, at least if the content that water is wet embedded in $\rP\_{10}$ is constrained by $s$ or her linguistic community having being in contact with water. Thus condition (D3$^{\}$) is also met. Furthermore, $\rP\_{10}$ entails $\rQ\_{10}$ (when $\rP\_{11}$ is in background information). Hence, condition (D1) is fulfilled. If one could also show that (D4) is satisfied in Water, in the sense that if $\rQ\_{10}$ were false $\rR\_{\textit{water}}$ would be true, one would have shown that the disjunctive template* is satisfied by Water. Wright (2000) takes (D4) to be fulfilled and concludes that the disjunctive template is satisfied by Water. Unfortunately, the claim that (D4) is satisfied in Water cannot easily be vindicated (cf. Wright 2003). Condition (D4) is satisfied in Water only if it is true that if $s$ (or $s$'s linguistic community) had not been embedded in an environment that contains water, the term 'water' would have referred to no natural kind. This is true only if the closest possible world $w$ in which this counterfactual's antecedent is true is like Boghossian's (1997) Dry Earth--namely, a world where no one has ever had any contact with any kind of watery stuff, and all apparent contacts with it are always due to multi-sensory hallucination. If $w$ is not Dry Earth, but Putnam's Twin Earth, however, the counterfactual turns out false, as in this possible world people usually have contact with some other watery stuff that they call 'water'. So, in this world 'water' refers to a natural kind, though not to H2O. Since determining which of Dry Earth or Twin Earth is modally closer to the actual world (supposing $s$ is in the actual world)--and so determining whether (D4) is satisfied in Water--is a potentially elusive task, the claim that Water instantiates the disjunctive template appears less than fully warranted.[20] An alternative dissolution of McKinsey paradox--also based on a diagnosis of non-transmissivity--seems to be available if one considers the proposition that $\rQ\_{11}$. $s$ (or $s$'s linguistic community) has been embedded in an environment containing some watery substance (cf. Wright 2003 and 2011). This alternative strategy assumes that $\rQ\_{11}$, rather than $\rQ\_{10}$, is a presupposition of $s$'s cognitive project of attaining introspective justification for $\rP\_{10}$. Even if Water doesn't instantiate the disjunctive template, a new diagnosis of what's wrong with McKinsey's paradox could rest on the claim that the different argument yielded by expanding Water with $\rQ\_{11}$ as conclusion--call it Water\--instantiates the disjunctive template. If $\rR\_{\textit{water}}$ is the same proposition as above, it is easy to see that Water\** satisfies conditions (D4), (D2), and (D3$^{\}$) of the disjunctive template. Furthermore (D1) is satisfied at least in the sense that it seems a priori* that $\rP\_{10}$ via $\rQ\_{10}$ entails $\rQ\_{11}$ (if $\rP\_{11}$ is in background information) (cf. Wright 2011). On this novel diagnosis of non-transmissivity, what would be paradoxical is that $s$ could earn justification for $\rQ\_{11}$ in virtue of her non-empirical justification for $\rP\_{10}$ and $\rP\_{11}$ and her knowledge of the a priori link from $\rP\_{10}$, $\rP\_{11}$ via $\rQ\_{10}$ to $\rQ\_{11}$. If one follows Wright's (2003) suggestion[21] that $s$ is entitled to accept $\rQ\_{11}$--namely, the presupposition that there is a watery substance that provides 'water' with its extension--Water becomes innocuously transmissive, and the apparent paradox surrounding Water vanishes. This is so at least if one grants that it is a priori that water is the watery stuff of our actual acquaintance, once it is presupposed that there is any watery stuff of our actual acquaintance. For useful criticism of responses of this type to McKinsey's paradox see Sainsbury (2000), Pritchard (2002), Brown (2003, 2004), McLaughlin (2003), McKinsey (2003), and Kallestrup (2011).
watsuji-tetsuro	## 1. Life and Career Watsuji Tetsuro (1889-1960) was one of a small group of philosophers in Japan during the twentieth century who brought Japanese philosophy to the attention of the world. In terms of his influence, exemplary scholarship, and originality he ranks with Nishida Kitaro, Tanabe Hajime, and Nishitani Keiji. The latter three were all members of the so-called Kyoto School, and while Watsuji is not usually thought of as being a member of this school, the influence and tone of his work clearly shows him to be a like-minded thinker. The Kyoto School, of which Nishida was the pioneering founder, is so identified because of its common focus: Nishida's important work, East/West comparative philosophy, and an on-going attempt to give expression to Japanese ideas and concepts by means of the clarity afforded by Western philosophical tools and techniques. Whereas the general emphasis of the Kyoto School is on epistemology, metaphysics and logic, Watsuji's primary focus came to be ethics, although his earlier studies of Schopenhauer, Nietzsche, and Kierkegaard ranged far beyond the ethical. And while his impressive output (an original nineteen volume collected works has been expanded by Yuasa Yasuo (1925-2005), his former student who became one of Japan's leading contemporary philosophers, to twenty-seven volumes) includes wide-ranging studies in the history of both Eastern and Western philosophy, his focus was not on Nishida and Tanabe, but on reconstructing the origins of Japanese culture more or less from the ground up. Hence, while many of his ideas, e.g. the centrality of the concept of 'nothingness' or 'emptiness,' and dialectical contradiction, show the influence of the Kyoto School thinkers, his own creative approach led him to formulate them differently, and to apply them to ethical, political, and cultural issues. Nevertheless, given the overlap of concepts, it would be a mistake to exclude him from some sort of honorary membership in the Kyoto School (see the entry on the Kyoto School). And given that there never was an actual 'school' at all, it is enough to say that Watsuji was an active force in Japanese philosophy in the twentieth century, along with Nishida, Tanabe, and later on, Nishitani. Watsuji was born in 1889, the second son of a physician, in Himeji City, in Hyogo Prefecture. He entered the prestigious First Higher School in Tokyo (now Tokyo University), in 1906, graduating in 1909. Later that same year, he entered Tokyo Imperial University, where his specialization was philosophy, in the Faculty of Literature. He married Takase Teru in 1912, and a daughter, Kyoko was born to them in 1914. As a student at Himeji Middle School, he displayed a passion for literature, especially Western literature, and "is said to have been fired with the ambition to become a poet like Byron" (Furukawai 1961, 217). He wrote stories and plays, and was coeditor of a literary magazine. When he entered the First Higher School in Tokyo, even though he had decided to pursue philosophy seriously, he remained "as deeply immersed in Byron as ever and attracted chiefly to things literary and dramatic" (Furukawa 1961, 218). He is said to have had several excellent teachers at this school, and his interests were considerably expanded in the arts and in literature. The Headmaster, Nitobe Inazo, was of particular importance. James Kodera writes that "Nitobe's book on Bushido, The Soul of Japan, began to awaken Tetsuro not only to the Eastern heritage but also to the study of ethics" (Kodera 1987, 6). Still, while this early glimpse into the forgotten depths of his own culture may have planted the seeds of later inquiries and insights, Watsuji gained sustenance and insight at this time in his academic career from his reading in Western Romanticism and Individualism. Without a doubt, the most important influence in Watsuji's early years was the brilliant novelist Natsume Soseki, considered a foremost interpreter of modern Japan. James Kodera writes that Watsuji found in Soseki "a human being struggling with the human condition in the particularities of early modern Japan, suddenly exposed to the West and yet struggling to sustain his own identity" (Kodera 1987, 6). Soseki was beginning to abandon his unqualified admiration of Western individualism, and was embarking on a critique of both individualism and the modern culture of the Western world. It was not only a move away from the values which he had come to associate with Western culture, but a move back to the values of his own Japanese cultural tradition. Watsuji never met Soseki while at the University of Tokyo, but did meet him in 1913 and became a member of a study group that met at Soseki's home. Soseki died three years later, when Watsuji was 27, and upon his death, Watsuji began to compose a lengthy reminiscence of him. His reflections were published in 1918 (in his Guzo saiko), and they marked Watsuji's own transformation from advocate of Western ways to critic of the West, turning toward a reconsideration of Japanese and Eastern cultural resources. It is perhaps telling that in a series that Soseki wrote for a popular Tokyo newspaper during 1912-1913, "Soseki depicted the plight of the modern individual as one of painful loneliness and helplessness" (Kodera 1987, 6). He saw egoism as the source of the plight, and has Ichiro, the hero of the serially appearing novel Wayfarer, conclude that "there is no bridge leading from one man to another; loneliness, loneliness, thou [are] mine home" (Kodera 1987, 6). A solution to the modern predicament of estrangement, loneliness and helplessness came to be found, for Watsuji, in our social interconnections. What modern Western society was losing with great rapidity, was still evident in Japanese society, where individuality was tempered with a strong social consciousness, and this more balanced sense of self could be found in the earliest of Japanese cultural documents. Soseki's proposed solution was to "follow Heaven, forsake the self" (Kodera 1987, 6), but Watsuji's more humanistic remedy for an egoism which inevitably led to estrangement, was to reinvest in one's social interconnectedness. We are not only by nature social beings, but we inevitably come into the world already in relationship: with our language and culture, traditions and expectations, parents, caregivers, and teachers. It is a myth of abstraction that we come into the world as isolated egos. Watsuji graduated from Tokyo University in 1912, but not without a frantic effort to write a second thesis in a very short span of time, because the topic of his first thesis was deemed unacceptable. Furukawa Tetsushi notes that "at the time the atmosphere in the Faculty of Philosophy was inimical to the study of a poet-philosopher like Nietzsche. Consequently, Watsuji's 'Nietzschean Studies' was rejected as a suitable graduation thesis" (Furukawa 1961, 219). In its place he was obliged to substitute a second thesis on Schopenhauer, which he entitled "Schopenhauer's Pessimism and Theory of Salvation." This thesis "was presented only just in time" for him to graduate (Furukawa 1961, 219). Both theses were eventually published. Watsuji enrolled in the graduate school of Tokyo Imperial University in 1912, the same year that he completed his undergraduate studies. His studies of Schopenhauer and Nietzsche in 1912, and of Kierkegaard in 1915 provide ample evidence of his interest in and competence in Western philosophy. At the same time, he continued to study the Romantic poets, Byron, Shelley, Tennyson and Keats, being torn between his literary and philosophical interests. In any case, perhaps because his literary attempts were "complete failures," he decided to give "up literary creation and devoted all his exertions to the writing of critical essays and philosophical treatises" (Furukawa 1961, 218). It was not long before he was in demand as a teacher, first as a lecturer at Toyo University in 1920, an instructor at Hosei University in 1922, an instructor at Keio University in 1922-23, and at the Tsuda Eigaku-juku from 1922-24 (Dilworth et al 1998, 221). But his real break came in 1925, when Nishida Kitaro and Hatano Seiichi offered him the position of lecturer in the Philosophy Department of the Faculty of Literature at Kyoto Imperial University, where he was to take on the responsibility for the courses in ethics. This put him at the center of the developing Kyoto School philosophy. As was the custom at the time with promising young scholars, he was sent to Germany in 1927 on a three-year scholarship. His reflections on that sojourn in Europe became the substance of his highly successful book, Fudo, which has been translated into English as Climate and Culture. In fact, Watsuji spent only fourteen months in Europe, being forced to return to Japan in the summer of 1928 because of the death of his father. In 1929, he also took on a part-time position at Ryukoku University, and in 1931, he became a professor at Kyoto Imperial University. In 1934, he was appointed professor in the Faculty of Literature at Tokyo Imperial University, and he continued to hold this important post until his retirement in 1949. Perhaps part of the reason that he is so often viewed as someone on the periphery of the Kyoto School philosophic tradition has to do with his work in Tokyo, geographically removed from the center of discussion and dialogue in Kyoto. One cannot help but be impressed by the extent of his published output, as well as by its remarkable diversity, spanning literature, the arts, philosophy, cultural theory, as well as Japanese, Chinese, Indian, and Western traditions. In addition to his studies of Schopenhauer (1912), Nietzsche (1913), and Kierkegaard (1915), in 1919, he published Koji junrei (Pilgrimages to the Ancient Temples in Nara ), his study of the temples and artistic treasures of Nara, a work that became exceptionally popular. In 1920 came Nihon kodai bunka (Ancient Japanese Culture), a study of Japanese antiquity, including Japan's most ancient writings, the Kojiki and Nihongi. In 1925, he published the first volume of Nihon seishinshi kenkyu (A Study of the History of the Japanese Spirit), with the second volume appearing in 1935. This study contained his investigation of Dogen's work (in Shamon Dogen, [The Monk Dogen ]), and it can be said that it was Watsuji who single-handedly brought Dogen's work out of nearly total obscurity, into the forefront of philosophical discussion. Also, in 1925, he published Kirisutokyo no bunkashiteki igi (The Significance of Primitive Christianity in Cultural History), followed by Genshi Bukkyo no jissen tetsugaku (The Practical Philosophy of Primitive Buddhism) in 1927. Returning from Europe in 1928, he continued at Kyoto Imperial University until his appointment as professor at Tokyo Imperial University in 1934. In 1929, he edited Dogen's Shobogenzo zuimonki. Watsuji received his degree of Doctor of Letters, based on his The Practical Philosophy of Primitive Buddhism in 1932. He wrote as well A Critique of Homer at about this time, a work that was not published until 1946. Also in 1932 he wrote Porisuteki ningen no rinrigaku (The Ethics of the Man of the Greek Polis), which was not published until 1948. In 1936, he completed his Koshi (Confucius). In 1935, he completed the important Ethics as the Study of Man (Ningen no gaku to shite no rinrigaku), to be followed by his three volume expansion of his views on ethics, Rinrigaku (Ethics) appearing successively in 1937, 1942, and 1949. In 1938, he published Jinkaku to jinruisei (Personality and Humanity), and in 1943, he published his two volume Sonno shiso to sono dento (The Idea of Reverence for the Emperor and the Imperial Tradition). This latter publication is one of the works for which Watsuji was branded a right-wing, reactionary thinker. In 1944, he published a volume of two essays, Nihon no shindo (The Way of the Japanese [Loyal] Subject; and Amerika no kokominsei (The Character of the American People), and in 1948, The Symbol of National Unity (Kokumin togo no shocho). His last publications were the best-selling Sakoku--Nihonno higeki (A Closed Country--The Tragedy of Japan) in 1950, Uzumoreta Nihon (Buried Japan) in 1951, Nihon rinri shisoshi (History of Japanese Ethical Thought) in two volumes in 1953, Katsura rikyu: seisaku katei no kosatsu (The Katsura Imperial Villa: Reflections on the Process of Its Construction) in 1955, and Nihon geijutsu kenkyu (A Study of Japanese Art), also published in 1955. By any standard, this is an impressive array of major publications, several of them extremely influential both with the world of scholarship, and among the general public as well. One cannot but be impressed by the breadth of Watsuji's interests, by the depth of his scholarship, and by his ability as a remarkably clear and graceful writer. Yet his Fudo (Climate and Culture), and his studies in ethics, particularly his Rinrigaku stand out as his two most influential publications. ## 2. The Philosophy of Watsuji The foundations of Watsuji's thought were the extensive studies in Western philosophy that he engaged in during his earlier years, up until 1917 or 1918, followed by his extensive studies in Japanese and Far Eastern philosophy and culture. Upon his return from Europe in 1928, and as a direct result of being among the very first to read Martin Heidegger's Sein und Zeit, Watsuji began work on Fudo (Fudo ningen-gakuteki kosatsu), translated into English as Climate and Culture. 'Fudo' means "wind and earth...the natural environment of a given land" (Watsuji 1988 [1961], 1). We are all inescapably environed by our land, its geography and topography, its climate and weather patterns, temperature and humidity, soils and oceans, its flora and fauna, and so on, in addition to the resultant human styles of living, related artifacts, architecture, food choices, and clothing. This is but a partial list, but even this sketchy list makes clear that Watsuji is calling attention to the many ways in which our environment, taken in the broad sense, shapes who we are from birth to death. Heidegger's emphasis was on time and the individual, and too little, according to Watsuji, on space and the social dimensions of human beings. When we add to our sense of climate as including not only the natural geographic setting of a people and the region's weather patterns, but also the social environment of family, community, society at large, lifestyle, and even the technological apparatus that supports community survival and interaction, then we begin to glimpse what Watsuji had in mind by climate, and how there exists a mutuality of influence from human to environment, and environment to human being which allows for the continued evolution of both. Climate is the entire interconnected network of influences that together create an entire people's attitudes and values. "History and nature," remarks Yuasa Yasuo, "like man's mind and body, are in an inseparable relationship" (Yuasa 1996, 168). Culture is that mutuality of influence, recorded over eons of time past, which continues to effect the cultural present of a people. Who we are is not simply what we think, or what we choose as individuals in our aloneness, but is also the result of the climatic space into which we are born, live, love, and die. Even before his travels in Europe (1927-28), Watsuji was convinced that one's environment was central in shaping persons and cultures. In Guzo Saiku (Revival of Idols), published in 1918, he had concluded that "all inquiries into the culture of Japan must in their final reduction go back to the study of her nature" (Furukawa 1961, 214). From about 1918 on, Watsuji's focus became the articulation of what it is that constitutes the Japanese spirit. Ancient Japanese Culture, which he wrote in 1920, is an attempt to revitalize Japan's oldest Chronicles (the Kojiki and Nihongi) using modern literary techniques as well as newly available archaeological evidence. He treated these collections of ancient stories, legends, poems, songs, and myths as literature, rather than as sacred scripture. Quoting from the Kojiki the imaginative story of creation, he then glosses this rich account: "'When the land was still young and as a piece of floating grease, drifting about as does a jellyfish, there came into existence a god, issuing from what grew up like a reed-bud...' [is] a superb...image of a piece of grease floating about without definite form like a thin, muddy substance far thicker than water, yet not solid, and of the image of a soft jellyfish with a formless form drifting on the water, and of the image of exuberant life of a reed-bud sprouting powerfully out of the muddy water of a swampy marsh. There is no other description that I know of that so graphically depicts the state of the world before creation" (Furukawa 1961, 221-22). It is a concrete and graphic depiction of the formlessness out of which all things arose, and is perhaps an early attempt at talk of nothingness, a central idea of the Kyoto School, and later on for Watsuji as well. Not only can one discern his early interest in the importance of nothingness in his thinking, but there is also indication that he was sensitive to the non-dual immediacy of experience, which Nishida described as pure experience (see Section 2.1 of the entry on Nishida Kitaro. As though discovering the roots of Nishida's pure experience, he writes that "the ancient poets, whose feelings still retain a virgin simplicity as a single undivided experience, are not yet troubled by this division of the subjective and the objective" (as found in Japan's oldest collection of poems, the Man'yoshu) (Furukawa 1961, 224). Natural beauty, for the poet, is as yet an undivided experience, a pure experience which is prior to the subject/object dichotomy, a total immersion in the moment without thought or reflection, in an ecstasy of feeling. In his Preface to Revival of Idols he warns that he intends not "a mere 'revival of the old,'" but rather what he wished to achieve "is nothing more or nothing less than to advance such life as lives in the everlasting New" (Furukawa 1961, 227). To revive the old, then, is to cause it to shine anew, but in the light of contemporary issues and concerns. It is to revive its meaning for us, here and now, and not merely to show that it had meaning in the past. As part of one's cultural climate, the past inevitably operates still in the present of every Japanese. It was this still active element that Watsuji sought to uncover, and to express in such a way as to allow others to share in the present infused with the past in the consciousness of their own lives, rather than in some unconscious and only partially developed way. Similarly, he attempted to reveal the relevance of primitive Christianity, primitive Buddhism, and Confucianism as cultural inheritances which continue to shape people in various 'climates' around the world. He made explicit what was implicit, and he sought to revitalize the active ingredients of cultural traditions for right 'now'. In the Preface to Climate and Culture, Watsuji states that when we come to consider both the natural and the human cultural climate in which we find ourselves, we render both nature and culture as "already objectified, with the result that we find ourselves examining the relation between object and object, and there is no link with subjective human existence" (Watsuji 1961, v). Watsuji distinguishes his interpretation of fudo, "literally 'Wind and Earth' in Japanese" (Watsuji 1961, 1) from what he maintained was its then conventional understanding simply as a term used for natural environment -- something that is a resource to be used, or an object separate from or merely alongside our being-in-the-world. The phenomena of climate must be seen "as expressions of subjective human existence and not of natural environment" (Watsuji 1961, v). He explains this using the example of the phenomenon of cold, a single climatic feature. Ordinarily, we think of 'us' and 'coldness' as objectively distinct and separate from us. Phenomenologically, however, we only come to know that it is cold after we actually feel it as cold. Coldness does not press upon us from the outside; rather, we are already out in the cold. As Heidegger emphasized, we 'ex-istere' outside of ourselves, and in this case, in the cold. It is not the cold which is outside of us, but we who are already out in the cold. And we feel this cold in common with other people. We all talk about the weather. To existere, then, means that we experience the cold with other 'I's. We experience coldness within ourselves, with others, and "in relation to the soil, the topographic and scenic features and so on of a given land" (Watsuji 1961, 5). In a telling, and poetically adept passage, he writes that a cold wind may be experienced as a sharp mountain blast, or a dry wind sweeping through a city at the end of winter, or "the spring breeze may be one which blows off cherry blossoms or which caresses the waves" (Watsuji 1961, 5). All weather is as much 'subjective' as it is 'objective.' Because of the cold, we must decide upon sources of heat for our houses, design and create appropriate clothing (for each of the seasons and conditions), seek proper ventilation, defend ourselves and our houses against special conditions (floods, monsoon rains, typhoons, tornadoes, earthquakes, volcanic eruptions, tsunamis, etc.), counteract excessive humidity in some way, and we must learn how to grow our food and eat in ways compatible with the climatic conditions, and our capabilities of farming and gathering. We apprehend ourselves in climate, revealing ourselves to ourselves as both social and historical beings. Here is the crux of Watsuji's insight, and of his criticism of Heidegger's Sein und Zeit: to emphasize our being in time is "to discover human existence on the level only of individual consciousness" (Watsuji 1961, 9). As temporal beings, we can exist alone, in isolative reflection. On the other hand, if we recognize the 'dual character' of human beings as existing in both time and space, as both individuals and social beings, "then it is immediately clear that space must be regarded as linked with time" (Watsuji 1961, 9). Space is inextricably linked with time, and the individual and social aspects of ourselves are inextricably linked as well, and our history and culture are linked to our climate. And while change is constant, and hence all structures are continuously evolving, this evolution is inextricably linked to our history, traditions, and cultural forms of expression. We discover ourselves in climate, and it is because of climate that we have come to express ourselves as we have: "climate...is the agent by which human life is objectivised" (Watsuji 1961, 14). In order to more faithfully convey Watsuji's notion of fudo, Augustin Berque suggests "milieu" as a more preferable translation than climate. Milieu more accurately captures the mutual co-constituting at the heart of fudo. For Watsuji, the notion of fudo is supposed to suggest that the spatial, environmental, and collective aspects of human existence are all intertwined (Berque, 1994, 498). Climate, or milieu, serves as the always present background to what becomes the foreground focus for Watsuji, the study of Japanese ethics in practice and in theory. Ethics is the study of the ways in which men and women, adults and children, the rulers and those ruled, have come to deal with each other in their specific climatic conditions. Ethics is the pattern of proper and effective social interaction. ## 3. Ethics Watsuji's objection to individualistic ethics, which he associated with virtually all Western thinkers to some degree, is that it loses touch with the vast network of interconnections that serves to make us human. We are individuals inescapably immersed in the space/time world, together with others. Individual persons, if conceived of in isolation from their various social contexts, do not and cannot exist except as abstractions. Our way of being in the world is an expression of countless people and countless actions performed in a particular 'climate,' which together have shaped us as we are. Indeed a human being is a unified structure of past, present, and future; each of us is an intersection of past and future, in the present 'now.' There is no possibility of the isolation of the ego, and yet many write as though there were. They are able to make a case of it, in part because they ignore the spatiality of ningen (human being), focussing on ningen's temporality. Watsuji believed that it was far more difficult to consider a human being as strictly an individual, when thought of as a being in space. Spatially, we move in a common field, and that field is cultural in that it is criss-crossed by roads and paths, and even by forms of communication such as messenger services, postal routes, newspapers, flyers, broadcasts over great distances, all in addition to everyday polite conversation. Watsuji makes a point of the legend of the isolated and hopelessly marooned Robinson Crusoe, for even Robinson Crusoe continued to be culturally connected, continuing to speak an inherited language, and improvising housing, food, and clothing based on past social experiences, and continuing to hope for rescue at the hands of unknown others. Watsuji rejects all such 'desert island' constructions as mere abstractions. Thomas Hobbes imagined a state of nature in which we are radically discrete individuals, at a time before significant social interconnections have been established. Watsuji counters that we are inescapably born into social relationships, beginning with one's mother, and one's caregivers. Our very beginnings are etched by the relational interconnections which keep us alive, educate us, and initiate us into the proper ways of social interaction. At the center of Watsuji's study of Japanese ethics is his analysis of the human person, in Japanese, ningen. In his Rinrigaku, he affirms that ethics is, in the final analysis, the study of human persons. Offering an etymological analysis, as he does so often, he displays the important complexity in the meaning of ningen. Ningen is composed of two characters, nin, meaning 'person' or 'human being,' and gen, meaning 'space' or 'between.' He cautions that it is imperative to recognize that a human being is not just an individual, but is also a member of many social groupings. We are individuals, and yet we are not just individuals, for we are also social beings; and we are social beings, but we are not just social beings, for we are also individuals. Many who interpret Watsuji forget the importance which he gave to this balanced and dual-nature of a human being. They read the words, but then go on to argue that he really gives priority to the collectivistic or social aspect of what it means to be a human being. That such an imbalance often occurs in Japanese society may be the reason for this conclusion. Yet it does not fit Watsuji's theoretical position, which is that we are, at one and the same time, both individual and social. In A Study of the History of the Japanese Spirit (1935) Watsuji cautions that "...the communion between man and man does not mean their becoming merely one. It is only through the fact that men are unique individuals that a cooperation between 'man and man' can be realized" (Watsuji 1935, 112). The tension between one's individual and one's social nature must not be slackened, or else the one is likely to overwhelm the other. He makes this point even clearer in discussing the creation of renga poetry, in the same volume. Renga poems are not created by a single individual but by a group of poets, with each individual verse linked to the next, and each verse the creation of a single individual, and yet each must cohere with the 'poetic sphere' as a whole. Watsuji concludes, "if there are self-centered persons in the company, a certain 'distortion' will be felt and group spirit itself will not be produced. When there are people, who, lacking individuality, are influenced only by others' suggestions, a certain 'lack of power' will be felt, and a creative enthusiasm will not appear. It is by means of attaining to Nothingness while each remains individual to the last, or in other words, by means of movements based on the great Void by persons each of whom has attained his own fulfilment, that the company will be complete and interest for creativity will be roused" (Watsuji 1935, 113). Individuality is not, and must not be lost, else the balance is destroyed, and creativity will not effectively arise. What is required is that we become selfless, no longer self-centered, and open to the communal sense of the whole group or society. It is a sense of individuality that is aware of social, public interconnections. One expresses one's individuality by negating the social group or by rebelling against various social expectations or requirements. To be an individual demands that one negate the supremacy of the group. On the other hand, to envision oneself as a member of a group is to negate one's individuality. But is this an instance of poor logic? One can remain an individual and as such join as many groups as one wishes. Or one can think of oneself as an individual and yet as a parent, a worker, an artist, a theatre goer, and so forth. Watsuji understood this, but his argument is that it is possible to think in such ways only if one has already granted logical priority to the individual qua individual. Whatever group one belongs to, one belongs to it as an individual, and this individuality is not quenchable, except through death, or inauthenticity. Nevertheless, Watsuji's conception of what he calls the 'negation of negation' has a quite different, and perhaps deeper emphasis. To extricate ourselves from one or another socio-cultural inheritance, perhaps the acceptance of the Shinto faith, one has to rebel against this socio-cultural form by affirming one's individuality in such a way as to negate its overt influence on oneself. This is to negate an aspect of one's history by affirming one's individuality. But the second negation occurs when one becomes a truly ethical human being, and one negates one's individual separateness by abandoning one's individual independence from others. What we have now is a forgetting of the self, as Dogen urged ("to study the way is to study the self, to study the self is to forget the self, to forget the self is to become enlightened by all things"), which yields a 'selfless' morality. To be truly human is not the asserting of one's individuality, but an annihilation of self-centeredness such that one is now identified with others in a nondualistic merging of self and others. Benevolence or compassion results from this selfless identification. This is our authentic 'home ground,' and it rekindles our awareness of our true and original nature. This home ground he calls 'nothingness,' about which more will be said below. Watsuji's analysis of gen is of equal interest. He makes much of the notion of 'betweenness,' or 'relatedness.' He traces gen (ken) back to its earlier form, aida or aidagara, which refers to the space or place in which people are located, and in which the various crossroads of relational interconnection are established. Watsuji's now famous former student, Yuasa Yasuo, observes that "this betweenness consists of the various human relationships of our life-world. To put it simply, it is the network which provides humanity with a social meaning, for example, one's being an inhabitant of this or that town or a member of a certain business firm. To live as a person means...to exist in such betweenness" (Yuasa 1987, 37). As individuals, we are private beings, but as social beings we are public beings. We enter the world already within a network of relationships and obligations. Each of us is a nexus of pathways and roads, and our betweenness is already etched by the natural and cultural climate that we inherit and live our lives within. With the social aspect of self being more foregrounded in Japan than in the West (especially at the time Watsuji was writing), it is imperative, therefore, that one know how to navigate these relational waters successfully, appropriately, and with relative ease and assurance. The study of these relational navigational patterns--between the individual and the family, self and society, as well as one's relationship to the milieu--is the study of ethics. Watsuji usually writes of ningen sonzai, and sonzai (existence) is composed of two characters, son (which means to preserve, to sustain over time), and zai (to stay in place, and in this case, to persevere in one's relationships). Ningen sonzai, then, refers to human nature as individual yet social, private as well as public, with our coming together in relationship occurring in the betweenness between us, which relationships we preserve and nourish to the fullest. Ethics has to do with the ways in which we, as human beings, respect, preserve, and persevere in the vast complexity of interconnections which etch themselves upon us as individuals, thereby forming our natures as social selves, and providing the necessary foundation for the creation of cooperative and workable societies. The Japanese word for ethics is rinri, which is composed of two characters, rin and ri. Rin means 'fellows,' 'company,' and specifically refers to a system of relations guiding human association. Ri means 'reason,' or 'principle,' the rational ordering of human relationships. These principles are what make it possible for human beings to live in a cooperative community. Watsuji refers to the ancient Confucian patterns of human interaction as between parent and child, lord and vassal, husband and wife, young and old, and friend and friend. Presumably, one also acquires a sense of the appropriate and ethical in all other relationships as one grows to maturity in society. If enacted properly these relationships, which occur in the betweenness between us, serve as the oil which lubricates interaction with others in such a way as to minimize abrasive occurrences, and to maximize smooth and positive relationships. One can think of the betweenness between each of us as a basho, an empty space, in which we can either reach out to the other in order to create a relationship of positive value, or to shrink back, or to lash out, making a bad situation worse. The space is pure potential, and what we do with it depends on the degree to which we can encounter the other in a fruitful and appropriate manner in that betweenness. Nevertheless, every encounter is already etched with the cultural traditions of genuine encounter; ideally positive expectation, good will, open-heartedness, cheerfulness, sincerity, fellow-feeling, and availability. Ethics "consists of the laws of social existence" writes Watsuji (Watsuji 1996, 11). ## 4. Emptiness The annihilation of the self, as the negation of negation "constitutes the basis of every selfless morality since ancient times," asserts Watsuji (Watsuji 1996, 225). The negating of the group or society, and the emptying of the individual in Watsuji's sense of the negation of each by the other pole of ningen, makes evident that both are ultimately 'empty,' causing one to reflect upon that which is ultimate, and at the base of both one's individuality and the groups with which one associates. The losing of self is a returning to one's authenticity, to one's home ground as that source from which all things derive, and by which they are sustained. It is the abandonment of the self as independent which paves the way for the nondual relation between the self and others that terminates in the activity of benevolence and compassion through a unification of minds. The ethics of benevolence is the development of the capacity to embrace others as oneself or, more precisely, to forget one's self such that the distinction between the self and other does not arise in this nondualistic awareness. One has now abandoned one's self, one's individuality, and become the authentic ground of the self and the other as the realization of absolute totality. Ethics is now a matter of spontaneous compassion, spontaneous caring, and concern for the whole. This is the birth of selfless morality, for which the only counterparts in the West are the mystical traditions and perhaps some forms of religiosity in which it is God who moves in us, and not we ourselves. The double negation referred to earlier whereby the individual is negated by the group aspect of self, and the group aspect is in turn negated by the individual aspect, is not to be taken as a complete negation that obliterates that which is negated. The negated are preserved, else there would be no true self-contradiction. This robust sense of the importance of self-contradiction shares much with the more developed sense of the identity of self-contradiction about which Nishida said so much (see the entry on Nishida Kitaro). What is stressed by both thinkers is that some judgments of logical contradiction are at best penultimate judgments, which may point us in the direction of a more comprehensive and accurate understanding of our own experience. Watsuji refers to 'wakaru ' (to understand), which is derived from 'wakeru ' (to divide); and in order to understand, one must already have presupposed something whole, that is to say, a system or unity. For example, in self-reflection, we make our own self, other. Yet the distinction reveals the original unity, for the self is other as objectified and of course is divided from the originally unified self. To think of a thing is to distinguish it from something else, and yet in order to make such a distinction, the two must already have had something in common. Thus, to emphasize the contradiction is to plunge into the world as many; to emphasize the context, or background, or matrix is to plunge into the world as one. Readers will be familiar with the logical formulation, often encountered in Zen but ubiquitous in Buddhism generally, that A is A; and A is not-A; therefore, A is A. There is a double negation in evidence here: an individual is an individual, and yet an individual is not individual unless one stands opposed to other individuals. That an individual stands opposed to others means that one is related to others as a member of a group or groups. Because an individual is a member of a group, one is both a member as an individual, and an otherwise isolated individual as a member of a group. We are both, in mutual interactive negation, and as such we are determined by the group or community, and yet we ourselves determine and shape the group or community. As such we are living self-contradictions, and, therefore, living identities of self-contradiction. Morality, for Watsuji, is a coming back to authentic unity through an initial opposition between the self and other, and then a re-establishing of betweenness between self and other, ideally culminating in a nondualistic connection between the self and others that actually negates any trace of difference or opposition in the emptiness of the home ground. This is the negation of negation, and it occurs in both time and space. It is not simply a matter of enlightenment as a private, individual experience, calling one to awareness of the interconnectedness of all things. Rather, it is a spatio-temporal series of interconnected actions, occurring in the betweenness between us, which leads us to an awareness of betweenness that ultimately eliminates the self and other, but of course, only from within a nondualist perspective. Dualistically comprehended, both the self and other are preserved. As the Zen saying goes, "not one, not two." As Taigen Dan Leighton explains, "nonduality is not about transcending the duality of form and emptiness. This deeper nonduality is not the opposite of duality, but the synthesis of duality and nonduality, with both included, and both seen as ultimately not separate, but as integrated"(Leighton 2004, 35). For Watsuji, emptiness is what makes the subject-object relation, or better, the relation both inherent in ningen and between ningen possible. What is left is betweenness itself in which human actions occur. In this sense, betweenness is emptiness, and emptiness is betweenness. Betweenness is the place where compassion arises and is acted out selflessly in the spatio-temporal theatre of the world. It is that which makes possible the variety of relationships of which human beings are capable. Watsuji's ningen views each person's ethical identity as integrally related to that of others and extends it beyond merely human relations. Self and other, self and nature, are viewed as inseparable from ethical identity and are related to, rather than viewed as opposed to or exclusive of each other. In Watsuji's ningen we find this nondualism lived out in ways that extend ethical identity beyond relations between human beings to encompass the world in which we live -- it is part of milieu or fudo. ## 5. The State Watsuji's theory of the state, and his vocal support of the Emperor system, garnered considerable criticism after the Second World War. La Fleur maintains that Watsuji's detractors were "dominant in Japanese intellectual life from 1945 to approximately 1975," while his admirers became "newly articulate since the decade of the 1980s" (LaFleur 1994, 453). The former group considered his position to be a dangerous one. He argued that the culmination of the double negation, which he conceived of as a single movement, was the restoration of the absolute totality. In other words, while the individual negates the group in order to be an authentic and independent individual, the second negation is to abandon one's independence as an individual in the full realization of totality. It is a 'surrender' to totality, moving one beyond the myriad specific groups to the one total and absolute whole. It would seem natural that this ultimate wholeness would be the home ground of nothingness, and in a way it is. But Watsuji argues that it is the state that takes on the authority of totality. It is the highest and best social structure thus far. The political implications of this position could easily result in a totalitarian state ethics. Indeed, Watsuji extolled the superiority of traditional Japanese culture because of its emphasis on self-negation, including the making of the ultimate sacrifice if required by the Emperor. In America's National Character (1944), he contrasted this willingness to an assumed selfishness or egocentrism found in the West, together with a utilitarian ethic of expediency, which he felt was rarely able to commit to self-surrender in aid of the state. What he saw as most exemplary in the Japanese way of life was the Bushido ideal of "the absolute negativity of the subject" (Odin 1996, 67), through which the totality of the whole is able to be achieved. There is no doubt that Watsuji's position could easily be interpreted as a totalitarian state ethics. Yet, insofar as Watsuji's analysis of Japanese ethics is an account of how the Japanese do actually act in the world, then it is little surprise that the Japanese errors of excess which culminated in the fascism of the Second World War period should be found somehow implicit and possible in Watsuji's acute presentation of Japanese cultural history. Perhaps the fault to be found lies not in his analysis per se, but rather in his all too sanguine collapsing of the descriptive and the prescriptive. That the Japanese way-in-the-world might include a totalitarian seed is something which demands a normative warning. Surely this is not what should be applauded as an aspect of the alleged superiority of Japanese culture, nor should Bushido in and of itself be taken as a blameless path to the highest of ethical achievements. The willingness to be loyal, whatever the rights and wrongs of the situation might be, in order to remain loyal to one's Lord, however evil or foolhardy he might be, is not an adequate or rational position, and it is surely not laudable ethically. It is, perhaps, the way the samurai saw themselves, as martial servants loyal to death. But the 'is' here is clearly not a moral 'ought'. What Watsuji adds to this picture which makes it extremely difficult to condemn him too harshly, and possibly to condemn him at all, is his adament insistence in the third volume of Rinrigaku that no one nation has charted the correct political and cultural path, and that the diversity of each is to be both encouraged and respected. He writes of unity in diversity, and rejects the idea that any nation has the right to culturally assimilate another. Each nation is shaped by its particular geography, climate, culture and history, and the resultant diversity is both to be protected and appreciated, and the notion of a universal state is, therefore, but an unwanted and dangerous delusion. We must know our own traditions, and to cherish those, but we must not extol them as superior out of our ignorance of other ways and cultural traditions. In fact, such ignorance, resulting from Japan's unwise self-imposed isolationism, Watsuji saw as the most tragic flaw in Japan's own history, and a cause of the Second World War. Japan knew so little of the outside world in the late nineteenth and early twentieth centuries, that it exaggerated its own worth and power, and vastly underestimated the importance of political and diplomatic involvement in the happenings of the rest of the world. Nationalism must not express itself at the expense of internationalism, and internationalism must not establish itself at the expense of nationalism. Here is another pair of seeming contradictions to be held together by the unity of mutual interaction; the one modifies the other, and that tension is not to be resolved. Internationalism must be a unity of independent and distinctive nations. Five months before the end of the Second World War, Watsuji organized a study group to re-think the Tokugawa Period (1600-1868), which resulted in his popular work, Closed Nation: Japan's Tragedy (Sakoku), published in 1950. Some critics reacted by dubbing this work "Watsuji's Tragedy" (LaFleur 2001, 1), yet La Fleur insists that Sakoku, together with a collection of other contemporaneous writings, provides a serious and important insight into Watsuji's later philosophical position. His focal thesis was that the tragedy of the Japanese involvement in the Second World War was a direct result of Japan's policy of national seclusion. Isolationism took Japan out of the events of the world's activities for two centuries, allowing the West to outstrip Japan in terms of science and technology. But even more important is Watsuji's insistence that the nationalists and militarists should have seen that Japan was in no position to win the war since both "men and materiel were in increasingly short supply" (LaFleur 2001, 3). Not that Watsuji opposed the war, but the point he was making was that the tragedy came to be because the earlier isolationist attitude had been revived from 1940 on. Japan was out of touch with the strengths and determination of its adversaries, and seemingly convinced that the character of the Japanese people would ensure success. Furthermore, coupled with this short-term return to a head-in-the-sand isolationism of the 1940s, was Watsuji's insistence that throughout most of Japan's history, the Japanese people had demonstrated an intense curiosity and a robust desire to learn, and in particular to learn from cultures quite different from their own: "No matter how far we go back in Japanese culture we will not find an age in which evidence of admiration of foreign cultures is not to be found" (Watsuji 1998, 250). Japan was withdrawing inwards at precisely the time when the West's expansionism and imperialism was gathering steam. LaFleur goes so far as to suggest that in a 1943 lecture to Navy officials, Watsuji attempted to warn those in charge of the dangerous course they continued to pursue. LaFleur speculates that this is why Watsuji felt no need to recant after the War had ended. The salient point of all of this is that the instances of isolationism in Japanese history are exceptions which run counter to what Watsuji saw as the dominant tendency of the Japanese to both welcome and encourage outside influence. Japan's national character throughout the bulk of its history displays a remarkable openness to and interest in other cultures, and a steadfast desire to learn from those cultures. He was advocating a return to this more 'normal' attitude towards the outside world, and not a return to Japanese nationalism or chauvinism. Watsuji presents us with a complex perspective. ## 6. Religion Watsuji's interest in religion was as a social phenomenon, that is, in religion as an aspect of the cultural environment. And yet in the Watsuji scholarship there remains disagreement about how religion -- Buddhism in particular -- is to be read in his work (Sevilla 2016, 608-609). William LaFleur remarks, for example, that Watsuji "embraces a religious solution...philosophically and methodologically" (LaFleur 1978, 238). While David Dilworth contends that "Watsuji's position was not essentially a Buddhistic or religious one"(Dilworth 1974, 17). While there is disagreement on LaFleur's claim that "it was in the Buddhist notion of emptiness that Watsuji found the principle that gives his system...coherence" (LaFleur 1978, 239), it cannot be denied that when Watsuji writes of emptiness as that final place or context in which all distinctions disappear, or empty, and yet from which they emerge, that this notion is at least partly rooted in his study of Buddhist philosophy and culture, and the concept of emptiness in particular. Watsuji "embraces a religious solution...philosophically and methodologically" (LaFleur 1978, 238).The Buddhist notion of pratitya-samutpada, or 'dependent co-origination' (or 'relationality,' 'conditioned co-production,' 'dependent co-arising,' 'co-dependent origination') implies that everything is 'empty' that is to say, "that everything is deprived of its substantiality, nothing exists independently, everything is related to everything else, nothing ranks as a first cause, and even the self is but a delusory construction" (Abe 1985, 153). The delusion of independent individuality can be overcome by recognizing our radical relational interconnectedness. At the same time, even this negation must be emptied or negated, hence our radical relational interconnectedness is possible only because true individuals have created a network in the betweenness between them. The result is a selfless awareness of that totality beyond all limited, social totalities, namely the emptiness or nothingness at the bottom of all things, whether individual or group. LaFleur summarises Watsuji's position: "the social side of human existence 'empties' the individuated side of any possible priority or autonomy and the individuated side does something exactly the same and exactly proportionate to and for the social side. There can be no doubt, I think, that for Watsuji, emptiness is the key to it all" (LaFleur 1978, 250). Watsuji's emptiness is more a recognition of underlying relatedness, and manifests as a place, a basho, that is the dynamic and creative origination of all relationships and all networks of interactions. However, it is important to keep in mind that even when Watsuji uses religious language (as in "it is a religious ecstasy of the great emptiness" (LaFleur 1978, 249), that ecstasy is not a mystical-like insight into one's union with God or the Absolute or a recognition of floating in 'other-power,' but a relational union of those persons involved in some communal forms, eventually culminating in the state. It is an ethical, or political, or communal ecstasy and not a religious identification with a transcendent Other - and as much as we hear the influence of Buddhism, we also hear, for example, echoed of the Hegelian concept of sublation (Sevilla 2016, 626). Watsuji gives no evidence of deep religiosity but expresses a profound and sometimes ecstatic ethical humanism, one which is, nonetheless, significantly Buddhist in conception. In denying the reality of the subject/object distinction, and affirming the emptiness of all things, Watsuji was able to argue that as the individual negates or rebels against society, thereby emptying society of an unchanging objective status, and as the second negation establishes the totality of society by emptying the individual, it reduces the individual ego to emptiness as well. Neither of the dual characteristics of ningen are either unchanging or ultimately real. What is ultimate is the emptiness which is revealed as their basis, for all things are empty, and yet, once this truth is realized, it is as individuals and societies that nothingness is expressed and revealed. Emptiness negates itself, or empties itself as the beings of this world. And it does this in the emptiness of betweenness, the empty space within which relations between individuals and societies form and continue to re-form. The state, as the culminating social organization thus far achieved, is the form of forms, "transcending all the other levels of social organization...giving to each of those protected forms a proper place" (LaFleur 1994, 457). The state is, ultimately, "the moral systematiser of all those organizations" (LaFleur, french, 457). However, the problem of the proper relation of the individual to society emerges, for Watsuji goes on to argue that "the state subsumes within itself all these forms of private life and continually turns them into the form of the public domain" (LaFleur 1994, 457). In attempting to move away from the selfishness of egoistic action, Watsuji has given primacy to the state over individual and group rights. As LaFleur states, "by valorizing the state as that entity that has the moral authority--in fact the moral obligation--to 'turn private things into the public domain, Watsuji provides a smooth rationale for a totalizing state" (LaFleur 1994, 458). Nevertheless, it must be kept in mind that his intent was not to advocate tyranny or fascism, but to seek out an ethical and social theory whereby human beings as human beings could interact easily and fruitfully in the space between them, creating as a result a society, and a world-wide association of societies which selflessly recognized the value of the individual and the crucial importance of the well-being of the whole. His solution may have been inadequate, and open to unwelcome misuse. His analysis of 'betweenness' shows it to be communality, and communality as a mutuality wherein each individual may affect every other individual and thereby affect the community or communities; and the community, as an historical expression of the whole may affect each individual. Ideally, what would result would be an enlightened sense of our interconnectedness with all human beings, regardless of race, color, religion, or creed, and a selfless, compassionate capacity to identify with others as though they were oneself. By reintroducing a vivial sense of our communitarian interconnectedness, and our spatial and bodily place in the betweenness between us, where we meet, love, and strive to live ethical lives together, Watsuji provides an ethical and political theory which might well prove to be helpful both to non-Japanese societies, and to a modern Japan itself which is torn between what it was, and what it is becoming.
weakness-will	## 1. Hare on the Impossibility of Weakness of Will Let us commence our examination of contemporary discussions of this issue in appropriately Socratic vein, with an account that gives expression to and builds on many of the intuitions that lead us to be sceptical about reports like (3) above. For the moral philosopher R. M. Hare--as for Socrates--it is impossible for a person to do one thing if he genuinely and in the fullest sense holds that he ought instead to do something else. (If, that is--to echo the earlier quote from Socrates--he "believes that there is another possible course of action, better than the one he is following.") This certainly seems to constitute a denial of the possibility of akratic or weak-willed action. In Hare's case it is a consequence of the general account of the nature of evaluative judgments which he defends (Hare 1952; see also Hare 1963). Hare is much impressed by what we vaguely referred to above as the "special character" of evaluative judgments: judgments, that is, such as that one course of action is better than another, or that one ought to do a certain thing. Such evaluative judgments seem to have properties that differentiate them from merely "descriptive" judgments such as that one thing is more expensive than another, or rounder than another (Hare 1952, p. 111). Evaluative judgments seem, in particular, to bear a special connection to action which no purely descriptive judgment possesses. Hare's analysis, then, takes off from something like the data we rehearsed earlier. Hare goes on to develop these data in the following way. He begins by identifying, as the fundamental distinctive feature of evaluative judgments--that which lends them a special character--that evaluative judgments are intended to guide conduct. (See, e.g., Hare 1952, p. 1; p. 29; p. 46; p. 125; p. 127; p. 142, pp. 171-2; Hare 1963, p. 67; p. 70.) The special function of evaluative judgments is to be action-guiding: that is, if you will, what evaluative judgments are for. Hare then puts a more precise gloss on what it is for a judgment to "guide conduct": an action-guiding judgment is one which entails an answer to the practical question "What shall I do?" (Hare 1952, p. 29; see Hare 1963, p. 54 for the terminology "practical question").[3] What is it that an action-guiding judgment must entail? That is, what constitutes an answer to the question "What shall I do?" Hare holds that no (descriptive) statement can constitute an answer to such a question (Hare 1952, p. 46). Rather, such a question is answered by a first-person command or imperative (Hare 1952, p. 79), which could be verbally expressed as "Let me do a" (Hare 1963, p. 55). To recap the argument thus far: it is the function of evaluative judgments like "I ought to do a" to guide conduct. Guiding conduct means entailing an answer to the question "What shall I do?" An answer to that question will take the form "Let me do a," where this is a first-person command or imperative. Therefore evaluative judgments entail such first-person imperatives (Hare 1952, p. 192). Now in general, if judgment J1 entails judgment J2, then assenting to J1 must involve assenting to J2: someone who professed to assent to J1 but who disclaimed J2 would be held not to have spoken correctly when he claimed to assent to J1 (Hare 1952, p. 172). So assenting to an evaluative judgment like "I ought to do a" involves assenting to the first-person command "Let me do a" (Hare 1952, pp. 168-9). We should inquire, then, what exactly is involved in sincerely assenting to a first-person command or imperative of this type. Just as sincere assent to a statement involves believing that statement, sincere assent to an imperative addressed to ourselves involves doing the thing in question: > > It is a tautology to say that we cannot sincerely assent to a > ... command addressed to ourselves, and at the same > time not perform it, if now is the occasion for performing it > and it is in our (physical and psychological) power to do so. > (Hare 1952, p. 20) > So: provided it is within my power to do a now, if I do not do a now it follows that I do not genuinely judge that I ought to do a now. Thus, as Hare states at the very opening of his book, a person's evaluative judgments are infallibly revealed by his actions and choices: > > If we were to ask of a person 'What are his moral > principles?' the way in which we could be most sure of a true > answer would be by studying what he did.... It would > be when ... he was faced with choices or decisions between > alternative courses of action, between alternative answers to the > question 'What shall I do?', that he would reveal in > what principles of conduct he really believed. (Hare 1952, > p. 1) > Note that Hare is not simply saying that a person's actions are the most reliable source of evidence as to his evaluative judgments, or that if a person did b the most likely hypothesis is that he judged b to be the best thing to do. Hare is saying, rather, that it follows from a person's having done b that he judged b best from among the options open to him at the time. On this view, then, akratic or weak-willed actions as we have understood them are impossible. There could not be a case in which someone genuinely and in the fullest sense held that he ought to do a now (where a was within his power) and yet did b. On Hare's view, "it becomes analytic to say that everyone always does what he thinks he ought to [if physically and psychologically able]" (Hare 1952, p. 169). But does everyone always do what he thinks he ought to, when he is physically and psychologically able? It may seem that this is simply not always the case (even if it is usually the case). Have you, dear reader, never failed to get up off the couch and turn off the TV when you judged it was really time to start grading those papers? Have you never had one or two more drinks than you thought best on balance? Have you never deliberately pursued a sexual liaison which you viewed as an overall bad idea? In short, have you never acted in a way which departed from your overall evaluation of your options? If so, let me be the first to congratulate you on your fortitude. While weak-willed action does seem somehow puzzling, or defective in some important way, it does nonetheless seem to happen. For Hare, however, any apparent case of akrasia must in fact be one in which the agent is actually unable to do a, or one in which the agent does not genuinely evaluate a as better--even if he says he does.[4] As an example of the first kind of case Hare cites Medea (Hare 1963, pp. 78-9), who (he contends) is powerless, literally helpless, in the face of the strong emotions and desires roiling her: she is truly unable (psychologically) to resist the temptations besieging her. A typical example of the second kind of case, on the other hand, would be one in which the agent is actually using the evaluative term "good" or "ought" only in what Hare calls an "inverted-commas" sense (Hare 1952, p. 120; pp. 124-6; pp. 164-5; pp. 167-171). In such cases, when the agent says (while doing b) "I know I really ought to do a," he means only that most people--or, at any rate, the people whose opinions on such matters are generally regarded as authoritative--would say he ought to do a. As Hare notes (Hare 1952, p. 124), to believe this is not to make an evaluative judgment oneself; rather, it is to allude to the value-judgments of other people. Such an agent does not himself assess the course of action he fails to follow as better than the one he selects, even if other people would. No doubt there are cases of the two types Hare describes; but they do not seem to exhaust the field. We can grant that there is the odd murderer, overcome by irresistible homicidal urges but horrified at what she is doing. But surely not every case that we might be tempted to describe as one of acting contrary to one's better judgment involves irresistible psychic forces. Consider, for example, the following case memorably put by J. L. Austin: > > I am very partial to ice cream, and a bombe is served divided into > segments corresponding one to one with the persons at High Table: I > am tempted to help myself to two segments and do so, thus succumbing > to temptation and even conceivably (but why necessarily?) going > against my principles. But do I lose control of myself? Do I raven, > do I snatch the morsels from the dish and wolf them down, impervious > to the consternation of my colleagues? Not a bit of it. We often > succumb to temptation with calm and even with finesse. > (Austin 1956/7, p. 198) > (I might add that it also seems doubtful that irresistible psychic forces kept you on the couch watching TV while those papers were waiting.) As for the "inverted-commas" case, this too surely happens: people do sometimes pay lip service to conventional standards which they themselves do not really accept. But again, it seems highly doubtful that this is true of all seeming cases of weak-willed action. It seems depressingly possible to select and implement one course of action while genuinely believing that it is an overall worse choice than some other option open to you. Has something gone wrong? We started with the unexceptionable-sounding thought that moral and evaluative judgments are intended to guide conduct; we arrived at a blanket denial of the possibility of akratic action which fits ill with observed facts. But if we are disinclined to follow Hare this far we should ask what the alternative is, for it may be even worse. For Hare, the answer is clear: our only other option is to repudiate the idea that moral and other evaluative judgments have a special character or nature, namely that of being action-guiding. For we should recall that Hare presents all his subsequent conclusions as simply following, through a series of steps, from that initial thought. "The reason why actions are in a peculiar way revelatory of moral principles is that the function of moral principles is to guide conduct," Hare continues in the passage quoted earlier (Hare 1952, p. 1). For Hare, then, the only way to escape his "Socratic" conclusion about weakness of will would be to give up the idea that evaluative judgments are intended to guide conduct, or to "have [a] bearing upon our actions" (Hare 1963, p. 169; see also Hare 1952, p. 46; p. 143; p. 163; pp. 171-2; and Hare 1963, p. 70; p. 99). The choices before us, then, as presented by Hare, are Hare's own view, or one which assigns no distinctive role in action or practical thought to evaluative judgments, treating them as just like any other judgment. We might call the first of these an extreme version of (judgment) internalism. (I use this polysemous label to refer, here, to the idea that certain judgments have an internal or necessary connection to motivation and to action.) By extension, we might usefully follow Michael Bratman in calling the second type of view "extreme externalism" (Bratman 1979, pp. 158-9). Extreme externalism also seems unsatisfactory, however. First, it seems unable to explain why there should be anything perplexing or problematical about action contrary to one's better judgment, why there should be any philosophical problem about its possibility or its analysis. On this kind of view, it seems, Joseph's choice ((3) above) should strike us as no more puzzling than Julie's or Jimmy's ((1) or (2)). As Hare puts it: > > On the view that we are considering, there is nothing odder about > thinking something the best thing to do in the circumstances, but > not doing it, than there is about thinking a stone the roundest > stone in the vicinity and not picking it up, but picking up some > other stone instead.... There will be nothing that requires > explanation if I choose to do what I think to be, say, the worst > possible thing to do and leave undone what I think the best thing to > do. (Hare 1963, pp. 68-9) > But our reactions to (1), (2), and (3) show that we do think there is something peculiar about action contrary to one's better judgment which renders such action hard to understand, or perhaps even impossible. An extreme externalist view thus seems to mischaracterize the status of akratic actions. Perhaps even more importantly, however, extreme externalism has dramatic implications for our understanding of intentional action in general--not just weak-willed action. For such a view implies that > > deliberation about what it would be best to do has no closer > relation to practical reasoning than, say, deliberation about what > it would be chic to do. If one happens to care about what it would > be chic to do, then a consideration of this matter may play an > important role in one's practical reasoning. If one does not care, > it will be irrelevant. The case is the same with reasoning about > what it would be best to do. (Bratman 1979, p. 158) > To adopt a general doctrine of this sort seems an awfully precipitous response to the possibility of akrasia. For it seems extremely plausible to assign to our overall evaluations of our options an important role in our choices. Man is a rational animal, the saying goes; that is--to offer one gloss on this idea--we act on reasons, and in the light of our assessments of the overall balance of reasons. When we engage in deliberation or reasoning about what to do, we often proceed by thinking about the reasons which favor our various options, and then bringing these together into an overall assessment which is, precisely, intended to guide our choice. Or, as Bratman puts it, we very often reason about what it is best to do as a way of settling the question of what to do. (He calls this "evaluative practical reasoning": Bratman 1979, p. 156.) "One's evaluations [thus] play a crucial role in the reasoning underlying full-blown action," Bratman holds (p. 170), and to be forced to deny this would be in his view "too high a price to pay" (p. 159). As Alfred Mele similarly puts it, "there is a real danger that in attempting to make causal and conceptual space for full-fledged akratic action one might commit oneself to the rejection of genuine ties between evaluative judgment and action" (1991, p. 34). But that would be to throw the baby out with the bathwater. If we want to resist Hare's conclusions, we must do so in a way which steers clear of the danger to which Mele alerts us. We must navigate between the Scylla of extreme internalism and the Charybdis of extreme externalism. ## 2. Davidson on the Possibility of Weakness of Will This is just what Donald Davidson set out to do in a rich, elegant, and incisive paper published in 1970 which has had a towering influence on the subsequent literature. Davidson's treatment aims to vindicate the possibility of weakness of will; to offer a novel analysis of its nature; to clarify its status as a marginal, somehow defective instance of agency which we rightly find dubiously intelligible; and to do all this within the contours of a general view of practical reasoning and intentional action which assigns a central and special role to our evaluative judgments. Let us see how he proposes to do these things. First, Davidson offers the following general characterization of weak-willed or incontinent action:[5] > > In doing b an agent acts incontinently if and only > if: (a) the agent does b intentionally; > (b) the agent believes there is an alternative action > a open to him; and (c) the agent judges that, all things > considered, it would be better to do a than to do > b. > We initially described weak-willed action as free, intentional action contrary to the agent's better judgment; it may be useful to see how Davidson's more precise definition matches up with that initial characterization. Davidson's condition (a) requires that the action in question be intentional.[6] Condition (b) seems intended to ensure that the action in question is free.[7] Part (c) of Davidson's definition represents what we have called the agent's "better judgment," that is, the overall evaluation of his options contrary to which the incontinent agent acts. Davidson notes that "there is no proving such actions exist; but it seems to me absolutely certain that they do" (p. 29). Why, then, is there a persistent tendency, both in philosophy and in ordinary thought, to deny that such actions are possible? Davidson's diagnosis is that two plausible principles which "derive their force from a very persuasive view of the nature of intentional action and practical reasoning" (p. 31) appear to entail that incontinence is impossible. He articulates those two principles as follows (p. 23): > > P1. If an agent wants to do a more than he wants to > do b and he believes himself free to do either a > or b, then he will intentionally do a if he does > either a or b intentionally. > > > > P2. If an agent judges that it would be better to do > a than to do b, then he wants to do a > more than he wants to do b. > > > P2, Davidson observes, "connects judgements of what it is better to do with motivation or wanting" (p. 23); he adds later that it "states a mild form of internalism" (p. 26). Davidson is proposing, contra the extreme externalist position, that our evaluative judgments about the merits of the options we deem open to us are not motivationally inert. While he admits that one could quibble or tinker with the formulation of P1 and P2 (pp. 23-4; p. 27; p. 31), he is confident that they or something like them give expression to a powerfully attractive picture of practical reasoning and intentional action, one which assigns an important motivational role to the agent's evaluative judgments.[8] The difficulty is, though, that P1 and P2--however attractive--together imply that an agent never intentionally does b when he judges that it would be better to do a (if he takes himself to be free to do either). And this certainly looks like a denial of the possibility of incontinent action. No wonder, then, that so many have been tempted to say that akratic action is impossible! Looking carefully, however, we can see that P1 and P2 do not imply the impossibility of incontinent actions as Davidson has defined them. For Davidson characterizes the agent who incontinently does b as holding, not that it would be better to do a than to do b, but that it would be better, all things considered, to do a than to do b. Is the "all things considered" just a rhetorical flourish? Or does it mark a genuine difference between these two judgments? If these are two different judgments, and one can hold the latter without holding the former, then incontinent action is possible even if P1 and P2 are true. In the rest of his paper Davidson sets out to vindicate that very possibility. The phrase "all things considered" is not, as it might seem, merely a minor difference in wording that allows weakness of will to get off on a technicality. Rather, that phrase marks an important contrast in logical form to which we would need to attend in any case in order properly to understand the structure of practical reasoning. For that phrase indicates a judgment that is conditional or relational rather than all-out or unconditional in form; and that difference is crucial.[9] We can better see the relational character of an all-things-considered judgment if we first look at evaluative judgments that play an important role in an earlier phase of practical reasoning, the phase where we consider what reasons or considerations favor doing a and what reasons or considerations favor doing b. (For simplicity, imagine a case in which an agent is choosing between only two mutually incompatible options, a and b.) These prima facie judgments, as Davidson terms them, take the form: > > PF: In light of r, a is prima > facie better than b. > In this schema r refers to a consideration, say that a would be relaxing, while b would be stressful. A PF judgment of this kind thus identifies one respect in which a is deemed superior to b, one perspective from which a comes out on top. We should pause to note three things about PF judgments. (a) A PF judgment is not itself a conclusion in favor of the overall superiority of a. Such "all-out" evaluative judgments have a simpler logical form, namely: > > AO: a is better than b. > (b) Indeed, no conclusion of the form AO follows logically from any PF judgment. (c) More strongly: the fact, taken by itself, that someone has made a certain PF judgment does not even supply her with sufficient grounds to draw the corresponding AO conclusion. For even if she makes one PF judgment which favors a over b, as in the case we imagined, she may also make other PF judgments which favor b over a (say, when r is the consideration that b would be lucrative, while a would be expensive). We do not want to say in that case that she has sufficient grounds to draw each of two incompatible conclusions (that a is better than b, and that b is better than a; these are incompatible provided the better-than relation is asymmetric, assumed here). We have contrasted PF judgments with AO or "all-out" evaluative judgments. PF judgments are relational in character: they point out a relation which holds between the consideration r and doing a. (We could call that relation the "favoring" relation.) That relation is not such as to permit us to "detach" (as Davidson puts it, p. 37) an unconditional evaluative conclusion in favor of doing a from PF and the supposition that r obtains. That is, we are not to understand PF judgments as having the form of a material conditional. Davidson's innovative suggestion is that judgments with this PF logical form are an appropriate way to model what happens in the early stages of practical reasoning, where we rehearse reasons for and against the options we are considering. And his stressing that no such PF judgment commits the agent to an overall evaluative conclusion in favor of a or b is useful in thinking about a case like Julie's ((1) above). We described Julie as knowing (and therefore believing) that b was more expensive than a, but opting for b nonetheless. We can imagine, then, that among the ingredients of Julie's practical reasoning was a PF judgment like this: > > In light of the fact that b is more expensive than a, > a is prima facie better than b. > But this PF judgment alone, as we have seen, does not commit her to the overall judgment that a is better than b. For she may also have made other PF judgments, such as > > In light of the fact that b would be much more > gastronomically exciting than a, b is prima > facie better than a. > But we would not then want to say Julie has sufficient grounds to conclude that a is better than b and to conclude that b is better than a. She does not have sufficient grounds to embrace a contradiction; her premises all seem consistent. So her various PF judgments, when considered separately, must not each commit her to a corresponding overall conclusion in favor of a or b. Practical reasoning, Davidson suggests, starts from judgments like these, each identifying one respect in which one of the options is superior. But in order to make progress in our practical reasoning we shall eventually need to consider how a compares to b not just with respect to one consideration, but in the light of several considerations taken together. That is, Julie will eventually need to consider how to fill in the blanks in a PF judgment like this: > > In light of the fact that b is more expensive > than a and the fact that b would be much > more gastronomically exciting than a, ... is prima > facie better than .... > This PF judgment is more comprehensive than the ones we attributed to Julie a moment ago, as it takes into account a broader range of considerations. (I take the label "comprehensive" from Lazar 1999.) Now in Julie's case we can surmise how she filled in those blanks: with "b is prima facie better than a." Julie's filling in the blanks in that way can naturally be taken as expressing the view that the much greater gastronomic excitement promised by b outweighs or overrides b's inferiority to a from a strictly financial standpoint. We can generalize our schema for PF judgments to account for the possibility of relativizing our comparative assessment of a and b not just to a single consideration, but to multiple considerations taken together or as a body: > > PFN: In light of <r1, ..., > rn> , a is prima facie > better than b. > Notice that PFN judgments are still relational in form: they assert that a relation (the "favoring" relation) holds between the set of considerations <r1, ..., rn> and doing a. Indeed, the relational character of a PFN judgment remains even if we make it as comprehensive as we can: if we expand the set <r1, ..., rn> to incorporate all the considerations the agent deems relevant to her decision. Following Davidson (p. 38), let us give the label e to that set. So even the following judgment: > > ATC: In light of e, a is prima > facie better than b. > is a relational or conditional judgment and not an all-out conclusion in favor of doing a. To make a judgment of the form ATC is not to draw an overall conclusion in favor of doing a. We may be better able to see this by considering an analogy from theoretical reason. Suppose Hercule Poirot has been called in to investigate a murder. We can imagine him assessing bits of evidence as he encounters them: > > In light of the fact that the murder weapon belongs to Colonel > Mustard, Mustard looks guilty; > > > In light of his having an alibi for the time of the murder, Mustard > looks not guilty; > > > and so on. These are theoretical analogues of the PF judgments relativized to single considerations which we looked at earlier. However, Poirot will eventually need to consider how these various bits of evidence add up; that is, he will eventually need to fill in the blanks in a more comprehensive PFN judgment like this: > > In light of > <e1, ..., en> , > ... looks to be the guilty party, > where <e1, ..., en> is a set of bits of pertinent evidence. Notice, though, that no such PFN judgment actually constitutes settling on a particular person as the culprit. For even if we put in a maximally large <e1, ..., en> consisting of all the evidence Poirot has seen, and imagine him thinking > > All the evidence I have seen points toward Colonel Mustard as the > guilty party, > to make this observation is manifestly not to conclude that Mustard is guilty. In the same way, an ATC or all-things-considered judgment, although comprehensive, is still relational in nature, and therefore distinct from an AO judgment in favor of a. That is, it is possible to make an ATC judgment in favor of a without making the corresponding AO judgment in favor of a. (This is the analogue of Poirot's position.) And this is the key to Davidson's solution to the problem of how weakness of will is possible. For ATC is, precisely, the agent's better judgment as Davidson construes it in his definition of incontinent action. P1 and P2 together imply that an agent who reaches an AO conclusion in favor of a will not intentionally do b. But the incontinent agent never reaches such an AO conclusion. With respect to a, he remains stuck at the Hercule Poirot stage: he sees that the considerations he has rehearsed, taken as a body, favor a, but he is unwilling or unable to make a commitment to a as the thing to do.[10] He makes only a relational ATC judgment in favor of a, contrary to which he then acts. What should we say about an agent who does this? Returning to the three features of prima facie or PF judgments which we noted earlier, features (a) and (b) hold even of the special subclass of PF judgments which are ATC judgments. Such judgments neither are equivalent to, nor logically imply, any AO judgment. So the incontinent agent who fails to draw the AO conclusion which corresponds to his ATC conclusion, and to perform the corresponding action, is not committing "a simple logical blunder" (p. 40). Notably, he does not contradict himself. He does, however, exhibit a defect in rationality, on Davidson's account. For feature (c) of PF judgments in general does not hold of the special subclass of such judgments which are ATC judgments. Drawing an ATC conclusion in favor of a does give one sufficient grounds to conclude that a is better sans phrase and, indeed, to do a. For Davidson proposes that the transition from an ATC judgment in favor of a to the corresponding AO judgment, and to doing a, is enjoined by a substantive principle of rationality which he dubs "the principle of continence." That principle tells us to "perform the action judged best on the basis of all available relevant reasons" (p. 41); and the incontinent agent violates this injunction. The principle of continence thus substantiates the idea that "what is wrong is that the incontinent man acts, and judges, irrationally, for this is surely what we must say of a man who goes against his own best judgement" (p. 41). He acts irrationally in virtue of violating this substantive principle, obedience to which is a necessary condition for rationality. We must put this point about the irrationality of incontinence with some care, however. For recall that an incontinent action must itself be intentional, that is, done for a reason. The weak-willed agent, then, has a reason for doing b, and does b for that reason. What he lacks--and lacks by his own lights--is a sufficient reason to do b, given all the considerations that he takes to favor a. As Davidson puts it, if we ask "what is the agent's reason for doing [b] when he believes it would be better, all things considered, to do another thing, then the answer must be: for this, the agent has no reason" (p. 42). And this is so even though he does have a reason for doing b (p. 42, n. 25). Because the agent has, by his own lights, no adequate reason for doing b, he cannot make sense of his own action: "he recognizes, in his own intentional behaviour, something essentially surd" (p. 42). So akratic action, while possible on Davidson's account, is nonetheless necessarily irrational; this is the sense in which it is a defective and not fully intelligible instance of agency, despite being a very real phenomenon. ## 3. The Debate After Davidson ### 3.1 Internalist and Externalist Strands Davidson has certainly presented an arresting theory of practical reasoning. But has he shown how weakness of the will is possible? Most philosophers writing after him, while acknowledging his pathbreaking work on the issue, think he has not. One principal difficulty which subsequent theorists have seized on is that Davidson's view can account for the possibility of action contrary to one's better judgment only if one's better judgment is construed merely as a conditional or prima facie judgment. Davidson's P1 and P2 in fact rule out the possibility of free intentional action contrary to an all-out or unconditional evaluative judgment.[11] But it seems that such cases exist. Michael Bratman, for instance, introduces us to Sam, who, in a depressed state, is deep into a bottle of wine, despite his acknowledged need for an early wake-up and a clear head tomorrow (1979, p. 156). Sam's friend, stopping by, says: > > Look here. Your reasons for abstaining seem clearly stronger than > your reasons for drinking. So how can you have thought that it would > be best to drink? > To which Sam replies: > > I don't think it would be best to drink. Do you think > I'm stupid enough to think that, given how strong my reasons > for abstaining are? I think it would be best to abstain. Still, > I'm drinking. > Sam's case certainly seems possible as described. Davidson's view, though, must reject it as impossible. Given his conduct, Sam can't think it best to abstain; at most, he thinks it all-things-considered best to abstain, a very different kettle of fish. But this seems false of Sam: there is no evidence that he has remained stuck at the Hercule Poirot stage with respect to the superiority of abstaining. He seems to have gone all the way to a judgment sans phrase that abstaining would be better; and yet he drinks. Ironically, this complaint makes Davidson out to be a bit like Hare. Like Hare, Davidson subscribes to an internalist principle (P2) which connects evaluative judgments with motivation and hence with action. (Indeed, in light of the difficulty raised here, one might wonder if Davidson is entitled to consider P2 a "mild" form of internalism (p. 26).) As with Hare, this internalist commitment rules out as impossible certain kinds of action contrary to one's evaluative judgment. Now Davidson, like Hare, does accept the possibility of certain phenomena in this neighborhood; but--as with Hare--critics think the cases permitted by his analysis simply do not exhaust the range of actual cases of weakness of will. The phenomenon seems to run one step ahead of our attempts to make room for it. Those writing after Davidson have tended to focus, then, on the question of the possibility and rational status of action contrary to one's unconditional better judgment.[12] Naturally, different theorists have plotted different courses through these shoals. Some tack more to the internalist side, wishing to preserve a strong internal connection between evaluation and action even at the risk of denying or seeming to deny the possibility of akratic action (or at least some understandings of it). Examples of some post-Davidson treatments which share a broadly internalist emphasis, even if they feature different flavors of internalism, are those of Bratman (1979), Buss (1997), Tenenbaum (1999; see also 2007, ch. 7), and Stroud (2003). The main danger for such approaches is that in seeking to preserve and defend a certain picture of the primordial role of evaluative thought in rational action--a picture critics are likely to dismiss as too rationalistic--such theorists may be led to reject common phenomena which ought properly to have constrained their more abstract theories. (See the opening of Wiggins 1979 for a forceful articulation of this criticism.) Other theorists, by contrast, are more drawn toward the externalist shoreline. They emphasize the motivational importance of factors other than the agent's evaluative judgment and the divergences that can result between an agent's evaluation of her options and her motivation to act. They are thus disinclined to posit any strong, necessary link between evaluative judgment and action. Michael Stocker, for instance, argues that the philosophical tradition has been led astray in assuming that evaluation dictates motivation. "Motivation and evaluation do not stand in a simple and direct relation to each other, as so often supposed," he writes. Rather, "their interrelations are mediated by large arrays of complex psychic structures, such as mood, energy, and interest" (1979, pp. 738-9). Similarly, Alfred Mele proposes as a fundamental and general truth--and one that underlies the possibility of akrasia--that "the motivational force of a want may be out of line with the agent's evaluation of the object of that want" (1987, p. 37). Mele goes on to offer several different reasons why the two can come apart: for example, rewards perceived as proximate can exert a motivational influence disproportionate to the value the agent reflectively attaches to them (1987, ch. 6). Such wants may function as strong causes even if the agent takes them to constitute weak reasons. With respect to these questions, the challenge sketched at the end of Section 1 above remains in full force. What is required is a view which successfully navigates between the Scylla of an extreme internalism about evaluative judgment which would preclude the possibility of weakness of will, and the Charybdis of an extreme externalism which would deny any privileged role to evaluative judgment in practical reasoning or rational action. For one's verdict about akrasia will in general be closely connected to one's more general views of action, practical reasoning, rationality, and evaluative judgment--as was certainly true of Davidson. Views that downplay the role of evaluative judgment in action and hence tack more toward the externalist side of the channel may more easily be able to accept the possibility and indeed the actuality of weakness of will. But they are subject to their own challenges. For example, suppose we follow Mele's image of akrasia and posit that a certain agent is caused to do x by motivation to do x which is dramatically out of kilter with her assessment of the merits of doing x. In what sense, then, is her doing x free, intentional, and uncompelled? Such an agent might seem rather to be at the mercy of a motivational force which is, from her point of view, utterly alien. Thus, worries about distinguishing akrasia from compulsion come back in full force in connection with proposals like these. (See fn. 7 above for relevant references; Buss and Tenenbaum press these worries against accounts like Mele's in particular.) Moreover, there is the danger, for accounts of this more externalist stripe, of taking too much of the mystery out of weakness of will. Even if akratic action is possible and indeed actual, it remains a puzzling, marginal, somehow defective instance of agency, one that we rightly find not fully intelligible. Views that do not assign a privileged place in rational deliberation and action to the agent's overall assessment of her options risk making akratic action seem no more problematic than Julie's or Jimmy's decisions, or Hare's agent who fails to pick up the roundest stone in the vicinity. ### 3.2 Weakness of Will as Potentially Rational The "externalist turn" toward downplaying the role of an agent's better judgment and emphasizing other psychic factors instead is connected to a second way in which some theorists writing after Davidson have dissented from his analysis. Davidson, as we saw, viewed akratic action as possible, but irrational. The weak-willed agent acts contrary to what she herself takes to be the balance of reasons; her choice is thus unreasonable by her own lights. On this picture, incontinent action is a paradigm case of practical irrationality. Many other theorists have agreed with Davidson on this score and have taken akrasia to be perhaps the clearest example of practical irrationality. But some writers (notably Audi 1990, McIntyre 1990, and Arpaly 2000) have questioned whether akratic action is necessarily irrational. Perhaps we ought to leave room, not just for the possibility of akratic action, but for the potential rationality of akratic action. The irrationality which is held necessarily to attach to akratic action derives from the discrepancy between what the agent judges to be the best (or better) thing to do, and what she does. That is, her action is faulted as irrational in virtue of not conforming to her better judgment. But--ask these critics--what if her better judgment is itself faulty? There is nothing magical about an agent's better judgment that ensures that it is correct, or even warranted; like any other judgment, it can be in error, or even unjustified. (Recall that by "better judgment" we meant, all along, only "a judgment as to which course of action is better," not "a superior judgment.") Where the agent's better judgment is itself defective, in doing what she deems herself to have insufficient reason to do, the agent may actually be doing what she has most reason to do. "Even though the akratic agent does not believe that she is doing what she has most reason to do, it may nevertheless be the case that the course of action that she is pursuing is the one that she has ... most reason to pursue" (McIntyre 1990, p. 385). In that sense the akratic agent may be wiser than her own better judgment. How, concretely, could an agent's better judgment go astray in this way? Perhaps her survey of what she took to be the relevant considerations did not include, or did not attach sufficient weight to, what were in fact significant reasons in favor of one of the possible courses of action. She may have overlooked these, or (wrongly) deemed them not to be reasons, or failed to appreciate their full force; and in that case her judgment of what it is best to do will be incorrect. Consider, for example, Jonathan Bennett's Huckleberry Finn (Bennett 1974, discussed in McIntyre 1990), who akratically fails to turn in his slave friend Jim to the authorities. Huck's judgment that he ought to do so, however, was based primarily on what he took to be the force of Miss Watson's property rights; it ignored his powerful feelings of friendship and affection for Jim, as well as other highly relevant factors. His "better judgment" was thus not in fact a very comprehensive judgment; it did not take into account the full range of relevant considerations. Or consider Emily, who has always thought it best that she pursue a Ph.D. in chemistry (Arpaly 2000, p. 504). When she revisits the issue, as she does periodically, she discounts her increasing feelings of restlessness, sadness, and lack of motivation as she proceeds in the program, and concludes that she ought to persevere. But in fact she has very good reasons to quit the program--her talents are not well suited to a career in chemistry, and the people who are thriving in the program are very different from her. If she impulsively, akratically quits the program, purely on the basis of her feelings, Emily is in fact doing just what she ought to do.[13] That her action conflicts with her better judgment does not significantly impugn its rationality, given all the considerations that do support her quitting the program. "A theory of rationality should not assume that there is something special about an agent's best judgment. An agent's best judgment is just another belief" (Arpaly 2000, p. 512). Emily's action conflicts, then, with one belief she has; but it coheres with many more of her beliefs and desires overall. So even though she may find her own action inexplicable or "surd," she is in fact acting rationally, although she does not know it. Contra Davidson, "we can ... act rationally just when we cannot make any sense of our actions" (Arpaly 2000, p. 513). It is unclear, however, whether these arguments and examples are likely to sway those who take akrasia to be a paradigm of practical irrationality. These dissenters stress the substantive merits of the course of action the akratic agent follows. But traditionalists may say that is beside the point: however well things turn out, the practical thinking of the akratic agent still exhibits a procedural defect. Someone who flouts her own conclusion about where the balance of reasons lies is ipso facto not reasoning well. Even if the action she performs is in fact supported by the balance of reasons, she does not think it is, and that is enough to show her practical reasoning to be faulty. The defenders of the traditional conception of akrasia as irrational thus wish to grant special rational authority (in this procedural sense) to the agent's better judgment, even if they admit that such a judgment can be substantively incorrect. By contrast, the dissenters "[do] not believe best judgments have any privileged role" (Arpaly 2000, p. 513). We see again the contrast between "internalist" and "externalist" tendencies in the debates over weakness of will. ### 3.3 Changing the Subject A final revisionist strand now emerging in the literature takes the agent's better judgment even farther out of the picture. In an outstandingly lucid and stimulating essay published in 1999 (see also his 2009, ch. 4), Richard Holton argued that weakness of will is not action contrary to one's better judgment at all. The literature has gone astray in understanding weakness of will in this way; weakness of will is actually quite a different phenomenon, in which the agent's better judgment plays no role.[14] For Holton, when ordinary people speak of weakness of will they have in mind a certain kind of failure to act on one's intentions. What matters for weakness of will, then, is not whether you deem another course of action superior at the time of action. It is whether you are abandoning an intention you previously formed. Weakness of will as the untutored understand it is not akrasia (if we reserve that term for action contrary to one's better judgment), but rather a certain kind of failure to stick to one's plans.[15] This understanding of weakness of will changes the subject in two ways. First, the state of the agent with which the weak-willed action is in conflict is not an evaluative judgment (as in akrasia) but a different kind of state, namely an intention. Second, it is not essential that there be synchronic conflict, as akrasia demands. You must act contrary to your present better judgment in order to exhibit akrasia; conflict with a previous better judgment does not indicate akrasia, but merely a change of mind. However, you can exhibit weakness of will as Holton understands it simply by abandoning a previously formed intention. Of course not all cases of abandoning or failing to act on a previously formed intention count as weakness of will. I intend to run five miles tomorrow evening. If I break my leg tomorrow morning and fail to run five miles tomorrow evening, I will not have exhibited weakness of will. How can we characterize which failures to act on a previously formed intention count as weakness of will? Holton's answer has two parts. First, he says, there is an irreducible normative dimension to the question whether someone's abandoning of an intention constituted weakness of will (Holton 1999, p. 259).[16] That is, there is no purely descriptive criterion (such as whether her action conflicted with her better judgment) which is sufficient for weakness of will; in order to decide whether a given case was an instance of weakness of will we must consider normative questions, such as whether it was reasonable for the agent to have abandoned or revised that intention, or whether she should have done so. In the case of my broken leg, for instance, it was clearly reasonable for me to abandon my intention; that is why I could not be charged with weakness of will in that case. Second, says Holton, we need to attend to an important subclass of our intentions to do something at a future time, namely contrary-inclination-defeating intentions, or, as he later terms them (Holton 2003), resolutions. Resolutions are intentions that are formed precisely in order to insulate one against contrary inclinations one expects to feel when the time comes. Thus one reason I might form an intention on Monday to run five miles on Tuesday--as opposed to leaving the issue open until Tuesday, for decision then--is to reduce the effect of feelings of lassitude to which I fear I may be subject when Tuesday rolls around. Then suppose Tuesday rolls around; I am indeed prey to feelings of lassitude; and I decide as a result not to run. Now I can be charged with weakness of will. Weakness of will involves, specifically, a failure to act on a resolution; this is sufficient to differentiate weakness of will from mere change of mind and even from caprice (which is a different species of unreasonable intention revision, according to Holton). As a later paper by Alison McIntyre shows (McIntyre 2006), understanding weakness of will in this way casts a fresh light on the issue of its rational status. The weak-willed agent abandons a resolution because of a contrary inclination of exactly the type which the resolution was expressly designed to defeat. Therefore, as McIntyre underlines, weak-willed action always involves a procedural rational defect:[17] a technique of self-management has been deployed but has failed (McIntyre 2006, p. 296). To that extent we have grounds to criticize weak-willed action simply in virtue of the second of the ways in which Holton wishes to distinguish weakness of will from a mere change of mind, without even resolving the potentially murky issue of whether the agent was reasonable in abandoning her intention. McIntyre holds, however, that it would be overstating the case to say that because weakness of will involves this procedural defect, it is always irrational (McIntyre 2006, p. 290; pp. 298-9; p. 302). She proposes rather that practical rationality has multiple facets and aims, and that failure in one respect or along one dimension does not automatically justify the especially severe form of rational criticism which we intend by the term "irrational." For example, consider an agent who succumbs to contrary inclination of exactly the type expected when the time comes to act on a truly stupid resolution. (Holton gives the example of resolving to go without water for two days just to see what it feels like: Holton 2003, p. 42.) There will indeed be a blemish on this agent's rational scorecard if he eventually gives in and drinks: he will have failed in his attempt at self-management. But wouldn't it be rationally far worse for him to stick to his silly resolution no matter what the cost? We can also re-examine the issue of the rationality of akrasia in light of this analysis of weakness of will; for we can distinguish between akratic and non-akratic cases of the latter. As McIntyre points out, resolutions typically rest on judgments about what it is best that one do at a (future) time t. If an agent fails to act on a previously formed resolution to do a at t, thus exhibiting weakness of will, we can distinguish the case in which he still endorses at t the judgment that it is best that he do a at t (even though he does not do it) from the case in which he abandons that judgment as well as his resolution. In the latter, non-akratic type of case, the agent in effect rationalizes his failure to live up to his resolution by deciding that it is not after all best that he do a at t. McIntyre points out that the traditional view that akrasia is always irrational seems to give us a perverse incentive to rationalize, since in that case we escape the grave charge of practical irrationality, being left only with the procedural practical defect present in all cases of weakness of will (McIntyre 2006, p. 291). But this seems implausible: are the two sub-cases so radically different in their rational status? Indeed, she argues, if anything, akratic weakness of will is typically rationally preferable to rationalizing weakness of will (McIntyre 2006, p. 287; pp. 309ff.). "In the presence of powerful contrary inclinations that bring about a failure to be resolute," she writes, "resisting rationalization and remaining clearheaded about one's reasons to act can constitute a modest accomplishment" (McIntyre 2006, p. 311). Have we witnessed the transformation of akrasia from impossible, to irrational, to downright admirable? ### 3.4 Recent Developments #### 3.4.1 Epistemic Akrasia One focus of renewed attention in recent years has been (so-called) epistemic akrasia. An epistemically akratic agent holds beliefs of the form "P, but my evidence doesn't support P." In an influential discussion, Sophie Horowitz posits an analogy between epistemic akrasia and practical akrasia: "Just as an akratic agent acts in a way she believes she ought not act, an epistemically akratic agent believes something that she believes is unsupported by her evidence" (Horowitz 2014, p. 718). Is epistemic akrasia possible? Some philosophers who are happy to countenance practical akrasia have answered in the negative with respect to its putative doxastic analogue (these include Hurley 1989, Adler 2002, and Owens 2002). One argument for this denial focuses on the notion of doxastic control. David Owens argues that we lack the requisite sort of doxastic control required for our beliefs to be formed freely and deliberately, "either in accordance with our judgement about what we should believe or against those judgements" (Owens 2002, p. 395). But such control, Owens argues, would be necessary in order for epistemic akrasia to be possible. A second argument for the impossibility of epistemic akrasia (discussed in Adler 2002) turns on what is seen as an important disanalogy between epistemic and practical reasoning. Conflict between two incompatible beliefs weakens one or both beliefs (since both can't be true), whereas conflict between two desires that can't both be satisfied need not weaken either desire. A practically akratic agent who has formed an all things considered judgment about what to do may thus still be in the grip of a desire to do something else. But this, Adler says, has no analogue in the epistemic realm: one can't remain in the grip of a belief if one views the evidence for an opposing belief as decisive. Thus, practical akrasia is "motivationally intelligible" in a way that epistemic akrasia is not (Adler 2002, p. 8). Neil Levy responds to this concern by arguing that while beliefs in some domains fit this model, the disanalogy Adler points to does not hold in domains where there is room for "ongoing, rational controversy," including philosophy (Levy 2004, p. 156). One may form a belief in favor of a philosophical view while nonetheless feeling the pull of incompatible views, just as one may retain a desire that one judges all things considered one shouldn't satisfy. Though Levy only claims that Adler's impossibility argument fails, Ribeiro (2011) uses similar considerations to motivate the claim that epistemic akrasia is not only possible, but actual. Some philosophers who think there could be an epistemic version of akrasia have raised questions about how closely it would parallel practical akrasia. Daniel Greco (2014), for instance, argues that while a divergence between one's moral emotions and one's urges might play an important role in understanding some cases of practical akrasia, there are no corresponding epistemic emotions that would help to illuminate epistemic akrasia (Greco 2014, p. 207). Practical and epistemic varieties of akrasia nonetheless have in common that they involve a kind of fragmentation or inner conflict; Greco uses this characterization to support the idea that both epistemic and practical akrasia are always irrational. (Feldman 2005 also sees epistemic akrasia as a paradigm of irrationality.) Just as the rational status of practical akrasia has become contested (see section 3.2 above), however, some theorists have now argued that epistemic akrasia could in fact be rational. For example, Coates (2012), Weatherson (2019), Wedgwood (2011), and Lasonen-Aarnio (2014) have argued for the rational permissibility of some beliefs of the characteristic akratic form "P, but my evidence doesn't support P." This discussion has been shaped by a growing interest in higher-order evidence--that is, evidence about what one's evidence supports. The notion of higher-order evidence complicates the picture of belief at work in arguments like Adler's (2002) above. Defenders of the rationality of epistemic akrasia have argued that beliefs of the form "P, but my evidence doesn't support P" can be rationally permissible when one has misleading higher-order evidence. In such cases, they contend, a person could have good grounds both for believing that P and for believing that her evidence doesn't support the conclusion that P. For instance, Horowitz describes a case of epistemic akrasia involving a detective, Sam, who stays up all night trying to identify a thief. He knows the evidence he is working with is good, and concludes on the basis of this evidence that the thief was Lucy. He calls his partner and tells her that he's cracked the case, only to have her remind him that his reasoning while sleepy is often poor. Sam hadn't thought about his previous track record, but he believes his partner that he's often wrong about what his evidence supports when he's tired. (Horowitz 2014, p. 719). Defenders would say that Sam could be rational in believing both that Lucy was the thief, and in believing that his evidence doesn't support that claim. Against this view, Horowitz (2014) maintains that higher-order evidence should affect our first-order attitudes whenever we expect our evidence to be truth-guiding (which rationally we almost always should); the conjunction involved in an akratic belief is thus rationally unstable (Horowitz 2014, p. 740). (See also Lasonen-Aarnio 2020 for the argument that epistemically akratic subjects, while sometimes rational, nonetheless manifest bad dispositions). #### 3.4.2 Addiction as Akrasia? There has also been growing interest in the question of whether addiction is perhaps best understood as a form of akrasia. Building on the arguments of Mele (2002), Nick Heather (2016) contends that addiction shares the core features of akratic action and should be understood as a special kind of akrasia, one in which agents consistently act against both their present judgments and their prior resolutions (Heather 2016 p. 133-4). Heather thus accepts Holton's (1999) distinction between akrasia and weakness of will, but argues that addiction paradigmatically involves both halves of this distinction. In contrast, Edmund Henden (2016) argues against classifying addiction as a form of weakness of will. He thinks that the phenomenology of addiction tells against such an assimilation. Addiction often involves habitual behavior, and relapses are often triggered by environmental cues that the addict is not conscious of at the time of action. Addicts may continue to take drugs even when they don't find it pleasurable to do so, and while retaining a strong sense of the disvalue of this course of action (Henden 2016, p. 122). In these respects, Henden contends that addiction is unlike giving into temptation, as paradigm examples of akrasia are often described. Henden also notes that weakness of will seems rationally criticizable in a way that addiction does not, which suggests they can't be the same phenomenon. Many addicts sincerely try very hard to abstain from using drugs. Though their effort may be insufficient for them to succeed in abstaining, it may nonetheless be sufficient relative to ordinary standards. That is, if someone who was not an addict made the same effort with respect to similar endeavors, it would be reasonable to expect her to succeed. The challenges that face addicts thus seem much more demanding than the challenges that face the average practical agent, including those prone to akrasia (Henden 2016, p. 123-4).
weber	## 1. Life and Career Maximilian Carl Emil "Max" Weber (1864-1920) was born in the Prussian city of Erfurt to a family of notable heritage. His father, Max Sr., came from a Westphalian family of merchants and industrialists in the textile business and went on to become a lawyer and National Liberal parliamentarian in Wilhelmine politics. His mother, Helene, came from the Fallenstein and Souchay families, both of the long illustrious Huguenot line, which had for generations produced public servants and academicians. His younger brother, Alfred, was an influential political economist and sociologist, too. Evidently, Max Weber was brought up in a prosperous, cosmopolitan, and cultivated family milieu that was well-plugged into the political, social, and cultural establishment of the German Burgertum [Roth 2000]. Also, his parents represented two, often conflicting, poles of identity between which their eldest son would struggle throughout his life - worldly statesmanship and ascetic scholarship. Educated mainly at the universities of Heidelberg and Berlin, Weber was trained in law, eventually writing his dissertation on medieval trading companies under Levin Goldschmidt and Rudolf von Gneist (and examined by Theodor Mommsen) and Habilitationsschrift on Roman law and agrarian history under August Meitzen. While contemplating a career in legal practice and public service, he received an important research commission from the Verein fur Sozialpolitik (the leading social science association under Gustav Schmoller's leadership) and produced the so-called East Elbian Report on the displacement of the German agrarian workers in East Prussia by Polish migrant labours. Greeted upon publication with high acclaim and political controversy, this early success led to his first university appointment at Freiburg in 1894 to be followed by a prestigious professorship in political economy at Heidelberg two years later. Weber and his wife Marianne, an intellectual in her own right and early women's rights activist, soon found themselves at the center of the vibrant intellectual and cultural life of Heidelberg. The so-called "Weber Circle" attracted such intellectual luminaries as Georg Jellinek, Ernst Troeltsch, and Werner Sombart and later a number of younger scholars including Marc Bloch, Robert Michels, and Gyorgy Lukacs. Weber was also active in public life as he continued to play an important role as a Young Turk in the Verein and maintain a close association with the liberal Evangelische-soziale Kongress (especially with the leader of its younger generation, Friedrich Naumann). It was during this time that he solidified his reputation as a brilliant political economist and outspoken public intellectual. All these fruitful years came to an abrupt halt in 1897 when Weber collapsed with a nervous-breakdown shortly after his father's sudden death (precipitated by a confrontation with Weber) [Radkau 2011, 53-69]. His routine as a teacher and scholar was interrupted so badly that he eventually withdrew from regular teaching duties in 1903, to which he would not return until 1919. Although severely compromised and unable to write as prolifically as before, he still managed to immerse himself in the study of various philosophical and religious topics. This period saw a new direction in his scholarship as the publication of miscellaneous methodological essays as well as The Protestant Ethic and the Spirit of Capitalism (1904-1905) testifies. Also noteworthy about this period is his extensive trip to America in 1904, which left an indelible trace in his understanding of modernity in general [Scaff 2011]. After this stint essentially as a private scholar, he slowly resumed his participation in various academic and public activities. With Edgar Jaffe and Sombart, he took over editorial control of the Archiv fur Sozialwissenschaften und Sozialpolitik, turning it into a leading social science journal of the day as well as his new institutional platform. In 1909, he co-founded the Deutsche Gesellschaft fur Soziologie, in part as a result of his growing unease with the Verein's conservative politics and lack of methodological discipline, becoming its first treasurer (he would resign from it in 1912, though). This period of his life, until interrupted by the outbreak of the First World War in 1914, brought the pinnacles of his achievements as he worked intensely in two areas - the comparative sociology of world religions and his contributions to the Grundriss der Sozialokonomik (to be published posthumously as Economy and Society). Along with the major methodological essays that he drafted during this time, these works would become mainly responsible for Weber's enduring reputation as one of the founding fathers of modern social science. With the onset of the First World War, Weber's involvement in public life took an unexpected turn. At first a fervent patriotic supporter of the war, as virtually all German intellectuals of the time were, he grew disillusioned with the German war policies, eventually refashioning himself as one of the most vocal critics of the Kaiser government in a time of war. As a public intellectual, he issued private reports to government leaders and wrote journalistic pieces to warn against the Belgian annexation policy and the unlimited submarine warfare, which, as the war deepened, evolved into a call for overall democratization of the authoritarian state (Obrigkeitsstaat) that was Wilhelmine Germany. By 1917, Weber was campaigning vigorously for a wholesale constitutional reform for post-war Germany, including the introduction of universal suffrage and the empowerment of parliament. When defeat came in 1918, Germany found in Weber a public intellectual leader, even possibly a future statesman, with unscathed liberal credentials who was well-positioned to influence the course of post-war reconstruction. He was invited to join the draft board of the Weimar Constitution as well as the German delegation to Versaille; albeit in vain, he even ran for a parliamentary seat on the liberal Democratic Party ticket. In those capacities, however, he opposed the German Revolution (all too sensibly) and the Versaille Treaty (all too quixotically) alike, putting himself in an unsustainable position that defied the partisan alignments of the day. By all accounts, his political activities bore little fruit, except his advocacy for a robust plebiscitary presidency in the Weimar Constitution. Frustrated with day-to-day politics, he turned to his scholarly pursuits with renewed vigour. In 1919, he briefly taught in turn at the universities of Vienna (General Economic History was an outcome of this experience) and Munich (where he gave the much-lauded lectures, Science as a Vocation and Politics as a Vocation), while compiling his scattered writings on religion in the form of massive three-volume Gesammelte Aufsatze zur Religionssoziologie [GARS hereafter]. All these reinvigorated scholarly activities came to an end in 1920 when he died suddenly of pneumonia in Munich (likely due to the Spanish flu). Max Weber was fifty-six years old. ## 2. Philosophical Influences Putting Weber in the context of philosophical tradition proper is not an easy task. For all the astonishing variety of identities that can be ascribed to him as a scholar, he was certainly no philosopher at least in the narrow sense of the term. His reputation as a Solonic legislator of modern social science also tends to cloud our appreciation of the extent to which his ideas were embedded in the intellectual context of the time. Broadly speaking, Weber's philosophical worldview, if not coherent philosophy, was informed by the deep crisis of the Enlightenment project in fin-de-siecle Europe, which was characterized by the intellectual revolt against positivist reason, a celebration of subjective will and intuition, and a neo-Romantic longing for spiritual wholesomeness [Hughes 1977]. In other words, Weber belonged to a generation of self-claimed epigones who had to struggle with the legacies of Darwin, Marx, and Nietzsche. As such, the philosophical backdrop to his thoughts will be outlined here along two axes -- epistemology and ethics. ### 2.1 Knowledge: Neo-Kantianism Weber encountered the pan-European cultural crisis of his time mainly as filtered through the jargon of German Historicism [Beiser 2011]. His early training in law had exposed him to the sharp divide between the reigning Labandian legal positivism and the historical jurisprudence championed by Otto von Gierke (one of his teachers at Berlin); in his later incarnation as a political economist, he was keenly interested in the heated "strife over methods" (Methodenstreit) between the positivist economic methodology of Carl Menger and the historical economics of Schmoller (his mentor during the early days). Arguably, however, it was not until Weber grew acquainted with the Baden or Southwestern School of Neo-Kantians, especially through Wilhelm Windelband, Emil Lask, and Heinrich Rickert (his one-time colleague at Freiburg), that he found a rich conceptual template suitable for the clearer elaboration of his own epistemological position. In opposition to a Hegelian emanationist epistemology, briefly, Neo-Kantians shared the Kantian dichotomy between reality and concept. Not an emanent derivative of concepts as Hegel posited, reality is irrational and incomprehensible, and the concept, only an abstract construction of our mind. Nor is the concept a matter of will, intuition, and subjective consciousness as Wilhelm Dilthey posited. According to Hermann Cohen, one of the early Neo-Kantians, concept formation is fundamentally a cognitive process, which cannot but be rational as Kant held. If our cognition is logical and all reality exists within cognition, then only a reality that we can comprehend in the form of knowledge is rational - metaphysics is thereby reduced to epistemology, and Being to logic. As such, the process of concept formation both in the natural (Natur-) and the cultural-historical sciences (Geisteswissenschaften) has to be universal as well as abstract, not different in kind but in their subject matters. The latter is only different in dealing with the question of values in addition to logical relationships. For Windelband, however, the difference between the two kinds of knowledge has to do with its aim and method as well. Cultural-historical knowledge is not concerned with a phenomenon because of what it shares with other phenomena, but rather because of its own definitive qualities. For values, which form its proper subject, are radically subjective, concrete and individualistic. Unlike the "nomothetic" knowledge that natural science seeks, what matters in historical science is not a universal law-like causality, but an understanding of the particular way in which an individual ascribes values to certain events and institutions or takes a position towards the general cultural values of his/her time under a unique, never-to-be-repeated constellation of historical circumstances. Therefore, cultural-historical science seeks "ideographic" knowledge; it aims to understand the particular, concrete and irrational "historical individual" with inescapably universal, abstract, and rational concepts. Turning irrational reality into rational concept, it does not simply paint (abbilden) a picture of reality but transforms (umbilden) it. Occupying the gray area between irrational reality and rational concept, then, its question became twofold for the Neo-Kantians. One is in what way we can understand the irreducibly subjective values held by the historical actors in an objective fashion, and the other, by what criteria we can select a certain historical phenomenon as opposed to another as historically significant subject matter worthy of our attention. In short, the issue was not only the values to be comprehended by the seeker of historical knowledge, but also his/her own values, which are no less subjective. Value-judgment (Werturteil) as well as value (Wert) became a keen issue. According to Rickert's definitive elaboration, value-judgment precedes values. He posits that the "in-dividual," as opposed to mere "individual," phenomenon can be isolated as a discrete subject of our historical inquiry when we ascribe certain subjective values to the singular coherence and indivisibility that are responsible for its uniqueness. In his theory of value-relation (Wertbeziehung), Rickert argues that relating historical objects to values can still retain objective validity when it is based on a series of explicitly formulated conceptual distinctions. They are to be made firmly between the investigator's values and those of the historical actor under investigation, between personal or private values and general cultural values of the time, and between subjective value-judgment and objective value-relations. In so positing, however, Rickert is making two highly questionable assumptions. One is that there are certain values in every culture that are universally accepted within that culture as valid, and the other, that a historian free of bias must agree on what these values are. Just as natural science must assume "unconditionally and universally valid laws of nature," so, too, cultural-historical science must assume that there are "unconditionally and universally valid values." If so, an "in-dividual" historical event has to be reduced to an "individual" manifestation of the objective process of history, a conclusion that essentially implies that Rickert returned to the German Idealist faith in the meaningfulness of history and the objective validity of the diverse values to be found in history. An empirical study in historical science, in the end, cannot do without a metaphysics of history. Bridging irrational reality and rational concept in historical science, or overcoming hiatus irrationalis (a la Emil Lask) without recourse to a metaphysics of history still remained a problem as acutely as before. While accepting the broadly neo-Kantian conceptual template as Rickert elaborated it, Weber's methodological writings would turn mostly on this issue. ### 2.2 Ethics: Kant and Nietzsche German Idealism seems to have exerted another enduring influence on Weber, discernible in his ethical worldview more than in his epistemological position. This was the strand of Idealist discourse in which a broadly Kantian ethic and its Nietzschean interlocution figure prominently. The way in which Weber understood Kant seems to have come through the conceptual template set by moral psychology and philosophical anthropology. In conscious opposition to the utilitarian-naturalistic justification of modern individualism, Kant viewed moral action as principled and self-disciplined while expressive of genuine freedom and autonomy. On this Kantian view, freedom and autonomy are to be found in the instrumental control of the self and the world (objectification) according to a law formulated solely from within (subjectification). Furthermore, such a paradoxical compound is made possible by an internalization or willful acceptance of a transcendental rational principle, which saves it from falling prey to the hedonistic subjectification that Kant found in Enlightenment naturalism and which he so detested. Kant in this regard follows Rousseau in condemning utilitarianism; instrumental-rational control of the world in the service of our desires and needs just degenerates into organized egoism. In order to prevent it, mere freedom of choice based on elective will (Willkur) has to be replaced by the exercise of purely rational will (Wille) [Taylor 1989, 364]. The so-called "inward turn" is thus the crucial benchmark of autonomous moral agency for Kant, but its basis has been fundamentally altered; it should be done with the purpose of serving a higher end, that is, the universal law of reason. A willful self-transformation is demanded now in the service of a higher law based on reason, or an "ultimate value" in Weber's parlance. Weber's understanding of this Kantian ethical template was strongly tinged by the Protestant theological debate taking place in the Germany of his time between (orthodox Lutheran) Albrecht Ritschl and Matthias Schneckenburger (of Calvinist persuasion), a context with which Weber became acquainted through his Heidelberg colleague, Troeltsch. Suffice it to note in this connection that Weber's sharp critique of Ritschl's Lutheran communitarianism seems reflective of his broadly Kantian preoccupation with radically subjective individualism and the methodical transformation of the self [Graf 1995]. All in all, one might say that "the preoccupations of Kant and of Weber are really the same. One was a philosopher and the other a sociologist, but there... the difference ends" [Gellner 1974, 184]. That which also ends, however, is Weber's subscription to a Kantian ethic of duty when it comes to the possibility of a universal law of reason. Weber was keenly aware of the fact that the Kantian linkage between growing self-consciousness, the possibility of universal law, and principled and thus free action had been irrevocably severed. Kant managed to preserve the precarious identification of non-arbitrary action and subjective freedom by asserting such a linkage, which Weber believed to be unsustainable in his allegedly Nietzschean age. According to Nietzsche, "will to truth" cannot be content with the metaphysical construction of a grand metanarrative, whether it be monotheistic religion or modern science, and growing self-consciousness, or "intellectualization" a la Weber, can lead only to a radical skepticism, value relativism, or, even worse, nihilism. According to such a Historicist diagnosis of modernity that culminates in the "death of God," the alternative seems to be either a radical self-assertion and self-creation that runs the risk of being arbitrary (as in Nietzsche) or a complete desertion of the modern ideal of self-autonomous freedom (as in early Foucault). If the first approach leads to a radical divinization of humanity, one possible extension of modern humanism, the second leads inexorably to a "dedivinization" of humanity, a postmodern antihumanism [Vattimo 1988, 31-47]. Seen in this light, Weber's ethical sensibility is built on a firm rejection of a Nietzschean divination and Foucaultian resignation alike, both of which are radically at odds with the Kantian ethic of duty. In other words, Weber's ethical project can be described as a search for non-arbitrary freedom (his Kantian side) in what he perceived as an increasingly post-metaphysical world (his Nietzschean side). According to Paul Honigsheim, Weber's ethic is that of "tragedy" and "nevertheless" [Honigsheim 2013, 113]. This deep tension between the Kantian moral imperatives and a Nietzschean diagnosis of the modern cultural world is apparently what gives such a darkly tragic and agnostic shade to Weber's ethical worldview. ## 3. History ### 3.1 Rationalization as a Thematic Unity Weber's main contribution as such, nonetheless, lies neither in epistemology nor in ethics. Although they deeply informed his thoughts to an extent still under-appreciated, his main concern lay elsewhere. He was after all one of the founding fathers of modern social science. Beyond the recognition, however, that Weber is not simply a sociologist par excellence as Talcott Parsons's quasi-Durkheimian interpretation made him out to be, identifying an idee maitresse throughout his disparate oeuvre has been debated ever since his own days and is still far from settled. Economy and Society, his alleged magnum opus, was a posthumous publication based upon his widow's editorship, the thematic architectonic of which is unlikely to be reconstructed beyond doubt even after its recent reissuing under the rubric of Max Weber Gesamtausgabe [MWG hereafter]. GARS forms a more coherent whole since its editorial edifice was the work of Weber himself; and yet, its relationship to his other sociologies of, for instance, law, city, music, domination, and economy, remains controvertible. Accordingly, his overarching theme has also been variously surmised as a developmental history of Western rationalism (Wolfgang Schluchter), the universal history of rationalist culture (Friedrich Tenbruck), or simply the Menschentum as it emerges and degenerates in modern rational society (Wilhelm Hennis). The first depicts Weber as a comparative-historical sociologist; the second, a latter-day Idealist historian of culture reminiscent of Jacob Burckhardt; and the third, a political philosopher on a par with Machiavelli, Hobbes, and Rousseau. Important as they are for in-house Weber scholarship, however, these philological disputes need not hamper our attempt to grasp the gist of his ideas. Suffice it for us to recognize that, albeit with varying degrees of emphasis, these different interpretations all converge on the thematic centrality of rationality, rationalism, and rationalization in making sense of Weber. At the outset, what immediately strikes a student of Weber's rationalization thesis is its seeming irreversibility and Eurocentrism. The apocalyptic imagery of the "iron cage" that haunts the concluding pages of the Protestant Ethic is commonly taken to reflect his fatalism about the inexorable unfolding of rationalization and its culmination in the complete loss of freedom and meaning in the modern world. The "Author's Introduction" (Vorbemerkung to GARS) also contains oft-quoted passages that allegedly disclose Weber's belief in the unique singularity of Western civilization's achievement in the direction of rationalization, or lack thereof in other parts of the world. For example: > > A child of modern European civilization (Kulturwelt) who > studies problems of universal history shall inevitably and justfiably > raise the question (Fragestellung): what combination of > circumstances have led to the fact that in the West, and here only, > cultural phenomena have appeared which - at least as we > like to think - came to have universal significance and > validity [Weber 1920/1992, 13: translation altered]? > Taken together, then, the rationalization process as Weber narrated it seems quite akin to a metahistorical teleology that irrevocably sets the West apart from and indeed above the East. At the same time, nonetheless, Weber adamantly denied the possibility of a universal law of history in his methodological essays. Even within the same pages of Vorbemerkung, he said, "rationalizations of the most varied character have existed in various departments of life and in all areas of culture" [Ibid., 26]. He also made clear that his study of various forms of world religions was to be taken for its heuristic value rather than as "complete analyses of cultures, however brief" [Ibid., 27]. It was meant as a comparative-conceptual platform on which to erect the edifying features of rationalization in the West. If merely a heuristic device and not a universal law of progress, then, what is rationalization and whence comes his uncompromisingly dystopian vision? ### 3.2 Calculability, Predictability, and World-Mastery Roughly put, taking place in all areas of human life from religion and law to music and architecture, rationalization means a historical drive towards a world in which "one can, in principle, master all things by calculation" [Weber 1919/1946, 139]. For instance, modern capitalism is a rational mode of economic life because it depends on a calculable process of production. This search for exact calculability underpins such institutional innovations as monetary accounting (especially double-entry bookkeeping), centralization of production control, separation of workers from the means of production, supply of formally free labour, disciplined control on the factory floor, and other features that make modern capitalism qualitatively different from all other modes of organizing economic life. The enhanced calculability of the production process is also buttressed by that in non-economic spheres such as law and administration. Legal formalism and bureaucratic management reinforce the elements of predictability in the sociopolitical environment that encumbers industrial capitalism by means of introducing formal equality of citizenship, a rule-bound legislation of legal norms, an autonomous judiciary, and a depoliticized professional bureaucracy. Further, all this calculability and predictability in political, social, and economic spheres was not possible without changes of values in ethics, religion, psychology, and culture. Institutional rationalization was, in other words, predicated upon the rise of a peculiarly rational type of personality, or a "person of vocation" (Berufsmensch) as outlined in the Protestant Ethic. The outcome of this complex interplay of ideas and interests was modern rational Western civilization with its enormous material and cultural capacity for relentless world-mastery. ### 3.3 Knowledge, Impersonality, and Control On a more analytical plateau, all these disparate processes of rationalization can be surmised as increasing knowledge, growing impersonality, and enhanced control [Brubaker 1991, 32-35]. First, knowledge. Rational action in one very general sense presupposes knowledge. It requires some knowledge of the ideational and material circumstances in which our action is embedded, since to act rationally is to act on the basis of conscious reflection about the probable consequences of action. As such, the knowledge that underpins a rational action is of a causal nature conceived in terms of means-ends relationships, aspiring towards a systematic, logically interconnected whole. Modern scientific and technological knowledge is a culmination of this process that Weber called intellectualization, in the course of which, the germinating grounds of human knowledge in the past, such as religion, theology, and metaphysics, were slowly pushed back to the realm of the superstitious, mystical, or simply irrational. It is only in modern Western civilization, according to Weber, that this gradual process of disenchantment (Entzauberung) has reached its radical conclusion. Second, impersonality. Rationalization, according to Weber, entails objectification (Versachlichung). Industrial capitalism, for one, reduces workers to sheer numbers in an accounting book, completely free from the fetters of tradition and non-economic considerations, and so does the market relationship vis-a-vis buyers and sellers. For another, having abandoned the principle of Khadi justice (i.e., personalized ad hoc adjudication), modern law and administration also rule in strict accordance with the systematic formal codes and sine ira et studio, that is, "without anger or passion." Again, Weber found the seed of objectification not in material interests alone, but in the Puritan vocational ethic (Berufsethik) and the life conduct that it inspired, which was predicated upon a disenchanted monotheistic theodicy that reduced humans to mere tools of God's providence. Ironically, for Weber, modern inward subjectivity was born once we lost any inherent value qua humans and became thoroughly objectified vis-a-vis God in the course of the Reformation. Modern individuals are subjectified and objectified all at once. Third, control. Pervasive in Weber's view of rationalization is the increasing control in social and material life. Scientific and technical rationalization has greatly improved both the human capacity for a mastery over nature and institutionalized discipline via bureaucratic administration, legal formalism, and industrial capitalism. The calculable, disciplined control over humans was, again, an unintended consequence of the Puritan ethic of rigorous self-discipline and self-control, or what Weber called "innerworldly asceticism (innerweltliche Askese)." Here again, Weber saw the irony that a modern individual citizen equipped with inviolable rights was born as a part of the rational, disciplinary ethos that increasingly penetrated into every aspect of social life. ## 4. Modernity ### 4.1 The "Iron Cage" and Value-fragmentation Thus seen, rationalization as Weber postulated it is anything but an unequivocal historical phenomenon. As already pointed out, first, Weber viewed it as a process taking place in disparate fields of human life with a logic of each field's own and varying directions; "each one of these fields may be rationalized in terms of very different ultimate values and ends, and what is rational from one point of view may well be irrational from another" [Weber 1920/1992, 27]. Second, and more important, its ethical ramification for Weber is deeply ambivalent. To use his own dichotomy, the formal-procedural rationality (Zweckrationalitat) to which Western rationalization tends does not necessarily go with a substantive-value rationality (Wertrationalitat). On the one hand, exact calculability and predictability in the social environment that formal rationalization has brought about dramatically enhances individual freedom by helping individuals understand and navigate through the complex web of practice and institutions in order to realize the ends of their own choice. On the other hand, freedom and agency are seriously curtailed by the same force in history when individuals are reduced to a "cog in a machine," or trapped in an "iron cage" that formal rationalization has spawned with irresistible efficiency and at the expense of substantive rationality. Thus, his famous lament in the Protestant Ethic: > > No one knows who will live in this cage (Gehause) in the > future, or whether at the end of this tremendous development entirely > new prophets will arise, or there will be a great rebirth of old ideas > and ideals, or, if neither, mechanized petrification, embellished with > a sort of convulsive self-importance. For the "last man" > (letzten Menschen) of this cultural development, it might > well be truly said: "Specialist without spirit, sensualist > without heart; this nullity imagines that it has attained a level of > humanity (Menschentums) never before achieved" [Weber > 1904-05/1992, 182: translation altered]. > Third, Weber envisions the future of rationalization not only in terms of "mechanized petrification," but also of a chaotic, even atrophic, inundation of subjective values. In other words, the bureaucratic "iron cage" is only one side of the modernity that rationalization has brought about; the other is a "polytheism" of value-fragmentation. At the apex of rationalization, we moderns have suddenly found ourselves living "as did the ancients when their world was not yet disenchanted of its gods and demons" [Weber 1919/1946, 148]. Modern society is, Weber seems to say, once again enchanted as a result of disenchantment. How did this happen and with what consequences? ### 4.2 Reenchantment via Disenchantment In point of fact, Weber's rationalization thesis can be understood with richer nuance when we approach it as, for lack of better terms, a dialectics of disenchantment and reenchantment rather than as a one-sided, unilinear process of secularization. Disenchantment had ushered in monotheistic religions in the West. In practice, this means that ad hoc maxims for life-conduct had been gradually displaced by a unified total system of meaning and value, which historically culminated in the Puritan ethic of vocation. Here, the irony was that disenchantment was an ongoing process nonetheless. Disenchantment in its second phase pushed aside monotheistic religion as something irrational, thus delegitimating it as a unifying worldview in the modern secular world. Modern science, which was singularly responsible for this late development, was initially welcomed as a surrogate system of orderly value-creation, as Weber found in the convictions of Bacon (science as "the road to true nature") and Descartes (as "the road to the true god") [Weber 1919/1946, 142]. For Weber, nevertheless, modern science is a deeply nihilistic enterprise in which any scientific achievement worthy of the name must "ask to be surpassed and made obsolete" in a process "that is in principle ad infinitum," at which point, "we come to the problem of the meaning of science." He went on to ask: "For it is simply not self-evident that something which is subject to such a law is in itself meaningful and rational. Why should one do something which in reality never comes to an end and never can?" [Ibid., 138: translation altered]. In short, modern science has relentlessly dismantled all other sources of value-creation, in the course of which its own meaning has also been dissipated beyond repair. The result is the "Gotterdammerung of all evaluative perspectives" including its own [Weber 1904/1949, 86]. Irretrievably gone as a result is a unifying worldview, be it religious or scientific, and what ensues is its fragmentation into incompatible value spheres. Weber, for instance, observed: "since Nietzsche, we realize that something can be beautiful, not only in spite of the aspect in which it is not good, but rather in that very aspect" [Weber 1919/1946, 148]. That is to say, aesthetic values now stand in irreconcilable antagonism to religious values, transforming "value judgments (Werturteile) into judgments of taste (Geschmacksurteile) by which what is morally reprehensible becomes merely what is tasteless" [Weber 1915/1946, 342]. Weber is, then, not envisioning a peaceful dissolution of the grand metanarratives of monotheistic religion and universal science into a series of local narratives and the consequent modern pluralist culture in which different cultural practices follow their own immanent logic. His vision of polytheistic reenchantment is rather that of an incommensurable value-fragmentation into a plurality of alternative metanarratives, each of which claims to answer the same metaphysical questions that religion and science strove to cope with in their own ways. The slow death of God has reached its apogee in the return of gods and demons who "strive to gain power over our lives and again ... resume their eternal struggle with one another" [Weber 1919/1946, 149]. Seen this way, it makes sense that Weber's rationalization thesis concludes with two strikingly dissimilar prophecies - one is the imminent iron cage of bureaucratic petrification and the other, the Hellenistic pluralism of warring deities. The modern world has come to be monotheistic and polytheistic all at once. What seems to underlie this seemingly self-contradictory imagery of modernity is the problem of modern humanity (Menschentum) and its loss of freedom and moral agency. Disenchantment has created a world with no objectively ascertainable ground for one's conviction. Under the circumstances, according to Weber, a modern individual tends to act only on one's own aesthetic impulse and arbitrary convictions that cannot be communicated in the eventuality; the majority of those who cannot even act on their convictions, or the "last men who invented happiness" a la Nietzsche, lead the life of a "cog in a machine." Whether the problem of modernity is accounted for in terms of a permeation of objective, instrumental rationality or of a purposeless agitation of subjective values, Weber viewed these two images as constituting a single problem insofar as they contributed to the inertia of modern individuals who fail to take principled moral action. The "sensualists without heart" and "specialists without spirit" indeed formed two faces of the same coin that may be called the disempowerment of the modern self. ### 4.3 Modernity contra Modernization Once things were different, Weber claimed. An unflinching sense of conviction that relied on nothing but one's innermost personality once issued in a highly methodical and disciplined conduct of everyday life - or, simply, life as a duty. Born in the crucible of the Reformation, this archetypal modern subjectivity drew its strength solely from within in the sense that one's principle of action was determined by one's own psychological need to gain self-affirmation. Also, the way in which this deeply introspective subjectivity was practiced, that is, in self-mastery, entailed a highly rational and radically methodical attitude towards one's inner self and the outer, objective world. Transforming the self into an integrated personality and mastering the world with tireless energy, subjective value and objective rationality once formed "one unbroken whole" [Weber 1910/1978, 319]. Weber calls the agent of this unity the "person of vocation" (Berufsmensch) in his religious writings, "personality" (Personlichkeit) in the methodological essays, "genuine politician" (Berufspolitiker) in the political writings, and "charismatic individual" in Economy and Society. The much-celebrated Protestant Ethic thesis was indeed a genealogical reconsturction of this idiosyncratic moral agency in modern times [Goldman 1992]. Once different, too, was the mode of society constituted by and in turn constitutive of this type of moral agency. Weber's social imagination revealed its keenest sense of irony when he traced the root of the cohesive integration, intense socialization, and severe communal discipline of sect-like associations to the isolated and introspective subjectivity of the Puritan person of vocation. The irony was that the self-absorbed, anxiety-ridden and even antisocial virtues of the person of vocation could be sustained only in the thick disciplinary milieu of small-scale associational life. Membership in exclusive voluntary associational life is open, and it is such membership, or "achieved quality," that guarantees the ethical qualities of the individuals with whom one interacts. "The old 'sect spirit' holds sway with relentless effect in the intrinsic nature of such associations," Weber observed, for the sect was the first mass organization to combine individual agency and social discipline in such a systematic way. Weber thus claimed that "the ascetic conventicles and sects ... formed one of the most important foundations of modern individualism" [Weber 1920/1946, 321]. It seems clear that what Weber was trying to outline here is an archetypical form of social organization that can empower individual moral agency by sustaining group disciplinary dynamism, a kind of pluralistically organized social life we would now call a "civil society" [Kim 2007, 57-94]. To summarize, the irony with which Weber accounted for rationalization was driven by the deepening tension between modernity and modernization. Weber's problem with modernity originates from the fact that it required a historically unique constellation of cultural values and social institutions, and yet, modernization has effectively undermined the cultural basis for modern individualism and its germinating ground of disciplinary society, which together had given the original impetus to modernity. The modern project has fallen victim to its own success, and in peril is the individual moral agency and freedom. Under the late modern circumstances characterized by the "iron cage" and "warring deities," then, Weber's question becomes: "How is it at all possible to salvage any remnants of 'individual' freedom of movement in any sense given this all-powerful trend" [Weber 1918/1994, 159]? ## 5. Knowledge Such an appreciation of Weber's main problematic, which culminates in the question of modern individual freedom, may help shed light on some of the controversial aspects of Weber's methodology. In accounting for his methodological claims, it needs to be borne in mind that Weber was not at all interested in writing a systematic epistemological treatise in order to put an end to the "strife over methods" (Methodenstreit) of his time between historicism and positivism. His ambition was much more modest and pragmatic. Just as "the person who attempted to walk by constantly applying anatomical knowledge would be in danger of stumbling" [Weber 1906/1949, 115; translation altered], so can methodology be a kind of knowledge that may supply a rule of thumb, codified a posteriori, for what historians and social scientists do, but it could never substitute for the skills they use in their research practice. Instead, Weber's attempt to mediate historicism and positivism was meant to aid an actual researcher make a practical value-judgment that is fair and acceptable in the face of the plethora of subjective values that one encounters when selecting and processing historical data. After all, the questions that drove his methodological reflections were what it means to practice science in the modern polytheistic world and how one can do science with a sense of vocation. In his own words, "the capacity to distinguish between empirical knowledge and value-judgments, and the fulfillment of the scientific duty to see the factual truth as well as the practical duty to stand up for our own ideals constitute the program to which we wish to adhere with ever increasing firmness" [Weber 1904/1949, 58]. Sheldon Wolin thus concludes that Weber "formulated the idea of methodology to serve, not simply as a guide to investigation but as a moral practice and a mode of political action" [Wolin 1981, 414]. In short, Weber's methodology was as ethical as it was epistemological. ### 5.1 Understanding (Verstehen) Building on the Neo-Kantian nominalism outlined above [2.1], thus, Weber's contribution to methodology turned mostly on the question of objectivity and the role of subjective values in historical and cultural concept formation. On the one hand, he followed Windelband in positing that historical and cultural knowledge is categorically distinct from natural scientific knowledge. Action that is the subject of any social scientific inquiry is clearly different from mere behaviour. While behaviour can be accounted for without reference to inner motives and thus can be reduced to mere aggregate numbers, making it possible to establish positivistic regularities, and even laws, of collective behaviour, an action can only be interpreted because it is based on a radically subjective attribution of meaning and values to what one does. What a social scientist seeks to understand is this subjective dimension of human conduct as it relates to others. On the other hand, an understanding(Verstehen) in this subjective sense is not anchored in a non-cognitive empathy or intuitive appreciation that is arational by nature; it can gain objective validity when the meanings and values to be comprehended are explained causally, that is, as a means to an end. A teleological contextualization of an action in the means-end nexus is indeed the precondition for a causal explanation that can be objectively ascertained. So far, Weber is not essentially in disagreement with Rickert. From Weber's perspective, however, the problem that Rickert's formulation raised was the objectivity of the end to which an action is held to be oriented. As pointed out [2.1 above], Rickert in the end had to rely on a certain transhistorical, transcultural criterion in order to account for the purpose of an action, an assumption that cannot be warranted in Weber's view. To be consistent with the Neo-Kantian presuppositions, instead, the ends themselves have to be conceived of as no less subjective. Imputing an end to an action is of a fictional nature in the sense that it is not free from the subjective value-judgment that conditions the researcher's thematization of a certain subject matter out of "an infinite multiplicity of successively and coexistently emerging and disappearing events" [Weber 1904/1949, 72]. Although a counterfactual analysis might aid in stabilizing the process of causal imputation, it cannot do away completely with the subjective nature of the researcher's perspective. In the end, the kind of objective knowledge that historical and cultural sciences may achieve is precariously limited. An action can be interpreted with objective validity only at the level of means, not ends. An end, however, even a "self-evident" one, is irreducibly subjective, thus defying an objective understanding; it can only be reconstructed conceptually based on a researcher's no less subjective values. Objectivity in historical and social sciences is, then, not a goal that can be reached with the aid of a correct method, but an ideal that must be striven for without a promise of ultimate fulfillment. In this sense, one might say that the so-called "value-freedom" (Wertfreiheit) is as much a methodological principle for Weber as an ethical virtue that a personality fit for modern science must possess. ### 5.2 Ideal Type The methodology of "ideal type" (Idealtypus) is another testimony to such a broadly ethical intention of Weber. According to Weber's definition, "an ideal type is formed by the one-sided accentuation of one or more points of view" according to which "concrete individual phenomena ... are arranged into a unified analytical construct" (Gedankenbild); in its purely fictional nature, it is a methodological "utopia [that] cannot be found empirically anywhere in reality" [Weber 1904/1949, 90]. Keenly aware of its fictional nature, the ideal type never seeks to claim its validity in terms of a reproduction of or a correspondence with reality. Its validity can be ascertained only in terms of adequacy, which is too conveniently ignored by the proponents of positivism. This does not mean, however, that objectivity, limited as it is, can be gained by "weighing the various evaluations against one another and making a 'statesman-like' compromise among them" [Weber 1917/1949, 10], which is often proposed as a solution by those sharing Weber's kind of methodological perspectivism. Such a practice, which Weber calls "syncretism," is not only impossible but also unethical, for it avoids "the practical duty to stand up for our own ideals" [Weber 1904/1949, 58]. According to Weber, a clear value commitment, no matter how subjective, is both unavoidable and necessary. It is unavoidable, for otherwise no meaningful knowledge can be attained. Further, it is necessary, for otherwise the value position of a researcher would not be foregrounded clearly and admitted as such - not only to the readers of the research outcome but also to the very researcher him/herself. In other words, Weber's emphasis on "one-sidedness" (Einseitigkeit) not only affirms the subjective nature of scientific knowledge but also demands that the researcher be self-consciously subjective. The ideal type is devised for this purpose, for "only as an ideal type" can subjective value - "that unfortunate child of misery of our science" - "be given an unambiguous meaning" [Ibid., 107]. Along with value-freedom, then, what the ideal type methodology entails in ethical terms is, on the one hand, a daring confrontation with the tragically subjective foundation of our historical and social scientific knowledge and, on the other, a public confession of one's own subjective value. Weber's methodology in the end amounts to a call for the heroic character-virtue of clear-sightedness and intellectual integrity that together constitute a genuine person of science - a scientist with a sense of vocation who has a passionate commitment to one's own specialized research, yet is utterly "free of illusions" [Lowith 1982, 38]. ## 6. Politics and Ethics Even more explicitly ethical than his methodology, Weber's political project also discloses his entrenched preoccupation with the willful resuscitation of certain character traits in modern society. At the outset, it seems undeniable that Weber was a deeply liberal political thinker especially in German context which is not well known for political liberalism. This means that his ultimate value as a political thinker was locked on individual freedom, that "old, general type of human ideals" [Weber 1895/1994, 19]. He was also a bourgeois liberal, and self-consciously so, in a time of great transformations that were undermining the social conditions necessary to support classical liberal values and bourgeois institutions, thereby compelling liberalism to search for a fundamental reorientation. To that extent, he belongs to that generation of liberal political thinkers in fin-de-siecle Europe who clearly perceived the general crisis of liberalism and sought to resolve it in their own liberal ways [Bellamy 1992, 157-216]. Weber's own way was to address the problem of classical liberal characterology that was, in his view, being progressively undermined by the indiscriminate bureaucratization of modern society. ### 6.1 Domination and Legitimacy Such an ethical subtext is legible even in Weber's stark realism that permeates his political sociology - or, a sociology of domination (Herrschaftssoziologie) as he called it [for the academic use of this term in Weber's time, see Anter 2016, 3-23]. For instance, utterly devoid of moral qualities that many of his contemporaries attributed to the state, it is defined all too thinly as "a human community that (successfully) claims the monopoly of the legitimate use of physical force within a given territory" [Weber 1919/1994, 310]. With the same brevity, he asserted that domination of the ruled by the ruler, or more literally, "lordship" (Herrschaft), is an immutable reality of political life even in a democratic state. That is why, for Weber, an empirical study of politics cannot but be an inquiry into the different modalities by which domination is effectuated and sustained. All the while, he also maintained that a domination worthy of sustained attention is about something far more than the brute fact of subjugation and subservience. For "the merely external fact of the order being obeyed is not sufficient to signify domination in our sense; we cannot overlook the meaning of the fact that the command is accepted as a valid norm" [Weber 1921-22/1978, 946]. In other words, it has to be a domination mediated through justification and interpretation in which the ruler's claim to authority, not mere threat of force or promise of benefits, is the reason for the obedience, not mere compliance, by the ruled. This bipolar emphasis on the factuality of coercive domination at the phenomenal level and the essentially noumenal nature of power (a la Rainer Forst) is what characterizes Weber's political realism [Forst 2012]. In terms of contemporary political realism, Weber seemed to hold that the primary concern of politics is the establishment of an orderly domination and its management within a given territory rather than the realization of such pre- or extra-political moral goals as justice (Kant) or freedom (Hegel) - thus the brevity with which the state is defined above. Sharing this Hobbesian outlook on politics, or what Bernard Williams calls the "First Political Question" (FPQ), enables Weber to square his diagnosis of agonistic value pluralism with an entrenched suspicion of natural-law foundation of liberalism to sustain a democratic politics that is uniquely his own [see 6.2 below]. He went beyond Ordorealism, however, when an evaluative perspective on politics is advocated without recourse to the moral commitments coming from outside the political sphere. The making of a workable political order cannot be authorized by virtue of its coming-into-being and has to satisfy what Williams called the "Basic Legitimation Demand" (BLD) to be an acceptable arrangement of social coordination. A legitimate political order is an institutionalized modus vivendi for collective life that "makes sense as an intelligible order" (MSIO) in the eyes of the beholder [Williams 2005, 1-17]. Since such an acceptance by those living under a particular arrangement depends on the political morality animating that particular community, the ruler's claim to authority can meet with success only when based on a reasonable fit with the local mores, values, and cultures [Cozzaglio and Greene 2019, 1025-26]. Like Machiavelli's Principe, then, Weber's Herren do not behave in a normless vacuum. They rule under certain political-normative constraints that turn on the congruence between the way their domination is justified and the way such a public justification is interpreted as acceptable to the ruled. Weber's concept of domination is as much noumenal as phenomenal. To that extent, it is little wonder that his name figures not only prominently but also uniquely in the pantheon of political realists [Galston 2010]. From this nuanced realist premise, Weber famously moved on to identify three ideal types of legitimate domination based on, respectively, charisma, tradition, and legal rationality. Roughly, the first type of legitimacy claim depends on how persuasively the leaders prove their charismatic qualities, for which they receive personal devotions and emotive followings from the ruled. The second kind of claim can be made successfully when certain practice, custom, and mores are institutionalized to (re)produce a stable pattern of domination over a long duration of time. In contrast to these crucial dependences on personality traits and the passage of time, the third type of authority is unfettered by time, place, and other forms of contingency as it derives its legitimacy from adherence to impersonal rules and general principles that can only be found by suitable legal-rational reasoning. It is, along with the traditional authority, a type of domination that is inclined towards the status quo in ordinary times as opposed to the charismatic authority that represents extraordinary, disruptive, and transformative forces in history. Weber's fame and influence as a political thinker are built most critically upon this typology and the ways in which those ideal types are deployed in his political writings. As such, Weber's sociology of domination has been suspected variously of its embedded normative biases. For one, his theory of legitimacy is seen as endorsing a cynical and unrealistic rejection of universal morality in politics that makes it hard to pass an objective and moral evaluative judgment on legitimacy claims, a charge that is commonly leveled at political realism at large. Under Weber's concept of legitimacy, anything goes, so to speak, as long as the ruler goes along with the local political morality of the ruled (even if it is formed independently of any coercive or corrosive interference by the ruler, thereby satisfying Williams's "critical theory principle"). Read in conjunction with his voluminous political writings, especially, it is criticized to this day as harbouring or foreshadowing, among others, Bonapartist caesarism, passive-revolutionary Fordist ideology, quasi-Fascist elitism, and even proto-Nazism (especially with respect to his robust nationalism and/or nihilistic celebration of power) [inter alia, Strauss 1950; Marcuse in Stammer (ed.) 1971; Mommsen 1984; Rehman 2013]. In addition to these politically heated charges, Weber's typology also reveals a crucial lacuna even as an empirical political sociology. That is to say, it allows scant, or ambiguous, a conceptual topos for democracy. In fact, it seems as though Weber is unsure of the proper place of democracy in his schema. At one point, democracy is deemed as a fourth type of legitimacy because it should be able to embrace legitimacy from below whereas his three ideal types all focus on that from above [Breuer in Schroeder (ed.) 1998, 2]. At other times, Weber seems to believe that democracy is simply non-legitimate, rather than another type of legitimate domination, because it aspires to an identity between the ruler and the ruled (i.e., no domination at all), but without assuming a hierarchical and asymmetrical relationship of power, his concept of legitimacy takes hardly off the ground. Thus, Weber could describe the emergence of proto-democracy in the late medieval urban communes only in terms of "revolutionary usurpation" [Weber 1921-22/1978, 1250], calling them the "first deliberately non-legitimate and revolutionary political association" [ibid., 1302]. Too recalcitrant to fit into his overall schema, in other words, these historical prototypes of democracy simply fall outside of his typology of domination as non- or not-legitimate at all. Overlapping but still distinguishable is Weber's yet another way of conceptualizing democracy, which had to do with charismatic legitimacy. The best example is the Puritan sect in which authority is legitimated only on the grounds of a consensual order created voluntarily by proven believers possessing their own quantum of charismatic legitimating power. As a result of this political corollary of the Protestant doctrine of universal priesthood, Puritan sects could and did "insist upon 'direct democratic administration' by the congregation" and thereby do away with the hierarchical distinction between those ruling and those ruled [ibid., 1208]. In a secularized version of this group dynamics, a democratic ballot would become the primary tool by which presumed charisma of the individual lay citizenry are aggregated and transmitted to their elected leader who become "the agent and hence the servant of his voters, not their chosen master" [ibid., 1128]. Rather than an outright non-legitimate or fourth type of domination, here, democracy comes across as an extremely rare subset of a diffused and institutionalized form of charismatic legitimacy. ### 6.2 Democracy, Partisanship, and Compromise All in all, the irony is unmistakable. It seems as though one of the most influential political thinkers of the twentieth century cannot come to clear terms with its zeitgeist in which democracy, in whatever form, shape and shade, emerged as the only acceptable ground for political legitimacy. Weber's such awkwardness is nowhere more compelling than in his advocacy for "leadership democracy" (Fuhrerdemokratie) during the constitutional politics of post-WWI Germany. If the genuine self-rule of the people is impossible, according to his unsentimental outlook on democracy, the only choice is one between leaderless and leadership democracy. When advocating a sweeping democratization of defeated Germany, thus, Weber envisioned democracy in Germany as a political marketplace in which strong charismatic leaders can be identified and elected by winning votes in a free competition, even battle, among themselves. Preserving and enhancing this element of struggle in politics is important since it is only through a dynamic electoral process that national leadership strong enough to control the otherwise omnipotent bureaucracy can be made. The primary concern for Weber in designing democratic institutions has, in other words, less to do with the realization of democratic ideals, such as freedom, equality, justice, or self-rule, than with cultivation of certain character traits befitting a robust national leadership. In its overriding preoccupation with the leadership qualities, Weber's theory of democracy contains ominous streaks that may vindicate Jurgen Habermas's infamous dictum that Carl Schmitt, "the Kronjurist of the Third Reich," was "a legitimate pupil of Weber's" [Habermas in Stammer (ed.) 1971, 66]. For a fair and comprehensive assessment, however, it should also be brought into purview that Weber's leadership democracy is not solely reliant upon the fortuitous personality traits of its leaders, let alone a caesaristic dictator. "[A] genuine charisma is radically different from the convenient presentation of the present 'divine right of king'... the very opposite is true of the genuinely charismatic ruler, who is responsible to the ruled" [1922/1978, 1114]. Such responsibility is conceivable because charisma is attributed to a leader through a process that can be described as "imputation" from below [Joose 2014, 271]. In addition to the free electoral competition led by the organized mass parties, Weber saw localized, yet public associational life as a breeding ground for such an imputation of charisma. When leaders are identified and selected at the level of, say neighborhood choral societies and bowling clubs [Weber 1910/2002], the alleged authoritarian elitism of leadership democracy comes across as more pluralistic in its valence, far from its usual identification with demagogic dictatorship and unthinking mass following. Insofar as a vibrant civil society functions as an effective medium for the horizontal diffusion of charismatic qualities among lay people, his notion of charismatic leadership can retain a strongly democratic tone to the extent that he also suggested associational pluralism as a sociocultural ground for the political education of the lay citizenry from which genuine leaders would hail. Weber's charismatic leadership has to be "democratically manufactured" [Green 2008, 208], in short, and such a formative political project is predicated upon a pluralistically organized civil society as well as such liberal institutions as universal suffrage, free elections, and organized parties. In this respect, however, it should be noted that Weber's take on civil society is crucially different from a communitarian-Tocquevillean outlook, and this contrast can be cast into sharper relief once put in terms of the contemporary democratic theory of partisanship [cf., inter alia, Rosenblum 2008; Muirhead 2014; White and Ypi 2016]. Like the contemporary advocates of partisanship, Weber is critical of the conventional communitarian view that simply equates civil society with voluntary associational life itself. For not all voluntary associations are conducive to democracy; some are in fact "bad" for its viability. Even in a "good" civil society, those "associative practice," or Vergesellschaftung in Weber's parlance [Weber 1910/2002], may cultivate the kind of civil virtues that regulate our private lives, but such social capital cannot be automatically transferred to the public realm as a useful set of civic virtues and skills for democratic politics. Political capital can be acquired by living political experiences daily. This realization led Weber as well as a growing number of contemporary democratic theorists to converge on an insistent call for the politicization of civil society in the form of not less, but better partisanship, making his politics of civil society crucially different from that of a communitarian-Tocquevillean persuasion [see Kim in Hanke, Scaff & Whimster (eds.) 2020]. Also different from this intensely political civil society is a liberal-Habermasian "public sphere," a rational-communicative haven in which the open exchange and fair deliberation of impartial opinions take place until reasonable consensus emerges. By contrast, Weber's civil society is to be an agonistic arena of organized rivalry, competition, and struggle on behalf of the irreducibly partial claims between which consensus - be that reasonable, overlapping, or bipartisan - may not always be found. Given the incommensurable value fragmentation of the modern politics and society, Weber would wholeheartedly embrace the so-called "circumstances of politics" under which deep disagreements are reasonable and permanent, agreeing that it is not necessarily a bad thing for democracy as long as those "permanent disagreements" remain peaceful [Waldron 1999]. From such an agonistic perspective, the best that can be expected is some kind of mixture of those partial claims - a compromise wherein lies the true meaning of political virtue. That is to say, although no "overlapping consensus" can be expected, it is precisely because all partisan claims are so partial that a political compromise can be made at least between good partisans. For neither too unprincipled (as in opportunistic power-seekers) nor too principled (as in moral zealots), good partisan citizens welcome a political compromise, albeit their passionate value convictions, because they know that some reasonable disagreements are permanent. Then, the kind of political capital expected to be accumulated in a good partisan civil society is a mixture of "principle and pragmatism" [Muirhead 2014, 41-42] - a political virtue much akin to Weber's syncretic ethics of conviction (Gesinnungsethik) and responsibility (Verantwortungsethik) [see 6.3 below]. Together, Weber's ethics also demand that the political leaders and public citizenry combine unflinching commitments to higher causes (which make them different from mere bureaucratic careerists) with sober realism that no political claim, including their own, can represent the whole truth (which makes them different from moral purists and political romantics). This syncretic ethic is the ultimate hallmark of those politicians with a sense of vocation who would fight for their convictions with fierce determination yet not without a "sense of pragmatic judgment" (Augenmass)that a compromise is unavoidable between incommensurable value positions, and all they can do in the end is to take robust responsibility for the consequences, either intended or unintended, of what they thought was a principled compromise. This is why Weber said: "The politician must make compromises ... the scholar may not cover them (Der Politiker muss Kompromisse machen ... der Gelehrte darf sie nicht decken)" [MWG II/10, 983; also see Bruun 1972 (2007, 244)]. It is this type of political virtue that Weber wants to instill at the citizenship as well as leadership level, and the site of this political education is a pluralistically organized civil society in which leaders and citizens can experience the dynamic and institutionalized politicization (re)produced by partisan politics. ### 6.3 Conviction and Responsibility What are, then, these two ethics of conviction and responsibility exactly that Weber wanted to foster through a "'chronic' political schooling" [Weber 1894/1994, 26]. According to the ethic of responsibility, on the one hand, an action is given meaning only as a cause of an effect, that is, only in terms of its causal relationship to the empirical world. The virtue lies in an objective understanding of the possible causal effect of an action and the calculated reorientation of the elements of an action in such a way as to achieve a desired consequence. An ethical question is thereby reduced to a question of technically correct procedure, and free action consists of choosing the correct means. By emphasizing the causality to which a free agent subscribes, in short, Weber prescribes an ethical integrity between action and consequences, instead of a Kantian emphasis on that between action and intention. According to the ethic of conviction, on the other hand, a free agent should be able to choose autonomously not only the means, but also the end; "this concept of personality finds its 'essence' in the constancy of its inner relation to certain ultimate 'values' and 'meanings' of life" [Weber 1903-06/1975, 192]. In this respect, Weber's central problem arises from the recognition that the kind of rationality applied in choosing a means cannot be used in choosing an end. These two kinds of reasoning represent categorically distinct modes of rationality, a boundary further reinforced by modern value fragmentation. With no objectively ascertainable ground of choice provided, then, a free agent has to create purpose ex nihilo: "ultimately life as a whole, if it is not to be permitted to run on as an event in nature but is instead to be consciously guided, is a series of ultimate decisions through which the soul - as in Plato - chooses its own fate" [Weber 1917/1949, 18]. This ultimate decision and the Kantian integrity between intention and action constitute the essence of what Weber calls an ethic of conviction. It is often held that the gulf between these two types of ethics is unbridgeable for Weber. Demanding an unmitigated integrity between one's ultimate value and political action, that is to say, the deontological ethic of conviction cannot be reconciled with that of responsibility which is consequentialist in essence. In fact, Weber himself admitted the "abysmal contrast" that separates the two. This frank admission, nevertheless, cannot be taken to mean that he privileged the latter over the former as far as political education is concerned. Weber keenly recognized the deep tension between consequentialism and deontology, but he still insisted that they should be forcefully brought together. The former recognition only lends urgency to the latter agenda. Resolving this analytical inconsistency in terms of certain "ethical decrees" did not interest Weber. Instead, he sought for a moral character that can manage this "combination" with a sheer force of will. In fact, he also called this synthetic ethic as that of responsibility without clearly distinguishing it from the merely consequentialist ethic it sought to overcome, thus creating an interpretive debate that continues to this day [de Villiers 2018, 47-78]. Be that as it may, his advocacy for this willful synthesis is incontrovertible, and he called such an ethical character a "politician with a sense of vocation" (Berufspolitiker) who combines a passionate conviction in supra-mundane ideals that politics has to serve and a sober rational calculation of its realizability in this mundane world. Weber thus concluded: "the ethic of conviction and the ethic of responsibility are not absolute opposites. They are complementary to one another, and only in combination do they produce the true human being who is capable of having a 'vocation for politics'" [Weber 1919/1994, 368]. This synthetic political virtue seems not only hard to achieve, but also without a promise of felicitous ending. Weber's synthesis demands a sober confrontation with the reality of politics, i.e., the ever-presence of "physical forces" and all the unintended consequences or collateral damages that come with the use of coercion. Only then may it be brought under ethical control by a superhuman deployment of passion and prudence, but, even so, Weber's political superhuman (Ubermensch) cannot circumvent the so-called "dirty-hands dilemma" [cf. Walzer 1973; Coady 2009]. For, even at the moment of triumph, the unrelenting grip of responsibility would never let him or her disavow the guilt and remorse for having employed the "physical forces," no matter how beneficial or necessary. It is a tragic-heroic ethic of "nevertheless (dennoch)" [see 2.2] and, as such, Weber's "tragicism" goes beyond politics [Honigsheim 2013, 115]. Science as a Vocation is a self-evident case in which the virtue of "value freedom" demands a scientist to confront the modern epistemological predicament of incommensurable value-fragmentation without succumbing to the nihilistic plethora of subjective values by means of a disciplined and willful devotion to the scholarly specialization and scientific objectivity [see 5.2]. From this ethical vintage point, The Protestant Ethic and the Spirit of Capitalism may as well be re-titled Labour as a Vocation. It was in this much earlier work (1904-5) that Weber first outlined the basic contours of the ethic of vocation (Berufsethik) and a person of vocation (Berufsmensch) and the way those work practices emerged historically in the course of the Reformation (and faded away subsequently). The Calvinist doctrine of predestination has amplified the innermost anxiety over one's own salvation, but such a subjective fear and trembling was channeled into a psychological reservoir for the most disciplined and methodical life conduct (Lenbensfuhrung), or labour in calling, that created the "spirit" of capitalism. Paradoxically combining subjective value commitments and objective rationality in the pursuit of those goals, in short, the making and unmaking of the Berufsmensch is where Weber's ethical preoccupations in politics, science, and economy converge [cf. Hennis 1988]. In the end, Weber's project is not about formal analysis of moral maxims, nor is it about substantive virtues that reflect some kind of ontic telos. It is too formal or empty to be an Aristotelean virtue ethics, and it is too concerned with moral character to be a Kantian deontology narrowly understood. The goal of Weber's ethical project, rather, aims at cultivating a character who can willfully bring together these conflicting formal virtues to create what he calls a "total personality" (Gesamtpersonlichkeit). It culminates in an ethical characterology or philosophical anthropology in which passion and reason are properly ordered by sheer force of individual will. As such, Weber's political virtue resides not simply in a subjective intensity of value commitment nor in a detached intellectual integrity and methodical purposefulness, but in their willful combination in a unified soul. In this abiding preoccupation with statecraft-cum-soulcraft, Weber was a moralist and political educator who squarely belonged to the venerable tradition that stretches back to the ancient Greeks down to Rousseau, Hegel, and Mill. ## 7. Concluding Remarks Seen this way, we find a remarkable consistency in Weber's thought. Weber's main problematic turned on the question of individual autonomy and freedom in an increasingly rationalized society. His dystopian and pessimistic assessment of rationalization drove him to search for solutions through politics and science, which broadly converge on a certain practice of the self. What he called the "person of vocation," first outlined famously in The Protestant Ethic, provided a bedrock for his various efforts to resuscitate a character who can willfully combine unflinching conviction and methodical rationality even in a society besieged by bureaucratic petrification and value fragmentation. It is also in this entrenched preoccupation with an ethical characterology under modern circumstances that we find the source of his enduring influences on twentieth-century political and social thought. On the left, Weber's articulation of the tension between modernity and modernization found resounding echoes in the "Dialectics of Enlightenment" thesis by Theodor Adorno and Max Horkheimer; Lukacs's own critique of the perversion of capitalist reason owes no less to Weber's problematization of instrumental rationality on which is also built Habermas's elaboration of communicative rationality as an alternative. Different elements in Weber's political thought, e.g., intense political struggle as an antidote to modern bureaucratic petrification, leadership democracy and plebiscitary presidency, stark realist outlook on democracy and power-politics, and value-freedom and value-relativism in political ethics, were selected and critically appropriated by such diverse thinkers on the right as Carl Schmitt, Joseph Schumpeter, Leo Strauss, Hans Morgenthau, and Raymond Aron. Even the postmodernist project of deconstructing the Enlightenment subjectivity finds, as Michel Foucault does, a precursor in Weber. All in all, across the vastly different ideological and methodological spectrum, Max Weber's thought will continue to be a deep reservoir of fresh inspiration as long as an individual's fate under (post)modern circumstances does not lose its privileged place in the political, social, cultural, and philosophical reflections of our time.
simone-weil	## 1. Philosophical Development Simone Weil was born in Paris on 3 February 1909. Her parents, both of whom came from Jewish families, provided her with an assimilated, secular, bourgeois French childhood that was cultured and comfortable. Weil and her older brother Andre--himself a math prodigy, founder of the Bourbaki group, and a distinguished mathematician at the Princeton Institute for Advanced Study--studied at prestigious Parisian schools. Weil's first philosophy teacher, at the Lycee Victor-Duruy, was Rene Le Senne; it was he who introduced her to the thesis--which she would maintain--that contradiction is a theoretical obstacle generative of nuanced, alert thinking. Beginning in October 1925, Weil studied at Henri IV Lycee in preparation for the entrance exams of the Ecole Normale Superieure. At Henri IV she studied under the philosopher and essayist Emile-Auguste Chartier (known pseudonymously as Alain), whose teacher was Jules Lagneau. Like Weil at this time, Alain was agnostic. In his classes he emphasized intellectual history: in philosophy this included Plato, Marcus Aurelius, Descartes, Spinoza, and Kant, and in literature, Homer, Aeschylus, Sophocles, and Euripides. Already sympathetic with the downtrodden and critical of French society, she gained the theoretical tools to levy critiques against her country and philosophical tradition in Alain's class. There, employing paradox and attention through the form of the essay (it is important to note that none of her writings was published as a book in her lifetime), she began intentionally developing what would become her distinct mode of philosophizing. It is therefore arguable that she is part of the Alain/Lagneau line of voluntarist, spiritueliste philosophy in France. In 1928 Weil began her studies at the Ecole Normale. She was the only woman in her class, the first woman having been first admitted in 1917. In 1929-1930 she worked on her dissertation on knowledge and perception in Descartes, and having received her agregation diploma, she served from late 1931 to mid-1934 as a teacher at lycees. Throughout this period, outside of her duties at each lycee where she instructed professionally, Weil taught philosophy to, lobbied for, and wrote on behalf of workers' groups; at times, moreover, she herself joined in manual labor. In her early thinking she prized at once the first-person perspective and radical skepticism of Descartes, the class-based solidarity and materialist analysis of Marx, and the moral absolutism and respect for the individual of Kant. Drawing from each, her early work can be read as an attempt to provide, with a view toward liberty, her own analysis of the fundamental causes of oppression in society. In early August 1932, Weil travelled to Germany in order to understand better the conditions fostering Nazism. German trade unions, she wrote to friends upon her return to France, were the single force in Germany able to generate a revolution, but they were fully reformist. Long periods of unemployment left many Germans without energy or esteem. At best, she observed frankly, they could serve as a kind of dead weight in a revolution. More specifically, by early 1933 she criticized the tendency of social organizations to engender bureaucracy, which elevated management and collective thinking over and against the individual worker. Against this tendency, she advocated for workers' understanding the physical labor they performed within the context of the whole organizational apparatus. In "Reflections Concerning the Causes of Liberty and Social Oppression" (1934), Weil presented both a summation of her early thought and a prefiguring of central elements in her thematic trajectory. The essay employs a Marxian method of analysis that pays attention to the oppressed, critiques her own position as an intellectual, privileges manual labor, and demands precise and unorthodox individual thinking that unites theory and practice against collective cliches, propaganda, obfuscation, and hyper-specialization. These ideas would provide a theoretical framework for her idiosyncratic practice of philosophy. Near the end of her life she wrote in a notebook: "Philosophy (including problems of cognition, etc.) is exclusively an affair of action and practice" (FLN 362). On 20 June, 1934, Weil applied for a sabbatical from teaching. She was to spend a year working in Parisian factories as part of its most oppressed group, unskilled female laborers. Weil's "year of factory work" (which amounted, in actuality, to around 24 weeks of laboring) was not only important in the development of her political philosophy but can also be seen as a turning point in her slow religious evolution. In Paris's factories, Weil began to see and to comprehend firsthand the normalization of brutality in modern industry. There, she wrote in her "Factory Journal", "[t]ime was an intolerable burden" (FW 225) as modern factory work comprised two elements principally: orders from superiors and, relatedly, increased speeds of production. While the factory managers continued to demand more, both fatigue and thinking (itself less likely under such conditions) slowed work. As a result, Weil felt dehumanized. Phenomenologically, her factory experience was less one of physical suffering per se, and more one of humiliation. Weil was surprised that this humiliation produced not rebellion but rather fatigue and docility. She described her experience in factories as a kind of "slavery". On a trip to Portugal in August 1935, upon watching a procession to honor the patron saint of fishing villagers, she had her first major contact with Christianity and wrote that > > > the conviction was suddenly borne in upon [her] that Christianity is > pre-eminently the religion of slaves, that slaves cannot help > belonging to it, and [she] among others. (1942 "Spiritual > Autobiography" in WFG 21-38, 26) > > > In comparison with her pre-factory "Testament", we see that in her "Factory Journal" Weil maintains the language of liberty, but she moves terminologically from "oppression" to "humiliation" and "affliction". Thus her conception and description of suffering thickened and became more personal at this time. Weil participated in the 1936 Paris factory occupations and, moreover, planned on returning to factory work. Her trajectory shifted, however, with the advent of the Spanish Civil War. On the level of geopolitics, she was critical of both civil and international war, and she approved of France's decision not to intervene on the Republican side. On the level of individual commitment, however, she obtained journalist's credentials and joined an international anarchist brigade. On 20 August, 1936, Weil, clumsy and nearsighted, stepped in a pot of boiling oil, severely burning her lower left leg and instep. Only her parents could persuade her not to return to combat. By late 1936 Weil wrote against French colonization of Indochina, and by early 1937 she argued against French claims to Morocco and Tunisia. In April 1937 she travelled to Italy. Within the basilica Santa Maria degli Angeli, inside the small twelfth-century Romanesque chapel where St. Francis prayed, Weil had her second significant contact with Christianity. As she would later describe in a letter, "[S]omething stronger than I compelled me for the first time to go down on my knees" (WFG 26). From 1937-1938 Weil revisited her Marxian commitments, arguing that there is a central contradiction in Marx's thought: although she adhered to his method of analysis and demonstration that the modern state is inherently oppressive--being that it is composed of the army, police, and bureaucracy--she continued to reject any positing of revolution as immanent or determined. Indeed, in Weil's middle period, Marx's confidence in history seemed to her a worse ground for judgment than Machiavelli's emphasis on contingency. During the week of Easter 1938, Weil visited the Benedictine abbey of Solesmes from Palm Sunday to the following Tuesday. At Solesmes she had her third contact with Christianity: suffering from headaches, Weil found a joy so pure in Gregorian chant that, by analogy, she gained an understanding > > > of the possibility of living divine love in the midst of affliction. > It goes without saying that in the course of these services the > thought of the passion of Christ entered into [her] being once and for > all. (WFG 26) > > > At Solesmes, she was also introduced to the seventeenth century poet George Herbert by a young Englishman she met there. She claimed to feel Christ's presence while reciting Herbert's poem "Love". As she fixed her full attention on the poem while suffering from her most intense headache, Weil came to see that her recitation had the virtue of prayer, saying, "Christ himself came down and took possession of me" (WFG 27). Importantly, she thought God "in his mercy" had prevented her from reading the mystics until that point; therefore, she could not say that she invented her unexpected contact with Christ (WFG 27). These events and writings in 1936-1938 exemplify the mutually informing nature of solidarity and spirituality in Weil's thought that began in August 1935. After the military alliance of Germany and Italy in May 1939, Weil renounced her pacifism. It was not that she felt she was wrong in holding such a position before, but rather that now, she argued, France was no longer strong enough to remain generous or merely defensive. Following the German Western offensive, she left Paris with her family in June 1940, on the last train. They settled eventually but temporarily in Marseilles, at the time the main gathering point for those attempting to flee France, and where Weil would work with the Resistance. In Vichy France Weil took up a practice she had long sought, namely, to apprentice herself to the life of agricultural laborers. In addition, in Marseilles she was introduced to the Dominican priest Joseph-Marie Perrin, who became a close friend as well as a spiritual interlocutor, and through whom she began to consider the question of baptism. In an effort to help Weil find a job as an agricultural laborer, Perrin turned to his friend Gustave Thibon, a Catholic writer who owned a farm in the Ardeche region. Thus in Fall 1941 Weil worked in the grape harvest. Importantly, however, she was not treated like the rest of the laborers; although she worked a full eight hours per day, she resided and ate at the house of her employers. She reportedly carried Plato's Symposium with her in the vineyards and attempted to teach the text to her fellow laborers. In 1942 Weil agreed to leave France in part so her parents would be in a safe place (they would not leave without her, she knew), but principally because she thought she might be more useful for France's war effort if she were in another country. Thus she went to New York via Morocco. In New York, as in Marseilles, she filled notebook after notebook with philosophical, theological, and mathematical considerations. New York, however, felt removed from the sufferings of her native France; the Free French movement in London felt one step closer to returning to France. In 1943 Weil was given a small office at 19 Hill Street in London. From this room she would write day and night for the next four months, sleeping around three hours each night. Her output in this period totaled around 800 printed pages, but she resigned from the Free French movement in late July (Petrement 1973 [1976: xx]). Weil died Tuesday, 24 August 1943. Three days later, the coroner pronounced her death a suicide--cardiac failure from self-starvation and tuberculosis. The accounts provided by her biographers tell a more complex story: Weil was aware that her fellow country-men and women in the occupied territory had to live on minimal food rations at this time, and she had insisted on the same for herself, which exacerbated her physical illness to the point of death (Von der Ruhr 2006: 18). On 30 August she was buried at Ashford's New Cemetery between the Jewish and the Catholic sections. Her grave was originally anonymous. For fifteen years Ashford residents thought it was a pauper's. ## 2. Social-Political Philosophy Always writing from the left, Weil continually revised her social-political philosophy in light of the rapidly changing material conditions in which she lived. However, she was consistent in her acute attention to and theorizing from the situation of the oppressed and marginalized in society. Her early essays on politics, a number of which were posthumously collected by Albert Camus in Oppression et liberte (OL), include "Capital and the Worker" (1932), "Prospects: Are We Heading for the Proletarian Revolution?" (1933), "Reflections concerning Technology, National Socialism, the U.S.S.R., and certain other matters" (1933) and, most importantly, "Reflections concerning the Causes of Liberty and Social Oppression" (1934 in OL 36-117). In her early writings, Weil attempted to provide an analysis of the real causes of oppression so as to inform militants in revolutionary action. Her concern was that, without this analysis, a socially enticing movement would lead only to superficial changes in the appearance of the means of production, not to new and freer forms of structural organization. The division of labor or the existence of material privilege alone is not a sufficient condition for Weil's concept of "oppression". That is, she recognized that there are some forms of social relations involving deference, hierarchy, and order that are not necessarily oppressive. However, the intervention of the struggle for power--which she, following Hobbes, sees as an inexorable feature of human society--generates oppression. > > > [P]ower [puissance] contains a sort of fatality which weighs > as pitilessly on those who command as on those who obey; nay more, it > is in so far as it enslaves the former that, through their agency, it > presses down upon the latter ("Reflections concerning the Causes > of Liberty and Social Oppression", OL 62). > > > Oppression, then, is a specific social organization that, as a consequence of the essentially unstable struggle for power, and principally the structure of labor, limits the individual from experiencing the world to the full extent of her or his full capability, a capability Weil describes as "methodical thinking" (pensee methodique). By divorcing the understanding from the application of a method, oppression, exercised through force, denies human beings direct contact with reality. Also informing her sense of oppression is her notion of "privilege", which includes not only money or arms, but also a corpus of knowledge closed to the working masses that, thereby, engenders a culture of specialists. Privilege thus exacerbates oppression in modern societies, which are held together not by shared goals, meaningful relations, or organically developing communities, but through a "religion of power" (OL 69). In this way, both power and prestige contribute to a modern reversal of means and ends. That is, elements like money, technology, and war--all properly speaking "means"--are, through the workings of power, treated as "ends" worthy of furtherance, enhancement, and multiplication without limit. In her early social-political thought Weil testified to what she called "a new species of oppression" ("Prospects", OL 9), namely, the bureaucracy of modern industry. Both her anarchism and Marxian method of analysis influenced how she problematized revolutionary struggle: the problem lies in forming a social organization that does not engender bureaucracy, an anonymous and institutionalized manifestation of force. The oppressiveness of modern bureaucracy, to which she responded through analysis and critique, includes cliched, official and obscurantist language--the "caste privileges" of intellectuals ("Technology, National Socialism, the U.S.S.R", OL 34)-- and a division of labor such that, in the case of most workers, labor does not involve, but in fact precludes, engaged thinking. Out of balance in this way, modern humans live as cogs in a machine: less like thinking individuals, and more like instruments crushed by "collectivities". Collectivity--a concept that centrally comprises industry, bureaucracy, and the state (the "bureaucratic organization par excellence" [OL 109]), but that also includes political parties, churches, and unions--by definition quashes individual subjectivity. Despite her critiques of oppression, prestige, and collectivity, in a polemical argument Weil is also critical of "revolution", which for her refers to an inversion of forces, the victory of the weak over the powerful. This, she says, is "the equivalent of a balance whose lighter scale were to go down" (OL 74). "Revolution" in its colloquial or deterministic sense, then, has itself become an "opiate", a word for which the laboring masses die, but which lies empty. For Weil, real revolution is precisely the re-organization of labor such that it subordinates the laborers neither to management (as in bureaucracy) nor to oppressive conditions (as in factory work). As she sees it, if revolution is to have meaning, it is only as a regulative, and not a positive, ideal (OL 53). In Weil's adaptation of this Kantian notion, such a regulative ideal involves an attention to reality, e.g., to the present political and working conditions, to human conditions such as the struggle for power. Only in this way can it provide a standard of analysis for action: the ideal allows for a dialectical relation between a revolutionary alternative and present praxis, which is always grounded in material conditions. Freedom, then, is a unity of thought and action. It is, moreover--and not unlike in Hegel's conceptualization--a condition of balanced relations to and interdependence with others (who check our sovereignty) and the world (which is limited and limiting). A freeing mode of production, as opposed to an alienating and oppressing one, would involve a meaningful relation to thinking and to others throughout the course of labor. When workers understand both the mechanical procedures and the efforts of other members of the collectivity, the collectivity itself becomes subject to individuals, i.e., means and ends are rightfully in a relation of equilibrium. The ontology behind Weil's notion of free labor draws on the Kantian emphasis on the individual as an end--especially, for Weil, in her or his capacity for thinking (with Plato, for whom knowing and doing are united). In addition, the teleology of labor in Weil's philosophy corresponds with the thought of Hegel and Marx, and is in opposition to Locke: in the individual's mixing with the material world, her interest lies in how this promotes liberty, rather than property. The uprooting Weil experienced in factory work introduced a shift in her conception of freedom: as she began to see the human condition as not just one of inexorable struggle, but of slavery, her notion of freedom shifted from a negative freedom from constraints to a positive freedom to obey. She referred to the latter, a particular kind of relational freedom, as "consent". She concluded her "inventory of modern civilization" (OL 116) with a call to introduce play into the bureaucratic machine and a call to think as individuals, denying the "social idol" by "refusing to subordinate one's own destiny to the course of history" and by taking up "critical analysis" (OL 117). In sum, by 1934 Weil pictured an ideal society, which she conceptualized as a regulative impossibility in which manual labor, understood and performed by thinking individuals, was a "pivot" toward liberty. Weil's "Factory Journal" from 1934-1935 suggests a broader shift in her social-political philosophy. During this period, her political pessimism deepened. In light of the humiliating work she conducted in the factories, her post-factory writings feature a terminological intensification: from "humiliation" and "oppression" to "affliction" (malheur), a concept informed by her factory experience of embodied pain combined with psychological agony and social degradation--and to which she would later add spiritual distress. In 1936 Weil advanced her political commitments in ways that foreshadowed her later social-political thought. She continued to eschew revolution and instead to work toward reform, namely, greater equality in the factory through a shift from a structure of subordination to one of collaboration. In addition, in response to the outbreak of the Spanish Civil War, she argued against fascism while favoring--against the Communist position--French non-intervention. She was opposed to authoritarian logic, and she saw going to war as a kind of surrender to the logic of power and prestige inherent in fascism. > > > One must choose between prestige and peace. And whether one claims to > believe in the fatherland, democracy, or revolution, the policy of > prestige means war. (1936 "Do We Have to Grease Our Combat > Boots", FW 258) > > > Weil's development of the concepts of power and prestige continued in her article "Let Us Not Start Another Trojan War" (1936, subtitled, and translated into English as, "The Power of Words" in SWA 238-258). Her thesis is that a war's destruction is inversely proportional to the official pretexts given for fighting it. Wars are absurd, she argued (against Clausewitz), because "they are conflicts with no definable objective" (SWA 239). Like the phantom of Helen that inspired ten years of fighting, ideologies (e.g., capitalism, socialism, fascism), as well as capitalized words such as "Nation" and "State", have taken on the role of the phantom in the modern world. Power relies on prestige--itself illusory and without limit because no nation thinks it has enough or is sure of maintaining its imagined glory, and therefore ever increases its means to wage war--so as to appear absolute and invincible. In response to these forces, Weil prescribed distinguishing between the imaginary and the real and, relatedly, defining words properly and precisely. Taken together, these prescriptions amount to a critique of ideology with its bloated political rhetoric. > > > [W]hen a word is properly defined it loses its capital letter and can > no longer serve either as a banner or as a hostile slogan; it becomes > simply a sign, helping us to grasp some concrete reality or concrete > objective, or method of activity. To clarify thought, to discredit the > intrinsically meaningless words, and to define the use of others by > precise analysis--to do this, strange though it may appear, might > be a way of saving human lives. (SWA 242) > > > Further, Weil desired a kind of equilibrium between forces instead of an endless pursuit of an illusion of absolute stability and security. Following Heraclitus, she saw struggle as a condition of life. What is required of the thinking individual, in turn, is to distinguish between worthwhile conflict, such as class struggle, and illusions of prestige, which often serve as the foundation of war. In her middle period, then, Weil maintained the contrariety between reality, limit, and equilibrium on the one hand and imagination, limitlessness, and collectivity on the other. In her 1937 "Note on Social Democracy", she defined politics as follows: > > > The material of the political art is the double perspective, ever > shifting between the real conditions of social equilibrium and the > movements of collective imagination. Collective imagination, whether > of mass meetings or of meetings in evening dress, is never correctly > related to the really decisive factors of a given social situation; it > is always beside the point, or ahead of it, or behind it. (SE 152) > > > In her late period Weil provided an explication of the all-pervasive and indiscriminate concept of "force" in the essay "The Iliad or the Poem of Force" (1940 in SWA 182-215). It is important to note Weil's locus of enunciation for this concept, temporally after the fall of France and spatially from a position of exile, including antisemitic marginalization, which is why the essay appeared in the December 1940 and January 1941 Cahiers du Sud under the anagrammatic pseudonym Emile Novis, taken up by Weil to avoid antisemitic confrontation and censorship. This essay was not only the most widely read of Weil's pieces during her lifetime, but it was also her first essay to appear in English, translated by Mary McCarthy and published in Dwight Macdonald's magazine Politics in 1945 (November, pp. 321-330). In her essay on the Iliad--a text Roberto Esposito calls "a phenomenology of force" (Esposito 1996 [2017: 46])--Weil further develops an understanding of force initially presented in her earlier, unfinished essay "Reflections on Barbarism" (1939): "I believe that the concept of force must be made central in any attempt to think clearly about human relations" (quoted in Petrement 1973 [1976: 361]). The protagonist of the Iliad, Weil writes in an original reading, is not Achilles or Hector, but force itself. Like her concept of "power" in her early writings, "force" reifies and dehumanizes no matter if one wields or undergoes it. Further, force includes not only coercion, but also prestige, which is to say that it has a social element. Two important implications follow. First, on the level of the individual, each "personality" (la personne) is informed by social values and thereby features the operations of force through accidental characteristics, such as the name of the family into which one is born or the embodied features that are considered physically attractive at a certain place, in a certain society, at a certain time. Second, on the level of the collectivity, force can destroy not only bodies, but also values and cultures, as is the case, Weil is at pains to point out, in French colonialism. In her mid-to-late writings, then, she saw neither Marx's notion of class nor Hegel's self-development of Geist, but force itself, as the key to history. She presents this concept as "the force that kills" and as a specific kind of violence "that does not kill just yet", though > > > [i]t will surely kill, it will possibly kill, or perhaps it merely > hangs, poised and ready, over the head of the creature it can > kill, at any moment, which is to say at every moment. (SWA > 184-185) > > > Weil's concept of force, then, is also a development from Hobbes and Hegel: it names that which renders the individual a slave. It was in London (1942-1943), during her work for the Free French, that Weil articulated her most robust late social-political philosophy. Her concepts of "labor" and "justice" thickened as she moved further toward Christianity and--against her early emphasis on the individual--toward the social. Weil's pieces d'occasion from this time period include "Draft for a Statement of Human Obligations" (1943), written in response to de Gaulle's State Reform Commission in its drafting of a new Declaration of the Rights of Man and Citizen, as well as "Note on the General Suppression of Political Parties" (1943), written because the Free French was considering recognizing political parties. In this piece Weil argues for the complete abolition of political parties. Drawing on Rousseau's concept of the general will, Weil contends that political parties subdue the independent, individual wills of which the general will is derivative and on which democracy depends. Most important from this period was her major work The Need for Roots (L'enracinement), which Weil called her second "magnum opus" (SL 186), and which Albert Camus published posthumously in 1949, with Gallimard, as the first of 11 volumes of Weil's he would promote. Weil wrote additional essays in London, the most conceptually important of which was "La Personne et le sacre" (1942-1943, translated into English as "Human Personality"), in which she critiques "rights" as reliant on force and poses as counter-terms "obligation" and "justice". She distinguishes between two conceptions of justice: natural (hence social and contingent) and supernatural (hence impersonal and eternal). The Need for Roots (1943 NR) adds "needs of the soul" as another counter-balance to rights. Overall, Weil presents not a law-based or rights-based, but a compassion-based morality, involving obligations to another that are discernible through attention, centered on and evolving toward a supernatural justice that is not of the world, but that can be in it. As a departure from Kant, and through Plato, whom Weil "came to feel... was a mystic" ("Spiritual Autobiography", WFG 28), the basis of her sense of (supernatural) justice was not human rationality, but a desire for the Good, which she believed all humans share, even if at times they forget or deny this. Importantly, given her critique of French colonialism, and despite her claim that obligations to another must be indiscriminate, i.e., universal, she did not want to universalize law in a Kantian fashion. Rather, her aim was for cultures to continue their own traditions, for the goal of rootedness (l'enracinement) is not to change cultural values per se, but more precisely, to change how individuals in those cultures read and orient themselves toward those values. Weil's concept of "roots" is crucial to her late political thought. With connotations of both vitality and vulnerability, "roots" conceptualizes human society as dynamic and living while attesting to the necessity of stability and security if growth and flourishing are to occur on the level of the individual. Beyond the organic metaphor on the level of the natural, her concept of roots serves as a kind of bridge between the reality of society and the ideal of supernatural justice. Roots do this by manifesting human subjection to material and historical conditions, including the need to participate in the life of a community, to feel a sense of connection to a place, and to maintain temporal links, e.g., to cultural history and to hopes for the future. In turn, a rooted community allows for the development of the individual with a view toward God or eternal values. As such, and contra her early and middle maintenance of a critical distance from any collectivity (or Great Beast, to use the Platonic metaphor she frequently employs), in her late thought Weil sees roots as allowing for a "new patriotism" based on compassion. The establishing of roots (l'enracinement) enables multiple relations to the world (e.g., on the level of the nation, the organically developing community, the school) that at once nourish the individual and the community. In addition to war and colonization, in The Need for Roots Weil points to money and to contemporary education as self-perpetuating forces that uproot human life. The longest section of the text describes modern uprootedness (deracinement), occurring when the imagined modern nation and money are the only binding forces in society, and she characterizes this condition as a threat to the human soul. Modern education is corrupted both by capitalism--such that it is nothing except "a machine for producing diplomas, in other words, jobs" (NR 118)--and by a Roman inheritance for cultivating prestige with respect to the nation: "It is this idolatry of self which they have bequeathed to us in the form of patriotism" (NR 137). As an alternative to modern uprootedness, Weil outlines a civilization based not on force, which turns a person into a thing, but on free labor, which in its engagement with and consent to necessary forces at play in the world, including time and death, allows for direct contact with reality. Moreover, Weil's writing in The Need for Roots is refracted through her religious experience. In this later period, then, she no longer conceptualizes labor in the mechanical terms of a "pivot" as she did in her early writings; it is now the "spiritual core" of "a well-ordered social life" (NR 295). (In her late social-political philosophy, in fact, a spiritual revolution is more important than an economic one.) Contra the Greeks, who devalued physical work, her conception of labor serves to mediate between the natural and the supernatural. In her late writings, labor is fully inflected by her experiences with Christianity. As such, labor's consent to and, moreover, working through natural forces (e.g., gravity) is in fact consent to God, who created the natural world. Labor's kenotic activity, as energy is expended daily, is a kind of imitatio Christi. Her social-political writings in London are markedly different from her early writings as an anarchist informed principally by Descartes, Marx, and Kant. While those influences remain, her later writings must be read through the lens of her Christian Platonism. She suggests that we must draw our spiritual life from our social environment. That is, while spirituality is individual vis-a-vis God, this spirituality occurs within a social context, namely, the collectivity, and principally, the nation. This is a reversal from the critical distance she had maintained from the collectivity, especially in her early emphasis on the individual's methodical thinking. ## 3. Epistemology Throughout her life, Weil argued that knowledge in and of the world demanded rigorous, balanced thinking, even if that difficulty and measurement led the thinker to near-impossible tasks. For her these tasks of thinking included, for instance, her attempt to synthesize perspectives of intricate Catholic doctrine on the threshold of the Church with wisdom from various traditions such as ancient Greek philosophy and tragedies, Hinduism, Buddhism, and Taoism. Following Aeschylus, she believed knowledge was gained through suffering. Shaped by her social-political and religious thought, Weil's epistemology would change over time, especially in light of her mystical experiences. In her dissertation, Weil attempts to think with Descartes in order to find foundational knowledge. Like Descartes, she argues for the existence of self, God, and world. Her cogito, however, was a decisive break from his: "I can, therefore I am" (Je puis, donc je suis) (1929-1930 "Science and Perception in Descartes", FW 59). The self has the power of freedom, Weil argues, but something else--the omnipotent, God--makes the self realize that it is not all-powerful. Self-knowledge, then, is capability always qualified by the acknowledgment that one is not God. She maintained a kind of Cartesian epistemology during her time as a teacher, early in her academic life. In Lectures on Philosophy (1978 LP), a collection of lecture notes taken by one of her students during the academic year 1933-1934 at a girl's lycee in Roanne, we find that Weil is initially critical of sensations as grounds for knowledge, and thereby critical of empiricist epistemology (LP 43-47). From her early writings onward she was to problematize the imagination as the "folle imagination", a barrier between mind and reality, meaning that the human knower is kept from things-in-themselves. Weil's epistemology, then, informed by her initial studies of Descartes, would take on inflections of Kant and Plato while positioning her against Aristotle. She was critical of any sensation that universalizes one's reading of the world, and she saw the imagination as thus extending the self, because it could not help but filter phenomena through its own categories, wishes, and desires, thereby reading the world on its own terms. Relatedly, Weil would develop an intersubjective epistemology. Knowing the truth requires not extending one's own limited perspective, but suspending or abandoning it such that reality--including the reality of the existence of others--could appear on its own terms. This suspension involves a practice of epistemic humility and openness to all ideas; intelligence for Weil demands the qualified use of language, acknowledgement of degrees, proportions, contingencies, and relations, as well as an ability to call the self into question. These epistemic practices are part of a broader recognition that the individual knower is limited. The progression of Weil's epistemology can be seen first in her conceptualization of contradiction. By Lectures on Philosophy, she affirmed a sense of contradiction beyond the logical conjunction "a is b" and "a is not-b". Indeed, she argues that contradiction can be a generative obstacle in that it requires the mind to expand its thinking in order to transcend the obstacle. Drawing on the mathematics of Eudoxus, she elaborates on this notion to claim that incommensurables can be reconciled when set on a kind of "higher plane". This is not, however, a type of Hegelian synthesis that can be intellectually apprehended. Instead, the contemplation of contradictions can lead the knower to a higher contemplation of truth-as-mystery. Thus, in her late epistemology Weil presents this concept of "mystery" as a certain kind of contradiction in which incommensurables appear linked in an incomprehensible unity. Mystery, as a conceptualization of contradiction, carries theological, or at least supernatural, implications. For instance, if contradiction is understood through formal logic, then the existence of affliction would seemingly prove the nonexistence of an omnipotent, wholly benevolent God; however, contradiction, understood as mystery, can itself serve as a mediation, allowing for the coexistence of affliction and God, seen most prominently on the Cross. Indeed, Christ is the religious solution to Weil's principal contradiction, that between the necessary and the good. Moreover, through incarnation that opens onto universality, Christ also manifests and solves the contradiction of the individual and the collectivity. Thus, in a modification of the Pythagorean idea of harmony, she claims that Christ allows for "the just balance of contraries" ("Spiritual Autobiography", WFG 33). More broadly, in her 1941 "The Pythagorean Doctrine", she argues that mathematics are a bridge between the natural and eternal (or between humans and God). That is, the Pythagoreans held an intellectual solution to apparent natural contradictions. Inspired by Pythagoras, she claimed that the very study of mathematics can be a means of purification in light of the principles of proportion and the necessary balancing of contraries--especially in geometry. She saw in the Pythagorean legacy and spirit a link between their mathematical insights and their distinctly religious project to penetrate the mysteries of the cosmos. As opposed to the suppression or dissolution of contradictions, as in systematic philosophy, in Weil's value-centered philosophy contradictions are to be presented honestly and tested on different levels; for her, they are "the criterion of the real" and correspond with the orientation of detachment (GG 98). For example, Christ's imperative to "love your enemies" contains a contradiction in value: love those who are detestable and who threaten the vulnerability of loving. For Weil, submitting to this union of contraries loosens one's attachments to particular, ego-driven perspectives and enables a "well-developed intellectual pluralism" (Springsted 2010: 97). She writes, "An attachment to a particular thing can only be destroyed by an attachment which is incompatible with it" (GG 101). With these philosophical moves in mind, Robert Chenavier argues that she has not a philosophy of perception, a phenomenology, but rather, borrowing Gaston Bachelard's phrase, a "dynamology of contradiction" (Chenavier 2009 [2012: 25]). Overall, Weil's presentation of contradiction is more Pythagorean or Platonic than it is Marxian: it takes up contradiction not through resolution on the level of things, but through dialectics on the level of thought, where mystery is the beginning and end-point of thought (Springsted 2010: 97). In her unfinished essay "Is There a Marxist Doctrine?", from her time in London, Weil writes, > > > Contradiction in matter is imaged by the clash of forces coming from > different directions. Marx purely and simply attributed to social > matter this movement towards the good through contradictions, which > Plato described as being that of the thinking creature drawn upwards > by the supernatural operation of grace. (OL 180) > > > She follows the Greek usage of "dialectics" to consider "the virtue of contradiction as support for the soul drawn upwards by grace" (or the good); Marx errs, she thought, in coupling such a movement with "materialism" (OL 181). A second central epistemological concept for Weil is "reading" (lecture). Reading is a kind of interpretation of what is presented to knowers by both their physical sensations and their social conditions; therefore, reading--as the reception and attribution of certain meanings in the world--is always mediated. In turn, readings are mediated through other readings, since our perception of meaning is undoubtedly involved in and affected by an intersubjective web of interpretations. Weil explains this through the metaphor, borrowed from Descartes, of a blind man's stick. We can read a situation through attention to another in order to expand our awareness and sensitivity, just as the blind man enlarges his sensibility through the use of his stick. But our readings of the world can also become more narrow and simplistic, as for instance, when a context of violence and force tempts us to see everyone we encounter as a potential threat. Moreover, readings are not free from power dynamics and can become projects of imposition and intervention; here her epistemology connects to her social-political philosophy, specifically to her concept of force. > > > We read, but also we are read by, others. Interferences in these > readings. Forcing someone to read himself as we read him (slavery). > Forcing others to read us as we read ourselves (conquest). (NB 43) > > > She connects reading to war and imagination: "War is a way of imposing another reading of sensations, a pressure upon the imagination of others" (NB 24). In her 1941 "Essay on the Concept of Reading" (LPW 21-27) Weil elaborates, "War, politics, eloquence, art, teaching, all action on others essentially consists in changing what they read" (LPW 26). In the same essay she develops "reading" in relation to the aforementioned epistemological concepts of appearance, the empirical world, and contradiction: > > > [A]t each instant of our life we are gripped from the outside, as it > were, by meanings that we ourselves read in appearances. That is why > we can argue endlessly about the reality of the external world, since > what we call the world are the meanings that we read; they are not > real. But they seize us as if they were external; that is real. Why > should we try to resolve this contradiction when the more important > task of thought in this world is to define and contemplate insoluble > contradictions, which, as Plato said, draw us upwards? (LPW 22) > > > It is important to note that for Weil, we are not simply passive in our readings. That is, we can learn to change our readings of the world or of others. An elevated transformation in reading, however, demands an apprenticeship in loving God through the things of this world--a kind of attentiveness that will also entail certain bodily involvements, labors, postures, and experiences. Particular readings result from particular ways of living. Ideally for her, we would read the natural as illuminated by the supernatural. This conceptualization of reading involves recognition on hierarchical levels, as she explains in her notebooks: > > > To read necessity behind sensation, to read order behind necessity, to > read God behind order. We must love all facts, not for their > consequences, but because in each fact God is there present. But that > is tautological. To love all facts is nothing else than to read God in > them. (NB 267) > > > Thus the world is known as a kind of text featuring several significations on several stages, levels, or domains. Weil's epistemology grounds her critique of modern science. In The Need for Roots she advocates for a science conducted "according to methods of mathematical precision, and at the same time maintained in close relationship with religious faith" (NR 288). Through contemplation of the natural world via this kind of science, the world could be read on multiple levels. The knower, reading thus, would understand that the order of the world is the same as a unity, but different on its myriad levels: > > > with respect to God [it] is eternal Wisdom; with respect to the > universe, perfect obedience; with respect to our love, beauty; with > respect to our intelligence, balance of necessary relations; with > respect to our flesh, brute force. (NR 288-289) > > > As in her social-political thought, in her epistemology Weil is a kind of anti-modern. She sees modern science and epistemology as a project of self-expansion that forgets limit and thinks the world should be subject to human power and autonomy. Labor (especially physical labor, such as farming), then, also assumes an epistemological role for her. By heteronomously subjecting the individual to necessity on a daily basis, it at once contradicts self-aggrandizement and allows for a more balanced reading: the intelligence qualifies itself as it reads necessary relations simultaneously on multiple levels. This reading on different levels, and inflected through faith, amounts to a kind of non-reading, in that it is detached, impersonal and impartial. Through these predicates "reading" is connected to her social thought, aesthetics, and religious philosophy. That is, in her later thought Weil's epistemology relies on a time-out-of-mind metaphysics for justification. Analogous to aesthetic taste, spiritual discernment--God-given and graceful--allows the abdicated self to read from a universal perspective at its most developed stage. Thus, she argued, one can love equally and indiscriminately, just as the sun shines or the rain falls without preference. ## 4. Ethics Weil's central ethical concept is "attention" (l'attention), which, though thematically and practically present in her early writings, reached its robust theoretical expression while she was in Marseilles in 1942. Attention is a particular kind of ethical "turn" in her conceptualization. Fundamentally, it is less a moral position or specific practice and more an orientation that nevertheless requires an arduous apprenticeship leading to a capacity of discernment on multiple levels. Attention includes discerning what someone is going through in her or his suffering, the particular protest made by someone harmed, the social conditions that engender a climate for suffering, and the fact that one is, by chance (hazard) at a different moment, equally a subject of affliction. Attention is directed not by will but by a particular kind of desire without an object. It is not a "muscular effort" but a "negative effort" (WFG 61), involving release of egoistic projects and desires and a growing receptivity of the mind. For Weil, as a Christian Platonist, the desire motivating attention is oriented toward the mysterious good that "draws God down" (WFG 61). In her essay "Reflections on the Right use of School Studies with a View to the Love of God" (1942 in WFG 57-65), Weil takes prayer, defined as "the orientation of all the attention of which the soul is capable toward God", as her point of departure (WFG 57). She then describes a kind of vigilance in her definition of attention: > > > Attention consists of suspending our thought, leaving it detached > [disponible], empty [vide], and ready to be > penetrated by the object; it means holding in our minds, within the > reach of this thought, but on the lower level and not in contact with > it, the diverse knowledge we have acquired which we are forced to make > use of. ... Above all our thought should be empty > [vide], waiting [en attente], not seeking anything > [ne rien chercher], but ready to receive in its naked truth > the object that is to penetrate it. (WFG 62) > > > The French makes more clear the connection between attention (l'attention) and waiting (attente). For Weil the problem with searching, instead of waiting en hupomene (di upomones), is precisely that one is eager to fill the void characterizing attente. As a result, one settles too quickly on some-thing: a counterfeit, falsity, idol. Because in searching or willing the imagination fills the void (le vide), it is crucial that attention be characterized by suspension and detachment. Indeed, the void by definition is empty (vide)--of idols, futural self-projections, consolations that compensate un-thinking, and attachments of collective and personal prestige. As such, its acceptance marks individual fragility and destructibility, that is, mortality. But this acceptance of death is the condition for the possibility of the reception of grace. (As explained below in regard to her religious philosophy, Weil's concept for the disposition characterized by these features of attention, with obvious theological resonances, is "decreation".) In attention one renounces one's ego in order to receive the world without the interference of one's limited and consumptive perspective. This posture of self-emptying, a stripping away of the "I" (depouillement)--ultimately for Weil an imitatio Christi in its kenosis--allows for an impersonal but intersubjective ethics. Indeed, if the primary orientation of attention is toward a mysterious and unknown God (often experienced as a desire for the Good), the secondary disposition is toward another person or persons, especially toward those going through affliction. > > > The soul empties itself [se vide] of all its own contents in > order to receive into itself the being it is looking at, just as he > is, in all his truth. Only he who is capable of attention can do this. > (WFG 65) > > > Weil recognizes and problematizes the fact that the autonomous self naturally imposes itself in its projects, as opposed to disposing itself to the other; for this reason, attention is rare but is required of any ethical disposition. The exemplary story of attention for Weil is the parable of the Good Samaritan, in which, on her reading, compassion is exchanged when one individual "turns his attention toward" another individual, anonymous and afflicted (WFG 90). > > > The actions that follow are just the automatic effect of this moment > of attention. The attention is creative. But at the moment when it is > engaged it is a renunciation. This is true, at least, if it is pure. > The man accepts to be diminished by concentrating on an expenditure of > energy, which will not extend his own power but will only give > existence to a being other than himself, who will exist independently > of him. (1942 "Forms of the Implicit Love of God" in WFG > 83-142, 90) > > > As such, attention not only gives human recognition and therefore meaningful existence to another, but it also allows the individual engaged in renunciation to take up a moral stance in response to her or his desire for good. It is important to distinguish Weil's ethics of attention from canonical conceptualizations of ethics. Attention is not motivated by a duty (although Weil thinks we are obligated to respond to others' needs, whether of soul or body [see "Draft for a Statement of Human Obligations" in SWA, esp. 224-225]) or assessed by its consequences. In addition, through its sense of phronesis (practical wisdom), which Weil assumed from Aristotle through Marx, attention is arguably closer to virtue ethics than it is to deontology or consequentialism. However, it is separated from the tradition of virtue ethics in important ways: it often emerges more spontaneously than virtue, which, for Aristotle, is cultivated through habituation as it develops into a hexis; Weil's emphasis on a "negative effort" suggests an active-passive orientation that militates against Aristotle's emphasis on activity (it is more a "turning" than a "doing", more orientation than achievement); it is excessive in its generosity, as opposed to being a mean, e.g., liberality; its supernatural inspiration contrasts with Aristotle's naturalism; it does not imply a teleology of realizing one's own virtuous projects--in fact, it is a suspension of one's own projects; finally, for Weil, Aristotle lacks a sense of the impersonal good toward which attention is oriented (and in this respect she is, again, inspired by Plato). Weil treats the connections among attention, void, and love, relying on her supernatural (Christian Platonic) metaphysics, in "Forms of the Implicit Love of God". > > > To empty ourselves [Se vider] of our false divinity, to deny > ourselves, to give up being the center of the world in imagination, to > discern that all points in the world are equally centers and that the > true center is outside the world, this is to consent to the rule of > mechanical necessity in matter and of free choice at the center of > each soul. Such consent is love. The face of this love, which is > turned toward thinking persons, is the love of our neighbor. (WFG > 100) > > > Attention can be seen as love, for just as attention consents to the existence of another, love requires the recognition of a reality outside of the self, and thus de-centers the self and its particularity. Contra the colloquial sense of love, it is because we do not love personally, but "it is God in us who loves them [les malheureux]" (WFG 93-94), that our love for others is "quite impersonal" and thereby universal (WFG 130). Weil allows, however, "one legitimate exception to the duty of loving universally", namely, friendship ("Last Thoughts", WFG 51). Friendship is "a personal and human love which is pure and which enshrines an intimation and a reflection of divine love" (WFG 131), a "supernatural harmony, a union of opposites" (WFG 126). The opposites that form the miraculous harmony are necessity/subordination (i.e., drawing from Thucydides' Melian dialogue, she sees it as an impossibility for one to want to preserve autonomy in both oneself and another; in the world the stronger exerts force through will) and liberty/equality (which is maintained through the desire of each friend to preserve the consent of oneself and of the other, a consent to be "two and not one" [WFG 135]). In other words, in friendship a particular, self-founded reading of the other is not forced. Distance is maintained, and in this way Weil's concept of friendship advances her previous critiques of the ethos of capitalism, bureaucracy, and colonialism--i.e., free consent of all parties is the essential ingredient of all human relations that are not degraded or abusive. Hence friendship is a model for ethics more generally--even, contra her claim, universally. > > > Friendship has something universal about it. It consists of loving a > human being as we should like to be able to love each soul in > particular of all those who go to make up the human race. (WFG > 135) > > > Weil's ethics of attention informs her later social-political philosophy and epistemology. In relation to her social-political thought, attention suspends the centrality of the self to allow for supernatural justice, which involves simultaneously turning attention to God and to affliction. Justice and the love of the neighbor are not distinct, for her. > > > Only the absolute identification of justice and love makes the > coexistence possible of compassion and gratitude on the one hand, and > on the other, of respect for the dignity of affliction > [malheur] in the afflicted--a respect felt by the > sufferer himself [les malheureux par lui-meme] and the > others. (WFG 85) > > > Additionally, attention, in its consent to the autonomy of the other, is an antidote to force. Important to Weil's epistemology, attention militates against readings that are based on imagination, unexamined perceptions, or functions of the collectivity (e.g., prestige). Attention here manifests as independent, detached thought. In its religious valence involving obedience and consent, attention also bears on Weil's epistemology in an additional way: it suggests that knowing the reality of the world is less an individual achievement or attainment of mastery and more a gift of grace. That is, attention is openness to what cannot be predicted and to what often takes us by surprise. In this way, attention resists the natural tendency of humans to seek control and dominance over others. At stake in this ethical mode, then, is the prevention of injustices that result from projects of self-expansion, including the French colonialism Weil criticized in her time. ## 5. Metaphysical and Religious Philosophy Although the metaphysics of Weil's later thought was both Christian and Platonic, and therefore graceful and supernatural, her turn to God occurred not despite but, rather, because of her attention to reality and contact with the world. It is not the case, then, that her spiritual turn and "theological commitment" (Springsted 2015: 1-2) severed her contact with her early materialism, solidarities, or Marxian considerations; rather, her spiritual turn occurred within this context, which would ground the religious philosophy she would subsequently articulate. In her late thought Weil presents an original creation theology. God, as purely good, infinite, and eternal, withdrew (or reduced Godself) so that something else (something less than fully good, finite, and spatio-temporally determinate) could exist, namely, the universe. Implicit in this outside-of-God universe is a contingency of forces. She calls this principle of contingency, the "web of determinations" (McCullough 2014: 124) contrary to God, "necessity". Necessity is "the screen" (l'ecran) placed between God and creatures. Here her creation metaphysics echoes Plato's distinction between the necessary and the good in the Timaeus. Her Christian inflection translates this to the "supreme contradiction" between creature and creator; "it is Christ who represents the unity of these contradictories" (NB 386). Weil is clear that God and the world is less than God alone, yet that only heightens the meaning of God's abdication. That is, out of love (for what would be the world and the creatures therein), God decided to be lesser. Because existence--God's very denial of Godself--is itself a mark of God's love, providence is therefore not found in particular interventions of God, but understood by recognizing the universe in all its contingency as the sum of God's intentions. In this theology rests an implication for creatures. If humans are to imitate God, then they must also renounce their autonomy (including imagined "centeredness" in the universe) and power out of love for God and therefore the world; she calls this "decreation" (decreation), which she describes paradoxically by "passive activity" (WFG 126) and, drawing on her readings of the Bhagavad Gita, "non-active action" (NB 124). Weil articulates her religious philosophy through a series of distinctions--of oppositions or contradictions. It is important to read the distinctions she makes not as positing dualisms, but as suggesting contraries that are, unsynthesized, themselves mediations through which the soul is drawn upward. Her concept of "intermediaries"--or metaxu (metaxu), the Greek term she employs in her notebooks--becomes explicit beginning in 1939. Through metaxu God is indirectly present in the world--for example, in beauty, cultural traditions, law, and labor--all of which place us into contact with reality. In terms of her periodization, her concept of mediation moved from relations of mind and matter on the level of the natural (found in the early Weil) to a mediation relating the natural intelligence to attention and love on the level of the supernatural (at work in the late Weil), thus encompassing her early view through a universal perspective. Given that reality is itself metaxu, Weil's late concept of mediation is more universal than the aforementioned specific examples suggest. For her the real (le reel) itself is an obstacle that represents contradiction, an obstacle felt, say, in a difficult idea, in the presence of another, or in physical labor; thus thought comes into contact with necessity and must transform contradiction into correlation or mysterious and crucifying relation, resulting in spiritual edification. Her explanation of this mediation is based on her idiosyncratic cosmology, especially its paradoxical claim that what is often painful reality, as distance from God, is also, as intermediate, connection to God. She illustrates this claim with the following metaphor: > > > Two prisoners whose cells adjoin communicate with each other by > knocking on the wall. The wall is the thing which separates them but > it is also their means of communication. It is the same with us and > God. Every separation is a link. (GG 145) > > > When developing her concept of the real, Weil is especially interested in distinguishing it from the imaginary. The imagination--problematized on epistemological grounds in her early thought--is criticized once again in her religious philosophy for its insidious tendency to pose false consolations that at once invite idolatry or self-satisfaction, both of which obviate real contemplation. This is why decreation, in which the individual withdraws her or his "I" and personal perspective so as to allow the real and others to give themselves, is crucial not only spiritually, but also epistemologically and ethically. It is, additionally and more radically, why Weil suggests that atheism can be a kind of purification insofar as it negates religious consolation that fills the void. For those for whom religion amounts to an imagined "God who smiles on [them]" (GG 9), atheism represents a necessary detachment. Crucially, for her, love is pure not in the name of a personal God or its particular image, but only when it is anonymous and universal. The real leads us back to the aforementioned concept of "necessity", for reality--essentially determinate, limited, contingent, and conditional--is itself a "network of necessity" (Veto 1971 [1994: 90]), such that necessity is a reflection of reality. Moreover, like "power" in her early period and "force" in her middle period, Weil's concept of necessity includes not only the physical forces of the created world, but also the social forces of human life. Through necessity a sense of slavery remains in her thought, for humans are ineluctably subject to necessity. In this way, enslavement to forces outside our control is essentially woven into the human condition. Time, contrary to God as eternal, along with space and matter, is first of all the most basic form of necessity. In its constant reminder of distance from God, and in the experiences of enduring and waiting, time is also painful. Weil's Christian Platonism comes to light in the two most poignant metaphors she uses to refer to time, namely, the (Platonic) Cave and the (Christian) Cross. In both cases, time is the weight or the pull of necessity, through which the soul, in any effort of the self, feels vulnerable, contingent, and unavoidably subject to necessity's mediation here below (ici-bas). Time, then, is both the Cave where the self pursues its illusory goals of expansion into the future and the Cross where necessity, a sign of God's love, pins the self, suffering and mortal, to the world. Importantly, by 1942 in New York, Weil's concept of time aligns with Plato's against what she sees as Christian emphasis on progress: > > > Christianity was responsible for bringing this notion of progress, > previously unknown, into the world; and this notion, become the bane > of the modern world, has de-Christianized it. We must abandon the > notion. We must get rid of our superstition of chronology in order to > find Eternity. (LPr 29) > > > For Weil progress does not carry normative implications of improvement, for the Good is eternal and non-existent (in that it is neither spatial nor temporal); time must be consented to and suffered, not fled. As for Christ on the cross, for creatures there is redemption not from but through suffering. She thus presents a supernatural use of, not a remedy for, affliction. One form of sin, then, is an attempt to escape time, for only God is time-out-of-mind. However, time, paradoxically, can also serve as metaxu. While monotony is dreadful and fatiguing when it is in the form of a pendulum's swinging or a factory's work, it is beautiful as a reflection of eternity, in the form of a circle (which unites being and becoming) or the sound of Gregorian chant. This beauty of the world suggests, when read from the detached perspective, order behind necessity, and God behind order. A second form of necessity is "gravity" (pesanteur), as distinct from supernatural "grace". Gravity signifies the forces of the natural world that subject all created beings physically, materially, psychologically, and socially, and thus functions as a downward "pull" on the attention, away from God and the afflicted. "Grace", on the other hand, is a counter-balance, the motivation by and goodness of God. Grace pierces the world of necessity and serves to orient, harmonize, and balance, thus providing a kind of "supernatural bread" for satiating the human void. Grace, entering the empirical world, disposes one to be purified by leaving the void open, waiting for a good that is real but that could never "exist" in a material sense (i.e., as subject to time, change, force, etc.). For Weil, natural/necessary gravity (force) and supernatural/spiritual grace (justice) are the two fundamental aspects of the created world, coming together most prominently in the crucifixion. The shape of the cross itself reflects this intersection of the horizontal (necessity) and the vertical (grace). Weil's concept of necessity bears on her late conceptualization of the subject. She connects seeing oneself as central to the world to seeing oneself as exempt from necessity. From this perspective, if something were to befall oneself, then the world would cease to have importance; therefore, the assertive and willful self concludes, nothing could befall oneself. Affliction contradicts this perspective and thus forcefully de-centers the self. Unlike her existentialist contemporaries such as Sartre, Weil did not think human freedom principally through agency; for her, humans are free not ontologically as a presence-to-self but supernaturally through obedience and consent. More than obedience, consent is the unity of necessity in matter with freedom in creatures. A creature cannot not obey; the only choice for the intelligent creature is to desire or not to desire the good. To desire the good--and here her stoicism through Marcus Aurelius and Spinoza emerges in her own appeal to amor fati--is a disposition that implies a consent to necessity and a love of the order of the world, both of which mean accepting divine will. Consent, therefore, is a kind of reconciliation in her dialectic between the necessary and the good. Consent does not follow from effort or will; rather, it expresses an ontological status, namely, decreation. In supernatural compassion one loves through evil: through distance (space) and through monotony (time), attending in a void and through the abdication of God. Thus, in regard to contradiction and mediation, just as the intelligence must grapple with mystery in Weil's epistemology, so too love must be vulnerable and defenseless in the face of evil in her metaphysical and religious philosophy. ## 6. Aesthetics Weil's metaphysic sheds light on her aesthetic philosophy, which is primarily Kantian and Platonic. For Weil beauty is a snare (a la Homer) set by God, trapping the soul so that God might enter it. Necessity presents itself not only in gravity, time, and affliction, but also in beauty. The contact of impersonal good with the faculty of sense is beauty; contact of evil with the faculty of sense is ugliness and suffering--both are contact with the real, necessary, and providential. Weil is a realist in regard to aesthetics in that she uses the language of being gripped or grasped by beauty, which weaves, as it were, a link among mind, body, world, and universe. Woven through the world, but beyond relying simply on the individual's mind or senses, beauty, in this linking, lures and engenders awareness of something outside of the self. In paradoxical terms, for Weil, following Kant, the aesthetic experience can be characterized as a disinterested interestedness; against Kant, her telos of such experience is Platonic, namely, to orient the soul to the contemplation of the good. Moreover, for Weil beauty is purposive only because it is derivative of the good, i.e., the order of the world is a function of God. In a revision of Kant's line that beauty is a finality without finality, then, she sees beauty not only as a kind of feeling or presence of finality, but also as a gesture toward, inclination to, or promise of supernatural, transcendental goodness. At the same time, Weil's concept of beauty is not only informed by her Platonism and thus marked by eternity, but it is also inspired by her experiences with Christ and hence features a kind of incarnation: the ideal can become a reality in the world. As such, beauty is a testament to and manifestation of the network of inflexible necessity that is the natural world--a network that, in some sense, the intelligence can grasp. It is in this way--not as an ontological category but, rather, as a sensible experience--that through beauty necessity becomes the object of love (i.e., in Kantian terms, beauty is regulative, not constitutive). Thus beauty is metaxu (an intermediary) attracting the soul to God, and as metaxu, beauty serves as a "locus of incommensurability" (Winch 1989: 173) between the fragile contingency of time (change, becoming, death) and an eternal reality. Because beauty is the order of the world as necessity, strictly speaking, "beauty" applies principally to the world as a whole, and therefore consent to beauty must be total. Thus the love of beauty functions as an "implicit love of God" [see WFG 99]. More specifically, on the level of the particular there are, secondarily, types of beauty that nevertheless demonstrate balance, order, proportion, and thus their divine provenance (e.g., those found in nature, art, and science). One's recognition of the beauty/order of the world has implications, once again, for the subject. Because beauty is to be contemplated at a distance and not consumed through the greedy will, it trains the soul to be detached in the face of something irreducible, and in this sense it is similar to affliction. Both de-center the self and demand a posture of waiting (attente). Contemplating beauty, then, means transcending the perspective of one's own project. Because beauty, as external to self, is to be consented to, it implies both that one's reality is limited and that one does not want to change the object of her/his mode of engagement. Furthermore, beauty has an element of the impersonal coming into contact with a person. Real interaction with beauty is decreative. True to this idea, Weil's aesthetic commitments are reflected in her style: in her sharp prose she scrutinizes her own thought while tending to exclude her own voice and avoid personal references; thus she performs "the linguistic decreation of the self" (Dargan 1999: 7). ## 7. Reception and Influence Although the French post-structuralists who succeeded Weil did not engage extensively with her thought, her concepts carry a legacy through her contemporaries domestically and her heirs internationally (see Rozelle-Stone 2017). In regard to her generation of French thinkers, the influence of "attention" can be seen in the writings of Maurice Blanchot; her Platonic sense of good, order, and clarity was taken up--and rejected--by both Georges Bataille and Emmanuel Levinas. Following Weil's generation, in his younger years Jacques Derrida took interest in her mysticism and, specifically, her purifying atheism, only to leave her behind almost entirely in his later references (Baring 2011). It is possible that Weil's limited influence on post-structuralists is derived not only from the fact that she was not influenced by Nietzsche and Heidegger to the extent of the four aforementioned French thinkers, but also, quite simply, because she did not survive World War II and hence did not write thereafter. It is also important to note that throughout her life, Weil's position as a woman philosopher contributed to ad hominem attacks against her person, not her thought; she was often perceived as "psychologically cold" as opposed to being engaged in "an ethical project with different assumptions" (Nelson 2017: 9). Across Europe and more recently, Weil's "negative politics"--that is, turning away from institutions and ideology and toward religious reflection--, in conjunction with Michel Foucault's concept of biopolitics, has been taken up by the political philosophies of Giorgio Agamben and Roberto Esposito (Ricciardi 2009). In doing so, Agamben (who wrote his dissertation on Weil's political thought and critique of personhood, ideas of which went on to shape Homo Sacer [see Agamben 2017]) and Esposito rely on Weil's concepts of decreation, impersonality, and force. Beyond the Continental scene, her Christian Platonism--especially her concepts of the good, justice, void, and attention--influenced Iris Murdoch's emphasis on the good, metaphysics, and morality, and was thereby part of a recent revival in virtue ethics (Crisp & Slote 1997). In addition, many have noted the "spiritual kinship" that is apparent between the religious and ethical philosophies of Weil and Ludwig Wittgenstein. For example, the view that belief in God is not a matter of evidence, logic, or proof is shared by these thinkers (Von der Ruhr 2006). The legacy of Weil's writings on affliction and beauty in relation to justice is also felt in Elaine Scarry's writings on aesthetics (Scarry 1999). T. S. Eliot, who wrote the introduction to The Need for Roots, cites Weil as an inspiration of his literature, as do W. H. Auden, Czeslaw Milosz, Seamus Heaney, Flannery O'Connor, Susan Sontag, and Anne Carson. The Anglo-American secondary literature on Weil has emphasized her concept of supernatural justice, including the philosophical tensions that inform her materialism and mysticism (Winch 1989; Dietz 1988; Bell 1993, 1998; Rhees 2000). Additional considerations treat her Christian Platonism (Springsted 1983; Doering & Springsted 2004). Recent English-language scholarship on Weil has included texts on her concept of force (Doering 2010), her radicalism (Rozelle-Stone & Stone 2010), and the relationship in her thought between science and divinity (Morgan 2005), between suffering and trauma (Nelson 2017), and between decreation and ethics (Cha 2017). Furthermore, her concepts have influenced recent contributions to questions of identity (Cameron 2007), political theology (Lloyd 2011), animality (Pick 2011), and international relations (Kinsella 2021).
economic-justice	## 1. Economics and Ethics The role of ethics in economic theorizing is still a debated issue. In spite of the reluctance of many economists to view normative issues as part and parcel of their discipline, normative economics now represents an impressive body of literature. One may, however, wonder if normative economics cannot also be considered a part of political philosophy. ### 1.1 Positive vs. Normative Economics In the first half of the twentieth century, most leading economists (Pigou, Hicks, Kaldor, Samuelson, Arrow etc.) devoted a significant part of their research effort to normative issues, notably the definition of criteria for the evaluation of public policies. The situation is very different nowadays. "Economists do not devote a great deal of time to investigating the values on which their analyses are based. Welfare economics is not a subject which every present-day student of economics is expected to study", writes Atkinson (2001, p. 195), who regrets "the strange disappearance of welfare economics". Normative economics itself may be partly guilty for this state of affairs, in view of its repeated failure to provide conclusive results and its long-lasting focus on impossibility theorems (see SS 4.1). But there has also been a persistent ambiguity about the status of normative propositions in economics. The subject matter of economics and its close relation to policy advice make it virtually impossible to avoid mingling with value judgments. Nonetheless, the desire to separate positive statements from normative statements has often been transformed into the illusion that economics could be a science only by shunning the latter. Robbins (1932) has been influential in this positivist move, in spite of a late clarification (Robbins 1981) that his intention was not to disparage normative issues, but only to clarify the normative status of (useful and necessary) interpersonal comparisons of welfare. It is worth emphasizing that many results in normative economics are mathematical theorems with a primary analytical function. Endowing them with a normative content may be confusing, because they are most useful in clarifying ethical values and do not imply by themselves that these values must be endorsed. "It is a legitimate exercise of economic analysis to examine the consequences of various value judgments, whether or not they are shared by the theorist." (Samuelson 1947, p. 220) The role of ethical judgments in economics has received recent and valuable scrutiny in Sen (1987), Hausman and McPherson (2006) and Mongin (2001b). ### 1.2 Normative Economics and Political Philosophy There have been many mutual influences between normative economics and political philosophy. In particular, Rawls' difference principle (Rawls 1971) has been instrumental in making economic analysis of redistributive policies pay some attention to the maximin criterion, which puts absolute priority on the worst-off, and not only to sum-utilitarianism. (It has taken more time for economists to realize that Rawls' difference principle applies to primary goods, not utilities.) Conversely, many concepts used by political philosophers come from various branches of normative economics (see below). There are, however, differences in focus and in methodology. Political philosophy tends to focus on the general issue of social justice, whereas normative economics also covers microeconomic issues of resource allocation and the evaluation of public policies in an unjust society (although there is now philosophical work on non-ideal theory). Political philosophy focuses on arguments and basic principles, whereas normative economics is more concerned with the effective ranking of social states than the arguments underlying a given ranking. The difference is thin in this respect, since the axiomatic analysis in normative economics may be interpreted as performing not only a logical decomposition of a given ranking or principle, but also a clarification of the underlying basic principles or arguments. But consider for instance the "leveling-down objection" (Parfit 1995), which states that egalitarianism is wrong because it considers that there is something good in achieving equality by leveling down (even when the egalitarian ranking says that, all things considered, leveling down is bad). This kind of argument has to do with the reasons underlying a social judgment, not with the content of the all things considered judgment itself. It is hard to imagine if and how the leveling-down objection could be incorporated in the models of normative economics. A final difference between normative economics and political philosophy, indeed, lies in conceptual tools. Normative economics uses the formal apparatus of economics, which gives powerful means to derive non-intuitive conclusions from simple arguments, although it also deprives the analyst of the possibility of exploring issues that are hard to formalize. There are now several general surveys of normative economics, some of which do also cover the intersection with political philosophy: Arrow, Sen and Suzumura (1997, 2002, 2011), Fleurbaey (1996), Hausman and McPherson (2006, rev. and augmented in Hausman et al. 2016), Kolm (1996), Moulin (1988, 1995, 1998), Roemer (1996), Young (1994). ## 2. Inequality and Poverty Measurement Focusing traditionally on income inequality and poverty, this field has been first built on the assumption that there is a given unidimensional measure of individual advantage. The gist of the analysis is then about the distribution of this particular notion of advantage. More recently, partly due to the emergence of data about living conditions, there has been growing interest in the measurement of inequality and poverty when individual situations are described by a multidimensional list of attributes or deprivations. This has generated the field of "multidimensional inequality and poverty measurement". So far most of this literature has remained disconnected from the welfare economics literature in which interpersonal comparisons of well-being in relation to individual preferences is a key issue. The typical multidimensional indices do not refer to well-being or preferences. But the connection is being made and the welfare economics literature will eventually blend with the inequality and poverty measurement literature. Excellent surveys of the unidimensional part of the theory include: Chakravarty (1990, 2009), Cowell (2000), Dutta (2002), Lambert (1989), Sen and Foster (1997), Silber (1999). The multidimensional approach is surveyed or discussed in Weymark (2006), Chakravarty (2009), Decancq and Lugo (2013), Aaberge and Brandolini (2015), Alkire et al. (2015), Duclos and Tiberti (2016). The link between inequality and poverty measurement and welfare economics is discussed in Decancq et al. (2015) and in several chapters of Adler and Fleurbaey (2016). For a comprehensive handbook on economic inequality, see Atkinson and Bourguignon (2000, 2015). ### 2.1 Indices of Inequality and Poverty The study of inequality and poverty indices started from a statistical, pragmatic perspective, with such indices as the Gini index of inequality or the poverty head count. Recent research has provided two valuable insights. First, it is possible to relate inequality indices to social welfare functions, so as to give inequality indices a more transparent ethical content. The idea is that an inequality index should not simply measure dispersion in a descriptive way, but would gain in relevance if it measured the harm to social welfare done by inequality. There is a simple method to derive an inequality index from a social welfare function, due to Kolm (1969) and popularized by Atkinson (1970) and Sen (1973). Consider a social welfare function which is defined on distributions of income and is symmetrical (i.e., permuting the income of two individuals leaves social welfare unchanged). For any given unequal distribution of income, one may compute the egalitarian distribution of income which would yield the same social welfare as the unequal distribution. This is called the "equally-distributed equivalent" (or "equal-equivalent") distribution. If the social welfare function is averse to inequality, the total amount of income in the equal-equivalent distribution is less than in the unequal distribution. In other words, the social welfare function condones some sacrifice of total income in order to reach equality. This drop in income, measured in proportion of the initial total income, may serve as a valuable index of inequality. This index may also be used in a picturesque decomposition of social welfare. Indeed, an ordinally equivalent measure of social welfare is then total income (or average income - it does not matter when the population is fixed) times one minus the inequality index. This method of construction of an index of inequality, often referred to as the ethical approach to inequality measurement, is most useful when the argument of the social welfare function, and the object of the measurement of inequality, is the distribution of individual well-being (which may or may not be measured by income). Then the social welfare function is indeed symmetrical (by requirement of impartiality) and its aversion to inequality reflects its underlying ethical principles. In other contexts, the method is more problematic. Consider the case when social welfare depends on individual well-being, and individual well-being depends on income with some individual variability due to differential needs. Then income equality may no longer be a valuable goal, because the needy individuals may need more income than others. Using this method to construct an index of inequality of well-being is fine, but using it to construct an index of inequality of incomes would be strange, although it would immediately reveal that income inequality is not always bad (when it compensates for unequal needs). Now consider the case when social welfare is the utilitarian sum of individual utilities, and all individuals have the same strictly concave utility function (strict concavity means that it displays a decreasing marginal utility). Then using this method to construct an index of income inequality is amenable to a different interpretation. The index then does not reflect a principled aversion to inequality in the social welfare function, since the social welfare function has no aversion to inequality of utilities. It only reflects the consequence of an empirical fact, the degree of concavity of individual utility functions. To call this the ethical approach, in this context, seems a misnomer. In the field of multidimensional inequality or poverty measurement, a key divide has separated the measures that evaluate the distribution in every dimension (such as income, health, asset deprivation...) before they aggregate over the dimensions, from the measures that first evaluate individual situations, all dimensions included, before aggregating over individuals. The latter has been praised (Decancq and Lugo 2013) for being closer to the standard individualistic approach in welfare economics, and for making it possible to have measures that are sensitive to the correlation between disadvantages (e.g., the positive correlation between income and health), and therefore more accurately record the multidimensional suffering of the worst off. An interesting feature of such measures is that a positive correlation among attributes of advantage or disadvantage worsens inequalities only if the elasticity of substitution between attributes, in the measure of individual multidimensional advantage, is greater than the inequality aversion in the social index. To illustrate, consider two canonical distributions, with two individuals and two attributes: (Perfect correlation) Ann: (1,1) and Bob: (2,2) (Anti-correlation) Ann: (1,2) and Bob: (2,1) If the attributes are deemed perfectly substitutable (meaning that only the sum, potentially weighted, of the attributes matters for the assessment of individual advantage), and the inequality aversion is zero (meaning that only the sum of individual indexes of advantage matters), the two distributions are considered equally good. But they also appear equally good if there is no possible substitution (only the value of the worst attribute matters) and inequality aversion is infinite (only the worst-off individual matters). In contrast, the first distribution appears worse if the attributes are substitutable and inequality aversion is strong (the first distribution is then unequal, unlike the second one); whereas it appears better if there is no possible substitution and inequality aversion is weak (only one individual has a bad attribute in the first distribution, whereas both have one in the second). The second valuable contribution of recent research in this field is the development of an alternative ethical approach through the axiomatic study of the properties of indices. The main ethical axioms deal with transfers. The Pigou-Dalton principle of transfers says that inequality decreases (or social welfare increases) when an even transfer is made from a richer to a poorer individual without reversing their pairwise ranking (although this may alter their ranking relative to other individuals). Since this condition is about even transfers, it is quite weak and other axioms have been proposed in order to strengthen the priority of the worst-off. The principle of diminishing transfers (Kolm 1976) says that a Pigou-Dalton transfer has a greater impact the lower it occurs in the distribution. The principle of proportional transfers (Fleurbaey and Michel 2001) says that an inefficient transfer in which what the donor gives and what the beneficiary receives is proportional to their initial positions increases social welfare. Similar transfer axioms have been adapted to the measurement of poverty. For instance, Sen (1976) proposed the condition saying that poverty increases when an even transfer is made from someone who is below the poverty line to a richer individual (below or above the line). The other axioms with which the axiomatic analysis has been made usually have a less obvious ethical appeal, and relate to decomposability of indices, scale invariance and the like. Characterization results have been obtained, which identify classes of indices satisfying particular lists of axioms. The two ethical approaches may be combined, when one takes as an axiom the condition that the index be derived from a social welfare function with particular features. ### 2.2 The Dominance Approach The multiplicity of indices, even when a restriction to special sub-classes may be justified by axiomatic characterization, raises a serious problem for applications. How can one make sure that a distribution is more or less unequal, or has more or less poverty, than another without checking an infinite number of indices? Although this may look like a purely practical issue, it has given rise to a broad range of deep results, relating the statistical concept of stochastic dominance to general properties of social welfare functions and to the satisfaction of transfer axioms by inequality and poverty indices. This approach, in particular, justifies the widespread use of Lorenz curves in the empirical studies of inequality. The Lorenz curve depicts the percentage of the total amount of whatever is measured, income, wealth or well-being, possessed by any given percentage of the poorest among the population. For instance, according to the Census Bureau, in 2006 the poorest 20%'s share of total income was 3.7%, the poorest 40%'s share was 13.1%, the poorest 60%'s share was 28.1%, the poorest 80%'s share was 50.6%, while the top 5%'s share was 22.2%. This indicates that the Lorenz curve is approximately as in the following figure. ![graph of lorenz curve](lorenz2006.png) Considerable progress has been made in the development of dominance techniques of the Lorenz type, with extensions to multidimensional inequality and to poverty measurement (Aaberge and Brandolini 2015). ### 2.3 Equality, Priority, Sufficiency Philosophical interest in the measurement of inequality has recently risen (Temkin 1993). Most of this philosophical literature, however, tends to focus on defining the right foundations for an aversion to inequality. In particular, Parfit (1995) proposes to give priority to the worse-off not because of their relative position compared to the better-off, but because and to the extent that they are badly off. This probably corresponds to defining social welfare by an additively separable social welfare function, with diminishing marginal social utility (a social welfare function is additively separable when it is the sum of separate terms, each of which depends only on one individual's well-being). Interestingly, if egalitarianism is defined in opposition to this "priority view" by the feature that it relies on judgments of relative positions, this means that egalitarian values cannot be correctly represented by a separable social welfare function. This seems to raise the ethical stakes concerning properties of decomposability of indices or separability of social welfare functions, which are usually considered in economics merely as convenient conditions simplifying the functional forms (although separability may also be justified by the subsidiarity principle, according to which unconcerned individuals need not have a say in a decision). The content and importance of the distinction between egalitarianism and prioritarianism remains a matter of debate (see, among many others, Tungodden 2003, and the contributions in Holtug and Lippert-Rasmussen 2007). It is also interesting to notice that philosophers are often at ease to work with the notion of social welfare (or social good, or inequality) as a numerical quantity with cardinal meaning, whereas economists typically restrict their interpretation of social welfare or inequality to a purely ordinal ranking of social states. Beside egalitarian and prioritarian positions one must also mention the "sufficiency view", defended e.g. by Frankfurt (1987) who argues that priority should be given only to those below a certain threshold. One may consider that this view supports the idea that poverty indices might summarize everything that is relevant about social welfare. ## 3. Welfare Economics Welfare economics is the traditional generic label of normative economics, but, in spite of substantial variations between authors, it now tends to be associated with a particular subcontinent of this domain, maybe as a result of the development of "non-welfarist" approaches and of approaches with a broader scope, such as the theory of social choice. Surveys on welfare economics in its restricted definition can be found in Graff (1957), Boadway and Bruce (1984), Chipman and Moore (1978), Samuelson (1981). Many topics of welfare economics are addressed in the more recent handbooks by Arrow, Sen and Suzumura (2002, 2011), Atkinson and Bourguignon (2000, 2015) and Adler and Fleurbaey (2016). Accessible introductions are given by (book-length) Adler (2019) and (article-length) Fleurbaey (2019). ### 3.1 Old and New Welfare Economics The proponents of a "new" welfare economics (Hicks, Kaldor, Scitovsky) have distanced themselves from their predecessors (Marshall, Pigou, Lerner) by abandoning the idea of making social welfare judgments on the basis of interpersonal comparisons of utility. Their problem was then that in absence of any kind of interpersonal comparisons, the only principle on which to ground their judgments was the Pareto principle, according to which a situation is a global improvement if it is an improvement for every member of the concerned population (there are variants of this principle depending on how individual improvement is defined, in terms of preferences or some notion of well-being, and depending on whether it is a strict improvement for all members or some of them stay put). Since most changes due to public policy hurt some subgroups for the benefit of others, the Pareto principle remains generally silent. The need for a less restrictive criterion of evaluation has led Kaldor (1939) and Hicks (1939) to propose an extension of the Pareto principle through compensation tests. According to Kaldor's criterion, a situation is a global improvement if ex post the gainers could compensate the losers. For Hicks' criterion, the condition is that ex ante the losers could not compensate the gainers (a change from situation A to situation B is approved by Hicks' criterion if the change from B to A is not approved by Kaldor's criterion). These criteria are much less partial than the Pareto principle, but they remain partial (that is, they fail to rank many pairs of alternatives). This is not, however, their main drawback. They have been criticized for two basic flaws. First, for plausible definitions of how the compensation transfers could be computed, these criteria may lead to inconsistent social judgments: the same criterion may simultaneously declare that a situation A is better than another situation B, and conversely. Scitovsky (1941) has proposed to combine the two criteria, but this does not prevent the occurrence of intransitive social judgments. Second, the compensation tests have a dubious ethical value. If the compensatory transfers are performed in Kaldor's criterion, then the Pareto criterion alone suffices since after compensation everybody gains. If the compensatory transfers are not performed, the losers remain losers and the mere possibility of compensation is a meager consolation to them. Such criteria are then typically biased in favor of the rich whose willingness to pay is generally high (i.e., they are willing to give a lot in order to obtain whatever they want, and therefore they can easily compensate the losers; when they do not actually pay the compensation, they can have the cake and eat it too). Cost-benefit analysis has more recently developed criteria which are very similar and rely on the summation of willingness to pay across the population. In spite of repeated criticism by specialists (Arrow 1951, Boadway and Bruce 1984, Sen 1979, Blackorby and Donaldson 1990), practitioners of cost-benefit analysis and some branches of economic theory (industrial organization, international economics) still commonly rely on such criteria. More sophisticated variants of cost-benefit analysis (Layard and Glaister 1994, Dreze and Stern 1987) avoid these problems by relying on weighted sums of willingness to pay or even on consistent social welfare functions. Adler (2012) offers a comprehensive study of the foundations of the social welfare function approach to cost-benefit analysis. Many specialists of public economics (e.g. Stiglitz 1987) have considered that the Pareto criterion was the core ethical principle on which economists should buttress their social evaluations, and that they should focus on denouncing all sources of inefficiency in social organizations and public policies. A subfield of welfare economics focused on the possibility of making social welfare judgments on the basis of national income. An increase in national income may reflect an increase in social welfare under some stringent assumptions, most conspicuously the assumption that the distribution of incomes is socially optimal. Although very restrictive, this kind of result has a lasting influence, in theory (international economics) and in practice (the salience of GDP growth in policy discussions). There exists a school of social indicators (see the Social Indicators Research journal) which fights this influence and the number of alternative indicators (of happiness, genuine progress, social health, economic well-being, etc.) has soared in the last decades (see e.g. Miringoff and Miringoff 1999, Frey and Stutzer 2002, Kahneman et al. 2004 and Gadrey and Jany-Catrice 2006). Bergson (1938) and Samuelson (1947, 1981) occupy a special position, which may be described as a third way between old and new welfare economics. From the former, they retain the goal of making complete and consistent social welfare judgments with the help of well-defined social welfare functions. The formula W(U1(x),...,Un(x)) is often named a "Bergson-Samuelson social welfare function" (x is the social state; Ui(x), for i=1,...,n, is individual i's utility in this state). With the latter, however, they share the idea that only ordinal non-comparable information should be retained about individual preferences. This may seem contradictory with the formula of the Bergson-Samuelson social welfare function, in which individual utility functions appear, and there has been a controversy about the possibility of constructing a Bergson-Samuelson social welfare function on the sole basis of individual ordinal non-comparable preferences (see in particular Arrow (1951), Kemp and Ng (1976), Samuelson (1977, 1987), Sen (1986) and a recent discussion in Fleurbaey and Mongin (2005)). Samuelson and his defenders are commonly considered to have lost the contest, but it may also be argued that their opponents have misunderstood them. Indeed, individual utility functions in the W(U1(x),...,Un(x)) formula are, according to Bergson and Samuelson, to be constructed out of individual preference orderings, on the basis of fairness principles. The logical possibility of such a construction has been repeatedly proven by Samuelson (1977), Pazner (1979), Mayston (1974, 1982). The fact that such a construction does not require any other information than ordinal non-comparable preferences is indisputable. Bergson and Samuelson acknowledged the need for interpersonal comparisons, but considered that these could be done, in an ethically relevant way, on the sole basis of non-comparable preference orderings. They failed, however, to be more specific about the fairness principles on which the construction could be justified. The theory of fair allocation (see SS 6) may fill the gap. ### 3.2 Harsanyi's Theorems Harsanyi may be viewed as the last representative of the old welfare economics, to which he made a major contribution in the form of two arguments. The first one is often called the "impartial observer argument". An impartial observer should decide for society as if she had an equal chance of becoming anyone in the considered population. This is a risky situation in which the standard decision criterion is expected utility. The computation of expected utility, in this equal probability case, yields an arithmetic mean of the utilities that the observer would have if she became anyone in the population. Harsanyi (1953) considers this to be an argument in favor of utilitarianism. The obvious weakness of the argument, however, is that not all versions of utilitarianism would measure individual utility in a way that may be entered in the computation of the expected utility of the impartial observer. In other words, ask a utilitarian to compute social welfare, and ask an impartial observer to compute her expected utility. There is little reason to believe that they will come up with similar conclusions, even though both compute a sum or a mean. For instance, a very risk-averse impartial observer may come arbitrarily close to the maximin criterion. This argument has triggered controversies, in particular with Rawls (1974), about the soundness of the maximin criterion in the original position, and with Sen (1977b). See Harsanyi (1976) and recent analyses in Weymark (1991), Mongin (2001a). There is a related, but different controversy about the consequences of the veil of ignorance in Dworkin's hypothetical insurance scheme (Dworkin 2000). Roemer (1985) argues that if individuals maximize their expected utility on the insurance market, they insure against states in which they have low marginal utility. If low marginal utility happens to be the consequence of some handicaps, then the hypothetical market will tax the disabled for the benefit of the others, a paradoxical but typical consequence of utilitarian policies. It is indeed well known that insurance markets have strange consequences when utilities are state-dependent (that is, when the utility of income is affected by random events). For a recent revival of this controversy, see Dworkin (2002), Fleurbaey (2008) and Roemer (2002a). Harsanyi's second argument, the "aggregation theorem", is about a social planner who, facing risky prospects, maximizes expected social welfare and wants to respect individual preferences about prospects. Harsanyi (1955) shows that these two conditions imply that social welfare must be a weighted sum of individual utilities, and concludes that this is another argument in favor of utilitarianism. Recent evaluation of this argument and its consequences may be found in Broome (1991), Weymark (1991). In particular, Broome uses the structure of this argument to conclude that social good must be computed as the sum of individual goods, although this does not preclude incorporating a good deal of inequality aversion in the measurement of individual good. Diamond (1967) has raised a famous objection against the idea that expected utility is a good criterion for the social planner. This criterion implies that if the social planner is indifferent between the distributions of utilities, for two individuals, (1,0) and (0,1), then he must also be indifferent between these two distributions and an equal probability of getting either distribution. This is paradoxical, this lottery being ex ante better since it gives equal prospects to individuals. Broome (1991) raises another puzzle. An even better lottery would yield either (0,0) or (1,1) with equal probability. It is better because ex ante it gives individuals the same prospects as the previous lottery, and it is more egalitarian ex post. The problem is that it seems quite hard to construct a social criterion which ranks these four alternatives as suggested here. Defining social welfare under uncertainty is still a matter of bafflement. See Deschamps and Gevers (1977), Ben Porath, Gilboa and Schmeidler (1997), Fleurbaey (2010). Things are even more difficult when probabilities are subjective and individual beliefs may differ. Harsanyi's aggregation theorem then transforms into an impossibility theorem. On this, see in particular Hylland and Zeckhauser (1979), Mongin (1995), Bradley (2005), Mongin and Pivato (2016, 2020), Dietrich (2021). ## 4. Social Choice The theory of social choice in its modern incarnation originated in Arrow's pioneering work. Arrow himself described it as a 'subversive' attempt to 'operationalize' the Bergson-Samuelson approach (Arrow 1983, 26), but it could equally be viewed as extending the tradition of earlier work on voting and committee decisions, going back to Condorcet and Black. It has developed into an immense literature, with many ramifications to a variety of subfields and topics. The social choice framework is, potentially, so general that one may think of using it to unify normative economics. In a restrictive definition, however, social choice is considered to deal with the problem of synthesizing heterogeneous individual preferences into a consistent ranking. Sometimes an even more restrictive notion of "Arrovian social choice" is used to name works which faithfully adopt Arrow's particular axioms. There are many surveys of social choice theory, in broad and restrictive senses: Arrow, Sen and Suzumura (1997, in particular chap. 3, 4, 7, 11, 15; 2002, in particular chap. 1, 2, 3, 4, 7, 10; 2011, in particular chap. 13, 14, 17-20), Sen (1970, 1977a, 1986, 1999, 2009), Anand, Puppe and Pattanaik (2009). See also the entry on social choice theory. ### 4.1 Arrow's Theorem In an attempt to construct a consistent social ranking of a set of alternatives on the basis of individual preferences over this set, Arrow (1951) obtained: 1) an impossibility theorem; 2) a generalization of the framework of welfare economics, covering all collective decisions from political democracy and committee decisions to market allocation; 3) an axiomatic method which set a standard of rigor for any future endeavor. The impossibility theorem roughly says that there is no general way to rank a given set of (more than two) alternatives on the basis of (at least two) individual preferences, if one wants to respect three conditions: (Weak Pareto) unanimous preferences are always respected (if everyone prefers A to B, then A is better than B); (Independence of Irrelevant Alternatives) any subset of two alternatives must be ranked on the sole basis of individual preferences over this subset; (No-Dictatorship) no individual is a dictator in the sense that his strict preferences are always obeyed by the ranking, no matter what they and the other individuals' preferences are. The impossibility holds when one wants to cover a great variety of possible profiles of individual preferences. When there is sufficient homogeneity among preferences, for instance when alternatives differ only in one dimension and individual preferences are based on the distance of alternatives to their preferred alternative along this dimension (think, for instance, of political options on the left-right spectrum), then consistent methods exist (the majority rule, in this example; Black 1958). Arrow's result clearly extends the scope of analysis beyond the traditional focus of welfare economics, and nicely illuminates the difficulties of democratic voting procedures such as the Condorcet paradox (consisting of the fact that majority rule may be intransitive). The analysis of voting procedures is wide domain. For recent surveys, see e.g. Saari (2001) and Brams and Fishburn (2002), as well as the entry on Arrow's Theorem. This analysis reveals a deep tension between rules based on the majority principle and rules which protect minorities by taking account of preferences in a more extended way (see Pattanaik 2002). Specialists of welfare economics once claimed that Arrow's result had no bearing on economic allocation (e.g. Samuelson 1967), and there is some ambiguity in Arrow (1951) about whether, in an economic context, the best application of the theorem is about individual self-centered tastes over personal consumptions, in which case it is indeed relevant to welfare economics, or about individual ethical values about general allocations. It is now generally considered that the formal framework of social choice can sensibly be applied to the Bergson-Samuelson problem of ranking allocations on the basis of individual tastes. Applications of Arrow's theorem to various economic contexts have been made (see the surveys by Le Breton 1997, Le Breton and Weymark 2011). ### 4.2 The Informational Basis Sen (1970a) proposes a further generalization of the social choice framework, by permitting consideration of information about individual utility functions, not only preferences. This enlargement is motivated by the impossibility theorem, but also by the ethical relevance of various kinds of data. Distributional issues obviously require interpersonal comparisons of well-being. For instance, an egalitarian evaluation of allocations needs a determination of who the worst-off are. It is tempting to think of such comparisons in terms of utilities. This has triggered an important body of literature which has greatly clarified the meaning of various kinds of interpersonal utility comparisons (of levels, differences, etc.) and the relation between them and various social criteria (egalitarianism, utilitarianism, etc.). This literature (esp. d'Aspremont and Gevers 1977) has also provided an important formal analysis of the concept of welfarism, showing that it contains two subcomponents. The first one is the Paretian condition that an alternative is equivalent to another when all individuals are indifferent between them. This excludes using non-welfarist information about alternatives, but does not exclude using non-welfarist information about individuals (one individual may be favored because of a physical handicap). The second one is an independence condition formulated in terms of utilities. It may be called Independence of Irrelevant Utilities (Hammond 1987), and says that the social ranking of any pair of alternatives must depend only on utility levels at these two alternatives, so that a change in the profile of utility functions which would leave the utility levels unchanged at the two alternatives should not alter how they are ranked. This excludes using non-welfarist information about individuals, but does not exclude using non-welfarist information about alternatives (one may be preferred because it has more freedom). The combination of the two conditions excludes all non-welfare information. Excellent surveys are given in d'Aspremont (1985), d'Aspremont and Gevers (2002), Bossert and Weymark (2004), Mongin and d'Aspremont (1998). In spite of the important clarification made by this literature, the introduction of utility functions essentially amounts to going back to old welfare economics, after new welfare economics and authors such as Bergson, Samuelson and Arrow failed to provide appealing solutions with data on consumer tastes only. A related issue is how the evaluation of individual well-being must be made, or, equivalently, how interpersonal comparisons must be performed. There is now a growing interest in exploring concrete ways of measuring individual well-being, as illustrated by happiness studies (section 7.5 below) and the handbook published by Adler and Fleurbaey (2016). Welfare economics traditionally relied on "utility", and the extended informational basis of social choice is mostly formulated with utilities (although the use of extended preference orderings is often shown to be formally equivalent: for instance, saying that Jones is better-off than Smith is equivalent to saying that it is better to be Jones than to be Smith, in some social state). But utility functions may be given a variety of substantial interpretations, so that the same formalism may be used to discuss interpersonal comparisons of resources, opportunities, capabilities and the like. In other words, one may separate two issues: 1) whether one needs more information than individual preference orderings in order to perform interpersonal comparisons; 2) what kind of additional information is ethically relevant (subjective utility or objective notions of opportunities, etc.). The latter issue is directly related to philosophical discussions about how well-being should be conceived and to the "equality of what" debate. This debate, which originates with Sen (1980), focuses on seeking the appropriate metric of advantage that should be used by theories of justice that are egalitarian or that give priority to the worst-off. The former issue is still debated. Extending the informational basis by introducing numerical indices of well-being (or equivalent extended orderings) is not the only conceivable extension. Arrow's impossibility is obtained with the condition of Independence of Irrelevant Alternatives, which may be logically analyzed, when the theorem is reformulated with utility functions as primitive data, as the combination of Independence of Irrelevant Utilities (defined above) with a condition of ordinal non-comparability, saying that the ranking of two alternatives must depend only on individuals' ordinal non-comparable preferences. Arrow's impossibility may be avoided by relaxing the ordinal non-comparability condition, and this is the above-described extension of the informational basis by relying on utility functions. But Arrow's impossibility may also be avoided by relaxing Independence of Irrelevant Utilities only. In particular, it makes sense to rank alternatives on the basis of how these alternatives are considered by individuals in comparison to other alternatives. For instance, when considering to make a transfer of consumption goods from Jones to Smith, it is not enough to know that Jones is against it and Smith is in favor of it (this is the only information usable under Arrow's condition). It is also relevant to know if both consider that Jones has a better bundle, or not, which involves considering other alternatives in which bundles are permuted, for instance. In this vein, Hansson (1973) and Pazner (1979) have proposed to weaken Arrow's axiom so as to make the ranking of two alternatives depend on the indifference curves of individuals at these two alternatives. In particular, Pazner relates this approach to Samuelson's (Samuelson 1977) and concludes that the Bergson-Samuelson social welfare function can indeed be constructed consistently in this way. Interpersonal comparisons may be sensibly made on the sole basis of indifference curves and therefore on the sole basis of ordinal non-comparable preferences. This requires broadening the concept of interpersonal comparisons in order to cover all kinds of comparisons, not just utility comparisons (see Fleurbaey and Hammond 2004). The concept of informational basis itself need not be limited to issues of interpersonal comparisons. Many conditions of equity, efficiency, separability, responsibility, etc. bear on the kind and quantity of information that is deemed relevant for the ranking of alternatives. The theory of social choice gives a convenient framework for a rigorous analysis of this issue. ### 4.3 Around Utilitarianism The theory of social choice with utility functions has greatly systematized our understanding of social welfare functions. For instance, it has shown how to construct a continuum of intermediate social welfare functions between sum-utilitarianism and the maximin criterion (or its lexicographic refinement, the leximin criterion, which ranks distributions of well-being by examining first the worst-off position, then the position which is just above the worst-off, and so on; for instance, the maximin is indifferent between the three distributions (1,2,5), (1,3,5) and (1,3,6), whereas the leximin ranks them in increasing order). Three other developments around utilitarian social welfare functions are worth mentioning. The first development is related to the application of theories of equality of opportunity, and involves the construction of mixed social welfare functions which combine utilitarianism and maximin. Suppose that there is a double partition of the population, such that one would like the social welfare function to display infinite inequality aversion within subgroups of the first partition, and zero inequality aversion within subgroups of the second partition. For instance, subgroups of the first partition consist of equally deserving individuals, for which one would like to obtain equality of outcomes, whereas subgroups of the second partition consist of individuals who have equal opportunities, so that inequalities among them do not matter. Van de gaer (1993) proposes to apply average utilitarianism within each subgroup of the second partition, and to apply the maximin criterion to the vector of average utilities obtained in this way. In other words, the average utilities measure the value of the opportunity sets offered to individuals, and one applies the maximin criterion to such values, in order to equalize the value of opportunity sets. Roemer (1998) proposes to apply the maximin criterion within each subgroup of the first partition, and then to apply average utilitarianism to the vector of minimum utilities obtained in this way. In other words, one tries to equalize outcomes for equally deserving individuals first, and then applies a utilitarian calculus. These may not be the only possible combinations of utilitarianism and maximin, but they are given an axiomatic justification which suggests that they are indeed salient, in Ooghe, Schokkaert and Van de gaer (2007) and Fleurbaey (2008). For a survey on the applications of Roemer's criterion, see Roemer (2002b) and, more recently, Ferreira and Peragine (2016), Roemer and Trannoy (2016), Ramos and Van de gaer (2016). A second interesting development deals with intergenerational ethics. With an infinite horizon, it is essentially impossible to combine the Pareto criterion and anonymity (permuting the utilities of some generations does not change social welfare) in a fully satisfactory way, even when utilities are perfectly comparable. This is a similar but more basic impossibility than Arrow's theorem. The intuition of the problem can be given with the following simple example. Consider the following sequence of utilities: (1,2,1,2,...). Permute the utility of every odd period with the next one. One then obtains (2,1,2,1,...). Then permute the utility of every even period with the next one. This yields (2,2,1,2,...). This third sequence Pareto-dominates the first one, even though it was obtained only through simple permutations. This impossibility is now better understood, and various results point to the "catching up" criterion as the most reasonable extension of sum-utilitarianism to the infinite horizon setting. This criterion, which does not rank all alternatives, applies when the finite-horizon sums of utilities, for two infinite sequences of utilities, are ranked in the same way for all finite horizons above some finite time. Interestingly, this topic has seen parallel and sometimes independent contributions by economists and philosophers (see e.g. Lauwers and Liedekerke 1997, Lauwers and Vallentyne 2003, Roemer and Suzumura 2007). A third development worth mentioning has to do with population ethics. Sum-utilitarianism appears to be overly populationist, since it implies the "repugnant conclusion" (Parfit 1984) that we should aim for an unhappy but sufficiently large population in preference to a small and happy one. Conversely, average utilitarianism is "Malthusian", preferring a happier population, no matter how small, to a less happy one, no matter how large. Here again there is an interesting tension, namely, between accepting all individuals whose utility is greater than zero, accepting equalization of utilities, and avoiding the "repugnant conclusion". This tension is shown in this way. Start with a given affluent population of any size. Add any number of individuals with positive but almost zero utilities. This does not reduce social welfare. Then equalize utilities, which again does not reduce social welfare. One then obtains, compared to the initial population, a larger population with lower utilities. One sees that these lower utilities may be arbitrarily low, if the added individuals are sufficiently numerous and have sufficiently low initial utilities, thus yielding the repugnant conclusion (see Arrhenius 2000). Average utilitarianism disvalues additional individuals whose utility is below the average, which is very restrictive for affluent populations. A less restrictive approach is that of critical-level utilitarianism, which disvalues only individuals whose utility level is below some fixed, low but positive threshold. For an extensive overview and a defense of critical-level utilitarianism, see Blackorby, Bossert and Donaldson (1997, 2005), as well as Broome (2004). For an original proposal relying on a different social welfare function (inspired by the Gini function), see Asheim and Zuber (2014). ## 5. Bargaining and Cooperative Games At the time when Arrow declared social choice to be impossible, Nash (1950) published a possibility theorem for the bargaining problem, which is the problem of finding an option acceptable to two parties, among a subset of alternatives. Interestingly, Nash relied on the axiomatic analysis just like Arrow, so that both can be given credit for the introduction of this method in normative economics. In the same decade, a similar contribution was made by Shapley (1953) to the theory of cooperative games. The development of such approaches has been impressive since then, but some questioning has emerged regarding the ethical relevance of this theory to issues of distributive justice. ### 5.1 Nash's and Other Solutions Nash (1950) adopted a welfarist framework, in which alternatives are described only by the utility levels they give to the two parties. His solution consists in choosing the alternative which, in the feasible set, maximizes the product of individual utility gains from the disagreement point (this point is the fallback option when the parties fail to reach an agreement). This solution is therefore related to a particular social welfare function which is somehow intermediate between sum-utilitarianism and the maximin criterion. Contrary to these, however, it is invariant to independent changes in utility zeros and scales, which means that it can be applied with utility functions which are defined only up to an affine transformation (i.e., no difference is made between utility function Ui and utility function ai Ui + bi), such as Von Neumann-Morgenstern utility functions. Nash uses this invariance property in his axiomatic characterization of the solution. He also uses another property, which holds for any solution maximizing a social welfare function, namely, that removing non-selected options does not alter the choice. This particular property is criticized in Kalai and Smorodinsky (1975), because it makes the solution ignore the relative size of the sacrifices made by the parties in order to reach a compromise. They propose another solution, which consists of equalizing the parties' sacrifice relative to the maximum gain they could expect in the available set of options. This solution, contrary to Nash's, guarantees that an enlargement of the set of options that is favorable to one party never hurts this party in the ultimate selection. It is very similar to Gauthier's (1986) "minimax relative concession" solution. Many other solutions to the bargaining problem have been proposed, but these two are by far the most prominent. The relevance of bargaining theory to the theory of distributive justice has been questioned. First, if the disagreement point is defined, as it probably should, in relation to the relative strength of parties in a "state of nature", then the scope for redistributive solidarity is very limited. One obtains a theory of "justice as mutual advantage" (Barry 1989, 1995) which is not satisfactory at the bar of any minimal conception of impartiality or equality. Second, the welfarist formal framework of the theory of bargaining is poor in information (Roemer 1986b, 1996). Describing alternatives only in terms of utility levels makes it impossible to take account of basic physical features of allocations. For instance, it is impossible to find out, from utility data alone, which of the alternatives is a competitive equilibrium with equal shares. As another illustration, Nash's and Kalai and Smorodinsky's solutions both recommend allocating an indivisible prize by a fifty-fifty lottery, whether the prize is symmetric (a one-dollar bill for either party) or asymmetric (a one-dollar bill if party 1 wins, ten dollars if party 2 wins). Extensive surveys of bargaining theory can be found in Peters (1992), Thomson (1999). ### 5.2 Axiomatic Bargaining and Cooperative Games The basic theory of bargaining focuses on the two-party case, but it can readily be extended to the case when a greater number of parties are at the table. However, when there are more than two parties, it becomes relevant to consider the possibility for subgroups (coalitions) to reach separate agreements. Such considerations lead to the broader theory of cooperative games. This broader theory is, however, more developed for the relatively easy case when coalition gains are like money prizes which can be allocated arbitrarily among coalition members (the "transferable utility case"). In this case, for the two-party bargaining problem the Nash and Kalai-Smorodinsky solutions coincide and give equal gains to the two parties. The Shapley value is a solution which generalizes this to any number of parties and gives a party the average value of the marginal contribution that this party brings to all coalitions which it can join. In other words, it rewards the parties in proportion to the increase in coalition gain that they bring about by gathering with others. Another important concept is the core. This notion generalizes the idea that no rational party would accept an agreement that is less favorable than the disagreement point. An allocation of the total population prize is in the core if the total amount received by any coalition is at least as great as the prize this coalition could obtain on its own. Otherwise, obviously, the coalition has an incentive to "block" the agreement. Interestingly, the Shapley value is not always in the core, except for "convex games", that is, games such that the marginal contribution of a party to a coalition increases when the coalition is bigger. The basics of this theory are very well presented in Moulin (1988) and Myerson (1991). Cooperative games are distinguished from non-cooperative games by the fact that the players can commit to an agreement, whereas in a non-cooperative game every player always seeks his interest and never commits to a particular strategy. The central concept of the theory of non-cooperative games is the Nash equilibrium (every player chooses his best strategy, taking others' strategies as given), which has nothing to do with Nash's bargaining solution. It has been shown, however, that Nash's bargaining solution can be obtained as the Nash equilibrium of a non-cooperative bargaining game, in which players make offers alternatively, and accept or reject the other's offer. ## 6. Fair allocation The theory of fair allocation studies the allocation of resources in economic models. The seminal contribution is Kolm (1972), where the criterion of equity as no-envy is extensively analyzed with the conceptual tools of general equilibrium theory. Later the theory borrowed the axiomatic method from bargaining theory, and it now covers a great variety of economic models and encompasses a variety of fairness concepts. There are several surveys of this theory: Thomson and Varian (1985), Moulin and Thomson (1997), Maniquet (1999), Thomson (2011). ### 6.1 Equity as No-Envy An allocation is envy-free if no individual would prefer having the bundle of another. An egalitarian distribution in which everyone has the same bundle is trivially envy-free, but is generally Pareto-inefficient, which means that there exist other feasible allocations that are better for some individuals and worse for none. A competitive equilibrium with equal shares (i.e., equal budgets) is the central example of a Pareto-efficient and envy-free allocation. It is envy-free since all agents have the same budget options, so that everyone could buy everyone's bundle. It is Pareto-efficient because, by an important theorem of welfare economics, any perfectly competitive equilibrium is Pareto-efficient (in absence of asymmetric information, externalities, public goods). A non-technical presentation of this theorem can be found in Hausman and McPherson (2006, 5.2). This concept of equity does not need any other information than individual ordinal preferences. It is not welfarist, in the sense that from utility data alone it is impossible to distinguish an envy-free allocation from an allocation with envy. Moreover, an envy-free allocation may be Pareto-indifferent (everyone is indifferent) to another allocation that has envy. On the other hand, this concept is strongly egalitarian, and it is quite natural to view it as capturing the idea of equality of resources (Dworkin 2000). When resources are multi-dimensional, for instance when there are several consumption goods, and when individual preferences are heterogeneous, it is not obvious to define equality of resources, but the no-envy criterion seems the best concept for this purpose. It guarantees that no individual will consider that another has a better bundle than his. It has been shown by Varian (1976) that, if preferences are sufficiently diverse and numerous (a continuum), then the competitive equilibrium with equal shares is the only Pareto-efficient and envy-free allocation. This concept can also be related to the idea of equality of opportunities (Kolm 1996). An allocation is envy-free if and only if the bundles granted to everyone could have been chosen by each individual in the same opportunity set, such as, for instance, the set containing all the bundles of the allocation under consideration. Along this vein, the concept of no-envy can also be shown to have a close connection with incentive considerations. A no-envy test is used in the theory of optimal taxation in order to make sure that no one would have interest to lie about one's preferences (Boadway and Keen 2000). Consider the condition that, when an allocation is selected and some individuals' preferences change so that their bundle goes up in their own preference ranking, then the selected allocation is still acceptable. A particular version of this condition plays a central role in the theory of incentives, under the name of Maskin monotonicity (see e.g. Jackson 2001), but it can also be given an ethical meaning, in terms of neutrality with respect to changes in preferences. Notice that envy-free allocations satisfy this condition, since after such a change of preferences every individual's bundle goes up in his ranking, thereby precluding any appearance of envy. Conversely, it turns out that this condition implies that the selected allocation must be envy-free, under the additional assumption that, in any selected allocation, individuals with identical preferences must have equivalent bundles. If one also requires the selection to be Pareto-efficient, then one obtains a characterization of the competitive equilibrium with equal shares (Gevers 1986). ### 6.2 Extensions Kolm's (1972) seminal monograph focused on the simple problem of distributing a bundle of non-produced commodities, and on equity as no-envy. Other economic problems, and other fairness concepts, have been studied later. Here is a non-exhaustive list of other economic problems that have been analyzed: sharing labor and consumption in the production of a consumption good; producing a public good and allocating the contribution burden among individuals; distributing indivisible commodities, with or without the possibility of making monetary compensations; matching pairs of individuals (men-women, employers-workers...); distributing compensations for differential needs; rationing on the basis of claims; distributing a divisible commodity when preferences are satiable. In the main stream of this theory, the problem is to select a good subset of allocations under perfect knowledge of the characteristics of the population and of the feasible set. There is also a branch which studies cost and surplus sharing, when the only information available are the quantities demanded or contributed by the population, and the cost or surplus may be distributed as a function of these quantities only (see Moulin 2002). The relevance of this literature for political philosophers should not be underestimated. Even models which seem to be devoted to narrow microeconomic allocation problems may turn out to be quite relevant, and some models are addressing issues already salient in political philosophy. This is the case in particular for the model of production of a private good when individuals have unequal skills, which is a rough description of a market economy, and for the model of differential needs. Both models are especially relevant for analyzing the issue of responsibility, talent and handicap, which is now prominent in egalitarian theories of justice. A survey on these two models in made in Fleurbaey and Maniquet (2011a), and a monograph connecting the various relevant fields of economic analysis to theories of responsibility-sensitive egalitarianism is in Fleurbaey (2008). Among the other concepts of fairness which have been introduced, two families are important. The first family contains principles of solidarity, which require individuals to be affected in the same way (they all gain or all lose) by some external shock (change in resources, technology, population size, population characteristics). For instance, if resources or technology improve, then it is natural to hope that everyone will benefit. The second family contains welfare bounds, which provide guarantees to everyone against extreme inequality. For instance, in the division of non-produced commodities, it is very natural to require that nobody should be worse-off than at the equal-split allocation (i.e. the allocation in which everyone gets the per capita amount of resources). Let us briefly describe some of the insights that are gained through this theory and seem relevant to political philosophy. A very important one is that there is a conflict between no-envy and solidarity (Moulin and Thomson 1988, 1997). This conflict is well illustrated by the fact that in a market economy, typically any change in technology benefits some agents and hurts others, even when the change is a pure progress which could benefit all. Solidarity principles are not obeyed by allocation rules which pass the no-envy test, and these principles point toward a different kind of distribution, named "egalitarian-equivalence" by Pazner and Schmeidler (1978). An allocation is egalitarian-equivalent when everyone is indifferent between his bundle in this allocation and the bundle he would have in an egalitarian economy defined in some simple way. For instance, the egalitarian economy may be such that everyone has the same bundle. In this case, an egalitarian-equivalent allocation is such that everyone is indifferent between his bundle and one particular bundle. In more sophisticated versions, the egalitarian economy is such that everyone has the same budget set, in some particular family of budget sets. Egalitarian-equivalence is a serious alternative to no-envy for the definition of equality of resources, and its superiority in terms of solidarity is quite significant, in relation to the next point. The second insight, indeed, is that no-envy itself is a combination of conflicting principles (Fleurbaey and Maniquet 2011a, Fleurbaey 2008). This conflict is made apparent in models with talents and handicaps. For instance, Pazner and Schmeidler (1974) found out that there may not exist envy-free and Pareto-efficient allocations in the context of production with unequal skills (when there are high-skilled individuals who are strongly averse to labor). This results from an incompatibility between a compensation principle saying that individuals with identical preferences should have equivalent bundles (suppressing inequalities due to skills), and a reward principle saying that individuals with the same skills should not envy each other (no preferential treatment on the basis of different preferences). Both principles are a logical implication of the no-envy test. This is obvious for the latter. For the former, notice that no-envy among individuals with the same preferences means that they must have bundles on the same indifference curve. Interestingly, the compensation principle is a logical consequence of solidarity principles and is therefore perfectly compatible with them. It is very well satisfied by egalitarian-equivalent allocation rules. In contrast, it is violated by Dworkin's hypothetical insurance which applies the no-envy test behind a veil of ignorance (see Dworkin 2000, Fleurbaey 2008, and SS 3.2). A recent philosophical analysis of the no-envy approach has been developed in Olson (2020). The relation between envy and theories of justice is also scrutinized in the entry on envy. The theory of fair allocations contains many positive results about the existence of fair allocations, for various fairness concepts, and this stands in contrast to Arrow's impossibility theorem in the theory of social choice. The difference between the two theories has often been interpreted as due to the fact that they perform different exercises (Sen 1986, Moulin and Thomson 1997). The theory of social choice, it is said, seeks a ranking of all options, while the theory of fair allocation focuses on the selection of a subset of allocations. This explanation is not convincing, since selecting a subset of fair allocations is formally equivalent to defining a full-blown albeit coarse ranking, with "good" and "bad" allocations. A more convincing explanation lies in the fact that the information used in fairness criteria is richer than allowed by Arrow's Independence of Irrelevant Alternatives (Fleurbaey, Suzumura and Tadenuma 2002). For instance, in order to check that an allocation is envy-free while another displays envy, it is not enough to know how individuals rank these two allocations in their preferences. One must know individual preferences over other alternatives involving permutations of bundles (an envious individual would prefer an allocation in which his bundle is permuted with one he envies). In this vein, one discovers that it is possible to extend the theory of fair allocation so as to construct fine-grained rankings of all allocations. This is very useful for the discussion of public policies in "second-best" settings, that is, in settings where incentive constraints make it impossible to reach Pareto-efficiency. With this extension, the theory of fair allocation can be connected to the theory of optimal taxation (Maniquet 2007), and becomes even more relevant to the political philosophy of redistributive institutions (Fleurbaey 2007). It turns out that the egalitarian-equivalence approach is very convenient for the definition of fine-grained orderings of allocations, which provides an additional argument in its favor. A detailed study of fair social orderings is made in Fleurbaey and Maniquet (2011b). ## 7. Related topics ### 7.1 Freedom and Rights Sen (1970b) and Gibbard (1974) propose, within the framework of social choice, paradoxes showing that it may not be easy to rank alternatives when some individuals have a special right to rank some alternatives that differ only in matters belonging to their private sphere, and when their preferences are sensitive to what happens in other individuals' private spheres. For instance, as an illustration of Gibbard's paradox, individuals have the right to choose the color of their shirt, but, in terms of social ranking, should A and B wear the same color or different colors, when A wants to imitate B and B wants to have a different color? There is a huge literature on this topic, and after Gaertner, Pattanaik and Suzumura (1992), who argue that no matter what choice A and B make, their rights to choose their own shirt is respected, a good part of it examines how to describe rights properly. The framework of game forms is an interesting alternative to the social choice model. Recent surveys can be found in Arrow, Sen and Suzumura (1997, vol. 2). A related but different literature has focused on measuring the freedom contained in a given menu from which a person may choose. Inspired by Sen's (1985) remarks on how to value capabilities (see section 7.2 below), this literature has offered characterizations of various measures, starting from simple counting methods (Pattanaik and Xu 1990) and later on including additional considerations such as quality and diversity (e.g., Arlegi & Nieto 2001, Barbera et al. 2004). Surveys of this literature can be found in Dowding and van Hees (2009) and Foster (2011). Apart from this formal analysis of rights and measures of freedom, economic theory is not very well connected to libertarian philosophy, since economic models show that, apart from the very specific context of perfect competition with complete markets, perfect information, no externalities and no public goods, the laissez-faire allocation is typically inefficient and arbitrarily unequal. Therefore libertarian philosophers do not find much help or inspiration in economic theory, and there is little cross-fertilization in this area. ### 7.2 Capabilities The capability approach, developed in Sen (1985, 1992), is a particular response to the "equality of what" debate, and is presented by Sen as the best way to think about the relevant interpersonal comparisons to be made for evaluations of social situations at the bar of distributive justice. It is often presented as intermediate between resourcist and welfarist approaches, but it is perhaps accurate to present it as more general. A "functioning" is any doing or being in the life of an individual. A "capability set" is the set of functioning vectors that an individual has access to. This approach has attracted a lot of interest in particular because it makes it possible to take into account all the relevant dimensions of life, in contrast with the resourcist and welfarist approaches which can be criticized as too narrow. Being so general, the approach needs to be specified in order to inspire original applications. The body of empirical literature that takes inspiration from the capability approach is now numerically impressive. As noted in Robeyns (2006) and Schokkaert (2007b), in many cases the empirical studies are essentially similar, except for terminology, to the sociological studies of living conditions. But there are more original applications, e.g., when an evaluation of development programs that takes account of capabilities is contrasted with cost-benefit analysis (Alkire 2002) or when a list of basic capabilities is enshrined in a theory of what a just society should provide to all citizens (Nussbaum 2000). More generally, all studies which seek to incorporate multiple dimensions of quality of life into the evaluation of individual and social situations can be considered, broadly speaking, as pertaining to this approach. Two central questions pervade the empirical applications. The first concerns the distinction between capabilities and functionings. The latter are easier to observe because individual achievements are more accessible to the statistician than pure potentialities. There is also the normative issue of whether the evaluation of individual situations should be based on capabilities only, viewed as opportunity sets, or should take account of achieved functionings as well. The second central question is the index problem, which has also been raised about Rawls' theory of primary goods. There are many dimensions of functionings and capabilities and not all of them are equally valuable. The definition of a proper system of weights has appeared problematic in connection to the difficulties of social choice theory. Recent surveys on this approach and its applications can be found in Alkire (2016), Basu and Lopez-Calva (2011), Kuklys (2005), Robeyns (2006), Robeyns and Van der Veen (2007), Schokkaert (2009). ### 7.3 Marxism Roemer (1982, 1986c) proposes a renewed economic analysis of Marxian concepts, in particular exploitation. He shows that, even if the theory of labor value is flawed as a causal theory of prices, it may be consistently used in order to measure exploitation and analyze the correlation between exploitation and the class status of individuals. However, he considers that this concept of exploitation is ethically not very appealing, since it roughly amounts to requiring individual consumption to be proportional to labor, and he suggests a different definition of exploitation, in terms of undue advantage due to unequal distribution of some assets. This leads him eventually to merge this line of analysis with the general stream of egalitarian theories of justice. The idea that consumption should be proportional to labor has also received some attention in the theory of fair allocation (Moulin 1990, Roemer & Silvestre 1993). See Roemer (1986a) for a collection of philosophical and economic essays on Marxism. There has been a recent wave of analysis of the concept of exploitation in economics, in particular under the impulsion of Veneziani and Yoshihara (2015, forthcoming). See also Veneziani (2013), Fleurbaey (2014) and Skillman (2014). For the references in philosophy, see the entry on exploitation. ### 7.4 Opinions In normative economics, theorists have often been wary of relying on concepts which are disconnected from the lay person's intuition. Questionnaire surveys, usually performed among students, have indeed given some disturbing results. Welfarist approaches have been questioned by the results of Yaari and Bar Hillel (1984), the Pigou-Dalton principle has been critically scrutinized by Amiel and Cowell (1992), the principles of compensation and reward have obtained mixed support in Schokkaert and Devooght (1998), etc. It is of course debatable how much theorists can learn from such results (Bossert 1998). Surveys of this questionnaire approach are available in Schokkaert and Overlaet (1989), Amiel and Cowell (1999), Schokkaert (1999), Gaertner and Schokkaert (2011). Philosophers have also performed similar inquiries (Miller 1992). ### 7.5 Altruism and Reciprocity It is standard in normative economics, as in political philosophy, to evaluate individual well-being on the basis of self-centered preferences, utility or advantage. Feelings of altruism, jealousy, etc. are ignored in order not to make the allocation of resources depend on the contingent distribution of benevolent and malevolent feelings among the population (see e.g. Goodin 1986, Harsanyi 1977). It may be worth mentioning here that the no-envy criterion discussed above has nothing to do with interpersonal feelings, since it is defined only with self-centered preferences. When an individual "envies" another in this particular sense, he simply prefers the other's consumption to his own, but no feeling is involved (he might even not be aware of the existence of the other individual). But positive economics is quite relevantly interested in studying the impact of individual feelings on behavior. Homo oeconomicus may be rational without being narrowly focused on his own consumption. The analysis of labor relations, strategic interactions, transfers within the family, generous gifts require a more complex picture of human relations (Fehr and Fischbacher 2002). Reciprocity, in particular, seems to be a powerful source of motivation, leading individuals to incur substantial costs in order to reward nice partners and punish faulty partners (Fehr and Gachter 2000). For an extensive survey of this branch of the economic literature, see Gerard-Varet, Kolm and Mercier-Ythier (2004). ### 7.6 Happiness studies The literature on happiness has surged in the last decade. The findings are well summarized in many surveys (see in particular Diener (1994, 2000), Diener et al. (1999), Frey and Stutzer (2002), Graham (2009), Kahneman et al. (1999), Kahneman and Krueger (2006), Layard (2005), Oswald (1997), Van Praag and Ferrer-i-Carbonell 2008), and reveal the main factors of happiness: personal temperament, health, social connections (in particular being married and employed). The impact of material wealth is debated, some arguing that it is more a matter of relative position than of absolute comfort, at least above a minimal level of affluence (Easterlin 1995, Clark et al. 2008), others arguing that there is a positive (but logarithmic: doubling income induces a constant increment on happiness) impact over the whole range of observed living standards (Deaton 2008, Sacks et al. 2010) . A hotly debated question is what to make of this approach in welfare economics. There is a wide variety of positions, from those who propose to measure and maximize national happiness (Diener 2000, Kahneman et al. 2004, Layard 2005) to those who firmly oppose this idea on various grounds (Burchardt 2006, Nussbaum 2008, among others). There seems to be a consensus on the idea that happiness studies suggest a welcome shift of focus, in social evaluation, from purely materialistic performances to a broader set of values. Above all, one can consider that the traditional suspicion among economists about the possibility to measure subjective well-being is being assuaged by the recent progress. However, the fact that subjective well-being can be measured does not imply that it ought to be taken as the metric of social evaluation. Surprisingly, the literature on happiness refers very little to the lively philosophical debates of the previous decades about welfarism, and in particular the criticisms raised by Rawls (1982) and Sen (1985) against utilitarianism. (Two exceptions are Burchardt (2006) and Schokkaert (2007). Layard (2005) also mentions and quickly rebuts some of the arguments against welfarism.) Nonetheless, one of the key elements of that earlier debate, namely, the fact that subjective adaptation is likely to hide objective inequalities, shows up in the data, challenging happiness specialists. Subjective well-being seems relatively immune in the long run to many aspects of objective circumstances, individuals displaying a remarkable ability to adapt. After most important life events, satisfaction returns to its usual level and the various affects return to their usual frequency. If subjective well-being is not so sensitive to objective circumstances, should we stop caring about inequalities, safety, and productivity? These issues and the possible uses of subjective well-being data for welfare analysis are discussed in detail in several chapters of Adler and Fleurbaey (2016), in particular the chapters by Fujiwara and Dolan, Bykvist, Haybron, Lucas, Graham, Clark, Decancq and Neumann. ### 7.7 Animals Normative economics, even more so than political and moral philosophy, has traditionally been anthropocentric, but growing interest in animal rights and animal interests is emerging, following the lead of influential philosophers (see the entry on the moral status of animals) and, e.g., Sunstein and Nussbaum 2004). Interestingly, specialized studies in animal welfare, in biology, adopt similar methods as welfare economics to estimate animal preferences and conceptualize various approaches to their needs and their well-being (e.g., Appleby et al. 2018). Many arguments being developed against the current farming practices are still based on anthropocentric considerations such as climate change and ecosystem services. But the idea of carving a place in the social welfare function for animals, alongside their fellow humans, is gaining ground (Johansson-Stenman 2018, Carlier and Treich 2020, Espinosa and Treich 2021; for the opposite view, see Eichner and Pethig 2006). Budolfson and Spears (2020) propose a way to compute an inclusive utilitarian social welfare function based on a distinction between well-being potential and relative realized well-being.
ramsey-economics	## 1. Production Possibilities in Ramsey's Formulation Ramsey's goal was practical: "How much of a nation's output should it save for the future?" The demographic profile over time was taken by him to be given, meaning that future numbers of people were seen as exogenous and predictable. We were therefore to imagine that economic policies have a negligible effect on reproductive behaviour (but see Dasgupta, 1969, for a study of the joint population/saving problem, using Classical Utilitarianism as the guiding principle). Parfit (1984) christened choices involving the same demographic profile, "Same Numbers Choices." The ingredients of Ramsey's theory are individuals' lifetime well-beings. Government House in his world maximizes the expected sum of the lifetime well-beings of all who are here today and all who will ever be born, subject to resource constraints. The optimum distribution of lifetime well-beings across generations is derived from that maximization exercise. Of course, the passage of time is not the same as the advance of generations. An individual's lifetime well-being is an aggregate of the flow of well-being she experiences, while intergenerational well-being is an aggregate of the lifetime well-beings of all who appear on the scene. It is doubtful that the two aggregates should have the same functional form. On the other hand, there is little evidence to suggest that we would be way off the mark in assuming they do have the same form. As a matter of practical ethics, it helps enormously to approximate by not distinguishing the functional form of someone's well-being over time from that of well-being across the generations. Ramsey adopted this short-cut. People were also taken to be identical, so we may as well assume that there is a single individual at each date. The move removes any distinction between time and generations. An alternative interpretation would have us imagine that the economy consists of a single dynasty, where parents in each generation leave bequests for their children (Meade 1966, adopted this interpretation). Ramsey also assumed, probably because the mathematics is simpler, that time is a continuous variable, not discrete. Let $t \ge 0$ denote time. In Ramsey's model there is no uncertainty (but see Levhari and Srinivasan, 1969, for one of the first of many extensions of the Ramsey model that incorporate uncertainty about future possibilities). The economy is endowed with a single, non-depreciating commodity that can be worked by labour to produce output at each date (Gale 1967 and Brock 1973, were among the first of many extensions of the Ramsey model that contain a heterogeneous collection of capital goods). The economy is assumed to be closed to international trade (opening the economy to trade involves only a minor extension to Ramsey's model). That means some of the output can be invested so as to add to the commodity's stock while the remainder can be consumed immediately. We call the stock of the commodity that serves to produce output, "capital." The problem is then to find the optimum allocation of output at each date between consumption and investment. Ramsey assumed that work is unpleasant. But because including the disutility of work in our account of his work here would add nothing of substance, we suppose that labour supply is an exogenously given constant (e.g., it is independent of the wages labour can demand). That enables us to suppress the supply of labour in both production and the factors affecting well-being. If $K$ is the stock of capital of the economy's one and only commodity, output is taken to be $F(K)$, where $F(0) = 0$ (i.e., output is zero if there is no capital), $dF(K)/dK \gt 0$ (i.e., the marginal product of capital is positive), and $d^2 F(K)/dK^2 \le 0$ (i.e., the marginal product of $K$ does not increase with $K$). $F(K)$ is a flow (production at a moment in time), in contrast to $K$, which is a stock (quantity of capital, period). Notice also that output depends solely on the stock of capital. No mention is made of possible improvements in the quality of capital or labour. Thus, there is no prospect of technological progress or accumulation of human capital in Ramsey's model (but see Mirrlees 1967, for one of the first of many extensions of the Ramsey model that include technological advances in production and human capital formation); nor are there any natural resources in the model (but see Dasgupta and Heal 1974, for one of the first of many extensions of the Ramsey model that include natural capital in production). Let $C(t)$ be consumption at $t$. It is a flow (units of consumption per moment). Similarly, we write $K(t)$ for the stock of capital at $t$. As $dK(t)/dt$ is the rate of change in the capital stock at $t$, it is "net investment at $t$," which too is a flow. And because the capital stock is assumed not to depreciate, gross investment equals net investment. In Ramsey's model anticipated output at each moment equals the sum of intended investment and intended consumption. Intentions are always realized. To put it in technical language, the economy is in equilibrium at each moment, which is another way of saying that at each moment intended saving equals intended investment. (The assumption needs no explanation in a model with a single agent, but has real bite in a world where savers are not the same agents as investors.) Capital is assumed to be always fully deployed, and labour (which is hidden in the production function $F(K)$) is taken to be fully employed. Output at $t$ is $F(K(t))$. It follows that the economy is driven by the dynamical equation \[\tag{1} \frac{dK(t)}{dt} = F(K(t)) - C(t) \] Equation (1) says that if consumption is $C(t)$, investment is what remains of output. So, Ramsey's problem can be cast equally as, "How much of a nation's output should it consume?" If consumption is less than output at $t$ (i.e., $C(t) \lt F(K(t))$, investment is positive (i.e., $dK(t)/dt \gt 0)$ and the stock of capital increases; but if consumption exceeds output at $t$, investment is negative, which means capital is eaten into and the stock declines (i.e., $dK(t)/dt \lt 0).$ We now imagine that Government House is advised by a "socially-concerned citizen," the person being someone who is trying to determine the right balance between the economy's consumption and investment at each date. We shall call that person the decision maker, or DM. Ramsey imagined that DM is a Classical-Utilitarian. ## 2. The Classical-Utilitarian Calculus Classical Utilitarianism identifies the good as the expected sum of well-being over time and across generations. Here is Sidgwick (1907: 414) on the matter: > > > It seems ... clear that the time at which a man exists cannot > affect the value of his happiness from the universal point of view; > and that the interests of posterity must concern a Utilitarian as much > as those of his contemporaries, except in so far as the effect of his > actions on posterity - and even the existence of human > beings to be affected - must necessarily be more > uncertain. (Italics added) To formalize this, we consider an arbitrary date $t$ at which DM is deliberating. Let $\tau$ denote dates not earlier than $t$ (i.e., $\tau \ge t)$. Ramsey considered a deterministic, infinitely lived world (but see Yaari, 1965, for the first of many extensions of the Ramsey model that incorporate the risk of individual or societal extinction). Well-being is assumed to be a numerical quantity. Let $U(t)$ be well-being at $t$, and let $V(t)$ be an aggregate measure of the flow of well-being across time and generations, as evaluated at time $t$. Ramsey followed Sidgwick in assuming that \[\tag{2} V(t) = \int^{\infty}\_t[U(\tau)]d\tau \] $V(t)$ is intergenerational well-being at $t$. Because Ramsey's world is deterministic, $V(t)$ is also the expected value of $V(t)$. So Sidgwick's criterion is the $V(t)$ in equation (2). Well-being at any given date is assumed to be a function solely of consumption at that date. We therefore write $U(t) = U(C(t))$. Ramsey assumed that marginal well-being is positive (i.e., $dU(C)/dC \gt 0)$ but diminishes with increasing consumption levels (i.e., $d^2 U(C)/dC^2 \lt 0)$. The latter property implies that $U(C)$ is a strictly concave function. (Edgeworth 1885, had routinized the idea that marginal well-being declines with increasing consumption.) Thus equation (2) can be written as \[\tag{3} V(t) = \int^{\infty}\_t [U(C(\tau))]d\tau \] Classical Utilitarianism, as reflected in equation (3), requires that if $U$ is a numerical measure of well-being, then so is $\alpha U+\beta$, where $\alpha$ is a positive number and $\beta$ is a number of either sign. Formally, we say that $U$ is unique up to "positive affine transformations." We confirm presently that the theory's recommendations are invariant under such transformations. ### 2.1 Zero Discounting of Future Well-Beings In equation (3), future values of $U$ are not discounted when viewed from the present moment, $t$. This particular move has provoked more debate among economists and philosophers than any other feature of Ramsey's theory of optimum saving. The debate has on occasion been shriller than even we economists are used to (see in particular Nordhaus 2007). At the risk of generalizing wildly, economists have favoured the use of positive rates to discount future well-beings (e.g., Arrow and Kurz 1970), whereas philosophers have insisted that the well-being of future people should be given the same weight as that of present people (e.g., Parfit 1984). What would Classical Utilitarianism with positive discounting of future well-beings look like? Let $\delta \gt 0$ be the rate at which it is deemed desirable to discount future well-beings (for simplicity we take the discount rate to be constant). Then, in place of equations (2)-(3), intergenerational well-being at $t$, would read as \[\tag{4} \begin{align} V(t) &= \int^{\infty}\_t [U(\tau)e^{-\delta(\tau -t)}]d\tau \\ &= \int^{\infty}\_t [U(C(\tau))e^{-\delta(\tau -t)}]d\tau, t \ge 0 \\ \end{align}\] In equation (4), $\delta$ the "time discount rate" and $e^{-\delta}$ the resulting "time discount factor." $\delta \gt 0$ implies $e^{-\delta} \lt 1$. That means $e^{-\delta(\tau -t)}$ tends to zero exponentially as $\tau$ tends to infinity. In the latter part of his paper Ramsey (1928: 553-555) did use equation (4) to study the problem of optimum saving, but he did not approve of the formulation. Instead, he wrote (p. 543) that to discount later $U$'s in comparison with earlier ones is "... ethically indefensible and arises merely from the weakness of the imagination." In a book that inaugurated the formal study of economic development, Harrod (1948: 40) followed suit by calling the practice a "... polite expression for rapacity and the conquest of reason by passion." Strong words, but to some economists, the Ramsey-Harrod stricture in a deterministic world reads like a Sunday pronouncement. Solow (1974a: 9) expressed this feeling exactly when he wrote, "In solemn conclave assembled, so to speak, we ought to act as if the [discount rate on future well-beings] were zero." But the matter cannot be settled without a study of production and consumption possibilities open to an economy. Consider the following tension between two sets of considerations: 1. Low rates of consumption by generations sufficiently far into the future would not be seen to be a bad thing by the current DM if future well-beings were discounted at a positive rate. So today's DM would recommend high consumption rates for now and the near future even if that meant generations in the distant future would live in penury. But if such a policy were followed, the demands of a further moral requirement to Classical Utilitarianism that DM may hold, namely, "intergenerational equity," would not be met. Therefore we should follow Ramsey and not discount future well-beings. 2. Write $dF(K)/dK$ as $F\_K$. From equation (1) it is simple to deduce that $F\_K$ is the rate of return on investment. In Ramsey's economy $F\_K \gt 0$, which means every unit of output that is saved yields more than a unit of future consumption, other things equal. For example, if DM were to reduce consumption at $t$ by a unit, the additional consumption that would be available in the briefest of periods later - we write that as $\Delta t$ - without affecting consumption at any future date would be $1+[dF(K(t))/dK(t)]\Delta t$. The productivity of capital is thus tied to the arrow of time, which creates a bias in favour of future generations. This bias gives bite to the adage, "We can do something for posterity, but what can posterity ever do for us?" The thought inevitably arises that perhaps the bias should be countered in DM's calculus if some attention were to be given by her to intergenerational equity in realized well-being as a supplement to Classical Utilitarianism. That in turn suggests that DM should abandon Ramsey and discount future well-beings at a positive rate. The force of each consideration has been demonstrated in the economics literature. It has been shown in the context of a simple model that if production requires produced capital and exhaustible resources, then optimum consumption declines to zero in the long run if future well-beings are discounted at a positive rate (Dasgupta and Heal 1974), but increases indefinitely if we follow Ramsey in not discounting future well-beings (Solow 1974b). The exercises tell us that the long-run features of optimum saving policies depend on the relative magnitudes of the rate at which future well-beings are discounted and the long-term productivity of capital assets. There is a more general point here, which was explored by Koopmans (1960, 1965, 1967, 1972) in a remarkable set of publications on the idea of economic development. In such complex exercises as those involving consumption and investment over a long time horizon, it is foolish to regard any ethical principle (e.g., Classical Utilitarianism) as sacrosanct. One can never know in advance what it may run up against. A more judicious tactic than Ramsey's would be to to play off one set of ethical assumptions against another in not-implausible worlds, see what their implications are for the distribution of well-being across generations, and then appeal to our intuitive senses before arguing over policy. Settling ex ante whether to use a positive rate to discount future well-beings could be a self-defeating move.[1] ## 3. The Problem of Optimum Saving Ramsey considered a world with an indefinite future. This could appear to be an odd move, but it has a strong rationale. Suppose DM were to choose a horizon of $T$ years. As she doesn't know when our world will end, she will want to specify the resources that should be left behind at $T$ in case the world doesn't terminate then. But to find a justification for the amount to leave behind at $T$, DM will need an assessment of the world beyond $T$. That would, however, amount to including the world beyond $T$. And so on. Denote a consumption stream from the present $(t = 0)$ to infinity as $\{C(t)\}.$ $K(0) \gt 0$ circumscribes the economy; it is the quantity of capital that society has inherited from the past. Mathematicians would call $K(0)$ an "initial condition." The problem Ramsey set himself was to determine the consumption stream $\{C(t)\}$ from 0 to infinity that DM would select if she were a Classical Utilitarian. ### 3.1 Undiscounted Utilitarianism Call a consumption stream $\{C(t)\}$ feasible if it satisfies equation (1) with initial condition $K(0)$. In Ramsey's deterministic world the Classical Utilitarian formulation of the problem of optimum national saving at date $t = 0$ is thus: "From the set of all feasible consumption streams, find that $\{C(t)\}$ which maximizes \[ V(0) = \int^{\infty}\_0 [U(C(t))]dt." \] We will call this optimization problem, Ramsey Mark I. There is a serious difficulty with Ramsey Mark I: it is not coherent. Infinite sums don't necessarily converge. For any $\{C(t)\}$ for which the infinite integral doesn't converge, $V(0)$ doesn't exist. If the integral is non-convergent for every feasible consumption streams $\{C(t)\}$, the maximization problem is meaningless: One cannot maximize something that appears to be a real-valued function $V(0)$ when in fact the function doesn't exist. The force of this observation can be seen in Example 1 (attributed to David Gale) Suppose as an extreme special case of the Ramsey economy, $F(K) = 0$ for all $K \ge 0$. Then equation (1) reduces to \[\tag{5} \frac{dK(t)}{dt} = - C(t) \] The economy described in equation (5) consists of a non-deteriorating piece of cake, of size $K(0) \gt 0$ at the initial date. It is obvious that every consumption stream $\{C(t)\}$ satisfying equation (5) tends to zero in the long run. Formally, $C(t) \rightarrow 0$ as $t \rightarrow \infty$. Because the $U$-function is unique up to positive affine transformations, we may without any loss of generality normalize it so that $U(0) \ne 0$. It is then obvious that for all feasible $\{C(t)\}$, $V(0)$ in Ramsey Mark I diverges to minus infinity if $U(0) \lt 0$, but diverges to plus infinity if $U(0) \gt 0$. That an optimum policy does not exist in the cake-eating model can be seen if we now recall that $U(C)$ has been assumed to be strictly concave. The assumption implies that any non-egalitarian distribution of consumption among the generations can be improved upon by a suitable redistribution. The ideal distribution would be equal consumption for all generations. The only consumption stream with the latter property is $C(t) = 0$ for all $t$. But that's the worst possible distribution. QED ### 3.2 Re-normalizing Undiscounted Utilitarianism The question arises whether there are circumstances in which there is a best consumption stream even though $V(0)$ does not converge for all consumption streams. Ramsey formulated the question by altering the way the saving problem is posed. Imagine that well-being is bounded above no matter how large consumption happens to be. Let $U$ be the numerical measure of well-being that DM chooses to work with. (All positive affine transformations of $U$ would be equally legitimate measures of well-being.) Let $B$ be the lowest upper bound of $U$. Ramsey christened it "Bliss". Because the rate of return on investment $(F\_K)$ in his model is positive, consumption would grow indefinitely and tend to infinity in the long run if saving rates were suitably chosen. That means there are possible paths of economic development in which $U(C(t))$ tend to $B$ in the long run. But that implies there are possible paths of economic development in which the short-fall of $U(C(t))$ from $B$ tends to zero in the long run. If the short-fall tends to zero fast enough, the undiscounted integral of the difference between $U(C(t))$ and $B$ would exist, and DM could seek to maximize the modified integral. So we have Ramsey Mark II, which reads as "From the set of all feasible consumption streams, find that $\{C(t)\}$ which maximizes \[ V(0) = \int^{\infty}\_0 [U(C(t))-B]dt." \] Notice that Mark II is a transformation of Mark I. The transformation amounts to re-normalizing the optimality criterion. Not only was the move from Mark I to Mark II on Ramsey's part ingenious, it also displayed his moral integrity. It would have been easy enough for him to ask DM instead to discount future consumption and expand the range of circumstances in which Utilitarianism provides an answer to the problem DM is attempting to solve. He chose not to do that. Ramsey's intuition in moving from Mark I to Mark II was powerful, but in a paper that initiated the modern literature on the Ramsey problem, Chakravarty (1962) observed that to rely exclusively on the condition Ramsey had identified as being necessary for a consumption stream to be the optimum (see below) can lead to absurd results (see below, Sect. 4). In effect Chakravarty observed that infinite integrals, even when cast in the re-normalized form in Ramsey Mark II, don't necessarily converge to finite values. ### 3.3 The Overtaking Criterion What was needed was to de-link the question whether infinite well-being integrals converge from the question whether optimum consumption streams exist. That insight was provided by Koopmans (1965) and von Weizsacker (1965). The latter author's re-statement of the problem of optimum saving was as follows: We say that the feasible consumption stream $\{C^\(t)\}$ is superior* to a feasible consumption stream $\{C(t)\}$ if there exists $T \gt 0$ such that for all $t \ge T$, \[\tag{6} \int^t\_0 [U(C^\(s))]ds \ge \int^t\_0 [U(C(s))]ds \] We call $\{C^\(t)\}$ optimum if it is superior to all other feasible consumption streams. The condition that is represented in inequality (6) is known as the Overtaking Criterion (OC), for that is what it is. OC avoids asking whether the integrals on either side of inequality (6) converge as $t \rightarrow \infty$. If they do, OC reduces to Classical Utilitarianism. But OC is able to respond to Ramsey's saving problem in a wider class of situations. In his work Koopmans (1965) identified a canonical economic model in which the $U$-function is bounded above and in which Ramsey Mark II is equivalent to an optimization problem that is posed in terms of OC. What are we to make of the ethics of discounting the well-beings of future generations? Ramsey (1928) began by dismissing it but then studied it at the tail end of his paper. DM could of course justify discounting future well-being if there is a possibiliy of future extinction. Sidgwick (1907) himself noted that in the passage quoted earlier. If Classical Utilitarianism is taken to commend the expected sum of well-beings, then the "hazard rate" at date $t$ (i.e., the probability of extinction at date $t$ conditional on society surviving until $t)$ would appear in the expression for expected well-being as a discount rate for well-being at $t$. The question remains whether Classical Utilitarianism would insist on zero-discounting of future utilities in a deterministic world. In a remarkable pair of works Koopmans (1960, 1972) exposed internal contradictions in ethical reasoning in a deterministic world in both Ramsey Mark I and Ramsey Mark II. He (and subsequently Diamond, 1965) showed that if relatively weak normative requirements are imposed on the concept of intergenerational well-being in a deterministic world, equal treatment of the $U$-function across generations has to be abandoned. We turn to that now. ### 3.4 Discounted Utilitarianism It transpires the mathematics is a lot simpler if, instead of assuming time is continuous, time is taken to be discrete. Thus we now assume that $t = 0,1,2,\ldots$ . Assume also that intergenerational well-being at $t = 0$ can be measured in terms of a numerical function $V$. The idea is to require the function, which is defined on infinite well-being streams, to satisfy properties that reflect ethical directives. Let $\{U(t)\}$ be an infinite well-being stream, that is, $\{U(t)\} = (U(0),U(1),\ldots ,U(t),\ldots)$. We say $V(\{U(t)\})$ is continuous if in an appropriate mathematical sense the values of $V$ for well-being streams $\{U(t)\}$ that don't differ much in the space of $\{U(t)\}$s are close to one another. A further condition on the $V$-function that is ethically attractive is "monotonicity". To define the notion let us say a well-being stream is "superior" to another if no generation enjoys less well-being along the former than along the latter and if there is at least one generation that enjoys greater well-being in the former than it does in the latter. We say that $V$ is monotonic if $V$ is larger for a well-being stream than it is for another if the former is superior to the latter. Both properties are attractive. Lexicographic orderings notwithstanding, there are not convincing arguments against continuity. Of course Rawls (1972) placed priority rules and the lexicographic orderings on the objects of interest in his conception of justice that come with them at the centre of his theory, but that has proved to have been one of his most contentious moves. The richness and depth of his analysis would not be lessened if small tradeoffs were admitted between the objects of justice. And it's hard to find reasons against monotonicity. Even Rawls, whose work was so pointed toward distributive justice, insisted on monotonicity. But it can be shown that any $V$-function that satisfies continuity and monotonicity must have generation discounting built into it. It would seem the real numbers are not rich enough to accommodate infinite well-being streams in a manner that respects continuity and monotonicity while awarding the well-beings of all generations equal weight. Proof of the proposition is in Diamond (1965), and was attributed by the author to Menahem Yaari. So we now introduce positive well-being discounting in the $V$-function and formulate Ramsey Mark III. Return once again to the formulation where time is continuous. As previously, we say a consumption stream $\{C(t)\}$ is feasible if it satisfies equation (1) with an initial capital stock of $K(0)$. Ramsey Mark III (Ramsey 1928, 553-555) is then: "From the set of all feasible consumption streams, find that $\{C(t)\}$ which maximizes \[ V(0) = \int^{\infty}\_0 [U(C(t))e^{-\delta t}]dt, \delta \gt 0." \] In Mark III the discount rate $\delta$ is a positive constant. That means the corresponding discount factor $e^{-\delta}$ is less than 1. The latter in turn can be shown to mean that in a wide range of economic models $e^{-\delta t}$ tends to zero at so fast a rate that Mark III has an answer. Let $\{C^\(t)\}$ be the solution of Ramsey Mark III. Heuristically it is useful to imagine that there is a DM at each date. The measure of intergenerational well-being for the DM at date $t$ is the $V(t)$ of equation (4). Notice that the ethical views of the successive DMs are congruent with one another. There is thus no need for the DMs to draw up an "intergenerational contract". The DM at every date will want to choose the level of consumption it deems to be optimum, aware that succeeding DMs will choose in accordance with what she had planned for them. In modern game theoretic parlance, Ramsey's optimum consumption stream $\{C^\(t)\}$ is a "non-cooperative" (Nash) equilibrium among the DMs. ## 4. The Ramsey Rule and Its Ramifications We now construct an informal version of the variational argument Ramsey used for determining $\{C^\(t)\}$ in Mark III. Loosely speaking, the DMs require the marginal rate of ethically indifferent substitution between consumption at any two brief periods of time to equal the marginal rate at which consumption can be transformed between those same pair of brief periods of time. Their equality (i.e., the right balance from among the "desirables" and the "feasibles") is a necessary* property of an optimum consumption stream. Ramsey constructed a mathematical expression of the property, but did not look for conditions that, taken together, are both necessary and sufficient. We will use a simple example, which is also in his paper, to show how a sufficient condition can be obtained. ### 4.1 The Variational Argument Write $dU/dC = U\_C$ and $d^2 U/dC^2 = U\_{CC}.$ Let $\{C(t)\}$ be a feasible consumption stream. We first deduce a formal expression for the marginal rate of ethically indifferent substitution between consumption at any two brief periods of time. Suppose the intention is to reduce consumption at some future date $t$ by a small quantity $\Delta C(t)$ and raise consumption at a nearby date $t+\Delta t$ while keeping consumption at all other dates the same as in $\{C(t)\}$. The loss in well-being that would follow from the move is $e^{-\delta t}U\_{C(t)}\Delta C(t)$. We now seek to determine the percentage increase in consumption that would be required at $t+\Delta t$ if $V(0)$ is to remain unchanged; because that's the marginal rate of ethically indifferent substitution between consumption at $t$ and consumption at $t+\Delta t$. Denote that rate by $\varrho(t)$. Then $\varrho(t)$ must be the percentage rate at which discounted marginal well-being declines at $t$. It also follows that $\varrho(t)$ is the rate the DM at $t = 0$ would use to discount a unit of consumption at $t$ so as to bring it to the present (because that's what is meant by the percentage rate at which discounted marginal well-being declines at $t$ - for a formal demonstration, see Dasgupta, 2008). Some economists call $\varrho(t)$ the consumption rate of interest (Little and Mirrlees 1974), others call it the social rate of discount (Arrow and Kurz 1970). $\varrho(t)$ is a fundamental object in social cost-benefit analysis. Let $\Delta$ be vanishingly small. Then, by definition \[\tag{7} \varrho(t) = - [d(e^{-\delta t}U\_{C(t)})/dt]/e^{-\delta t}U\_{C(t)} \] So as to simplify the notation let $g(C(t))$ denote the percentage rate of growth in $C(t)$ (i.e. $g(C(t)) = [dC(t)/dt]/C(t)$, which can be negative), and let $\sigma(C)$ denote the elasticity of marginal well-being (i.e., $\sigma(C) = -CU\_{CC}/U\_C \gt 0)$. Equation (7) then simplifies to \[\tag{8} \varrho(t) = \delta + \sigma(C(t))g(C(t)) \] Because $\{C^\(t)\}$ is by assumption the optimum, no feasible deviation from $\{C^\(t)\}$ can increase $V(0)$. That means the consumption rate of interest $(\varrho(t))$ must equal the social rate of return on investment $(F\_{K(t)})$ at every $t$. To see why, suppose in some vanishingly small interval of time $F\_{K(t)} \gt \varrho(t)$. Then $V(0)$ could be increased by consuming a unit less at $t$ and enjoying the return of $(1+F\_{K(t)})$ soon after. Alternatively, if $F\_{K(t)} \lt \varrho(t), V(0)$ could be increased by consuming a unit more at $t$ and reducing consumption soon after by an amount equal to the return $(1+F\_{K(t)})$. But that means the consumption rate of interest $\varrho(t)$ equals the social rate of return $F\_{K(t)}$ along $\{C^\(t)\}$ at every date. Using equation (8) we have, \[\tag{9} \delta + \sigma(C(t))g(C(t)) = F\_{K(t)} \] Equation (9) is the Ramsey Rule. It is a necessary condition for optimality in Ramsey Mark III* and is unarguably the most famous equation in intertemporal welfare economics. The rule is a formal statement of the requirement of $\{C^\(t)\}$, that the marginal rate of substitution between consumption at two nearby dates (the left-hand-side of eq. 9) equals the marginal rate of transformation between consumption at those same pair of nearby dates (the right-hand-side of eq. (9). It is simple to confirm that equation (9) is invariant under positive affine transformations of the $U$-function. ### 4.2 Incompleteness in Ramsey's Analysis Presently we will specify a $U$-function for which $\sigma$ is independent of $C$. For the moment we merely suppose that $\sigma$ is constant. In that case the Ramsey Rule reads as \[\tag{10} \delta + \sigma g(C(t)) = F\_{K(t)} \] In Ramsey Mark III, $K(0)$ is given as an inheritance from the past. That means $F\_{K(0)}$ is given as an initial condition, it is not a choice for the DM at $t = 0$. Moreover $\delta$ and $\sigma$ are parameters, both reflecting ethical values. The DM can therefore determine $g(C(0))$ from equation (10). But that's the optimum percentage rate of growth in consumption at the initial date. The Ramsey Rule gives the DM an equation for determining the initial growth rate of consumption, but it does not say what the initial level* of consumption ought to be. Below we show by way of an example that there are an infinity of feasible consumption paths satisfying the Ramsey Rule. It follows that the DM at $t = 0$ needs a further condition to determine $C^\(0)$. Example 2* (the linear economy) Assume \[\begin{align} \tag{11a} F(K) &= \mu K, \mu \gt 0 \\ \tag{11b} U(C) &= - C^{-(\sigma -1)}, \sigma \gt 1 \end{align}\] From equation (11a) it follows that $F\_K = \mu$, which means the rate of return on investment is constant. From equation (11b) it follows that $\sigma$ is the elasticity of marginal well-being. Notice also that $U(C) \rightarrow -\infty$ as $C \rightarrow 0$ and that, under the chosen normaliztion of the $U$-function, $U(C) \rightarrow 0$ as $C \rightarrow \infty$. Using equation (11a) in equation (1) yields, \[\tag{12} \frac{dK(t)}{dt} = \mu K(t) - C(t) \] Write $m = (\mu -\delta)/\sigma$. Applying equations (11a-b) to equation (10) reduces the Ramsey Rule to \[\tag{13} \frac{dC(t)}{dt} = [(\mu - \delta)/\sigma]C(t) = mC(t) \] Equation (13) says that if $\mu \lt \delta , C(t)$ declines to 0 at an exponential rate. Empirically, the pausible case to consider is $\mu \gt \delta$, which is what we shall do here. It means that the rate of return on investment $(\mu)$ exceeds the rate at which time is discounted $(\delta)$. And that in turn means $m \gt 0$. Integrating equation (13) yields \[\tag{14} C(t) = C(0)e^{mt} \] Equation (14) says $C(t)$ grows exponentially at the rate $m$. We reconfirm a point that was made previously, that although equation (14) reveals the rate of growth optimum consumption at the initial date (i.e., $t = 0)$, it doesn't reveal the initial level of consumption (i.e., $C(0)$). That's the indeterminacy in the Ramsey Rule. The simplest way to determine the optimum initial consumption, $C^\(0)$, is to observe from equation (14) that if $C^\(t)$ grows indefinitely at the rate $m$, so should $K(t)$ be required to grow at that same rate. The reason is that if the growth rate of $K(t)$ were to be less than $m$, capital would be eaten into, which means the stock would be exhausted in finite time. The economy would then cease to exist $(V(0)$ would be minus infinity if the future trajectory of the economy were to be thus.) If on the other hand the growth rate of $K(t)$ were to exceed $m$, there would be over-accumulation of capital, in the sense that consumption would be lower at every date than it needs be. The situation would resemble one where DM throws away a part of the initial capital stock $K(0)$ and then settles on a saving behaviour that satisfies the Ramsey Rule. Exponential growth in our linear economy (eq. 11a) tells us that the saving rate should be constant. Let us define the saving rate, $s$, as the proportion of output (GDP) that is invested at each instant. Then equation (1) can be re-written as \[\tag{15} \frac{dK(t)}{dt} = s\mu K(t) \] Equation (15) says that intended saving equals intended investment. Integrating equation (15) yields \[\tag{16} K(t) = K(0)e^{s\mu t} \] But we are insisting that both $K(t)$ and $C(t)$ should grow at the same rate. Equations (14) and (16) therefore imply \[\tag{17} m = \frac{\mu -\delta}{\sigma} = s\mu \] The saving rate in equation (17) is the optimum. So we write it as $s^\$. Thus \[\tag{18} s^\ = \frac{m}{\mu} = \frac{\mu -\delta}{\sigma\mu} \lt 1 \] Equations (16)-(18) tell us that the optimum rate of growth of consumption, $g^\$, is \[\tag{19} g^\ = \frac{\mu -\delta}{\sigma} \gt 0 \] Notice also that if $\delta = 0$, equation (18) reduces to \[\tag{20} s^\* = \frac{1}{\sigma} \] Equation (20) offers as elegant a simplified answer as there could be to the question with which Ramsey started his paper. ### 4.3 The Transversality Condition The linear technology (eq. 11a) and the iso-elastic $U$-function (eq. 11b) allowed us to recognise immediately that if a consumption stream satisfying the Ramsey Rule is to be the optimum, both capital and consumption should grow at the same exponential rate, $m$. Identifying a sufficient condition for optimality in more general models is a lot more difficult. What we need is a condition on the long-run features of a consumption stream satisfying the Ramsey Rule that can ensure it is the optimum. von Weizsacker (1965) showed that the required condition relates to the long-run behaviour of the social value of capital associated with that consumption stream. We now formalize the condition. Let $U$ be the unit of account. Consider a consumption stream $\{C(t)\}$. It follows that $U\_{C(t)}$ is the social worth of a marginal unit of consumption. Write $P(t)$ for $U\_{C(t)}. P(t)$ is called the (spot) accounting price of consumption. Because $e^{-\delta t}P(t)$ is the discounted value of $P(t)$, it is called the present-value accounting price of consumption. If $\{C(t)\}$ satisfies the Ramsey Rule in Mark III, $e^{-\delta t}P(t)$ is also the present-value accounting price of a unit of capital stock. von Weizsacker (1965) showed that a sufficient condition for the optimality of $\{C(t)\}$ is $e^{-\delta t}P(t)K(t) \rightarrow A$ as t $\rightarrow \infty$, where $A$ is a (finite) non-negative number. In words, a necessary and sufficient condition for $\{C(t)\}$ to be the optimum is (i) that it satisfies the Ramsey Rule, and (ii) that the present-value of the economy's stock of capital is finite. Condition (ii), which is widely known as the "transversality condition," eliminates those feasible consumption streams that satisfy the Ramsey Rule but along which there is excessive saving. A simple calculation confirms that in Example 2 the transversality condition is satisfied if the saving rate is $s^\$ (eq. 18). ### 4.4 Numerical Estimates of the Optimum Rate of Saving Equation (18) says that $s^\$ is an increasing function of the return on investment $(\mu)$, a decreasing function of the time rate of discount $(\delta)$, and a decreasing function of the elasticity of marginal well-being $(\sigma)$. Each of these properties is intuitively obvious: (1) The higher is the rate of return on investment $(\mu)$, the greater is the gain to future generations from a marginal increase in saving by initial generations. That says the optimum rate of saving should be an increasing function of $\mu$, other things equal. (2) The larger is the value of the time discount rate $(\delta)$ chosen by DM, the lower is the weight that she awards to the well-being of future generations. That implies higher optimum consumption levels for early generations (Sect. 2.1), which in turn implies that the optimum rate of saving is lower, other things equal. (3) As the return on investment is positive $(\mu \gt 0)$, the arrow of time displays a bias in favour of future generations (Sect. 2.1). But the larger is the chosen value of $\sigma$, the more DM displays concerns over equity in consumption across the generations. Therefore, the larger is that concern, the higher is the optimum rate of consumption to be enjoyed by initial generations. So we should expect the optimum rate of saving to be a decreasing function of $\sigma$, other things equal. It is instructive to consider stylized figures for the parameters on the right-hand-sides of equations (18) and (19), respectively. Although stylized, they are figures for the pair of ethical parameters $\sigma$ and $\delta$ that economists who have written on the economics of climate change have assumed in their work. To be sure, the welfare economics of climate change has demanded more complicated models than the model that is represented in equations (1) and (11a), but as we confirm below, it has not offered any additional theoretical insights. In what follows we take a year to be the unit of time and assume that $\mu = 0.05$ (i.e., 5% a year). Along the optimum, the consumption rate of interest equals the rate of return on investment (the Ramsey Rule), which means that the optimum consumption rate of interest equals a constant 5% a year. A figure of 5% a year for $\mu$ implies a capital-output ratio $(1/\mu)$ of 20 years, which is far higher than the estimates of capital-output ratios from inter-industry studies that economists in various parts of the world have arrived at (Behrman 2001); a representative figure for 1/$\mu$ in that literature is 3 years. But their estimates have been based on a definition of "capital" that is confined to "produced" capital, such as factories, roads, ports, and buildings. Human capital (education, health, knowledge) is missing from them, as is natural capital (ecosystems, sub-soil resources). Ramsey's model, as encapsulated in equation (11a), embraces all forms of capital goods. No doubt his formulation requires a heroic (read, impossible!) feat of aggregation, but when all capital goods that enter production are taken into account, we should expect an aggregate capital-output ratio (which we should call the (inclusive) wealth-output ratio), to be a lot higher than 3 years; perhaps even higher than 20 years (Arrow et al., 2012, 2013). Large categories of capital goods are absent from the national economic accounts that inform we economists' understanding of production and consumption possibilities (Dasgupta, 2019). It would thus seem there is still a long way to go before we can reach a good approximation of what we should bequeath to our descendants. Example 3 (taken from the economics of climate change) We now turn our attention to the values of the two ethical parameters in equation (11b) that were chosen by three economists in their study of the economics of climate change. \[\begin{align} \tag\{Cline (1992)} \sigma = 1.5 \quad &\text{and} \quad \delta = 0 \\ \tag\{Nordhaus (1994)} \sigma = 1 \quad &\text{and} \quad \delta = 0.03 \text{ (3% a year)} \\ \tag\{Stern (2007)} \sigma = 1 \quad &\text{and} \quad \delta = 0.001 \text{ (0.1% a year)} \end{align}\] (NB: $\sigma = 1$ corresponds to the logarithmic well-being function, that is $U(C) =$ log$C$, and can be obtained as a limit of the functional form of $U(C)$ in equation (11b) as $\sigma \rightarrow 1.)$ We impose those parameter values to find that the optimum saving rate $s^\$ (eq. 18) and the optimum rate of growth of consumption (eq. 19) are, in turn: \[\begin{align} \tag{21a} s^\* = 67\% \quad &\text{and} \quad g^\* = 3.3\% \text{ a year (Cline)} \\ \tag{21b} s^\* = 40\% \quad &\text{and} \quad g^\* = 2.0\% \text{ a year (Nordhaus)} \\ \tag{21c} s^\* = 98\% \quad &\text{and} \quad g^\* = 4.9\% \text{ a year (Stern)} \end{align}\] ### 4.5 Commentary A national saving rate of 40% (eq. 21b) is no doubt high by the standards of contemporary western economies, but there are countries that in recent years have achieved 40-45% saving rates (China is a prominent example). A figure of 67% for $s^\$ (eq. 21a) is higher than the saving rate in any country, but is not beyond belief. The truly outlandish figures is 98% (eq. 21c). It is outlandish especially because the figure is the optimum saving rate no matter how small $K(0)$ happens to be. Admittedly, the model here (eqs. 11a-b) is phenomenally stylized, but it does bring out sharply the observation of Koopmans (1965), that it is foolish to assume $\delta = 0$ (or close to 0) without first checking its possible consequences for the distribution of well-being across the generations. Equation (19) has shown that the optimum growth rate of consumption is bounded above by $\mu$, which explains why $g^\$ is less than 5% a year for each of the three parametric specifications we have considered. The specifications come from three studies in the welfare economics of global climate change, in which the authors worked with models that are a lot more complex than Ramsey's. And yet their findings are exactly what his formulation would point to (Dasgupta 2008), namely, that other things equal, the lower is the chosen value of $\delta$ and/or the larger is the damage to future well-being that is expected to be caused by global climate change, the greater is the investment level DM should recommend to avert climate change or soften the effects of that change on human well-being. The often shrill debate (e.g., Nordhaus 2007) on the extent to which global investment should be directed at reducing the deletarious effects of climate change was spurred by differences in model specification among climate-change economists. The linear technology (eq. 11a) and the iso-elastic $U$-function (eq. 11b), when taken together, have offered deep insights even though we have restricted the discussion to pen-and-paper calculations here. The functional forms are not believable; nevertheless, Ramsey made use of them. His paper showed that unbelievably simplified models, provided their construction is backed by strong intuition, can illuminate questions that are seemingly impossible to frame, let alone to answer quantitatively. That has been Ramsey's enduring gift to theoretical economics.
well-being	## 1. The Concept Popular use of the term 'well-being' usually relates to health. A doctor's surgery may run a 'Women's Well-being Clinic', for example. Philosophical use is broader, but related, and amounts to the notion of how well a person's life is going for that person. A person's well-being is what is 'good for' them. Health, then, might be said to be a constituent of my well-being, but it is not plausibly taken to be all that matters for my well-being. One correlate term worth noting here is 'self-interest': my self-interest is what is in the interest of myself, and not others. The philosophical use of the term also tends to encompass the 'negative' aspects of how a person's life goes for them. So we may speak of the well-being of someone who is, and will remain in, the most terrible agony: their well-being is negative, and such that their life is worse for them than no life at all. The same is true of closely allied terms, such as 'welfare', which covers how a person is faring as a whole, whether well or badly, or 'happiness', which can be understood--as it sometimes was by the classical utilitarians from Jeremy Bentham onwards, for example--to be the balance between good and bad things in a person's life. But note that philosophers also use such terms in the more standard 'positive' way, speaking of 'ill-being', 'ill-faring', or, of course, 'unhappiness' to capture the negative aspects of individuals' lives. Most philosophical discussion has been of 'goods' rather than 'bads', but recently more interest has been shown in the latter (e.g. Kagan 2015; Bradford 2021). 'Happiness' is often used, in ordinary life, to refer to a short-lived state of a person, frequently a feeling of contentment: 'You look happy today'; 'I'm very happy for you'. Philosophically, its scope is more often wider, encompassing a whole life. And in philosophy it is possible to speak of the happiness of a person's life, or of their happy life, even if that person was in fact usually pretty miserable. The point is that some good things in their life made it a happy one, even though they lacked contentment. But this usage is uncommon, and may cause confusion. Over the last few decades, so-called 'positive psychology' has hugely increased the attention paid by psychologists and other scientists to the notion of 'happiness'. Such happiness is usually understood in terms of contentment or 'life-satisfaction', and is measured by means such as self-reports or daily questionnaires. Is positive psychology about well-being? As yet, conceptual distinctions are not sufficiently clear within the discipline. But it is probably fair to say that many of those involved, as researchers or as subjects, are assuming that one's life goes well to the extent that one is contented with it--that is, that some kind of hedonistic account of well-being is correct. Some positive psychologists, however, explicitly reject hedonistic theories in preference to Aristotelian or 'eudaimonist' accounts of well-being, which are a version of the 'objective list' theory of well-being discussed below. A leader in the field, Martin Seligman, for example, has suggested that, rather than happiness, positive psychology should concern itself with positive emotion, engagement, relationships, meaning and accomplishment ('Perma') (Seligman 2011). When discussing the notion of what makes life good for the individual living that life, it is preferable to use the term 'well-being' instead of 'happiness'. For we want at least to allow conceptual space for the possibility that, for example, the life of a plant may be 'good for' that plant. And speaking of the happiness of a plant would be stretching language too far. (An alternative here might be 'flourishing', though this might be taken to bias the analysis of human well-being in the direction of some kind of natural teleology.) In that respect, the Greek word commonly translated 'happiness' (eudaimonia) might be thought to be superior. But, in fact, eudaimonia seems to have been restricted not only to conscious beings, but to human beings: non-human animals cannot be eudaimon. This is because eudaimonia suggests that the gods, or fortune, have favoured one, and the idea that the gods could care about non-humans would not have occurred to most Greeks. It is occasionally claimed that certain ancient ethical theories, such as Aristotle's, result in the collapse of the very notion of well-being. On Aristotle's view, if you are my friend, then my well-being is closely bound up with yours. It might be tempting, then, to say that 'your' well-being is 'part' of mine, in which case the distinction between what is good for me and what is good for others has broken down. But this temptation should be resisted. Your well-being concerns how well your life goes for you, and we can allow that my well-being depends on yours without introducing the confusing notion that my well-being is constituted by yours. There are signs in Aristotelian thought of an expansion of the subject or owner of well-being. A friend is 'another self', so that what benefits my friend benefits me. But this should be taken either as a metaphorical expression of the dependence claim, or as an identity claim which does not threaten the notion of well-being: if you really are the same person as I am, then of course what is good for you will be what is good for me, since there is no longer any metaphysically significant distinction between you and me. Well-being is a kind of value, sometimes called 'prudential value', to be distinguished from, for example, aesthetic value or moral value. What marks it out is the notion of 'good for'. The serenity of a Vermeer painting, for instance, is a kind of goodness, but it is not 'good for' the painting. It may be good for us to contemplate such serenity, but contemplating serenity is not the same as the serenity itself. Likewise, my giving money to a development charity may have moral value, that is, be morally good. And the effects of my donation may be good for others. But it remains an open question whether my being morally good is good for me; and, if it is, its being good for me is still conceptually distinct from its being morally good. A great deal of attention has been paid in philosophy to the issue of moral 'normativity', less so to that of prudential normativity (recent exceptions are Dorsey 2021; Fletcher 2021). The most common view of well-being is 'invariabilism', according to which there is a single account of well-being for all individuals for whom life can go well or badly (Lin 2018). Some have argued, however, that we should develop a variabilist view, according to which, for example, there might be one theory of well-being for adults and another for children (see Skelton 2015). According to Benatar (2006), existence for any individual with well-being is always of overall negative value. On well-being and death, see Bradley (2006). ## 2. Moore's Challenge There is something mysterious about the notion of 'good for'. Consider a possible world that contains only a single item: a stunning Vermeer painting. Leave aside any doubts you might have about whether paintings can be good in a world without viewers, and accept for the sake of argument that this painting has aesthetic value in that world. It seems intuitively plausible to claim that the value of this world is constituted solely by the aesthetic value of the painting. But now consider a world which contains one individual living a life that is good for them. How are we to describe the relationship between the value of this world, and the value of the life lived in it for the individual? Are we to say that the world has a value at all? How can it, if the only value it contains is 'good for' as opposed to just 'good'? And yet we surely do want to say that this world is better ('more good') than some other empty world. Well, should we say that the world is good, and is so because of the good it contains 'for' the individual? This fails to capture the idea that there is in fact nothing of value in this world except what is good for the individual. Thoughts such as these led G.E. Moore to object to the very idea of 'good for' (Moore 1903, pp. 98-9). Moore argued that the idea of 'my own good', which he saw as equivalent to what is 'good for me', makes no sense. When I speak of, say, pleasure as what is good for me, he claimed, I can mean only either that the pleasure I get is good, or that my getting it is good. Nothing is added by saying that the pleasure constitutes my good, or is good for me. But the distinctions I drew between different categories of value above show that Moore's analysis of the claim that my own good consists in pleasure is too narrow. Indeed Moore's argument rests on the very assumption that it seeks to prove: that only the notion of 'good' is necessary to make all the evaluative judgements we might wish to make. The claim that it is good that I get pleasure is, logically speaking, equivalent to the claim that the world containing the single Vermeer is good. It is, so to speak, 'impersonal', and leaves out of account the special feature of the value of well-being: that it is good for individuals. One way to respond both to Moore's challenge, and to the puzzles above, is to try, when appropriate, to do without the notion of 'good' (see Kraut 2011) and make do with 'good for', alongside the separate and non-evaluative notion of reasons for action. Thus, the world containing the single individual with a life worth living, might be said to contain nothing good per se, but a life that is good for that individual. And this fact may give us a reason to bring about such a world, given the opportunity. ## 3. Scanlon's Challenge Moore's book was published in Cambridge, England, at the beginning of the twentieth century. At the end of the same century, a book was published in Cambridge, Mass., which also posed some serious challenges to the notion of well-being: What Do We Owe to Each Other?, by T.M. Scanlon. Moore's ultimate aim in criticizing the idea of 'goodness for' was to attack egoism. Likewise, Scanlon has an ulterior motive in objecting to the notion of well-being--to attack so-called 'teleological' or end-based theories of ethics, in particular, utilitarianism, which in its standard form requires us to maximize well-being. But in both cases the critiques stand independently. One immediately odd aspect of Scanlon's position that 'well-being' is an otiose notion in ethics is that he himself seems to have a view on what well-being is. It involves, he believes, among other things, success in one's rational aims, and personal relations. But Scanlon claims that his view is not a 'theory of well-being', since a theory must explain what unifies these different elements, and how they are to be compared. And, he adds, no such theory is ever likely to be available, since such matters depend so much on context. Scanlon does, however, implicitly make a claim about what unites these values: they are all constituents of well-being, as opposed to other kinds of value, such as aesthetic or moral. Nor is it clear why Scanlon's view of well-being could not be developed so as to assist in making real-life choices between different values in one's own life. Scanlon suggests that we often make claims about what is good in our lives without referring to the notion of well-being, and indeed that it would often be odd to do so. For example, I might say, 'I listen to Alison Krauss's music because I enjoy it', and that will be sufficient. I do not need to go on to say, 'And enjoyment adds to my well-being'. But this latter claim sounds peculiar only because we already know that enjoyment makes a person's life better for them. And in some circumstances such a claim would anyway not be odd: consider an argument with someone who claims that aesthetic experience is worthless, or with an ascetic. Further, people do use the notion of well-being in practical thinking. For example, if I am given the opportunity to achieve something significant, which will involve considerable discomfort over several years, I may consider whether, from the point of view of my own well-being, the project is worth pursuing. Scanlon argues also that the notion of well-being, if it is to be philosophically acceptable, ought to provide a 'sphere of compensation'--a context in which it makes sense to say, for example, that I am losing one good in my life for the sake of gain over my life as a whole. And, he claims, there is no such sphere. For Scanlon, giving up present comfort for the sake of future health 'feels like a sacrifice'. But this does not chime with my own experience. When I donate blood, this feels to me like a sacrifice, albeit a minor one. But when I visit the dentist, it feels to me just as if I am weighing present pains against potential future pains. And we can weigh different components of well-being against one another. Consider a case in which you are offered a job which is highly paid but many miles away from your friends and family. Scanlon denies that we need an account of well-being to understand benevolence, since we do not have a general duty of benevolence, but merely duties to benefit others in specific ways, such as to relieve their pain. But, from the philosophical perspective, it may be quite useful to use the heading of 'benevolence' in order to group such duties. And, again, comparisons may be important: if I have several pro tanto duties of benevolence, not all of which can be fulfilled, I shall have to weigh the various benefits I can provide against one another. And here the notion of well-being will again come into play. Further, if morality includes so-called 'imperfect' duties to benefit others, that is, duties that allow the agent some discretion as to when and how to assist, the lack of any overarching conception of well-being is likely to make the fulfillment of such duties problematic. ## 4. Theories of Well-being ### 4.1 Hedonism On one view, human beings always act in pursuit of what they think will give them the greatest balance of pleasure over pain. This is 'psychological hedonism', and will not be my concern here. Rather, I intend to discuss 'evaluative hedonism' or 'prudential hedonism', according to which well-being consists in the greatest balance of pleasure over pain. This view was first, and perhaps most famously, expressed by Socrates and Protagoras in the Platonic dialogue, Protagoras (Plato 1976 [C4 BCE], 351b-c). Jeremy Bentham, one of the most well-known of the more recent hedonists, begins his Introduction to the Principles of Morals and Legislation thus: 'Nature has placed mankind under the governance of two sovereign masters, pain and pleasure. It is for them alone to point out what we ought to do'. In answer to the question, 'What does well-being consist in?', then, the hedonist will answer, 'The greatest balance of pleasure over pain'. We might call this substantive hedonism. A complete hedonist position will involve also explanatory hedonism, which consists in an answer to the following question: 'What makes pleasure good, and pain bad?', that answer being, 'The pleasantness of pleasure, and the painfulness of pain'. Consider a substantive hedonist who believed that what makes pleasure good for us is that it fulfills our nature. This theorist is not an explanatory hedonist. Hedonism--as is demonstrated by its ancient roots--has long seemed an obviously plausible view. Well-being, what is good for me, might be thought to be naturally linked to what seems good to me, and pleasure does, to most people, seem good. And how could anything else benefit me except in so far as I enjoy it? The simplest form of hedonism is Bentham's, according to which the more pleasantness one can pack into one's life, the better it will be, and the more painfulness one encounters, the worse it will be. How do we measure the value of the two experiences? The two central aspects of the respective experiences, according to Bentham, are their duration, and their intensity. Bentham tended to think of pleasure and pain as a kind of sensation, as the notion of intensity might suggest. One problem with this kind of hedonism, it has often been claimed, is that there does not appear to be a single common strand of pleasantness running through all the different experiences people enjoy, such as eating hamburgers, reading Shakespeare, or playing water polo. Rather, it seems, there are certain experiences we want to continue, and we might be prepared to call these--for philosophical purposes--pleasures (even though some of them, such as diving in a very deep and narrow cave, for example, would not normally be described as pleasurable). Hedonism could survive this objection merely by incorporating whatever view of pleasure was thought to be plausible. A more serious objection is to the evaluative stance of hedonism itself. Thomas Carlyle, for example, described the hedonistic component of utilitarianism as the 'philosophy of swine', the point being that simple hedonism places all pleasures on a par, whether they be the lowest animal pleasures of sex or the highest of aesthetic appreciation. One might make this point with a thought experiment. Imagine that you are given the choice of living a very fulfilling human life, or that of a barely sentient oyster, which experiences some very low-level pleasure. Imagine also that the life of the oyster can be as long as you like, whereas the human life will be of eighty years only. If Bentham were right, there would have to be a length of oyster life such that you would choose it in preference to the human. And yet many say that they would choose the human life in preference to an oyster life of any length. Now this is not a knockdown argument against simple hedonism. Indeed some people are ready to accept that at some length or other the oyster life becomes preferable. But there is an alternative to simple hedonism, outlined famously by J.S. Mill, using his distinction (itself influenced by Plato's discussion of pleasure at the end of his Republic (Plato 1992 [C4 BCE], 582d-583a)) between 'higher' and 'lower' pleasures (1863 [1998], ch. 2). Mill added a third property to the two determinants of value identified by Bentham, duration and intensity. To distinguish it from these two 'quantitative' properties, Mill called his third property 'quality'. The claim is that some pleasures, by their very nature, are more valuable than others. For example, the pleasure of reading Shakespeare, by its very nature, is more valuable than any amount of basic animal pleasure. And we can see this, Mill suggests, if we note that those who have experienced both types, and are 'competent judges', will make their choices on this basis. A long-standing objection to Mill's move here has been to claim that his position can no longer be described as hedonism proper (or what I have called 'explanatory hedonism'). If higher pleasures are higher because of their nature, that aspect of their nature cannot be pleasantness, since that could be determined by duration and intensity alone. And Mill anyway speaks of properties such as 'nobility' as adding to the value of a pleasure. Now it has to be admitted that Mill is sailing close to the wind here. But there is logical space for a hedonist position which allows properties such as nobility to determine pleasantness, and insists that only pleasantness determines value. But one might well wonder how nobility could affect pleasantness, and why Mill did not just come out with the idea that nobility is itself a good-making property. Above I noted the plausibility of the claim that nothing can benefit me if I don't enjoy it. Some non-hedonists have denied this, while accepting the so-called 'experience requirement' on well-being. They suggest that what matters is valuable consciousness, and consciousness can be valuable for non-hedonic reasons (see Kraut 2018; Kriegel 2019). But there is a yet more weighty objection both to hedonism and to the view that well-being consists only in conscious states: the so-called 'experience machine'. Imagine that I have a machine that I could plug you into for the rest of your life. This machine would give you experiences of whatever kind you thought most valuable or enjoyable--writing a great novel, bringing about world peace, attending an early Rolling Stones' gig. You would not know you were on the machine, and there is no worry about its breaking down or whatever. Would you plug in? Would it be wise, from the point of your own well-being, to do so? Robert Nozick thinks it would be a big mistake to plug in: 'We want to do certain things ... we want to be a certain way ... plugging into an experience machine limits us to a man-made reality' (Nozick 1974, p. 43). One can make the machine sound more palatable, by allowing that genuine choices can be made on it, that those plugged in have access to a common 'virtual world' shared by other machine-users, a world in which 'ordinary' communication is possible, and so on. But this will not be enough for many anti-hedonists. A further line of response begins from so-called 'externalism' in the philosophy of mind, according to which the content of mental states is determined by facts external to the experiencer of those states. Thus, the experience of really writing a great novel is quite different from that of apparently writing a great novel, even though 'from the inside' they may be indistinguishable. But this is once again sailing close to the wind. If the world can affect the very content of my experience without my being in a position to be aware of it, why should it not directly affect the value of my experience? The strongest tack for hedonists to take is to accept the apparent force of the experience machine objection, but to insist that it rests on 'common sense' intuitions, the place in our lives of which may itself be justified by hedonism. This is to adopt a strategy similar to that developed by 'two-level utilitarians' in response to alleged counter-examples based on common-sense morality. The hedonist will point out the so-called 'paradox of hedonism', that pleasure is most effectively pursued indirectly. If I consciously try to maximize my own pleasure, I will be unable to immerse myself in those activities, such as reading or playing games, which do give pleasure. And if we believe that those activities are valuable independently of the pleasure we gain from engaging in them, then we shall probably gain more pleasure overall. These kinds of stand-off in moral philosophy are unfortunate, but should not be brushed aside (for a balanced discussion of the experience machine, see Lin 2016). They raise questions concerning the epistemology of ethics, and the source and epistemic status of our deepest ethical beliefs, which we are further from answering than many would like to think. Certainly the current trend of quickly dismissing hedonism on the basis of a quick run-through of the experience machine objection is not methodologically sound. ### 4.2 Desire Theories The experience machine is one motivation for the adoption of a desire theory (for a good introduction to the view, see Heathwood 2016; 2019). When you are on the machine, many of your central desires are likely to remain unfilled. Take your desire to write a great novel. You may believe that this is what you are doing, but in fact it is just a hallucination. And what you want, the argument goes, is to write a great novel, not the experience of writing a great novel. Historically, however, the reason for the current dominance of desire theories lies in the emergence of welfare economics. Pleasure and pain are inside people's heads, and also hard to measure--especially when we have to start weighing different people's experiences against one another. So economists began to see people's well-being as consisting in the satisfaction of preferences or desires, the content of which could be revealed by the choices of their possessors. This made possible the ranking of preferences, the development of 'utility functions' for individuals, and methods for assessing the value of preference-satisfaction (using, for example, money as a standard). The simplest version of a desire theory one might call the present desire theory, according to which someone is made better off to the extent that their current desires are fulfilled. This theory does succeed in avoiding the experience machine objection. But it has serious problems of its own. Consider the case of the angry adolescent. This boy's mother tells him he cannot attend a certain nightclub, so the boy holds a gun to his own head, wanting to pull the trigger and retaliate against his mother. Recall that the scope of theories of well-being should be the whole of a life. It is implausible that the boy will make his life go as well as possible by pulling the trigger. We might perhaps interpret the simple desire theory as a theory of well-being-at-at-a-particular-time. But even then it seems unsatisfactory. From whatever perspective, the boy would be better off if he put the gun down. We should move, then, to a comprehensive desire theory, according to which what matters to a person's well-being is the overall level of desire-satisfaction in their life as a whole. A summative version of this theory suggests, straightforwardly enough, that the more desire-fulfilment in a life the better. But it runs into Derek Parfit's case of addiction (1984, p. 497). Imagine that you can start taking a highly addictive drug, which will cause a very strong desire in you for the drug every morning. Taking the drug will give you no pleasure; but not taking it will cause you quite severe suffering. There will be no problem with the availability of the drug, and it will cost you nothing. But what reason do you have to take it? A global version of the comprehensive theory ranks desires, so that desires about the shape and content of one's life as a whole are given some priority. So, if I prefer not to become a drug addict, that will explain why it is better for me not to take Parfit's drug. But now consider the case of the orphan monk. This young man began training to be a monk at the earliest age, and has lived a very sheltered life. He is now offered three choices: he can remain as a monk, or become either a cook or a gardener outside the monastery, at a grange. He has no conception of the latter alternatives, so chooses to remain a monk. But surely it might be possible that his life would be better for him were he to live outside? So we now have to move to an informed desire version of the comprehensive theory (Sobel 1994). According to the informed desire account, the best life is the one I would desire if I were fully informed about all the (non-evaluative) facts. But now consider a case made famous by John Rawls (1971: 432; see Stace 1944: 238): the grass-counter. Imagine a brilliant Harvard mathematician, fully informed about the options available to her, who develops an overriding desire to count the blades of grass on the lawns of Harvard. Like the experience machine, this case is another example of philosophical 'bedrock'. Some will believe that, if she really is informed, and not suffering from some neurosis, then the life of grass-counting will be the best for her. Note that on the informed desire view the subject must actually have the desires in question for well-being to accrue to her. If it were true of me that, were I fully informed I would desire some object which at present I have no desire for, giving me that object now would not benefit me. Any theory which claimed that it would amounts to an objective list theory with a desire-based epistemology. All these problem cases for desire theories appear to be symptoms of a more general difficulty. Recall again the distinction between substantive and formal theories of well-being. The former state the constituents of well-being (such as pleasure), while the latter state what makes these things good for people (pleasantness, for example). Substantively, a desire theorist and a hedonist may agree on what makes life good for people: pleasurable experiences. But formally they will differ: the hedonist will refer to pleasantness as the good-maker, while the desire theorist must refer to desire-satisfaction. (It is worth pointing out here that if one characterizes pleasure as an experience the subject wants to continue, the distinction between hedonism and desire theories becomes quite hard to pin down.) The idea that desire-satisfaction is a 'good-making property' is somewhat odd. As Aristotle says (Metaphysics, 1072a, tr. Ross): 'desire is consequent on opinion rather than opinion on desire'. In other words, we desire things, such as writing a great novel, because we think those things are independently good; we do not think they are good because they will satisfy our desire for them. ### 4.3 Objective List Theories The threefold distinction I am using between different theories of well-being has become standard in contemporary ethics (Parfit 1984: app. I). There are problems with it, however, as with many classifications, since it can blind one to other ways of characterizing views (see Kagan 1992; Hurka 2019). Objective list theories are usually understood as theories which list items constituting well-being that consist neither merely in pleasurable experience nor in desire-satisfaction. Such items might include, for example, knowledge or friendship. But it is worth remembering, for example, that hedonism might be seen as one kind of 'list' theory, and all list theories might then be opposed to desire theories as a whole. What should go on the list (Moore 2000)? It is important that every good should be included. As Aristotle put it: 'We take what is self-sufficient to be that which on its own makes life worthy of choice and lacking in nothing. We think happiness to be such, and indeed the thing most of all worth choosing, not counted as just one thing among others' (Nicomachean Ethics, 1097b, tr. Crisp). In other words, if you claim that well-being consists only in friendship and pleasure, I can show your list to be unsatisfactory if I can demonstrate that knowledge is also something that makes people better off. What is the 'good-maker', according to objective list theorists? This depends on the theory. One, influenced by Aristotle and recently developed by Thomas Hurka (1993; see Bradford 2017), is perfectionism, according to which what makes things constituents of well-being is their perfecting human nature. (On the history of modern perfectionism, see Brink 2019.) If it is part of human nature to acquire knowledge, for example, then a perfectionist should claim that knowledge is a constituent of well-being. But there is nothing to prevent an objective list theorist's claiming that all that the items on her list have in common is that each, in its own way, advances well-being. How do we decide what goes on the list? All we can work on is the deliverance of reflective judgement--intuition, if you like. But one should not conclude from this that objective list theorists are, because they are intuitionist, less satisfactory than the other two theories. For those theories too can be based only on reflective judgement. Nor should one think that intuitionism rules out argument. Argument is one way to bring people to see the truth. Further, we should remember that intuitions can be mistaken. Indeed, as suggested above, this is the strongest line of defence available to hedonists: to attempt to undermine the evidential weight of many of our natural beliefs about what is good for people. One common objection to objective list theories is that they are elitist, since they appear to be claiming that certain things are good for people, even if those people will not enjoy them, and do not even want them. One strategy here might be to adopt a 'hybrid' account, according to which certain goods do benefit people independently of pleasure and desire-satisfaction, but only when they do in fact bring pleasure and/or satisfy desires. Another would be to bite the bullet, and point out that a theory could be both elitist and true. It is also worth pointing out that objective list theories need not involve any kind of objectionable authoritarianism or perfectionism. First, one might wish to include autonomy on one's list, claiming that the informed and reflective living of one's own life for oneself itself constitutes a good. Second, and perhaps more significantly, one might note that any theory of well-being in itself has no direct moral implications. There is nothing logically to prevent one's holding a highly elitist conception of well-being alongside a strict liberal view that forbade paternalistic interference of any kind with a person's own life (indeed, on some interpretations, J.S. Mill's position is close to this). One not implausible view, if desire theories are indeed mistaken in their reversal of the relation between desire and what is good, is that the debate is really between hedonism and objective list theories. And, as suggested above, what is most at stake here is the issue of the epistemic adequacy of our beliefs about well-being. The best way to resolve this matter would consist, in large part at least, in returning once again to the experience machine objection, and seeking to discover whether that objection really stands. ## 5. Well-being and Morality ### 5.1 Welfarism Well-being obviously plays a central role in any moral theory. A theory which said that it just does not matter would be given no credence at all. Indeed, it is very tempting to think that well-being, in some ultimate sense, is all that can matter morally. Consider, for example, Joseph Raz's 'humanistic principle': 'the explanation and justification of the goodness or badness of anything derives ultimately from its contribution, actual or possible, to human life and its quality' (Raz 1986, p. 194). If we expand this principle to cover non-human well-being, it might be read as claiming that, ultimately speaking, the justificatory force of any moral reason rests on well-being. This view is welfarism. Act-utilitarians, who believe that the right action is that which maximizes well-being overall, may attempt to use the intuitive plausibility of welfarism to support their position, arguing that any deviation from the maximization of well-being must be grounded on something distinct from well-being, such as equality or rights. But those defending equality may argue that egalitarians are concerned to give priority to those who are worse off, and that we do see here a link with concern for well-being. Likewise, those concerned with rights may note that we have rights to certain goods, such as freedom, or to the absence of 'bads', such as suffering (in the case of the right not to be tortured, for example). In other words, the interpretation of welfarism is itself a matter of dispute. But, however it is understood, it does seem that welfarism poses a problem for those who believe that morality can require actions which benefit no one, and harm some, such as, for example, punishments intended to give individuals what they deserve. ### 5.2 Well-being and Virtue Ancient ethics was, in a sense, more concerned with well-being than a good deal of modern ethics, the central question for many ancient moral philosophers being, 'Which life is best for one?'. The rationality of egoism--the view that my strongest reason is always to advance my own well-being--was largely assumed. This posed a problem. Morality is naturally thought to concern the interests of others. So if egoism is correct, what reason do I have to be moral? One obvious strategy to adopt in defence of morality is to claim that a person's well-being is in some sense constituted by their virtue, or the exercise of virtue, and this strategy was adopted in subtly different ways by the three greatest ancient philosophers, Socrates, Plato, and Aristotle (for a modern defence of the view, see Bloomfield (2014)). At one point in his writings, Plato appears to allow for the rationality of moral self-sacrifice: the philosophers in his famous 'cave' analogy in the Republic (519-20) are required by morality to desist from contemplation of the sun outside the cave, and to descend once again into the cave to govern their fellow citizens. In the voluminous works of Aristotle, however, there is no recommendation of sacrifice. Aristotle believed that he could defend the virtuous choice as always being in the interest of the individual. Note, however, that he need not be described as an egoist in a strong sense--as someone who believes that our only reasons for action are grounded in our own well-being. For him, virtue both tends to advance the good of others, and (at least when acted on) advances our own good. So Aristotle might well have allowed that the well-being of others grounds reasons for me to act. But these reasons will never come into conflict with reasons grounded in my own individual well-being. His primary argument is the notorious and perfectionist 'function argument', according to which the good for some being is to be identified through attention to its 'function' or characteristic activity. The characteristic activity of human beings is to exercise reason, and the good will lie in exercising reason well--that is, in accordance with the virtues. This argument, which is stated by Aristotle very briefly and relies on assumptions from elsewhere in his philosophy and indeed that of Plato, appears to conflate the two ideas of what is good for a person, and what is morally good. I may agree that a 'good' example of humanity will be virtuous, but deny that this person is doing what is best for them. Rather, I may insist, reason requires one to advance one's own good, and this good consists in, for example, pleasure, power, or honour. But much of Aristotle's Nicomachean Ethics is taken up with portraits of the life of the virtuous and the vicious, which supply independent support for the claim that well-being is constituted by virtue. In particular, it is worth noting the emphasis placed by Aristotle on the value to a person of 'nobility' (to kalon), a quasi-aesthetic value which those sensitive to such qualities might not implausibly see as a constituent of well-being of more worth than any other. In this respect, the good of virtue is, in the Kantian sense, 'unconditional'. Yet, for Aristotle, virtue or the 'good will' is not only morally good, but good for the individual.
weyl	## 1. Life and Achievements Hermann Weyl was born on 9 November 1885 in the small town of Elmshorn near Hamburg. In 1904 he entered Gottingen University, where his teachers included Hilbert, Klein and Minkowski. Weyl was particularly impressed with Hilbert's lectures on number theory and resolved to study everything he had written. Hilbert's work on integral equations became the focus of Weyl's (1908) doctoral dissertation, written under Hilbert's direction. In this and in subsequent papers Weyl made important contributions to the theory of self-adjoint operators. Virtually all of Weyl's many publications during his stay in Gottingen until 1913 dealt with integral equations and their applications. After Weyl's (1910b) habilitation, he became a Privatdozent and was thereby entitled to give lectures at the University of Gottingen. Weyl chose to lecture on Riemann's theory of algebraic functions during the winter semester of 1911-12. These lectures became the basis of Weyl's (1913) first book Die Idee der Riemannschen Flache (The Concept of a Riemann Surface). This work, in which function theory, geometry and topology are unified, constitutes the first modern and comprehensive treatment of Riemann surfaces. The work also contains the first construction of an abstract manifold. Emphasizing that the `points of the manifold' can be quite arbitrary, Weyl based his definition of a general two-dimensional manifold or surface on an extension of the neighbourhood axioms that Hilbert (1902) had proposed for the definition of a plane. The work is indicative of Weyl's exceptional gift for harmoniously uniting into a coherent whole a patchwork of distinct mathematical fields. In 1913 Weyl was offered, and accepted, a professorship at the Eidgenossische Technische Hochschule--ETH (Swiss Federal Institute of Technology)--in Zurich. Weyl's years in Zurich were extraordinarily productive and resulted in some of his finest work, especially in the foundations of mathematics and physics. When he arrived in Zurich in the fall of 1913, Einstein and Grossmann were struggling to overcome a difficulty in their effort to provide a coherent mathematical formulation of the general theory of relativity. Like Hilbert, Weyl appreciated the importance of a close relationship between mathematics and physics. It was therefore only natural that Weyl should become interested in Einstein's theory and the potential mathematical challenges it might offer. Following the outbreak of the First World War, however, in May 1915 Weyl was called up for military service. But Weyl's academic career was interrupted only briefly, since in 1916 he was exempted from military duties for reasons of health. In the meantime Einstein had accepted an offer from Berlin and had left Zurich in 1914. Einstein's departure had weakened the theoretical physics program at the ETH and (as reported by Frei and Stammbach (1992, 26) the administration hoped that Weyl's presence would alleviate the situation. But Weyl needed no external prompting to work in, and to teach, theoretical physics: his interest in the subject in general and, above all, in the theory of relativity, gave him more than sufficient motivation in that regard. Weyl decided to lecture on the general theory of relativity in the summer semester of 1917, and these lectures became the basis of his famous book Raum-Zeit-Materie (Space-Time-Matter) of 1918. During 1917-24, Weyl directed his energies equally to the development of the mathematical and philosophical foundations of relativity theory, and to the broader foundations of mathematics. It is in these two areas that his philosophical erudition, nourished from his youth, manifests itself most clearly. The year 1918, the same year in which Space-Time-Matter appeared, also saw the publication of Das Kontinuum (The Continuum), a work in which Weyl constructs a new foundation for mathematical analysis free of what he had come to see as fatal flaws in the set-theoretic formulation of Cantor and Dedekind. Soon afterwards Weyl embraced Brouwer's mathematical intuitionism; in the early 1920s he published a number of papers elaborating on and defending the intuitionistic standpoint in the foundations of mathematics. It was also during the first years of the 1920s that Weyl came to appreciate the power and utility of group theory, initially in connection with his work on the solution to the Riemann-Helmholtz-Lie problem of space. Weyl analyzed this problem, the Raumproblem, in a series of articles and lectures during the period 1921-23. Weyl (1949b, 400) noted that his interest in the philosophical foundations of the general theory of relativity had motivated his analysis of the representations and invariants of the continuous groups: > I can say that the wish to understand what really > is the mathematical substance behind the formal apparatus of > relativity theory led me to the study of the representations and > invariants of groups; and my experience in this regard is probably not > unique. This newly acquired appreciation of group theory led Weyl to what he himself considered his greatest work in mathematics, a general theory of the representations and invariants of the classical Lie groups (Weyl 1924a, 1924f, 1925, 1926a, 1926b, 1926c). Later Weyl (1939) wrote a book, The Classical Groups: Their Invariants and Representations, in which he returned to the theory of invariants and representations of semisimple Lie groups. In this work he realized his ambition "to derive the decisive results for the most important of these groups by direct algebraic construction, in particular for the full group of all non-singular linear transformations and for the orthogonal group." Weyl applied his work in group theory and his earlier work in analysis and spectral theory to the new theory of quantum mechanics. Weyl's mathematical analysis of the foundations of quantum mechanics showed that regularities in a physical theory are most fruitfully understood in terms of symmetry groups. Weyl's (1928) book Gruppentheorie und Quantenmechanik (Group Theory and Quantum Mechanics) deals not only with the theory of quantum mechanics but also with relativistic quantum electrodynamics. In this work Weyl also presented a very early analysis of discrete symmetries which later stimulated Dirac to predict the existence of the positron and the antiproton. During his years in Zurich Weyl received, and turned down, numerous offers of professorships by other universities--including an invitation in 1923 to become Felix Klein's successor at Gottingen. It was only in 1930 that he finally accepted the call to become Hilbert's successor there. His second stay in Gottingen was to be brief. Repelled by Nazism, "deeply revolted," as he later wrote, "by the shame which this regime had brought to the German name," he left Germany in 1933 to accept an offer of permanent membership of the newly founded Institute for Advanced Study in Princeton. Before his departure for Princeton he published The Open World (1932); his tenure there saw the publication of Mind and Nature (1934), the aforementioned The Classical Groups (1939), Algebraic Theory of Numbers (1940), Meromorphic Functions and Analytic Curves (1943), Philosophy of Mathematics and Natural Science (1949; an enlarged English version of a 1927 work Philosophie der Mathematik und Naturwissenschaften), and Symmetry (1952). In 1951 he formally retired from the Institute, remaining as an emeritus member until his death, spending half his time there and half in Zurich. He died in Zurich suddenly, of a heart attack, on 9 December 1955. ## 2. Metaphysics Weyl was first and foremost a mathematician, and certainly not a "professional" philosopher. But as a German intellectual of his time it was natural for him to regard philosophy as a pursuit to be taken seriously. In Weyl's case, unusually even for a German mathematician, it was idealist philosophy that from the beginning played a significant role in his thought. Kant, Husserl, Fichte, and, later, Leibniz, were at various stages major influences on Weyl's philosophical thinking. As a schoolboy Weyl had been impressed by Kant's "Critique of Pure Reason." He was especially taken with Kant's doctrine that space and time are not inherent in the objects of the world, existing as such and independently of our awareness, but are, rather, forms of our intuition As he reports in Insight and Reflection, (Weyl 1955), his youthful enthusiasm for Kant crumbled soon after he entered Gottingen University in 1904. There he read Hilbert's Foundations of Geometry, a tour-de-force of the axiomatic method, in comparison to which Kant's "bondage to Euclidean geometry" now appeared to him naive. After this philosophical reverse he lapsed into an indifferent positivism for a while. But in 1912 he found a new and exciting source of philosophical enlightenment in Husserl's phenomenology.[1] It was also at about this time that Fichte's metaphysical idealism came to "capture his imagination." Although Weyl later questioned idealist philosophy, and became dissatisfied with phenomenology, he remained faithful throughout his life to the primacy of intuition that he had first learned from Kant, and to the irreducibility of individual consciousness that had been confirmed in his view by Fichte and Husserl. Weyl never provided a systematic account of his philosophical views, and sorting out his overall philosophical position is no easy matter. Despite the importance of intuition and individual consciousness in Weyl's philosophical outlook, it would nevertheless be inexact to describe his outlook as being that of a "pure" idealist, since certain "realist" touches seem also to be present, in his approach to physics, at least. His metaphysics appears to rest on three elements, the first two of which may be considered "idealist", and the third "realist": these are, respectively, the Ego or "I", the (Conscious) Other or "Thou", and the external or "objective" world. It is the first of these constituents, the Ego, to which Weyl ascribes primacy. Indeed, in Weyl's words, > > > The world exists only as met with by an ego, as one appearing to > a consciousness; the consciousness in this function does not > belong to the world, but stands out against the being as the > sphere of vision, of meaning, of image, or however else one may call > it. (Weyl 1934, 1) The Ego alone has direct access to the given, that is, to the raw materials of the existent which are presented to consciousness with an immediacy at once inescapable and irreducible. The Ego is singular in that, from its own standpoint, it is unique. But in an act of self-reflection, through grasping (in Weyl's words) "what I am for myself", the Ego comes to recognize that it has a function, namely as "conscious-existing carrier of the world of phenomena." It is then but a short step for the Ego to transcend its singularity through the act of defining an "ego" to be an entity performing that same function for itself. That is, an ego is precisely what I am for myself (in other words, what the Ego is for itself)--again a "conscious-existing carrier of the world of phenomena"--and yet other than myself. "Thou" is the term the Ego uses to address, and so to identify, an ego in this sense. "Thou" is thus the Ego generalized, the Ego refracted through itself. The Ego grasps that it exists within a world of Thous, that is, within a world of other Egos similar to itself. While the Ego has, of necessity, no direct access to any Thou, it can, through analogy and empathy, grasp what it is to be Thou, a conscious being like oneself. By that very fact the Ego recognizes in the Thou the same luminosity it sees in itself. The relationship of the Ego with the external world, the realm of "objective" reality, is of an entirely different nature. There is no analogy that the Ego can draw--as it can with the Thou--between itself and the external world, since that world (presumably) lacks consciousness. The external world is radically other, and opaque to the Ego[2]. Like Kant's noumenal realm, the external world is outside the immediacy of consciousness; it is, in a word, transcendent. Since this transcendent world is not directly accessible to the Ego, as far as the latter is concerned the existence of that world must arise through postulation, "a matter of metaphysics, not a judgment but an act of acknowledgment or belief."[3] Indeed, according to Weyl, it is not strictly necessary for the Ego to postulate the existence of such a world, even given the existence of a world of Thous: For as long as I do not proceed beyond what is given, or, more exactly, what is given at the moment, there is no need for the substructure of an objective world. Even if I include memory and in principle acknowledge it as valid testimony, if I furthermore accept as data the contents of the consciousness of others on equal terms with my own, thus opening myself to the mystery of intersubjective communication, I would still not have to proceed as we actually do, but might ask instead for the 'transformations' which mediate between the images of the several consciousnesses. Such a presentation would fit in with Leibniz's monadology. (Weyl 1949, 117.) But once the existence of the transcendent world is postulated, its opacity to the Ego can be partly overcome by constructing a representation of it through the use of symbols, the procedure called by Weyl symbolic construction, (or constructive cognition)[4] and which he regarded as the cornerstone of scientific explanation. He outlines the process as follows (Weyl 1934, 53): 1. Upon that which is given, certain reactions are performed by which the given is in general brought together with other elements capable of being varied arbitrarily. If the results to be read from these reactions are found to be independent of the variable auxiliary elements they are then introduced as attributes inherent in the things themselves (even if we do not actually perform those reactions on which their meaning rests, but only believe in the possibility of their being performed). 2. By the introduction of symbols, the judgements are split up and a part of the manipulations is made independent of the given and its duration by being shifted onto the representing symbols which are time resisting and simultaneously serve the purpose of preservation and communication. Thereby the unrestricted handling of notions arises in counterpoint to their application, ideas in a relatively independent manner confront reality. 3. Symbols are not produced simply "according to demand" wherever they correspond to actual occurrences, but they are embedded into an ordered manifold of possibilities created by free construction and open towards infinity. Only in this way may we contrive to predict the future, for the future is not given actually. Weyl's procedure thus amounts to the following. In step 1, a given configuration is subjected to variation. One then identifies those features of the configuration that remain unchanged under the variation--the invariant features; these are in turn, through a process of reification, deemed to be properties of an unchanging substrate--the "things themselves". It is precisely the invariance of such features that renders them (as well as the "things themselves") capable of being represented by the "time resisting" symbols Weyl introduces in step 2. As (written) symbols these are communicable without temporal distortion and can be subjected to unrestricted manipulation without degradation. It is the flexibility conferred thereby which enables the use of symbols to be conformable with reality. Nevertheless (step 3) symbols are not haphazardly created in response to immediate stimuli; they are introduced, rather, in a structured, yet freely chosen manner which reflects the idea of an underlying order--the "one real world"--about which not everything is, or can be, known--it is, like the future, "open towards infinity". Weyl observes that the reification implicit in the procedure of symbolic construction leads inevitably to its iteration, for "the transition from step to step is made necessary by the fact that the objects at one step reveal themselves as manifestations of a higher reality, the reality of the next step" (Weyl (1934), 32-33). But in the end "systematic scientific explanation will finally reverse the order: first it will erect its symbolical world by itself, without any reference, then, skipping all intermediate steps, try to describe which symbolical configurations lead to which data of consciousness" (ibid.). In this way the symbolic world becomes (mistakenly) identified with the transcendent world. It is symbolic construction which, in Weyl's vision, allows us access to the "objective" world presumed to underpin our immediate perceptions; indeed, Weyl holds that the objective world, being beyond the grasp (the "lighted circle") of intuition, can only be presented to us in symbolic form[5]. We can see a double dependence on the Ego in Weyl's idea of symbolic construction to get hold of an objective world beyond the mental. For not only is that world "constructed" by the Ego, but the materials of construction, the symbols themselves, as signs intended to convey meaning, have no independent existence beyond their graspability by a consciousness. By their very nature these symbols cannot point directly to an external world (even given an unshakable belief in the existence of that world) lying beyond consciousness. Weyl's metaphysical triad thus reduces to what might be called a polarized dualism, with the mental (I, Thou) as the primary, independent pole and objective reality as a secondary, dependent pole[6]. In Weyl's view mathematics simply lies - as it did for Brouwer - within the Ego's "lighted circle of intuition" and so is, in principle at least, completely presentable to that intuition. But the nature of physics is more complicated. To the extent that physics is linked to the transcendent world of objective reality, it cannot arise as the direct object of intuition, but must, like the transcendent world itself, be presented in symbolic form; more exactly, as the result of a process of symbolic construction. It is this which, in Weyl's vision, allows us access to to the "objective" world presumed to underpin our immediate perceptions. Weyl's conviction that the objective world can only be presented to us through symbolic construction may serve to explain his apparently untroubled attitude towards the highly counterintuitive nature of quantum theory. Indeed, the claims of numerous physicists that the quantum microworld is accessible to us only through abstract mathematical description provides a vindication of Weyl's thesis that objective reality cannot be grasped directly, but only through the mediation of symbols. In his later years Weyl attempted to enlarge his metaphysical triad (I, Thou, objective world) to a tetrad, by a process of completion, as it were, to embrace the "godhead that lives in impenetrable silence", the objective counterpart of the Ego, which had been suggested to him by his study of Eckhart. But this effort was to remain uncompleted. During his long philosophical voyage Weyl stopped at a number of ports of call: in his youth, Kantianism and positivism; then Husserlian phenomenological idealism; later Brouwerian intuitionism and finally a kind of theological existentialism. But apart from his brief flirtation with positivism (itself, as he says, the result of a disenchantment with Kant's "bondage to Euclidean geometry"), Weyl's philosophical orientation remained in its essence idealist (even granting the significant realist elements mentioned above). Nevertheless, while he continued to acknowledge the importance of phenomenology, his remarks in Insight and Reflection indicate that he came to regard Husserl's doctrine as lacking in two essential respects: first, it failed to give due recognition to the (construction of) transcendent external world, with which Weyl, in his capacity as a natural scientist, was concerned; secondly, and perhaps in Weyl's view even more seriously, it failed to engage with the enigma of selfhood: the fact that I am the person I am. Grappling with the first problem led Weyl to identify symbolic construction as providing sole access to objective reality, a position which brought him close to Cassirer in certain respects; while the second problem seems to have led him to existentialism and even, through his reading of Eckhart, to a kind of religious mysticism. ## 3. Work in the foundations and philosophy of mathematics Towards the end of his Address on the Unity of Knowledge, delivered at the 1954 Columbia University bicentennial celebrations, Weyl enumerates what he considers to be the essential constituents of knowledge. At the top of his list[7] comes > > > ...intuition, mind's ordinary act of seeing what is > given to it. (Weyl 1954, 629) In particular Weyl held to the view that intuition, or insight--rather than proof--furnishes the ultimate foundation of mathematical knowledge. Thus in his Das Kontinuum of 1918 he says: > > In the Preface to Dedekind (1888) we read that "In science, > whatever is provable must not be believed without proof." This > remark is certainly characteristic of the way most mathematicians > think. Nevertheless, it is a preposterous principle. As if such an > indirect concatenation of grounds, call it a proof though we may, can > awaken any "belief" apart from assuring ourselves through > immediate insight that each individual step is correct. In all cases, > this process of confirmation--and not the proof--remains > the ultimate source from which knowledge derives its authority; it is > the "experience of truth". (Weyl 1987, 119) Weyl's idealism naturally inclined him to the view that the ultimate basis of his own subject, mathematics, must be found in the intuitively given as opposed to the transcendent. Nevertheless, he recognized that it would be unreasonable to require all mathematical knowledge to possess intuitive immediacy. In Das Kontinuum, for example, he says: > > The states of affairs with which mathematics deals are, apart from the > very simplest ones, so complicated that it is practically impossible > to bring them into full givenness in consciousness and in this way to > grasp them completely. (Ibid., 17) Nevertheless, Weyl felt that this fact, inescapable as it might be, could not justify extending the bounds of mathematics to embrace notions, such as the actual infinite, which cannot be given fully in intuition even in principle. He held, rather, that such extensions of mathematics into the transcendent are warranted only by the fact that mathematics plays an indispensable role in the physical sciences, in which intuitive evidence is necessarily transcended. As he says in The Open World[8]: > ... if mathematics is taken by itself, one should > restrict oneself with Brouwer to the intuitively cognizable truths > ... nothing compels us to go farther. But in the natural sciences > we are in contact with a sphere which is impervious to intuitive > evidence; here cognition necessarily becomes symbolical construction. > Hence we need no longer demand that when mathematics is taken into the > process of theoretical construction in physics it should be possible to > set apart the mathematical element as a special domain in which all > judgments are intuitively certain; from this higher standpoint which > makes the whole of science appear as one unit, I consider Hilbert to be > right. (Weyl 1932, 82). In Consistency in Mathematics (1929), Weyl characterized the mathematical method as > the a priori construction of the possible in opposition to > the a posteriori description of what is actually > given.[9] > The problem of identifying the limits on constructing "the possible" in this sense occupied Weyl a great deal. He was particularly concerned with the concept of the mathematical infinite, which he believed to elude "construction" in the naive set-theoretical sense [10]. Again to quote a passage from Das Kontinuum: > > No one can describe an infinite set other than by indicating > properties characteristic of the elements of the set.... The > notion that a set is a "gathering" brought together by > infinitely many individual arbitrary acts of selection, assembled and > then surveyed as a whole by consciousness, is nonsensical; > "inexhaustibility" is essential to the infinite. (Weyl > 1987, 23) But still, as Weyl attests towards the end of The Open World, "the demand for totality and the metaphysical belief in reality inevitably compel the mind to represent the infinite as closed being by symbolical construction". The conception of the completed infinite, even if nonsensical, is inescapable. ### 3.1 Das Kontinuum Another mathematical "possible" to which Weyl gave a great deal of thought is the continuum. During the period 1918-1921 he wrestled with the problem of providing the mathematical continuum--the real number line--with a logically sound formulation. Weyl had become increasingly critical of the principles underlying the set-theoretic construction of the mathematical continuum. He had come to believe that the whole set-theoretical approach involved vicious circles[11] to such an extent that, as he says, "every cell (so to speak) of this mighty organism is permeated by contradiction." In Das Kontinuum he tries to overcome this by providing analysis with a predicative formulation--not, as Russell and Whitehead had attempted, by introducing a hierarchy of logically ramified types, which Weyl seems to have regarded as excessively complicated--but rather by confining the comprehension principle to formulas whose bound variables range over just the initial given entities (numbers). Accordingly he restricts analysis to what can be done in terms of natural numbers with the aid of three basic logical operations, together with the operation of substitution and the process of "iteration", i.e., primitive recursion. Weyl recognized that the effect of this restriction would be to render unprovable many of the central results of classical analysis--e.g., Dirichlet's principle that any bounded set of real numbers has a least upper bound[12]--but he was prepared to accept this as part of the price that must be paid for the security of mathematics. As Weyl saw it, there is an unbridgeable gap between intuitively given continua (e.g. those of space, time and motion) on the one hand, and the "discrete" exact concepts of mathematics (e.g. that of natural number[13]) on the other. The presence of this chasm meant that the construction of the mathematical continuum could not simply be "read off" from intuition. It followed, in Weyl's view, that the mathematical continuum must be treated as if it were an element of the transcendent realm, and so, in the end, justified in the same way as a physical theory. It was not enough that the mathematical theory be consistent; it must also be reasonable. Das Kontinuum embodies Weyl's attempt at formulating a theory of the continuum which satisfies the first, and, as far as possible, the second, of these requirements. In the following passages from this work he acknowledges the difficulty of the task: > > > > ... the conceptual world of mathematics is so foreign to what > the intuitive continuum presents to us that the demand for > coincidence between the two must be dismissed as absurd. (Weyl 1987, > 108) > > > > > ... the continuity given to us immediately by intuition (in the > flow of time and of motion) has yet to be grasped mathematically as a > totality of discrete "stages" in accordance with that > part of its content which can be conceptualized in an exact way. > (Ibid., > 24)[14] > > > > > > Exact time- or space-points are not the ultimate, underlying atomic > elements of the duration or extension given to us in experience. On > the contrary, only reason, which thoroughly penetrates what is > experientially given, is able to grasp these exact ideas. And only in > the arithmetico- analytic concept of the real number belonging to the > purely formal sphere do these ideas crystallize into full > definiteness. (Ibid., 94) > > > > > When our experience has turned into a real process in a real world > and our phenomenal time has spread itself out over this world and > assumed a cosmic dimension, we are not satisfied with replacing the > continuum by the exact concept of the real number, in spite of the > essential and undeniable inexactness arising from what is given. > (Ibid., 93) > > > As these quotations show, Weyl had come to accept that it was in principle impossible to furnish the continuum as presented to intuition with an exact mathematical formulation : so, with reluctance, he lowered his sights. In Das Kontinuum his goal was, first and foremost, to establish the consistency of the mathematical theory of the continuum by putting the arithmetical notion of real number on a firm logical basis. Once this had been achieved, he would then proceed to show that this theory is reasonable by employing it as the foundation for a plausible account of continuous process in the objective physical world.[15] In SS6 of Das Kontinuum Weyl presents his conclusions as to the relationship between the intuitive and mathematical continua. He poses the question: Does the mathematical framework he has erected provide an adequate representation of physical or temporal continuity as it is actually experienced? In posing this question we can see the continuing influence of Husserl and phenomenological doctrine. Weyl begins his investigation by noting that, according to his theory, if one asks whether a given function is continuous, the answer is not fixed once and for all, but is, rather, dependent on the extent of the domain of real numbers which have been defined up to the point at which the question is posed. Thus the continuity of a function must always remain provisional; the possibility always exists that a function deemed continuous now may, with the emergence of "new" real numbers, turn out to be discontinuous in the future. [16] To reveal the discrepancy between this formal account of continuity based on real numbers and the properties of an intuitively given continuum, Weyl next considers the experience of seeing a pencil lying on a table before him throughout a certain time interval. The position of the pencil during this interval may be taken as a function of the time, and Weyl takes it as a fact of observation that during the time interval in question this function is continuous and that its values fall within a definite range. And so, he says, > > This observation entitles me to assert that during a certain period > this pencil was on the table; and even if my right to do so is not > absolute, it is nevertheless reasonable and well-grounded. It is > obviously absurd to suppose that this right can be undermined by > "an expansion of our principles of definition"--as if > new moments of time, overlooked by my intuition could be added to this > interval, moments in which the pencil was, perhaps, in the vicinity of > Sirius or who knows where. If the temporal continuum can be > represented by a variable which "ranges over" the real > numbers, then it appears to be determined thereby how narrowly or > widely we must understand the concept "real number" and > the decision about this must not be entrusted to logical deliberations > over principles of definition and the like. (Weyl 1987, 88) To drive the point home, Weyl focuses attention on the fundamental continuum of immediately given phenomenal time, that is, as he characterizes it, > > ... to that constant form of my experiences of consciousness by > virtue of which they appear to me to flow by successively. (By > "experiences" I mean what I experience, exactly as I > experience it. I do not mean real psychical or even physical processes > which occur in a definite psychic-somatic individual, belong to a real > world, and, perhaps, correspond to the direct experiences.) > (Ibid., 88) In order to correlate mathematical concepts with phenomenal time in this sense Weyl grants the possibility of introducing a rigidly punctate "now" and of identifying and exhibiting the resulting temporal points. On the collection of these temporal points is defined the relation of earlier than as well as a congruence relation of equality of temporal intervals, the basic constituents of a simple mathematical theory of time. Now Weyl observes that the discrepancy between phenomenal time and the concept of real number would vanish if the following pair of conditions could be shown to be satisfied: 1. The immediate expression of the intuitive finding that during a certain period I saw the pencil lying there were construed in such a way that the phrase "during a certain period" was replaced by "in every temporal point which falls within a certain time span OE". [Weyl goes on to say parenthetically here that he admits "that this no longer reproduces what is intuitively present, but one will have to let it pass, if it is really legitimate to dissolve a period into temporal points.") 2. If $P$ is a temporal point, then the domain of rational numbers to which $l$ belongs if and only if there is a time point $L$ earlier than $P$ such that \[ OL = l{.}OE \] can be constructed arithmetically in pure number theory on the basis of our principles of definition, and is therefore a real number in our sense. (Ibid., 89) Condition 2 means that, if we take the time span $OE$ as a unit, then each temporal point $P$ is correlated with a definite real number. In an addendum Weyl also stipulates the converse. But can temporal intuition itself provide evidence for the truth or falsity of these two conditions? Weyl thinks not. In fact, he states quite categorically that > > ... everything we are demanding here is obvious nonsense: to > these questions, the intuition of time provides no answer--just > as a man makes no reply to questions which clearly are addressed to > him by mistake and, therefore, are unintelligible when addressed to > him. (Ibid., 90) The grounds for this assertion are by no means immediately evident, but one gathers from the passages following it that Weyl regards the experienced continuous flow of phenomenal time as constituting an insuperable barrier to the whole enterprise of representing the continuum as experienced in terms of individual points, and even to the characterization of "individual temporal point" itself. As he says, > The view of a flow consisting of points and, therefore, > also dissolving into points turns out to be mistaken: precisely what > eludes us is the nature of the continuity, the flowing from point to > point; in other words, the secret of how the continually enduring > present can continually slip away into the receding past. Each one of > us, at every moment, directly experiences the true character of this > temporal continuity. But, because of the genuine primitiveness of > phenomenal time, we cannot put our experiences into words. So we shall > content ourselves with the following description. What I am conscious > of is for me both a being-now and, in its essence, something which, > with its temporal position, slips away. In this way there arises the > persisting factual extent, something ever new which endures and changes > in consciousness. (Ibid., 91-92) Weyl sums up what he thinks can be affirmed about "objectively presented time"--by which he presumably means "phenomenal time described in an objective manner"--in the following two assertions, which he claims apply equally, mutatis mutandis, to every intuitively given continuum, in particular, to the continuum of spatial extension. (Ibid., 92): 1. An individual point in it is non-independent, i.e., is pure nothingness when taken by itself, and exists only as a "point of transition" (which, of course, can in no way be understood mathematically); 2. It is due to the essence of time (and not to contingent imperfections in our medium) that a fixed temporal point cannot be exhibited in any way, that always only an approximate, never an exact determination is possible. The fact that single points in a true continuum "cannot be exhibited" arises, Weyl asserts, from the fact that they are not genuine individuals and so cannot be characterized by their properties. In the physical world they are never defined absolutely, but only in terms of a coordinate system, which, in an arresting metaphor, Weyl describes as "the unavoidable residue of the eradication of the ego." This metaphor, which Weyl was to employ more than once[17], again reflects the continuing influence of phenomenological doctrine in his thinking : here, the thesis that the existent is given in the first instance as the contents of a consciousness. ### 3.2 Weyl and Brouwerian Intuitionism By 1919 Weyl had come to embrace Brouwer's views on the intuitive continuum. Given the idealism that always animated Weyl's thought, this is not surprising, since Brouwer assigned the thinking subject a central position in the creation of the mathematical world[18]. In his early thinking Brouwer had held that that the continuum is presented to intuition as a whole, and that it is impossible to construct all its points as individuals. But later he radically transformed the concept of "point", endowing points with sufficient fluidity to enable them to serve as generators of a "true" continuum. This fluidity was achieved by admitting as "points", not only fully defined discrete numbers such as 1/9, $e$, and the like--which have, so to speak, already achieved "being"--but also "numbers" which are in a perpetual state of "becoming" in that the entries in their decimal (or dyadic) expansions are the result of free acts of choice by a subject operating throughout an indefinitely extended time. The resulting choice sequences cannot be conceived as finished, completed objects: at any moment only an initial segment is known. Thus Brouwer obtained the mathematical continuum in a manner compatible with his belief in the primordial intuition of time--that is, as an unfinished, in fact unfinishable entity in a perpetual state of growth, a "medium of free development". In Brouwer's vision, the mathematical continuum is indeed "constructed", not, however, by initially shattering, as did Cantor and Dedekind, an intuitive continuum into isolated points, but rather by assembling it from a complex of continually changing overlapping parts. Brouwer's impact looms large in Weyl's 1921 paper, On the New Foundational Crisis of Mathematics. Here Weyl identifies two distinct views of the continuum: "atomistic" or "discrete"; and "continuous". In the first of these the continuum is composed of individual real numbers which are well-defined and can be sharply distinguished. Weyl describes his earlier attempt at reconstructing analysis in Das Kontinuum as atomistic in this sense: > > Existential questions concerning real numbers only become > meaningful if we analyze the concept of real number in this > extensionally determining and delimiting manner. Through this > conceptual restriction, an ensemble of individual points is, so to > speak, picked out from the fluid paste of the continuum. The continuum > is broken up into isolated elements, and the flowing-into-each other of > its parts is replaced by certain conceptual relations between these > elements, based on the "larger-smaller" relationship. This > is why I speak of the atomistic conception of the > continuum. (Weyl 1921, 91) By this time Weyl had repudiated atomistic theories of the continuum, including that of Das Kontinuum.[19] While intuitive considerations, together with Brouwer's influence, must certainly have fuelled Weyl's rejection of such theories, it also had a logical basis. For Weyl had come to regard as meaningless the formal procedure--employed in Das Kontinuum--of negating universal and existential statements concerning real numbers conceived as developing sequences or as sets of rationals. This had the effect of undermining the whole basis on which his theory had been erected, and at the same time rendered impossible the very formulation of a "law of excluded middle" for such statements. Thus Weyl found himself espousing a position[20] considerably more radical than that of Brouwer, for whom negations of quantified statements had a perfectly clear constructive meaning, under which the law of excluded middle is simply not generally affirmable. Of existential statements Weyl says: > An existential statement--e.g., "there is an > even number"--is not a judgement in the proper sense at > all, which asserts a state of affairs; existential states of affairs > are the empty invention of logicians. (Weyl [1921], 97) Weyl termed such pseudostatements "judgment abstracts", likening them, with typical literary flair, to "a piece of paper which announces the presence of a treasure, without divulging its location." Universal statements, although possessing greater substance than existential ones, are still mere intimations of judgments, "judgment instructions", for which Weyl provides the following metaphorical description: > If knowledge be compared to a fruit and the realization of > that knowledge to the consumption of the fruit, then a universal > statement is to be compared to a hard shell filled with fruit. It is, > obviously, of some value, however, not as a shell by itself, but only > for its content of fruit. It is of no use to me as long as I do not > open it and actually take out a fruit and eat it. (Ibid., > 98) Above and beyond the claims of logic, Weyl welcomed Brouwer's construction of the continuum by means of sequences generated by free acts of choice, thus identifying it as a "medium of free Becoming" which "does not dissolve into a set of real numbers as finished entities". Weyl felt that Brouwer, through his doctrine of Intuitionism[21], had come closer than anyone else to bridging that "unbridgeable chasm" between the intuitive and mathematical continua. In particular, he found compelling the fact that the Brouwerian continuum is not the union of two disjoint nonempty parts--that it is, in a word, indecomposable. "A genuine continuum," Weyl says, "cannot be divided into separate fragments."[22] In later publications he expresses this more colourfully by quoting Anaxagoras to the effect that a continuum "defies the chopping off of its parts with a hatchet." Weyl also agreed with Brouwer that all functions everywhere defined on a continuum are continuous, but here certain subtle differences of viewpoint emerge. Weyl contends that what mathematicians had taken to be discontinuous functions actually consist of several continuous functions defined on separated continua. In Weyl's view, for example, the "discontinuous" function defined by $f(x) = 0$ for $x \lt 0$ and $f(x) = 1$ for $x \ge 0$ in fact consists of the two functions with constant values and 1 respectively defined on the separated continua $\{x: x \lt 0\}$ and $\{x: x \ge 0\}$. (The union of these two continua fails to be the whole of the real continuum because of the failure of the law of excluded middle: it is not the case that, for be any real number $x$, either $x \lt 0$ or $x \ge 0$.) Brouwer, on the other hand, had not dismissed the possibility that discontinuous functions could be defined on proper parts of a continuum, and still seems to have been searching for an appropriate way of formulating this idea.[23] In particular, at that time Brouwer would probably have been inclined to regard the above function $f$ as a genuinely discontinuous function defined on a proper part of the real continuum. For Weyl, it seems to have been a self-evident fact that all functions defined on a continuum are continuous, but this is because Weyl confines attention to functions which turn out to be continuous by definition. Brouwer's concept of function is less restrictive than Weyl's and it is by no means immediately evident that such functions must always be continuous. Weyl defined real functions as mappings correlating each interval in the choice sequence determining the argument with an interval in the choice sequence determining the value "interval by interval" as it were, the idea being that approximations to the input of the function should lead effectively to corresponding approximations to the input. Such functions are continuous by definition. Brouwer, in contrast, considers real functions as correlating choice sequences with choice sequences, and the continuity of these is by no means obvious. The fact that Weyl refused to grant (free) choice sequences--whose identity is in no way predetermined--sufficient individuality to admit them as arguments of functions betokens a commitment to the conception of the continuum as a "medium of free Becoming" even deeper, perhaps, than that of Brouwer. There thus being only minor differences between Weyl's and Brouwer's accounts of the continuum, Weyl accordingly abandoned his earlier attempt at the reconstruction of analysis, and joined Brouwer. He explains: > > I tried to find solid ground in the impending state of dissolution of > the State of analysis (which is in preparation, although still only > recognized by few)without forsaking the order on which it is founded, > by carrying out its fundamental principle purely and honestly. And I > believe I was successful--as far as this is possible. For > this order is itself untenable, as I have now convinced > myself, and Brouwer--that is the revolution!... It would > have been wonderful had the old dispute led to the conclusion that the > atomistic conception as well as the continuous one can be carried > through. Instead the latter triumphs for good over the former. It is > Brouwer to whom we owe the new solution of the continuum > problem. History has destroyed again from within the provisional > solution of Galilei and the founders of the differential and the > integral calculus. (Weyl 1921, 98-99) Weyl's initial enthusiasm for intuitionism seems later to have waned. This may have been due to a growing belief on his part that the mathematical sacrifices demanded by adherence to intuitionistic doctrine (e.g., the abandonment of the least upper bound principle, and other important results of classical analysis) would prove to be intolerable to practicing mathematicians. Witness this passage from Philosophy of Mathematics and Natural Science: > > > Mathematics with Brouwer gains its highest intuitive clarity. He > succeeds in developing the beginnings of analysis in a natural manner, > all the time preserving the contact with intuition much more closely > than had been done before. It cannot be denied, however, that in > advancing to higher and more general theories the inapplicability of > the simple laws of classical logic eventually results in an almost > unbearable awkwardness. And the mathematician watches with pain the > greater part of his towering edifice which he believed to be built of > concrete blocks dissolve into mist before his eyes. (Weyl [1949], > 54) Nevertheless, it is likely that Weyl remained convinced to the end of his days that intuitionism, despite its technical "awkwardness", came closest, of all mathematical approaches, to capturing the essence of the continuum. ### 3.3 Weyl and Hilbert Weyl's espousal of the intuitionistic standpoint in the foundations of mathematics in 1920-21 inevitably led to friction with his old mentor Hilbert. Hilbert's conviction had long been that there were in principle no limitations on the possibility of a full scientific understanding of the natural world, and, analogously, in the case of mathematics, that once a problem was posed with the required precision, it was, at least in principle, soluble. In 1904 he was moved to respond to Emil du Bois-Reymond's famous declaration concerning the limits of science, ignoramus et ignorabimus ("we are ignorant and we shall remain ignorant"): > > We hear within us the perpetual call. There is the problem. Seek the > solution. You can find it by pure reason, for in mathematics there is > no > ignorabimus.[24] > Hilbert was unalterably opposed to any restriction of mathematics "by decree", an obstacle he had come up against in the early stages of his career in the form of Leopold Kronecker's (the influential 19th century German mathematician) anathematization of all mathematics venturing beyond the finite. In Brouwer's intuitionistic program--with its draconian restrictions on what was admissible in mathematical argument, in particular, its rejection of the law of excluded middle, "pure" existence proofs, and virtually the whole of Cantorian set theory--Hilbert saw the return of Kroneckerian constaints on mathematics (and also, perhaps, a trace of du Bois-Reymond's "ignorabimus") against which he had struggled for so long. Small wonder, then, that Hilbert was upset when Weyl joined the Brouwerian camp.[25] Hilbert's response was to develop an entirely new approach to the foundations of mathematics with the ultimate goal of establishing beyond doubt the consistency of the whole of classical mathematics, including arithmetic, analysis, and Cantorian set theory. With the attainment of that goal, classical mathematics would be placed securely beyond the destructive reach of the intuitionists. The core of Hilbert's program was the translation of the whole apparatus of classical mathematical demonstration into a simple, finitistic framework (which he called "metamathematics") involving nothing more, in principle, than the straightforward manipulation of symbols, taken in a purely formal sense, and devoid of further meaning.[26] Within metamathematics itself, Hilbert imposed a standard of demonstrative evidence stricter even than that demanded by the intuitionists, a form of finitism rivalling (ironically) that of Kronecker. The demonstration of the consistency of classical mathematics was then to be achieved by showing, within the constraints of strict finitistic evidence insisted on by Hilbert, that the formal metamathematical counterpart of a classical proof in that system can never lead to an assertion evidently false, such as $0 = 1$. Hilbert's program rested on the insight that, au fond, the only part of mathematics whose reliability is entirely beyond question is the finitistic, or concrete part: in particular, finite manipulation of surveyable domains of distinct objects including mathematical symbols presented as marks on paper. Mathematical propositions referring only to concrete objects in this sense Hilbert called real, concrete, or contentual propositions, and all other mathematical propositions he distinguished as possessing an ideal, or abstract character. (Thus, for example, $2 + 2 = 4$ would count as a real proposition, while there exists an odd perfect number would count as an ideal one.) Hilbert viewed ideal propositions as akin to the ideal lines and points "at infinity" of projective geometry. Just as the use of these does not violate any truths of the "concrete" geometry of the usual Cartesian plane, so he hoped to show that the use of ideal propositions--even those of Cantorian set theory--would never lead to falsehoods among the real propositions, that, in other words, such use would never contradict any self-evident fact about concrete objects. Establishing this by strictly concrete, and so unimpeachable means was thus the central aim of Hilbert's program. Hilbert may be seen to have followed Kant in attempting to ground mathematics on the apprehension of spatiotemporal configurations; but Hilbert restricted these configurations to concrete signs (such as inscriptions on paper). Hilbert regarded consistency as the touchstone of existence, and so for him the important thing was the fact that no inconsistencies can arise within the realm of concrete signs, since correct descriptions of concrete objects are always mutually compatible. In particular, within the realm of concrete signs, actual infinity cannot generate inconsistencies since, again along with Kant, he held that this concept cannot correspond to any concrete object. Hilbert's view seems accordingly to have been that the formal soundness of mathematics issues ultimately, not from a logical source, but from a concrete one[27], in much the same way as the consistency of truly reported empirical statements is guaranteed by the concreteness of the external world[28]. Weyl soon grasped the significance of Hilbert's program, and came to acknowledge its "immense significance and scope"[29]. Whether that program could be successfully carried out was, of course, still an open question. But independently of this issue Weyl was concerned about what he saw as the loss of content resulting from Hilbert's thoroughgoing formalization of mathematics. "Without doubt," Weyl warns, "if mathematics is to remain a serious cultural concern, then some sense must be attached to Hilbert's game of formulae." Weyl thought that this sense could only be supplied by "fusing" mathematics and physics so that "the mathematical concepts of number, function, etc. (or Hilbert's symbols) generally partake in the theoretical construction of reality in the same way as the concepts of energy, gravitation, electron, etc."[30] Indeed, in Weyl's view, "it is the function of mathematics to be at the service of the natural sciences". But still: > > The propositions of theoretical physics... lack that feature > which Brouwer demands of the propositions of mathematics, namely, that > each should carry within itself its own intuitively comprehensible > meaning.... Rather, what is tested by confronting theoretical > physics with experience is the system as a whole. It seems that we > have to differentiate between phenomenal knowledge or > insight--such as is expressed in the statement: "This leaf > (given to me in a present act of perception) has this green color > (given to me in this same perception)"--and theoretical > construction. Knowledge furnishes truth, its organ is > "seeing" in the widest sense. Though subject to error, it > is essentially definitive and unalterable. Theoretical construction > seems to be bound only to one strictly formulable rational principle, > concordance, which in mathematics, where the domain of sense data > remains untouched, reduces to consistency; its organ is creative > imagination. (Weyl 1949, 61-62) Weyl points out that, just as in theoretical physics, Hilbert's account of mathematics "already... goes beyond the bounds of intuitively ascertainable states of affairs through... ideal assumptions." (Weyl 1927, 484.) If Hilbert's realm of contentual or "real" propositions--the domain of metamathematics--corresponds to that part of the world directly accessible to what Weyl terms "insight" or "phenomenal knowledge", then "serious" mathematics--the mathematics that practicing mathematicians are actually engaged in doing--corresponds to Hilbert's realm of "ideal" propositions. Weyl regarded this realm as the counterpart of the domain generated by "symbolic construction", the transcendent world focussed on by theoretical physics. Hence his memorable characterization: > > The set-theoretical approach is the stage of naive realism which is > unaware of the transition from the given to the transcendent. Brouwer > represents idealism, by demanding the reduction of all truth to the > intuitively given. In [Hilbert's] formalism, finally, consciousness > makes the attempt to "jump over its own shadow", to leave > behind the stuff of the given, to represent the > transcendent--but, how could it be otherwise?, only through the > symbol. (Weyl 1949, 65-66) In Weyl's eyes, Hilbert's approach embodied the "symbolic representation of the transcendent, which demands to be satisfied", and so he regarded its emergence as a natural development. But by 1927 Weyl saw Hilbert's doctrine as beginning to prevail over intuitionism, and in this an adumbration of "a decisive defeat of the philosophical attitude of pure phenomenology, which thus proves to be insufficient for the understanding of creative science even in the area of cognition that is most primal and most readily open to evidence--mathematics."[31] Since by this time Weyl had become convinced that "creative science" must necessarily transcend what is phenomenologically given, he had presumably already accepted that pure phenomenology is incapable of accounting for theoretical physics, let alone the whole of existence. But it must have been painful for him to concede the analogous claim in the case of mathematics. In 1932, he asserts: "If mathematics is taken by itself, one should restrict oneself with Brouwer to the intuitively cognizable truths ... nothing compels us to go farther." If mathematics could be "taken by itself", then there would be no need for it to justify its practices by resorting to "symbolic construction", to employ symbols which in themselves "signify nothing"--nothing, at least, accessible to intuition. But, unlike Brouwer, Weyl seems finally to have come to terms with the idea that mathematics could not simply be "taken by itself", that it has a larger role to play in the world beyond its service as a paradigm, however pure, of subjective certainty. The later impact of Godel's incompleteness theorems on Hilbert's program led Weyl to remark in 1949:[32] > > The ultimate foundations and the ultimate meaning of mathematics > remain an open problem; we do not know in what direction it will find > its solution, nor even whether a final objective answer can be > expected at all. "Mathematizing" may well be a creative > activity of man, like music, the products of which not only in form > but also in substance defy complete objective rationalization. The > undecisive outcome of Hilbert's bold enterprise cannot fail to affect > the philosophical interpretation. (Weyl 1949, 219) The fact that "Godel has left us little hope that a formalism wide enough to encompass classical mathematics will be supported by a proof of consistency" seems to have led Weyl to take a renewed interest in "axiomatic systems developed before Hilbert without such ambitious dreams", for example Zermelo's set theory, Russell's and Whitehead's ramified type theory and Hilbert's own axiom systems for geometry (as well, possibly, as Weyl's own system in Das Kontinuum, which he modestly fails to mention). In one of his last papers, Axiomatic Versus Constructive Procedures in Mathematics, written sometime after 1953, he saw the battle between Hilbertian formalism and Brouwerian intuitionism in which he had participated in the 1920s as having given way to a "dextrous blending" of the axiomatic approach to mathematics championed by Bourbaki and the algebraists (themselves mathematical descendants of Hilbert) with constructive procedures associated with geometry and topology. It seems appropriate to conclude this account of Weyl's work in the foundations and philosophy of mathematics by allowing the man himself to have the last word: > > This history should make one thing clear: we are less certain than > ever about the ultimate foundations of (logic and) mathematics; like > everybody and everything in the world today, we have our > "crisis". We have had it for nearly fifty years. Outwardly > it does not seem to hamper our daily work, and yet I for one confess > that it has had a considerable practical influence on my mathematical > life: it directed my interests to fields I considered relatively > "safe", and it has been a constant drain on my enthusiasm > and determination with which I pursued my research work. The > experience is probably shared by other mathematicians who are not > indifferent to what their scientific endeavours mean in the contexts > of man's whole caring and knowing, suffering and creative existence in > the world. (Weyl 1946, 13) ## 4. Contributions to the Foundations of Physics ### 4.1 Spacetime Geometries and Weyl's Unified Field Theory Weyl's clarification of the role of coordinates, invariance or symmetry principles, his important concept of gauge invariance, his group-theoretic results concerning the uniqueness of the Pythagorean form of the metric, his generalization of Levi-Civita's concept of parallelism, his development of the geometry of paths, his discovery of the causal-inertial method which prepared the way to empirically determine the spacetime metric in a non-circular, non-conventional manner, his deep analysis of the concept of motion and the role of Mach's Principle, are but a few examples of his important contributions to the philosophical and mathematical foundations of modern spacetime theory. Weyl's book, Raum-Zeit-Materie, beautifully exemplifies the fruitful and harmonious interplay of mathematics, physics and philosophy. Here Weyl aims at a mathematical and philosophical elucidation of the problem of space and time in general. In the preface to the great classical work of 1923, the fifth German edition, after mentioning the importance of mathematics to his work, Weyl says: > > > Despite this, the book does not disavow its basic, philosophical > orientation: its central focus is conceptual analysis; > physics provides the experiential basis, mathematics the sharp > tools. In this new edition, this tendency has been further > strengthened; although the growth of speculation was trimmed, the > supporting foundational ideas were more intuitively, more carefully > and more completely developed and analyzed. > > #### 4.1.1 Weyl's metric-independent construction of the symmetric linear connection Extending and abstracting from Gauss's treatment of curved surfaces in Euclidian space, Riemann constructed an infinitesimal geometry of $n$-dimensional manifolds. The coordinate assignments $x^{k}(p)\ [k \in \{1, \ldots ,n\}$] of the points $p$ in such an $n$-dimensional Riemannian manifold are quite arbitrary, subject only to the requirement of arbitrary differential coordinate transformations.[33] Riemann's assumption that in an infinitesimal neighbourhood of a point, Euclidean geometry and hence Pythagoras's theorem holds, finds its formal expression in Riemann's equation \[\tag{1} ds^2 = \sum\_{i,j} g\_{ij}(x^{k}(p))dx^{i}dx^{j}\ [\text{where } g\_{ij}(x^{k}(p)) = g\_{ji}(x^{k}(p))] \] for the square of the length $ds$ of an infinitesimal line element that leads from the point $p = x(p) = (x^{1}(p), \ldots ,x^{n}(p))$ to an arbitrary infinitely near point $p' = x(p') = (x^{1}(p) + dx^{1}(p), \ldots ,x^{n}(p) + dx^{n}(p))$. The assumption that Euclidean geometry holds in the infinitesimally small means that the $dx^{i}(p)$ transform linearly under arbitrary coordinate transformations. Using the Einstein summation convention[34], equation (1) can simply be written as \[\tag{2} ds^{2} = g\_{ij}(x^{k}(p))dx^{i}dx^{j}. \] Riemann assumed the validity of the Pythagorean metric only in the infinitely small. Riemannian geometry is essentially a geometry of infinitely near points and conforms to the requirement that all laws are to be formulated as field laws. Field laws are close-action-laws which relate the field magnitudes only to infinitesimally neighbouring points in space.[35] The value of some field magnitude at each point depends only on the values of other field magnitudes in the infinitesimal neighbourhoods of the corresponding points. The field magnitudes consist of partial derivatives of position functions at some point, and this requires the knowledge of the behavior of the position functions only with respect to the neighbourhood of that point. To construct a field law, only the behavior of the world in the infinitesimally small is required.[36] Riemann's ideas were brought to a concrete realization fifty years later in Einstein's general theory of relativity. The basic idea underlying the general theory of relativity was Einstein's recognition that the metric field, which has such powerful real effects on matter, cannot be a rigid once and for all given geometric structure of the spacetime, but must itself be something real, that not only has effects on matter, but is in turn also affected by matter. Riemann had already suggested that analogous to the electromagnetic field, the metric field reciprocally interacts with matter. Einstein came to this idea of reciprocity between matter and field independently of Riemann, and in the context of his theory of general relativity, applied this principle of reciprocity to four dimensional spacetime. Thus Einstein could adopt Riemann's infinitesimal geometry with the important difference: given the causal requirements of Einstein's theory of special relativity, Riemann's quadratic form is not positive definite but indefinite; it has signature 1.[37] Weyl (1922a) says: > > > All our considerations until now were based on the assumption, that > the metric structure of space is something that is fixed and > given. Riemann already pointed to another possibility which was > realized through General Relativity. The metrical structure of the > extensive medium of the external world is a field of physical reality, > which is causally dependent on the state of matter. > > And in another place Weyl (1918b) remarks: > > > The metric is not a property of the world [spacetime] in itself, > rather spacetime as a form of appearance is a completely formless > four-dimensional continuum in the sense of analysis situs, but the > metric expresses something real, something which exists in the world, > which exerts through centrifugal and gravitational forces physical > effects on matter, and whose state is conversely conditioned through > the distribution and nature of matter. > > After Einstein applied Riemannian geometry to his theory of general relativity, Riemannian geometry became the focus of intense research. In particular, G. Ricci and T. Levi-Civita's so-called Absolute Differential Calculus developed and clarified the Riemannian notions of an affine connection and covariant differentiation. The decisive step in this development, however, was T. Levi-Civita's discovery in 1917 of the concept of infinitesimal parallel vector displacement, and the fact that such parallel vector displacement is uniquely determined by the metric field of Riemannian geometry. Levi-Civita's construction of infinitesimal parallel transport on a manifold required the process of embedding the manifold into a flat higher-dimensional metric space. In 1918, Weyl generalized Levi-Civita's concept of parallel transport by means of an intrinsic construction that does not require the process of such an embedding, and is therefore independent of a metric. Weyl's intrinsic construction results in a metric-independent, symmetric linear connection. Weyl simply referred to the latter as affine connection.[38] Weyl defines what he means by an affine connection in the following way: A point $p$ on the manifold $M$ is affinely connected with its immediate neighborhood, if and only of for every tangent vector $v\_{p}$ at $p$, a tangent vector $v\_{q}$ at $q$ is determined to which the tangent vector $v\_{p}$ gives rise under parallel displacement from $p$ to the infinitesimally neighboring point $q$. This definition merely says that a manifold is affinely connected if it admits the process of infinitesimal parallel displacement of a vector. Weyl's next definition characterizes the essential nature of infinitesimal parallel displacement. The definition says that at any arbitrary point of the manifold there exists a geodesic coordinate system such that the components of any vector at that point are not altered by an infinitesimal parallel displacement with respect to it. This is a geometrical way of expressing Einstein's requirement that the gravitational field can always be made to vanish locally. According to Weyl (1923b, 115), it characterizes the nature of an affine connection on the manifold. A manifold which is an affine manifold is homogeneous in this sense. Moreover, manifolds do not exist whose affine structure is of a different nature. The transport of a tangent vector $v\_{p}$ at $p$ to an infinitesimally nearby point $q$ results in the tangent vector $v\_{q}$ at $q$, namely, \[\tag{3} v\_{q} = v\_{p} + dv\_{p}. \] This infinitesimal tangent vector transport Weyl defines as infinitesimal parallel displacement if and only if there exists a coordinate system $\overline{x}$, called a geodesic coordinate system for the neighborhood of $p$, relative to which the transported tangent vector $\overline{v}\_{q}$ at $q$ possesses the same components as the original tangent vector $\overline{v}\_{p}$ at $p$; that is, \[\tag{4} \overline{v}^{\,i}\_q - \overline{v}^{\,i}\_p = d\overline{v}^{\,i}\_p = 0. \] ![figure](fig1.png) Figure 1: Parallel transport in a geodesic coordinate system $\overline{x}$ For an arbitrary coordinate system $x$ the components $dv^{\,i}\_p$ vanish whenever $v^{\,i}\_p$ or $dx^{\,i}\_p$ vanish. Consequently, $dv^{\,i}\_p$ is bi-linear in $v^{\,i}\_p$ and $dx^{\,i}\_p$; that is, \[\tag{5} dv^{\,i}\_p = -\Gamma^{\,i}\_{jk}(x^i(p))v^{\,i}\_p dx^{\,k}\_p, \] where, in the case of four dimensions, the $4^{3} = 64$ coefficients $\Gamma^{\,i}\_{jk} (x^{i}(p))$ are coordinate functions, that is, functions of $x^{i}(p)$ $(i = 1, \ldots, 4)$, and the minus sign is introduced to agree with convention. ![figure](fig2.png) Figure 2: Parallel transport in an arbitrary coordinate system $x$ It is important to understand that there is no intrinsic notion of infinitesimal parallel displacement on a differentiable manifold. A notion of "parallelism" is not something that a manifold would possess merely by virtue of being a smooth manifold; additional structure has to be introduced which resides on the manifold and which permits the notion of infinitesimal parallelism. A manifold is an "affine manifold" $(M, \Gamma)$ if in addition to its manifold structure (differential topological structure) it is also endowed with an affine structure $\Gamma$ that assigns to each of its points 64 coefficients $\Gamma^{i}\_{jk} (x^{i}(p))$ satisfying the symmetry condition $\Gamma^{i}\_{jk} (x^{i}(p)) = \Gamma^{i}\_{kj} (x^{i}(p))$. An $n$-dimensional manifold $M$, which is an affinely connected manifold, Weyl (1918b) interprets physically as an $n$-dimensional world (spacetime) filled with a gravitational field. Weyl says, "...the affine connection appears in physics as the gravitational field..." Since there exists at each spacetime point a geodesic coordinate system in which the components $\Gamma^{i}\_{jk}$ of the symmetric linear connection vanish, the gravitational field can be made to vanish at each point of the manifold. The classical theory of physical geometry, developed by Helmholtz, Poincare and Hilbert, regarded the concept of "metric congruence" as the only basic relation of geometry, and constructed physical geometry from this one notion alone in terms of the relative positions and displacements of physical congruence standards. Although Einstein's general theory of relativity championed a dynamical view of spacetime geometry that is very different from the classical theory of physical geometry, Einstein initially approached the problem of the structure of spacetime from the metrical point of view. It was Weyl (1923b) who emphasized and developed the metric-independent construction of the symmetric linear connection and who pointed out the rationale for doing so. In both the non-relativistic and relativistic contexts, it is the symmetric linear connection, and not the metric, which plays the essential role in the formulation of all physical laws that are expressed in terms of differential equations. It is the symmetric linear connection that relates the state of a system at a spacetime point to the states at neighboring spacetime events and enters into the differentials of the corresponding magnitudes. In both Newtonian physics and the theory of general relativity, all dynamical laws presuppose the projective and affine structure and hence the Law of Inertia. In fact, the whole of tensor analysis with its covariant derivatives is based on the affine concept of infinitesimal parallel displacement and noton the metric. Weyl's metric independent construction not only led to a deeper understanding of the mathematical characterization of gravity, it also prepared the way for new constructions and generalizations in differential geometry and the general theory of relativity. In particular, it led to 1. The development of the geometry of paths, first introduced by Weyl in 1918. 2. Weyl's discovery of the causal-inertial method which prepared the way to empirically determine the spacetime metric in a non-circular, non-conventional manner. 3. Weyl's generalization of Riemannian geometry in his attempt to unify gravity and electromagnetism. 4. Weyl's introduction of the concept of gauge in the context of his attempt to unify gravity and electromagnetism. For more detail on Weyl's metric independent construction of the affine connection (linear symmetric connection), see the supplement. #### 4.1.2 Projective Geometry or the Geometry of Paths Weyl's metric-independent construction of the affine structure led to the development of differential projective geometries or the geometries of paths. The interest in projective geometry is in the paths, that is, in the continuous set of points of the image set of curves, rather than in the possible parameter descriptions of curves. A curve has one degree of freedom; it depends on one parameter, and its image set or path is a one-dimensional continuous set of points of the manifold. One represents a curve on a manifold $M$ as a smooth map (i.e., $C^{\infty})$ $\gamma$ from some open interval $I = (-\varepsilon, \varepsilon)$ of the real line $\mathbb{R}$ into $M$. ![figure](fig3.png) Figure 3: A curve on the manifold $M$ is the smooth map $\gamma : I \subset \mathbb{R} \rightarrow M$ It is important to understand that what one means by a "curve" is the map (the parametric description) itself, and not the set of its image points, the path. Consequently, two curves are mathematically considered to be different curves if they are given by different maps (different parameter descriptions), even if their image set, that is, their path, is the same. If we change a curve's parameter description we change the curve but not its image set (its path), the points it passes through. A path is therefore sometimes defined as an equivalence class of curves under arbitrary parameter transformations. Hence, projective geometry may be defined as an equivalence class of affine geometries. A geodesic curve in flat space is a straight line. Its tangent at one point is parallel to the tangent at previous or subsequent points. A straight line in Euclidean space is the only curve that parallel-transports its own tangent vector. This notion of parallel transport of the tangent vector also characterizes geodesic curves in curved space. That is, a curve $\gamma$ in curved space, which parallel-transports its own tangent vector along all of its points, is called a geodesic curve. Given a manifold with an affine structure and some arbitrary local coordinate system, the coordinate functions (components) $\gamma^{i}$ of a geodesic curve $\gamma$ satisfy the second-order non-linear differential equations \[\tag{6} \frac{d^2\gamma^i}{ds^2} + \Gamma^{i}\_{jk} \frac{d\gamma^j}{ds} \frac{d\gamma^k}{ds} = 0. \] One may characterize the projective geometry $\Pi$ on an affine manifold either in terms of an equivalence class of geodesic curves under arbitrary parameter diffeomorphisms[39], thereby eliminating all the parameter descriptions and hence all possible notions of distance along the curves satisfying (6),[40] or one may take the process of autoparallelism of directionsas fundamental in defining a projective structure. ![figure](fig4.png) Figure 4: A path $\xi$ is an equivalence class [$\gamma$] of curves under all parameter diffeomorphisms $\mu: \mathbb{R} \rightarrow \mathbb{R}; \lambda \mapsto \mu(\lambda)$ Weyl took the latter approach. According to Weyl, the infinitesimal process of parallel displacements of vectors contains, as a special case, the infinitesimal displacement of a direction into its own direction. Such an infinitesimal autoparallelism of directions is characteristic of the projective structure of an affinely connected manifold. Infinitesimal Autoparallelism of a Direction: An infinitesimal autoparallelism of a direction $R$ at an arbitrary point $p$ consists in the parallel displacement of $R$ at $p$ to a neighbouring point $p'$ which lies in the direction $R$ at $p$. A curve is geodesic if and only if its tangent direction $R$ experiences infinitesimal autoparallelism when moved along all the points of the curve. This characterization of a geodesic curve constitutes an abstraction from affine geometry. Through this abstraction, a geodesic curve is definable exclusively in terms of autoparallelism of tangent directions, and not tangent vectors. Roughly speaking, an affine geometry is essentially a projective geometry with the notion of distance defined along the curves. By eliminating all possible notions of distance along curves, or equivalently, all the parameter descriptions of the curves, one abstracts the projective geometry form affine geometry. As mentioned above, a projective geometry $\Pi$ may be defined as an equivalence class of affine geometries, that is, an equivalence class of projectively related affine connections [$\Gamma$]. Weyl presented the details of his approach to projective geometry, which uses the notion of autoparallelism of direction, in a set of lectures delivered in Barcelona and Madrid in the spring of 1922 (Weyl (1923a); see also Weyl (1921c)). Weyl began with the following necessary and sufficient condition for the invariance of the projective structure $\Pi$ under a transformation $\Gamma \rightarrow \overline{\Gamma}$ of the affine structure: Projective Transformation: A transformation $\Gamma \rightarrow \overline{\Gamma}$ preserves the projective structure $\Pi$ of a manifold with an affine structure $\Gamma$, and is called a projective transformation, if and only if \[\tag{7} (\overline{\Gamma} - \Gamma)^i\_{jk} v^{\,j}v^{\,k} \propto v^{\,i}, \] where $v^{\,i}$ is an arbitrary vector. Weyl's definition says that a change in the affine structure of the manifold $M$ preserves the projective structure $\Pi$ of $M$ if the vectors $v^{i}\_{q}$ and $\overline{v}^{i}\_{q}$ at $q$ that result from the vector $v^{i}\_{p}$ at $p$ by parallel transport under $\Gamma$ and $\overline{\Gamma}$ respectively, differ at most in length but not in direction.[41] A spacetime manifold $M$ is a "projective manifold" (M, $\Pi)$, if in addition to its manifold structure (differential topological structure), it is also endowed with a projective structure $\Pi$ that assigns to each of its manifold points 64 coefficients $\Pi^{\,i}\_{jk} x^{i}(p))$ satisfying certain symmetry conditions.[42] These projective coefficients characterize the equivalence class [$\Gamma$] of projectively equivalent connections, that is, connections equivalent under the projective transformation (7). In physical spacetime the projective structure has an immediate intuitive significance according to Weyl. The real world is a non-empty spacetime filled with an inertial-gravitational field, which Weyl calls the guiding field $(F$uhrungsfeld)[43]. It is an indubitable fact, according to Weyl (1923a, 13), that a body which is let free in a certain spacetime direction (time-like direction) carries out a uniquely determined natural motion from which it can only be diverted through an external force. The process of autoparallelism of direction appears, thus, as the tendency of persistence of the spacetime direction of a free particle whose motion is governed by, what Weyl calls, the guiding field$(F$uhrungsfeld). This natural motion occurs on the basis of an effective infinitesimal tendency of persistence, that parallelly displaces the spacetime direction $R$ of a body at an arbitrary point $p$ on its trajectory to a neighbouring point $p'$ that lies in the direction $R$ at $p$. If external forces exert themselves on a body, then a motion results which is determined through the conflict between the tendency of persistence due to the guiding field and the force. The tendency of persistence of the guiding field is a type of constraining guidance, that the inertial-gravitational field exerts on every body. Weyl (1923b, 219) says: > > > Galilei's inertial law shows, that there exists a type of > constraining guidance in the world [spacetime] that imposes > on a body that is let free in some definite world direction a unique > natural motion from which it can only be diverted through external > forces; this occurs on the basis of an effective infinitesimal > tendency of persistence from point to point, that > auto-parallelly transfers the world direction $r$ of > the body at an arbitrary point $P$ to an infinitesimally close > neighboring point $P'$, that lies in the direction > $r$ at $P$. > > > #### 4.1.3 Conformal Geometry, Weyl Geometry, and Weyl's Unified Field Theory Shortly after the completion of the general theory of relativity in 1915, Einstein, Weyl, and others began to work on a unified field theory. It was natural to assume at that time[44] that this task would only involve the unification of gravity and electromagnetism. In Einstein's geometrization of gravity, the Newtonian gravitational potential, and the Newtonian gravitational force, are respectively replaced by the components of the metric tensor $g\_{ij}(x)$, and the components of the symmetric linear connection $\Gamma^{i}\_{jk}(x)$. In the general theory of relativity the gravitational field is thus accounted for in terms of the curvature of spacetime, but the electromagnetic field remains completely unrelated to the spacetime geometry. Einstein's mathematical formulation of his theory of general relativity does not, however, provide room for the geometrization of the other long range force field, the electromagnetic field.[45] It was therefore natural to ask whether nature's only two long range fields of force have a common origin. Consequently, it was quite natural to suggest that the electromagnetic field might also be ascribed to some property of spacetime, instead of being merely something embedded in spacetime. Since, however, the components $g\_{ij}(x)$ of the metric tensor are already sufficiently determined by Einstein's field equations, this would require setting up a more general differential geometry than the one which underlies Einstein's theory, in order to make room for incorporating electromagnetism into spacetime geometry. Such a generalized differential geometry would describe both long range forces, and a new theory based on this geometry would constitute a unified field theory of electromagnetism and gravitation. In 1918, Weyl proposed such a theory. In Weyl (1918a, 1919a), and in the third edition (1920) of Raum-Zeit-Materie, Weyl presented his ingenious attempt to unify gravitation and electromagnetism by constructing a gauge-invariant geometry (see below), or what he called a purely infinitesimal 'metric' geometry. Since the conformal structure $C$ (see below) of spacetime does not determine a unique symmetric linear connection $\Gamma$ but only an equivalence class $K = [\Gamma]$ of conformally equivalent symmetric linear connections, Weyl was able to show that this degree of freedom in a conformal structure of spacetime provides just enough room for the geometrization of the electromagnetic potentials. The resulting geometry, called a Weyl geometry, is an intermediate geometric structure that lies between the conformal and Riemannian structures.[46] The metric tensor field that is locally described by \[\tag{8} ds^{2} = g\_{ij}(x(p))dx^{i}dx^{j}, \] is characteristic of a Riemannian geometry. That geometry requires of the symmetric linear connection $\Gamma$ that the infinitesimal parallel transport of a vector always preserves the length of the vector. Therefore, the metric field in Riemannian geometry determines a unique symmetric linear connection, a "metric connection" that satisfies the length-preserving condition of parallel transport. This means that the metric field, locally represented by (8), is invariant under parallel transport. The coefficients of this unique symmetric linear metricconnection are given by \[\tag{9} \Gamma^i\_{jk} = \frac{1}{2} g^{ir}(g\_{rj,k} + g\_{kr,j} - g\_{jk,r}). \] If $v\_{p}$ is a vector at $p \in M$, its length \[\tag{10} \lvert v\_p \rvert^2 = g\_{ij}(x(p))v^{\,i}\_p v^{\,j}\_p. \] Moreover, the angle between two vectors $v\_{p}$ and $w\_{p}$ at $p\in M$ is given by \[\tag{11} \cos \theta = \frac{g\_{ij}(x(p))v^{\,i}\_p w^{\,j}\_p}{\lvert v\_p \rvert \lvert w\_p \rvert}. \] While in Riemannian geometry the parallel transport of length is path independent, that is, it is possible to compare the lengths of any two vectors, even if they are located at two finitely different points, a vector suffers a path-dependent change in direction under parallel transport; that is, it is not possible to define the angle between two vectors, located at different points, in a path-independent way. Consequently, the angle between two vectors at a given point is invariant under parallel transport if and only if both vectors are transported along the same path. In particular, a vector which is carried around a closed circuit by a continual parallel displacement back to the starting point, will have the same length, but will not in general return to its initial direction. ![figure](fig5.png) Figure 5: The parallel transport of a vector by a two-dimensional creature, from $A \rightarrow B \rightarrow C \rightarrow A$ around a geodesic triangle on a two-dimensional surface $S^{2}$, ends up pointing in a different direction upon returning to $A$. For a closed loop which circumscribes an infinitesimally small portion of space, the rotation of the vector per unit area constitutes the measure of the local curvature of space. Consequently, whether or not finite parallel displacement of direction is integrable, that is, path-independent, depends on whether or not the curvature tensor vanishes. According to Weyl, Riemannian geometry, is not a pure or genuine infinitesimal differential (metric) geometry, since it permits the comparison of length at a finite distance. In his seminal 1918 paper entitled Gravitation und Elektrizitat (Gravitation and Electricity) Weyl (1918a) says: > > > However, in the Riemannian geometry described above, there remains a > last distant-geometric [ferngeometrisches] element--without any > sound reason, as far as I can see; the only cause of this appears to > be the development of Riemannian geometry from the theory of > surfaces. The metric permits the comparison of length of two vectors > not only at the same point, but also at any arbitrarily separated > points. A true near-geometry (Nahegeometrie), however, may > recognize only a principle of transferring a length at a point to an > infinitesimal neighbouring point, and then it is no more > reasonable to assume that the transfer of length from a point to a > finitely distant point is integrable, then it was to assume that the > transfer of direction is integrable. > > > Weyl wanted a metric geometry which would not permit distance comparison of length between two vectors located at finitely different points. In a pure infinitesimal geometry, Weyl argued, if attention is restricted to a single point of the manifold, then some standard of length or gauge must be chosen arbitrarily before the lengths of vectors can be determined. Therefore, all that is intrinsic to the notion of a pure infinitesimal metric differential geometry is the ability to determine the ratios of the lengths of any two vectors and the angle between any two vectors, at a point. Such a pure infinitesimal metric manifold must have at least a conformal structure $C$. The defining characteristic of a conformal spacetime structure is given by the equation \[\tag{12} 0 = ds^2 = g\_{ij}(x(p))dx^i dx^{\,j}, \] which determines the light cone at $p$. A gauge transformation of the metric is a map \[ g\_{ij}(x(p)) \rightarrow \lambda(x(p))g\_{ij}(x(p)) = \overline{g}\_{ij}(x(p)), \] which preserves the metric up to a positive and smooth but otherwise arbitrary scalar factor or gauge function $\lambda(x(p))$. In the case of a pseudo-Riemannian structure such a gauge transformation leaves the light cones unaltered. The angle between two vectors at $p$ is given by (11). Clearly, the gauge transformation $\overline{g}\_{ij}(x(p)) = \lambda(x(p))g\_{ij}(x(p))$ is angle preserving, that is, conformal. Two metrics which are related by a conformal gauge transformation are called conformally equivalent. A conformal structure does not determine the length of any one vector at a point. Only the relative lengths, the ratio of lengths, of any two vectors $\bfrac{\lvert v\_p \rvert}{\lvert w\_p \rvert}$ is determined. Weyl exploited these features of the conformal structure, and suggested that given a conformal structure, a gauge could be chosen at each point in a smooth but otherwise arbitrary manner, such that the metric (8) at any point of the manifold is conventional or undetermined to the extent that the metric \[\tag{13} d\overline{s}^2 = \lambda(x(p))g\_{ij}(x(p))dx^{i}dx^{\,j} \] is equally valid. However, a conformal structure by itself does not determine a unique symmetric linear connection; it only determines an equivalence class of conformally equivalent connections $K = [\Gamma]$, namely, connections which preserve the conformal structure $C$ during parallel transport. The difference between any two conformally equivalent symmetric linear connections $\overline{\Gamma}^{\,i}\_{jk}$, $\Gamma^i\_{jk} \in [\Gamma]$ is given by \[\tag{14} \overline{\Gamma}^{\,i}\_{jk} - \Gamma^i\_{jk} = \frac{1}{2}(\delta^{\,i}\_j \theta\_k + \delta^{\,i}\_k \theta\_j - g\_{jk}g^{ir}\theta\_{r}), \] where \[\tag{15} \theta\_{j}(x(p))dx^{\,j} \] is an arbitrary one-form field. Since the conformal structure determines only an equivalence class of conformally equivalent symmetric linear connections $K = [\Gamma]$, the affine connection in this type of geometry is not uniquely determined, and the parallel transport of vectors is not generally well defined. Moreover, the ratio of the lengths of two vectors located at different points is not determined even in a path-dependent way. According to Weyl, it is a fundamental principle of infinitesimal geometry that the metric structure on a manifold $M$ determines a unique affine structure on $M$. As was pointed out earlier, this principle is satisfied in Riemannian geometry where the metric determines a unique symmetric linear connection, namely, the metric connection according to (9). Evidently this fundamental principle of infinitesimal geometry is not satisfied for a structure which is merely a conformal structure, since the conformal structure only determines an equivalence class of conformally equivalent symmetric connections. Weyl showed that besides the conformal structure an additional structure is required in order to determine a unique symmetric linear connection from the equivalence class $K = [\Gamma]$ of conformally equivalent symmetric linear connections. Weyl showed that this additional structure is provided by the length connection or gauge field $A\_{j}$ that governs the congruent displacement of lengths. Weyl called this additional structure the "metric connection" on a manifold; however, we shall use the term "length connection" instead, in order to avoid confusion with the modern usage of the term "metric connection", which today denotes the symmetric linear connection that is uniquely determined by a Riemannian metric tensor according to (9). Weyl's Length Connection: A point $p$ is length connected with its infinitesimal neighborhood, if and only if for every length at $p$, there is determined at every point $q$ infinitesimally close to $p$ a length to which the length at $p$ gives rise when it is congruently displaced from $p$ to $q$. This definition merely says that a manifold is "length connected" if it admits the process of infinitesimal congruent displacement of length. The only condition imposed on the concept of congruent displacement of length is the following: Congruent Displacement of Length: With respect to a choice of gauge for a neighborhood of $p$, the transport of a length $l\_{p}$ at $p$ to an infinitesimally neighboring point $q$ constitutes a congruent displacement if and only if there exists a choice of gauge for the neighborhood of $p$ relative to which the transported length $\overline{l}\_{q}$ has the same value as $\overline{l}\_{p}$; that is \[\tag{16} \overline{l}\_q - \overline{l}\_p = d\overline{l}\_p = 0. \] Weyl called such a gauge at $p$ a geodesic gauge at $p$.[47] Weyl's proof of the following theorem closely parallels the proof of theorem A.3 in the supplement on Weyl's metric independent construction of the affine connection. Theorem 4.1: If for every point $p$ in a neighborhood $U$ of $M$, there exists a choice of gauge such that the change in an arbitrary length at $p$ under congruent displacement to an infinitesimally near point $q$ is given by \[\tag{17} d\overline{l}\_p = 0, \] then locally with respect to any other choice of gauge, \[\tag{18} dl = -lA\_j(x(p))dx^{\,j}, \] and conversely. Making use of \[\begin{align} dv^{\,i}\_p &= -\Gamma^i\_{jk}(x(p))v^{\,j}\_pdx^k \\ l\_p &= g\_{ij}(x(p))v^{\,i}\_p v^{\,j}\_p \\ dl\_p &= -l\_p A\_j x(p)dx^{\,j}, \end{align}\] Weyl (1923a, 124-125) shows that the conformal structure supplemented with the structure of a length connection or gauge field $A\_{j}(x)$ singles out a unique connection from the equivalence class $K = [\Gamma]$ of conformally equivalent connections.[48] This unique connection, which is called the Weyl connection, is given by \[\begin{align} \Gamma^{\,i}\_{jk} &= \frac{1}{2}g^{ir}(g\_{rj,k} + g\_{kr,j} - g\_{jk,r}) +\frac{1}{2} g^{ir}(g\_{rj}A\_{k} + g\_{kr}A\_{j} - g\_{jk}A\_{r}) \\ \tag{19} &= \frac{1}{2}g^{ir}(g\_{rj,k} + g\_{kr,j} - g\_{jk,r}) +\frac{1}{2}(\delta^{\,i}\_j A\_{k} + \delta^{\,i}\_k A\_{j} - g\_{jk}g^{ir}A\_{r}), \end{align}\] which is analogous to (14). The first term of the Weyl connection is identical to the metric connection (9) of Riemannian geometry, whereas the second term represents what is new in a Weyl geometry. The Weyl connection is invariant under the gauge transformation \[\begin{align} \overline{g}\_{ij}(x) &= e^{\theta(x)}g\_{ij}(x) \\ \tag{20} \overline{A}\_j(x) &= A\_j(x) - \partial\_j \theta(x), \end{align}\] where the gauge function is $\lambda(x) = e^{\theta(x)}$. Thus, a conformal structure plus length connection or gauge field $A\_{j}(x)$ determines a Weyl geometryequipped with a unique Weyl connection. Therefore, the fundamental principle of infinitesimal geometry also holds in a Weyl geometry; that is, the metric structure of a Weyl geometry determines a unique affine connection, namely, the Weyl connection. In Weyl's physical interpretation of his purely infinitesimal metric geometry (Weyl geometry), the gauge field $A\_{j}(x)$ is identified with the electromagnetic four potential, and the electromagnetic field tensor is given by \[\tag{21} F\_{jk}(x) = \partial\_{j} A\_{k}(x) - \partial\_{k} A\_{j}(x). \] A spacetime that is formally characterizable as a Weyl geometry, would not only have a curvature of direction (Richtungskrummung) but would also have a curvature of length (Streckenkrummung). Because of the latter property the formal characterization of the congruent displacement of length would be non-integrable, that is, path-dependent, in a Weyl geometry. ![figure](fig6.png) Figure 6: In a Weyl geometry parallel displacement of a vector along different paths not only changes its direction but also its length Suppose physical spacetime corresponds to a Weyl geometry. Then two identical clocks $A$ and $B$ at an event $p$ with a common unit of time, that is, a timelike vector of given length $l\_{p}$, which are separated and moved along different world lines to an event $q$, will not only differ with respect to the elapsed time (first clock effect (i.e., relativistic effect)), but in general the clocks will differ with respect to their common unit of time (rate of ticking) at $q$ (second clock effect). That is, congruent time displacement in a Weyl geometry is such that two congruent time intervals at $p$ will not in general be congruent at $q$, when congruently displaced in parallel along different world lines from $p$ to $q$, that is, $l^{A}\_{q} \ne l^{B}\_{q}$. This means that a twin who travels to a distant star and then returns to earth would not only discover that the other twin on earth had aged much more, but also that all the clocks on earth tick at a different rate. Hence, in the presence of a non-vanishing electromagnetic field $F\_{jk}(x)$ the clock rates will not in general be the same; that is, there will be a second clock effect in addition to the relativistic effect (first clock effect). Thus, $l^{A}\_{q} = l^{B}\_{q}$ if and only if the curl of $A\_{j}(x)$ vanishes, that is, if and only if the electromagnetic field tensor $F\_{jk}(x)$ vanishes, namely, \[ F\_{jk}(x) = \partial\_{j} A\_{k}(x) - \partial\_{k} A\_{j}(x) = 0. \] In that case the second term of the Weyl connection vanishes and (19) reduces to the metric connection (9) of Riemannian geometry. In a Weyl geometry there are no ideal absolute "meter sticks" or "clocks". For example, the rate at which any clock measures time is a function of its history. However, as Einstein pointed out in a Nachtrag (addendum) to Weyl (1918a), it is precisely this situation which suggests that Weyl's geometry conflicts with experience. In Weyl's geometry, the frequency of the spectral lines of atomic clocks would depend on the location and past histories of the atoms. But experience indicates otherwise. The spectral lines are well-defined and sharp; they appear to be independent of an atom's history. Atomic clocks define units of time, and experience shows they are integrably transported. Thus, if we assume that the atomic time and the gravitational standard time are identical, and that the gravitational standard time is determined by the Weyl geometry, then the electromagnetic field tensor is zero. But if that is the case, then a Weyl geometry reduces to the standard Riemannian geometry that underlies general relativity, since the vanishing of Weyl's Streckenkrummung(length curvature) is necessary and sufficient for the existence of a Riemannian metric $g\_{ij}$. When quantum theory was developed a few years later it became clear that Weyl's theory was in conflict with experience in an even more fundamental way since there is a direct relation between clock rates and masses of particles in quantum theory. A particle with a certain rest mass $m$ possesses a natural frequency which is a function of its rest mass, the speed of light $c$, and Planck's constant $h$. This means that in a Weyl geometry not only clocks would depend on their histories but also the masses of particles. For example, if two protons have different histories then they would also have different masses in a Weyl geometry. But this violates the quantum mechanical principle that particles of the same kind--in this case, protons--have to be exactly identical. However, in 1918 it was still possible for Weyl to defend his theory in the following way. In response to Einstein's criticism Weyl noted that atoms, clocks and meter sticks are complex objects whose real behavior in arbitrary gravitational and electromagnetic fields can only be inferred from a dynamical theory of matter. Since no detailed and reliable dynamical models were available at that time, Weyl could argue that there is no reason to assume that, for example, clock rates are correctly modelled by the length of a timelike vector. Weyl (1919a, 67) said: > > > At first glance it might be surprising that according to the purely > close-action geometry, length transfer is non-integrable in the > presence of an electromagnetic field. Does this not clearly contradict > the behaviour of rigid bodies and clocks? The behaviour of these > measurement instruments, however, is a physical process whose course > is determined by natural laws and as such has nothing to do with the > ideal process of 'congruent displacement of spacetime > distance' that we employ in the mathematical construction of the > spacetime geometry. The connection between the metric field and the > behaviour of rigid rods and clocks is already very unclear in the > theory of Special Relativity if one does not restrict oneself to > quasi-stationary motion. Although these instruments play an > indispensable role in praxis as indicators of the metric field, (for > this purpose, simpler processes would be preferable, for example, the > propagation of light waves), it is clearly incorrect > to define the metric field through the data that are directly > obtained from these instruments. > > > Weyl elaborated this idea by suggesting that the dynamical nature of such time keeping systems was such that they continually adapt to the spacetime structure in such a way that their rates remain constant. He distinguished between quantities that remain constant as a consequence of such dynamical adjustment, and quantities that remain constant by persistence because they are isolated and undisturbed. He argued that all quantities that maintain a perfect constancy probably do so as a result of dynamical adjustment. Weyl (1921a, 261) expressed these ideas in the following way: > > > What is the cause of this discrepancy between the idea of congruent > transfer and the behaviour of measuring-rods and clocks? I > differentiate between the determination of a magnitude in Nature by > "persistence" (Beharrung) and by > "adjustment" (Einstellung). I shall make the > difference clear by the following illustration: We can give to the > axis of a rotating top any arbitrary direction in space. This > arbitrary original direction then determines for all time the > direction of the axis of the top when left to itself, by means of a > tendency of persistence which operates from moment to > moment; the axis experiences at every instant a parallel > displacement. The exact opposite is the case for a magnetic needle in > a magnetic field. Its direction is determined at each instant > independently of the condition of the system at other instants by the > fact that, in virtue of its constitution, the system adjustsitself in an unequivocally determined manner to the field in > which it is situated. A priori we have no ground for > assuming as integrable a transfer which results purely from the > tendency of persistence. ...Thus, although, for example, > Maxwell's equations demand the conservational equation > $de\,/\,dt =0$ for the charge $e$ of an > electron, we are unable to understand from this fact why an electron, > even after an indefinitely long time, always possesses an unaltered > charge, and why the same charge $e$ is associated with all > electrons. This circumstance shows that the charge is not determined > by persistence, but by adjustment, and that there can exist only > one state of equilibrium of the negative electricity, to > which the corpuscle adjusts itself afresh at every instant. For the > same reason we can conclude the same thing for the spectral lines of > atoms. The one thing common to atoms emitting the same frequency is > their constitution, and not the agreement of their frequencies on the > occasion of an encounter in the distant past. Similarly, the length > of a measuring-rod is obviously determined by adjustment, for I could > not give this measuring-rod in this field-position > any other length arbitrarily (say double or treble length) in place > of the length which it now possesses, in the manner in which I can at > will pre-determine its direction. The theoretical possibility of a > determination of length by adjustment is given as a consequence of > the world-curvature, which arises from the metrical field > according to a complicated mathematical law. As a result of its > constitution, the measuring-rod assumes a length which possesses this > or that value, in relation to the radius of curvature of the > field. > > > Weyl's response to Einstein's criticism that a Weyl geometry conflicts with experience, took advantage of the fact that the underlying dynamical laws of matter which govern clocks and rigid rods, were not known at that time. Weyl could thus argue that it is at least theoretically possible that there exists an underlying dynamics of matter, such that a Weyl geometry, according to which length transfer is non-integrable, nonetheless coheres with observable experience, according to which length transfer appears to be integrable. However, as was clearly pointed out by Wolfgang Pauli, Weyl's plausible defence comes at a cost.[49] Pauli (1921/1958, 196) argued that Weyl's defence of his theory deprives it of its inherent convincing power from a physical point of view. > > > Weyl's present attitude to this problem is the > following: The ideal process of the congruent transference of > world lengths ... has nothing to do with the real behaviour of > measuring rods and clocks; the metric field must not be defined by > means of information taken from these measuring instruments. In > this case the quantities $g\_{ik}$ and $\varphi\_{i}$ are, be > definition, no longer observable, in contrast to the line elements of > Einstein's theory. This relinquishment seems to have very > serious consequences. While there now no longer exists a direct > contradiction with experiment, the theory appears nevertheless to have > been robbed of its inherent convincing power, from a physical point of > view. For instance, the connexion between electromagnetism and world > metric is not now essentially physical, but purely formal. For there > is no longer an immediate connection between the electromagnetic > phenomena and the behaviour of measuring rods and clocks. There is > only an interrelation between the former and the ideal process which > is mathematically defined as congruent transference of > vectors. Besides, there exists only formal, and not physical, evidence > for a connection between world metric and > electricity.[50] > > > > Pauli concluded his critical assessment of Weyl's theory with the following statement: > > Summarizing, we can say that Weyl's theory has not succeeded in > getting any nearer to solving the problem of the structure > of matter. As will be argued in more detail ... there is, > on the contrary, something to be said for the view that a solution of > this problem cannot at all be found in this way. > > > It should be noted, however, that Weyl's defence of his theory implicitly addresses an important methodological consideration concerning the relation between theory and evidence. As Pauli puts it above, according to Weyl "the metric field must not be defined by means of information taken from these measuring instruments [rigid rods and ideal clocks]". That is, Weyl rejects Einstein's operational standpoint which gives operational significance to the metric field in terms of the observable behaviour of ideal rigid rods and ideal clocks.[51] Unlike light propagation and freely falling (spherically symmetric, neutral) particles, rigid rods and ideal clocks are relativistically ill defined probative systems, and are thus unsuitable for the determination of the inherent structures of spacetime postulated by the theory of relativity. Weyl (1918a) clearly recognized this when he said in response to Einstein's critique "because of the problematic behaviour of yardsticks and clocks I have in my book Space-Time-Matter restricted myself for the specific measurement of the $g\_{ik}$, exclusively to the observation of the arrival of light signals." It is interesting to note parenthetically that in the first edition of his book Weyl thought that it was possible to have an intrinsic method of comparing the lengths of arbitrary spacetime intervals with an interval between two fiducial spacetime events, by using light signals only. It was Lorentz who pointed out to Weyl that not only the world lines of light rays but also the world lines of material bodies are required for an intrinsic method of comparing lengths. Not only did Weyl correct this mistake in subsequent editions, but already in 1921, Weyl (1921c) discovered the causal-inertial method for determining the spacetime metric (see SS4.3) by proving an important theorem that shows that the spacetime metric is already fully determined by the inertial and causal structure of spacetime. Weyl (1949a, 103) remarks "... therefore mensuration need not depend on clocks and rigid bodies but ... light signals and mass points moving under the influence of inertia alone will suffice." It is clear that Weyl regarded the use of clocks and rigid rods as an undesirable makeshift within the context of the special and general theory. Since neither spatial nor temporal intervals are invariants of spacetime, the invariant spacetime interval $ds$ cannot be directly ascertained by means of standard clocks and rigid rods. In addition, the latter presuppose quantum theoretical principles for their justification and therefore lie outside the relativistic framework because the laws which govern their physical processes are not known.[52] Weyl (1929c, 233) abandoned his unified field theory only with the advent of the quantum theory of the electron. He did so because in that theory a different kind of gauge invariance associated with Dirac's theory of the electron was discovered, which more adequately accounted for the conservation of electric charge. Weyl's contributions to quantum mechanics, and his construction of a new principle of gauge invariance, are discussed in SS4.5.3.[53] Weyl's unified field theory was revived by Dirac (1973) in a slightly modified form, which incorporated a real scalar field $\beta(x)$. Dirac also argued that the time intervals measured by atomic clocks need not be identified with the lengths of timelike vectors in the Weyl geometry.[54] ### 4.2 The Riemann-Helmholtz-Lie Problem of Space Prior to the works of Gauss, Grassmann and Riemann, the study of geometry tended to emphasize the employment of empirical intuitions and images of the three dimensional physical space. Physical space was thought of as having definite metrical attributes. The task of the geometer was to take physical mensuration devices in that space and work with them. Under the influence of Gauss and Grassmann, Riemann's great philosophical contribution consisted in the demonstration that, unlike the case of a discrete manifold, where the determination of a set necessarily implies the determination of its quantity or cardinal number, in the case of a continuous manifold, the concept of such a manifold and of its continuity properties, can be separated form its metrical structure. Using modern terminology, Riemann separated a manifold's local differential topological structure from its metrical structure. Thus Riemann's separation thesis gave rise to the space problem, or as Weyl called it, das Raumproblem: how can metric relations be determined on a continuous manifold $M$? A metric manifold is a manifold on which a distance function $f : M \times M \rightarrow \mathbb{R}$ is defined. Such a distance function must satisfy the following minimal conditions: for all $p, q, r \in M$, 1. $f(p, q) \ge 0$, and if $f(p,q) = 0$, then $p = q$, 2. $f(p, q) = f(q,p)$, (symmetry) 3. $f(p, q) + f(q,r) \ge f(p,r)$ (triangle inequality). In his famous inaugural lecture at Gottingen, entitled Uber die Hypothesen, welche der Geometrie zu Grunde liegen (About the hypotheses which lie at the basis of geometry), Riemann (1854) examined how metric relations can be determined on a continuous manifold; that is, what specific form $f : M \times M \rightarrow \mathbb{R}$ should have. Consider the coordinates $x^{i}(p)$ and $x^{i} (p) + dx^{i} (p)$ of two neighboring points $p, q \in M$. The measure of the distance $ds = f(p,q)$ must be some function $F\_{p}$ at $p$ of the differential increments $dx^{i}(p)$; that is, \[\tag{22} ds = F\_{p}(dx^{1}(p), \ldots ,dx^{n}(p)). \] Riemann states that $F\_{p}$ should satisfy the following requirements: Functional Homogeneity: If $\lambda \gt 0$ and $ds = F\_{p}(dx(p))$, then \[\tag{23} \lambda ds = \lambda F\_{p}(dx(p)) = F\_{p}(\lambda dx(p)). \] Sign Invariance: A change in sign of the differentials should leave the value of $ds$ invariant. Sign invariance is satisfied by every positive homogeneous function of degree $2m$ $(m = 1, 2, 3, \ldots)$. In the simple case $m = 1$, and the length element $ds$ is the square root of a homogeneous function of second degree, which can be expressed in the standard form \[\tag{24} ds = \left[ \sum\_{i=1}^n (dx^{\,i}(p))^{2} \right]^{\bfrac{1}{2}}. \] That is, at each point of $M$ there exists a coordinate system (defined up to an element of the orthogonal group $O(n)$ in which the square root of the homogeneous function of second degree can be expressed in the above standard form. Riemann's well-known general expression for the measure of length at $p \in M$ with respect to an arbitrary coordinate system is given by \[\tag{25} ds^{2} = g\_{ij}(x(p))dx^{i}(p)dx^{\,j}(p), \] where the components of the metric tensor satisfy the symmetry condition $g\_{ij} = g\_{ji}$. The assumption that $ds^{2} = F^{2}\_{p}$ is a quadratic differential form is not only the simplest one, but also the preferred one for other important reasons. Riemann himself was well aware of other possibilities; for example, the possibility that $ds$ could be the 4th root of a homogeneous polynomial of 4th order in the differentials. But Riemann restricted himself to the special case $m = 1$ because he was pressed for time and because he wanted to give specific geometric interpretations of his results. As Weyl points out Riemann's own answer to the space problem is inadequate since Riemann's mathematical justification for the restriction to the Pythagorean case are not very compelling. The first satisfactory justification of the Pythagorean form of the Riemannian metric, although limited in scope because it presupposed the full homogeneity of Euclidean space, was provided by the investigations of Hermann von Helmholtz. Helmholtz diverged from Riemann's analytic approach and made use merely of the fundamental concept of geometry, namely, the concept of congruent mapping, and characterized the geometric structure of space by requiring of space the full homogeneity of Euclidean space. His analysis was thereby restricted to the cases of constant positive, zero, or negative curvature. Abstracting from our experience of the movement of rigid bodies, Helmholtz was able to mathematically derive Riemann's distance formula from a number of axioms about rigid body motion in space. Helmholtz (1868) argued that Riemann's hypothesis that the metric structure of space is determined locally by a quadratic differential form, is really a consequence of the facts (Tatsachen) of rigid-body motion. Considering the general case of $n$ dimensions, and using Lie groups and Lie algebras, Sophus Lie, (Lie (1886/1935, 1890a,b)), later developed and improved Helmholtz's justification. However, the Helmholtz-Lie treatment of, and solution to, the problem of space, lost its relevance with the arrival of Einstein's theory of general relativity. As Weyl (1922b) points out, instead of a three-dimensional continuum we must now consider a four-dimensional continuum, the metric of which is not positive definite but is given instead by an indefinite quadratic form. In addition, Helmholtz's presupposition of metric homogeneity no longer holds, since we are now dealing with an inhomogeneous metric field that causally depends on the distribution of matter. Consequently, Weyl provided a reformulation of the space problem that is compatible with the causal and metric structures postulated by the theory of general relativity. But Weyl went further. Such a reformulation should not only incorporate Riemann's infinitesimal standpoint, as required by Einstein's general theory, it should also cohere with Weyl's requirements of a pureinfinitesimal geometry developed earlier in the context of Weyl's construction of a unified field theory. More precisely, Weyl generalized the so-called Riemann-Helmholtz-Lie problem of space in two ways: First, he allowed for indefinite metrics in order to encompass the general theory of relativity. Secondly, he considered metrics with variable gauge $\lambda(x(p))$ together with an associated length connection, in order to obtain a purely infinitesimal geometry. Thus each member of a general class of geometries under consideration is locally determined relative to a choice of variable gauge by two structural fields (Strukturfelder): (1) a possibly indefinite Finsler metric field[55] $F\_{p}(dx)$, and (2) a length connection that is determined by a 1-form field $\theta\_{i}dx^{i}$. Weyl's task was to prove: > > > If the geometry satisfies the Postulate of Freedom, (the > nature of space imposes no restrictions on admissible metrical > relations), and determines a unique, symmetric, linear connection > $\Gamma$, then the Finsler metric field > $F\_{p}(dx)$ must be a Riemannian metric field of > some > signature.[56] > > > > In a Riemannian space the concept of parallel displacement is defined by two conditions: 1. The components of a vector remain unchanged during an infinitesimal parallel displacement in a suitably chosen coordinate system (geodesic coordinate).[57] This condition is satisfied if \[\begin{matrix} dv^{\,i}\_{p} = - \Gamma^{i}\_{jk} v^{\,j}\_{p} dx^{k}\_{p} & \text{and} & \Gamma^{i}\_{jk} = \Gamma^{i}\_{kj}\,. \end{matrix}\] 2. The length of a vector remains unchanged during an infinitesimal parallel displacement. It follows from these conditions that a Riemannian space possesses a definite symmetric linear connection--a symmetric linear metric connection[58]--which is uniquely determined by the Pythagorean-Riemannian metric. Weyl calls this: The Fundamental Postulate of Riemannian Geometry: Among the possible systems of parallel displacements of a vector to infinitely near points, that is among the possiblesets of symmetric linear connection coefficients, there exists one and only one set, and hence one and only one system of parallel displacement, which is length preserving. In his lectures[59] on the mathematical analysis of the problem of space delivered in 1922 at Barcelona and Madrid, Weyl sketched a proof demonstrating that the following is also true: Uniqueness of the Pythagorean-Riemannian Metric: Among all the possible infinitesimal metrics that can be put on a differentiable manifold, the Pythagorean-Riemannian metric is the only type of metric that uniquely determines a symmetric linear connection. Weyl begins his proof with two natural assumptions. First, the nature of the metric should be coordinate independent. If $ds$ is given by an expression $F\_{p}(dx^{1}, \ldots ,dx^{n})$ with respect to a given system of coordinates, then with respect to another system of coordinates, $ds$ is given by a function that is related to $F\_{p}(dx^{1}, \ldots ,dx^{n})$ by a linear, homogeneous transformation of its arguments $dx^{i}$. Second, it is reasonable to assume that the nature of the metric is the same everywhere, in the sense that at every point of the manifold, and with respect to every coordinate system for a neighborhood of the point in question, $ds$ is represented by an element of the equivalence class $[F]$ of functions generated by any one such function, say $F\_{p}(dx^{1}, \ldots ,dx^{n})$, by all linear, homogeneous transformations of its arguments $dx^{i}$. For the case in which $F\_{p}$ is Pythagorean in form, namely the square root of a positive-definite quadratic form, there exists just one possible equivalence class [$F$], because every function that is the square root of a positive-definite quadratic form can be transformed to the standard expression \[\tag{26} F = \left[(dx^{1})^{2} + \cdots + (dx^{n})^{2}\right]^{\bfrac{1}{2}} \] by means of a linear, homogeneous transformation. To every possible equivalence class [$F$] of homogeneous functions, there corresponds a type of metrical space. The Pythagorean-Riemannian space, for which $F^{2}\_{p} = (dx^{1})^{2} + \cdots + (dx^{n})^{2}$, is one among several types of possible metrical spaces. The problem, therefore, is to single out the equivalence class $[F]$, where $F$ corresponds to $F^{2}\_{p} = (dx^{1})^{2} + \cdots + (dx^{n})^{2}$, from the other possibilities, and to provide arguments for this preference. By the term 'metric' Weyl means anyinfinitesimal distance function $F\_{p} \in [F]$, where the equivalence class $[F]$ represents a type of metric structure or metric field. Any such type of metric field structure has a microsymmetry group $G\_{p}$ at each $p \in M$. Definition 4.1 (Microsymmetry Group) A microsymmetry of a structural field (Strukturfeld) at a point $p \in M$ is a local diffeomorphism that takes $p \in M$ into $p$ and preserves the structural field at $p \in M$. The microsymmetry group of a field at $p \in M$ is the group of its microsymmetries at $p \in M$ under the operation of composition. A microsymmetry group $G\_{p}$, at $p \in M$, of a metric structure, is a set of invertible, linear maps of the tangent space $T(M\_{p})$ onto itself, which preserve the infinitesimal distance function at $p \in M$. For every $p \in M, G\_{p}$ is isomorphic to one and the same abstract group. For a Riemannian type of metric structure the congruent linear maps of the tangent space T$(M\_{p})$ onto itself form a group $G\_{p}$ which is isomorphic to the orthogonal group $O(n)$. The Pythagorean-Riemannian metric at $p$ is therefore determined through the concrete realization of the orthogonal group at $p$ which leaves the fundamental quadratic differential form at $p$ invariant. Thus the Pythagorean-Riemannian type of metric is characterized by the abstract microsymmetry group $O(n)$. For a metric which is not of the Pythagorean-Riemannian metric type, the abstract microsymmetry group $G\_{p}$ will be different from $O(n)$ and will be some other subgroup of $GL(n)$. At each point of the manifold the microsymmetry group will be a concrete realization of this subgroup of $GL(n)$. Weyl now states what he calls The Postulate of Freedom: If only the nature (of whatever type) of the metric is specified, that is, if only the corresponding abstract microsymmetry group $G\_{p}$ is specified, and the metric in question is otherwise left arbitrary, then the mutual orientations of the corresponding microsymmetry groups at different points are also left arbitrary. Weyl emphasizes that the Postulate of Freedom provides the general framework for a concise formulation of The Hypothesis of Dynamical Geometry: Whatever the nature or type of the metric may be--provided it is the same everywhere--the variations in the mutual orientations of the concrete microsymmetry groups from point to point are causally determined by the material content that fills space. In contrast with Helmholtz's analysis, which presupposes the homogeneity of space, the Postulate of Freedom allows for the possibility of replacing Helmholtz's homogeneity requirement with the possibility of subjecting the metric field to arbitrary, infinitesimal change. To assert this dynamical possibility does not require that the nature of the metric be specified. Next, Weyl points out that what has been provided so far is merely an explication of the concepts metric, length connection and symmetric linear connection. Some claim which goes beyond conceptual analysis has to be made, according to Weyl, in order to prove that among the various types of possible metrical structures that can be put on a differentiable manifold representing physical space, the Pythagorean-Riemannian form is unique. Weyl suggests the following hypothesis: Weyl's Hypothesis: Whatever determination the essentially free length connection at some point $p$ of the manifold may realize with the points in its infinitesimal neighborhood, there always exists among the possible systems of parallel displacements of the tangent space $T(M\_{p})$, one and only one, which is at the same time a system of infinitesimal congruent transport. Weyl shows that this hypothesis does in fact single out metrics of the Pythagorean-Riemannian type by proving the following theorem: Theorem 4.2 If a specific length connection is such that it determines a unique symmetric linear connection, then the metric must be of the Pythagorean-Riemannian form (for some signature). Thus the Postulate of Freedom and Weyl's Hypothesis together entail the existence, at each $p \in M$, of a non-degenerate quadratic form that is unique up to a choice of gauge at $p \in M$, and that is invariant under the action of the microsymmetry group $G\_{p}$ that is isomorphic to an orthogonal group of some signature. This formulation suggests, according to Weyl, an intuitive contrast between Euclidean 'distance-geometry' and the 'near-geometry' (Nahegeometrie) or 'field-geometry' of Riemann. Weyl (1949a, 88) compared Euclidean 'distance-geometry' to a crystal "built up from uniform unchangeable atoms in the rigid and unchangeable arrangement of a lattice", and the latter [Riemannian field-geometry] to a liquid, "consisting of the same indiscernible unchangeable atoms, whose arrangement and orientation, however, are mobile and yielding to forces acting upon them." The nature of the metric field, that is the natureof the metric everywhere, is the same and is, therefore, absolutely determined. It reflects according to Weyl, the a priori structure of space or spacetime. In contrast, what is posteriori, that is, accidental and capable of continuous change being causally dependent on the material content that fills space, are the mutual orientations of the metrics at different points. Hence, the demarcation between the $a$ priori and the a posteriori has shifted, according to Weyl: Euclidean geometry is still preserved for the infinitesimal neighborhood of any given point, but the coordinate system in which the metrical law assumes the standard form $ds^{2} =\sum^{n}\_{i=1}(dx^{i})^{2}$ is in general different from place to place. Weyl's a priori and a posteriori distinction must not be confused with Kant's distinction. Weyl (1949a, 134) remarks: "In the case of physical space it is possible to counterdistinguish aprioristic and aposterioristic features in a certain objective sense without, like Kant, referring to their cognitive source or their cognitive character." Weyl makes the same remark in (Weyl, 1922b, 266). See also the discussion in SS4.5.8. In the context of his group-theoretical analysis, Weyl (1922b, p. 266) makes the following interesting and important statement: > > > I remark from an epistemological point of view: it is not correct to > say that space or the world [spacetime] is in itself, prior to any > material content, merely a formless continuous manifold in the sense > of analysis situs; the nature of the metric [its > infinitesimal Pythagorean-Riemannian character] is characteristic of > space in itself, only the mutual orientation of the metrics at the > various points is contingent, a posteriori and dependent on the > material content. > > > Within the context of general relativity, empty spacetime is impossible, if 'empty' is understood to mean not merely empty of all matter but also empty of all fields. At another place, Weyl (1949a, Engl. edn, 172) says: > > > Geometry unites organically with the field theory; space is not > opposed to things (as it is in substance theory) like an empty vessel > into which they are placed and which endows them with far-geometrical > relationships. No empty space exists here; the assumption that the > field omits a portion of the space is absurd. > > > According to Weyl, the metric field does not cease to exist in a world devoid of matter but is in a state of rest: As a rest field it would possess the property of metric homogeneity; the mutual orientations of the orthogonal groups characterizing the Pythagorean-Riemannian nature of the metric everywhere would not differ from point to point. This means that in a matter-empty universe the metric is fixed. Consequently, the set of congruence relations on spacetime is uniquely determined. Since the metric uniquely determines the symmetric linear connection, the homogeneous metric field (rest field) determines an integrable affine structure. Therefore, a flat Minkowski spacetime consistent with the complete absence of matter is endowed with an integrable connection and thus determines all (hypothetical) free motions. According to Weyl, there exists in the absence of matter a homogeneous metric field, a structural field (Strukturfeld), which has the character of a rest field, and which constitutes an all pervasive background that cannot be eliminated. The structure of this rest field determines the extension of the spacetime congruence relations and determines Lorentz invariance. The rest field possesses no net energy and makes no contribution to curvature. The contrast with Helmholtz and Lie is this: both of them require homogeneity and isotropy for physical space. From a general Riemannian standpoint, the latter characteristics are valid only for a matter-empty universe. Such a universe is flat and Euclidean, whereas a universe that contains matter is inhomogeneous, anisotropic and of variable curvature. It is important to note here that the validity of Weyl's assertion that the metric field does not cease to exist but is in a state of rest, has its source in the mathematical fact that the metric field is a $G$-structure. A $G$-structure may be flat or non-flat; but a $G$-structure can never vanish. Consequently, geometric fields characterizable as $G$-structures, such as the projective, conformal, affine and metric structures, do not vanish.[60] ### 4.3 Weyl's Causal-Inertial Method for determining the Spacetime Metric #### 4.3.1 Weyl's Field Ontology of Spacetime Geometry Riemann searched for the most general type of an $n$-dimensional manifold. On this manifold, Euclidean geometry turns out to be a special case resulting from a certain form of the metric. Weyl takes this general structure, the manifold structure, which has certain continuity and order properties, as basic, but leaves the determination of the other geometrical structures, such as the projective, conformal, affine and metric structures, open. The metrical axioms are no longer dictated, as they were for Kant, by pure intuition. According to Weyl (1949a, 87), for Riemann the metric is not, as it was for Kant, "part of the static homogeneous form of phenomena, but of their ever changing material content". Weyl (1931a, 338) says: > > > We differentiate now between the amorphous continuum and its metrical > structure. The first has retained its a priori > character,[61] > ... whereas the structural field [Strukturfeld] is completely > subjected to the power-play of the world; being a real entity, > Einstein prefers to call it the ether. > > > There is no indication in Riemann's work on gravitation and electromagnetism that would indicate that he anticipated the conceptual revolution underlying Einstein's theory. However, Weyl's interpretation of Riemann's work suggests that Riemann foresaw something like its possibility in the following sense: > > > By formally separating the post-topological structures such as the > affine, projective, conformal and metric structures from the > manifold, so that these structures are no longer rigidly tied to it, > Riemann deprived them of their formal geometric rigidity and, on the > basis of his infinitesimal geometric standpoint or > "near-geometry", allowed for the possibility of > interpreting them as mathematical representations of flexible, > dynamical physical structural fields [Strukturfelder] on the manifold > of spacetime, geometrical fields that reciprocally interact with > matter. > > > Riemann's separation thesis together with his adoption of the infinitesimal standpoint, were prerequisite steps for the development of differential geometry as the mathematics of differentiable geometric fields on manifolds. When interpreted physically, these mathematical structures or geometrical fields correspond, as Weyl says, to physical structural fields (Strukturfelder). Analogous to the electromagnetic field, these structural fields act on matter and are in turn acted on by matter. Weyl (1931a, 337) remarks: > > > I now come to the crucial idea of the theory of General Relativity. > Whatever exerts as powerful and real effects as does the > metric structure of the world, cannot be a rigid, once and for > all, fixed geometrical structure of the world, but must > itself be something real which not only exerts effects on > matter but which in turn suffers them through matter. Riemann > already suggested for space the idea that the structural field, like > the electromagnetic field, reciprocally interacts with matter. > > > Weyl (1931a, 338) continues: > > > We already explained with the example of inertia, that the structural > field [Strukturfeld] must, as a close-action [Nahewirkung], be > understood infinitesimally. How this can occur with the metric > structure of space, Riemann abstracted from Gauss's theory of curved > surfaces. > > > The various geometrical fields are not "intrinsic" to the manifold structure of spacetime. The manifold represents an amorphous four-dimensional differentiable continuum in the sense of analysis situs and has no properties besides those that fall under the concept of a manifold. The amorphous four-dimensional differentiable manifold possesses a high degree of symmetry. Because of its homogeneity, all points are alike; there are no objective geometric properties that enable one to distinguish one point from another. This full homogeneity or symmetry of space must be described by its group of automorphisms, the one-to-one mappings of the point field onto itself which leave all relations of objective significance between points undisturbed. If a geometric object $F$, that is a point set with a definite relational structure is given, then those automorphisms of space that leave $F$ invariant, constitute a group and this group describes exactly the symmetry which $F$ possesses. For instance, to use an example by Weyl (1938b) (see also Weyl (1949a, 72-73) and Weyl (1952)), if $R(p\_{1},p\_{2},p\_{3})$ is a ternary relation that asserts $p\_{1},p\_{2},p\_{3}$ lie on a straight line, then we require that any three points, satisfying this relation $R$, are mapped by an automorphism into three other points $p\_{1}',p\_{2}',p\_{3}'$, fulfilling the same relation. The group of automorphisms of the $n$-dimensional number space contains only the identity map, since all numbers of $\mathbb{R}^{n}$ are distinct individuals. It is essentially for this reason that the real numbers are used for coordinate descriptions. Whereas the continuum of real numbers consists of individuals, the continua of space, time, and spacetime are homogeneous. Spacetime points do not admit of an absolute characterization; they can be distinguished, according to Weyl, only by "a demonstrative act, by pointing and saying here-now". In a little book entitled Riemanns geometrische Ideen, ihre Auswirkung und ihre Verknupfung mit der Gruppentheorie, published posthumously in 1988, Weyl (1988, 4-5) makes this interesting comment: > > > Coordinates are introduced on the Mf [manifold] in the most direct > way through the mapping onto the number space, in such a way, that > all coordinates, which arise through one-to-one continuous > transformations, are equally possible. With this the coordinate > concept breaks loose from all special constructions to which it was > bound earlier in geometry. In the language of relativity this means: > The coordinates are not measured, their values are not read off from > real measuring rods which react in a definite way to physical fields > and the metrical structure, rather they are a priori placed in the > world arbitrarily, in order to characterize those physical fields > including the metric structure numerically. The metric structure > becomes through this, so to speak, freed from space; it becomes an > existing field within the remaining structure-less space. Through > this, space as form of appearance contrasts more clearly with its > real content: The content is measured after the form is arbitrarily > related to coordinates. > > > By mapping a given spacetime homeomorphically onto the real number space, providing through the arbitrariness of the mapping, what Weyl calls, a qualitatively non-differentiated field of free possibilities--the continuum of all possible coincidences--we represent spacetime points by their coordinates corresponding to some coordinate system. The four-dimensional arithmetical space can be utilized as a four-dimensional schema for the localization of events of all possible "here-nows". Physical dynamical quantities in spacetime, such as the geometrical structural fields on the four-dimensional spacetime continuum, are describable as functions of a variable point which ranges over the four-dimensional number space $\mathbb{R}^{4}$. Instead of thinking of the spacetime points as real substantival entities, and any talk of fields as just a convenient way of describing geometrical relations between points, one thinks of the geometrical fields such as the projective, conformal causal, affine and metric fields, as real physical entities with dynamical properties, such as energy, momentum and angular momentum, and the field points as mere mathematical abstractions. Spacetime is not a medium in the sense of the old ether concept. No ether in that sense exists here. Just as the electromagnetic fields are not states of a medium but constitute independent realities which are not reducible to anything else, so, according to Weyl, the geometrical fields are independent irreducible physical fields.[62] A class of geometric structural fields of a given type is characterized by a particular Lie group. A geometric structural field belonging to a given class has a microsymmetry group (see definition 4.1) at each point $p \in M$ which is isomorphic to the Lie group that is characteristic of the class. In relativity theory, this microsymmetry group is isomorphic to the Lorentz group and leaves invariant a pseudo-Riemannian metric of Lorentzian signature. The different types of geometric, structural fields may be represented from a modern mathematical point of view as cross sections of appropriate fiber bundles over the manifold $M$; that is, the amorphous manifold $M$ has associated with it various geometric fields in terms of a mapping of a certain kind (called a cross section) from the manifold $M$ to the corresponding bundle space over $M$.[63] In particular, Einstein's general theory of relativity postulates a physical field, the metrical field, which, mathematically speaking, may be characterized as a cross section of the bundle of non-degenerate, second-order, symmetric, covariant tensors of Lorentz signature over $M$. Weyl (1931a, 336) says of this world structure: > > > However this structure is to be exactly and completely described and > whatever its inner ground might be, all laws of nature show that it > constitutes the most decisive influence on the evolution of physical > events: the behavior of rigid bodies and clocks is almost exclusively > determined through the metric structure, as is the pattern of the > motion of a force-free mass point and the propagation of a light > source. And only through these effects on the concrete natural > processes can we recognize this structure. > > > The views of Weyl are diametrically opposed to geometrical conventionalism and some forms of relationalism. According to Weyl, we discover through the behavior of physical phenomena an already determined metrical structure of spacetime. The metrical relations of physical objects are determined by a physical field, the metric field, which is represented by the second rank metric tensor field. Contrary to geometric conventionalism, spacetime geometry is not about rigid rods, ideal clocks, light rays or freely falling particles, except in the derivative sense of providing information about the physically real metric field which, according to Weyl, is as physically real as is the electromagnetic field, and which determines and explains the metrical behavior of congruence standards under transport. The metrical field has physical and metrical significance, and the metrical significance does not consist in the mere articulation of relations obtaining between, say, rigid rods or ideal clocks. The special and general, as well as the non-relativistic spacetime theories postulate various structural constraints which events are held to satisfy. When interpreted physically, these mathematical structures or constraints correspond to physical structural fields (Strukturfelder). Analogous to the electromagnetic field, these structural fields act on matter and are, within the context of the general theory of relativity, in turn acted on by matter. An $n$-dimensional manifold $M$ whose sole properties are those that fall under the concept of a manifold, Weyl (1918b) physically interprets as an $n$-dimensional empty world, that is, a world empty of both matter and fields. On the other hand, an $n$-dimensional manifold $M$ that is an affinely connected manifold, Weyl physically interprets as an $n$-dimensional world filled with a gravitational field, and an $n$-dimensional manifold $M$ endowed with a projective structure represents an $n$-dimensional non-empty world filled with an inertial-gravitational field, or what Weyl calls the guiding field Fuhrungsfeld). In a similar vein, an $n$-dimensional manifold $M$ that possesses a conformal structure of Lorentz type, represents a non-empty $n$-dimensional world filled with a causal field. Finally, an $n$-dimensional manifold $M$ endowed with a metrical structure may be interpreted physically as an $n$-dimensional non-empty world filled with a metric field. #### 4.3.2 The Causal and Inertial Field uniquely determine the Metric The mathematical model of physical spacetime is the four-dimensional pseudo-Riemannian manifold. Weyl (1921c) distinguished between two primitive substructures of that model: the conformal and projective structures and showed that the conformal structure, modelling the causal field governing light propagation, and the projective structure, modelling the inertial or guiding field governing all free (fall) motions, uniquely determine the metric. That is, Weyl (1921c) proved Theorem 4.3 The projective and conformal structure of a metric space determine the metric uniquely. A metric $g$ on a manifold determines a first-order conformal structure on the manifold, namely, an equivalence class of conformally related metrics \[\tag{27} [g] = \{\overline{g} \mid \overline{g} = e^{\theta} g \}. \] A metric $g$ also uniquely determines a symmetric linear connection $\Gamma$ on the manifold. Under a conformal transformation \[\tag{28} g \rightarrow e^{\theta} g = \overline{g}, \] the change of the components of the symmetric linear connection is given by (14), that is, \[\tag{29} \Gamma^{i}\_{jk} \rightarrow \overline{\Gamma}^{i}\_{jk} = \Gamma^{i}\_{jk} + \frac{1}{2}(\delta^{i}\_{j}\theta\_{k} + \delta^{i}\_{k}\theta\_{j} - g\_{jk}g^{ir}\theta\_{r}). \] Thus the set of all arbitrary conformal transformations of the metric induces an equivalence class $K$ of conformally related symmetric linear connections. This equivalence class $K$ constitutes a second-order conformal structure on the manifold and the difference between any two connections in the equivalence class is given by (29). Weyl shows that a conformal transformation (29) preserves the projective structure and hence is a projective transformation (that is, a conformal transformation which also satisfies (7)), if and only if $\theta\_{j} = 0$, in which case the conformal and projective structures are compatible. Weyl remarks after the proof: > > > If it is possible for us, in the real world, to discern causal > propagation, and in particular light propagation, and if moreover, we > are able to recognize and observe as such the motion of free mass > points which follow the guiding field, then we are able to read off > the metric field from this alone, without reliance on clocks and > rigid rods. > > > Elsewhere, Weyl (1949a, 103) says: > > > As a matter of fact it can be shown that the metrical structure of > the world is already fully determined by its inertial and causal > structure, that therefore mensuration need not depend on clocks and > rigid bodies but that light signals and mass points moving under the > influence of inertia alone will suffice. > > > The use of clocks and rigid rods is, within the context of either theory, an undesirable makeshift for two reasons. First, since neither spatial nor temporal intervals are invariants of the four-dimensional spacetime of the special theory of relativity and the general theory of relativity, the invariant spacetime interval $ds$ cannot be directly ascertained by means of standard clocks and rigid rods. Second, the concepts of a rigid body and a periodic system (such as pendulums or atomic clocks) are not fundamental or theoretically self-sufficient, but involve assumptions that presuppose quantum theoretical principles for their justification and thus lie outside the present conceptual relativistic framework. Therefore, methodological and ontological considerations decidedly favor Weyl's causal-inertial method for determining the spacetime metric. From the physical point of view, Weyl emphasized the roles of light propagation and free (fall) motion in revealing the conformal-causal and the projective structures respectively. However, from the mathematical point of view, Weyl did not use these two structures directly in order to derive from them and their compatibility relation, the metric field. Rather, Weyl regarded the metric and affine structures as fundamental and showed that the conformal and the projective structures respectively arise from those structures by mathematical abstraction. ![figure](fig7.png) Figure 7: Weyl took the metric and affines structures as fundamental and showed that the conformal and projective structures respectively arise from them by mathematical abstraction. #### 4.3.3 The Ehlers, Pirani, Schild Construction of the Causal-Inertial Method Ehlers et al. (1972) generalized Weyl's causal-inertial method by deriving the metric field directly from the conformal and projective fields and derived a unique pseudo-Riemannian spacetime metric solely as a consequence of a set of natural, physically well-motivated, constructive, "geometry-free" axioms concerning the incidence and differential properties of light propagation and free (fall) motion. Ehlers, Pirani and Schild adopt Reichenbach's (1924) term, constructive axiomatics to describe the nature of their approach. The "geometry-free" axioms are propositions about a few general qualitative assumptions concerning free (fall) motion and light propagation that can be verified directly through experience in a way that does not presuppose the full blown edifice of the general theory of relativity. From these axioms, the theoretical basis of the theory is reconstructed step by step. The constructive axiomatic approach to spacetime structure is roughly this: 1. Primitive Notions. The constructive axiomatic approach is based on a triple of sets \[ \langle M, \mathcal{P}, \mathcal{L}\rangle \] of objects corresponding respectively to the notions of events, particle paths and light rays, which are taken as primitive. The set $M$ of events is assumed to have a Hausdorff topology with a countable basis in order to state local axioms through the use of such terms as "neighborhood". Members of the sets $\mathcal{P} = \{P, Q, P\_{1}, Q\_{1}, \ldots \}$ and $\mathcal{L} = \{ L, N, L\_{1}, N\_{1}, \ldots \}$ are subsets of $M$ that represent the possible or actual paths of massive particles and light rays in spacetime. 2. Differential Structure. The differential structure is not presupposed; rather through the first few axioms a differential-manifold structure is introduced on the set of events $M$ that is sufficient for the localization of events by means of local coordinates, such as radar coordinates. Once $M$ is given a differential-manifold structure through the introduction of local radar coordinates by means of particles and light rays (such that any two radar coordinates are smoothly related to one another), one can do calculus on $M$ and one may speak of tangent and direction spaces. It is important to emphasize that the members of $\mathcal{P}$ represent possible or actual paths of arbitrary massive particles that may have some internal structure such as higher order gravitational and electromagnetic multipole moments and that may therefore interact in complicated ways with various physical fields. In order to constructively establish the projective structure of spacetime, it is necessary to single out a subset of $\mathcal{P}$, namely $\mathcal{P}\_{f}$, the set of possible or actual paths of spherically symmetric, electrically neutral particles (that is, the world lines of freely falling particles). However, the set $\mathcal{P}\_{f} \subset \mathcal{P}$, can be properly characterized only after a coordinate system (differential structure) is available. Consequently, one must employ arbitrary particles in the statement of those axioms that lead to the local differential structure of spacetime. 3. Second-Order Conformal Structure. The Law of Causality asserts the existence of a unique first-order conformal structure on spacetime (27), that is, a field of infinitesimal light cones. Only null one-directions are determined. Therefore no special choice of parameters along light rays is determined by this structure. The first-order conformal structure can be measured using only the local differential-topological structure. Moreover, by a purely mathematical process involving only differentiation, the first-order conformal structure determines a second-order conformal structure, namely, an equivalence class $K$ of conformally related symmetric linear connections. 4. Projective Structure. The motions of freely falling particles governed by the guiding field reveal the geodesics of spacetime, that is, the geodesics corresponding to an equivalence class $\Pi$ of projectively equivalent symmetric linear connections. Only geodesic one-directions are determined, that is, no special choice of parameters is involved in characterizing free fall motion. 5. Compatibility between the Conformal and Projective Structures. That the conformal and projective structures are compatible is suggested by high energy experiments, according to Ehlers, Pirani and Schild: "A massive particle $(m \gt 0)$, though always slower than a photon, can be made to chase a photon arbitrarily closely." Ehlers, Pirani and Schild therefore assume an axiom of compatibility between the conformal and projective structures, and this leads to a Weyl space: If the projective and conformal structures are compatible, then the intersection \[ \Pi \cap K = \Gamma\ \text{ (Weyl connection)} \] of the equivalence class $K$ of conformally equivalent symmetric linear connections, and the equivalence class $\Pi$ of projectively equivalent symmetric linear connections, contains a unique symmetric linear connection, a Weyl connection. Thus light propagation and free (fall) motion reveal on spacetime a unique Weyl connection which determines the parallel transport of vectors, preserving their timelike, null or spacelike character, and for any pair of non-null vectors, the Weyl connection leaves invariant the ratio of their lengths and the angle between them, provided the vectors are transported along the same path. 6. Pseudo-Riemannian Metric. Since length transfer is non-integrable (i.e., path-dependent) in a Weyl space, a Weyl geometry reduces to a pseudo-Riemannian geometry if and only if Weyl's length-curvature (Streckenkrummung) tensor equals zero, in which case the length of a vector is path-independent under parallel transport, and there exists no second clock effect. #### 4.3.4 The Philosophical Significance of the Causal-Inertial Method Can it be argued that Ehlers, Pirani and Schild's generalization of Weyl's causal-inertial method for determining the spacetime metric constitutes a convention-free, and - in relevant respects - theory-independent body of evidence that can adjudicate between spacetime geometries, and hence between spacetime theories that postulate them? As Weyl showed, we can empirically determine the metric field, provided certain epistemic conditions are satisfied, that is, provided we can measure the conformal-causal structure, and provided "we are able to recognize and observe as such the motion of free mass points which follow the guiding field." Criticisms of Ehlers, Pirani and Schild's constructive axiomatics suggest that the causal-inertial method is not convention-free and that it is ineffective epistemologically in providing a possible solution to the controversy between geometrical realism and conventionalism in favor of realism. Basically, all of the charges laid against Ehlers, Pirani and Schild's constructive axiomatics concentrate on the roles which massive particles play in their construction. One of the constructive axioms employed by Ehlers, Pirani and Schild, the projective axiom, is a statement of the infinitesimal version of the Law of Inertia, the law of free (fall) motion which contains Newton's first law of motion as a special case in the absence of gravitation. Since Ehlers, Pirani and Schild do not provide an independent, non-circular criterion by which to characterize free (fall) motion, their approach has been charged with circularity by philosophers such as Grunbaum (1973), Salmon (1977), Sklar (1977) and Winnie (1977). The problem is a familiar one; how to introduce a class of preferred motions, that is, how to characterize that particular path structure that would govern the motions of free particles $\mathcal{P}\_{f}$, that is, neutral, spherically symmetric, non-rotating test bodies, while avoiding the circularity problem surrounding the notion of a free particle: The only way of knowing when no forces act on a body is by observing that it moves as a free particle along the geodesics of spacetime. But how, without already knowing the geodesics or the projective structure of spacetime is it possible to determine which particles are free and which are not? And to determine the projective structure of spacetime it is necessary to use free particles. Coleman and Korte (1980) have addressed these and related difficulties by providing a non-conventional procedure for the empirical determination of the projective structure.[64] It is worth emphasizing that Weyl's approach to differential geometry, in which the affine, projective and conformal structures are treated in their own right rather than as mere aspects of the metric, was instrumental for his discovery of the non-circular and non-conventional geodesic method for the empirical determination of the spacetimet metric. The old notion of a 'geodesic path' had its inception in the context of classical metrical geometry and 'geodesicity' was characterized in terms of extremal paths of curves, which presupposed a metric. It was Weyl's metric-independent construction of the symmetric linear connection that led him to introduce the geometry of paths and the metric-independent characterization of a geodesic path in terms of the process of autoparallelism of its tangent direction. ### 4.4 The Laws of Motion, Mach's Principle, and Weyl's Cosmological Postulate Weyl provided a general conceptual/mathematical clarification of the concept of motion that applies to any spacetime theory that is based on a differential manifold. In particular, Weyl's penetrating analysis shows that Einstein's understanding of the role and significance of Mach's Principle for the general theory of relativity and cosmology is actually inconsistent with the basic principles of general relativity. Weyl's major contribution to cosmology is known as "Weyl's Hypothesis". The name was coined by Weyl (1926d) himself in an article in the Encyclopedia of Britannica.[65] According to Weyl's Postulate, the worldlines of all galaxies are non-intersecting diverging geodesics that have a common origin in the distant past. From this system of worldlines Weyl derived a common cosmic time. On the basis of his postulate, Weyl (1923c, Appendix III) was also the first to show that there is an approximately linear relation between the redshift of galactic spectra and distance. Weyl had basically discovered Hubble's Law six years prior to Hubble's formulation of it in 1929. Another contribution to cosmology is Weyl's (1919b) spherically symmetric static exact solution to Einstein's linearized[66] field equations. #### 4.4.1 The Laws of Motion and Mach's Principle There are essentially two ways to understand Mach's Principle: (1) Mach's Principle rejects the absolute character of the inertial structure of spacetime, and (2) Mach's Principle rejects the inertial structure of spacetime per se. Version (2) might be characterized as Leibnizian relativity or body relationalism; that is, one understands by relative motion the motion of bodies with respect only to other observable bodies or observable bodily reference frames. The relative motion of a body with respect to absolute space or to the inertial structure of space (Newton) or spacetime is ruled out on epistemological and/or metaphysical grounds. In the context of his general theory of relativity, what Einstein is objecting to in Newtonian Mechanics, and by implication, the theory of special relativity, is the absolute character of the inertial structure; he is not asserting its fictitious character. That is, the general theory of relativity incorporates Mach's Principle as expressed in version (1) by treating the inertial structure as dynamical and not as absolute. However, Einstein also tried to extend and generalize the special theory of relativity by incorporating version (2) of Mach's Principle into the general theory of relativity. Einstein was deeply influenced by Mach's empiricist programme and accepted Mach's insistence on the primacy of observable facts of experience: only observable facts of experience may be invoked to account for the phenomena of motion. As a consequence, Einstein restricted the concept of relative motions to relative motions between bodies. Newton thought that the plane of Foucault's pendulum remains aligned with respect to absolute space. Since the fixed stars are at rest with respect to absolute space the plane of Foucault's pendulum remains aligned to them as well, and rotates relative to the earth. But according to Einstein, Newton's intermediary notion of absolute space is as questionable as it is unnecessary in explaining the behaviour of Foucault's pendulum. Not absolute space, but the actually existing masses of the fixed stars of the whole cosmos guide the plane of Foucault's pendulum. Einstein (1916) argued that the general theory of relativity removes from the special theory of relativity and Newton's theory an inherent epistemological defect. The latter is brought to light by Mach's paradox, namely, Einstein's example of two fluid bodies, $A$ and $B$, which are in constant relative rotation about a common axis. With regard to the extent to which each of the spheres bulges at its equator, infinitely many different states are possible although the relative rotation of the two bodies is the same in every case. Einstein considered the case in which $A$ is a sphere and $B$ is an oblate spheroid. The paradox consists in the fact that there is no readily discernible reason that accounts for the fact that one of the bodies bulges and the other does not. According to Einstein, an epistemological satisfactory solution to this paradox must be based on 'an observable fact of experience'. Einstein wanted to implement a Leibnizian-Machian relationalconception of motion according to which all motion is to be interpreted as the motion of some bodies in relation to other bodies. Einstein wished to extend the body-relative concept of uniform inertial motion to the concept of a body-relative accelerated motion. #### 4.4.2 Weyl's Critique of Einstein's Machian Ideas Weyl was very critical of Einstein's attempt to incorporate version (2) of Mach's Principle into the theory of general relativity and relativistic cosmology because he considered the Leibnizian-Machian relational conception of motion--according to which all motion is to be interpreted as the motion of some bodies in relation to other bodies--to be an incoherent notion within the context of the general theory of relativity. In a paper entitled Massentragheit und Kosmos. Ein Dialog [Inertial Mass and Cosmos. A Dialogue] Weyl (1924b) articulates his overall position on the concept of motion and the role of Mach's Principle in general relativity and cosmology.[67] Weyl defines Mach's Principle as follows: M (Mach's Principle): The inertia of a body is determined through the interaction of all the masses in the universe. Weyl (1924b) then makes the observation that the kinematic principle of relative motion is by itself without any content, unless one also makes the additional physical causal assumption that C (Physical Causality): All events or processes are uniquely causally determined through matter, that is, through charge, mass and the state of motion of the elementary particles of matter.[68] The underlying motivation for assumption $\mathbf{C}$ of physical causality is essentially Mach's empiricist programme, namely, Mach's insistence on the primacy of observable facts of experience. Addressing Einstein's formulation of Mach's paradox, Weyl (1924b) says: Only if we conjoin the kinematic principle of relative motion with the physical assumption $\mathbf{C}$ does it appear groundless or impossible on the basis of the kinematic principle that in the absence of any external forces a stationary body of fluid has the form of a sphere "at rest", while on the other hand it has the form of a "rotating" flattened ellipsoid. Weyl rejects principle $\mathbf{C}$ of physical causality because he denies the feasibility of $\mathbf{M}$ (Mach's Principle), as defined above, on $a$ priori[69] grounds. According to Weyl (1924b) A: The concept of relative motion of several isolated bodies with respect to each other is as untenable according to the theory of general relativity as is the concept of absolute motion of a single body. Weyl notes that what we seem to observe as the rotation of the stars, is in reality not the rotation of the stars themselves but the rotation of the "star compass" [Sternenkompass] which consists of light signals from the stars that meet our eyes at our present location from a certain direction. It is crucial, Weyl reminds us, to be cognisant of the existence of the metric field between the stars and our eyes. This metric field determines the propagation of light, and, like the electromagnetic field, it is capable of change and variation. Weyl (1924b) says that "the metric field is no less important for the direction in which I see the star then is the location of the star itself." How is it possible, Weyl asks, to compare within the context of the general theory of relativity, the state of motions of two separate bodies? Of course, Weyl notes, prior to the general theory of relativity, during Mach's time, one could rely on a rigid frame of reference such as the earth, and indefinitely extend such a frame throughout space. One could then postulate the relative motion of the stars with respect to this frame. However, under the hands of Einstein the coordinate system has lost its rigidity to such a degree, that it can always "cling to the motion of all bodies simultaneously"; that is, whatever the motions of the bodies are, there exists a coordinate system such that all bodies are at rest with respect to that coordinate system. Weyl then clarifies and illustrates the above with the plasticine example, which Weyl (1949a, 105) elsewhere describes as follows: > > > Incidentally, without a world structure the concept of relative > motion of several bodies has, as the postulate of general relativity > shows, no more foundation than the concept of absolute motion of a > single body. Let us imagine the four-dimensional world as a mass of > plasticine traversed by individual fibers, the world lines of the > material particles. Except for the condition that no two world lines > intersect, their pattern may be arbitrarily given. The plasticine can > then be continuously deformed so that not only one but all fibers > become vertical straight lines. Thus no solution of the problem is > possible as long as in adherence to the tendencies of Huyghens and > Mach one disregards the structure of the world. But once the inertial > structure of the world is accepted as the cause for the dynamical > inequivalence of motions, we recognize clearly why the situation > appeared so unsatisfactory. ... Hence the solution is attained > as soon as we dare to acknowledge the inertial structure as a > real thing that not only exerts effects upon matter but in > turn suffers such effects. > > > ![figure](fig8.png) Figure 8: Weyl's plasticine example Applying these considerations to the fixed stars and assuming that it is possible that the (conformal) metrical field which determines the cones of light propagation (light cones) at each point of the plasticine, is carried along by the continuous transformation of the plasticine, then both the earth and the fixed stars will be at rest with respect to the plasticine's coordinate system. Yet despite this the "star compass" is rotating with respect to the earth, exactly as we observe! Employing the concept of the microsymmetry group (definition 4.1), Coleman and Korte (1982) have analyzed Weyl's plasticine example in the following way: Consider a space-time manifold equipped only with a differentiable structure, the plasticine of Weyl's example. Then our spacetime does not have an affine, conformal, projective or metric structure defined on it. In such a world it is possible do define curves and paths; however, there are no preferred curves or paths. Since there is only the differentiable structure, one may apply any diffeomorphism; that is, alldiffeomorphisms preserve this structure; consequently, in the absence of a post-differential-topological structure, the microsymmetry group at any event $p$ is an infinite-parameter group isomorphic to the group of all invertible formal power series in four variables. If there is no post-differentiable topological geometric field in the neighbourhood of a point, then all of these infinite parameters may be chosen freely within rather broad limits. Clearly then, given an infinite number of parameters, one can, as Weyl says, straighten out an arbitrary pattern of world lines (fibers) in the neighbourhood of any event. Now suppose that there exists a post-differentiable topological geometric field, namely, the projective structure at any event of spacetime. Then the microsymmetry group that preserves that structure is a 20-parameter Lie group (see Coleman and Korte (1981)). Thus instead of an infinity of degrees of freedom, only twenty degrees of freedom may be used to actively deform the neighbouring region of spacetime. The fact that only a finite number of parameter are available prevents an arbitrary realignment of the worldlines of material bodies in the neighbourhood of any given event. Other post-differential topological geometrical field structures are similarly restrictive. For example, the microsymmetry group of the conformal structure, which determines the causal structure of spacetime, permits 7 degrees of freedom (6 Lorentz transformations and a dilatation), and permits four more degrees of freedom in second order. Consequently, the existence of the conformal metrical field which determines at each point the cones of light propagation would prevent an arbitrary realignment of light-like fibers, that is, it would be impossible to realign the earth and the fixed stars such that both are at rest with the coordinate system of the plasticine. Weyl's plasticine example shows that the Leibnizian-Machian view of relative motion, namely the view according to which all motion must be defined as motion relative to bodies, is self-defeating in the general theory of relativity. The fact that a stationary, homogeneous elastic sphere will, when set in rotation, bulge at the equator and flatten at the poles is, according to Weyl (1924b), to be accounted for in the following way. The complete physical system consisting of both the body and the local inertial-gravitational field is not the same in the two situations. The cause of the effect is the state of motion of the body with respect to the local inertial-gravitational field, the guiding field, and is not, indeed as Weyl's plasticine example shows, cannot be the state of motion of the body relative to other bodies. To attribute the effect as Einstein and Mach did to the rotation of the body with respect to the other bodies in the universe is, according to Weyl, to endorse a remnant of the unjustified monopoly of the older body ontology, namely, the sovereign right of material bodies to play the role of physically real and acceptable causal agents.[70] #### 4.4.3 Coordinate Transformation Laws of Acceleration Weyl's view that there must be an inertial structure fieldon spacetime, which governs material bodies in free motion, follows from the mathematical nature of the coordinate-transformation laws for acceleration. In a world equipped with only a differential structure, it is possible to do calculus; one can define curves and paths and differentiate, etc. However, as was already pointed out, in such a world, the world of Weyl's plasticine example, there would be no preferred curves or paths. Consequently, the motion of material bodies would not be predictable. However, experience overwhelmingly indicates that the acceleration of a massive body cannot be freely chosen. In particular, consider a simple type of particle, a monopole (unstructured) particle. Experience overwhelmingly tells us that such a particle is characterized by the fact that at any event on its world line, its velocity at that event is sufficient to determine its acceleration at that event. Predictability of motion, therefore, entails that corresponding to every type of massive monopole, there exists a geometric structure field, or what Weyl calls a Strukturfeld that governs the motion of that type of particle. The basic reason which explains this brute fact of experience is a simple mathematical fact about how the acceleration of bodies transforms under a coordinate transformation. Moreover, this simple mathematical fact, involving no more than the basic techniques of partial differentiation, holds in all relativistic, non-relativistic, curved or flat, dynamic or non-dynamic spacetime theories that are based on a local differential topological structure, the minimal structure required for the possibility of assigning arbitrary local coordinates on a differential manifold. Transformation law for acceleration: The transformation law for acceleration is linear, but is not homogeneous in the acceleration variable. As an example consider the transformation laws for the 4-velocity and the 4-acceleration. Recall that a curve in the four-dimensional spacetime manifold $M$ is a map $\gamma : \mathbb{R} \rightarrow M$. For convenience we restrict our attention to those curves which satisfy $\gamma(0) = p$. If we set $\gamma^{i} = x^{i} \circ \gamma(0)$, then the components of the 4-velocity and 4-acceleration at $p \in M$ are respectively given by \[\tag{30} \gamma^i\_1 =\_{def} \frac{d}{dt}\gamma^i(0), \] \[\tag{31} \gamma^i\_2 =\_{def} \frac{d^2}{dt^2}\gamma^i(0), \] The transformation laws of the 4-velocity components $\gamma^{i}\_{1}$ and of the 4-acceleration components $\gamma^{i}\_{2}$ under a change of coordinate chart from $(U,x)\_{p}$ to $(\overline{U}, \overline{x})\_p$, follow from their pointwise definition. From \[ \overline{\gamma}^i(t) = \overline{X}^i(\gamma^i(t)), \] where $\overline{X}^{i} = \overline{x}^i \circ x^{- 1}$, one obtains the transformation law for the 4-velocity and the 4-acceleration respectively: \[\begin{align} \tag{32} \overline{\gamma}^i\_1 &= \overline{X}^i\_j \gamma^{\,j}\_1 \\ \tag{33} \overline{\gamma}^i\_2 &= \overline{X}^i\_j \gamma^{\,j}\_2 + \overline{X}^i\_{jk} \gamma^{\,j}\_1 \gamma^k\_1. \end{align}\] The $\overline{X}^{i}\_{j}$ and $\overline{X}^{i}\_{jk}$ denote the first and second partial derivatives of $\overline{X}^{i}(x^{i})$ at $x^{i} (p)$, namely, \[ \frac{\partial \overline{x}^i}{\partial x^{j}} \text{ and } \frac{\partial \overline{x}^i}{\partial x^{j} \partial x^{k} } \] The expression $\overline{X}^{i}\_{jk}\gamma^{\,j}\_{1}\gamma^{k}\_{1}$ in equation (33) represents the inhomogeneous term of the transformation of the 4-acceleration. The inhomogeneity of the transformation law entails that a 4-acceleration that is zero with respect to one coordinate system is not zero with respect to another coordinate system. This means that there does not exist a unique standard of zero 4-acceleration that is intrinsic to the differential topological structure of spacetime. Moreover, even the difference of the 4-accelerations of two bodies at the same spacetime point has no absolute meaning, unless their 4-velocities happen to be the same. This shows that while the differential topological structure of spacetime gives us sufficient structure to do calculus and to derive the transformation laws for 4-velocities and 4-accelerations by way of simple differentiation, it does not provide sufficient structure with which to determine a standard of zero 4-acceleration. Therefore, as Weyl repeatedly emphasized, no solution to the problem of motion is possible, unless "we dare to acknowledge the inertial structure as a real thing that not only exerts effects upon matter but in turn suffers such effects". In other words there must exist a structure in addition to the differential topological structure in the form of a geometric structure field, or in Weyl's words, geometrisches Strukturfeld, which constitutes the inertial structure of spacetime, and which provides the standard of zero 4-acceleration. Since this field provides the standard of zero 4-acceleration we can call it a geodesic 4-acceleration field, or simply, geodesic acceleration field. A particle in free motion is one that is exclusively governed by this geodesic acceleration field. An acceleration field, geodesic or non-geodesic, can be constructed in the following way. Since the terms that are independent of the 4-acceleration depend on both the spacetime location and on the corresponding 4-velocity of the particle, it is necessary to specify a geometric field standard for zero 4-acceleration that also depends on those independent variables. The transformation law for a 4-acceleration field can be obtained from (33) by replacing $\overline{\gamma}^{i}\_{2}$ by $\overline{A}^{i}\_{2}(\overline{x}^{i},\overline{\gamma}^{i}\_{1})$ and $\gamma^{j}\_{2}$ by $A^{j}\_{2}(x^{i}, \gamma^{i}\_{1}$) to yield \[ \overline{A}^{i}\_{2}(\overline{x}^{i}, \overline{\gamma}^{i}\_{1}) = \overline{X}^{i}\_{j} A^{j}\_{2}(x^{i},\gamma^{i}\_{1}) + \overline{X}^{i}\_{jk}\gamma^{j}\_{1}\gamma^{k}\_{1}. \] The important special case for which the function $A^{i}\_{2}(x^{i}, \gamma^{i}\_{1})$ is a geodesic 4-acceleration field corresponds to the affine structure of spacetime. For this special case the function $A^{i}\_{2}(x^{i}, \gamma^{i}\_{1}$) is denoted by $\Gamma^{i}\_{2}(x^{i}, \gamma^{i}\_{1}$) and is given by \[ \Gamma^{i}\_{2}(x^{i},\gamma^{i}\_{1}) = -\Gamma^{i}\_{jk}(x^{i},\gamma^{i}\_{1}) \gamma^{j}\_{1}\gamma^{k}\_{1}. \] The familiar transformation law for the affine structure (geodesic 4-acceleration field) is then given by \[\tag{34} \overline{\Gamma}^{i}\_{2}(\overline{x}^{1},\overline{\gamma}^{i}\_{1}) = \overline{X}^{i}\_{j}\Gamma^{j}\_{2}(x^{i},\gamma^{i}\_{1}) +\overline{X}^{i}\_{jk}\gamma^{j}\_{1}\gamma^{k}\_{1}. \] Note that the inhomogeneous term $\overline{X}^{i}\_{jk}\gamma^{j}\_{1}\gamma^{k}\_{1}$ of the geodesic 4-acceleration field is identical to the inhomogeneous term of the transformation law (33) for the 4-acceleration of body motion. The differences \[\tag{35} \overline{\gamma}^{i}\_{2} - \overline{\Gamma}^{i}\_{2}(\overline{x}^{i}, \overline{\gamma}^{i}\_{1}) = \overline{X}^{i}\_{j}(\gamma^{j}\_{2} - \Gamma^{j}\_{2}(x^{i},\gamma^{i}\_{1})) \] then transform linearly and homogeneously; consequently, the vanishing or non-vanishing of body accelerations relative to the standard of zero acceleration provided by the geodesic 4-acceleration field (the affine structure), is coordinate independent. That is, the 4-accelerations of bodies and the corresponding 4-forces, are tensorial quantities in concordance with experience. The above argument for the necessity of geometric fields also holds for 3-velocity and 3-acceleration, denoted respectively by $\xi^{\alpha}\_{1}$ and $\xi^{\alpha}\_{2}$. The transformation law for the 3-acceleration is much more complicated than that of the 4-acceleration. However, analogous to the case of 4-acceleration, the transformation law of 3-acceleration is linear and is inhomogeneous in the 3-acceleration variable $\xi^{\alpha}\_{2}$. Consequently, there does not exist a unique standard of zero 3-acceleration that is intrinsic to the differential topological structure of spacetime. The standard of zero 3-acceleration must be provided by a geodesic 3-acceleration field or geodesic directing field, or what Weyl calls the guiding field. The guiding field is also referred to as the projective structure of spacetime and is denoted by $\Pi^{\alpha}\_{2}(x^{i}, \xi^{\alpha}\_{1}$). It is a function of spacetime location and the 3-velocity, both variables of which are independent of the 3-acceleration, as is required. Since the transformation law of the projective structure $\Pi^{\alpha}\_{2}(x^{i}, \xi^{\alpha}\_{1}$) has the same inhomogeneous form as the 3-acceleration $\xi^{\alpha}\_{2}$, the difference \[\tag{36} \xi^{\alpha}\_{2} - \Pi^{\alpha}\_{2}(x^{i}, \xi^{\alpha}\_{1}) \] also transforms linearly and homogeneously. The components $\gamma^{i}\_{2}$ and $\xi^{\alpha}\_{2}$ of the 4-acceleration and 3-acceleration can be thought of as the dynamic descriptors of a material body. On the other hand, the components $\Gamma^{i}\_{2}(x^{i},\gamma^{i}\_{1}$) and $\Pi^{\alpha}\_{2}(x^{i}, \xi^{\alpha}\_{1}$) of the geodesic acceleration field, and the geodesic directing field, respectively, are field quantities. The differences \[\tag{37} \gamma^{i}\_{2} - \Gamma^{i}\_{2}(x^{i},\gamma^{i}\_{1}) \] and \[\tag{38} \xi^{\alpha}\_{2} -\Pi^{\alpha}\_{2}(x^{i}, \xi^{\alpha}\_{1}) \] denote the components of a coordinate independent field-body relation.[71] Weyl (1924b) remarks: > > > We have known since Galileo and Newton, that the motion of a body > involves an inherent struggle between inertia and force. According to > the old view, the inertial tendency of persistence, the > "guidance", which gives a body its natural inertial > motion, is based on a formal geometric structure of the spacetime > (uniform motion in a straight line) which resides once and for all in > spacetime independently of any natural processes. This assumption > Einstein rejects; because whatever exerts as powerful effects as > inertia--for example, in opposition to the molecular forces of > two colliding trains it rips apart their freight cars--must be > something real which itself suffers effect from matter. Moreover, > Einstein recognized that the guiding field's variability and > dependence on matter is revealed in gravitational effects. Therefore, > the dualism between guidance and force is maintained; but > > > > (G) Guidance is a physical field, like the > electromagnetic field, which stands in mutual interaction > with matter. Gravitation belongs to the guiding field and not to > force. Only thus is it possible to explain the equivalence > between inertial and gravitational mass. > > > To move from the old conception to the new conception (G) means, according to Weyl (1924b) > > to replace the geometric difference between uniform and > accelerated motion with the dynamic difference between > guidance and force. Opponents of Einstein asked the question: > Since the church tower receives a jolt in its motion relative to the > train just as the train receives a jolt in its motion relative to the > church tower, why does the train become a wreckage and not the church > tower which it passes? Common sense would answer: because the train > is ripped out of the pathway of the guiding field, but the church > tower is not. ... As long as one ignores the guiding field one > can neither speak of absolute nor of relative motion; only if one > gives due consideration to the guiding field does the concept of > motion acquire content. The theory of relativity, correctly > understood, does not eliminate absolute motion in favour of relative > motion, rather it eliminates the kinematic concept of motion and > replaces it with a dynamic one. The worldview for which Galileo > fought is not undermined by it [relativity]; to the contrary, it is > more concretely interpreted. > > > #### 4.4.4 Weyl's Field-Body Relationist Ontology and Newton's Laws of Motion It is now possible to provide a reformulation of Newton's laws of motion which explicitly takes account of Weyl's field-body-relationalist spacetime ontology, and his analysis of the concept of motion. The law of inertia is an empirically verifiable statement[72] which says > > The Law of Inertia: There exists on spacetime a > unique projective structure $\Pi\_{2}$ or equivalently, a > unique geodesic directing field $\Pi\_{2}$. > > > Free motion is defined with reference to the projective structure $\Pi\_{2}$ as follows: > > Definition of Free Motion: A possible or actual > material body is in a state of free motion during any part of its > history just in case its motion is exclusively governed by the > geodesic directing field (projective structure), that is, just in > case the corresponding segment of its world path is a solution path > of the differential equation determined by the unique projective > structure of spacetime. > > > Newton's second law of motion may be reformulated as follows: > > The Law of Motion: With respect to any coordinate > system, the world line path of a possible or actual material body > satisfies an equation of the form > > > > \[ > m(\xi^{\alpha}\_{2} - \Pi^{\alpha}\_{2}(x^{i}, \xi^{\alpha}\_{1})) > = F^{\alpha}(x^{i},\xi^{\alpha}\_{1}), > \] > > > > where $m$ is a scalar constant characteristic of the material > body called its inertial mass, and > $F^{\alpha}(x^{i},\xi^{\alpha}\_{1}$) is the > 3-force acting on the body. > > > To emphasize, the Law of Inertia and the Law of Motion, as formulated above, apply to all, relativistic or non-relativistic, curved or flat, dynamic or non-dynamic, spacetime theories. The reason for the general character of these laws consists in the fact that they require for their formulation only the local differential topological structure of spacetime, a structure which is common to all spacetime theories. In addition, as was noted earlier in SS4.2, the affine and projective spacetime structures are G-structures. Consequently, they may be flat or non-flat; but they can never vanish. In theories prior to the advent of general relativity, the affine and projective structures were flat. It was common practice, however, to use coordinate systems that were adapted to these flat G-structures. And since in such adapted coordinate systems the components of the affine and projective structures vanish, it was difficult to recognize and to appreciate the existence of these structures, and their important role in providing a coherent account of motion. #### 4.4.5 Mie's Pure Field Theory, Weyl's 'Agens Theory' and Wormhole Theory of Matter We saw that Weyl forcefully advocated a field-body ontological dualism, according to which matter and the guiding field are independent physical realities that causally interact with each other: matter uniquely generates the various states of the guiding field, and the guiding field in turn acts on matter. Weyl did not always subscribe to this ontological dualist position. For a short period, from 1918 to 1920, he advocated a pure field theory of matter , developed in 1912 by Gustav Mie, in the context of Einstein's special theory of relativity: Pure Field Theory of Matter: The physical field has an independent reality that is not reducible to matter; rather, the physical field is constitutive of all matter in the sense that the mass (quantity of matter) of a material particle, such as an electron, consists of a large field energy that is concentrated in a very small region of spacetime. Mie's theory of matter is akin to the traditional geometric view of matter: matter is passive and pure extension. Weyl (1921b) remarks that he adopted the standpoint of the classical pure field theory of matter in the first three editions of Weyl (1923b) because of its beauty and unity, but then gave it up. Weyl (1931a) points out in the Rouse Ball Lecture that since the theory of general relativity geometrized a physical entity, the gravitational field, it was natural to try to geometrize the whole of physics. Prior to the advent of quantum physics one was justified in regarding gravitation and electromagnetism as the only basic entities of nature and to seek their unification by geometrizing both. One could hope, following the example of Gustav Mie, to construct elementary material particles as knots of energy in the gravitational-electromagnetic field, that is, tiny demarcated regions in which the field magnitudes attain very high values. Already in a letter to Felix Klein,[73] toward the end of 1920, Weyl indicated that he had finally freed himself completely from Mie's theory of matter. It now appeared to him that the classical field theory of matter is not the key to reality. In the Rouse Ball Lecture Weyl adduces two reasons for this. First, due to quantum mechanics, there are, in addition to electromagnetic waves, matter waves (Materiewellen) represented by Schrodinger's wave function $\psi$. And Pauli and Dirac recognized that $\psi$ is not a scalar but a magnitude with several components. Thus, from the point of view of the classical field theory of matter not two but three entities would have to be unified. Moreover, given the transformation properties of the wave function, Weyl says it is certain that the magnitude $\psi$ cannot be reduced to gravitation or electromagnetism. Weyl saw clearly, that this geometric view of matter or physics--which to a certain extent had also motivated his earlier construction and manner of presentation of a pure infinitesimal geometry--was untenable in light of the new developments in atomic physics. The second reason, Weyl says, consists in the radical new interpretation of the wave function, which replaces the concept of intensity with that of probability. It is only through such a statistical interpretation that the corpuscular and atomistic aspect of nature is properly recognized. Instead of a geometrictreatment of the classical field theory of matter, the new quantum theory called for a statistical treatment of matter.[74] Already in 1920, Weyl (1920) addressed the relationship between causal and statistical approaches to physics.[75] The theory of general relativity, as well as early developments in atomic physics, clearly tell us, Weyl (1921b) suggests, that matter uniquely determines the field, and that there exist deeper underlying physical laws with which modern physics, such as quantum theory, is concerned, which specify "how the field is affected by matter". That is, experience tells us that matter plays the role of a causal agent which uniquely determines the field, and which therefore has an independent physical reality that cannot be reduced to the field on which it acts. Weyl (1921b, 1924e) refers to his theory of matter as the Agenstheorie der Materie(literally, agent-theory of matter): Matter-Field Dualism (Weyl's Agens Theory of Matter): Matter and field are independent physical realities that causally interact with each other: matter uniquely generates the various states of the field, and the field in turn acts on matter. To excite the field is the essential primary function of matter. The field's function is to respond to the action of matter and is thus secondary. The secondary role of the field is to transmit effects (from body to body) caused by matter, thereby in return affecting matter. The view, that matter uniquely determines the field, was a necessary postulate of an opposing ontological standpoint, according to Weyl. The postulate essentially says that Matter is the only thing which is genuinely real. According to this ontological view, held to a certain degree by the younger Einstein and others who advocated a form of Machian empiricism, the field is relegated to play the role of a feeble extensive medium which transmits effects from body to body.[76] According to this opposing ontological view, the field laws, that is, certain implicit differential connections between the various possible states of the field, on the basis of which the field alone is capable of transmitting effects caused by matter, can essentially have no more significance for reality than the laws of geometry could, according to earlier views. But as we saw earlier, Weyl held that no satisfactory solution can be given to the problem of motion as long as we adhere to the Einstein-Machian empiricist position that relegates the field to the role of a feeble extensive medium, and which does not acknowledge that the guiding field is physically real. However, from the standpoint of Weyl's agens theory of matter, a satisfactory answer to Mach's paradox can be given: the reason why a stationary, homogeneous elastic sphere will bulge at the equator and flatten at the poles, when set in rotation, is due to the fact that the complete physical system consisting of both the body and the guiding field, differs in the rotating case from the stationary one. The local guiding field is the real cause of the inertial forces. Weyl lists two reasons in support for his agens theory of matter. First, the agens theory of matter is the only theory which coheres with the basic experiences of life and physics: matter generates the field and all our actions ultimately involve matter. For example, only through matter can we change the field. Secondly, in order to understand the fact of the existence of charged material particles, we have two possibilities: either we follow Mie and adopt a pure field theory of matter, or we elevate the ontological status of matter and regard it as a real singularity of the field and not merely as a high concentration of field energy in a tiny region of spacetime. Since Mie's approach is necessarily limited to the framework of the theory of special relativity, and since there is no room in the general theory of relativity for a generalization and modification of the classical field laws, as envisaged by Mie in the context of the special theory of relativity, Weyl adopted the second possibility. He was motivated to do so by his recognition that the field equation of an electron at rest contains a finite mass term $m$ that appears to have nothing to do with the energy of the associated field. Weyl's subsequent analysis of mass in terms of electromagnetic field energy provided a definition of mass and a derivation of the basic equations of mechanics, and led Weyl to the invention of the topological idea of wormholes in spacetime. Weyl did not use the term 'wormholes'; it was John Wheeler who later coined the term 'wormhole' in 1957. Weyl spoke of one-dimensional tubes instead. "Inside" these tubes no space exists, and their boundaries are, analogous to infinite distance, inaccessible; they do not belong to the field. In a chapter entitled "Hermann Weyl and the Unity of Knowledge" Wheeler (1994) says, > > > Another insight Weyl gave us on the nature of electricity is > topological in character and dates from 1924. We still do not know > how to assess it properly or how to fit it into the scheme of > physics, although with each passing decade it receives more > attention. The idea is simple. Wormholes thread through space as air > channels through Swiss cheese. Electricity is not electricity. > Electricity is electric lines of force trapped in the topology of > space. > > > #### 4.4.6 Relativistic Cosmology and Weyl's Postulate A year after Einstein (1916) had established the field equations of his new general theory of relativity, Einstein (1917) applied his theory for the first time to cosmology. In doing so, Einstein made several assumptions: Cosmological Principle: Like Newton, Einstein assumed that the universe is homogeneous and isotropic in its distribution of matter. Static Universe: Einstein assumed, as did Newton and most cosmologists at that time, that the universe is static on the large scale. Mach's Principle: Einstein believed that the metric field is completely determined through the masses of bodies. The metric field is determined through the energy-momentum tensor of the field equations. The cosmological principle continuous to play an important role in cosmological modelling to this day. However, Einstein's second assumption that the universe is static was in conflict with his field equations, which permitted models of the universe that were homogeneous and isotropic, but not static. In this regard, Einstein's difficulties were essentially the same that Newton had faced: A static Newtonian model involving an infinite container with an infinite number of stars was unstable; that is, local regions would collapse under gravity. Because Einstein was committed to Mach's Principle he faced a problem concerning the boundary conditions for infinite space containing finiteamount of matter.[77] Einstein recognized that it was impossible to choose boundary conditions such that the ten potentials of the metric $g\_{ij}$ are completely determined by the energy-momentum tensor $T\_{ij}$, as required by Mach's Principle. That is, the boundary conditions "flat at infinity" entail a global inertial frame that is tied to empty flat space at infinity, and hence is unrelated to the mass-energy content of space, contrary to Mach's Principle, according to which only mass-energy can influence inertia. Einstein thought that he could solve the difficulties of an unstable non-static universe with boundary conditions at infinity that do not satisfy Mach's Principle, by introducing the cosmological term $\Lambda$ into his field equations. He showed that for positive values of the cosmological constant, his modified field equation admitted a solution for a static[78] universe in which space is curved, unbounded and finite; that is, space is a hyper surface of a sphere in four dimensions. Einstein's spatially closed universe is often referred to as Einstein's "cylinder" world: with two of the spatial dimensions suppressed, the model universe can be pictured as a cylinder where the radius $A$ represents the space and the axis the time coordinate. ![figure](fig9.png) Figure 9: Einstein Universe According to Einstein's Machian convictions, since inertia is determined only by matter, there can be no inertial structure or field in the absence of matter. Consequently, it is impossible, Einstein conjectured, to find a solution to the field equations--that is, to determine the metric $g\_{ij}$--if the energy-momentum tensor $T\_{ij}$ representing the mass-energy content of the universe is zero. The non-existence of 'vacuum solutions' for a static universe demonstrated, Einstein thought, that Mach's Principle had been successfully incorporated into his theory of general relativity. Einstein also believed that his solution was unique because of the assumptions of isotropy and homogeneity.[79] However, Einstein was mistaken. In 1917, the Dutch astronomer Willem de Sitter published another solution to Einstein's field equations containing the cosmological constant. De Sitter's solution showed that Einstein's solution is not a unique solution of his field equations. In addition, since de Sitter's universe is empty it provided a direct counter-example to Einstein's hope that Mach's Principle had been successfully incorporated into his theory.[80] There are cosmologists who, like Einstein, are favourably disposed towards some version of Mach's Principle, and who believe that the local laws, which are satisfied by various physical fields, are determined by the large scale structure of the universe. On the other hand, there are those cosmologists who, like Weyl, take a conservative approach; they take empirically confirmed locallaws and investigate what these laws might imply about the universe as a whole. Our understanding of the large scale structure of the universe, Weyl emphasized, must be based on theories and principles which are verified locally. Einstein's general theory is a local field theory; like electromagnetism, it is a close action theory.[81] Weyl (1924b) says: > > > It appears to me that one can grasp the concrete physical content of > the theory of relativity without taking a position regarding the > causal relationship between the masses of the universe and inertia. > > > And, referring to (G), (see citation at the end of SS4.4.3), which says that "Guidance is a physical field, like the electromagnetic field, which stands in mutual interaction with matter. Gravitation belongs to the guiding field and not to force", Weyl (1924b) says: > > > What I have so far presented and briefly formulated in the two > sentences of G, that alone impacts on physics and underlies the > actual individual investigations of problems of the theory of > relativity. Mach's Principle, according to which the fixed stars > intervene with mysterious power in earthly events, goes far beyond > this [G] and is until now pure speculation; it merely has > cosmological significance and does not become important for natural > science until astronomical observations reach the totality of the > cosmos [Weltganze], and not merely one island of stars > [Sterneninsel]. We could leave the question unanswered if I did not > have to admit that it is tempting to construct, on the basis of the > theory of relativity, a picture of the totality of the cosmos. > > > Weyl's claim is that because general relativity is an inherently local field theory, its validity and soundness is essentially independent of global cosmological considerations. However, if we wish to introduce such global considerations into our local physics, then we can do so only on the basis of additional assumptions, such as, for example, the Cosmological Principle, already mentioned. In 1923 Weyl (1923b, SS39) introduced another cosmological assumption, namely, the so-called Weyl Postulate. De Sitter's solution and the new astronomical discoveries in the early 1920's, which suggested that the universe is not static but expanding, led to a drastic change in thinking about the nature of the universe and an increased scepticism towards Einstein's model of a static universe. In 1923, Weyl (1923b, SS39) notes in the fifth edition of Raum Zeit Materie, that despite its attractiveness, Einstein's cosmology suffers from serious defects. Weyl begins by pointing out that spectroscopic results indicate that the stars have an age. Weyl continued, > > > all our experiences about the distribution of stars show that the > present state of the starry sky has nothing to do with a > "statistical final state." The small velocities > of the stars is due to a common origin rather than some equilibrium; > incidentally, it appears, based on observation, that the more distant > configurations are from each other, the greater the velocities on > average. Instead of uniform distribution of matter, astronomical > facts lead rather to the view that individual clouds of stars glide > by in vast empty space. > > > Weyl further points out that de Sitter showed that Einstein's cosmological equations of gravity have "a very simple regular solution" and that an empty spacetime, namely, "a metrically homogeneous spacetime of non-vanishing curvature," is compatible with these equations after all. Weyl says that de Sitter's solution, which on the whole is not static, forces us to abandon our predilection for a static universe. The Einstein and the de Sitter universe are both spacetimes with two separate fringes, the infinitely remote past and the infinitely remote future. Dropping two of its spatial dimensions we imagine Einstein's universe as the surface of a straight cylinder of a certain radius and de Sitter's universe as a one sheeted hyperboloid. Both surfaces are surfaces of infinite extent in both directions. Both the Einstein universe and the de Sitter universe spread from the eternal past to the eternal future. However, unlike de Sitter's universe, in Einstein's universe "the metrical relations are such that the light cone issuing from a world point is folded back upon itself an infinite number of times. An observer should therefore see infinitely many images of a star, showing him the star in states between which an eon has elapsed, the time needed by the light to travel around the sphere of the world." Weyl (1930) says: > > > ... I start from de Sitter's solution: the world, according to > its metric constitution, has the character of a four-dimensional > "sphere" (hyperboloid) > > > > \[\tag{39} > x^{2}\_{1} + x^{2}\_{2} + x^{2}\_{3} + x^{2}\_{4} - x^{2}\_{5} = a^{2} > \] > > > > in a five-dimensional quasi-euclidean space, with the line element > > > > \[\tag{40} > ds^{2} = dx^{2}\_{1} + dx^{2}\_{2} + dx^{2}\_{3} + dx^{2}\_{4} - dx^{2}\_{5}. > \] > > > > The sphere has the same degree of metric homogeneity as the world of > the special theory of relativity, which can be conceived as a > four-dimensional "plane" in the same space. The plane, > however, has only one connected infinitely distant > "seam," while it is the most prominent topological > property of the sphere to be endowed with two--the infinitely > distant past and the infinitely distant future. In this sense one may > say that space is closed in de Sitter's solution. On the other hand, > however, it is distinguished from the well-known Einstein solution, > which is based on a homogeneous distribution of mass, by the fact > that the null cone of future belonging to a world-point does not > overlap with itself; in this causal sense, the de Sitter space is > open. > > > On this hyperboloid, a single star (nebula or galaxy, in later contexts) $A$, also called "observer" by Weyl, traces a geodesic world line, and from each point of the star's world line a light cone opens into the future and fills a region $D$, which Weyl calls the domain of influence of the star. In de Sitter's cosmology this domain of influence covers only half of the hyperboloid and Weyl suggests that it is reasonable to assume that this half of the hyperboloid corresponds to the real world. ![figure](fig10.png) Figure 10: De Sitter's hyperboloid with domain of influence $D$ covering half of the hyperboloid and world lines of stars. There are innumerable stars or geodesics, according to Weyl, that have the same domain of influence as the arbitrarily chosen star $A$; they form, he says, a system that has been causally interconnected since eternity. Such a system of causally interconnected stars Weyl describes as stars of a common origin that lies in an infinitely remote past. The sheaf of world-lines of such a system of stars converges, in the direction of the infinitely remote past, on an infinitely small part of the total extent of the hyperboloid, and diverges in the direction of the future on an ever increasing extent of the hyperboloid. Weyl's choice of singling out a particular sheaf of non-intersecting timelike geodesics as constituting the cosmological substratum is the content of Weyl's Postulate. Weyl (1923b, 295) says: > > > The hypothesis is suggestive, that all the celestial bodies which we > know belong to such a single system; this would explain the small > velocities of the stars as a consequence of their common origin. > > > The transition from a static to a dynamic universe opens up the possibility of a disorderly universe where galaxies could collide, that is, their world lines might intersect. Roughly speaking, Weyl's Postulate states that the actual universe is an orderly universe. It says that the world lines of the galaxies form a 3-sheaf of non-intersecting[82] geodesics orthogonal to layers of spacelike hypersurfaces. ![figure](fig11.png) Figure 11: Weyl's Postulate Since the relative velocities of matter is small in each collection of galaxies extending over an astronomical neighbourhood, one can approximate a "smeared-out" motion of the galaxies and introduce a substratum or fluid which fills space and in which the galaxies move like "fundamental particles".[83] Weyl's postulate says that observers associated with this smeared-out motion constitute a privileged class of observers of the universe. Since geodesics do not intersect, according to Weyl's Postulate, there exist one and only one geodesic which passes through each spacetime point. Consequently, matter possesses a unique velocity at any spacetime point. Therefore, the fluid may be regarded as a perfect fluid; and this is the essential content of Weyl's Postulate. Since the geodesics of the galaxies are orthogonal to a layer of spacelike hypersrfaces according to Weyl's Postulate, one can introduce coordinates $(x^{0}, x^{1}, x^{2}, x^{3})$ such that the spacelike hypersurfaces are given by $x^{0} =$ constant, and the spacelike coordinates $x^{\alpha}$ $(\alpha = 1, 2, 3)$ are constant along the geodesics of each galaxy. Therefore, the spacelike coordinates $x^{\alpha}$ are co-moving coordinates along the geodesics of each galaxy. The orthogonality condition permits a choice of the time coordinate $x^{0}$ such that the metric or line element has the form \[\begin{align} ds^{2} &= (dx^{0})^{2} - g\_{\alpha \beta}dx^{\alpha}dx^{\beta} \\ \tag{41} &= c^{2}dt^{2} - g\_{\alpha \beta}dx^{\alpha}dx^{\beta}, \end{align}\] where $ct = x^{0}, x^{0}$ is called the cosmic time, and $t$ is the proper time of any galaxy. The spacelike hypsersurfaces are therefore the surfaces of simultaneity with respect to the cosmic time $x^{0}$. The Cosmological Principle in turn tells us that these hypersurfaces of simultaneity are homogeneous and isotropic. Independently, Robertson and Walker, were subsequently able to give a precise mathematical derivation of the most general metric by assuming Weyl's Postulate and the Cosmological Principle. #### 4.4.7 Discovering Hubble's Law Weyl's introduction of his Postulate made it possible for him to provide the first satisfactory treatment of the cosmological redshift. Consider a light source, say a star $A$, which emits monochromatic light that travels along null geodesics $L, L',\ldots$ to an observer $O$. Let $s$ be the proper time of the light source, and let $\sigma$ be the proper time of the observer $O$. Then to every point $s$ on the world line of the light source $A$ there corresponds a point on the world line of the observer $O$, namely, $\sigma = \sigma(s)$. ![figure](fig12.png) Figure 12: A body or star $A$ emits monochromatic light which travels along null geodesics $L, L',\ldots$ to an observer $O$. Consequently, if one of the generators of the light cone issuing from $A$'s world line at $A$'s proper time $s\_{0}$--the null geodesic $L$--reaches observer $O$ at the observer's proper time $\sigma(s\_{0})$, then \[\tag{42} d\sigma = \left.\frac{d\sigma(s)}{ds}\right\|\_{s\_0} ds. \] Therefore, the frequency $\nu\_{A}$ of the light that would be measured by some hypothetical observer on $A$ is related to the frequency $\nu\_{O}$ measured on $O$ by \[\tag{43} \frac{\nu\_{O}}{\nu\_{A}} = \frac{d\sigma(s)}{ds}. \] According to Weyl (1923c) this relationship holds in arbitrary spacetimes and for arbitrary motions of source and observer. Weyl (1923b, Anhang III) then applied this relationship to de Sitter's world and showed, to lowest order, that the redshift is linearin distance; that is, Weyl theoretically derived, what was later called Hubble's redshift law. Using Slipher's redshift data Weyl estimated a Hubble constant six years prior to Hubble. Weyl (1923b, Anhang III) remarks: > > > It is noteworthy that neither the elementary nor Einstein's cosmology > lead to such a redshift. Of course, one cannot claim today, that our > explanation hits the right mark, especially since the views about the > nature and distance of the spiral nebulae are still very much in need > of further clarification. > > > In 1933 Weyl gave a lecture in Gottingen in which Weyl (1934b) recalls > > > According to the Doppler effect the receding motion of the stars is > revealed in a redshift of their spectral lines which is proportional > to distance. In this form, where De Sitter's solution of the > gravitational equation is augmented by an assumption concerning the > undisturbed motion of the stars, I had predicted the redshift in the > year 1923. > > > ### 4.5 Quantum Mechanics and Quantum Field Theory #### 4.5.1 Group Theory During the period 1925-1926 Weyl published a sequence of groundbreaking papers (Weyl (1925, 1926a,b,c)) in which he presented a general theory of the representations and invariants of the classical Lie groups. In these celebrated papers Weyl drew together I. Schur's work on invariants and representations of the $n$-dimensional rotation group, and E. Cartan's work on semisimple Lie algebras. In doing so, Weyl utilized different fields of mathematics such as, tensor algebra, invariant theory, Riemann surfaces and Hilbert's theory of integral equations. Weyl himself considered these papers his greatest work in mathematics. The central role that group theoretic techniques played in Weyl's analysis of spacetime was one of several factors which led Weyl to his general theory of the representations and invariants of the classical Lie groups. It was in the context of Weyl's investigation of the space-problem (see SS4.2) that Weyl came to appreciate the value of group theory for investigating the mathematical and philosophical foundations of physical theories in general, and for dealing with fundamental questions motivated by the general theory of relativity, in particular. A motivation of quite another sort, which led Weyl to his general representation theory, was provided by Study when he attacked Weyl specifically, as well as other unnamed individuals, by accusing them "of having neglected a rich cultural domain (namely, the theory of invariants), indeed of having completely ignored it".[84] Weyl (1924c) replied immediately providing a new foundation for the theory of invariants of the special linear groups $SL(n, \mathbb{C})$ and its most important subgroups, the special orthogonal group $SO(n, \mathbb{C})$ and the special symplectic group $SSp(\bfrac{n}{2}, \mathbb{C})$ (for $n$ even) based on algebraic identities due to Capelli. In a footnote, Weyl (1924c) sarcastically informed Study that "even if he [Weyl] had been as well versed as Study in the theory of invariants, he would not have used the symbolic method in his book Raum, Zeit, Materie and even with the last breath of his life would not have mentioned the algebraic completeness theorem for invariant theory". Weyl's point was that in the context of his book Raum-Zeit-Materie, the kernel-index method of tensor analysis is more appropriate than the methods of the theory of algebraic invariants.[85] While this account of events leading up to Weyl's groundbreaking papers on group theory seems reasonable enough, Hawkins (2000) has suggested a fuller account, which brings into focus Weyl's deep philosophical interest in the mathematical foundations of the theory of general relativity by drawing attention to Weyl (1924d) on tensor symmetries, which, according to Hawkins, played an important role in redirecting Weyl's research interests toward pure mathematics.[86] Weyl (1949b, 400) himself noted that his interest in the philosophical foundations of the general theory of relativity motivated his analysis of the representations and invariants of the continuous groups: "I can say that the wish to understand what really is the mathematical substance behind the formal apparatus of relativity theory led me to the study of representations and invariants of groups; and my experience in this regard is probably not unique". Weyl's paper (Weyl (1924a)), and the first chapter Weyl (1925) of his celebrated papers on representation theory, have the same title: "The group theoretic foundation of the tensor calculus". Hawkins (1998) says, Weyl > > > had obtained through the theory of groups, and in particular through > the theory of group representations--as augmented by his own > contributions--what he felt was a proper mathematical > understanding of tensors, tensor symmetries, and the reason they > represent the source of all linear quantities that might arise in > mathematics or physics. Once again, he had come to appreciate the > importance of the theory of groups--and now especially the > theory of group representation--for gaining insight into > mathematical questions suggested by relativity theory. Unlike his > work on the space problem ...Weyl now found himself drawing upon > far more than the rudiments of group theory. ... And of course > Cartan[87] > had showed that the space problem could also be resolved with the aid > of results about representations. In short, the representation theory > of groups had proved itself to be a powerful tool for answering the > sort of mathematical questions that grew out of Weyl's involvement > with relativity theory. > > > Somewhat later, Weyl (1939) wrote a book, entitled The Classical Groups, Their Invariants and Representations, in which he returned to the theory of invariants and representations of the semisimple Lie groups. In this work, he satisfied his ambition "to derive the decisive results for the most important of these groups by direct algebraic construction, in particular for the full group of all non-singular linear transformations and for the orthogonal group". He intentionally restricted the discussion of the general theory and devoted most of the book to the derivation of specific results for the general linear, the special linear, the orthogonal and the symplectic groups. #### 4.5.2 Weyl's philosophical critique of Cartan's approach to geometry As far back as the 1920s, the great French mathematician and geometer Elie Cartan had recognized that the notions of parallelism and affine connection admit of an important generalization in the sense that (1) the spaces for which the notion of infinitesimal parallel transport is defined need not be the tangent spaces that intrinsically arise from the differential structure of a Riemannian manifold $M$ at each of its points; rather, the spaces are general spaces that are not intrinsically tied to the differential manifold structure of $M$, and (2) relevant groups operate on these general spaces directly and not on the manifold $M$, and therefore groups play a dominant and independent role. Weyl (1938a) published a critical review of Cartan's (1937) book in which Cartan further developed his notion of moving frames("reperes mobiles") and generalized spaces ("espaces generalises"). However, Weyl (1988) expressed some of his reservations to Cartan's approach as early as 1925; and four years later Weyl (1929e) presented a more detailed critique. Cartan's approach to differential geometry is in response to the fact that Euclidean geometry was generalized in two ways resulting in essentially two incompatible approaches to geometry.[88] The first generalization occurred with the discovery of non-Euclidean geometries and with Klein's (1921) subsequent Erlanger program in 1872, which provided a coherent group theoretical framework for the various non-Euclidean geometries. The second generalization of Euclidean geometry occurred when Riemann (1854) discovered Riemannian geometry. The two generalizations of Euclidean geometry essentially constitute incompatible approaches to applied geometry. In particular, while Klein's Erlanger program provides an appropriate group theoretical framework for Einstein's theory of special relativity, it is Riemannian geometry, and not Klein's group theoretic approach, which provides the appropriate underlying geometric framework for Einstein's theory of general relativity. As Cartan observes: > > > General relativity threw into physics and philosophy the antagonism > that existed between the two principle directors of geometry, Riemann > and Klein. The space-times of classical mechanics and of special > relativity are of the type of Klein, those of general relativity are > of the type of > Riemann.[89] > > > > Cartan eliminated the incompatibility between the two approaches by synthesizing Riemannian geometry and Klein's Erlanger program through a further generalization of both, resulting in what Cartan called, generalized spaces (or generalized geometries). In his Erlanger program, Klein provided a unified approach to the various "global" geometries by showing that each of the geometries is characterized by a particular group of transformations: Euclidean geometry is characterized by the group of translations and rotations in the plane; the geometry of the sphere $S^{2}$ is characterized by the orthogonal group $O(3)$; and the geometry of the hyperbolic plane is characterized by the pseudo-orthogonal group $O(1, 2)$. In Klein's approach each geometry is a (connected) manifold endowed with a group of automorphisms, that is, a Lie group $G$ of "motions" that acts transitively on the manifold, such that two figures are regarded as congruent if and only if there exists an element of the appropriate Lie group $G$ that transforms one of the figures into the other. A generalized geometry in Klein's sense shifts the emphasis from the underlying manifold or space to the group. Thus a Klein geometry (space) consists of, (1) a smooth manifold, (2) a Lie group $G$ (the principal group of the geometry), and (3) a transitive action of $G$ on the manifold. Besides being "global", a Klein geometry (space) is completely homogeneous in the sense that its points cannot be distinguished on the basis of geometric relations because the transitive group action preserves such relations. As Weyl (1949b) describes it, Klein's approach to the various "global" geometries is very suited to Einstein's theory of special relativity: > > > According to Einstein's special relativity theory the > four-dimensional world of the spacetime points is a Klein space > characterized by a definite group $\Gamma$; and that group is the > ... group of Euclidean similarities--with one very > important difference however. The orthogonal transformations, i.e., > the homogeneous linear transformations which leave > > > > \[ > x^{2}\_{1} + x^{2}\_{2} + x^{2}\_{3} + x^{2}\_{4} > \] > > > > unchanged have to be replaced by the Lorentz transformations leaving > > > > \[ > x^{2}\_{1} + x^{2}\_{2} + x^{2}\_{3} - x^{2}\_{4} > \] > > > > invariant. > > > However, with the advent of Einstein's general theory of relativity the emphasis shifted from global homogeneous geometric structures to local inhomogeneous structures. Whereas Klein spaces are global and fully homogeneous, the Riemannian metric structure underlying Einstein's general theory is local and inhomogeneous. A general Riemannian space admits of no isometry other than the identity. Referring to Cartan (1923a), Weyl (1929e) says that Cartan's generalization of Klein geometries consists in adapting Klein's Erlanger program to infinitesimal geometry by applying Klein's Erlanger program to the tangent plane rather than to the manifold itself.[90] > > > Cartan developed a general scheme of infinitesimal geometry in which > Klein's notions were applied to the tangent plane and not to the > $n$-dimensional manifold $M$ itself. > > > ![figure](fig13.png) Figure 13: Cartan's generalization Figure 13 above, adapted from Sharpe (1997), may help in clarifying the discussion. The generalization of Euclidean geometry to a Riemannian space (the left vertical blue arrow) says: 1. A general Riemannian space approximates Euclidean space only locally; that is, at each point $p \in M$ there exists a tangent space $T(M\_{p})$ that arises intrinsically from the underlying differential structure of $M$. 2. In addition, a Riemannian space is inhomogeneous through the introduction of curvature. Analogously, Cartan's generalization of a Klein space to a Cartan space (the right vertical blue arrow) says: 1. Cartan's generalized space $\Sigma(M)$ approximates a Klein space only locally; that is, at each point $p \in M$ there exists a "Tangent Space", that is, a Klein space $\Sigma(M\_{p})$. Note that a Klein space $\Sigma(M\_{p})$ is itself a generalized space (in the sense of Cartan) with zero curvature; it possesses perfect homogeneity. 2. In addition, Cartan's generalized space $\Sigma(M)$ is inhomogeneous by the introduction of curvature. ![figure](fig14.png) Figure 14: Cartan's generalized space Cartan's generalized space $\Sigma$(M) is the space of all "Tangent Spaces" (i.e., all Klein spaces $\Sigma(M\_{p}))$ and contains a mixture of homogeneous and inhomogeneous spaces (see figure 14). Finally, Cartan's generalization of Riemannian space (lower horizontal red arrow) (figure 13) turns on the recognition that the "Tangent Space" in Cartan's sense is not the same, or need not be the same, as the ordinary tangent space that arises naturally from the underlying differential structure of a Riemannian manifold. Cartan's "Tangent Space" $\Sigma(M\_{p})$ at $p \in M$ denotes what is known as a fiber in modern fiber bundle language, where the manifold $M$ is called the base space of the fiber bundle. In Weyl (1929e, 1988) and to a lesser extent in Weyl (1938a), Weyl objected to Cartan's approach by noting that Cartan's "Tangent Space", namely the Klein space $\Sigma(M\_{p})$ associated with each point of the manifold $M$, does not arise intrinsically from the differential structure of the manifold the way the ordinary tangent vector space does. Weyl therefore noted that it is necessary to impose certain non-intrinsic embedding conditions on $\Sigma(M\_{p})$ that specify how the "Tangent Space" $\Sigma(M\_{p})$ is associated with each point of the manifold $M$. Paraphrasing Weyl, the situation is as follows: We assume that we can associate a copy $\Sigma(M\_{p})$ of a given Klein space with each point $p$ of the manifold $M$ and that the displacement of the Klein space $\Sigma(M\_{p})$ at $p \in M$ to the Klein space $\Sigma(M\_{p'})$ associated with an infinitely nearby point $p'\in M$, constitutes an isomorphic representation of $\Sigma(M\_{p})$ on $\Sigma(M\_{p'})$ by means of an infinitesimal action of the group $G$. In choosing an admissible frame of reference $f$ for each Klein space $\Sigma(M\_{p})$, their points are represented by normal coordinates $\xi$. Any two frames $f,f'$ are related by a group element $s \in G$, and a succession of transformations $f \rightarrow f'$ and $f' \rightarrow f''$ by $s \in G$ and $t \in G$ respectively, relates $f$ and $f''$ by the group composition $t \circ s \in G$. Nothing so far has been said about how specifically the "Tangent Space" $\Sigma(M\_{p})$ is connected to the manifold. Since $\Sigma(M\_{p})$ is supposed to be a generalization of the ordinary tangent space which arises intrinsically from the local differential structure of $M$, Weyl suggests that certain embedding conditions have to be imposed on the normal coordinates $\xi$ of the Klein space $\Sigma(M\_{p})$. Embedding Condition 1: We must first designate a point as the center of $\Sigma(M\_{p})$ and then require that they coincide or cover the point $p \in M$. This leads, Weyl says, to a restriction in the choice of a normal coordinate system $\xi$ on $\Sigma(M\_{p})$. And because $G$ acts transitively, a normal coordinate system $\xi$ on $\Sigma(M\_{p})$ can be chosen such that the normal coordinates $\xi$ vanish at the center, that is, $\xi^{1} = \xi^{2} = \cdots = 0$. The group $G$ is therefore restricted to the subgroup $G\_{0}$ of all representations of $G$ which leave the center invariant. Embedding Condition 2: The notion of a tangent plane also requires that there is a one-to-one linear mapping between the line elements of $\Sigma(M\_{p})$ starting from 0, with the line elements of $M$ starting from $p$. This means that the number of dimension of the Klein space $\Sigma(M\_{p})$ has the same number of dimension as the manifold $M$. Embedding Condition 3: The infinitesimal displacement $\Sigma(M\_{p}) \rightarrow \Sigma(M\_{p'})$ will carry an infinitesimal vector at the center of $\Sigma(M\_{p})$, which is in one-to-one correspondence with a vector at $p \in M$, to the center of $\Sigma(M\_{p'})$. No further conditions need be imposed according to Weyl. If we displace $\Sigma(M\_{p})$ by successive steps around a curve $\gamma$ back to the point $p \in M$ then the final position of $\Sigma(M\_{p})$ is obtained from its original poition or orientation by a certain automorphism $\Sigma(M\_{p}) \rightarrow \Sigma(M\_{p})$. This automorphism is Cartan's generalization of Riemann's concept of curvature along the curve $\gamma$ on $M$. According to Weyl, the "Tangent Space" $\Sigma(M\_{p})$ is not uniquely determined by the differential structure of $M$. If $G$ were the affine group, Weyl says, then the conditions above would fully specify the normal coordinate system $\xi^{\alpha}$ on $\Sigma(M\_{p})$ as a function of the chosen local coordinates $x^{i}$ on $M$. Since this is not the case if $G$ is a more extensive group than the affine group, Weyl concludes that the "Tangent Space" $\Sigma(M\_{p})$ "is not as yet uniquely determined by the nature of $M$, and so long as this is not accomplished we can not say that Cartan's theory deals only with the manifold $M$." Weyl adds: > > > Conversely, the tangent plane in $p$ in the ordinary sense, that > is, the linear manifold of line elements in $p$, is a centered > affine space; its group $G$ is not a matter of convention. This > has always appeared to me to be a deficiency of the theory .... > > > The reader may wish to consult Ryckman (2005, 171-173), who argues "that a philosophical contention, indeed, phenomenological one, underlies the stated mathematical reasons that kept him [Weyl] for a number of years from concurring with Cartan's "moving frame" approach to differential geometry". In 1949 Weyl explicitly acknowledged and praised Cartan's approach. Unlike his earlier critical remarks, he now considered it to be a virtue that the frame of reference in $\Sigma(M\_{p})$ is independent of the choice of coordinates on $M$. Weyl (1949b) says of the traditional approach and Cartan's new approach to geometry: > > > Hence we have here before us the natural general basis on which that > notion rests. The infinitesimal trend in geometry initiated by Gauss' > theory of curved surfaces now merges with that other line of thought > that culminated in Klein's Erlanger program. > > > > > It is not advisable to bind the frame of reference in $\Sigma\_{p}$ > to the coordinates $x^{i}$ covering the neighborhood of $p$ in > $M$. In this respect the old treatment of affinely connected > manifolds is misleading. ... [I]n the modern development of > infinitesimal geometry in the large, where it combines with topology > and the associated Klein spaces appear under the name of fibres, it > has been found best to keep the reperes, the > frames of the fibre spaces, independent of the coordinates of the > underlying manifold. > > > Moreover, in 1949, Weyl also emphasizes that it is necessary to employ Cartan's method if one wishes to fit Dirac's theory of the electron into general relativity. Weyl (1949b) says: > > > When one tries to fit Dirac's theory of the electron into general > relativity, it becomes imperative to adopt the Cartan method. For > Dirac's four $\psi$-components are relative to a Cartesian (or rather > a Lorentz) frame. One knows how they transform under transition from > one Lorentz frame to another (spin representation of the Lorentz > group); but this law of transformation is of such a nature that it > cannot be extended to arbitrary linear transformations mediating > between affine frames. > > > Weyl is here referring to his three important papers, which appeared in 1929--the same year in which he had published his detailed critique of Cartan's method--in which he investigates the adaptation of Dirac's theory of the special relativistic electron to the theory of general relativity, and where he develops the tetrad or Vierbein formalism for the representation of local two-component spinor structures on Lorentz manifolds. #### 4.5.3 Weyl's New Gauge Principle and Dirac's Special Relativistic Electron Only a year after Pauli's review article in 1921, in which Pauli had argued that Weyl's defence of his unified field theory deprives it of its inherent convincing power from a physical point of view, Schrodinger (1922) suggested the possibility that Weyl's 1918 gauge theory could suitably be employed in the quantum mechanical description of the electron.[91] Similar proposals were subsequently made by Fock (1926) and London (1927). With the advent of the quantum theory of the electron around 1927/28 Weyl abandoned his gauge theory of 1918. He did so because in the new quantum theory a different kind of gauge invariance associated with Dirac's theory of the electron was discovered which, as had been suggested by Fock (1926) and London (1927), more adequately accounted for the conservation of electric charge.[92] Why did Weyl hold on to his gauge theory for almost a decade despite a preponderance of compelling empirical arguments that were mounted against it by Einstein, Pauli and others?[93] In one of Weyl's (1918/1998) last letters to Einstein concerning his unified field theory, Weyl made it clear that it was mathematics and not physics that was the driving force behind his unified field theory.[94] > > > Incidentally, you must not believe that it was because of physics > that I introduced the linear differential form d$\varphi$ in addition > to the quadratic form. I wanted rather to eliminate this > "inconsistency" which always has been a bone of > contention to > me.[95] > And then, to my surprise, I realized that it looked as if it might > explain electricity. You clap your hands above your head and shout: > But physics is not made this way! > > > As London (1927, 376-377) remarks, one must admire Weyl's immense courage in developing his gauge invariant interpretation of electromagnetism and holding on to it on the mere basis of purely formal considerations. London observes that the principle of equivalence of inertial and gravitational mass, which prompted Einstein to provide a geometrical interpretation of gravity, was at least a physical fact underlying gravitational theory. In contrast, an analogous fact was not known in the theory of electricity; consequently, it would seem that there was no compelling physical reason to think that rigid rods and ideal clocks would be under the universal influence of the electromagnetic field. To the contrary, London says, experience strongly suggests that atomic clocks exhibit sharp spectral lines that are unaffected by their history in the presence of a magnetic field, contrary to Weyl's non-integrability assumption. London concludes, that in the face of such elementary empirical facts it must have been an unusually clear metaphysical conviction which prevented Weyl from abandoning his idea that nature ought to make use of the beautiful geometrical possibilities that a pure infinitesimal geometry offers. In 1955, shortly before his death, Weyl wrote an addendum[96] to his 1918 paper Gravitation und Elektrizitat, in which he looks back at his early attempt to find a unified field theory and explains why he reinterpreted his gauge theory of 1918, a decade later. > > > This work stands at the beginning of attempts to construct a > "unified field theory" which subsequently were continued > by many, it seems to me, without decisive results. As is known, the > problem relentlessly occupied Einstein in particular, until his end. > ... The strongest argument for my theory appeared to be that > gauge invariance corresponds to the principle of the conservation of > electric charge just as coordinate invariance corresponds to the > conservation theorem of energy-impulse. Later, quantum theory > introduced the Schrodinger-Dirac potential $\psi$ of the > electron-positron field; the latter revealed an experimentally based > principle of gauge invariance which guaranteed the conservation of > charge and which connected the $\psi$ with the electromagnetic > potentials $\varphi\_{i}$ in the same way that my > speculative theory had connected the gravitational potentials > $g\_{ik}$ with $\varphi\_{i}$, where, > in addition, the $\varphi\_{i}$ are measured in known > atomic rather than unknown cosmological units. I have no doubts that > the principle of gauge invariance finds its correct place here and > not, as I believed in 1918, in the interaction of electromagnetism > and gravity. > > > By the late 1920s Weyl's methodological approach to gauge theory underwent an "empirical turn". In contrast to $a$ priori geometrical reasoning, which guided his early unification attempts--Weyl calls it a "speculative theory" in the above citation--by 1928/1929 Weyl emphasized experimentally-based principles which underlie gauge invariance.[97] In early 1928 P. A. M. Dirac provided the first physically compelling theoretical account of the dynamics of an electron in the presence of an electric field. The components $\psi^{i} (x)$ of Dirac's four-component wave function or spinor field in Minkowski space, $\psi(x) = (\psi^{1}(x), \psi^{2}(x), \psi^{3}(x), \psi^{4}(x))$, are complex-valued functions that satisfy Dirac's first-order partial differential equation and provide probabilistic information about the electron's dynamical behaviour, such as angular momentum and location. Prior to the appearance of spinor fields $\psi$ in Dirac's equation, it was generally thought that scalars, vectors and tensors provided an adequate system of mathematical objects that would allow one to provide a mathematical description of reality independently of the choice of coordinates or reference frames.[98] For example, spin zero particles $(\pi$ mesons, $\alpha$ particles) could be described by means of scalars; spin 1 particles (deuterons) by vectors, and spin 2 particles (hypothetical gravitons) by tensors. However, the most frequently occurring particles in Nature are electrons, protons, and neutrons. They are spin $\bfrac{1}{2}$ particles, called fermions that are properly described by mathematical objects called spinors, which are neither scalars, vectors or tensors.[99] Weyl referred to the $\psi(x)$ in Dirac's equation as the "Dirac quantity" and von Neumann called it the "$\psi(x)$-vector". Both von Neumann and Weyl, and others, immediately recognized that Dirac had introduced something that was new in theoretical physics. v. Neumann (1928, 876) remarks: > > > ... $\psi$ does by no means have the relativistic transformation > properties of a common four-vector. ... The case of a quantity > with four components that is not a four-vector is a case which has > never occurred in relativity theory; the Dirac $\psi$-vector is the > first example of this type. > > > (Weyl (1929c)) notes that the spinor representation of the orthogonal group $O(1, 3)$ cannot be extended to a representation of the general linear group $GL(n)$, $n = 4$, with the consequence that it is necessary to employ the Vierbein, tetrad or Lorentz-structure formulation of the theory of general relativity in order to incorporate Dirac's spinor fields $\psi(x)$: > > > The tensor calculus is not the proper mathematical instrument to use > in translating the quantum-theoretic equations of the electron over > into the general theory of relativity. Vectors and terms > [tensors] are so constituted that the law which defines the > transformation of their components from one Cartesian set of axes to > another can be extended to the most general linear transformation, to > an affine set of axes. That is not the case for quantity $\psi$, > however; this kind of quantity belongs to a representation of the > rotation group which cannot be extended to the affine group. > Consequently we cannot introduce components of $\psi$ relative to an > arbitrary coordinate system in general relativity as we can for the > electromagnetic potential and field strengths. We must rather > describe the metric at a point $p$ by local Cartesian axes > $e(a)$ instead of by the $g\_{pq}$. > The wave field has definite components > $\psi^{+}\_{1}, \psi^{+}\_{2}, \psi^{-}\_{1}, \psi^{-}\_{2}$ > relative to such axes, and we know how they transform on transition to > any other Cartesian axes in $p$. > > > Impressed by the initial success of Dirac's equation of the spinning electron within the special relativistic context, Weyl adapted Dirac's special relativistic theory of the electron to the general theory of relativity in three groundbreaking papers (Weyl (1929b,c,d)). A complete exposition of this formalism is presented in (Weyl (1929b)). O'Raifeartaigh (1997) says of this paper: > > > Although not fully appreciated at the time, Weyl's 1929 paper has > turned out to be one of the seminal papers of the century, both from > the philosophical and from the technical point of view. > > > In this ground braking paper, as well as in (Weyl (1929c,d)), Weyl explicitly abandons his earlier attempt to unify electromagnetism with the theory of general relativity. In his early attempt he associated the electromagnetic vector potential $A\_{j}(x)$ with the additional connection coefficients that arise when a conformal structure is reduced to a Weyl structure (see SS4.1). The important concept of gauge invariance, however, is preserved in his 1929 paper. Rather than associating gauge transformations with the scale or gauge of the spacetime metric tensor, Weyl now associates gauge transformations with the phase of the Dirac spinor field $\psi$ that represents matter. In the introduction of (Weyl (1929b)), which presents in detail the new formalism, Weyl describes his reinterpretation of the gauge principle as follows: > > > The Dirac field-equations for $\psi$ together with the Maxwell > equations for the four potentials $f\_{p}$ of the > electromagnetic field have an invariance property which, from a > formal point of view, is similar to the one that I called gauge > invariance in my theory of gravitation and electromagnetism of 1918; > the equations remain invariant when one makes the simultaneous > replacements > > > > \[\begin{array}{ccc} > \psi \text{ by } e^{i\lambda}\psi > & \text{and} & > f\_p \text{ by } f\_p - \dfrac{\partial\lambda}{\partial x^p}, > \end{array}\] > > > > where $\lambda$ is understood to be an arbitrary function of > position in the four-dimensional world. Here the factor > $\bfrac{e}{ch}$, where $- e$ is the charge of the electron, $c$ > is the speed of light, and $\bfrac{h}{\pi}$ is the quantum of > action, has been absorbed in $f\_{p}$. The connection of this > "gauge invariance" to the conservation of electric charge > remains untouched. But an essential difference, which is significant > for the correspondence to experience, is that the exponent of the > factor multiplying $\psi$ is not real but purely imaginary. $\psi$ > now assumes the role that $ds$ played in Einstein's old > theory. It seems to me that this new principle of gauge invariance, > which follows not from speculation but from experiment, compellingly > indicates that the electromagnetic field is a necessary > accompanying phenomenon, not of gravitation, but of the material wave > field represented by $\psi$. Since gauge invariance includes an > arbitrary function $\lambda$ it has the character of > "general" relativity and can naturally only be understood > in that context. > > > Weyl then introduces his two-component spinor theory in Minkowski space. Since one of his aims is to adapt Dirac's theory to the curved spacetime of general relativity, Weyl develops a theory of local spinor structures for curved spacetime.[100] He achieves this by providing a systematic formulation of local tetrads or Vierbeins (orthonormal basis vectors). Orthonormal frames had already been introduced as early as 1900 by Levi-Civita and Ricci. Somewhat later, Cartan had shown the usefulness of employing local orthonormal-basis vector fields, the so-called "moving frames" in his investigation of Riemannian geometry in the 1920s. In addition, Einstein (1928) had used tetrads or Vierbeins in his attempt to unify gravitation and electricity by resorting to distant parallelism with torsion. In Einstein's theory, the effects of gravity and electromagnetism are associated with a specialized torsion of spacetime rather than with the curvature of spacetime. Since the curvature vanishes everywhere, distant parallelism is a feature of Einstein's theory. However, distant parallelism appeared to Weyl to be quite unnatural from the viewpoint of Riemannian geometry. Weyl expressed his criticism in all three papers (Weyl (1929b,c,d)) and he contrasted the way in which Vierbeins are employed in his own work with the way they were used by Einstein. In the introduction Weyl (1929b) says: > > > I prefer not to believe in distant parallelism for a number of > reasons. First, my mathematical attitude resists accepting such an > artificial geometry; it is difficult for me to understand the force > that would keep the local tetrads at different points and in rotated > positions in a rigid relationship. There are, I believe, two > important physical reasons as well. In particular, by loosening the > rigid relationship between the tetrads at different points, the gauge > factor $e^{i\lambda}$, which remains arbitrary > with respect to the quantity $\psi$, changes from a constant to an > arbitrary function of spacetime location; that is, only through the > loosening of the rigidity does the actual gauge invariance become > understandable. Secondly, the possibility to rotate the tetrads at > different points independently from each other, is as we shall see, > equivalent to the symmetry of the energy-momentum tensor or with the > validity of its conservation law. > > > Every tetrad uniquely determines the pseudo-Riemannian spacetime metric $g\_{ij}$. However, the converse does not hold since the tetrad has 16 independent components whereas the spacetime metric, $g\_{ij} = g\_{ji}$, has only 10 independent components. The extra 6 degrees of freedom of the tetrads that are not determined by the metric may be represented by the elements of a 6-parameter internal Lorentz group. That is, the local tetrads are determined by the spacetime metric up to local Lorentz transformations. The tetrad formalism made it possible, therefore, for Weyl to derive, as a special case of Noether's second theorem[101], the energy-momentum conservation laws for general coordinate transformations and the internal Lorentz transformations of the tetrads. Moreover, Weyl had always emphasized the strong analogy between gravitation and electricity. The tetrad formalism and the conservation laws both made explicit and supported this analogy. Weyl introduced the final section of his seminal 1929 paper saying "We now come to the critical part of the theory", and presented a derivation of electromagnetism from the new gauge principle. The initial step in Weyl's derivation exploits the intrinsic gauge freedom of his two-component theory of spinors for Minkowski space, namely \[ \psi(x) \rightarrow e^{i\lambda}\psi(x), \] where the gauge factor is a constant. Since Weyl wished to adapt his theory to the curved spacetime of general relativity, the above phase transformation must be generalized to accommodate localtetrads. That is, each spacetime point has its own tetrad and therefore its own point-dependent gauge factor. The phase transformation is thus given by \[ \psi(x) \rightarrow e^{i\lambda(x)}\psi(x), \] where the $\lambda(x)$ is a function of spacetime. Weyl says: > > > We come now to the critical part of the theory. In my view the origin > and the necessity for the electromagnetic field lie in the following > justification. The components $\psi\_{1},\psi\_{2}$ > are, in fact, not uniquely determined by the tetrad but only to the > extent that they can still be multiplied by an arbitrary > "gauge-factor" $e^{i\lambda}$ of > absolute value 1. The transformation of the $\psi$ induced by a > rotation of the tetrad is determined only up to such a factor. In the > special theory of relativity one must regard this gauge factor as a > constant, since we have here only a single point-independent tetrad. > This is different in the general theory of relativity. Every point > has its own tetrad, and hence its own arbitrary gauge factor, because > the gauge factor necessarily becomes an arbitrary function of > position through the removal of the rigid connection between tetrads > at different points. > > > Today, the concept of gauge invariance plays a central role in theoretical physics. Not until 1954 did Yang and Mills (1954) generalize Weyl's electromagnetic gauge concept to the case of the non-Abelian group $O(3)$.[102] Although Weyl's reinterpretation of gauge invariance had been preceded by suggestions from London and Fock, it was Weyl, according to O'Raifeartaigh and Straumann (2000), > > > who emphasized the role of gauge invariance as a symmetry > principle from which electromagnetism can be derived. It took > several decades until the importance of this symmetry > principle--in its generalized form to non-Abelian gauge groups > developed by Yang, Mills, and others--also became fruitful for a > description of the weak and strong interactions. The mathematics of > the non-Abelian generalization of Weyl's 1929 paper would have been > an easy task for a mathematician of his rank, but at the time there > was no motivation for this from the physics side. > > > It is interesting in this context to consider the following remarks by Yang. Referring to Einstein's objection to Weyl's 1918 gauge theory, Yang (1986, 18) asked, "what has happened to Einstein's original objection after quantum mechanics inserted an $-i$ into the scale factor and made it into a phase factor?" Yang continuous: > > > Apparently no one had, after 1929, relooked at Einstein's objection > until I did in 1983. The result is interesting and deserves perhaps > to be a footnote in the history of science: Let us take Einstein's > Gedankenexperiment .... When the two clocks come back, because > of the insertion of the factor $-i$, they would not have > different scales but different phases. That would not influence their > rates of time-keeping. Therefore, Einstein's original objection > disappears. But you can ask a further question: Can one measure their > phase difference? Well, to measure a phase difference one must do an > interference experiment. Nobody knows how to do an interference > experiment with big objects like clocks. However, one can do > interference experiments with electrons. So let us change Einstein's > Gedankenexperiment to one of bringing electrons back along two > different paths and ask: Can one measure the phase difference? The > answer is yes. That was in fact a most important development in 1959 > and 1960 when Aharonov and Bohm realized--completely > independently of Weyl--that electromagnetism has some meaning > which was not understood > before.[103] > > > > We end the discussion on Weyl's gauge theory by quoting the following remarks by Dyson (1983). > > > A more recent example of a great discovery in mathematical physics > was the idea of a gauge field, invented by Hermann Weyl in 1918. This > idea has taken only 50 years to find its place as one of the basic > concepts of modern particle physics. Quantum chromodynamics, the most > fashionable theory of the particle physicists in 1981, is > conceptually little more than a synthesis of Lie's group-algebras > with Weyl's gauge fields. > > > > > The history of Weyl's discovery is quite unlike the history of Lie > groups and Grassmann algebras. Weyl was neither obscure nor > unrecognized, and he was working in 1918 in the most fashionable area > of physics, the newborn theory of general relativity. He invented > gauge fields as a solution of the fashionable problem of unifying > gravitation with electromagnetism. For a few months gauge fields were > at the height of fashion. Then it was discovered by Weyl and others > that they did not do what was expected of them. Gauge fields were in > fact no good for the purpose for which Weyl invented them. They > quickly became unfashionable and were almost forgotten. But then, > very gradually over the next fifty years, it became clear that gauge > fields were important in a quite different context, in the theory of > quantum electrodynamics and its extensions leading up to the recent > development of quantum chromodynamics. The decisive step in the > rehabilitation of gauge fields was taken by our Princeton colleague > Frank Yang and his student Bob Mills in 1954, one year before Hermann > Weyl's death [Yang and Mills, 1954]. There is no evidence that Weyl > ever knew or cared what Yang and Mills had done with his brain-child. > > > > > > So the story of gauge fields is full of ironies. A fashionable idea, > invented for a purpose which turns out to be ephemeral, survives a > long period of obscurity and emerges finally as a corner-stone of > physics. > > > #### 4.5.4 Weyl's two-component Neutrino theory It is remarkable that Weyl's (1929b) two-component spinor formalism led him to anticipate the existence of particles that violate conservation of parity, that is, left-right symmetry. In 1929 left-right symmetry was taken for granted and considered a basic fact of all the laws of Nature. Weyl formulated the four-component Dirac spinor $\psi$ in terms of a two-component left-handed Weyl spinor $\psi\_{L}$ and a two-component right-handed Weyl spinor $\psi\_{R}$: \[\begin{align} \psi &= (\psi^{1}, \psi^{2}, \psi^{3}, \psi^{4})^{T} \\ &= (\psi^{1}\_{L}, \psi^{2}\_{L}, \psi^{1}\_{R}, \psi^{2}\_{R})^{T} \\ &= (\psi\_L, \psi\_R)^T \end{align}\] The four-component Dirac spinor, formulated in terms of the two Weyl spinors \[ \psi = \left[\matrix{\psi\_L \\ \psi\_R}\right] \] preserves parity; it applies to all massive spin $\bfrac{1}{2}$ particles (fermions) and all massive fermions are known to obey parity conservation. However, a single Weyl spinor, either $\psi\_{L}$ or $\psi\_{R}$, does not preserve parity. Weyl noted that instead of the four-component Dirac spinor "two components suffice if the requirement of left-right symmetry (parity) is dropped". A little later he added, "the restriction 2 removes the equivalence of left and right. It is only the fact that left-right symmetry actually appears in Nature that forces us to introduce a second pair of $\psi$-components". Weyl's two-spinor version of the Dirac equation is a coupled system of equations requiring both Weyl spinors $\psi\_{L}$ and $\psi\_{R}$ in order to preserve parity. Weyl considerd massless particles in his two-spinor version of the Dirac equation. In this case, the equations of the two-spinor version of Dirac's equation decouple, yielding an equation for $\psi\_{L}$ and for $\psi\_{R}$. These equations are independent of each other, and the equation for the 2-component left-handed Weyl spinor $\psi\_{L}$ is called Weyl's equation; it is applicable to the massless particle called the neutrino[104], a spin $\bfrac{1}{2}$ particle, that was discovered in 1956. Yang (1986, 12) remarks > > > Now I come to another piece of work of Weyl's which dates back to > 1929, and is called Weyl's two-component neutrino theory. He invented > this theory in 1929 in one of his very important articles ... as > a mathematical possibility satisfying most of the requirements of > physics. But it was rejected by him and by subsequent physicists > because it did not satisfy left-right symmetry. With the realisation > that left-right symmetry was not exactly right in 1957 it became > clear that this theory of Weyl's should immediately be re-looked at. > So it was and later it was verified theoretically and experimentally > that this theory gave, in fact, the correct description of the > neutrino. > > > #### 4.5.5 The Theory of Groups and Quantum Mechanics During the interval from 1924-26, in which Weyl was intensely occupied with the pure mathematics of Lie groups, the essentials of the formal apparatus of the new revolutionary theory of quantum mechanics had been completed by Heisenberg, Schrodinger and others. As if to make up for lost time, Weyl immediately returned from pure mathematics to theoretical physics, and applied his new group theoretical results to quantum mechanics. As Yang (1986, 9, 10) describes it, > > > In the midst of Weyl's profound research on Lie groups there > occurred a great revolution in physics, namely the development of > quantum mechanics. We shall perhaps never know Weyl's initial > reaction to this development, but he soon got into the act and studied > the mathematical structure of the new mechanics. There resulted a > paper of 1927 and later a book, this book together with Wigner's > articles and Gruppen Theorie und Ihre Anwendung auf die Quanten > Mechanik der Atome were instrumental in introducing group theory > into the very language of quantum mechanics. > > > Mehra and Rechenberg (2000, 482) note in this context: "Actually, we have mentioned in previous volumes Weyl's early reactions to both matrix mechanics (in 1925) and wave mechanics (in early 1926), and they were very enthusiastic. Therefore, we have to assume quite firmly that it was only his deep involvement with the last stages of his work on the theory of semisimple continuous groups that prevented Weyl 'to get in the act' immediately." Weyl was particularly well positioned to handle some of the mathematical and foundational problems of the new theory of quantum mechanics. Almost every aspect of his mathematical expertise, in particular, his recent work on group theory and his very early work on the theory of singular differential-integral equations (1908-1911), provided him with the precise tools for solving many of the concrete problems posed by the new theory: the theory of Hilbert space, singular differential equations, eigenfunction expansions, the symmetric group, and unitary representations of Lie groups. Weyl's (1927) paper, referred to by Yang above, is entitled Quantenmechanik und Gruppentheorie (Quantum Mechanics and Group Theory). In it, Weyl provides an analysis of the foundations of quantum mechanics and he emphasizes the fundamental role Lie groups play in that theory.[105] Weyl begins the paper by raising two questions: (1) how do I arrive at the self-adjoint operators, which represent a given quantity of a physical system whose constitution is known, and (2), what is the physical interpretation of these operators and which physical consequences can be derived from them? Weyl suggests that while the second question has been answered by von Neumann, the first question has not yet received a satisfactory answer, and Weyl proposes to provide one with the help of group theory. In a way, Weyl's 1927 paper was programmatic in character; nearly all the topics of that paper were taken up again a year later in his famous book (Weyl (1928)) entitled Gruppentheorie und Quantenmechanik (The Theory of Groups and Quantum Mechanics). The book emerged from the lecture notes taken by a student named F. Bohnenblust of Weyl's lectures given in Zurich during the winter semester 1927-28. A revised edition of that book appeared in 1931. In the preface to the first edition Weyl says: > > > Another time I venture on stage with a book that belongs only half to > my professional field of mathematics, the other half to physics. The > external reason is not very different from that which led some time > ago to the origin of the book Raum Zeit Materie. In the > winter term 1927/28 Zurich was suddenly deprived of all > theoretical physics by the simultaneous departures of Debye and > Schrodinger. I tried to fill the gap by changing an already > announced lecture course on group theory into one on group theory and > quantum mechanics. > > ... > > Since I have for some years been deeply occupied with the theory of > the representation of continuous groups, it appeared to me at this > point to be a fitting and useful project, to provide an organically > coherent account of the knowledge in this field won by mathematicians, > on such a scale and in such a form, that is suitable for the > requirements of quantum physics. > > > Weyl's book is one of the first textbooks on the new theory of quantum mechanics. As Weyl indicates in the preface it was necessary for him to include a short account of the foundation of quantum theory in order to be able to show how the theory of groups finds its application in that theory. If the book fulfils its purpose, Weyl suggests, then the reader should be able to learn from it the essentials of both the theory of groups and quantum theory. Weyl's aim was to explain the mathematics to the physicists and the physics to the mathematicians. However, as Yang (1986, 10) points out, referring to Weyl's book: > > > Weyl was a mathematician and a philosopher. He liked to deal with > concepts and the connection between them. His book was very famous, > and was recognized as profound. Almost every theoretical physicist > born before 1935 has a copy of it on his bookshelves. But very few > read it: Most are not accustomed to Weyl's concentration on the > structural aspects of physics and feel uncomfortable with his > emphasis on concepts. The book was just too abstract for most > physicists. > > > Weyl's book (Weyl (1931b, 2 edn)) is remarkably complete for such an early work and covers many topics. Chapters I and III are mainly concerned with preliminary mathematical concepts. The first chapter provides an account of the theory of finite dimensional Hilbert spaces and the third chapter is an exposition of the unitary representation theory of finite groups and compact Lie groups. Chapter II is entitled Quantum Theory; it is the earliest systematic and comprehensive account of the new quantum theory. Chapter IV, entitled Application of the Theory of Groups to Quantum Mechanics, is divided into four parts. In part A, entitled The Rotation Group, Weyl provides a systematic explanatory account of the theory of atomic spectra in terms of the unitary representation theory of the rotation group, followed by a discussion of the selection and intensity rules. Part B is entitled The Lorentz Group. After discussing the spin of the electron and its role in accounting for the anomalous Zeeman effect, Weyl presents Dirac's theory of the relativistic quantum mechanics of the electron and develops in detail the theory of an electron in a spherically symmetric field, including an analysis of the fine structure of the spectrum. In part C, entitled The Permutation Group, Weyl applies the Pauli exclusion principle to explicate the periodic table of the elements. Next, Weyl develops the second quantization of the Maxwell and Dirac fields required for the analysis of many body relativistic systems. Weyl noted in the preface to the second edition that his treatment is in accordance with the recent work of Heisenberg and Pauli. It is now customary to include such a topic under the heading of relativistic quantum field theory. The final part of Chapter IV, part D, is entitled Quantum Kinematics; it provides an exposition of part II of Weyl's (1927) paper, mentioned earlier. Chapter V, entitled The Symmetric Permutation Group and the Algebra of Symmetric Transformations, is for the most part pure mathematics. It is widely regraded to be the most difficult part of the Weyl's book. Overall, Weyl's treatment is quite modern except for the confusion regarding the positive electron (anti-electron) that at that time was identified with the proton rather than with the positron, which was discovered a few years later. Weyl was quite concerned about the identification of the proton with the positive electron because his analysis of the discrete symmetries $\mathbf{C}, \mathbf{P}, \mathbf{T}$ and $\mathbf{CPT}$ led him to conclude that the mass of the positive electron should equal the mass of the electron.[106] #### 4.5.6 Weyl's Early Discussion of the Discrete Symmetries $\mathbf{C}, \mathbf{P}, \mathbf{T}$ and $\mathbf{CPT}$ Weyl (1931b, 2 edn) analyzed Dirac's relativistic theory of the electron (Dirac (1928a,b)). Although this theory correctly accounted for the spin of the electron, there was however a problem because in addition to the positive-energy levels, Dirac's theory predicted the existence of an equal number of negative-energy levels. Dirac (1930) reinterpreted the theory by assuming that all of the negative-energy levels were normally occupied. The Pauli Exclusion Principle, which asserts that it is impossible for two electrons to occupy the same quantum state, would prevent an electron with positive energy from falling into a negative- energy state. Dirac's theory also predicted that one of the negative-energy electrons could be raised to a state of positive energy, thereby creating a 'hole' or unoccupied negative-energy state. Such a hole would behave like a particle with a positive energy and a positive charge, that is, like a positive electron. Because the only fundamental particles that were known to exist at that time were the electron and the proton, one was justifiably reluctant to postulate the existence of new particles that had not yet been observed experimentally; consequently, it was suggested that the positive electron should be identified with the proton. However, Weyl was quite concerned about the identification of the proton with the anti-electron. In the preface to the second German edition of his book Gruppentheorie und Quantenmechanik, Weyl (1928, 2 edn, 1931, VII) wrote > > > The problem of the proton and the electron is discussed in connection > with the symmetry properties of the quantum laws with respect to the > interchange of right and left, past and future, and positive and > negative electricity. At present no acceptable solution is in sight; > I fear, that in the context of this problem, the clouds are rolling > together to form a new, serious crisis in quantum physics. > > > Weyl had good reasons for his concern. He analyzed the invariance of the Maxwell-Dirac equations under the discrete symmetries that correspond to the transformations now called $\mathbf{C}, \mathbf{P}, \mathbf{T}$ and $\mathbf{CPT}$ both for the case of relativistic quantum mechanics and for the case of relativistic quantum field theory, and concluded in both cases that the mass of the anti-electron should be the same as the mass of the electron. That the mass of the proton was so different from the mass of the electron, therefore, appeared to Weyl to constitute a new serious crisis in physics. In a lecture presented at the Centenary for Hermann Weyl held at the ETH in Zurich, Yang (1986, 10) says of the above quote from Weyl's preface to the second edition of Gruppentheorie und Quantenmechanik: > > > This was a most remarkable passage in retrospect. The symmetry that > he mentioned here, of physical laws with respect to the interchange > of right and left, had been introduced by Weyl and Wigner > independently into quantum physics. It was called parity > conservation, denoted by the symbol $P$. The symmetry between > the past and future was something that was not well understood in > 1930. It was understood later by Wigner, was called time reversal > invariance, and was denoted by the symbol $T$. The symmetry with > respect to positive and negative electricity was later called charge > conjugation invariance $C$. It is a symmetry of physical laws > when you change positive and negative signs of electricity. Nobody, > to my knowledge, absolutely nobody in the year 1930, was in any way > suspecting that these symmetries were related in any manner. I will > come back to this matter later. What had prompted Weyl in 1930 to > write the above passage is a great mystery to me. > > > It would seem that Yang's comment is misleading since it suggests that Weyl did not have a good reason for his remark. In fact, however, Weyl's statement was firmly based on a detailed analysis of the discrete symmetries $\mathbf{C}, \mathbf{P}, \mathbf{T}$ and $\mathbf{CPT}$. Coleman and Korte (2001) have shown in detail that Weyl's treatment of these symmetries is the same as that used today except for the fact that the symmetry $\mathbf{T}$ is treated by Weyl as linear and unitary, rather than as antilinear and antiunitary. Weyl had presented in 1931 a complete analysis, in the context of the quantized Maxwell-Dirac field equations, of the discrete symmetries that are now called $\mathbf{C}, \mathbf{P}, \mathbf{T}$ and $\mathbf{CPT}$. His transformations $\mathbf{C}$ and $\mathbf{P}$ are the same as those used today. His transformations $\mathbf{T}$ and $\mathbf{CPT}$ are also very close to those used today except that Weyl's transformations were linear and unitary rather than antilinear and antiunitary. Moreover, Weyl drew two very important conclusions from his analysis of these discrete symmetries. First, Weyl announced that the important question of the arrow of time had been solved because the field equations were not invariant under his time-reversal transformation $\mathbf{T}$. Second, Weyl pointed out that the invariance of the field equations under his charge-conjugation transformation $\mathbf{C}$ implied that the mass of the 'anti-electron' is necessarily the same as that of the electron; moreover, Weyl's result is the primary reason that Dirac (1931, 61) abandoned the assignment of the proton to the role of the anti-electron. Many years later Dirac (1977, 145) recalled: > > > Well, what was I to do with these holes? The best I could think of > was that maybe the mass was not the same as the mass of the electron. > After all, my primitive theory did ignore the Coulomb forces between > the electrons. I did not know how to bring those into the picture, > and it could be that in some obscure way these Coulomb forces would > give rise to a difference in the masses. > > > > > Of course, it is very hard to understand how this difference could be > so big. We wanted the mass of the proton to be nearly 2000 times the > mass of the electron, an enormous difference, and it was very hard to > understand how it could be connected with just a sort of perturbation > effect coming from Coulomb forces between the electrons. > > > > > However, I did not want to abandon my theory altogether, and so I put > it forward as a theory of electrons and protons. Of course I was very > soon attacked on this question of the holes having different masses > from the original electrons. I think the most definite attack came > from Weyl, who pointed out that mathematically the holes would have > to have the same mass as the electrons, and that came to be the > accepted view. > > > At another place Dirac (1971, 52-55) remarks: > > > But still, I thought there might be something in the basic idea and > so I published it as a theory of electrons and protons, and left it > quite unexplained how the protons could have such a different mass > from the electrons. > > > > > This idea was seized upon by Herman [sic] Weyl. He said boldly that > the holes had to have the same mass as the electrons. Now Weyl was a > mathematician. He was not a physicist at all. He was just concerned > with the mathematical consequences of an idea, working out what can > be deduced from the various symmetries. And this mathematical > approach led directly to the conclusion that the holes would have to > have the same mass as the electrons. Weyl just published a blunt > statement that the holes must have the same mass as the electrons and > did not make any comments on the physical implications of this > assertion. Perhaps he did not really care what the physical > implications were. He was just concerned with achieving consistent > mathematics. > > > Dirac's characterization of Weyl's unconcern for physics seems unfair in light of Weyl's own statement in the preface of the second edition of his book, cited earlier, where he expresses the fear "that in the context of this problem, the clouds are rolling together to form a new, serious crisis in quantum physics"; Weyl did care about the physics. Weyl's analysis did have a significant impact on the development of the Maxwell-Dirac theory; however, as Coleman and Korte (2001) have argued, Weyl's early analysis of the transformations $\mathbf{C}, \mathbf{P}, \mathbf{T}$ and $\mathbf{CPT}$ was, for the most part, lost to subsequent researchers and had to be essentially re-invented. However, it should be noted in this context that Schwinger (1988, 107-129) was greatly influenced by Weyl's book. Schwinger makes particular reference to Weyl's work on the discrete symmetries and says that this work "... was the starting point of my own considerations concerning the connection between spin and statistics, which culminated in what is now referred to as the TCP--or some permutation thereof--theorem". #### 4.5.7 Weyl's Philosophical Views about Quantum Mechanics Weyl analyzed the foundations of both the general theory of relativity and the theory of quantum mechanics. For both theories, he provided a coherent exposition of the mathematical structure of the theory, elegant characterizations of the entities and laws postulated by the theory and a lucid account of how these postulates explain the most significant, more directly observable, lower-level phenomena. In both cases, he was also concerned with the constructive aspects of the theory, that is, with the extent to which the higher-level postulates of the theory are necessary. There is no doubt that with regard to the general theory of relativity, Weyl held strong philosophical views. Some of these views are couched in a phenomenological language and reveal Husserl's influence on Weyl. Ryckman's (2005) study The Reign of Relativityprovides an extensive account of Weyl's orientation to Husserl's phenomenology. On the other hand, many of Weyl's philosophical views are couched in an unequivocal empiricist-realist language. For example, Weyl rejected Poincare's geometrical conventionalism and forcefully argued that the spacetime metric field is physically real, that it is a physically real structural field (Strukturfeld), which is determined by the physically real causal (conformal) structure and the physically real inertial (projective) structure or guiding field (Fuhrungsfeld) of spacetime. He was not deterred in putting forward such ontological claims about the metric structure of spacetime despite the fact that a complete epistemologically satisfactory solution to the measurement problem for the spacetime metric field was not then available. In the same manner, Weyl forcefully advanced a field-body-relationalist ontology of spacetime structure. He argued that a Leibnizian or Einstein-Machian form of relationalism that is based on a pure body ontology, is not tenable, indeed is incoherent within the context of general relativity, and he presented a reductio argument, the plasticine example, to underscore the necessity of the existence of a physically real guiding field in addition to the existence of bodies. However, in contrast to Weyl's many philosophical views with regard to spacetime theories, Weyl's philosophical positions regarding the status of quantum mechanics, while not absent, are not as transparent. There are passages, such as the following ((Weyl, 1931b, 2 edn, 44), which argue for the reality of photons. > > > The intensity of the monochromatic radiation that is used to generate > the photoelectric effect has no influence on the speed with which the > electrons are ejected from the metal but affects only the frequency > of this process. Even with intensities so weak that on the classical > theory hours would be required before the electromagnetic energy > passing through a given atom would attain to an amount equal to that > of a photon, the effect begins immediately, the points at which it > occurs being distributed irregularly over the entire metal plate. > This fact is a proof of the existence of light quanta that is no less > meaningful than the flashes of light on the scintillation screen are > for the corpuscular-discontinuous nature of $\alpha$-rays. > > > On the other hand, Weyl's (1931b) discussion of the problem of 'directional quantization' in the old quantum theory and of the way that this problem is 'resolved' in the new quantum theory appears to have a distinctly instrumentalist flavour. In a number of places, he describes the essence of the dilemma posed by quantum mechanics with a dispassionate precision. Consider, for example, the following (Weyl, 1931b, 2 edn, 67): > > Natural science has a constructive character. The phenomena with which > it deals are not independent manifestations or qualities which can be > read off from nature, but can only be determined by means of an > indirect method, through interaction with other bodies. Their implicit > definition is bound up with definite natural laws which underlie the > interactions. Consider, for example, the introduction of the Galilean > concept of mass which essentially comes down to the following indirect > definition: "Each body possesses a momentum, that is, a vector > $m\overline{v}$ which has the same direction as its velocity > $\overline{v}$--the scalar factor $m$ is called its mass. The > law of momentum holds, according to which the sum of the momenta > before a reaction between several bodies is the same as the sum of > their momenta after the reaction." By applying this law to the > observed collision phenomena, one obtains data for the determination > of the relative masses. The scientific consensus was, however, that > such constructive phenomena can nevertheless be attributed to the > things themselves even if the manipulations, which alone can lead > to their recognition, are not being carried out. In Quantum Theory > we encounter a fundamental limitation to this epistemological position > of the constructive natural science. > > > It is difficult for many people to accept quantum mechanics as an ultimate theory without at the same time giving up some form of realism and adopting something like an instrumentalist view of the theory. It is clear that Weyl was fully aware of this state of affairs, and yet in all of his published work, he refrained from making any bold statements of his views on the fundamental questions about quantum reality. He did not vigorously participate in the debate between Einstein and Schrodinger and the Copenhagen School nor did he offer decisive views concerning, for example, the Einstein, Podolsky, Rosen thought experiment or Schrodinger's Cat. Since Weyl held strong philosophical views within the context of the general theory of relativity, it is therefore only natural that one might have expected him to take a stand with respect to Schrodinger's cat and whether or not one should be fully satisfied with a theory according to which the cat is neither alive or dead but is in a superposition of these two states. The reason for Weyl's seeming reticence concerning the ontological/epistemological questions about quantum reality was already hinted at in note 5 of SS2, where it was suggested that Weyl was not especially bothered by the counterintuitive nature of quantum mechanics because he held the view that "objective reality cannot be grasped directly, but only through the use of symbols". Although Weyl (1948, 1949a, 1953) did express his philosophical views about quantum theory, he did so cautiously. Weyl (1949a, 263) summarizes some of the features of quantum mechanics that he considered of "paramount philosophical significance": the measurement problem, the incompatibility of quantum physics with classical logic, quantum causality, the non-local nature of quantum mechanics, the Leibniz-Pauli Exclusion Principle[107], and the irreducible probabilistic nature of quantum mechanics. At the end of the summary Weyl remarks: > > > It must be admitted that the meaning of quantum physics, in spite of > all its achievements, is not yet clarified as thoroughly as, for > instance, the ideas underlying relativity theory. The relation of > reality and observation is the central problem. We seem to need a > deeper epistemological analysis of what constitutes an experiment, a > measurement, and what sort of language is used to communicate its > result. > > > #### 4.5.8 Science as Symbolic Construction According to Weyl (1948, 295), both the theory of general relativity and quantum mechanics force upon us the realization that "instead of a real spatio-temporal material being what remains for us is only a construction in pure symbols". If it is necessary, Weyl (1948, 302) says, that our scientific grasp of an objective world must not depend on sense qualities, because of their inherent subjective nature, then it is for the same reason necessary to eliminate space and time. And Descartes gave us the means to do this with his discovery of analytic geometry. As Weyl (1953, 529) observes, when Newton explained the experienced world through the movements of solid particles in space, he rejected sense qualities for the construction of the objective world, but he held on to, and used an intuitively given objective space for the construction of a real world that lies behind the appearances. It was Leibniz who recognized the phenomenal character (Phenomenalitat) of space and time as consisting in the mere ordering of phenomena; however, space and time themselves do not have an independent reality. It is the freely created pure numbers, that is, pure symbols, according to Weyl, which serve as coordinates, and which provide the material with which to symbolically construct the objective world. In symbolically constructing the objective world we are forced to replace space and time through a pure arithmetical construct. Instead of spacetime points, $n$-tuples of pure numbers corresponding to a given coordinate system are used. Weyl (1948, 303) says: > > > ... the laws of physics are viewed as arithmetic laws between > numerical values of variable magnitudes, in which spatial points and > moments of time are represented through their numerical coordinates. > Magnitudes such as the temperature of a body or the field strength of > an electric field, which have at each spacetime point a definite > value, appear as functions of four variables, the spacetime > coordinates $x, y, z, t$. > > > In systematic theorizing we construct a formal scaffold that consists of mere symbols, according to Weyl (1948, 311), without explaining initially what the symbols for mass, charge, field strength, etc., mean; and only toward the end do we describe how the symbolic structure connects directly with experience. > > > It is certain, that on the symbolic side, not space and time but four > independent variables $x, y, z, t$ appear; > one speaks of space, as one does of sounds and colours, only on the > side of conscious experience. A monochromatic light signal ... has > now become a mathematical formula in which a certain symbol $F$, > called electromagnetic field strength, is expressed as a pure > arithmetically constructed function of four other symbols > $x, y, z, t$, called spacetime coordinates. > > > At another place Weyl (1949a, 113) says: > > > Intuitive space and intuitive time are thus hardly the adequate > medium in which physics is to construct the external world. No less > than the sense qualities must the intuitions of space and time be > relinquished as its building material; they must be replaced by a > four-dimensional continuum in the abstract arithmetical sense. > > > Weyl's point is that while space and time exist within the realm of conscious experience, or, according to Kant, as $a$ prioriforms underlying all of our conscious experiences, they are unsuited as elements with which to construct the objective world and must be replaced by means of a purely arithmetical symbolic representation. All that we are left with, according to Weyl, is symbolic construction. If this still needed any confirmation, Weyl (1948, 313) says, it was provided by the theory of relativity and quantum theory.[108] For ease of reference we repeat a citation of Weyl (1988, 4-5) in SS4.3.1: > > > Coordinates are introduced on the Mf [manifold] in the most direct > way through the mapping onto the number space, in such a way, that > all coordinates, which arise through one-to-one continuous > transformations, are equally possible. With this the coordinate > concept breaks loose from all special constructions to which > it was bound earlier in geometry. In the language of relativity this > means: The coordinates are not measured, their > values are not read off from real measuring rods which react > in a definite way to physical fields and the metrical structure, > rather they are a priori placed in the world arbitrarily, in > order to characterize those physical fields including the metric > structure numerically. The metric structure becomes through > this, so to speak, freed from space; it becomes an existing field > within the remaining structure-less space. Through this, space as > form of appearance contrasts more clearly with its real content: The > content is measured after the form is arbitrarily related to > coordinates.[109] > > > > The last two sentences in the above quote suggest that, (a) Weyl embraces something close to Kant's position, according to which space and time are "a priori forms of appearances", or that (b) Weyl adheres to a position called spacetime substantivalism, according to which, in addition to body and fields and their relations, there also exists a 'container', the spacetime manifold, and this manifold, its points and the manifold differential-topological relations are physically real. However, this interpretation would contradict Weyl's basic thesis that in the symbolic construction of the objective word we are left with nothing but symbolic arithmetic functional relations. Weyl's phrases, do not denote either a physically real container or something like Kant's a priori form of intuition. They merely denote a conceptual or formal scaffolding, a logical space, as it were, whose points are represented by purely formal coordinates $(n$-tuple of pure numbers). It is such a formal space which is employed by the theorist in the initial stages of constructing an objective world. To emphasize, in modelling the objective world the theorist begins by constructing a formal scaffold which consists of mere symbols and formal coordinates, without explaining initially what the symbols for mass, charge, field strength, etc., mean; only toward the end does the theorist describe how the symbolic structure connects directly with experience ((Weyl, 1948, 311)). The four-dimensional space-time continuum must be replaced by a four-dimensional coordinate space $\mathbb{R}^{4}$. However, the sheer arbitrariness with which we assign coordinates does not affect the objective relations and features of the world itself. To the contrary, it is only relative to a symbolic construction or modelling by means of an assignment of coordinates that the state of the world, its relations and properties, can be objectively determined by means of distinct, reproducible symbols. While our immediate experiences are subjective and absolute, our symbolic construction of the objective world is of necessity relative. Weyl (1949a, 116) says: > > > Whoever desires the absolute must take the subjectivity and > egocentricity into the bargain; whoever feels drawn toward the > objective faces the problem of relativity. > > > Weyl (1949a, 75) notes, "The objectification, by elimination of the ego and its immediate life of intuition, does not fully succeed, and the coordinate system remains as the necessary residue of the ego-extinction." However, this residue of ego involvement is subsequently rendered harmless through the principle of invariance. The transition from one admissible coordinate system to another can be mathematically described, and the natural laws and measurable quantities must be invariant under such transformations. This, Weyl (1948, 336) says, constitutes the general principle of relativity. Weyl (1949a, 104) says: > > > ... Only such relations will have objective meaning as are > independent of the mapping chosen and therefore remain invariant > under deformations of the map. Such a relation is, for instance, the > intersection of two world lines. If we wish to characterize a special > mapping or a special class of mappings, we must do so in terms of the > real physical events and of the structure revealed in them. That is > the content of the postulate of general relativity. > According to the special theory of relativity, it > is possible in particular to construct a map of the world such that > (1) the world line of each mass point which is subject to no external > forces appears as a straight line, and (2) the light cone issuing > from an arbitrary world point is represented by a circular cone with > vertical axis and a vertex angle of 90deg. In this > theory the inertial and causal structure and hence also the metrical > structure of the world have the character of rigidity, they are > absolutely fixed once and for all. It is impossible objectively, > without resorting to individual exhibition, to make a narrower > selection from among the 'normal mappings' satisfying the > above conditions (1) and (2). > > > Weyl (1949a, 115) provides an illustration, which shows how a measurement by observer $B$ of the angular distance $\delta$ between two stars $\Sigma$ and $\Sigma^{\}$ can be constructed in the four-dimensional number space, and can be expressed as an invariant.[110] ![figure](fig15.png) Figure 15: Measurement of the angular distance $\delta$ by an observer $B$ between two stars In figure 15 the stars and observer are represented by their world lines, and the past light cone $K$ issuing from the observation event $O$ intersects the world lines of the stars $\Sigma$ and $\Sigma^{\}$ in $E$ and $E^{\}$ respectively. The light rays emitted at $E$ and $E^{\}$, which arrive at the observation event $O$, are null geodesics laying on the past light cone and are respectively denoted by $\Lambda$ and $\Lambda^{\}$. This construction of the numerical quantity of the angle $\delta$ observed by $B$ at $O$, which is describable in the form of purely arithmetical relations, is invariant under arbitrary coordinate transformations and constitutes an objective fact of the world.[111] On the other hand, the angles between two stars determine the objectively* indescribable subjective experience of the observer. Moreover, Weyl says, "there is no difference in our experiences to which there does not correspond a difference in the underlying objective situation." And that difference is itself invariant under arbitrary coordinate transformations. In other words, an observer's subjective experiences supervene on the invariant relationships and structures of a symbolically constructed objective world. Perhaps no statement captures the contrast between the objective-symbolic and the subjective-intuitive more vividly then Weyl's famous statement > > > The objective world simply $is$, it does not happen. > Only to the gaze of my consciousness, crawling upward along the life > line of my body, does a section of this world come to life as a > fleeting image in space which continuously changes in > time.[112] > > > >
whewell	## 1. Biography Whewell was born in 1794, the eldest child of a master-carpenter in Lancaster. The headmaster of his local grammar school, a parish priest, recognized Whewell's intellectual abilities and persuaded his father to allow him to attend the Heversham Grammar School in Westmorland, some twelve miles to the north, where he would be able to qualify for a closed exhibition to Trinity College, Cambridge. In the 19th century and earlier, these "closed exhibitions" or scholarships were set aside for the children of working class parents, to allow for some social mobility. Whewell studied at Heversham Grammar for two years, and received private coaching in mathematics. Although he did win the exhibition it did not provide full resources for a boy of his family's means to attend Cambridge; so money had to be raised in a public subscription to supplement the scholarship money. He thus came up to Trinity in 1812 as a "sub-sizar" (scholarship student). In 1814 he won the Chancellor's prize for his epic poem "Boadicea," in this way following in the footsteps of his mother, Elizabeth Whewell, who had published poems in the local papers. Yet he did not neglect the mathematical side of his training; in 1816 he proved his mathematical prowess by placing as both second Wrangler and second Smith's Prize man. The following year he won a college fellowship. He was elected to the Royal Society in 1820, and ordained a priest (as required for Trinity Fellows) in 1825. He took up the Chair in Mineralogy in 1828, and resigned it in 1832. In 1838 Whewell became Professor of Moral Philosophy. Almost immediately after his marriage to Cordelia Marshall on 12 October 1841, he was named Master of Trinity College upon the recommendation of the Prime Minister Robert Peel. He was Vice-Chancellor of the University in 1842 and again in 1855. In 1848 he played a large role in establishing the Natural and Moral Sciences Triposes at the University. His first wife died in 1855, and he later married Lady Evalina Affleck, the widowed sister of his friend Robert Ellis. Lady Affleck died in 1865. Whewell had no children. He died, after being thrown from his horse, on 6 March 1866. (More details about Whewell's life and times can be found in Snyder 2011.) ## 2. Philosophy of Science: Induction According to Whewell, all knowledge has both an ideal, or subjective dimension, as well as an objective dimension. He called this the "fundamental antithesis" of knowledge. Whewell explained that "in every act of knowledge ... there are two opposite elements, which we may call Ideas and Perceptions" (1860a, 307). He criticized Kant and the German Idealists for their exclusive focus on the ideal or subjective element, and Locke and the "Sensationalist School" for their exclusive focus on the empirical, objective element. Like Francis Bacon, Whewell claimed to be seeking a "middle way" between pure rationalism and ultra-empiricism. Whewell believed that gaining knowledge requires attention to both ideal and empirical elements, to ideas as well as sensations. These ideas, which he called "Fundamental Ideas," are "supplied by the mind itself"--they are not (as Mill and Herschel protested) merely received from our observations of the world. Whewell explained that the Fundamental Ideas are "not a consequence of experience, but a result of the particular constitution and activity of the mind, which is independent of all experience in its origin, though constantly combined with experience in its exercise" (1858a, I, 91). Consequently, the mind is an active participant in our attempts to gain knowledge of the world, not merely a passive recipient of sense data. Ideas such as Space, Time, Cause, and Resemblance provide a structure or form for the multitude of sensations we experience. The Ideas provide a structure by expressing the general relations that exist between our sensations (1847, I, 25). Thus, the Idea of Space allows us to apprehend objects as having form, magnitude, and position. Whewell held, then, that observation is "idea-laden;" all observation, he noted, involves "unconscious inference" using the Fundamental Ideas (see 1858a, I, 46). Each science has a Particular Fundamental idea which is needed to organize the facts with which that science is concerned; thus, Space is the Fundamental Idea of geometry, Cause the Fundamental Idea of mechanics, and Substance the Fundamental Idea of chemistry. Moreover, Whewell explained that each Fundamental Idea has certain "conceptions" included within it; these conceptions are "special modifications" of the Idea applied to particular types of circumstances (1858b, 187). For example, the conception of force is a modification of the Idea of Cause, applied to the particular case of motion (see 1858a, I, 184-5 and 236). Thus far, this discussion of the Fundamental Ideas may suggest that they are similar to Kant's forms of intuition, and indeed there are some similarities. Because of this, some commentators have argued that Whewell's epistemology is a type of Kantianism (see, for example, Butts 1973; Buchdahl 1991). However, this interpretation ignores several crucial differences between the two views. Whewell did not follow Kant in drawing a distinction between "precepts," or forms of intuition, such as Space and Time, and the categories, or forms of thought, in which Kant included the concepts of Cause and Substance. Moreover, Whewell included as Fundamental Ideas many ideas which function not as conditions of experience but as conditions for having knowledge within their respective sciences: although it is certainly possible to have experience of the world without having a distinct idea of, say, Chemical Affinity, we could not have any knowledge of certain chemical processes without it. Unlike Kant, Whewell did not attempt to give an exhaustive list of these Fundamental Ideas; indeed, he believed that there are others which will emerge in the course of the development of science. Moreover, and perhaps most importantly for his philosophy of science, Whewell rejected Kant's claim that we can only have knowledge of our "categorized experience." The Fundamental Ideas, on Whewell's view, accurately represent objective features of the world, independent of the processes of the mind, and we can use these Ideas in order to have knowledge of these objective features. Indeed, Whewell criticized Kant for viewing external reality as a "dim and unknown region" (see 1860a, 312). Further, Whewell's justification for the presence of these concepts in our minds takes a very different form than Kant's transcendental argument. For Kant, the categories are justified because they make experience possible. For Whewell, though the categories do make experience (of certain kinds) possible, the Ideas are justified by their origin in the mind of a divine creator (see especially his discussion of this in his 1860a). And finally, the type of necessity which Whewell claimed is derived from the Ideas is very different from Kant's notion of the synthetic a priori (For a nuanced view of the relation between the views of Kant and Whewell, see Ducheyne 2011.) I return to these last two points in the section on Necessary Truth below. I turn now to a discussion of the theory of induction Whewell developed with his antithetical epistemology. From his earliest thoughts about scientific method, Whewell was interested in developing an inductive theory. At their philosophical breakfasts at Cambridge, Whewell, Babbage, Herschel and Jones discussed how science had stagnated since the heady days of the Scientific Revolution in the 17th century. It was time for a new revolution, which they pledged to bring about. The cornerstone of this new revolution was to be the promotion of a Baconian-type of induction, and all four men began their careers endorsing an inductive scientific method against the deductive method then being advanced by political economist David Ricardo, logician Richard Whately, and their followers (see Snyder 2011). (Although the four friends agreed about the importance of an inductive scientific method, Herschel and Jones would late take issue with some aspects of Whewell's version, notably his antithetical epistemology.) Whewell's first explicit, lengthy discussion of induction is found in his Philosophy of the Inductive Sciences, founded upon their History, which was originally published in 1840 (a second, enlarged edition appeared in 1847, and the third edition appeared as three separate works published between 1858 and 1860). He called his induction "Discoverers' Induction" and explained that it is used to discover both phenomenal and causal laws. Whewell considered himself to be a follower of Bacon, and claimed to be "renovating" Bacon's inductive method; thus one volume of the third edition of the Philosophy is entitled Novum Organon Renovatum. Whewell followed Bacon in rejecting the standard, overly-narrow notion of induction that holds induction to be merely simple enumeration of instances. Rather, Whewell explained that, in induction, "there is a New Element added to the combination [of instances] by the very act of thought by which they were combined" (1847, II, 48). This "act of thought" is a process Whewell called "colligation." Colligation, according to Whewell, is the mental operation of bringing together a number of empirical facts by "superinducing" upon them a conception which unites the facts and renders them capable of being expressed by a general law. The conception thus provides the "true bond of Unity by which the phenomena are held together" (1847, II, 46), by providing a property shared by the known members of a class (in the case of causal laws, the colligating property is that of sharing the same cause). Thus the known points of the Martian orbit were colligated by Kepler using the conception of an elliptical curve. Often new discoveries are made, Whewell pointed out, not when new facts are discovered but when the appropriate conception is applied to the facts. In the case of Kepler's discovery, the observed points of the orbit were known to Tycho Brahe, but only when Kepler applied the ellipse conception was the true path of the orbit discovered. Kepler was the first one to apply this conception to an orbital path in part because he had, in his mind, a very clear notion of the conception of an ellipse. This is important because the fundamental ideas and conceptions are provided by our minds, but they cannot be used in their innate form. Whewell explained that "the Ideas, the germs of them at least, were in the human mind before [experience]; but by the progress of scientific thought they are unfolded into clearness and distinctness" (1860a, 373). Whewell referred to this "unfolding" of ideas and conceptions as the "explication of conceptions." Explication is a necessary precondition to discovery, and it consists in a partly empirical, partly rational process. Scientists first try to clarify and make explicit a conception in their minds, then attempt to apply it to the facts they have precisely examined, to determine whether the conception can colligate the facts into a law. If not, the scientist uses this experience to attempt a further refinement of the conception. Whewell claimed that a large part of the history of science is the "history of scientific ideas," that is, the history of their explication and subsequent use as colligating concepts. Thus, in the case of Kepler's use of the ellipse conception, Whewell noted that "to supply this conception, required a special preparation, and a special activity in the mind of the discoverer. ... To discover such a connection, the mind must be conversant with certain relations of space, and with certain kinds of figures" (1849, 28-9). Once conceptions have been explicated, it is possible to choose the appropriate conception with which to colligate phenomena. But how is the appropriate conception chosen? According to Whewell, it is not a matter of guesswork. Nor, importantly, is it merely a matter of observation. Whewell explained that "there is a special process in the mind, in addition to the mere observation of facts, which is necessary" (1849, 40). This "special process in the mind" is a process of inference. "We infer more than we see," Whewell claimed (1858a, I, 46). Typically, finding the appropriate conception with which to colligate a class of phenomena requires a series of inferences, thus Whewell noted that discoverers's induction is a process involving a "train of researches" (1857/1873, I, 297). He allowed any type of inference in the colligation, including enumerative, eliminative and analogical. Thus Kepler in his Astronomia Nova (1609) can be seen as using various forms of inference to reach the ellipse conception (see Snyder 1997a). When Augustus DeMorgan complained, in his 1847 logic text, about certain writers using the term "induction" as including "the use of the whole box of [logical] tools," he was undoubtedly referring to his teacher and friend Whewell (see Snyder 2008). After the known members of a class are colligated with the use of a conception, the second step of Whewell's discoverers' induction occurs: namely, the generalization of the shared property over the complete class, including its unknown members. Often, as Whewell admitted, this is a trivially simple procedure. Once Kepler supplied the conception of an ellipse to the observed members of the class of Mars' positions, he generalized it to all members of the class, including those which were unknown (unobserved), to reach the conclusion that "all the points of Mars' orbit lie on an ellipse with the sun at one focus." He then performed a further generalization to reach his first law of planetary motion: "the orbits of all the planets lie on ellipses with the sun at one focus." I've mentioned that Whewell thought of himself as renovating Bacon's inductive philosophy. His inductivism does share numerous features with Bacon's method of interpreting nature: for instance the claims that induction must involve more than merely simple enumeration of instances, that science must proceed by successive steps of generalization, that inductive science can reach unobservables (for Bacon, the "forms," for Whewell, unobservable entities such as light waves or properties such as elliptical orbits or gravitational forces). (For more on the relation between Whewell and Bacon see Snyder 2006; McCaskey 2014.) Yet, surprisingly, the received view of Whewell's methodology in the 20th century tended to describe Whewell as an anti-inductivist in the Popperian mold. (see, for example, Ruse 1975; Niiniluoto 1977; Laudan 1980; Butts 1987; Buchdahl 1991). That is, it was claimed that Whewell endorsed a "conjectures and refutations" view of scientific discovery. However, it is clear from the above discussion that his view of discoverers' induction does not resemble the view asserting that hypotheses can be and are typically arrived at by mere guesswork. Indeed, Whewell explicitly rejected the hypothetico-deductive claim that hypotheses discovered by non-rational guesswork can be confirmed by consequentialist testing. For example, in his review of his friend Herschel's Preliminary Discourse on the Study of Natural Philosophy, Whewell argued, against Herschel, that verification is not possible when a hypothesis has been formed non-inductively (1831, 400-1). Nearly thirty years later, in the last edition of the Philosophy, Whewell referred to the belief that "the discovery of laws and causes of phenomena is a loose hap-hazard sort of guessing," and claimed that this type of view "appears to me to be a misapprehension of the whole nature of science" (1860a, 274). In other mature works he noted that discoveries are made "not by any capricious conjecture of arbitrary selection" (1858a, I, 29) and explained that new hypotheses are properly "collected from the facts" (1849, 17). In fact, Whewell was criticized by his contemporary David Brewster for not agreeing that discoveries, including Newton's discovery of the universal gravitation law, were typically made by accident. Why has Whewell been misinterpreted by so many modern commentators? One reason has to do with the error of reading certain terms used by Whewell in the 19th century as if they held the same meaning they have in the 20th and 21st. Thus, since Whewell used the terms "conjectures" and "guesses," some writers believe that he shares Popper's methodology. Whewell made mention, for instance, of the "happy guesses" made by scientists (1858b, 64) and claimed that "advances in knowledge" often follow "the previous exercise of some boldness and license in guessing" (1847, II, 55). But Whewell generally used these terms to connote a conclusion that is not (yet) conclusively confirmed. The Oxford English Dictionary tells us that prior to the 20th century the term "conjecture" was used to connote not a hypothesis reached by non-rational means, but rather one which is "unverified," or which is "a conclusion as to what is likely or probable" (as opposed to the results of demonstration). The term was used this way by Bacon, Kepler, Newton, and Dugald Stewart, writers whose work was well-known to Whewell. In other places where Whewell used the term "conjecture" he suggests that what appears to be the result of guesswork is actually what we might call an "educated guess," i.e., a conclusion drawn by (weak) inference. Whewell described Kepler's discovery, which seems so "capricious and fanciful" as actually being "regulated" by his "clear scientific ideas" (1857/1873, I, 291-2). Finally, Whewell's use of the terminology of guessing sometimes occurs in the context of a distinction he draws between the generation of a number of possible conceptions, and the selection of one to superinduce upon the facts. Before the appropriate conception is found, the scientist must be able to call up in his mind a number of possible ones (see 1858b, 79). Whewell noted that this calling up of many possibilities "is, in some measure, a process of conjecture." However, selecting the appropriate conception with which to colligate the data is not conjectural (1858b, 78). Thus Whewell claimed that the selection of the conception is often "preluded by guesses" (1858b, xix); he does not, that is, claim that the selection consists in guesswork. When inference is not used to select the appropriate conception, the resulting theory is not an "induction," but rather a "hasty and imperfect hypothesis." He drew such a distinction between Copernicus' heliocentric theory, which he called an induction, and the heliocentric system proposed by Aristarchus in the third century BCE to which he referred as a hasty and imperfect hypothesis (1857/1873, I, 258). Thus Whewell's philosophy of science cannot be described as the hypothetico-deductive view. It is an inductive method; yet it clearly differs from the more narrow inductivism of Mill. Whewell's view of induction has the advantage over Mill's of allowing the inference to unobservable properties and entities, and for this reason is a more accurate view of how science works. (For more detailed arguments against reading Whewell as a hypothetico-deductivist, see Snyder 2006; Snyder 2008; McCaskey 2014). ## 3. Philosophy of Science: Confirmation On Whewell's view, once a theory is invented by discoverers' induction, it must pass a variety of tests before it can be considered confirmed as an empirical truth. These tests are prediction, consilience, and coherence (see 1858b, 83-96). These are characterized by Whewell as, first, that "our hypotheses ought to fortel [sic] phenomena which have not yet been observed" (1858b, 86); second, that they should "explain and determine cases of a kind different from those which were contemplated in the formation" of those hypotheses (1858b, 88); and third that hypotheses must "become more coherent" over time (1858b, 91). I start by discussing the criterion of prediction. Hypotheses ought to foretell phenomena, "at least all phenomena of the same kind," Whewell explained, because "our assent to the hypothesis implies that it is held to be true of all particular instances. That these cases belong to past or to future times, that they have or have not already occurred, makes no difference in the applicability of the rule to them. Because the rule prevails, it includes all cases" (1858b, 86). Whewell's point here is simply that since our hypotheses are in universal form, a true hypothesis will cover all particular instances of the rule, including past, present, and future cases. But he also makes the stronger claim that successful predictions of unknown facts provide greater confirmatory value than explanations of already-known facts. Thus he held the historical claim that "new evidence" is more valuable than "old evidence." He believed that "to predict unknown facts found afterwards to be true is ... a confirmation of a theory which in impressiveness and value goes beyond any explanation of known facts" (1857/1873, II, 557). Whewell claimed that the agreement of the prediction with what occurs (i.e., the fact that the prediction turns out to be correct), is "nothing strange, if the theory be true, but quite unaccountable, if it be not" (1860a, 273-4). For example, if Newtonian theory were not true, he argued, the fact that from the theory we could correctly predict the existence, location and mass of a new planet, Neptune (as did happen in 1846), would be bewildering, and indeed miraculous. An even more valuable confirmation criterion, according to Whewell, is that of "consilience." Whewell explained that "the evidence in favour of our induction is of a much higher and more forcible character when it enables us to explain and determine [i.e., predict] cases of a kind different from those which were contemplated in the formation of our hypothesis. The instances in which this have occurred, indeed, impress us with a conviction that the truth of our hypothesis is certain" (1858b, 87-8). Whewell called this type of evidence a "jumping together" or "consilience" of inductions. An induction, which results from the colligation of one class of facts, is found also to colligate successfully facts belonging to another class. Whewell's notion of consilience is thus related to his view of natural classes of objects or events. To understand this confirmation criterion, it may be helpful to schematize the "jumping together" that occurred in the case of Newton's law of universal gravitation, Whewell's exemplary case of consilience. On Whewell's view, Newton used the form of inference Whewell characterized as "discoverers' induction" in order to reach his universal gravitation law, the inverse-square law of attraction. Part of this process is portrayed in book III of the Principia, where Newton listed a number of "propositions." These propositions are empirical laws that are inferred from certain "phenomena" (which are described in the preceding section of book III). The first such proposition or law is that "the forces by which the circumjovial planets are continually drawn off from rectilinear motions, and retained in their proper orbits, tend to Jupiter's centre; and are inversely as the squares of the distances of the places of those planets from that centre." The result of another, separate induction from the phenomena of "planetary motion" is that "the forces by which the primary planets are continually drawn off from rectilinear motions, and retained in their proper orbits, tend to the sun; and are inversely as the squares of the distances of the places of those planets from the sun's centre." Newton saw that these laws, as well as other results of a number of different inductions, coincided in postulating the existence of an inverse-square attractive force as the cause of various classes of phenomena. According to Whewell, Newton saw that these inductions "leap to the same point;" i.e., to the same law. Newton was then able to bring together inductively (or "colligate") these laws, and facts of other kinds of events (e.g., the class of events known as "falling bodies"), into a new, more general law, namely the universal gravitation law: "All bodies attract each other with a force of gravity which is inverse as the squares of the distances." By seeing that an inverse-square attractive force provided a cause for different classes of events--for satellite motion, planetary motion, and falling bodies--Newton was able to perform a more general induction, to his universal law. What Newton found was that these different kinds of phenomena--including circumjovial orbits, planetary orbits, as well as falling bodies--share an essential property, namely the same cause. What Newton did, in effect, was to subsume these individual "event kinds" into a more general natural kind comprised of sub-kinds sharing a kind essence, namely being caused by an inverse-square attractive force. Consilience of event kinds therefore results in causal unification. More specifically, it results in unification of natural kind categories based on a shared cause. Phenomena that constitute different event kinds, such as "planetary motion," "tidal activity," and "falling bodies," were found by Newton to be members of a unified, more general kind, "phenomena caused to occur by an inverse-square attractive force of gravity" (or, "gravitational phenomena"). In such cases, according to Whewell, we learn that we have found a "vera causa," or a "true cause," i.e., a cause that really exists in nature, and whose effects are members of the same natural kind (see 1860a, p. 191). Moreover, by finding a cause shared by phenomena in different sub-kinds, we are able to colligate all the facts about these kinds into a more general causal law. Whewell claimed that "when the theory, by the concurrences of two indications ... has included a new range of phenomena, we have, in fact, a new induction of a more general kind, to which the inductions formerly obtained are subordinate, as particular cases to a general population" (1858b, 96). He noted that consilience is the means by which we effect the successive generalization that constitutes the advancement of science (1847, II, 74). (For early discussions of consilience see Laudan 1971 and, especially, Hesse 1968 and Hesse 1971; for more on consilience and its relation to realism and natural kinds, see Snyder 2005 and Snyder 2006. On consilience and classification, see Quinn, 2017. For more on Whewell and scientific kinds, see Cowles 2016.) Whewell discussed a further, related test of a theory's truth: namely, "coherence." In the case of true theories, Whewell claimed, "the system becomes more coherent as it is further extended. The elements which we require for explaining a new class of facts are already contained in our system....In false theories, the contrary is the case" (1858b, 91). Coherence occurs when we are able to extend our hypothesis to colligate a new class of phenomena without ad hoc modification of the hypothesis. When Newton extended his theory regarding an inverse-square attractive force, which colligated facts of planetary motion and lunar motion, to the class of "tidal activity," he did not need to add any new suppositions to the theory in order to colligate correctly the facts about particular tides. On the other hand, Whewell explained, when phlogiston theory, which colligated facts about the class of phenomena "chemical combination," was extended to colligate the class of phenomena "weight of bodies," it was unable to do so without an ad hoc and implausible modification (namely, the assumption that phlogiston has "negative weight") (see 1858b, 92-3). Thus coherence can be seen as a type of consilience that happens over time; indeed, Whewell remarked that these two criteria--consilience and coherence--"are, in fact, hardly different" (1858b, 95). ## 4. Philosophy of Science: Necessary Truth A particularly intriguing aspect of Whewell's philosophy of science is his claim that empirical science can reach necessary truths. Explaining this apparently contradictory claim was considered by Whewell to be the "ultimate problem" of philosophy (see the important paper by Morrison 1997). Whewell accounted for it by reference to his antithetical epistemology. Necessary truths are truths which can be known a priori; they can be known in this way because they are necessary consequences of ideas which are a priori. They are necessary consequences in the sense of being analytic consequences. Whewell explicitly rejected Kant's claim that necessary truths are synthetic. Using the example "7 + 8 = 15," Whewell claimed that "we refer to our conceptions of seven, of eight, and of addition, and as soon as we possess the conceptions distinctly, we see that the sum must be 15." That is, merely by knowing the meanings of "seven," and "eight," and "addition," we see that it follows necessarily that "7 + 8 = 15" (1848, 471). Once the Ideas and conceptions are explicated, so that we understand their meanings, the necessary truths which follow from them are seen as being necessarily true. Thus, once the Idea of Space is explicated, it is seen to be necessarily true that "two straight lines cannot enclose a space." Whewell suggested that the first law of motion is also a necessary truth, which was knowable a priori once the Idea of Cause and the associated conception of force were explicated. This is why empirical science is needed to see necessary truths: because, as we saw above, empirical science is needed in order to explicate the Ideas. Thus Whewell also claimed that, in the course of science, truths which at first required experiment to be known are seen to be capable of being known independently of experiment. That is, once the relevant Idea is clarified, the necessary connection between the Idea and an empirical truth becomes apparent. Whewell explained that "though the discovery of the First Law of Motion was made, historically speaking, by means of experiment, we have now attained a point of view in which we see that it might have been certainly known to be true independently of experience" (1847, I, 221). Science, then, consists in the "idealization of facts," the transferring of truths from the empirical to the ideal side of the fundamental antithesis. He described this process as the "progressive intuition of necessary truths." Although they follow analytically from the meanings of ideas our minds supply, necessary truths are nevertheless informative statements about the physical world outside us; they have empirical content. Whewell's justification for this claim is a theological one. Whewell noted that God created the universe in accordance with certain "Divine Ideas." That is, all objects and events in the world were created by God to conform to certain of his ideas. For example, God made the world such that it corresponds to the idea of Cause partially expressed by the axiom "every event has a cause." Hence in the universe every event conforms to this idea, not only by having a cause but by being such that it could not occur without a cause. On Whewell's view, we are able to have knowledge of the world because the Fundamental Ideas which are used to organize our sciences resemble the ideas used by God in his creation of the physical world. The fact that this is so is no coincidence: God has created our minds such that they contain these same ideas. That is, God has given us our ideas (or, rather, the "germs" of the ideas) so that "they can and must agree with the world" (1860a, 359). God intends that we can have knowledge of the physical world, and this is possible only through the use of ideas which resemble those that were used in creating the world. Hence with our ideas--once they are properly "unfolded" and explicated--we can colligate correctly the facts of the world and form true theories. And when these ideas are distinct, we can know a priori the axioms which express their meaning. An interesting consequence of this interpretation of Whewell's view of necessity is that every law of nature is a necessary truth, in virtue of following analytically from some idea used by God in creating the world. Whewell drew no distinction between truths which can be idealized and those which cannot; thus, potentially, any empirical truth can be seen to be a necessary truth, once the ideas and conceptions are explicated sufficiently. For example, Whewell suggests that experiential truths such as "salt is soluble" may be necessary truths, even if we do not recognize this necessity (i.e., even if it is not yet knowable a priori) (1860b, 483). Whewell's view thus destroys the line traditionally drawn between laws of nature and the axiomatic propositions of the pure sciences of mathematics; mathematical truth is granted no special status. In this way Whewell suggested a view of scientific understanding which is, perhaps not surprisingly, grounded in his conception of natural theology. Since our ideas are "shadows" of the Divine Ideas, to see a law as a necessary consequence of our ideas is to see it as a consequence of the Divine Ideas exemplified in the world. Understanding involves seeing a law as being not an arbitrary "accident on the cosmic scale," but as a necessary consequence of the ideas God used in creating the universe. Hence the more we idealize the facts, the more difficult it will be to deny God's existence. We will come to see more and more truths as the intelligible result of intentional design. This view is related to the claim Whewell had earlier made in his Bridgewater Treatise (1833), that the more we study the laws of nature the more convinced we will be in the existence of a Divine Law-giver. (For more on Whewell's notion of necessity, see Fisch 1985; Snyder 1994; Morrison 1997; Snyder 2006; Ducheyne 2009.) ## 5. The Relation Between Scientific Practice, History of Science, and Philosophy of Science An issue of interest to philosophers of science today is the relation between knowledge of the actual practice and history of science and developing a philosophy of science. Whewell is interesting to examine in relation to this issue because he claimed to be inferring his philosophy of science from his study of the history and practice of science. His large-scale History of the Inductive Sciences (first edition published 1837) was a survey of science from ancient to modern times. He insisted upon completing this work before writing his Philosophy of the Inductive Sciences, founded upon their history. Moreover, Whewell sent proof-sheets of the History to his many scientist-friends to ensure the accuracy of his accounts. Besides knowing about the history of science, Whewell had first-hand knowledge of scientific practice: he was actively involved in science in several important ways. In 1825 he traveled to Berlin and Vienna to study mineralogy and crystallography with Mohs and other acknowledged masters of the field. He published numerous papers in the field, as well as a monograph, and is still credited with making important contributions to giving a mathematical foundation to crystallography. He also made contributions to the science of tidal research, pushing for a large-scale world-wide project of tidal observations; he won a Royal Society gold medal for this accomplishment. (For more on Whewell's contributions to science, see Becher 1986; Ruse 1991; Ducheyne 2010a; Snyder 2011; Honenberger 2018). Whewell acted as a terminological consultant for Faraday and other scientists, who wrote to him asking for new words. Whewell only provided terminology when he believed he was fully knowledgeable about the science involved. In his section on the "Language of Science" in the Philosophy, Whewell makes this position clear (see 1858b, p. 293). Another interesting aspect of his intercourse with scientists becomes clear in reading his correspondence with them: namely, that Whewell constantly pushed Faraday, Forbes, Lubbock and others to perform certain experiments, make specific observations, and to try to connect their findings in ways of interest to Whewell. In all these ways, Whewell indicated that he had a deep understanding of the activity of science. So how is this knowledge of science and its history important for his work on the philosophy of science? Some commentators have claimed that Whewell developed an a priori philosophy of science and then shaped his History to conform to his own view (Stoll 1929; Strong 1955). It is true that he started out, from his undergraduate days, with the project of reforming the inductive philosophy of Bacon; indeed this early inductivism led him to the view that learning about scientific method must be inductive (i.e., that it requires the study of the history of science). Yet it is clear that he believed his study of the history of science and his own work in science were needed in order to flesh out the details of his inductive position. Thus, as in his epistemology, both a priori and empirical elements combined in the development of his scientific methodology. Ultimately, Whewell criticized Mill's view of induction developed in the System of Logic not because Mill had not inferred it from a study of the history of science, but rather on the grounds that Mill had not been able to find a large number of appropriate examples illustrating the use of his "Methods of Experimental Inquiry." As Whewell noted, Bacon too had been unable to show that his inductive method had been exemplified throughout the history of science. Thus it appears that what was important to Whewell was not whether a philosophy of science had been, in fact, inferred from a study of the history of science, but rather, whether a philosophy of science was inferable from it. That is, regardless of how a philosopher came to invent her theory, she must be able to show it to be exemplified in the actual scientific practice used throughout history. Whewell believed that he was able to do this for his discoverers' induction. ## 6. Moral Philosophy Whewell's moral philosophy was criticized by Mill as being "intuitionist" (see Mill 1852). Whewell's moral philosophy is intuitionist in the sense of claiming that humans possess a faculty ("conscience") which enables them to discern directly what is morally right or wrong. Yet his view differs from that of earlier philosophers such as Shaftesbury and Hutcheson, who claimed that this faculty is akin to our sense organs and thus spoke of conscience as a "moral sense." Whewell's position is more similar to that of intuitionists such as Cudworth and Clarke, who claimed that our moral faculty is reason. Whewell maintained that there is no separate moral faculty, but rather that conscience is just "reason exercised on moral subjects." Because of this, Whewell referred to moral rules as "principles of reason" and described the discovery of these rules as an activity of reason (see 1864, 23-4). These moral rules "are primary principles, and are established in our minds simply by a contemplation of our moral nature and condition; or, what expresses the same thing, by intuition" (1846, 11). Yet, what he meant by "intuition" was not a non-rational mental process, as Mill claimed. On Whewell's view, the contemplation of the moral principles is conceived as a rational process. Whewell noted that "Certain moral principles being, as we've said, thus seen to be true by intuition, under due conditions of reflection and thought, are unfolded into their application by further reflection and thought"(1864, 12-13). Morality requires rules because reason is our distinctive property, and "Reason directs us to Rules" (1864, 45). Whewell's morality, then, does not have one problem associated with the moral sense intuitionists. For the moral sense intuitionist, the process of decision-making is non-rational; just as we feel the rain on our skin by a non-rational process, we just feel what the right action is. This is often considered the major difficulty with the intuitionist view: if the decision is merely a matter of intuition, it seems that there can be no way to settle disputes over how we ought to act. However, Whewell never suggested that decision-making in morality is a non-rational process. On the contrary, he believed that reason leads to common decisions about the right way to act (although our desires/affections may get in the way): he explained "So far as men decide conformably to Reason, they decide alike" (see 1864, 43). Thus the decision on how we ought to act should be made by reason, and so moral disputes can be settled rationally on Whewell's view. Mill also criticized Whewell's claim that moral rules are necessary truths which are self-evident. Mill took this to mean that there can be no progress in morality--what is self-evident must always remain so--and thus to the further conclusion that Whewell believed the current rules of society to be necessary truths. Such a view would tend to support the status quo, as Mill rightly complained. (Thus he, rather unfairly. accused Whewell of justifying evil practices such as cruelty to animals, forced marriages, and even slavery.) But Mill was wrong to attribute such a view to Whewell. Whewell did claim that moral rules are necessary truths, and invested them with the epistemological status of self-evident "axioms" (see 1864, 58). However, as noted above, Whewell's view of necessary truth is a progressive one. This is as much the case in morality as in science. The realm of morality, like the realm of physical science, is structured by certain Fundamental Ideas: in this case, Benevolence, Justice, Truth, Purity, and Order (see 1852, xxiii). These moral ideas are conditions of our moral experience; they enable us to perceive actions as being in accordance with the demands of morality. Like the ideas of the physical sciences, the ideas of morality must be explicated before the moral rules can be derived from them (see 1860a, 388). There is a progressive intuition of necessary truth in morality as well as in science. Hence it does not follow that because the moral truths are axiomatic and self-evident that we currently know them (see 1846, 38-9). Indeed, Whewell claimed that "to test self-evidence by the casual opinion of individual men, is a self-contradiction" (1846, 35). Nevertheless, Whewell did believe that we can look to the dictates of positive law of the most "morally advanced" societies as a starting point in our explication of the moral ideas. Although he surely had a biased view of what societies were "morally advanced," he was not therefore suggesting that the laws of those societies are the standard of morality. Just as we examine the phenomena of the physical world in order to explicate our scientific conceptions, we can examine the facts of positive law and the history of moral philosophy in order to explicate our moral conceptions. Only when these conceptions are explicated can we see what axioms or necessary truths of morality truly follow from them. Mill was therefore wrong to interpret Whewell's moral philosophy as a justification of the status quo or as constituting a "vicious circle." Rather, Whewell's view shares some features of Rawls's later use of the notion of "reflective equilibrium." (For more on Whewell's moral philosophy, and his debate with Mill over morality, see Snyder 2006, chapter four.)
cambridge-platonists	## Benjamin Whichcote The oldest member of the group, Benjamin Whichcote, is usually considered to be the founding father of Cambridge Platonism. During the Civil War period, Whichcote was appointed Provost of King's College, Cambridge, and he served as Vice Chancellor of the University in 1650. However, he was removed from his post at King's College at the Restoration in 1660, and was obliged to seek employment elsewhere, as a clergyman in London. The interruption to his academic career may explain why he never published any philosophical treatises as such. The main source for his philosophical views are his posthumously-published sermons and aphorisms. Whichcote's tolerant, optimistic and rational outlook set the intellectual tone for Cambridge Platonism. Whichcote's philosophical views are grounded in his repudiation of Calvinist theology. He held that God being supremely perfect is necessarily good, wise and loving. Whichcote regarded human nature as rational and perfectible, and he believed that it is through reason as much as revelation that God communicates with man. 'God is the most knowable of any thing in the world' (Patrides, 1969, p.58). Without reason we would have no means of demonstrating the existence of God, and no assurance that revelation is from God. By reason Whichcote did not mean the disputatious logic of the schools but discursive, demonstrative and practical reason enlightened by contemplation of the divine. He held that moral principles are immutable absolutes which exist independently of human minds and institutions, and that virtuous conduct is grounded in reason. Whichcote's Aphorisms amount to a manual of practical ethics which amply illustrates his conviction that the fruit reason is not 'bare knowledge' but action, or knowledge which 'doth go forth into act'. It is through reason that we gain knowledge of the natural world, and recognise natural phenomena as 'the EFFECTS OF GOD'. Although Whichcote's published writings do not discuss natural philosophy as such, his recognition of the demonstrative value of natural philosophy for the argument from design anticipates the use of natural philosophy in the apologetics of Cudworth and More. ## Culverwell, Smith and Sterry Whichcote's optimism about human reason and his conviction that philosophy properly belongs within the domain of religion, are shared by the other Cambridge Platonists, all of whom affirmed the compatibility of reason and faith. The fullest statement of this position is Henry More's The Apology of Henry More (1664) which sets out rules for the application of reason in religious matters, stipulating the use of only those 'Philosophick theorems' which are 'solid and rational in themselves, nor really repugnant to the word of God'. Like Whichcote, Peter Sterry, John Smith and Nathaniel Culverwell are known only through posthumously published writings. The first published treatise by any of the Cambridge Platonists was Nathaniel Culverwell's An Elegant and Learned Discourse of the Light of Nature of 1652. Like the other Cambridge Platonists Culverwell emphasises the freedom of the will and proposes an innatist epistemology, according to which the mind is furnished with 'clear and indelible Principles' and reason an 'intellectual lamp' placed in the soul by God to enable it to understand God's will promulgated in the law of nature. These innate principles of the mind also include moral principles. The soul is a divine spark, which derives knowledge by inward contemplation, not outward observation. Like the other Cambridge Platonists, Culverwell held that goodness is intrinsic to all things, and that moral principles do not depend on the will of God. He nevertheless underscored the importance of natural law as the foundation of moral obligation.. John Smith taught mathematics at Queen's College until his premature death in 1652. His posthumously published Select Discourses (1659) discusses a number of metaphysical and epistemological issues relating to religious belief - the existence of God, immortality of the soul and the rationality of religion. Smith outlines a hierarchy of four grades of cognitive ascent from sense combined with reason, through reason in conjunction with innate notions, and, thirdly, through disembodied, self-reflective reason; and finally divine love. Peter Sterry's only philosophical work, his posthumously-published A Discourse of the Freedom of the Will (1675), is the most visionary of all the writings of the Cambridge Platonists. Sterry was deeply involved with events outside Cambridge as chaplain first to the Parliamentary leader, Lord Brooke, and then to Oliver Cromwell. After the death of Cromwell he retired to a Christian community in East Sheen. In his Discourse Sterry argues that to act freely consists in acting in accordance with ones nature, appropriately to ones level of being, be that plant, animal or intellectual entity. Human liberty is grounded in the divine essence and entails liberty of the understanding and of the will. ## Henry More A life-long fellow of Christ's College, Cambridge, Henry More was the most prolific of the Cambridge Platonists. He was also the most directly engaged in contemporary philosophical debate: not only did he enter into correspondence with Descartes (between 1648 and 1649) but he also wrote against Hobbes, and was one of the earliest English critics of Spinoza (whom he attacks in Demonstrationum duarum propositionum ... confutatio and Epistola altera both published in his Opera omnia, 1671). Although he eventually became a critic of Cartesianism, he initially advocated the teaching of Cartesianism in English Universities. More's published writings included, besides philosophy, poetry, theology and bible commentary. His main philosophical works are his An Antidote Against Atheism (1653), his Of the Immortality of the Soul (1659), Enchiridion metaphysicum (1671), and Enchiridion ethicum (1667). Like the other Cambridge Platonists, More used philosophy in defence of theism against the claims of rational atheists. The most important statement of More's theological position his An Explanation of the Grand Mystery of Godliness appeared in 1660. In opposition to Calvinist pessimistic voluntarism, this propounds a moral, rational providentialism in which he vindicates the goodness and justice of God by invoking the Origenist doctrine of the pre-existence of the soul. It also makes the case for religious toleration. In his philosophical writings, More elaborated a philosophy of spirit which explained all the phenomena of mind and of the physical world as the activity of spiritual substance controlling inert matter. More conceived of both spirit and body as spatially extended, but defined spiritual substance as the obverse of material extension: where body is inert and solid, but divisible; spirit is active and penetrable, but indivisible. It was in his correspondence with Descartes that he first expounded his view that all substance, whether material or immaterial, is extended. He went on to argue space is infinite, anticipating that other native of Grantham, Isaac Newton, and that God who is an infinite spirit is an extended being (res extensa). In Enchiridion metaphysicum, he argues that the properties of space are analogous to the attributes of God (infinity, immateriality, immobility etc.). Within the category of spiritual substance More includes not just the souls of living creatures and God himself but the main intermediate causal agent of the cosmos, the Spirit of Nature (or 'Hylarchic Principle'). Conceived as the interface between the divine and the material, the Spirit of Nature is a 'Superintendant Cause' which combines efficient and teleological causality to ensure the smooth-running of the universe according to God's plan. It can also be understood as encapsulating 'certain general Modes and Lawes of Nature' (More, A Collection, Preface, p. xvi) since it is the Spirit of Nature that is responsible for uniting individual souls with bodies, and for ensuring the regular operation of non-animate nature. More sought, by this hypothesis, to account for phenomena that apparently defy the laws of mechanical physics (for example the inter-vortical trajectory of comets, the sympathetic vibration of strings and tidal motion). More underpinned his soul-body dualism by his theory of 'vital congruity' which explains soul-body interaction as a sympathetic attraction between soul and body engineered by the operation the Spirit of Nature. The most consistent theme of his philosophical writings, are arguments for demonstrating the existence and providential nature of God. The foundation stone of More's philosophical and apologetic enterprise is his philosophy of spirit, especially his arguments for the existence of incorporeal causal agents, that is, souls or spirits. Furthermore, More attempted to answer materialists like Thomas Hobbes whom he regarded as an atheist. More's strategy was to show that the same arguments that materialists use demonstrate the existence and properties of body, also support the obverse, the existence of incorporeal substances. In this way More sought to demonstrate that the idea of incorporeal substance, or spirit, was as intelligible as that of corporeal substance, i.e. body. Like Plato (in Laws 10), More argues that the operations of the nature cannot be explained simply in terms of the chance collision of material particles. Rather we must posit some other source of activity, which More identifies as 'spirit'. It is a short step, he argues, from grasping the concept of spirit, to accepting the idea of an infinite spirit, namely God. More underpins these a priori arguments for the existence of spirit, with a wide range of a posteriori arguments, taken from observed phenomena of nature to demonstrate the actions of spirit. Through this excursus into observational method he accumulated a wide variety of data ranging from experiments conducted by Robert Boyle and members of the Royal Society, to supernatural effects including cases of witchcraft and demons. He was censured by Boyle for misappropriating his experiments to endorse his hypothesis of the Spirit of Nature. Although his belief in evil spirits appears inconsistent with his otherwise rational philosophy, it should be remembered that belief in witchcraft was (a) not unusual in his time, and (b) was entirely consistent with the theory of spirit according to which to deny the existence of spirits good or evil, leads, logically to the denial of the existence of God. As he put it, alluding to James I's defence of episcopacy, "That saying is no less true in Politicks 'No Bishop, no King,' than this in Metaphysicks, 'No Spirit, no God' " (More, 1662, Antidote, p. 142). His most well-known fellow-believer was Royal Society member, Joseph Glanvill (1636-1680), whose Sadducismus triumphatus, More edited. More also published a short treatise on ethics entitled Enchiridion Ethicum (1667, translated as An Account of Virtue), which was probably intended for to be used as a textbook. Indebted to Descartes' theory of the passions this argues that knowledge of virtue is attainable by reason, and the pursuit of virtue entails the control of the passions by the soul. Motivation to good is supplied by rightly-directed emotion, while virtue is achieved by the exercise free will or autoexousy (More uses the same term as Cudworth), that is the 'Power to act or not act within ourselves'. Anticipating Shaftesbury's concept of moral sense More posits a special faculty of the soul combining reason and sensation which he calls the 'Boniform Faculty'. More used a number of different genres for conveying his philosophical ideas to non-specialist readers. The most popular among these were his Philosophical Poems (1647) and his Divine Dialogues (1668). In Conjectura cabbalistica (1653), he presented core themes of his philosophy in the form of an exposition of occulted truths contained in the first book of Genesis. Subsequently he undertook a detailed study of the Jewish Kabbalist texts translated and published by Knorr von Rosenroth in Kabbala denudata (1679). These studies were based on the belief, then current, that kabbalistic writings contained, in symbolic form, original truths of philosophy, as well as of religion. Kabbalism therefore exemplified the compatibility of philosophy and faith. In addition to philosophy More published several studies of biblical prophecy (e.g. Apocalypsis apocalypseos, 1680, Paralipomena prophetica, 1685). In 1675, More prepared a Latin translation of his works, Opera omnia which ensured his philosophy reached a European audience as well as an English one. ## Ralph Cudworth Like his friend Henry More, Ralph Cudworth spent his entire career as a teacher at the University of Cambridge, ending up as Master of Clare College. Cudworth published only one major work of philosophy in his lifetime, The True Intellectual System of the Universe (1678). Among the papers he left at his death, were the treatises published posthumously as A Treatise Concerning Eternal and Immutable Morality (1731) and his A Treatise of Freewill (1848). These papers also included two further manuscript treatises on the topic of 'Liberty and Necessity', which have never been printed. Cudworth's True Intellectual System propounds an anti-determinist system of philosophy grounded in his conception of God as a fully perfect being, infinitely wise and good. The created world reflects the perfection, wisdom and goodness of its creator. It must, therefore be orderly, intelligible, and organised for the best. This anti-voluntarist understanding of God's attributes is also the foundation of epistemology and ethics, since God's wisdom and goodness are the guarantors of truth and of moral principles. By contrast, a philosophy founded on a voluntaristic conception of the deity would have no ground of certainty or of morality because it would depend on the arbitrary will of God who could, by arbitrary fiat, decree non-sense to be true and wrong to be right. It follows that misconceptions of God's attributes, which emphasise his power and will, result by definition in false philosopical systems with sceptical and atheistic implications. Much of The True Intellectual System amounts to an extended consensus gentium argument for belief in God, demonstrable from an analysis of ancient sources which showed that most ancient philosophers were theists, and therefore that theism is compatible with philosophy. Among the non-theists, Cudworth (1678, p. 165) identifies four schools of atheistic philosophy, each of which is a type of materialism: Hylopathian atheism or materialism, 'Atomical or Democritical' atheism, Cosmo-plastic atheism (which makes the world-soul the highest numen), and Hylozoic atheism (which attributes life to matter). Each of these ancient brands of atheism has its latter-day manifestations in philosophers such as Hobbes (an example of a Hylopathian atheist) and Spinoza (a latter-day Hylozoist). The philosophy constitutive of Cudworth's intellectual system combines atomist natural philosophy with Platonic metaphysics. Cudworth conceives this as having originated with Moses from whom it was transmitted via Pythagoras to Greek and other philosophers, including Descartes whom Cudworth regarded as a reviver of Mosaic atomism. For Cudworth, as for Plato, soul is ontologically prior to the physical world. Since motion, life, thought and action cannot be explained in terms of material particles, haphazardly jolted together, there must be some guiding originator, namely soul or spirit. In order to account for movement, life and orderliness in the operations of nature, Cudworth proposed his hypothesis of 'the Plastick Life of Nature'. Similar in conception to More's Spirit of Nature, Cudworth's Plastic Nature is a formative principle which acts as an intermediary between the divine and the natural world, as the means whereby God imprints His presence on his creation and makes His wisdom and goodness manifest (and therefore intelligible) throughout created nature. The Platonist principle that mind precedes the world lies at the foundation of Cudworth's epistemology which is discussed in A Treatise of Eternal and Immutable Morality. This is the most fully developed theory of knowledge by any of the Cambridge Platonists, and the most extensive treatment of innatism by any seventeenth-century philosopher. For Cudworth, as for Plato, ideas and moral principles 'are eternal and self-subsistent things'. The external world is, intrinsically, intelligible, since it bears the imprint of its creator in the order and relationship of its component parts, as archetype to ectype. Cognition depends on the same principles, for just as the created world is a copy of the divine archetype, so also human minds contain the imprint of Divine wisdom and knowledge. Since the human mind mirrors the mind of God, it is ready furnished with ideas and the ability to reason. Cognition therefore entails recollection and the ideas of things with which the mind thinks are therefore 'anticipations'- for which Cudworth adopts the Stoic term prolepsis. Cognition is not a passive process, but involves the active participation of the mind. Although innate knowledge is the only true knowledge, Cudworth does not reject sense knowledge because sensory input is essential for knowledge of the body and the external world. However, raw sense data is not, by itself, knowledge since it requires mental processing in order to become knowledge. As Cudworth puts it, we cannot understand the book of nature unless we know how to read. Cudworth's theory of the mind as active is matched by an anti-determinist ethics of action, according to which the soul freely directs itself towards the good. In A Treatise Cudworth argues not only that ideas exist independently of human minds, but also the principles of morality are eternal and immutable. In a concerted attack on Hobbesian moral relativism, Cudworth, argues that the criteria of right and wrong, good and evil, justice and injustice are not a matter of convention, but are founded in the goodness and justice of God. Like Plato in the Euthyphro, Cudworth argues that it is not God's will that determines goodness, but that God wills things because they are good. The exercise of virtue is not, however, a passive process, but requires the free exercise of the individual will. Cudworth sets out his theory of free will in three treatises on 'Liberty and Necessity', only one of which has been published, and that posthumously - A Treatise of Freewill (1848). According to Cudworth, the will is not a faculty of the soul, distinct from reason, but a power of the soul which combines the functions of both reason and will in order to direct the soul towards the good. Cudworth's use of the terms 'hegemonikon' (taken from Stoicism) and 'autexousion' (taken from Plotinus) underlines the fact that the exercise of will entails the power to act. It is internal direction, not external compulsion that induces us to act either morally or immorally. Without the freedom (and therefore power) to of act, there would be no moral responsibility. Moral conduct is active, not passive. Virtuous action is therefore a matter of active internal self-determination, rather than determination from without. In A Treatise of Freewill, Cudworth elaborates his conception of the hegemonikon, an integrative power of the soul, which combines the the higher intellectual functions of the soul, will and reason with the lower, animal, appetites of the soul. Furthermore, Cudworth conceives of the hegemonikon not simply as the soul but the whole person, 'that which is properly we ourselves' (Cudworth, 1996, p. 178). Cudworth's concept of hegemonikon lays the basis for a concept of self identity founded in a subject that is at once thinking, autonomous and end-directed. Cudworth did not (as far as is known) develop a political philosophy. However, the political implications of his ethical theory set him against Hobbes, but also, in many ways anticipate John Locke. ### Legacy Among the immediate philosophical heirs of the Cambridge Platonists, mention should be made of Henry More's pupil, Anne Conway (1631-1679), one of the very few female philosophers of the period. Her Principles of the Most Ancient and Modern Philosophy (1692) includes a critique of More's dualistic philosophy of spirit, proposing instead a metaphysical monism that anticipates Leibniz. Another figure linked to More was John Norris (1657-1712) who was to become the leading English exponent of the philosophy of Malebranche. Whichcote's philosophical wisdom was admired by Anthony Ashley Cooper, third Earl of Shaftesbury who published his Select Sermons in 1698. Shaftesbury's tutor, John Locke was the intimate friend of Cudworth's philosophical daughter, Damaris Masham. The Cambridge Platonists have yet to receive full recognition as philosophers. Evidence from publication and citation suggests that their philosophical influence was more far-reaching than is normally recognised in modern histories of philosophy. Culverwell's Discourse was reprinted four times, including at Oxford. The impact of Cudworth on Locke has yet to be fully investigated. Richard Price, and Thomas Reid were both indebted to Cudworth. As the first philosophers to write primarily and consistently in the English language (preceded only by Sir Kenelm Digby), their impact is still felt in English philosophical terminology, through their coinage of such familiar terms as 'materialism', 'consciousness', and 'Cartesianism'. The intellectual legacy of the Cambridge Platonists extends not just to philosophical debate in seventeenth-century England and to Scottish Enlightenment thought, but into the European Enlightenment and beyond. Leibniz certainly read Cudworth and More, whose works were known beyond the English-speaking world thanks to Latin translations More's Opera omnia appeared in 1675-9, and Cudworth's entire printed works were translated into Latin by Johann Lorenz Mosheim and published in Jena in 1733. Cudworth's theory of Plastic Nature was taken up in vitalist debates in the French enlightenment. Although their critique of Descartes, Hobbes, Spinoza has ensured that the Cambridge Platonists are never completely ignored in philosophical history, they deserve to be considered an important strand in English seventeenth-century philosophy.
whitehead	## 1. Life and Works The son of an Anglican clergyman, Whitehead graduated from Cambridge in 1884 and was elected a Fellow of Trinity College that same year. His marriage to Evelyn Wade six years later was largely a happy one and together they had a daughter (Jessie) and two sons (North and Eric). After moving to London, Whitehead served as president of the Aristotelian Society from 1922 to 1923. After moving to Harvard, he was elected to the British Academy in 1931. His moves to both London and Harvard were prompted in part by institutional regulations requiring mandatory retirement, although his resignation from Cambridge was also done partly in protest over how the University had chosen to discipline Andrew Forsyth, a friend and colleague whose affair with a married woman had become something of a local scandal. In addition to Russell, Whitehead influenced many other students who became equally or more famous than their teacher, examiner or supervisor himself. For example: mathematicians G. H. Hardy and J. E. Littlewood; mathematical physicists Edmund Whittaker, Arthur Eddington, and James Jeans; economist J. M. Keynes; and philosophers Susanne Langer, Nelson Goodman, and Willard van Orman Quine. Whitehead did not, however, inspire any school of thought during his lifetime, and most of his students distanced themselves from parts of his teachings that they considered anachronistic. For example: Whitehead's conviction that pure mathematics and applied mathematics should not be separated, but cross-fertilize each other, was not shared by Hardy, but seen as a remnant of the fading mixed mathematics tradition; after the birth of the theories of relativity and quantum physics, Whitehead's method of abstracting some of the basic concepts of mathematical physics from common experiences seemed antiquated compared to Eddington's method of world building, which aimed at constructing an experiment matching world from mathematical building blocks; when, due to Whitehead's judgment as one of the examiners, Keynes had to rewrite his fellowship dissertation, Keynes raged against Whitehead, claiming that Whitehead had not bothered to try to understand Keynes' novel approach to probability; and Whitehead's main philosophical doctrine--that the world is composed of deeply interdependent processes and events, rather than mostly independent material things or objects--turned out to be largely the opposite of Russell's doctrine of logical atomism, and his metaphysics was dispelled by the logical positivists from their dream land of pure scientific philosophy. Despite the fact that he did not inspire any school of thought during his life, his influence on contemporaries was significant. The late Associate Justice of the United States Supreme Court, Felix Frankfurter, wrote: > > > From knowledge gained through the years of the personalities who in > our day have affected American university life, I have for some time > been convinced that no single figure has had such a pervasive > influence as the late Professor Alfred North Whitehead. (New York > Times, January 8, 1948) > > > Moreover, Whitehead's philosophical views posthumously inspired the movement of process philosophy and theology, and today Whitehead's ideas continue to be felt and are revalued in varying degrees in all of the main areas in which he worked. One of the most important factors in recent Whitehead scholarship is the gradual publishing of volumes of The Edinburgh Critical Edition of the Complete Works of Alfred North Whitehead. Two volumes of The Harvard Lectures of Alfred North Whitehead, mainly containing student notes of lectures and seminars given by Whitehead at Harvard from September 1924 until May 1927, have already been published by Edinburgh University Press in 2017 and 2021 respectively (see the Primary Literature section of the Bibliography below), and four more volumes are on their way. Their publication is a stimulus for Whitehead researchers all over the world, as is clear from the 2020 book edited by Brian Henning and Joseph Petek, Whitehead at Harvard, 1924-1925. A short chronology of the major events in Whitehead's life is below. \| \| \| \| --- \| --- \| \| 1861 \| Born February 15 in Ramsgate, Isle of Thanet, Kent, England. \| \| 1880 \| Enters Trinity College, Cambridge, with a scholarship in mathematics. \| \| 1884 \| Elected to the Apostles, the elite discussion club founded by Tennyson in the 1820s; graduates with a B.A. in Mathematics; elected a Fellow in Mathematics at Trinity. \| \| 1890 \| Meets Russell; marries Evelyn Wade. \| \| 1903 \| Elected a Fellow of the Royal Society as a result of his work on universal algebra, symbolic logic, and the foundations of mathematics. \| \| 1910 \| Resigns from Cambridge and moves to London. \| \| 1911 \| Appointed Lecturer at University College London. \| \| 1912 \| Elected President of both the South-Eastern Mathematical Association and the London branch of the Mathematical Association for the year 1913. \| \| 1914 \| Appointed Professor of Applied Mathematics at the Imperial College of Science and Technology. \| \| 1915 \| Elected President of the Mathematical Association for the two-year period 1915-1917. \| \| 1921 \| Meets Albert Einstein. \| \| 1922 \| Elected President of the Aristotelian Society for the one-year period 1922-1923. \| \| 1924 \| Appointed Professor of Philosophy at Harvard University. \| \| 1931 \| Elected a Fellow of the British Academy. \| \| 1937 \| Retires from Harvard. \| \| 1945 \| Awarded Order of Merit. \| \| 1947 \| Dies December 30 in Cambridge, Massachusetts, USA. \| More detailed information about Whitehead's life can be found in the comprehensive two-volume biography A.N. Whitehead: The Man and His Work (1985, 1990) by Victor Lowe and J.B. Schneewind. Paul Schilpp's The Philosophy of Alfred North Whitehead (1941) also includes a short autobiographical essay, in addition to providing a comprehensive critical overview of Whitehead's thought and a detailed bibliography of his writings. Other helpful introductions to Whitehead's work include Victor Lowe's Understanding Whitehead (1962), Stephen Franklin's Speaking from the Depths (1990), Thomas Hosinski's Stubborn Fact and Creative Advance (1993), Elizabeth Kraus' The Metaphysics of Experience (1998), Robert Mesle's Process-Relational Philosophy (2008), John Cobb's Whitehead Word Book (2015) and Jay McDaniel's What is Process Thought? (2021). Recommendable for the more advanced Whitehead student are Ivor Leclerc's Whitehead's Metaphysics (1958), Wolfe Mays' The Philosophy of Whitehead (1959), Donald Sherburne's A Whiteheadian Aesthetics (1961), Charles Hartshorne's Whitehead's Philosophy (1972), Lewis Ford's The Emergence of Whitehead's Metaphysics (1984), George Lucas' The Rehabilitation of Whitehead (1989), David Griffin's Whitehead's Radically Different Postmodern Philosophy (2007), Steven Shaviro's Without Criteria (2009), Isabelle Stengers' Thinking with Whitehead (2011), and Didier Debaise's Speculative Empiricism: Revisiting Whitehead (2017a). For a chronology of Whitehead's major publications, readers are encouraged to consult the Primary Literature section of the Bibliography below. ## 2. Mathematics and Logic Whitehead began his academic career at Trinity College, Cambridge where, starting in 1884, he taught for a quarter of a century. In 1890, Russell arrived as a student and during the 1890s the two men came into regular contact with one another. According to Russell, > > > Whitehead was extraordinarily perfect as a teacher. He took a personal > interest in those with whom he had to deal and knew both their strong > and their weak points. He would elicit from a pupil the best of which > a pupil was capable. He was never repressive, or sarcastic, or > superior, or any of the things that inferior teachers like to be. I > think that in all the abler young men with whom he came in contact he > inspired, as he did in me, a very real and lasting affection. (1956: > 104) > > > By the early 1900s, both Whitehead and Russell had completed books on the foundations of mathematics. Whitehead's 1898 A Treatise on Universal Algebra had resulted in his election to the Royal Society. Russell's 1903 The Principles of Mathematics had expanded on several themes initially developed by Whitehead. Russell's book also represented a decisive break from the neo-Kantian approach to mathematics Russell had developed six years earlier in his Essay on the Foundations of Geometry. Since the research for a proposed second volume of Russell's Principles overlapped considerably with Whitehead's own research for a planned second volume of his Universal Algebra, the two men began collaboration on what eventually would become Principia Mathematica (1910, 1912, 1913). According to Whitehead, they initially expected the research to take about a year to complete. In the end, they worked together on the project for a decade. According to Whitehead--inspired by Hermann Grassmann--mathematics is the study of pattern: > > > mathematics is concerned with the investigation of patterns of > connectedness, in abstraction from the particular relata and the > particular modes of connection. (1933 [1967: 153]) > > > In his Treatise on Universal Algebra, Whitehead took a generalized algebra--called 'universal algebra'--to be the most appropriate tool for this study or investigation, but after meeting Giuseppe Peano during the section devoted to logic at the First International Congress of Philosophy in 1900, Whitehead and Russell became aware of the potential of symbolic logic to become the most appropriate tool to rigorously study mathematical patterns. With the help of Whitehead, Russell extended Peano's symbolic logic in order to be able to deal with all types of relations and, consequently, with all the patterns of relatedness that mathematicians study. In his Principles of Mathematics, Russell gave an account of the resulting new symbolic logic of classes and relations--called 'mathematical logic'--as well as an outline of how to reconstruct all existing mathematics by means of this logic. After that, instead of only being a driving force behind the scenes, Whitehead became the public co-author of Russell of the actual and rigorous reconstruction of mathematics from logic. Russell often presented this reconstruction--giving rise to the publication of the three Principia Mathematica volumes--as the reduction of mathematics to logic, both qua definitions and qua proofs. And since the 1920s, following Rudolf Carnap, Whitehead and Russell's project as well as similar reduction-to-logic projects, including the earlier project of Gottlob Frege, are classified under the header of 'logicism'. However, Sebastian Gandon has highlighted in his 2012 study Russell's Unknown Logicism that Russell and Whitehead's logicism project differed in at least one important respect from Frege's logicism project. Frege adhered to a radical universalism, and wanted the mathematical content to be entirely determined from within the logical system. Russell and Whitehead, however, took into account the consensus, or took a stance in the ongoing discussions among mathematicians, with respect to the constitutive features of the already existing, 'pre-logicized' branches of mathematics, and then evaluated for each branch which of several possible types of relations were best suited to logically reconstruct it, while safeguarding its topic-specific features. Contrary to Frege, Whitehead and Russell tempered their urge for universalism to take into account the topic-specificity of the various mathematical branches, and as a working mathematician, Whitehead was well positioned to compare the pre-logicized mathematics with its reconstruction in the logical system. For Russell, the logicism project originated from the dream of a rock-solid mathematics, no longer governed by Kantian intuition, but by logical rigor. Hence, the discovery of a devastating paradox--later called 'Russell's paradox'--at the heart of mathematical logic was a serious blow for Russell, and kicked off his search for a theory to prevent paradox. He actually came up with several theories, but retained the ramified theory of types in Principia Mathematica. Moreover, the 'logicizing' of arithmetic required extra-logical patchwork: the axioms of reducibility, infinity, and choice. None of this patchwork could ultimately satisfy Russell. His original dream evaporated and, looking back later in life, he wrote: "The splendid certainty which I had always hoped to find in mathematics was lost in a bewildering maze" (1959: 157). Whitehead originally conceived of the logicism project as an improvement upon his algebraic project. Indeed, Whitehead's transition from the solitary Universal Algebra project to the joint Principia Mathematica project was a transition from universal algebra to mathematical logic as the most appropriate symbolic language to embody mathematical patterns. It entailed a generalization from the embodiment of absolutely abstract patterns by means of algebraic forms of variables to their embodiment by means of propositional functions of real variables. Hardy was quite right in his review of the first volume of Principia Mathematica when he wrote: "mathematics, one may say, is the science of propositional functions" (quoted by Grattan-Guinness 1991: 173). Whitehead saw mathematical logic as a tool to guide the mathematician's essential activities of intuiting, articulating, and applying patterns, and he did not aim at replacing mathematical intuition (pattern recognition) with logical rigor. In the latter respect, Whitehead, from the start, was more like Henri Poincare than Russell (cf. Desmet 2016a). Consequently, the discovery of paradox at the heart of mathematical logic was less of a blow to Whitehead than to Russell and, later in life, now and again, Whitehead simply reversed the Russellian order of generality and importance, writing that "symbolic logic" only represents "a minute fragment" of the possibilities of "the algebraic method" (1947 [1968: 129]). For a more detailed account of the genesis of Principia Mathematica and Whitehead's place in the philosophy of mathematics, cf. Smith 1953, Code 1985, Grattan-Guinness 2000 and 2002, Irvine 2009, Bostock 2010, Desmet 2010, N. Griffin et al. 2011, N. Griffin & Linsky 2013. Following the completion of Principia, Whitehead and Russell began to go their separate ways (cf. Ramsden Eames 1989, Desmet & Weber 2010, Desmet & Rusu 2012). Perhaps inevitably, Russell's anti-war activities and Whitehead's loss of his youngest son during World War I led to something of a split between the two men. Nevertheless, the two remained on relatively good terms for the rest of their lives. To his credit, Russell comments in his Autobiography that when it came to their political differences, Whitehead > > > was more tolerant than I was, and it was much more my fault than his > that these differences caused a diminution in the closeness of our > friendship. (1956: 100) > > > ## 3. Physics Even with the publication of its three volumes, Principia Mathematica was incomplete. For example, the logical reconstruction of the various branches of geometry still needed to be completed and published. In fact, it was Whitehead's task to do so by producing a fourth Principia Mathematica volume. However, this volume never saw the light of day. What Whitehead did publish were his repeated attempts to logically reconstruct the geometry of space and time, hence extending the logicism project from pure mathematics to applied mathematics or, put differently, from mathematics to physics--an extension which Russell greeted with enthusiasm and saw as an important step in the deployment of his new philosophical method of logical analysis. At first, Whitehead focused on the geometry of space. When Whitehead and Russell logicized the concept of number, their starting point was our intuition of equinumerous classes of individuals--for example, our recognition that the class of dwarfs in the fairy tale of Snow White (Doc, Grumpy, Happy, Sleepy, Bashful, Sneezy, Dopey) and the class of days in a week (from Monday to Sunday) have 'something' in common, namely, the something we call 'seven.' Then they logically defined (i) classes C and C' to be equinumerous when there is a one-to-one relation that correlates each of the members of C with one member of C', and (ii) the number of a class C as the class of all the classes that are equinumerous with C. When Whitehead logicized the space of physics, his starting point was our intuition of spatial volumes and of how one volume may contain (or extend over) another, giving rise to the (mereo)logical relation of containment (or extension) in the class of volumes, and to the concept of converging series of volumes--think, for example, of a series of Russian dolls, one contained in the other, but idealized to ever smaller dolls. Whitehead made all this rigorous and then, crudely put, defined the points from which to further construct the geometry of space. There is a striking resemblance between Whitehead's construction of points and the construction of real numbers by Georg Cantor, who had been one of Whitehead and Russell's main sources of inspiration next to Peano. Indeed, Whitehead defined points as equivalence classes of converging series of volumes, and Cantor defined real numbers as equivalence classes of converging series of rational numbers. Moreover, because Whitehead's basic geometrical entities of geometry are not (as in Euclid) extensionless points but volumes, Whitehead can be seen as one of the fathers of point-free geometry; and because Whitehead's basic geometrical relation is the mereological (or part-whole) relation of extension, he can also be seen as one of the founders of mereology (and even, when we take into account his later work on this topic in part IV of Process and Reality, of mereotopology). "Last night", Whitehead wrote to Russell on 3 September 1911, > > > the idea suddenly flashed on me that time could be treated in exactly > the same way as I have now got space (which is a picture of beauty, by > the bye). (Unpublished letter kept in The Bertrand Russell Archives at > McMaster University) > > > Shortly after, Whitehead must have learned about Einstein's Special Theory of Relativity (STR) because in a letter to Wildon Carr on 10 July 1912, Russell suggested to the Honorary Secretary of the Aristotelian Society that Whitehead possibly might deliver a paper on the principle of relativity, and added: "I know he has been going into the subject". Anyhow, in the early years of the second decade of the twentieth century, Whitehead's interest shifted from the logical reconstruction of the Euclidean space of classical physics to the logical reconstruction of the Minkowskian space-time of the STR. A first step to go from space to space-time was the replacement of (our intuition of) spatial volumes with (our intuition of) spatio-temporal regions (or events) as the basis of the construction (so that, for example, a point of space-time could be defined as an equivalence class of converging spatio-temporal regions). However, whereas Whitehead had constructed the Euclidean distance based on our intuition of cases of spatial congruence (for example, of two parallel straight line segments being equally long), he now struggled to construct the Minkowskian metric in terms of a concept of spatio-temporal congruence, based on a kind of merger of our intuition of cases of spatial congruence and our intuition of cases of temporal congruence (for example, of two candles taking equally long to burn out). So, as a second step, Whitehead introduced a second relation in the class of spatio-temporal regions next to the relation of extension, namely, the relation of cogredience, based on our intuition of rest or motion. Whitehead's use of this relation gave rise to a constant k, which allowed him to merge spatial and temporal congruence, and which appeared in his formula for the metric of space-time. When Whitehead equated k with c2 (the square of the speed of light) his metric became equal to the Minkowskian metric. Whitehead's most detailed account of this reconstruction of the Minkowskian space-time of the STR was given in his 1919 book, An Enquiry concerning the Principles of Natural Knowledge, but he also offered a less technical account in his 1920 book, The Concept of Nature. Whitehead first learned about Einstein's General Theory of Relativity (GTR) in 1916. He admired Einstein's new mathematical theory of gravitation, but rejected Einstein's explanation of gravitation for not being coherent with some of our basic intuitions. Einstein explained the gravitational motion of a free mass-particle in the neighborhood of a heavy mass as due to the curvature of space-time caused by this mass. According to Whitehead, the theoretical concept of a contingently curved space-time does not cohere with our measurement practices; they are based on the essential uniformity of the texture of our spatial and temporal intuition. In general, Whitehead opposed the modern scientist's attitude of dropping the requirement of coherence with our basic intuitions, and he revolted against the issuing bifurcation of nature into the world of science and that of intuition. In particular, as Einstein's critic, he set out to give an alternative rendering of the GTR--an alternative that passed not only what Whitehead called "the narrow gauge", which tests a theory's empirical adequacy, but also what he called "the broad gauge", which tests its coherence with our basic intuitions. In 1920, first in a newspaper article (reprinted in Essays in Science and Philosophy), and then in a lecture (published as Chapter VIII of Concept of Nature), Whitehead made public an outline of his alternative to Einstein's GTR. In 1921, Whitehead had the opportunity to discuss matters with Einstein himself. And finally, in 1922, Whitehead published a book with a more detailed account of his alternative theory of gravitation (ATG)--The Principle of Relativity. According to Whitehead, the Maxwell-Lorentz theory of electrodynamics (unlike Einstein's GTR) could be conceived as coherent with our basic intuitions--even in its four-dimensional format, namely, by elaborating Minkowski's electromagnetic worldview. Hence, Whitehead developed his ATG in close analogy with the theory of electrodynamics. He replaced Einstein's geometric explanation with an electrodynamics-like explanation. Whitehead explained the gravitational motion of a free mass-particle as due to a field action determined by retarded wave-potentials propagating in a uniform space-time from the source masses to the free mass-particle. It is important to stress that Whitehead had no intention of improving the predictive content of Einstein's GTR, only the explanatory content. However, Whitehead's replacement of Einstein's explanation with an alternative explanation entailed a replacement of Einstein's formulae with alternative formulae; and these different formulae implied different predictions. So it would be incorrect to say that Whitehead's ATG is empirically equivalent to Einstein's GTR. What can be claimed, however, is that for a long time Whitehead's theory was experimentally indistinguishable from Einstein's theory. In fact, like Einstein's GTR, Whitehead's ATG leads to Newton's theory of gravitation as a first approximation. Also (as shown by Eddington in 1924 and J. L. Synge in 1952) Einstein's and Whitehead's theories of gravitation lead to an identical solution for the problem of determining the gravitational field of a single, static, and spherically symmetric body--the Schwarzschild solution. This implies, for example, that Einstein's GTR and Whitehead's ATG lead to the exact same predictions not only with respect to the precession of the perihelion of Mercury and the bending of starlight in the gravitational field of the sun (as already shown by Whitehead in 1922 and William Temple in 1924) but also with respect to the red-shift of the spectral lines of light emitted by atoms in the gravitational field of the sun (contrary to Whitehead's own conclusion in 1922, which was based on a highly schematized and soon outdated model of the molecule). Moreover (as shown by R. J. Russell and Christoph Wassermann in 1986 and published in 2004) Einstein's and Whitehead's theories of gravitation also lead to an identical solution for the problem of determining the gravitational field of a single, rotating, and axially symmetric body--the Kerr solution. Einstein's and Whitehead's predictions become different, however, when considering more than one body. Indeed, Einstein's equation of gravitation is non-linear while Whitehead's is linear; and this divergence qua mathematics implies a divergence qua predictions in the case of two or more bodies. For example (as shown by G. L. Clark in 1954) the two theories lead to different predictions with respect to the motion of double stars. The predictive divergence in the case of two bodies, however, is quite small, and until recently experimental techniques were not sufficiently refined to confirm either Einstein's predictions or Whitehead's, for example, with respect to double stars. In 2008, based on a precise timing of the pulsar B1913+16 in the Hulse-Taylor binary system, Einstein's predictions with respect to the motion of double stars were confirmed, and Whitehead's refuted (by Gary Gibbons and Clifford Will). The important fact from the viewpoint of the philosophy of science is not that, since the 1970s, now and again, a physicist rose to claim the experimental refutation of Whitehead's ATG, but that for decades it was experimentally indistinguishable from Einstein's GTR, hence refuting two modern dogmas. First, that theory choice is solely based on empirical facts. Clearly, next to facts, values--especially aesthetic values--are at play as well. Second, that the history of science is a succession of victories over the army of our misleading intuitions, each success of science must be interpreted as a defeat of intuition, and a truth cannot be scientific unless it hurts human intuition. Surely, we can be scientific without taming the authority of our intuition and without engaging in the disastrous race to disenchant nature and humankind. For a more detailed account of Whitehead's involvement with Einstein's STR and GTR, cf. Palter 1960, Von Ranke 1997, Herstein 2006 and Desmet 2011, 2016b, and 2016c. In 1925, in one of his Lowell lectures (published as Chapter VIII of Science and the Modern World), Whitehead made public a popular outline of his alternative to Bohr's solar system view of material atoms, but he never published a more detailed and technical account of this alternative quantum theory (AQT). Whitehead called his AQT, 'the theory of primates.' In it, primates are not apes, of course, but standing (or stationary) electromagnetic waves (or vibrations). These waves are different from the traveling electromagnetic waves outside atoms. They are not light waves, but standing waves of charge density inside atoms, in fact, constituting atoms. Whitehead was up to date with respect to the old quantum mechanics of Planck, Einstein and Bohr. He was as familiar with Jeans' Report on Radiation and the Quantum-Theory (1914) as with Eddington's Report on the Relativity Theory of Gravitation (1918). And prior to his departure to Harvard, on 12 July 1924, Whitehead chaired a symposium--"The Quantum Theory: How far does it modify the mathematical, the physical, and the psychological concepts of continuity?"--which was part of a joint session of the Aristotelian Society and the Mind Society. Unfortunately, he did not himself deliver a lecture at that symposium, and the most technical account of his AQT can be found in the student notes of lectures Whitehead gave at Harvard, especially in the fall semester of the academic year 1924-1925. Whitehead's lectures, however, were not clear-cut presentations, but instances of philosophy in the making. Moreover, his philosophy students lacked the necessary knowledge of mathematical physics to fully comprehend Whitehead when he talked physics. Consequently, the student notes of Whitehead's Harvard lectures dealing with quantum theory are low quality notes. Ronny Desmet, however, has made a first attempt to correct some of the mistakes and to reconstruct Whitehead's AQT in his 2020 paper, "Whitehead's Highly Speculative Lectures on Quantum Theory" (Henning & Petek 2020: 154-181), and in his 2022 paper, "Whitehead's 1925 View of Function and Time" (to be published). In his AQT, Whitehead conceived of all electrons and protons as compositions of standing electromagnetic waves of charge density, which he called 'primates' or 'primordial elements.' In Science and the Modern World, Whitehead wrote: "We shall conceive each primordial element as a vibratory ebb and flow of an underlying energy, or activity" (1925 [1967: 35]). When a material atom emits energy, one of its composing standing electromagnetic waves is transformed into a travelling electromagnetic wave, that is, into light. The energy emitted by an atom is the energy of the emitted light, and the action of the emitted light is the product of its energy and its period (which is the inverse of its frequency). Experiment had revealed that the amounts of energy and action of atomic light emission do not form a continuous, but a discrete spectrum; they are quantized physical variables. In fact, a smallest amount of action exists, the quantum of action, given by Planck's constant, and all measured amounts of action of atomic light emission are integer multiples of this quantum of action. Whitehead conceived of electrons and protons as composed of standing electromagnetic waves of which the action is also always a multiple of the quantum of action. And as he conceived of the atomic energy emission as a transformation of these standing into traveling electromagnetic waves, it became clear to him that this emission actually involves the decay of a primate, and that the loss of the primate's quantized energy and action explains why the emitted light has similarly quantized energy and action. Contrary to his ATG, Whitehead's AQT soon became obsolete. Experiments revealed that there are more elementary particles than electrons and protons. And, of course, the new quantum mechanics (of Schrodinger, Heisenberg, Dirac, Pauli, and many others) emerged. As Whitehead focused on philosophy when at Harvard, his knowledge of physics began to be out of date. His own theory, however, showed some similarities with the new quantum mechanics. Whitehead's standing electromagnetic waves, in a sense, foreshadowed Schrodinger's wave mechanics. And whereas the discovery of the quantum of action was the "physicist's nightmare" (Whitehead 2021:93), Whitehead was happy with the increasing importance of the notion of 'action' in physics. Action is an activity through time, and hence confirms Whitehead's idea that processes, and not instantaneous configurations of inactive masses, are fundamental in nature. Moreover, since there is a minimum quantum of action, Whitehead interpreted the equation, action equals the product of energy and period (which is a duration), by saying: "concentration of time increases energy" (2021: 93). This statement of Whitehead, in a sense, foreshadowed Heisenberg's uncertainty principle with respect to energy and time, which was formulated two years later. As his AQT thus foreshadowed the new quantum mechanics in two respects, one wonders what he thought of this new theory when it emerged. Charles Hartshorne wrote that Whitehead definitely saw "Heisenberg's famous article of 1927 on the Uncertainty Principle" (2010: 28). Hartshorne was sure about this, because it was he himself who showed it to Whitehead. But there is (as yet) no evidence that Whitehead seriously studied this paper in particular, or the new quantum mechanics in general. Whitehead said to his students: "A definite creative act is looked on as handing over to physics the ultimate unit" (2021: 41), where the ultimate unit is Planck's constant, or, in other words, the quantum of action. Clearly, Whitehead thought of the quantum of action as an abstract aspect of his ultimate metaphysical building blocks, his elementary creative acts or 'actual occasions,' and his AQT, like the Maxwell-Lorentz theory of electrodynamics and Whitehead's ATG, were still at the back of his mind when he developed his metaphysics at Harvard. This may well be the reason why his metaphysics--his process philosophy--has been conceived by many authors as an interesting framework to ontologically interpret the mathematical formalism of quantum mechanics. And even authors who are not directly inspired by Whitehead, often come up with interpretations of quantum mechanics that are strikingly Whiteheadian. A good and recent example is Carlo Rovelli's relational interpretation. Rovelli writes: > > > In the world described by quantum mechanics there is no reality except > in the relations between physical systems. It isn't things that > enter into relations but, rather, relations that ground the notion of > "thing". The world of quantum mechanics is not a world of > objects: it is a world of events. Things are built by the happenings > of elementary events: as the philosopher Nelson Goodman wrote in the > 1950s, in a beautiful phrase, "An object is a monotonous > process." A stone is a vibration of quanta that maintains its > structure for a while, just as a marine wave maintains its identity > for a while before melting again into the sea. ... We, like waves > and like all objects, are a flux of events; we are processes, for a > brief time monotonous ... (2017: 115-116) > > > And Rovelli adds that in the speculative world of quantum gravity: > > > There is no longer space which contains the world, and no > longer time during the course of which events occur. There > are elementary processes ... continuously interact[ing] with each > other. Just as a calm and clear Alpine lake is made up of a rapid > dance of a myriad of minuscule water molecules, the illusion of being > surrounded by continuous space and time is the product of a > long-sighted vision of a dense swarming of elementary processes. > (2017: 158) > > > ## 4. Philosophy of Science Whitehead's reconstruction of the space-time of the STR and his ATG make clear (i) that his main methodological requirement in the philosophy of science is that physical theories should cohere with our intuitions of the relatedness of nature (of the relations of extension, congruence, cogredience, causality, etc.), and (ii) that his paradigm of what a theory of physics should be like is the Maxwell-Lorentz theory of electrodynamics. And indeed, in his philosophy of science, Whitehead rejects David Hume's "sensationalist empiricism" (1929c [1985: 57]) and Isaac Newton's "scientific materialism" (1925 [1967: 17]). Instead Whitehead promotes (i) a radical empiricist methodology, which relies on our perception, not only of sense data (colors, sounds, smells, etc.) but also of a manifold of natural relations, and (ii) an electrodynamics-like worldview, in which the fundamental concepts are no longer simply located substances or bits of matter, but internally related processes and events. "Modern physical science", Whitehead wrote, > > > is the issue of a coordinated effort, sustained for more than three > centuries, to understand those activities of Nature by reason of which > the transitions of sense-perception occur. (1934 [2011: 65]) > > > But according to Whitehead, Hume's sensationalist empiricism has undermined the idea that our perception can reveal those activities, and Newton's scientific materialism has failed to render his formulae of motion and gravitation intelligible. Whitehead was dissatisfied with Hume's reduction of perception to sense perception because, as Hume discovered, pure sense perception reveals a succession of spatial patterns of impressions of color, sound, smell, etc. (a procession of forms of sense data), but it does not reveal any causal relatedness to interpret it (any form of process to render it intelligible). In fact, all "relatedness of nature", and not only its causal relatedness, was "demolished by Hume's youthful skepticism" (1922 [2004: 13]) and conceived as the outcome of mere psychological association. Whitehead wrote: > > > Sense-perception, for all its practical importance, is very > superficial in its disclosure of the nature of things. ... My > quarrel with [Hume] concerns [his] exclusive stress upon > sense-perception for the provision of data respecting Nature. > Sense-perception does not provide the data in terms of which we > interpret it. (1934 [2011: 21]) > > > Whitehead was also dissatisfied with Newton's scientific materialism, > > > which presupposes the ultimate fact of an irreducible brute matter, or > material, spread through space in a flux of configurations. In itself > such a material is senseless, valueless, purposeless. It just does > what it does do, following a fixed routine imposed by external > relations which do not spring from the nature of its being. > (1925 [1967: 17]) > > > Whitehead rejected Newton's conception of nature as the succession of instants of spatial distribution of bits of matter for two reasons. First: the concept of a "durationless" instant, "without reference to any other instant", renders unintelligible the concepts of "velocity at an instant" and "momentum at an instant" as well as the equations of motion involving these concepts (1934 [2011: 47]). Second: the concept of self-sufficient and isolated bits of matter, having "the property of simple location in space and time" (1925 [1967: 49]), cannot "give the slightest warrant for the law of gravitation" that Newton postulated (1934 [2011: 34]). Whitehead wrote: > > > Newton's methodology for physics was an overwhelming success. > But the forces which he introduced left Nature still without meaning > or value. In the essence of a material body--in its mass, motion, > and shape--there was no reason for the law of gravitation. (1934 > [2011: 23]) > > > > There is merely a formula for succession. But there is an absence of > understandable causation for that formula for that succession. (1934 > [2011: 53-54]) > > > "Combining Newton and Hume", Whitehead summarized, > > > we obtain a barren concept, namely, a field of perception devoid of > any data for its own interpretation, and a system of interpretation > devoid of any reason for the concurrence of its factors. (1934 [2011: > 25]) > > > "Two conclusions", Whitehead wrote, > > > are now abundantly clear. One is that sense-perception omits any > discrimination of the fundamental activities within Nature. ... > The second conclusion is the failure of science to endow its formulae > for activity with any meaning. (1934 [2011: 65]) > > > The views of Newton and Hume, Whitehead continued, are "gravely defective. They are right as far as they go. But they omit ... our intuitive modes of understanding" (1934 [2011: 26]). In Whitehead's eyes, however, the development of Maxwell's theory of electromagnetism constituted an antidote to Newton's scientific materialism, for it led him to conceive the whole universe as "a field of force--or, in other words, a field of incessant activity" (1934 [2011: 27]). The theory of electromagnetism served Whitehead to overcome Newton's "fallacy of simple location" (1925 [1967: 49]), that is, the conception of nature as a universe of self-sufficient isolated bits of matter. Indeed, we cannot say of an electromagnetic event that it is > > > here in space, and here in time, or here in space-time, in a perfectly > definite sense which does not require for its explanation any > reference to other regions of space-time. (1925 [1967: 49]) > > > The theory of electromagnetism "involves the entire abandonment of the notion that simple location is the primary way in which things are involved in space-time" because it reveals that, "in a certain sense, everything is everywhere at all times" (1925 [1967: 91]). "Long ago", Whitehead wrote, Faraday already remarked "that in a sense an electric charge is everywhere", and: > > > the modification of the electromagnetic field at every point of space > at each instant owing to the past history of each electron is another > way of stating the same fact. (1920 [1986: 148]) > > > The lesson that Whitehead learned from the theory of electromagnetism is unambiguous: > > > The fundamental concepts are activity and process. ... The notion > of self-sufficient isolation is not exemplified in modern physics. > There are no essentially self-contained activities within limited > regions. ... Nature is a theatre for the interrelations of > activities. All things change, the activities and their > interrelations. ... In the place of the procession of [spatial] > forms (of externally related bits of matter, modern physics) has > substituted the notion of the forms of process. It has thus swept away > space and matter, and has substituted the study of the internal > relations within a complex state of activity. (1934 [2011: > 35-36]) > > > But overcoming Newton was insufficient for Whitehead because Hume "has even robbed us of reason for believing that the past gives any ground for expectation of the future" (1934 [2011: 65]). According to Whitehead, > > > science conceived as resting on mere sense-perception, with no other > sources of observation, is bankrupt, so far as concerns its claims to > self-sufficiency. (1934 [2011: 66]) > > > In fact, science conceived as restricting itself to the sensationalist methodology can find neither efficient nor final causality. It "can find no creativity in Nature; it finds mere rules of succession" (1934 [2011: 66]). "The reason for this blindness", according to Whitehead, "lies in the fact that such science only deals with half of the evidence provided by human experience" (1934 [2011: 66]). Contrary to Hume, Whitehead held that it is untrue to state that our perception, in which sense perception is only one factor, discloses no causal relatedness. Inspired by the radical empiricism of William James and Henri Bergson, Whitehead gave a new analysis of perception. According to Whitehead, our perception is a symbolic interplay of two pure modes of perception, pure sense perception (which Whitehead ultimately called "perception in the mode of presentational immediacy"), and a more basic perception of causal relatedness (which he called "perception in the mode of causal efficacy"). According to Whitehead, taking into account the whole of our perception instead of only pure sense perception, that is, all perceptual data instead of only Hume's sense data, implies also taking into account the other half of the evidence, namely, our intuitions of the relatedness of nature, of "the togetherness of things". He added: > > > the togetherness of things involves some doctrine of mutual immanence. > In some sense or other ... each happening is a factor in the > nature of every other happening. (1934 [2011: 87]) > > > Hume demolished the relatedness of nature; Whitehead restored it, founded the "doctrine of causation ... on the doctrine of immanence", and wrote: "Each occasion presupposes the antecedent world as active in its own nature. ... This is the doctrine of causation" (1934 [2011: 88-89]). Whitehead also noticed that, in a sense, physicists are even more reductionist than Hume. In practice they rely on sense data, but in theory they abstract from most of the data of our five senses (sight, hearing, smell, taste, and touch) to focus on the colorless, soundless, odorless, and tasteless mathematical aspects of nature. Consequently, in a worldview inspired not by the actual practices of physicists, but by their theoretical speculations, nature--methodologically stripped from its 'tertiary' qualities (esthetical, ethical, and religious values)--is further reduced to the scientific world of 'primary' qualities (mathematical quantities and interconnections such as the amplitude, length, and frequency of mathematical waves), and this scientific world is bifurcated from the world of 'secondary' qualities (colors, sounds, smells, etc.). Moreover, the former world is supposed, ultimately, to fully explain the latter world (so that, for example, colors end up as being nothing more than electromagnetic wave-frequencies). Whitehead spoke of the "bifurcation of nature into two systems of reality" (1920 [1986: 30]) to denote the strategy--originating with Galileo, Descartes, Boyle and Locke--of bifurcating nature into the essential reality of primary qualities and the non-essential reality of "psychic additions" or secondary qualities, ultimately to be explained away in terms of primary qualities. Whitehead sided with Berkeley in arguing that the primary/secondary distinction is not tenable (1920 [1986: 43-44]), that all qualities are "in the same boat, to sink or swim together" (1920 [1986: 148]), and that, for example, > > > the red glow of the sunset should be as much part of nature as are the > molecules and electric waves by which men of science would explain the > phenomenon. (1920 [1986: 29]) > > > Whitehead described the philosophical outcome of the bifurcation of nature as follows: > > > The primary qualities are the essential qualities of substances whose > spatio-temporal relationships constitute nature. ... The > occurrences of nature are in some way apprehended by minds ... > But the mind in apprehending also experiences sensations which, > properly speaking, are qualities of the mind alone. These sensations > are projected by the mind so as to clothe appropriate bodies in > external nature. Thus the bodies are perceived as with qualities which > in reality do not belong to them, qualities which in fact are purely > the offspring of the mind. Thus nature gets credit which should in > truth be reserved for ourselves: the rose for its scent: the > nightingale for his song: and the sun for his radiance. The poets are > entirely mistaken. They should address their lyrics to themselves, and > should turn them into odes of self-congratulation on the excellency of > the human mind. Nature is a dull affair, soundless, scentless, > colourless; merely the hurrying of material, endlessly, meaninglessly. > (1925 [1967: 54]) > > > "The enormous success of the scientific abstractions", Whitehead wrote, "has foisted onto philosophy the task of accepting them as the most concrete rendering of fact" and, he added: > > > Thereby, modern philosophy has been ruined. It has oscillated in a > complex manner between three extremes. There are the dualists, who > accept matter and mind as on an equal basis, and the two varieties of > monists, those who put mind inside matter, and those who put matter > inside mind. But this juggling with abstractions can never overcome > the inherent confusion introduced by the ascription of misplaced > concreteness to the scientific scheme. (1925 [1967: 55]) > > > Whitehead's alternative is fighting "the Fallacy of Misplaced Concreteness"--the "error of mistaking the abstract for the concrete"--because "this fallacy is the occasion of great confusion in philosophy" (1925 [1967: 51]). The fallacy of misplaced concreteness is committed each time abstractions are taken as concrete facts, and "more concrete facts" are expressed "under the guise of very abstract logical constructions" (1925 [1967: 50-51]). This fallacy lies at the root of the modern philosophical confusions of scientific materialism and progressive bifurcation of nature. Indeed, the notion of simple location in Newton's scientific materialism is an instance of the fallacy of misplaced concreteness--it mistakes the abstraction of in essence unrelated bits of matter as the most concrete reality from which to explain the relatedness of nature. And the bifurcating idea that secondary qualities should be explained in terms of primary qualities is also an instance of this fallacy--it mistakes the mathematical abstractions of physics as the most concrete and so-called primary reality from which to explain the so-called secondary reality of colors, sounds, etc. In light of the rise of electrodynamics, relativity, and quantum mechanics, Whitehead challenged scientific materialism and the bifurcation of nature "as being entirely unsuited to the scientific situation at which we have now arrived", and he clearly outlined the mission of philosophy as he saw it: > > > I hold that philosophy is the critic of abstractions. Its function is > the double one, first of harmonising them by assigning to them their > right relative status as abstractions, and secondly of completing them > by direct comparison with more concrete intuitions of the universe, > and thereby promoting the formation of more complete schemes of > thought. It is in respect to this comparison that the testimony of > great poets is of such importance. Their survival is evidence that > they express deep intuitions of mankind penetrating into what is > universal in concrete fact. Philosophy is not one among the sciences > with its own little scheme of abstractions which it works away at > perfecting and improving. It is the survey of the sciences, with the > special object of their harmony, and of their completion. It brings to > this task, not only the evidence of the separate sciences, but also > its own appeal to concrete experience. (1925 [1967: 87]) > > > For more details on Whitehead's philosophy of science, cf. Hammerschmidt 1947, Lawrence 1956, Bright 1958, Palter 1960, Mays 1977, Fitzgerald 1979, Plamondon 1979, Eastman & Keeton (eds) 2004, Bostock 2010, Athern 2011, Deroo & Leclercq (eds) 2011, Henning et al. (eds) 2013, Segall 2013, McHenry 2015, Desmet 2016d, Eastman & Epperson & Griffin (eds) 2016, Eastman 2020, Lestienne 2020. ## 5. Philosophy of Education While in London, Whitehead became involved in many practical aspects of tertiary education, serving as President of the Mathematical Association, Dean of the Faculty of Science and Chairman of the Academic Council of the Senate at the University of London, Chairman of the Delegacy for Goldsmiths' College, and several other administrative posts. Many of his essays about education date from this time and appear in his book, The Aims of Education and Other Essays (1929a). At its core, Whitehead's philosophy of education emphasizes the idea that a good life is most profitably thought of as an educated or civilized life, two terms which Whitehead often uses interchangeably. As we think, we live. Thus it is only as we improve our thoughts that we improve our lives. The result, says Whitehead, is that "There is only one subject matter for education, and that is Life in all its manifestations" (1929a: 10). This view in turn has corollaries for both the content of education and its method of delivery. (a) With regard to delivery, Whitehead emphasizes the importance of remembering that a "pupil's mind is a growing organism ... it is not a box to be ruthlessly packed with alien ideas" (1929a: 47). Instead, it is the purpose of education to stimulate and guide each student's self-development. It is not the job of the educator simply to insert into his students' minds little chunks of knowledge. Whitehead conceives of the student's educational process of self-development as an organic and cyclic process in which each cycle consists of three stages: first the stage of romance, then the stage of precision, and finally, the stage of generalization. The first stage is all about "free exploration, initiated by wonder", the second about the disciplined "acquirement of technique and detailed knowledge", and the third about "the free application of what has been learned" (Lowe 1990: 61). These stages, continually recurring in cycles, determine what Whitehead calls "The Rhythm of Education" (cf. 1929a: 24-44). In the context of mathematics, Whitehead's three stages can be conceived of as the stage of undisciplined intuition, the stage of logical reasoning, and the stage of logically guided intuition. By skipping stage one, and never arriving at stage three, bad math teachers deny students the major motivation to love mathematics: the joy of pattern recognition. That education does not involve inserting into the student's mind little chunks of knowledge is clear from the description of culture that Whitehead offers as the opening of the first and title essay of The Aims of Education: > > > Culture is activity of thought, and receptiveness of beauty and humane > feeling. Scraps of information have nothing to do with it. (1929a: > 1) > > > On the contrary, Whitehead writes, > > > we must beware of what I call 'inert ideas'--that is > to say, ideas that are merely received into the mind without being > utilized, or tested, or thrown into fresh combinations, (1929a: > 1-2) > > > and he holds that "education is the acquisition of the art of the [interconnection and] utilization of knowledge" (1929a: 6), and that ideas remain disconnected and non-utilized unless they are related > > > to that stream, compounded of sense perceptions, feelings, hopes, > desires, and of mental activities adjusting thought to thought, which > forms our life. (1929a: 4) > > > This point--the point where Whitehead links the art of education to the stream of experience that forms our life--is the meeting point of Whitehead's philosophy of education with his philosophy of experience, which is also called: 'process philosophy.' According to Whitehead's process philosophy, the stream of experience that forms our life consists of occasions of experience, each of which is a synthesis of many feelings having objective content (what is felt) and subjective form (how it is felt); also, the synthesis of feelings is not primarily controlled by their objective content, but by their subjective form. According to Whitehead's philosophy of education, the attempt to educate a person by merely focusing on objective content--on inert ideas, scraps of information, bare knowledge--while disregarding the subjective form or emotional pattern of that person's experience can never be successful. The art of education has to take into account the subjective receptiveness and appreciation of beauty and human greatness, the subjective emotions of interest, joy and adventure, and "the ultimate motive power" (1929a: 62), that is, the sense of importance, values and possibilities (cf.1929a: 45-65). (b) With regard to content, Whitehead holds that any adequate education must include a literary component, a scientific component, and a technical component. According to Whitehead: > > > Any serious fundamental change in the intellectual outlook of human > society must necessarily be followed by an educational revolution. > (1929a: 116) > > > In particular, the scientific revolution and the fundamental changes it entailed in the seventeenth and subsequent centuries have been followed by an educational revolution that was still ongoing in the twentieth century. In 1912, Whitehead wrote: > > > We are, in fact, in the midst of an educational revolution caused by > the dying away of the classical impulse which has dominated European > thought since the time of the Renaissance. ... What I mean is the > loss of that sustained reference to classical literature for the sake > of finding in it the expression of our best thoughts on all subjects. > ... There are three fundamental changes ... Science now > enters into the very texture of our thoughts ... Again, > mechanical inventions, which are the product of science, by altering > the material possibilities of life, have transformed our industrial > system, and thus have changed the very structure of Society. Finally, > the idea of the World now means to us the whole round world of human > affairs, from the revolutions of China to those of Peru. ... The > total result of these changes is that the supreme merit of immediate > relevance to the full compass of modern life has been lost to > classical literature. (1947 [1968: 175-176]) > > > Whitehead listed the scientific and industrial revolutions as well as globalization as the major causes for the educational reforms of the nineteenth and twentieth century. These fundamental changes indeed implied new standards for what counts as genuine knowledge. However, together with these new standards emerged a romantic anxiety--the anxiety that the new standards of genuine knowledge, education, and living might impoverish human experience and damage both individual and social wellbeing. Hence arose the bifurcation of culture into the culture of "natural scientists" and the culture of "literary intellectuals" (cf. Snow 1959), and the many associated debates in the context of various educational reforms--for example, the 1880s debate in Victorian England, when Whitehead was a Cambridge student, between T. H. Huxley, an outspoken champion of science, defending the claims of modern scientific education, and Matthew Arnold, a leading man of letters, defending the claims of classical literary education. As for Whitehead, in whom the scientific and the romantic spirit merged, one cannot say that he sided with either Huxley or Arnold. He took his distance from those who, motivated by the idea that the sciences embody the ultimate modes of thought, sided with Huxley, but also from those who, motivated by conservatism, that is, by an anachronistic longing for a highly educated upper class and an elitist horror of educational democratization, sided with Arnold (cf. 1947 [1968: 23-24]). Next to not taking a stance in the debate on which is the ultimate mode of thought, the scientific or the literary, hence rejecting the antithesis between scientific and literary education, Whitehead also rejected the antithesis between thought and action (cf. 1947 [1968: 172]) and hence, between a liberal, that is, mainly intellectual and theoretical, education, and a technical, that is, mainly manual and practical, education (cf. 1929a: 66-92). In other words, according to Whitehead, we can identify three instead of two cultures but, moreover, we must refrain from promoting any one of these three at the expense of the other two. He writes: > > > My point is, that no course of study can claim any position of ideal > completeness. Nor are the omitted factors of subordinate importance. > The insistence in the Platonic culture on disinterested intellectual > appreciation is a psychological error. Action and our implication in > the transition of events amid the inevitable bond of cause to effect > are fundamental. An education which strives to divorce intellectual or > aesthetic life from these fundamental facts carries with it the > decadence of civilisation. (1929a: 73) > > > > Disinterested scientific curiosity is a passion for an ordered > intellectual vision of the connection of events. But the ... > intervention of action even in abstract science is often overlooked. > No man of science wants merely to know. He acquires knowledge to > appease his passion for discovery. He does not discover in order to > know, he knows in order to discover. The pleasure which art and > sciences can give to toil is the enjoyment which arises from > successfully directed intention. (1929a: 74) > > > > The antithesis between a technical and a liberal education is > fallacious. There can be no technical education which is not liberal, > and no liberal education which is not technical: that is, no education > which does not import both technique and intellectual vision. (1929a: > 74) > > > > There are three main methods which are required in a national system > of education, namely, the literary curriculum, the scientific > curriculum, the technical curriculum. But each of these curricula > should include the other two ... each of these sides ... > should be illuminated by the others. (1929a: 75) > > > For more details and an extensive bibliography on Whitehead's philosophy of education, cf. Part VI of Volume 1 of the Handbook of Whiteheadian Process Thought (Weber & Desmond 2008: 185-214). For some recent Whiteheadian contributions to Educating for an Ecological Civilization, see Ford & Rowe (eds.) 2017. ## 6. Metaphysics Facing mandatory retirement in London, and upon being offered an appointment at Harvard, Whitehead moved to the United States in 1924. Given his prior training in mathematics, it was sometimes joked that the first philosophy lectures he ever attended were those he himself delivered in his new role as Professor of Philosophy. As Russell comments, "In England, Whitehead was regarded only as a mathematician, and it was left to America to discover him as a philosopher" (1956: 100). A year after his arrival, he delivered Harvard's prestigious Lowell Lectures. The lectures formed the basis for Science and the Modern World (1926). The 1927/28 Gifford Lectures at the University of Edinburgh followed shortly afterwards and resulted in the publication of Whitehead's most comprehensive (but difficult to penetrate) metaphysical work, Process and Reality (1929c). And in the Preface of the third major work composing his mature metaphysical system, Adventures of Ideas (1933), Whitehead stated: > > > The three books--Science and The Modern World, Process and > Reality, Adventures of Ideas--are an endeavor to express a > way of understanding the nature of things, and to point out how that > way of understanding is illustrated by ... human experience. Each > book can be read separately; but they supplement each other's > omissions or compressions. (1933 [1967: vii]) > > > Whitehead's philosophy of science "has nothing to do with ethics or theology or the theory of aesthetics" (1922 [2004: 4]). Whitehead in his London writings was "excluding any reference to moral or aesthetic values", even though he was already aware that "the values of nature are perhaps the key to the metaphysical synthesis of existence" (1920 [1986: 5]). Whitehead's metaphysics, on the contrary, not only take into account science, but also art, morals and religion. Whitehead in his Harvard writings did not exclude anything, but aimed at a "synoptic vision" (1929c [1985: 5]) to which values are indeed the key. In his earlier philosophy of science, Whitehead revolted against the bifurcation of nature into the worlds of primary and secondary qualities, and he promoted the harmonization of the abstractions of mathematical physics with those of Hume's sensationalist empiricism, as well as the inclusion of more concrete intuitions offered by our perception--our intuitions of causality, extension, cogredience, congruence, color, sound, smell, etc. Closely linked to this completion of the scientific scheme of thought, Whitehead developed a new scientific ontology and a new theory of perception. His scientific ontology is one of internally related events (instead of merely externally related bits of matter). His theory of perception (cf. Symbolism: its Meaning and Effect) holds that our perception is always perception in the mixed mode of symbolic reference, which usually involves a symbolic reference of what is given in the pure mode of presentational immediacy to what is given in the pure mode of causal efficacy: > > > symbolic reference, though in complex human experience it works both > ways, is chiefly to be thought of as the elucidation of percepta in > the mode of causal efficacy by ... percepta in the mode of > presentational immediacy. (1929c [1985: 178]) > > > According to Whitehead, the failure to lay due emphasis on the perceptual mode of causal efficacy implies the danger of reducing the scientific method to Hume's sensationalist empiricism, and ultimately lies at the basis of the Humean failure to acknowledge the relatedness of nature, especially the causal relatedness of nature. Indeed, "the notion of causation arose because mankind lives amid experiences in the mode of causal efficacy" (1929c [1985: 175]). According to Whitehead, "symbolic reference is the interpretative element in human experience" (1929c [1985: 173]), and "the failure to lay due emphasis on symbolic reference ... has reduced the notion of 'meaning' to a mystery" (1929c [1985: 168]), and ultimately lies at the basis of Newton's failure to give meaning to his formulae of motion and gravitation. In his later metaphysics, Whitehead revolted against the bifurcation of the world into the objective world of facts (as studied by science, even a completed science, and one not limited to physics, but stretching from physics to biology to psychology) and the subjective world of values (aesthetic, ethic, and religious), and he promoted the harmonization of the abstractions of science with those of art, morals, and religion, as well as the inclusion of more concrete intuitions offered by our experience--stretching from our mathematical and physical intuitions to our poetic and mystic intuitions. Closely linked to this completion of the metaphysical scheme of thought (cf. Part I of Process and Reality), Whitehead refined his earlier ontology, and generalized his earlier theory of perception into a theory of feelings. Whitehead's ultimate ontology--the ontology of 'the philosophy of organism' or 'process philosophy'--is one of internally related organism-like elementary processes (called 'actual occasions' or 'actual entities') in terms of which he could understand both lifeless nature and nature alive, both matter and mind, both science and religion--"Philosophy", Whitehead even writes, "attains its chief importance by fusing the two, namely, religion and science, into one rational scheme of thought" (1929c [1985: 15]). His theory of feelings (cf. part III of Process and Reality) claims that not only our perception, but our experience in general is a stream of elementary processes of concrescence (growing together) of many feelings into one--"the many become one, and are increased with one" (1929c [1985: 21])--and that the process of concrescence is not primarily driven by the objective content of the feelings involved (their factuality), but by their subjective form (their valuation, cf. 1929c [1985: 240]). Whitehead's ontology cannot be disjoined from his theory of feelings. The actual occasions ontologically constituting our experience are the elementary processes of concrescence of feelings constituting the stream of our experience, and they throw light on the what and the how of all actual occasions, including those that constitute lifeless material things. This amounts to the panexperientialist claim that the intrinsically related elementary constituents of all things in the universe, from stones to human beings, are experiential. Whitehead writes: "each actual entity is a throb of experience" (1929c [1985: 190]) and "apart from the experiences of subjects there is nothing, nothing, nothing, bare nothingness" (1929c [1985: 167])--an outrageous claim according to some, even when it is made clear that panexperientialism is not the same as panpsychism, because "consciousness presupposes experience, and not experience consciousness" (1929c [1985: 53]). The relational event ontology that Whitehead developed in his London period might serve to develop a relational interpretation of quantum mechanics, such as Rovelli's (cf. supra) or one of the many proposed by Whitehead scholars (cf. Stapp 1993 and 2007, Malin 2001, Hattich 2004, Epperson 2004, Epperson & Zafiris 2013). But then this ontology has to take into account the fact that quantum mechanics suggests that reality is not only relational, but also granular (the results of measuring its changes do not form continuous spectra, but spectra of discrete quanta) and indeterminist (physicists cannot predetermine the result of a measurement; they can only calculate for each of the relevant discrete quanta, that is, for each of the possible results of the measurement, the probability that it becomes the actual result). In Whitehead's London writings, there is no "atomic structure" of events (1920 [1986: 59]). The events constituting the passage of nature, and underlying the abstraction of a continuous space-time, were continuous events. In his first academic year at Harvard, however, Whitehead changed his mind. Henceforth he conceived of the passage of nature as atomic, as constituted by atomic events. In The Emergence of Whitehead's Metaphysics: 1925-1929, Lewis Ford had already given an account of the change in Whitehead's thought from continuous to atomic becoming in the spring of 1925 (1984: 51-65). In "From Physics to Philosophy, and from Continuity to Atomicity" (Henning & Petek 2020: 132-153), Ronny Desmet confirmed Ford's account by means of a close reading of The Harvard Lectures of Alfred North Whitehead 1924-1925. Moreover, Desmet made clear that the main reason for Whitehead's change of mind was not the rise of quantum mechanics, nor the fact that the atomicity of becoming provided a solution to Zeno's paradox of becoming, nor even that it was needed to include contingency and freedom in his philosophy. The main raison was that this atomicity prevented his philosophy to contradict our immediate experience of the irreversibility of time. But no matter its reason, the introduction of the atomicity of becoming in Whitehead's philosophy implied the granularity and indeterminism of reality that are needed, next to its relationality, to develop a feasible interpretation of quantum mechanics within the frame of this philosophy. In Whitehead's Process and Reality, "the mysterious quanta of energy have made their appearance" (1929c [1985: 78]), "the ultimate metaphysical truth is atomism" (1929c [1985: 35]), and events are seen as networks (or 'societies') of elementary and atomic events, called 'actual occasions' or 'actual entities.' Whitehead writes: > > > I shall use the term 'event' in the more general sense of > a nexus of actual occasions ... An actual occasion is the > limiting type of an event with only one member. (1929c [1985: 73]) > > > Each actual occasion determines a quantum of extension--"the atomized quantum of extension correlative to the actual entity" (1929c [1985: 73])--and it is by means of the relation of extensive connection in the class of the regions constituted by these quanta that Whitehead attempted to improve upon his earlier construction of space-time (cf. Part IV of Process and Reality). The atomicity of events in quantum mechanics dovetails with the atomicity of the stream of experience as conceived by William James, hence reinforcing Whitehead's claim that each actual entity is an elementary process of experience. Whitehead writes: > > > The authority of William James can be quoted in support of this > conclusion. He writes: "Either your experience is of no content, > of no change, or it is of a perceptible amount of content or change. > Your acquaintance with reality grows literally by buds or drops of > perception. Intellectually and on reflection you can divide these into > components, but as immediately given, they come totally or not at > all". (1929c [1985: 68]) > > > Whitehead's conclusion reads: "actual entities are drops of experience, complex and interdependent" (1929c [1985: 18]), and he expresses that reality grows by drops, which together form the extensive continuum, by writing: "extensiveness becomes, but 'becoming is not itself extensive", and "there is a becoming of continuity, but no continuity of becoming" (1929c [1985: 35]). In Whitehead's London writings, he aims at logically reconstructing Einstein's STR and GTR, which are both deterministic theories of physics, and his notion of causality (that each occasion presupposes the antecedent world as active in its own nature) does not seem to leave much room for any creative self-determination. In his Harvard writings, however, Whitehead considers deterministic interaction as an abstract limit in some circumstances of the creative interaction that governs the becoming of actual entities in all circumstances, and he makes clear that his notion of causality includes both determination by the antecedent world (efficient causation of past actual occasions) and self-determination (final causation by the actual occasion in the process of becoming). Whitehead writes: > > > An actual entity is at once the product of the efficient past, and is > also, in Spinoza's phrase, causa sui. Every philosophy > recognizes, in some form or other, this factor of self-causation. > (1929a: 150) > > > Again: "Self-realization is the ultimate fact of facts. An actuality is self-realizing, and whatever is self-realizing is an actuality" (1929a: 222). Introducing indeterminism also means introducing potentiality next to actuality, and indeed, Whitehead introduces pure potentials, also called 'eternal objects,' next to actual occasions: > > > The eternal objects are the pure potentials of the universe, and the > actual entities differ from each other in their realization of > potentials. (1929c [1985: 149]) > > > Eternal objects can qualify (characterize) the objective content and the subjective form of the feelings that constitute actual entities. Eternal objects of the objective species are pure mathematical patterns: "Eternal objects of the objective species are the mathematical Platonic forms" (1929c [1985: 291]). An eternal object of the objective species can only qualify the objective content of a feeling, and "never be an element in the definiteness of a subjective form" (idem). Eternal objects of the subjective species, on the other hand, include sense data and values. > > > A member of the subjective species is, in its primary character, an > element in the definiteness of the subjective form of a feeling. It is > a determinate way in which a feeling can feel. (idem) > > > But it can also become an eternal object contributing to the definiteness of the objective content of a feeling, for example, when a smelly feeling gives rise to a feeling of that smell, or when an emotionally red feeling is felt by another feeling, and red, an element of the subjective form of the first feeling, becomes an element of the objective content of the second feeling. Whitehead's concept of self-determination cannot be disjoined from his idea that each actual entity is an elementary process of experience, and hence, according to Whitehead, it is relevant both at the lower level of indeterminist physical interactions and at the higher level of free human interactions. Indeed, each actual entity is a concrescence of feelings of the antecedent world, which do not only have objective content, but also subjective form, and as this concrescence is not only determined by the objective content (by what is felt), but also by the subjective form (by how it is felt), it is not only determined by the antecedent world that is felt, but also by how it is felt. In other words, each actual entity has to take into account its past, but that past only conditions and does not completely determine how the actual entity will take it into account, and "how an actual entity becomes constitutes what that actual entity is" (1929c [1985: 23]). How does this relate to eternal objects? How an actual entity takes into account its antecedent world involves "the realization of eternal objects [or pure potentials] in the constitution of the actual entity in question" (1929c [1985: 149]), and this is partly decided by the actual entity itself. In fact, "actuality is the decision amid potentiality" (1929c [1985: 43]). Another way of stating the same is that "the subjective form ... has the character of a valuation" and > > > according as the valuation is a 'valuation up' or 'a > valuation down,' the importance of the eternal object [or pure > potentials] is enhanced, or attenuated. (1929c [1985: > 240-241]) > > > According to Whitehead, self-determination gives rise to the probabilistic laws of science as well as human freedom. We cannot decide what the causes are of our present moment of experience, but--to a certain extent--we can decide how we take them into account. In other words, we cannot change what happens to us, but we can choose how we take it. Because our inner life is constituted not only by what we feel, but also by how we feel what we feel, not only by objective content, but also by subjective form, Whitehead argues that outer compulsion and efficient causation do not have the last word in our becoming; inner self-determination and final causation do. Whitehead completes his metaphysics by introducing God (cf. Part V of Process and Reality) as one of the elements to further understand self-determination (and that it does not result in chaos or mere repetition, but promotes order and novelty) and final causation (and that it ultimately aims at "intensity of feeling" (1929c [1985: 27]) or "depth of satisfaction" (1929c [1985: 105])). According to Whitehead: "God is the organ of novelty" and order (1929c [1985: 67]); > > > Apart from the intervention of God, there could be nothing new in the > world, and no order in the world. The course of creation would be a > dead level of ineffectiveness, with all balance and intensity > progressively excluded by the cross currents of incompatibility; > (1929c [1985: 247]) > > > and "God's purpose in the creative advance is the evocation of intensities" (1929c [1985: 105]). Actually, this last quote from Process and Reality is the equivalent of an earlier quote from Religion in the Making--"The purpose of God is the attainment of value in the world" (1926b [1996: 100])--and a later quote from Adventures of Ideas--"The teleology of the Universe is directed to the production of Beauty" (1933 [1967: 265]). Each actual occasion does not only feel its antecedent world (its past), but God as well, and it is the feeling of God which constitutes the initial aim for the actual occasion's becoming--"His [God's] tenderness is directed towards each actual occasion, as it arises" (1929c [1985: 105]). Again, however, the actual occasion is "finally responsible for the decision by which any lure for feeling is admitted to efficiency" (1929c [1985: 88]), even if that lure is divine. In other words, each actual occasion is "conditioned, though not determined, by an initial subjective aim supplied by the ground of all order and originality" (1929c [1985: 108]). For more details on Whitehead's metaphysics, cf. the books listed in section 1 as well as Emmet 1932, Johnson 1952, Eisendrath 1971, Lango 1972, Connelly 1981, Ross 1983, Ford 1984, Nobo 1986, McHenry 1992, Jones 1998, Basile 2009, Nordsieck 2015, Debaise 2017b, Dombrowski 2017, Stengers 2020, and Raud 2021. ## 7. Religion As Whitehead's process philosophy gave rise to the movement of process theology, most philosophers think that his take on religion was merely positive. This commonplace is wrong. Whitehead wrote: > > > Religion is by no means necessarily good. It may be very evil. > (1926 [1996: 17]) > > > > In considering religion, we should not be obsessed by the idea of its > necessary goodness. This is a dangerous delusion. (1926 [1996: > 18]) > > > > Indeed history, down to the present day, is a melancholy record of the > horrors which can attend religion: human sacrifice, and in particular, > the slaughter of children, cannibalism, sensual orgies, abject > superstition, hatred as between races, the maintenance of degrading > customs, hysteria, bigotry, can all be laid at its charge. Religion is > the last refuge of human savagery. The uncritical association of > religion with goodness is directly negatived by plain facts. > (1926 [1996: 37]) > > > This being said, Whitehead didn't hold that religion is merely negative. To him, religion can be "positive or negative, good or bad" (1926 [1996: 17]). So after highlighting that the necessary goodness of religion is a dangerous delusion in Religion in the Making, Whitehead abruptly adds: "The point to notice is its transcendent importance" (1926 [1996: 18]). In Science and the Modern World, Whitehead expresses this transcendent importance of religion as follows: > > > Religion is the vision of something which stands beyond, behind, and > within, the passing flux of immediate things; something which is real, > and yet waiting to be realized; something which is a remote > possibility, and yet the greatest of present facts; something that > gives meaning to all that passes, and yet eludes all apprehension; > something whose possession is the final good, and yet is beyond all > reach; something which is the ultimate ideal, and the hopeless quest. > (1925 [1967: 191-192]) > > > And after pointing out that religion is the last refuge of human savagery in Religion in the Making, Whitehead abruptly adds: "Religion can be, and has been, the main instrument for progress" (1926 [1996: 37-38]). In Science and the Modern World this message reads: > > > Religion has emerged into human experience mixed with the crudest > fantasies of barbaric imagination. Gradually, slowly, steadily the > vision recurs in history under nobler form and with clearer > expression. It is the one element in human experience which > persistently shows an upward trend. It fades and then recurs. But when > it renews its force, it recurs with an added richness and purity of > content. The fact of the religious vision, and its history of > persistent expansion, is our one ground for optimism. > (1925 [1967: 192]) > > > With respect to the relationship between science and religion, Whitehead's view clearly differs from Stephen Jay Gould's view that religion and science do not overlap. Gould wrote: > > > The lack of conflict between science and religion arises from a lack > of overlap between their respective domains of professional > expertise--science in the empirical constitution of the universe, > and religion in the search for proper ethical values and the spiritual > meaning of our lives. (1997) > > > Whitehead, on the contrary, wrote: "You cannot shelter theology from science, or science from theology" (1926 [1996: 79]). And: "The conflict between science and religion is what naturally occurs in our minds when we think of this subject" (1926 [1967: 181]). However, Whitehead did not agree with those who hold that the ideal solution of the science-religion conflict is the complete annihilation of religion. Whitehead, on the contrary, held that we should aim at the integration of science and religion, and turn the impoverishing opposition between the two into an enriching contrast. According to Whitehead, both religion and science are important, and he wrote: > > > When we consider what religion is for mankind, and what science is, it > is no exaggeration to say that the future course of history depends > upon the decision of this generation as to the relation between them. > (1925 [1967: 181]) > > > Whitehead never sided with those who, in the name of science, oppose religion with a misplaced and dehumanizing rhetoric of disenchantment, nor with those who, in the name of religion, oppose science with a misplaced and dehumanizing exaltation of existent religious dogmas, codes of behavior, institutions, rituals, etc. As Whitehead wrote: "There is the hysteria of depreciation, and there is the opposite hysteria which dehumanizes in order to exalt" (1927 [1985: 91]). Whitehead, on the contrary, urged both scientific and religious leaders to observe "the utmost toleration of variety of opinion" (1925 [1967: 187]) as well as the following advice: > > > Every age produces people with clear logical intellects, and with the > most praiseworthy grip of the importance of some sphere of human > experience, who have elaborated, or inherited, a scheme of thought > which exactly fits those experiences which claim their interest. Such > people are apt resolutely to ignore, or to explain away, all evidence > which confuses their scheme with contradictory instances. What they > cannot fit in is for them nonsense. An unflinching determination to > take the whole evidence into account is the only method of > preservation against the fluctuating extremes of fashionable opinion. > This advice seems so easy, and is in fact so difficult to follow > (1925 [1967: 187]). > > > Whitehead's advice of taking the whole evidence into account implies taking the inner life of religion into account and not only its external life: > > > Life is an internal fact for its own sake, before it is an external > fact relating itself to others. The conduct of external life is > conditioned by environment, but it receives its final quality, on > which its worth depends, from the internal life which is the > self-realization of existence. Religion is the art and the theory of > the internal life of man, so far as it depends on the man himself and > on what is permanent in the nature of things. > > > > This doctrine is the direct negation of the theory that religion is > primarily a social fact. Social facts are of great importance to > religion, because there is no such thing as absolutely independent > existence. You cannot abstract society from man; most psychology is > herd-psychology. But all collective emotions leave untouched the awful > ultimate fact, which is the human being, consciously alone with > itself, for its own sake. > > > > Religion is what the individual does with his own solitariness. > (1926 [1996: 15-16]) > > > Whitehead's advice also implies the challenge to continually reshape the outer life of religion in accord with the scientific developments, while remaining faithful to its inner life. When taking into account science, religion runs the risk of collapsing. Indeed, while reshaping its outer life, religion can only avoid implosion by remaining faithful to its inner life. "Religions commit suicide", according to Whitehead, when do they not find "their inspirations ... in the primary expressions of the intuitions of the finest types of religious lives" (1926 [1996: 144]). And he writes: > > > Religion, therefore, while in the framing of dogmas it must admit > modifications from the complete circle of our knowledge, still brings > its own contribution of immediate experience. (1926 [1996: > 79-80]) > > > On the other hand, when religion shelters itself from the complete circle of knowledge, it also faces "decay" and, Whitehead adds, "the Church will perish unless it opens its window" (1926 [1996: 146]). So there really is no alternative. But that does not render the task at hand any easier. Whitehead lists two necessary, but not sufficient, requirements for religious leaders to reshape, again and again, the outer expressions of their inner experiences: First, they should stop exaggerating the importance of the outer life of religion. Whitehead writes: > > > Collective enthusiasms, revivals, institutions, churches, rituals, > bibles, codes of behavior, are the trappings of religion, its passing > forms. They may be useful, or harmful; they may be authoritatively > ordained, or merely temporary expedients. But the end of religion is > beyond all this. (1926 [1996: 17]) > > > Secondly, they should learn from scientists how to deal with continual revision. Whitehead writes: > > > When Darwin or Einstein proclaim theories which modify our ideas, it > is a triumph for science. We do not go about saying that there is > another defeat for science, because its old ideas have been abandoned. > We know that another step of scientific insight has been gained. > > > > Religion will not regain its old power until it can face change in the > same spirit as does science. Its principles may be eternal, but the > expression of those principles requires continual development. This > evolution of religion is in the main a disengagement of its own proper > ideas in terms of the imaginative picture of the world entertained in > previous ages. Such a release from the bonds of imperfect science is > all to the good. (1925 [1967: 188-189]) > > > In this respect, Whitehead offers the following example: > > > The clash between religion and science, which has relegated the earth > to the position of a second-rate planet attached to a second-rate sun, > has been greatly to the benefit of the spirituality of religion by > dispersing [a number of] medieval fancies. (1925 [1967: 190]) > > > On the other hand, Whitehead is well aware that religion more often fails than succeeds in this respect, and he writes, for example, that both > > > Christianity and Buddhism ... have suffered from the rise of > ... science, because neither of them had ... the requisite > flexibility of adaptation. (1926 [1996: 146]) > > > If the condition of mutual tolerance is satisfied, then, according to Whitehead: "A clash of doctrines is not a disaster--it is an opportunity" (1925 [1967: 186]). In other words, if this condition is satisfied, then the clash between religion and science is an opportunity on the path toward their integration or, as Whitehead puts it: > > > The clash is a sign that there are wider truths and finer perspectives > within which a reconciliation of a deeper religion and a more subtle > science will be found. (1925 [1967: 185]) > > > According to Whitehead, the task of philosophy is "to absorb into one system all sources of experience" (1926 [1996: 149]), including the intuitions at the basis of both science and religion, and in Religion in the Making, he expresses the basic religious intuition as follows: > > > There is a quality of life which lies always beyond the mere fact of > life; and when we include the quality in the fact, there is still > omitted the quality of the quality. It is not true that the finer > quality is the direct associate of obvious happiness or obvious > pleasure. Religion is the direct apprehension that, beyond such > happiness and such pleasure remains the function of what is actual and > passing, that it contributes its quality as an immortal fact to the > order which informs the world. (1926 [1996: 80]) > > > The first aspect of this dual intuition that "our existence is more than a succession of bare facts" (idem) is that the quality or value of each of the successive occasions of life derives from a finer quality or value, which lies beyond the mere facts of life, and even beyond obvious happiness and pleasure, namely, the finer quality or value of which life is informed by God. The second aspect is that each of the successive occasions of life contributes its quality or value as an immortal fact to God. In Process and Reality, Whitehead absorbed this dual religious intuition in terms of the bipolar--primordial and consequent--nature of God. God viewed as primordial does not determine the becoming of each actual occasion, but conditions it (cf. supra--the initial subjective aim). He does not force, but tenderly persuades each actual occasion to actualize--from "the absolute wealth of potentiality" (1929c [1985: 343)--value-potentials relevant for that particular becoming. "God", according to Whitehead, "is the poet of the world, with tender patience leading it by his vision of truth, beauty, and goodness" (1929c [1985: 346]). "The ultimate evil in the temporal world", Whitehead writes, > > > lies in the fact that the past fades, that time is a "perpetual > perishing." ... In the temporal world, it is the empirical > fact that process entails loss. (1929c [1985: 340]) > > > In other words, from a merely factual point of view, "human life is a flash of occasional enjoyments lighting up a mass of pain and misery, a bagatelle of transient experience" (1925 [1967: 192]). According to Whitehead, however, this is not the whole story. On 8 April 1928, while preparing the Gifford Lectures that became Process and Reality, Whitehead wrote to Rosalind Greene: > > > I am working at my Giffords. The problem of problems which bothers me, > is the real transitoriness of things--and yet!!--I am > equally convinced that the great side of things is weaving something > ageless and immortal: something in which personalities retain the > wonder of their radiance--and the fluff sinks into utter > triviality. But I cannot express it at all--no system of words > seems up to the job. (Unpublished letter archived by the Whitehead > Research Project) > > > Whitehead's attempt to express it in Process and Reality reads: > > > There is another side to the nature of God which cannot be omitted. > ... God, as well as being primordial, is also consequent ... > God is dipolar. (1929c [1985: 345]) > > > > The consequent nature of God is his judgment on the world. He saves > the world as it passes into the immediacy of his own life. It is the > judgment of a tenderness which loses nothing that can be saved. (1929c > [1985: 346]) > > > > The consequent nature of God is the fluent world become > 'everlasting' ... in God. (1929c [1985: 347]) > > > Whitehead's dual description of God as tender persuader and tender savior reveals his affinity with "the Galilean origin of Christianity" (1929c [1985: 343]). Indeed, his > > > theistic philosophy ... does not emphasize the ruling Caesar, or > the ruthless moralist, or the unmoved mover. It dwells upon the tender > elements in the world, which slowly and in quietness operate by love. > (idem) > > > One of the major reasons why Whitehead's process philosophy is popular among theologians, and gave rise to process theology, is the fact that it helps to overcome the doctrine of an omnipotent God creating everything out of nothing. This creatio ex nihilo doctrine implies God's responsibility for everything that is evil, and also that God is the only ultimate reality. In other words, it prevents the reconciliation of divine love and human suffering as well as the reconciliation of the various religious traditions, for example, theistic Christianity and nontheistic Buddhism. In yet other words, the creatio ex nihilo doctrine is a stumbling block for theologians involved in theodicy or interreligious dialogue. Contrary to it, Whitehead's process philosophy holds that there are three ultimate (but inseparable) aspects of total reality: God (the divine actual entity), the world (the universe of all finite actual occasions), and the creativity (the twofold power to exert efficient and final causation) that God and all finite actual occasions embody. The distinction between God and creativity (that God is not the only instance of creativity) implies that there is no God with the power completely to determine the becoming of all actual occasions in the world--they are instances of creativity too. In this sense, God is not omnipotent, but can be conceived as "the fellow-sufferer who understands" (1929c [1985: 351]). Moreover, the Whiteheadian doctrine of three ultimates--the one supreme being or God, the many finite beings or the cosmos, and being itself or creativity--also implies a religious pluralism that holds that the different kinds of religious experience are (not experiences of the same ultimate reality, but) diverse modes of experiencing diverse ultimate aspects of the totality of reality. For example: > > > One of these [three ultimates], corresponding with what Whitehead > calls "creativity", has been called > "Emptiness" ("Sunyata") or > "Dharmakaya" by Buddhists, "Nirguna Brahman" > by Advaita Vedantists, "the Godhead" by Meister Eckhart, > and "Being Itself" by Heidegger and Tillich (among > others). It is the formless ultimate reality. The other > ultimate, corresponding with what Whitehead calls "God", > is not Being Itself but the Supreme Being. It is in-formed > and the source of forms (such as truth, beauty, and justice). It has > been called "Amida Buddha", "Sambhogakaya", > "Saguna Brahman", "Ishvara", > "Yaweh", "Christ", and "Allah". > (D. Griffin 2005: 47) > > > > [Some] forms of Taoism and many primal religions, including Native > American religions [...] regard the cosmos as sacred. By > recognizing the cosmos as a third ultimate, we are able to see that > these cosmic religions are also oriented toward something truly > ultimate in the nature of things. (D. Griffin 2005: 49) > > > The religious pluralism implication of Whitehead's doctrine of three ultimates has been drawn most clearly by John Cobb. In "John Cobb's Whiteheadian Complementary Pluralism", David Griffin writes: > > > Cobb's view that the totality of reality contains three > ultimates, along with the recognition that a particular tradition > could concentrate on one, two, or even all three of them, gives us a > basis for understanding a wide variety of religious experiences as > genuine responses to something that is really there to be experienced. > "When we understand global religious experience and thought in > this way", Cobb emphasizes, "it is easier to view the > contributions of diverse traditions as complementary". (D. > Griffin 2005: 51) > > > ## 8. Whitehead's Influence Whitehead's key philosophical concept--the internal relatedness of occasions of experience--distanced him from the idols of logical positivism. Indeed, his reliance on our intuition of the extensive relatedness of events (and hence, of the space-time metric) was at variance with both Poincare's conventionalism and Einstein's interpretation of relativity: his reliance on our intuition of the causal relatedness of events, and of both the efficient and the final aspects of causation, was an insult to the anti-metaphysical dogmas of Hume and Russell; his method of causal explanation was also an antipode of Ernst Mach's method of economic description; his philosophical affinity with James and Bergson as well as his endeavor to harmonize science and religion made him liable to the Russellian charge of anti-intellectualism; and his genuine modesty and aversion to public controversy made him invisible at the philosophical firmament dominated by the brilliance of Ludwig Wittgenstein. At first--because of Whitehead's Principia Mathematica collaboration with Russell as well as his application of mathematical logic to abstract the basic concepts of physics--logical positivists and analytic philosophers admired Whitehead. But when Whitehead published Science and the Modern World, the difference between Whitehead's thought and theirs became obvious, and they grew progressively more dissatisfied over the direction in which Whitehead was moving. Susan Stebbing of the Cambridge school of analysis is only one of many examples that could be evoked here (cf. Chapman 2013: 43-49), and in order to find a more positive reception of Whitehead's philosophical work, one has to turn to opponents of analytic philosophy such as Robin George Collingwood (for example, Collingwood 1945). The differences with logical positivism and analytic philosophy, however, should not lead philosophers to neglect the affinities of Whitehead's thought with these philosophical currents (cf. Shields 2003, Desmet & Weber 2010, Desmet & Rusu 2012, Riffert 2012). Despite signs of interest in Whitehead by a number of famous philosophers--for example, Hannah Arendt, Maurice Merleau-Ponty and Gilles Deleuze--it is fair to say that Whitehead's process philosophy would most likely have entered oblivion if the Chicago Divinity School and the Claremont School of Theology had not shown a major interest in it. In other words, not philosophers but theologians saved Whitehead's process philosophy from oblivion. For example, Charles Hartshorne, who taught at the University of Chicago from 1928 to 1955, where he was a dominant intellectual force in the Divinity School, has been instrumental in highlighting the importance of Whitehead's process philosophy, which dovetailed with his own, largely independently developed thought. Hartshorne wrote: > > > The century which produced some terrible things produced a scientist > scarcely second in genius and character to any that ever lived, > Einstein, and a philosopher who, I incline to say, is similarly second > to none, unless it be Plato. To make no use of genius of this order is > hardly wise; for it is indeed a rarity. A mathematician sensitive to > so many of the values in our culture, so imaginative and inventive in > his thinking, so eager to learn from the great minds of the past and > the present, so free from any narrow partisanship, religious or > irreligious, is one person in hundreds of millions. He can be > mistaken, but even his mistakes may be more instructive than most > other writers' truth. (2010: 30) > > > After mentioning a number of other theologians next to Hartshorne as part of "the first wave of ... impressive Whitehead-inspired scholars", Michel Weber--in his Introduction to the two-volume Handbook of Whiteheadian Process Thought--writes: > > > In the sixties emerged John B. Cobb, Jr. and Shubert M. Ogden. > Cobb's Christian Natural Theology remains a landmark in > the field. The journal Process Studies was created in 1971 by > Cobb and Lewis S. Ford; the Center for Process Studies was > established in 1973 by Cobb and David Ray Griffin in Claremont. The > result of these developments was that Whiteheadian process scholarship > has acquired, and kept, a fair visibility ... (Weber & > Desmond 2008: 25) > > > Indeed, inspired mainly by Cobb and Griffin, many other centers, societies, associations, projects and conferences of Whiteheadian process scholarship have seen the light of day all over the world. An important society is the European Society for Process Thought with its annual summer schools and its book series, European Studies in Process Thought (nine titles). Another active society is the Deutsche Whitehead Gesellschaft. Thirty six of today's most active process centers, however, are located in the People's Republic of China. They were founded between 2002 and 2018 by The Institute for Postmodern Development of China, which also founded the Cobb Eco-Academy in 2018, in Zhejiang. The Beijing Whitehead International Kindergarten is just one example of Chinese involvement with Whitehead. Out of Belgium, Michel Weber has created in 2001 the Whitehead Psychology Nexus and the Chromatiques whiteheadiennes scholarly societies, and he has been the driving force behind several book series, including the series published by Weber's own publishing company, Les editions Chromatika (forty four titles at present), and the Process Thought Series published by Ontos Verlag / De Gruyter (twenty seven titles). The already mentioned Handbook of Whiteheadian Process Thought is part of the latter book series. In it, 101 internationally renowned Whitehead scholars give an impressive overview of the 2008 status of their research findings in an enormous variety of domains (cf. Armour 2010). Missing in the Handbook, however, are most Whitehead scholars reading Whitehead through Deleuzian glasses--especially Isabelle Stengers, whose 2011 book, Thinking with Whitehead, cannot be ignored. The Lure of Whitehead, edited by Nicholas Gaskill and A. J. Nocek in 2014, can largely remedy that shortcoming. Important for Whitehead scholarship, is the fact that the Handbook is in the process of being transformed into an online Whitehead encyclopedia, which will contain additional entries, and of which the entries will be kept up to date. This is being done by the Whitehead Research Project, that is, by the people responsible for editing and publishing the book series, Contemporary Whitehead Studies (fifteen titles), and The Edinburgh Critical Edition of the Complete Works of Alfred North Whitehead, of which two volumes of The Harvard Lectures of Alfred North Whitehead have already seen the light of day. The Whitehead Research Project website also refers to a Whitehead Reading Group, contains a Research Blog, and provides access to the Whitehead Research Library, which holds a wealth of mostly unpublished materials, such as letters from and to Whitehead (cf. Whitehead Research Project [OIR]). The oldest Whiteheadian book series is the SUNY Series in Constructive Postmodern Thought (thirty two titles), and the two most recent series are published by the Process Century Press (cf. Process Century Press [OIR]). The most recent title in its Theological Exploration Series (nine titles) is James & Whitehead on Life after Death by David Griffin (2022), and in the Series Preface of its successful Toward Ecological Civilization Series (twenty two titles), John Cobb writes: > > > We live in the ending of an age. But the ending of the modern period > differs from the ending of previous periods, such as the classical or > the medieval. The amazing achievements of modernity make it possible, > even likely, that its end will also be the end of civilization, of > many species, or even of the human species. At the same time, we are > living in an age of new beginnings that give promise of an ecological > civilization. Its emergence is marked by a growing sense of urgency > and deepening awareness that the changes must go to the roots of what > has led to the current threat of catastrophe. > > > > In June 2015, the 10th Whitehead International Conference was held in > Claremont, CA. Called "Seizing an Alternative: Toward an > Ecological Civilization", it claimed an organic, relational, > integrated, nondual, and processive conceptuality is needed, and that > Alfred North Whitehead provides this in a remarkably comprehensive and > rigorous way. We proposed that he could be "the philosopher of > ecological civilization". With the help of those who have come > to an ecological vision in other ways, the conference explored this > Whiteheadian alternative, showing how it can provide the shared vision > so urgently needed. > > > Cobb refers to the tenth International Whitehead Conference (IWC). The IWCs are sponsored by the International Process Network. The IWC has been held at locations around the globe since 1981--the 12th one was in 2019, in Brasilia, Brazil, and the 13th one, "Whitehead and the History of Philosophy," will be in July 2023, in Munich, Germany. The IWC is an important venture in global Whiteheadian thought, as key Whiteheadian scholars from a variety of disciplines and countries come together for the continued pursuit of critically engaging a process worldview. Finally, in 2019, the Cobb Institute--A Community for Process & Practice was formed, which has, among other interesting initiatives, weekly John Cobb & Friends online gatherings to hear presentations from prominent scholars across the world. Recently, for example, Jeremy Lent gave a presentation on his award winning bestseller, The Web of Meaning--Integrating science and traditional wisdom to find our place in the universe (2021). The Cobb & Friends Gatherings YouTube Channel contains a sample of recordings of these presentations, among which the Lent presentation.
william-auvergne	## 1. Life As with so many important medieval figures, we know little about William's early years. According to one manuscript source, he was born in Aurillac, in the province of Auvergne in south-central France. His birth date is unknown, but since he was a professor of theology at the University of Paris by 1225, a position rarely attained before the age of 35, he is not likely to have been born later than 1190, and scholars have placed his probable birth date some time between 1180 and 1190. He may have come from a poor background, as the Dominican, Stephen of Bourbon (died c. 1261), tells a story of William begging as a young child (Valois 1880, 4). He was a canon of Notre Dame and a master of theology by 1223, and is mentioned in bulls of Pope Honorius III in 1224 and 1225. The story of his elevation to the episcopacy in 1228 paints a picture of a man of great determination and self-confidence. The previous year the Bishop of Paris, Bartholomeus, had died, and the canons of the chapter of Notre Dame met to select his successor. The initial selection of a cantor named Nicholas did not secure unanimous agreement and was contested, in particular, by William. Nicholas excused himself, and the canons went on to choose the Dean of the Cathedral. William again contested the election and went to Rome to appeal to the Pope to vacate it. He made a favorable impression, for the Pope, impressed by his "eminent knowledge and spotless virtue," as he put it (Valois 1880, 11), both ordained him priest and made him Bishop of Paris, a position he retained until his death in 1249. It was not long before the Pope would have regrets. In February 1229 a number of students were killed by the forces of the queen-regent, Blanche of Castile, when they intervened in a drunken student riot during Carnival. Outraged, the masters and students appealed to William for redress of their rights, but William failed to take action. The students and masters went on strike, dispersing from Paris and appealing to Pope Gregory IX. It was apparently during this period that William gave the Dominicans their first chair in theology at the university. The Pope, who was to remark that he regretted "having made this man," rebuked William, appointed a commission to settle the dispute, and ordered William to reinstate the striking masters. Nevertheless, William went on to receive important missions from the Pope in subsequent years, acting, for example, as a papal representative in peace negotiations between France and England in 1231. In 1239 William was closely involved in the condemnation of the Talmud. The Pope had asked him for his response to a list of heresies in the Talmud proposed by a converted Jew, Nicholas. William's response led to a papal bull ordering the confiscation of sacred books from synagogues in 1240, and to their burning in 1242 (for details, see Valois 1880, Smith 1992 and de Mayo 2007). William died at the end of March (the exact date is unsure) in 1249, and was buried in the Abbey of St. Victor. ## 2. Works William's most important philosophical writings form part of a vast seven-part work he calls the Magisterium divinale et sapientiale, a title Teske translates as Teaching on God in the Mode of Wisdom. The Magisterium is generally taken to consist of the following seven works in the order given: * De Trinitate, sive de primo principio (On the Trinity, or the First Principle) * De universo (On the Universe) * De anima (On the Soul) * Cur Deus homo (Why God Became Man) * De fide et legibus (On Faith and Laws) * De sacramentis (On the Sacraments) * De virtutibus et moribus (On Virtues and Morals). William did not compose the parts of the Magisterium in their rational order, and appears to have developed his conception of its structure as he wrote its various parts. He viewed it as having two principal parts. The first part is comprised of De Trinitate, De universo, and De anima. In this, the most philosophical part of the Magisterium, William proceeds by "the paths of proofs" or philosophizing rather than by an appeal to the authority of revelation. He intends to combat a range of philosophical errors incompatible, as he sees it, with the Christian faith. In On the Trinity he develops his metaphysics, attacking, among others, errors regarding creation, divine freedom and the eternity of the world, and follows it with a philosophical treatment of the Trinity. William's vast work, On the Universe, is divided into two principal parts. The first part (Ia, i.e. prima pars) is concerned with the corporeal universe and is in turn divided into three parts, the first (Ia-Iae, i.e. prima pars primae partis) arguing for a single first principle and for the oneness of the universe, the second (IIa-Iae) taking up the question of the beginning of the universe and its future state, and the third (IIIa-Iae) treating the governance of the universe through God's providence. The second principal part (IIa) is concerned with the spiritual universe, and is divided into three parts concerning, respectively, the intelligences or spiritual beings posited by Aristotle and his followers (Ia-IIae), the good and holy angels (IIa-IIae), and the bad and wicked angels, that is, demons (IIIa-IIae). On the Soul is concerned with the human soul. It is divided into seven parts concerning, respectively, its existence, its essence, its composition, the number of souls in a human being, how the human soul comes into being, the state of the soul in the body, and the soul's relation to God, with a focus on the human intellect. The works comprising the second part of the Magisterium are more expressly theological in nature and appeal to the authority of revelation. Nevertheless, William continually recurs to his philosophical doctrines in all his works, and even his most theological writings contain material of considerable philosophical interest. In particular, his treatises On Virtues and Morals and On Faith and Laws are of great importance for his moral philosophy. Scholars agree that On the Trinity is the earliest work in the Magisterium and that it was probably written in the early 1220s. The other works can be partially dated by references to contemporaneous events, though the sheer size of many individual works suggests they were probably written over a period of years. The whole Magisterium itself was probably written during a period extending from the early 1220s to around 1240, with On the Universe written in the 1230s and On the Soul completed by around 1240. Besides the works contained in the Magisterium, other works of philosophical importance are De bono et malo I (On Good and Evil), where William develops a theory of value; De immortalitate animae (On the Immortality of the Soul), a companion to the treatment of immortality in On the Soul; and De gratia et libero arbitrio (On Grace and Free Choice), a treatment of the Pelagian heresy. In addition, William wrote some biblical commentaries, a number of other major theological works, and a large number of sermons (for details, see Ottman 2005). The more than 550 sermons that can be attributed to William have recently been critically edited by F. Morenzoni; they comprise four volumes in the Corpus Christianorum series (Opera homiletica, CCCM 230-230C). With the exception of the sermons, most of William's works have not been critically edited; several are extant only in manuscript form. The standard edition, which includes all the works that comprise the Magisterium, is the 1674 Orleans-Paris Opera omnia (OO). (The sermons contained in this edition are by William Perrauld, not William of Auvergne.) As might be expected, the texts in this old edition are often in need of correction; Teske has conjectured many corrections in his translations. ## 3. The Character of William's Philosophical Works The reader of William's philosophical works is struck by their difference in character from the mainstream of philosophical and theological writing of the early thirteenth century. Whereas William's contemporaries tend to compose works as a series of linked questions, each treated according to the question method (a paradigm of which is Aquinas's later Summa theologiae), William instead follows the practice of Avicenna and Avicebron and composes treatises much more akin to a modern book. Indeed, William even models his writing style to some degree on the Latin translations of these authors. These features, together with William's frequent use of analogies, metaphors and examples drawn from everyday life, his long-winded, rambling prose and frequent digressions, his harsh assessments of opponents as imbeciles or morons, and his urging that proponents of dangerous doctrines be wiped out by fire and sword, make for an inimitable and immediately recognizable style. One of the costs of this style, however, is that at times it can be hard to tell precisely what William thinks. Too often he lets metaphors or analogies bear the argumentative burden in lieu of more careful analysis. And while William's inventive mind could generate arguments for a position with ease, their quality is uneven: often he proposes problematic arguments without appearing to recognize the difficulties they contain, as may be apparent below to the discerning reader, while at other times he shows an acute awareness of the fundamental problems an issue raises. ## 4. Sources William was one of the first thinkers in the Latin West to begin to engage seriously with Aristotle's writings on metaphysics and natural philosophy and with the thought of Islamic and Jewish thinkers, especially Avicenna (Ibn Sina, 980-1037) and Avicebron (Solomon Ibn Gabirol, 1021/2-1057/8) (see e.g. Bertolacci 2012, Caster 1996c, de Vaux 1934 and Teske 1999). The works of these thinkers, which had been coming into circulation in Latin translations from around the middle of the twelfth century, presented both philosophical sophistication and theological danger. Indeed, in 1210, public and private lecturing on Aristotle's works on natural philosophy and commentaries on them was forbidden at Paris, and in 1215 the papal legate, Cardinal Robert of Courcon forbade "masters in arts from 'reading,' i.e., lecturing, on Aristotle's books on natural philosophy along with the Metaphysics and Summae of the same (probably certain works of Avicenna and perhaps of Alfarabi)" (Wippel 2003, 66). This prohibition remained in effect until it unofficially lapsed around 1231. Nevertheless, it must be emphasized that personal study of these texts had not been forbidden, and William clearly worked hard to master them. He came to be keenly aware of the incompatibility of much of the teaching of these works with Christian doctrine, yet at the same time he found them to be a source of philosophical inspiration, and his thought brims with ideas drawn from Avicenna and Avicebron. His attitude to these thinkers is expressed well in a passage in On the Soul, where, regarding Aristotle, he writes that: > > though on many points one must contradict Aristotle, as is really > right and proper--and this holds for all the statements by which > he contradicts the truth--he should be accepted in all those > statements in which he is found to have held the right view. (OO II > suppl., 82a; Teske 2000, 89) > Of course, William was also strongly influenced by Christian thinkers. In particular, his writings are permeated by the thought of Saint Augustine, whom William refers to as "one of the more noble of Christian thinkers," if perhaps in a less obvious way than by the thought of Avicenna. Indeed, as the list in Valois 1880, 198-206, attests, William employs a remarkable range of sources and must be reckoned one of the most well-read thinkers of his day. The focus of the following outline is William's views on metaphysics and the soul, but even within these limits the vast range of his thought has required that many issues in these areas not be treated. In a future supplement, William's views on value theory and morality will be addressed. Teske's English translations are quoted whenever possible; otherwise, translations are my own (i.e. N. Lewis's). ## 5. Metaphysics William is the first thinker in the Latin West to develop a systematic metaphysics (as ontology) based on the concepts of being (esse) and essence (essentia). Influenced by Boethius and Avicenna, he develops this metaphysics in his early work, On the Trinity, as a preliminary to his account of the Trinity, and he returns to it in other works, especially in On the Universe. The central theme of his metaphysics is that all existing things, besides God, are composites of essence, on the one hand, and their being or existence, on the other, which is said to be acquired or partaken from God, the source of all being. God, in contrast, involves no composition of essence and being, but is his being. Inspired by Boethius's distinction in De hebdomadibus between what is good by participation and what is good by substance, William works these ideas out in terms of a distinction between a being by participation (ens participatione) and a being by substance or essence (ens substantia / ens per essentiam). By the essence of a thing William means what is signified by its definition or species name. Under the influence of Avicenna, he often refers to essence as a thing's quiddity (quidditas, literally, "whatness"), which he identifies with Boethius's notion of quod est or "that which is." Analogously, he identifies a thing's being (esse) or existence with the notion of quo est or "by which it is." In the case of a being by participation, its essence and features incidental to the essence are distinct from its being; William describes its being as clothed by them or as what would be left if they were stripped away. Such a being (ens) is said to participate in or acquire its being (esse). At times William, following Avicenna, speaks of its being as accidental to it, by which he just emphasizes that its being is not part of the essence. A being (ens) by substance, in contrast, does not participate in or acquire being but is its very being: "there is also the being whose essence is for it being (esse) and whose essence we predicate when we say, 'It is,' so that it itself and its being (esse) ... are one thing in every way" (Switalski 1976, 17; Teske and Wade 1989, 65). In the second chapter of On the Trinity, William also notes that the term "being" (esse) may be used to mean a thing's essence (similar to Avicenna's esse proprium), but this sense of "being" plays a minor role in his thought. William argues that there must be a being by essence; otherwise, no being could be intelligible. Thus, if there were just beings by participation, there would have to be either a circle of beings each of which participates in the being of what is prior to it in the circle, with the absurd consequence that a thing would ultimately participate in its own being; or else there would have to be an infinite series of beings, each member of which participates in the being of what is prior in the series. But then, for a member of the series--say, A--to be, would be for it to have or participate in the being of B; and for B to be, in turn, would be for it to have or participate in the being of C, and so on. But if this were so, the being of no member of the series could be intelligible, since any attempt to spell it out would result in an account of the form: A's having B's having C's having ad infinitum. Since being is intelligible, William concludes that there is a being by essence. Besides the distinctions between being by participation and being by essence, William mentions sixteen other distinctions, including the distinctions between being of need (esse indigentiae) and being of sufficiency (esse sufficientiae), possible being (esse possibile) and being necessary through itself (esse necesse per se ipsum), false and true being, and flowing and lasting being. William takes these distinctions to be coextensive with the distinction between being by participation and being by essence. Using a form of argument drawn from Avicenna, he argues that given the first member of each distinction, we can prove the existence of the second (see On the Trinity, ch. 6). The distinction between possible being (possibile esse) and necessary being through itself (necesse esse per se ipsum), which William draws from Avicenna, plays a prominent role in William's thought. William's notion of a possible is not that of something that exists in some possible world, but is rather the notion of something whose essence neither requires nor rules out its having being. William treats such possibles as in some sense prior to being. In the case of those that have being and thus actually exist, he holds that the possible "and its being (esse), which does not belong to it essentially, are really two. The one comes to the other and does not fall within its definition or quiddity. Being (ens) [i.e., a being] in this way is, therefore, composite and also resolvable into its possibility, or quiddity, and its being (esse)" (Switalski 1976, 44; Teske and Wade 1989, 87). Some commentators see in such remarks the idea that a real or mind-independent distinction obtains between being and essence in possible beings. In contrast, the being and essence of a being necessary through itself are not two; such a being does not participate in being but is its being and cannot but exist. William argues there can be only one being by substance or being necessary through itself. Speaking from a metaphysical point of view, he follows Avicenna and calls it the First (primum). Speaking as a Christian, he identifies it with God, the creator, as he thinks is indicated by Biblical references to God as "he who is": "being," he says, is a proper name of God (see e.g. On the Trinity, ch. 4). The uniqueness of a being by substance is, William argues, a consequence of its absolute simplicity, since nothing could differentiate two or more absolutely simple beings. The absolute simplicity of a being by substance, in turn, is a consequence of the fact that it cannot be caused (not, at least by an external causation; William argues that the being by substance is the Christian Trinity of persons and he admits a kind of internal causation in the Trinity). For everything that is caused is other than its being and thus is a being by participation, whereas a being by essence or substance is its being. But a composite or non-simple being must have an external cause, and must therefore be a being by participation. Therefore, the being by substance, that is to say, God, cannot be composite, but must be absolutely simple. God's absolute simplicity also entails that he is neither a universal (i.e., neither a genus nor species), nor an individual falling under such. For, William argues, genera and species are themselves composite, and the individuals that fall under them have a quiddity or definition and therefore compositeness. Nevertheless, William thinks God is an individual, since he holds that all that exists is either a universal or an individual (OO I, 855ab). William appeals to God's power to show God has will and knowledge. He argues that God must be omnipotent: he cannot be forced or impeded and has a "two-way" power (potentia), identified by William with what Aristotle calls a rational power. This is a power that "extends to both opposites, namely, to make and not to make" (Switalski 1976, 57; Teske and Wade 1989, 99), in contrast with something, such as fire, which when "it encounters what can be heated is not able to heat or not to heat . . . but it necessarily has only the power to heat" (Switalski 1976, 54; Teske and Wade 1989, 97). However, because a two-way power is poised between acting and not-acting, it must be inclined to acting by something, and William thinks this can only be through a choice. Therefore, God must have will and choice; and since "it is impossible that there be will and choice where there is no knowledge" (Switalski 1976, 59; Teske and Wade 1989, 101), God must also have knowledge. William goes on to argue that God must indeed be omniscient, since "his wisdom is absolute and free, not bound in any way to things, nor dependent upon them" (Switalski 1976, 61; Teske and Wade 1989, 102). William takes his claim that God has a two-way power to imply that God could have known or willed otherwise than he did. But, as William realizes, this raises the difficulty that if God had known or willed otherwise, it would seem that he would have been different, and this seems to be impossible given his absolute simplicity. In response, William frequently emphasizes that by the very same act of knowing or will by which God in fact knows or wills one thing, he could have known or willed something else (OO I, 780a; Teske 2007, 105). Thus, when we say "God knows or wills X," we must distinguish the act of knowing or willing, which is identical with God and utterly the same in every possible situation, from the object X to which it is related. ## 6. God's Relation to Creatures ### 6.1 God as the Source of Being According to William, everything participates in or acquires its being from God. But just what he means by this is less than clear. At times, as in the following passage from On the Universe, he seems to suggest that God is literally present in created things as their being: > > the creator is next to and most present to each of his creatures; in > fact, he is most interior to each of them. And this can be seen by you > through the subtraction or stripping away of all the accidental and > substantial conditions and forms. For when you subtract from each of > the creatures all these, last of all there is found being or entity, > and on this account its giver. (OO I, 625b; Teske 1998a, 100) > Yet William realizes that such statements may give an incorrect impression of his thought. Thus, at one point in On the Universe he writes that "every substance includes, if it is proper to say so (si dici fas est), the creator's essence within itself" (OO I, 920b), and he goes on to note that it is difficult to state clearly the sense in which the creator is in creatures. In other passages he emphasizes God's transcendence of creatures: "the first being (esse) is for all things the being (esse) by which they are ... It is one essence, pure, solitary, separate from and unmixed with all things." Often he appeals to an analogy with light: the first being fills "all things like light cast over the universe. By this filling or outpouring it makes all things reflect it. This is their being (esse), namely to reflect it" (Switalski 1976, 45; Teske and Wade 1989, 89). William perhaps intends in such remarks to exploit the medieval view that a point of light (lux) and the light emitted from that point (lumen), though distinct, nonetheless are also in some deep sense the same. ### 6.2 Against the Manichees Regarding the procession of creatures from God, William is concerned to attack a range of errors. In On the Universe Ia-Iae he attacks the error of the Manichees. Though he refers to Mani, the third-century originator of the Manichee sect, he seems chiefly concerned with the Cathars of his day. The Manichees deny God is the ultimate source of all things and instead posit two first principles, one good and one evil. Only in this way, they think, can we explain the presence of both good and evil in the world. William thinks a fundamental motivation for this view is the principle that "from one of two contraries the other cannot come of itself" (OO I, 602a; Teske 1998a, 54), and hence that evil cannot come from good but must stem from its own first principle. He replies that although one contrary cannot of itself (per se) be from the other, it is perfectly possible for it to be from the other incidentally (per accidens), and he gives everyday examples to illustrate. Thus, "drunkenness comes from wine. In that case it is evident that an evil comes from a good, unless one would say that wine is not good" (OO I, 602a; Teske 1998a, 54). Likewise, there is a sense in which evil is ultimately from God, though incidentally and not as something God aims at or intends as such. The evil of sin, for example, is a consequence of God's creating creatures with free will and permitting their misuse of it. William offers a host of arguments against the possibility of two first principles. On metaphysical grounds, for example, he objects that a first principle must be a being necessary through itself, and that he has shown that a plurality of such beings is impossible. And he gives arguments to the effect that the notion of a creative first evil principle is incoherent. Such a being, for example, could not do good for anything, and hence could not have created anything, for in doing so it would have done good for what it created (see also Bernstein 2005 and Teske 1993a). ### 6.3 Causation, and Creation by Intermediaries William claims "the creator alone properly and truly is worthy of the title 'cause,' but other things are merely messengers and bearers of the last things received as if sent from the creator" (OO I, 622a; Teske 1998a, 92). We speak of creatures as causes, but William says this is "ad sensum" (Switalski 1976, 79; Teske and Wade 1989, 117), that is, in respect of how things appear in sense experience. Both medieval and contemporary commentators have on the basis of such remarks attributed to William the denial that creatures are genuine causes (see Reilly 1953). Certainly he denies that any creature can exercise fully independent causal agency, a view he thinks "the philosophers" have adopted. But this claim is compatible with admission of genuine secondary causes that are reliant for their exercise of causation on the simultaneous activity of prior causes, and ultimately on that of the first cause, God. Is William simply making this point but using the term "cause" in a more restricted manner than his contemporaries? A negative answer is suggested by William's rationale for his usage. He thinks of causation as the giving of being, and he thinks that anything that receives being, and therefore all creatures, cannot itself give being but can merely transmit it. Moreover, his metaphors of creatures as riverbeds or windows through which the divine causal influence flows strongly suggest that he really does wish to hold that creatures themselves do not really bring anything about but are simply the conduits through which divine causality flows (see Miller 1998 and 2002). Whether in fact William admits genuine causal agency among creatures, he clearly thinks that God alone can create from nothing. Thus, he rejects Avicenna's so-called "emanationist" doctrine of creation, according to which, as he puts it, God creates from nothing only one thing, the first intelligence (or spiritual being), which in turn creates the next intelligence, and so on until the tenth or agent intelligence is reached, the creator of souls and the sensible world. William thinks one ground of this doctrine is Avicenna's principle that "from what is one in so far as it is one there cannot in any way come anything but what is one" (ex uno, secundum quod unum, non potest esse ullo modorum nisi unum) (OO I, 618b; Teske 1998a, 82), and hence from God, who is absolutely one, can at most come one thing. In fact, William agrees with this principle and applies it in the context of Trinitarian theology, but he thinks Avicenna has misapplied it by using it to explain creation. Instead, William follows the Jewish thinker Avicebron (whom he thought was a Christian) and holds that creatures come from God not insofar as he is one, but "through his will and insofar as he wills, just as a potter does not shape clay vessels through his oneness, but through his will" (OO I, 624a; Teske 1998a, 96). ### 6.4 Against the Necessity of Creation William thinks the will is necessarily free; thus the fact that God creates through his will means that creation is an exercise of God's free will. This point serves to undermine another aspect of Avicenna's doctrine of the procession of creatures from God. For Avicenna posits not only a series of creators, but also holds that this procession is utterly necessary: the actual world must exist and could not have been other than it is. William thinks this conclusion stems from a failure to grasp God's freedom in creation. Avicenna and others imposed "not only necessity, but natural servitude upon the creator, supposing that he operates in the manner of nature" (OO I, 614b; Teske 1998a, 72). Thus, they think the universe issued from God as brightness issues from the sun or heat from fire, and that God no more had it in his power to do otherwise than do the sun or heat. Against this William argues that it is a consequence of God's free creation that he had it in his power to create otherwise than he did. ### 6.5 Against the Eternity of the World One of the most contentious doctrines presented by Greek and Islamic philosophy to thinkers in the early thirteenth century is that of the eternity of the world, the view that the world had no beginning but existed over an infinite past. This view seems to contradict the opening words of the Bible: "In the beginning God created heaven and earth." In response, a number of thinkers in the early thirteenth century, including the Franciscan Alexander of Hales, argued that Aristotle, at least, had been misunderstood and had never proposed the eternity of the world. But William, like his Oxford contemporary, Robert Grosseteste, would have none of this: "Whatever may be said and whoever may try to excuse Aristotle, it was undoubtedly his opinion that the world is eternal and that it did not begin to be, and he held the same view concerning motion. Avicenna held this after him" (OO I, 690b; Teske 1998a, 117). The importance of this issue to William is indicated by the space he devotes to it. His treatment of the question of the eternity aeternitas of the world in On the Universe IIa-Iae is one of the longest discussions of the issue in the middle ages, and the first substantial treatment to be made in the light of Greek and Islamic thought. William thinks, in fact, that to call the world eternal in the aforementioned sense is not to use the term "eternal" in its strictest sense. Accordingly, he starts by explaining different senses of the term in the first extended treatment of the distinction between eternity and time in the middle ages. He then presents and refutes arguments that the world has no beginning, and provides his own positive arguments that the world must have a temporal beginning, a first instant of its existence. William's chief aim in his treatment of time and eternity is to emphasize the fundamental differences between eternity and time. He is especially concerned to attack a conception of eternity as existence at every time, a conception he attributes to Aristotle. In his view, eternity in a strict sense is proper to the creator alone and is his very being, though "the name 'eternity' says more than his being, namely, the privations of beginning and ending, as well as of flux and change, and this both in act and in potency" (OO I, 685b-686a; Teske 1998a, 109). The fundamental difference between time and eternity is that "as it is proper and essential to time to flow or cease to be, so it is proper and essential to eternity to remain and stand still" (OO I, 683b; Teske 1998a, 102). William, like other medieval thinkers, tends to describe eternity in terms of what it is not. It has nothing that flows and hence no before or after or parts. Its being is whole at once, not because it all exists at one time, but because it all exists with nothing temporally before or after it. William speaks of duration in eternity, but he holds that the term has a different (unfortunately unexplained) sense in this context than when applied to time. William had read Aristotle's Physics and agrees with Aristotle that time and motion are coextensive (OO I, 700a). Yet he does not propose Aristotle's definition of time as the number of motion in respect of before and after. Rather, in his account of the essential nature of time he describes time simply as being that flows and does not last, "that is, it has nothing of itself that lasts in act or potency" (OO I, 683a; Teske 1998a, 102). Echoing Aristotle and Augustine, he holds that time has the weakest being, and that "of all things that are said to be in any way, time is the most remote from eternity, and this is because it only touches eternity by the very least of itself, the now itself or a point of time" (OO I, 748a). William introduces his account of the eternity of the world in On the Universe by reference to Aristotle, but he devotes the bulk of his discussion to Avicenna's arguments for this doctrine. In parallel material in On the Trinity he claims that at the root of Avicenna's view lies the principle that "if the one essence is now as it was before when nothing came forth from it, something will not now come forth from it" (Switalski 1976, 69; Teske and Wade 1989, 108). That is, nothing new is caused to exist unless there is a change in the cause. Thus God, a changeless being, cannot start to create the world after not creating it; and therefore he must have created it from eternity and without a beginning. Therefore, the world must itself lack a beginning. William replies that Avicenna's principle leads to an infinite regress; for if something's operating after not operating requires a change in it, this change will itself require a new cause, and that new cause will itself require a new cause, and so on ad infinitum. William instead holds that it is possible that God, without changing in any way, start to operate after not operating. What William means by this is that the effect of God's eternal will is a world with a temporal beginning. In general, he holds that verbs such as "create" express nothing in the creator himself but rather something in things. We might put "God begins to create the world" to have the import of: "God by an eternal willing creates the world, and the world has a beginning." William emphasizes that God does not will that the world exist without qualification, but rather that it exist with a given temporal structure, namely, a beginning (Switalski 1976, 67-68; Teske and Wade 1989, 107). This might suggest that he thinks God might in fact have created a world without a beginning, but chose not to do so, as Aquinas would hold. But in fact, William argues that this option was not open to God: a world without a beginning is simply not a creatable item. William attempts to establish this point by a host of arguments, many of which are found in his writings for the first time in the Latin West and repeated by later thinkers such as Bonaventure and Henry of Ghent. On the whole, these arguments either allege that the notion of a created thing itself entails that what is created has a temporal beginning or point to alleged paradoxes stemming from the supposition of an infinite past. For example, William argues (OO I, 696b; Teske 1998a, 133) that because the world in itself is merely a possible, in itself it is in non-being and non-being is therefore natural to it. Hence its non-being, being natural to it, is prior to its being, and therefore in being created it must receive being after non-being, and thus have a beginning. Among arguments involving infinity, William argues that if the whole of past time is infinite then the infinite must have been traversed to arrive at the present; but since it is impossible to traverse the infinite (OO I, 697b; Teske 1998a, 136), the past must be finite. This line of argument is frequently found in later thinkers. William also presents arguments that the view that a continuum, such as time, is infinite results in paradoxes (OO I, 698a-700b). These arguments, which William develops at some length, constitute an important early medieval engagement with puzzles relating to infinity. ### 6.6 Providence Part IIIa-Iae of On the Universe is a lengthy treatise on God's providence (providentia). William believes the denial of God's providence, and in particular the denial that the good will be rewarded and the evil punished, is an error "so harmful and so pernicious for human beings that it eliminates from human beings by their roots all concern for moral goodness, all the honor of the virtues, and all hope of future happiness" (OO I, 776a; Teske 2007, 92). According to William, God's providence (providentia) or providing for the universe differs from foreknowledge (praescientia) in that, unlike foreknowledge, it embraces only goods and not also evils (see OO I, 754b; Teske 2007, 30). William is especially concerned to argue that God's providence is not just general in nature, but extends to all individual creatures. God's care for the universe and its constituents is a matter of God's knowing and paying attention to all things. From God's point of view, nothing happens by chance. William introduces final causality into his explanation by describing the universe as a teleological or goal-directed order, in which each thing has a function or purpose established by God. God's care is his concern that each thing fulfills its appointed function. Thus, God is not necessarily concerned with what we might think is best for individual things: flies get eaten by spiders, but this in fact is an end for which they were created. And ultimately animals were created for the sake of humans, as William points out (for details on animals and providence, see Wei 2020, 49-74). In the case of humans, however, who are the apex of creation, their end is to experience happiness in union with God, and God's providential care of human beings is aimed at this. But if this is so, why do the evil prosper and the good suffer? What are we to make of natural disasters, of pain, suffering, death, and the like? What, in short, are we to make of evil in the world? Although William does not take up the logical compatibility of evil with an all-good, knowing and powerful god, he does attempt to explain at length how a range of particular kinds of evil are compatible with God's providence and care for human beings. His explanation, in general, is not novel and has roots in Augustine. He distinguishes the evils perpetrated by free agents and for which they are therefore to blame from other forms of evil, and argues that these other evils are in fact not really evil but good. Pain, for example, provides many benefits, among which is that it mortifies and extinguishes "the lusts for pleasures and carnal and worldly desires" (OO I, 764a; Teske 2007, 58). In many cases, what we call evil is in fact God's just punishment, which William implies always has beneficial effects. As for the blameworthy evils done by free agents, William thinks that God permits these, but that it is incorrect to say they come from God or are intended by him. They are rather the consequence of rational creatures' misuse of the free will with which they were created, and the moral balance will be set right in the future life, if not this one. If the evils of this life appear to conflict with God's providence, God's providence and foreknowledge themselves seem to entail that everything occurs of necessity and hence that there is no free will. This concern leads William to consider a number of doctrines that aim to show that all events happen of necessity. In addition to foreknowledge or providence, he considers the views that all events are the necessary causal outcomes of the motion and configuration of the heavenly bodies, or that all events result from the heimarmene, i.e. the "interconnected and incessantly running series of causes and the necessary dependence of one upon another" (OO I, 785a; Teske 2007, 120); or finally, that all things happen through fate (fatum sive fatatio). William rejects all these grounds for the necessity of events (esp. on fate, see Sannino 2016). In an interesting discussion of foreknowledge in part IIIa-Iae, chapter 15, he considers the argument that since it is necessary that whatever God foresees will come about will come about, because otherwise God could be deceived or mistaken, then since God foresees each thing that will come about, it is necessary that it will come about. William replies that the proposition "It is necessary that whatever God foresees (providit) will come about will come about" has two interpretations, "because the necessity can refer to individuals or to the whole" (OO I, 778b; Teske 2007, 99). The argument for the necessity of all events equivocates between these two readings and is therefore invalid. William notes that the distinction he makes is close to the distinction between a divided and composite sense, but he is critical of those who instead distinguish an ambiguity between a de re (about the thing) and de dicto (about what is said) reading. ## 7. The Soul William's treatise On the Soul is one of the most substantial Latin works on the topic from before the middle of the thirteenth century. Although William begins this treatise with Aristotle's definition of the soul as the perfection of an organic body potentially having life, his conception of the soul is heavily influenced by Avicenna and decidedly Platonic in character. He interprets Aristotle's definition to mean that the body is an instrument of the soul (as he thinks the term "organic" indicates) and that "a body potentially having life" must refer to a corpse, since a body prior to death actually has life. And although he calls a soul, with Aristotle, a form, unlike Aristotle he holds that a soul is in fact a substance, something individual, singular and a "this something" (hoc aliquid), not at all dependent on the body for its existence. William adopts the standard view that souls serve to vivify or give life to bodies, and thus he posits not just souls of human beings (i.e., rational souls), but also those of plants and animals (vegetative and sensitive souls respectively), as they all have living bodies. But in the case of the human soul, William also identifies the soul with the human being, the referent of the pronoun "I." ### 7.1 The Existence and Substantiality of Souls Among William's many arguments for the existence of souls is the argument that bodies are instruments of souls and the power to carry out operations through an instrument cannot belong to the instrument itself. Thus, there must be something other than the bodies of living things that operates by means of bodies, and this is their soul. In the case of the human or rational soul, he argues that its existence follows in particular from the fact that understanding and knowledge are found in a human being but not in the whole body or any part of it. Since these are operations of a living substance, there must be a living incorporeal substance in which they are, and this is what people mean when they speak of their soul. These arguments serve not only to show the existence of souls, but also that they are substances. In the case of the human soul, William argues, the operations of understanding and knowing must be treated as proper operations of the soul, and this requires that the soul itself be a substance. To the objection that the soul itself does not literally understand or know but is instead that thanks to which a human being can understand and know, William will argue that the soul and human being, which he equates with the referent of the pronoun "I," are one and the same thing. More generally, the substantiality of souls--and not just of human souls--is indicated by the fact that bodies are their instruments. ### 7.2 The Incorporeality of Souls William argues at length that a soul cannot be a body, a conclusion he applies not just to human or rational souls, but also to those of plants and animals. In the case of human souls, he uses, among others, Avicenna's "flying-man" argument to establish this point, arguing that a human being flying in the air who lacks and never had the use of his senses would know that he exists but deny that he has a body. Therefore "it is necessary that he have being that does not belong to the body and, for this reason, it is necessary that the soul not be a body" (OO II suppl., 83a; Teske 2000, 91) (For details, see Hasse 2000). More generally, William argues that no body could perform the soul's function of giving life to the body. For example, if a body were to give life, it would itself have to be a living body, and thus it would itself have to have a soul that rendered it alive. This soul, in turn, would have to be a living body and thus have its own soul, and so on ad infinitum. Thus, the doctrine that souls are bodies leads to an infinite regress. ### 7.3 The Human Soul and the Human Being Perhaps the most striking aspect of William's teaching about the soul is his identification of the human soul with the human being and his denial that the human body constitutes in whole or part the human being. Instead, William describes the body variously as a house, the soul its inhabitant; as a prison, the soul its captive; as a cloak, the soul its wearer, and so on. Nevertheless, the definition of a human being--he perhaps has in mind the Platonic conception of the human being as "a soul using a body"--involves a reference to the body, and it this, William thinks, that has led people to believe that the body is a part of a human being. But this is an error akin to thinking that because a horseman is defined by reference to a horse, the horse must therefore be a part of the horseman. William is aware, of course, that at times we do seem to ascribe to ourselves, and thus to the human being, the acts of the body or things that happen to the body, but he holds that these are non-literal modes of speaking, akin to a rich man's complaint that he has been struck by hail, when in fact it was his vineyard that was struck. William, as usual, offers numerous arguments for the identification of the soul and human being. He argues that the body cannot be a part of a human being, since the soul and body no more produce a true unity than, for example, a carpenter and his ax, and a human being is a true unity. And he notes that since the soul is subject to reward and punishment in the afterlife for actions we correctly attribute to the human being or ourselves, it must be identified with the human being. He also points to cases where our use of names or demonstratives referring to human beings must be understood as referring to their souls, and concludes that this shows that human beings are their souls. ### 7.4 The Simplicity of the Soul According to William, souls are among the simplest of created substances. Nevertheless, like all created substances, they must involve some kind of composition. This composition cannot be a bodily composition, of course, since they are not bodies and thus cannot be divided into bodily parts. According to Avicebron, as well as some of William's contemporaries, souls are composite in the sense that they are composites of form and matter, though not a matter that involves physical dimensions, but rather what some called "spiritual matter." This view is part of the doctrine now termed "universal hylomorphism," according to which every created substance is a composite of form and matter (see Weisheipl 1979). William is one of the first to reject this doctrine; he is later followed by Aquinas. In support of his rejection, in On the Universe he makes one of his rare references to Averroes, citing with approval his claim that prime matter is a potentiality only of perceptible substances and hence is not found in spiritual substances. He also argues that there is no need to posit matter in angels or souls in order to explain their receptivity in cognition, as some had thought necessary (OO I, 851b-852b). Rather, souls and other spiritual substances are "pure immaterial forms" without matter. William also denies a real plurality of powers in the soul. He holds instead that each power of the soul is identical with the soul and not a part of it. When we speak of a power of the soul, we are really speaking of the soul considered as the source or cause of a certain kind of operation. Thus, the power to understand characteristic of a human or rational soul is the soul considered as a source or cause of the operation of understanding. What sort of composition, then, is to be found in souls and other spiritual beings? William holds that every being other than God > > is in a certain sense composed of that which is (quod est) > and of that by which it is (quo est) or its being or entity > ... since being or entity accrues and comes to each thing apart > from its completed substance and account. The exception is the first > principle, to which alone [being] is essential and one with it in the > ultimate degree of unity. (OO I, 852a) > Here William uses Boethius's language of quod est and quo est to make the point that in souls, as in all created substances, there is, to use Avicennian terms, a composition of being and essence, a doctrine to be developed by Aquinas. ### 7.5 Denial of a Plurality of Souls in the Human Body Besides rejecting universal hylomorphism, William also rejects the doctrine of a plurality of souls in a human being. This doctrine, which was popular among Franciscan thinkers, holds that the body of a human being is informed not just by a single soul, the rational soul, but also by a vegetative and a sensitive soul. This multiplicity of souls, it is held, serves to explain how human beings have the vital operations characteristic of plants and animals in addition to peculiarly human ones, and also explains the development of the fetus's vital functions. Against this doctrine, William claims to have shown earlier in On the Soul that not only the operation of understanding, but also the sensitive operations of seeing and hearing "are essential and proper to the soul itself" (OO II suppl., 108b; Teske 2000, 164) and "hence, it cannot be denied except through insanity that one and the same soul carries out such operations." Rather than posit a multiplicity of souls, William holds that higher souls incorporate the kinds of operations attributed to lower ones. The rational soul is infused into the body directly by God when the fetus reaches a suitable stage of organization, and the prior animal or sensitive soul in the fetus ceases to be. In this case too a view along similar lines was to be developed by Aquinas. Scholars have noted, however, that while William rejects a plurality of souls in the human body, he does subscribe to a version of the doctrine of a plurality of substantial forms in the sense that he treats the human body as itself a substance independent of its association with the soul, and hence having its own corporeal form, while also being informed by the human or rational soul, which renders it a living thing. (For details see Bazan 1969.) ### 7.6 The Immortality of the Soul According to William, the immortality of the human soul would be evident if not for the fact that the soul has, as it were, been put to sleep by the corruption of the body stemming from the punishment for sin. In our current state, however, William thinks the human soul's immortality can be shown by means of philosophical arguments, to which he devotes considerable attention, both in On the Soul and in a shorter work, On the Immortality of the Soul. He believes the error that the soul is naturally mortal "destroys the foundation of morality and of all religion" (Bulow 1897, 1; Teske 1991, 23), since those who believe in the mortality of the soul will have no motive for acting morally and honoring God. Thus, he notes that if the soul is not immortal, the honor of God is pointless in this life, since in the present life it "involves much torment and affliction for the soul" (Bulow 1897, 3; Teske 1991, 25) and receives no reward. From a metaphysical point of view, William argues that the soul does not depend on the body for its being, and therefore the destruction of the body does not entail the non-existence of the soul. The rational soul's independence from the body is due to the fact it is a self-subsistent substance whose proper operations do not involve or require a human body. Since William takes the souls of plants and animals to be incorporeal substances, it might be thought that he would treat them as immortal too. But William holds that these souls cease to exist upon the death of the plant or animal. This is because all their proper operations, unlike those of the rational soul, depend on the body, and thus there would be no point for their continued existence after the destruction of the body. ### 7.7 The Powers Characteristic of the Human Soul Characteristic of human souls, or human beings, are the intellective and motive powers, that is to say, the intellect and will. William thinks that of these, the will is by far the more noble power, and he is accordingly puzzled by the fact that neither Aristotle nor Avicenna have much to say about it. These authors do, however, discuss the intellective power at great length, and William accordingly devotes considerable attention to combating errors he sees in their teachings on the intellect. #### 7.7.1 The Motive Power: the Will William develops his account of the will using the metaphor of a king in his kingdom. The will "holds the position in the whole human being and in the human soul of an emperor and king," while "the intellective or rational power holds the place and function of counselor in the kingdom of the human soul" (OO II suppl. 95a; Teske 2000, 126, 129). William treats acts of will--expressed in the indicative mood by "I will" (volo) or "I refuse" (nolo)--as a kind of command that cannot be disobeyed, as distinguished from mere likings expressed in the form "I would like" (vellem) or "I would rather not' (nollem). He holds that as a king must understand in order to make use of his counselors' advice, so the will must not simply be an appetitive power but must also have its own capacity to apprehend. Likewise, the intellect must have its own desires. But he notes in On Virtues, that like a king the will need not heed the advice of its counselor; it may instead "give itself over to the advice of its slaves, that is, the inner powers and the senses"; in so doing it "gives itself over into becoming a slave of its slaves and is like a king ... who follows the will and advice of senseless children" (OO I, 122a). William frequently emphasizes the difference between voluntary action and the actions of brute animals. Animals operate in a "servile" manner, in the sense that they respond to their passions without having control over these responses. A human being, in contrast, has power over the suggestion of its desires; its will is "most free and is in every way in its own power and dominion" (OO II suppl., 94a; Teske 2000, 123). This is why human beings, unlike animals, can be imputed with blame or merit for their deeds. According to William, the freedom of the will consists in the fact that the will "can neither be forced in any way to its proper and first operation, which is to will or refuse, nor prevented from the same; and I mean that it is not possible for someone to will something and entirely refuse that he will it or to refuse to will something, entirely willing that he will it" (OO I, 957aA). In other words, William defines force and prevention of the will in terms of higher-order volitions and refusals: for the will to be forced is for it to refuse to will X and nonetheless be made to will X, and for it to be prevented is for it to will to will X and be made to refuse X. In a number of works, William argues that neither force nor prevention of the will is possible: if someone refuses to will X, this must be because he takes X to be bad and thus he must refuse X rather than will it; and if someone wills to will X, this must be because he takes X to be good, and thus he must will X rather than refuse it. #### 7.7.2 The Intellective Power William's discussions of the intellective power are driven by a concern to attack errors he sees in his contemporaries and Greek and Islamic thinkers. While his reasons for rejecting these errors are clear enough, if at times based on a confused understanding of his sources, his own views on the nature of cognition are less well developed. Yet it is clear that, with the exception of our knowledge of first principles and the concepts they involve and certain cases of supernaturally revealed knowledge, William wishes to present a naturalistic account of cognition in this life in which God plays no special role. In both On the Soul and On the Universe William attacks at length theories of cognition that posit an agent intellect (intellectus agens) or agent intelligence (intelligentia agens). He incorrectly ascribes the latter theory to Aristotle, although in fact it is Avicenna's teaching; he ascribes the former theory to certain unnamed contemporaries. These theories view cognition as involving the active impression of intelligible signs or forms--what we might call concepts--on a receptive or passive recipient. The agent intellect or agent intelligence is taken to perform this function of impressing intelligible forms on our material intellect, which is so-called because like matter it is receptive of forms, albeit intelligible forms. The chief difference between the agent intellect and agent intelligence, as far as the theory of cognition is concerned, is that the former is treated as a part of the human soul, while the latter is taken to be a spiritual substance apart from the human soul, identified with Avicenna's tenth intelligence. William notes how the agent intellect or agent intelligence is viewed by analogy with light or the sun; as light or the sun through its light serves to render actual merely potentially existing colors, so the agent intellect or intelligence serves to render actual intelligible forms existing potentially in the material intellect. William as usual offers a large number of objections to these theories. He attacks as inappropriate the analogies with light or the sun, and argues that the theory of the agent and material intellects is incompatible with the simplicity of the soul, as it posits these intellects as two distinct parts of the soul, one active and one receptive. In the case of both the agent intellect and the agent intelligence, he argues that their proponents will be forced to hold the absurd result that human beings know everything that is naturally knowable, whereas in fact we must study and learn and observe in order to acquire much of our knowledge. As for his own views on the intellective power, William holds that, like all the powers of the soul, the intellective power in the present life of misery has been corrupted. Therefore an account of it must distinguish its operation in its pure, uncorrupted state from its operation in the present life. In its pure state it is capable of knowing, without reliance on the senses, everything naturally knowable, even sensible things, and thus presumably also is capable of possessing the concepts in terms of which such knowledge is formulated. In its present state, however, most of its knowledge and repertoire of concepts stems in some manner from sense experience. The knowledge of which the intellective power is naturally capable extends not just to universals, but also to singulars. For the intellect is naturally directed at the true as the will is at the good, and has as its end knowledge of the first truth, the creator, who is singular. William also notes our knowledge of ourselves and the importance of singular knowledge for our dealings with the world. Unfortunately, he provides no theory of the nature of singular cognition, and this issue must be left on the agenda until taken up by thinkers such as Scotus and Ockham in the late thirteenth and early fourteenth centuries. William proposes a representationalist theory of cognition. He holds that it is not things or states of affairs themselves that are in the intellective power but rather signs or intelligible forms that serve to represent them. He takes these signs to be mental habits, that is to say, not bare potentialities but potentialities most ready to be actualized. Propositions are those complexes formed from signs through which states of affairs are represented to the mind. Although William speaks of these signs as likenesses, he is well aware of the difficulty in thinking of concepts as such, and he concludes that in fact there need not be a likeness between a sign and that of which it is a sign (OO II suppl. 214b; Teske 2000, 454). A key question William confronts is how we acquire concepts and knowledge in the present life. In On the Soul he draws an important distinction. In the case of the first rules of truth and morality, he speaks of God as a mirror or book of all truths in which human beings naturally and without an intermediary, and thus without reliance on sense experience, read these rules. Likewise, God impresses or inscribes on our intellective power the intelligible signs in terms of which these first rules or principles are formulated. Thus, William posits at least some innate concepts; he notes, for example, in On Virtues and Morals that the concept of the true (veri) is innate (OO I, 124a). God also reveals to some prophets in this life "hidden objects of knowledge to which the created intellect cannot attain except by the gift and grace of divine revelation" (OO II suppl., 211b; Teske 2000, 445). Does William therefore hold, as some have held (Gilson 1926), that at least in the case of these rules God in some special manner illuminates the human intellect? This might be suggested by his references to the intellective power as spiritual vision and his frequent use of the language of vision and illumination in his accounts of cognition. And yet, according to William, the first rules of truth and morality are "known through themselves" or self-evident. They are "lights in themselves ... visible through themselves without the help of something else" (OO II suppl., 210b; Teske 2000, 443), and they serve to illuminate to the intellect the conclusions drawn from them. God's role, it would seem, is to impress upon or supply to our intellect, in some way, these principles, but William gives no indication that he takes God to shed a special light on these principles in order that they may be known: once possessed they are, as it were, self illuminating. The first rules of truth and morality and the concepts in terms of which they are formulated form only a small subset, however, of our cognitive repertoire. The remainder of our concepts and knowledge, leaving to one side the special divine revelation made to prophets, stems in some way from our dealings with the sensible world. But the problem William faces is to explain how this is so. The problem, he says, is that "sensation ... brings to the intellect sensible substances and intellectual ones united to bodies [i.e., souls]. But it does not imprint [pingit] upon it their intelligible forms, because it does not receive such forms of them" (OO II suppl., 213a; Teske 2000, 449). William's response to this problem is to hold that the senses have the role simply of stimulating the intellective power in such a way that it "forms on its own and in itself intelligible forms for itself" (OO I, 914b), a doctrine he draws from Augustine. According to this account, the intellect is fundamentally active in the acquisition of intelligible forms. William speaks of the intellect as able, under the prompting of the senses, to consider the substances that underlie sensible accidents, and speaks of the intellect as "abstracting" from sense experience in the sense that it forms concepts of so-called "vague individuals" by stripping away perceptible features that serve to distinguish one individual from another. In these cases, William says, "the intellect is occasionally inscribed by these forms that are more separate and more appropriate to its nature" (OO II suppl., 213b; Teske 2000, 450). A third cognitive process William mentions, conjunction or connection, is concerned not with the acquisition of intelligible forms, but rather with how knowledge of one thing brings with it knowledge of another. In particular, William argues that knowledge of a cause brings with it knowledge of the effect. William is particularly concerned, however, that his account of the intellect's generation of intelligible forms will require a division of the intellect into active and passive parts--the very distinction between an agent and material or receptive intellect he has devoted so much energy to attacking. The need to make this division of the intellect into two part is because it is impossible for the same thing to be both active and passive in the same respect, and yet in treating the intellect as indivisible it looks as though William is in fact committed to its being both active and passive in the same respect in the formation and reception of intelligible forms. William's attempts to resolve this problem are repeated in a number of works and couched in highly metaphorical language that poses severe problems of interpretation. The intellective power is both "a riverbed and fountain of the scientific and sapiential waters ... Hence, in one respect it overflows and shines forth on itself, and in another respect it receives such outpouring or radiance" (OO II, suppl. 216b; Teske 2000, 457). William holds that because on this account the intellective power is not active and passive in the same respect, he is not forced to divide the intellect into two distinct parts, one active and one passive. ## 8. William's Influence Scholars have devoted little attention to William's influence on later writers. Nevertheless, it is clear that he gave rise to no school of thought, and subsequent thinkers seem to have picked and chosen from his works the parts they would accept and the parts they would reject. Thus, William's arguments that the world must have a beginning probably influenced thinkers such as Bonaventure and Henry of Ghent, while his rejection of universal hylomorphism and his metaphysics of being and essence may have influenced Aquinas, who presents similar views. There is also little doubt that some of William's views were the subject of criticism, as for example his apparent denial of genuine secondary causation (see Reilly 1953). The large body of manuscripts in which William's works survive suggests he was being read throughout the middle ages; Ottman 2005, for example, lists 44 manuscripts known to contain On the Universe. William's works were also published in a number of printed editions in the 16th and 17th centuries.
william-champeaux	## 1. Life and Works History is written by the winners. Most of what we know of William of Champeaux's life and work has been refracted down to us through the prism of a man who hated him. Peter Abelard lost almost every battle with William, his teacher and political enemy, yet he tells us that William was a discredited, defeated, jealous, and resentful man. Abelard claims to have humiliated William in debate, driving him from the Paris schools. He alleges that, in defeat, William cast himself in the role of monastic reformer only to advance his political career by an unearned reputation for piety. Still, even Abelard recognized that William was no fraud, calling him "first in reputation and in fact," and relocating to study under his tutelage (HC; trans. Radice 1974). On the scholarly front, Abelard presents only half the story. He brags of forcing William to abandon a firmly-held realist theory of universals, but, rather than come over to Abelard's vocalist or nominalist cause, William developed a second, more sophisticated, realist view. So what might have appeared as an expression of intellectual honesty and academic rigor on William's part, Abelard presents as somehow shameful. What Abelard leaves out is how much he learned from William. Now that more of William's work has become available, it is clear that William prodded Abelard to give up naive vocalism and develop the complex semantic theory we now associate with Abelard's nominalism. The extent of William's and Abelard's involvement on opposite sides of the major political struggles of the day is also beginning to come to light. William was a man of considerable influence in the monastic reform movement. Abelard was a man of considerably less influence allied with the opposite faction. It is probably true that Abelard attracted the better students and so precipitated William's move from Paris to St. Victor. But far from retiring in shame, William became a statesman, ambassador, and confidant to the Pope, all the while retaining enough political clout to prevent Abelard from either holding his former chair as canon of Notre Dame or establishing a school in Paris. One recent biographer of Abelard points out that in 1119, while Abelard was recovering from his castration and was soon to be facing charges of heresy, William was a bishop acting as papal negotiator to the court of Emperor Henry V. Unfortunately for William's reputation, it was Abelard whose writings captured the imagination of readers in later centuries (Clanchy 1997: 296ff). Very little is known about William's early life and education. The date usually assigned for his birth is 1070, but this may be off by as much as a decade. In 1094 leaders of the monastic reform movement appointed William as Master of Notre Dame. If this date is correct, he would have been only 24 at the time, and this was a crucial moment for the reformers. In 1094 King Philip I had been excommunicated for his "illegal marriage," and any person appointed to be Master at Notre Dame would have to have been an established scholar with considerable political influence, something not usually found in 24 year-olds. Thus, a date of 1060 or earlier may not be unlikely for William's birth (Iwakuma forthcoming c). Little is known of William's education before he took this important post. He is known to have studied with Anselm of Laon and a certain Manegold, but it is unclear whether this was the theologian, Manegold of Lautenbach, or the grammarian, Manegold of Chartres. It is even suspected that William studied with the vocalist Roscelin of Compiegne. Early in the first decade of the twelfth century, probably in 1104, William became archdeacon of Paris. In 1108 the reformist party received a setback when King Luis VI, King Philip's son by his "illegal marriage" ascended the throne. William resigned as archdeacon to move to the abbey of St. Victor and the Paris suburbs, where he continued to teach and remained an influential reformer. Both Hugh of St. Victor and Bernard of Clairvaux were among William's proteges from this period at St. Victor. In 1113, he was appointed bishop of Chalons-sur-Marne. He continued to act as papal legate and negotiator and remained influential in the reform movement, particularly through his patronage of Bernard of Clairvaux. His continued scholarly reputation is demonstrated in the somewhat ludicrous tale of Rupert of Duetz, who, in 1117, set himself on a quest to challenge William of Champeaux and Anselm of Laon to intellectual combat (Clanchy 1997: 143). On January 18, 1122, William took the habit of a Cistercian and died at Clairvaux eight days later. William's corpus of writings is not widely available and a great deal more textual work needs to be done before any assessment can be made about his broader philosophical significance. Nevertheless, his basic philosophical commitments and some of his characteristic views are known. The texts we do have indicate that he started his career as a realist in matters of logic and metaphysics and that his commitment to realism grew stronger with the appearance of the vocalists and early nominalists. Thanks to Abelard's criticisms, William is best known for his realist theories of universals. In response to the vocalist challenge, he seems to have held something like the Cratylus theory of language: the nature of words is intimately tied to the nature of the things they name. In logic, some vocalists claimed that the force of inference arises from words or ideas, but William argued that this is to be found instead in some thing or relation between things, which he identified as the locus or medium of argument. He interpreted Aristotle's Categories as ten general things, not ten general words as the vocalists maintained. These and other claims suggest a broadly realist philosophical outlook. It is possible that fragments and excerpts of William's work passed down to us by his near contemporaries are those in which he was debating with the vocalists, and so we do not know whether they represent William's deepest commitments or only brief forays into an important debate. William's views are known mostly through references in contemporary works, especially those of Abelard. Though several of William's works have been identified, very few have been edited, and none has been translated into English. The Introductions -- Introductiones dialecticae secundum Wilgelmum (ISW) and Introductiones dialecticae secundum magistrum G. Paganellum (IGP) -- are very early works, possibly dating from William's arrival in Paris in 1094. Edited with these two Introductions are two brief discussions of media or loci, and the beginning of an early commentary on Porphyry, all likely by William (Iwakuma 1993; ISW is also in de Rijk 1967). These texts defend some of William's characteristic doctrines, but in 1094 the vocalist movement had not yet begun to exert any real influence in Paris, and so it is likely that he had not yet been forced to articulate his views in response to serious criticism. Some of William's later works have also been identified. Although no editions have been published, several articles contain extensive excerpts. William is known to have written commentaries on Porphyry's Isagoge, Aristotle's Categories and De Interpretatione, Boethius' De Differentiis Topicis, and Cicero's De Inventione and Rhetorica ad Herennium. Some of his theological writings, along with those of other leading French theologians, were compiled in the later twelfth century under the title Liber Pancrisis. These have been edited as the 'Sententiae' of William of Champeaux (Lottin 1956). The Liber Pancrisis is thought to be a compilation of the best Parisian theology of the period, and so it is likely to represent William's most advanced views. ## 2. Universals The history of philosophy remembers William for his two theories of universals, material essence realism and indifference realism. These theories emerge from the writings of Peter Abelard, where they are paired with Abelard's decisive arguments against them. Indeed, Abelard's critique was so powerful that when John of Salisbury wrote his famous catalogue of twelfth-century theories of universals, William is not even mentioned (Metalogicon II 17-20; Abelard LI Por.: 10-16, trans. Spade 1994: 29-37; see also John of Salisbury, Peter Abelard, and King 2004) Material essence realism proposes that there are ten most general things or essences: one most general thing corresponding to each of Aristotle's ten categories. These essences are universal things: > It should be seen that there are ten common things which are > the foundations of all other things and are called the most general > things - as for example this common thing, substance, which is > dispersed through all substances, and this thing, quantity, which is > dispersed in all quantities and so on. And just as there are ten common > things which are the foundations of all other things, so also there are > ten words which, thanks to the things they signify, are said to be the > foundations of all other words (C8; Marenbon 1997: 38). These ten genera exist and are to some degree unformed. They are formed into subalternate genera and species by the addition of differentia. Species are formed into individuals by the addition of accidental forms: "a species is nothing other than a formed genus, an individual nothing other than a formed species" (P3; Marenbon 2004: 33; Iwakuma 2004: 309; Fredborg 1977: 35). This is how the view gets its name. The genus exists as the matter. The difference forms the genus "into a sub-altern genus which in turn becomes the matter for an inferior sub-altern genus or species. The addition of accidental forms divides the species into discrete individuals" [et fiant res discretae in actu rerum] (C8; Iwakuma 1999: 103). The individuals in a species or genus thus share a single material essence. Everything in the created world is an accidentally differentiated individual, but this is an incidental feature of the created world, and by no means a commitment to concretism. Universal essences exist; they are simply never found except as accidentally differentiated, qua individuals. > In actuality genera and species have their being in > individual things. I can, however, consider by reason the same thing > which is individuated with its accidents removed from its make-up, and > consider the pure simple thing, and the thing considered in this way is > the same as that which is in the individual. And so I understand it as > a universal. For it does not go against nature for it to be a > pure thing if it were to happen that all its accidents were removed. > But because it will never happen in actuality that any thing exists > without accidents, so neither in actuality will that pure universal > thing be found. (P3; Marenbon 2004: 33) This mental exercise of stripping away forms does not merely produce a universal concept. It reveals the underlying metaphysical reality. There is a real universal thing corresponding to our universal concepts. It is this principle of a single universal substance individuated by accidents that Abelard reduces to absurdity. When faced with Abelard's arguments against material essence realism, William gave up his belief in universal essences but refused to accept that universals are simply words or concepts. Indifference realism rejects the core principle of material essence realism by rejecting the notion that there are shared essences and holding that individuals are completely discrete from one another. The words 'one' and 'same' are ambiguous, William says: "when I say Plato and Socrates are the same I might attribute identity of wholly the same essence or I might simply mean that they do not differ in some relevant respect." William's newfound ambiguity is the seed of his second theory of universals. The stronger sense of 'one' and 'same' applies to Peter/Simon and Saul/Paul (we would say Cicero/Tully), who "are one and the same according to identity" (Sen 236.123). As for Plato and Socrates: > We call them the same in that they are men [in hoc quod > sunt homines], ['same'] pertaining with regard to > humanity. Just as one is rational, so is the other; just as one is > mortal, so is the other. But if we wanted to make a true confession, it > is not the same humanity in each one, but a similar [humanity], > since they are two men. (Sen 236.115-120) So, although Plato and Socrates have no common material--matter, form, or universal essence--they are still said to be the same because they do not differ. This leads to the claim that Abelard finds so disturbing. Each individual is both universal and particular: > > Those things which per se are considered many and wholly > diverse in essence are one considered in general or specific nature. > That is, they do not differ in being man (esse > hominem). > > > > > One Man is many men, taken particularly. Those which are one > considered in a special nature are many considered particularly. That > is to say, without accidents they are considered one per > indifference, with accidents many. > > > > > It should never be said that many men make one Man. Rather it should > be said that many men agree in being, in what it is to be a man > (in esse hoc quod est esse homo). Nevertheless, they are > wholly diverse in essence (P14; Iwakuma 1999: 119). > > > Indifference realism is not a complete departure from material essence realism because William still accepts accidental individuation. When the accidents are stripped away, Plato and Socrates are still one and the same although in the weaker sense of 'one' and 'same'. They do not share a material essence, though they each have the same state or status of being a man. William's indifference realism holds that when individuating accidents are stripped away from two individuals, what you are left with may be numerically distinct but it is not individually discernible (you can't tell which one is Socrates). What is left then are pure things--there are no individuating characteristics. In this sense, each thing is itself a universal. ## 3. Logic and Philosophy of Language With so few of William's works identified and even fewer properly edited and published, any serious discussion of his views on logic is years away. What follows is a brief presentation of William's thoughts on various issues. In many cases, these are gleaned from references in other texts, mostly by Abelard. These references indicate that William was a wide-ranging and serious thinker, but much as with the Pre-Socratics, we have just seen just the tip of the iceberg as far as his actual writings are concerned. ### 3.1 Signification Logic is the art of discerning truth from falsehood and of making and judging arguments (ISW I 1.1; IGP I 1.1). The study of logic must therefore begin with the study of words. The Introductions, ISW and IGP, first define sounds, significant sounds, words, phrases (orationes), and sentences, then proceed to detailed discussion of complex hypothetical and categorical sentences and syllogisms. William's approach here became a model for twelfth-century logic textbooks. William accepts the standard medieval definition of signification: a sound is significant if it generates an understanding in the mind of a hearer. Judging by William's discussion, there was some debate at the end of the eleventh century as to whether signification required that a spoken word actually, as opposed to merely potentially, generate an understanding in the mind of a hearer. Actual signification is too strong a criterion, but every sound potentially generates an understanding, even if only in the mind of the person uttering it. William argues that once a word is imposed--and a convention established--the word is significant because it is apt to signify whenever it is uttered (Iwakuma 1999 p109; forthcoming b). William's views on the conditions for imposing words are not yet fully known. The vocalists seem to have held that absent a linguistic convention, there is no connection between any sound and what it is imposed to signify, a view shared by Abelard (see Guilfoy 2002). On the other hand William held that "there is such an affinity between words and things that words draw their properties from things and so the nature of words is shown more clearly through the nature of things" (P14; Marenbon 1996: 6). William clearly invites controversy by claiming that the only significant sounds, that is, the only words, are those that are imposed to name presently existing things: > > > That word is significant which is imposed on an existing thing, like > 'Man'; that word is not significant which is imposed on a > non-existing thing, like 'chimaera', 'blictrix', and > 'hircocervus' (IGP I 2.2). > > > > A significant word is one whose significatum is found among > existing things (ISW I 1.4). > > > To claim that 'chimaera' is the equivalent to 'blictrix' implies that 'chimaera' is equally meaningless. The view that significant words must name existing things presents obvious and difficult cases because we can utter true or false sentences about things that never existed or that no longer exist. In such cases, William claims that words have a figurative signification. 'Chimaeras are imaginary' figuratively expresses the sentence 'Some mind has the imagination of a chimaera' (Abelard D 136.32; Kneale and Kneale 1962: 207). 'Homer is a poet' is properly understood figuratively as 'Homer's work, which he wrote in his role as poet, exists' (H9; Iwakuma 1999: 113; Abelard D 136.14ff, 168.11ff). Abelard attributes to William the view that words signify all things that they were imposed to name (Abelard D 112.24). Other texts indicate that William might have believed this. An anonymous text attributes to William the view that 'all' signifies all things simultaneously (H13; Iwakuma 1999: 111). However, it is also possible that Abelard misunderstood or misrepresented William (Iwakuma 1999: 107). It is reported that William held a similar view of infinite terms. William is said to have argued that an infinite term, e.g., 'non-man', signifies all those things that are not men (H13; Iwakuma 1999: 109). However, William's own discussion of infinite terms is available. In the Introductions William argues that the signification of infinite terms can be taken affirmatively, negatively, or correctly (ISW I 4.2). Taken affirmatively 'non-man' positively signifies every thing that is not a man: each rock, flower, and squirrel. The affirmative account allows one to substitute 'stone' for 'non-man' in syllogisms with absurd results. Taking infinite terms negatively solves this problem by claiming that infinite terms do not signify any existing thing, but William cannot accept this because it would mean that infinite terms are not significant. Rather, he argues that the imposition of infinite terms is related to the imposition of their correlative terms. > 'Animal' and 'man' signify the > very same thing, animal and man, by imposition [ponendo], > 'non-animal' and 'non-man' by remotion > [removendo]. But because infinite terms signify by remotion, > there is nothing which can be concluded by imposition. So it is not > possible to conclude 'stone' from 'non-animal' > (ISW I 4.2). Because the remotive imposition of the infinite term 'non-animal' is related to the positive imposition of the term 'animal', the infinite term does not signify stone in the way the word 'stone' would. So although William holds that the same thing is signified, the mode of signification would preclude substitution of positive and infinite terms. Abelard criticizes William for "doing such abuse to language" that he allowed 'rational' and 'rational animal' to signify the same thing (Abelard D 541.24). Again, another anonymous source confirms that William thought that a definition and the term defined, that is 'rational mortal animal' and 'man', signify the same thing (Green-Pedersen 1974, frag. 6). Both are consistent with William's theories of universals and may reflect William's commitment to words signifying presently existing things. William may well have thought that 'rational' and 'rational animal' have the same signification because each thing signified by 'rational' is signified by 'rational animal', and conversely. Such a view would involve a fairly simplistic theory of meaning, but this alone does not mean it was not William's view. Some aspects of William's theory of signification are considerably more complex. He developed a theory of two modes of signification of quality terms: > > One mode by imposition another by representation. 'White' > and 'Black' signify the denoted substance by imposition, > because they are imposed on this substance. They signify the > qualities whiteness and blackness by representation. The substance > they signify according to imposition is signified secondarily. They > principally designate those qualities which they signify by > representation.(C8; Iwakuma 1999 p.107) Abelard confirms that William made this sort of distinction. As Abelard describes it, William held that 'white' signifies the white individual and also whiteness. 'White' signifies the subject (fundamentum) by denoting (nominando) it. It signifies whiteness by determining it in the subject (determinando circa fundamentum) (Abelard LI Top: 272.14). But at the present time, not enough material is available to begin to flesh out William's account of signification by denotation and signification by determination. ### 3.2 Multiple Senses of Sentences William had several different views on the interpretation of different kinds of sentences. That he held the views is not much in dispute; they are attributed to him by several contemporaries and some are found in his own works. But the possible philosophical relation between them is a matter of speculation. William held that sentences have two senses, a grammatical sense and a dialectical sense. Taken in the grammatical sense, the verb in sentences such as 'Socrates is white' marks an intransitive copulation of essence: the same thing is denoted by the subject and predicate. In the dialectical sense, the verb in 'Socrates is white' marks a predication of inherence: what is denoted by the predicate inheres in what is denoted by the subject. To the grammarian, 'Socrates is white' and 'Socrates is whiteness' have different senses and different truth conditions. 'Socrates is white' says that Socrates is a subject of whiteness. 'Socrates is whiteness' says that Socrates is essentially whiteness. To the dialectician, however, the two sentences have the same sense, since they both say that whiteness inheres in Socrates. The dialectical sense is more general (generalior, largior, superior) because it does not distinguish between essential and accidental inherence. But the grammatical sense is more precise (determinatior). Taken in the dialectical sense, 'Socrates is white' and 'Socrates is whiteness' would both be true. So, while the dialectical sense has some use for dialecticians, sentences are true and false in the grammatical sense (Abelard LI Top: 271-273; GP 1974 Frag 9; de Rijk 1967 II.I: 183-85). According to William, maximal propositions contain or signify all the propositions under them. Thus, the maxim, 'Of whatever the species is predicated the genus is also predicated' signifies or contains the sentences 'If it is a man it is an animal', 'If it is a rock it is a substance', etc. Only the most general form is truly a maximal proposition, but it is the more precise (certior) sentences contained under the maxim that provide the inferential force for particular arguments. William may have developed this view in response to the vocalists. Abelard explains that multiple senses are needed given William's view that things, not words, warrant inference; only the more precise formulations can signify the things themselves (Abelard LI Top: 231.26-238.34). An anonymous author confirms Abelard's understanding of William's view. The argument, 'Socrates is a man therefore Socrates is an animal' is warranted by the topical maxim noted above. It is not the maxim itself but the more precise version contained in it, 'If it is a man it is an animal', that provides the force for this particular inference. According to this text, William argued that it is not the words (voces) or the sense (intellectus) of the more precise sentence that provides the inferential force, but the fact that the more precise sentence signifies relations (habitudines) between things (Green-Pedersen 1974: Frag 8;10;12). William divides the standard A, E, I, and O sentences according to what he calls the matter of the sentence materia propositionis). The matter of the sentence is determined by the relationship between what is signified as the subject and the predicate. Each standard-form sentence is found in one of three matters: natural, contingent, or remotive. In a sentence of natural matter the predicate inheres universally in the subject: 'All men are animals'. In a sentence of contingent matter the predicate inheres in the subject, but not universally: 'All men are white'. In sentences of remotive matter the predicate in no way inheres in the subject: 'All men are stones'. The validity of direct inferences between contraries, subalterns, etc. depends on the matter of the sentences involved. In all matter, contraries cannot both be true. So regardless of the matter, if an A sentence is true the corresponding E sentence is false. In natural and remotive matter, if either the A or E sentence is false the other must be true: if 'No men are animals' is false then 'All men are animals' must be true. However, in contingent matter both sentences may be false. It is possible for both 'All men are white' and 'No men are white' to be false. William gives a similar treatment for contradictories, subcontraries, and subalterns (ISW I 3-3.3; II 1.1-1.3; IGP I 5.4-5.6). ### 3.3 Argument, Conditionals, and Loci Contemporary logicians recognize a difference between arguments and their corresponding conditionals. But they accept that an argument: > > (1) Premise 1; Premise 2; therefore, Conclusion > and its corresponding conditional: > > > (2) If (Premise 1 and Premise 2) then Conclusion are equivalent by the deduction theorem. (2) in turn is interderivable with: > > (3) If Premise 1 then (if Premise 2 then Conclusion) These logical relationships were all in dispute during the twelfth century. (Iwakuma 2004b presents these distinctions and discusses William's role in the debate.) Logicians at that time attributed much more significance to the distinction between arguments and conditionals, and most did not accept what we would now recognize as the formal rules of material implication. Along these lines, several views were attributed to William as characteristic of his logical program, but his arguments in the available texts do not reflect what others say about him. So, once again, the extent of William's influence is unclear. William accepted that (1) and (2) are equivalent, but not for any metalogical reason. He thought that syllogisms were phrases (orationes) that contain the senses of several other sentences. The view attributed to William is that syllogisms and arguments are, as he puts it "subcontinuatives" [subcontinuativa] containing other sentences. 'Socrates is a man; therefore, Socrates is an animal' contains the sense of the sentences 'Socrates is a man', 'Socrates is an animal' and, 'If Socrates is a man, then Socrates is an animal' (B9 2.1 ed. Iwakuma 2004b: 102; this is the same text as Green-Pedersen 1974, frag 2). William calls the contained sentences 'continuatives [continuativa]'. He also treats all syllogisms as conditionals of type (2) rather than type (3). There is no argument for his preference, at least not in the Introductions, and none of the available sources give his reasons. At present all that can be said is that William seems to have had a hand in the origins of a debate that raged during the twelfth century and was not revived until the twentieth. William introduces several novelties in his discussion of loci. Most significantly, he limits his discussion of the traditional Boethian topics to the loci from the whole, part, opposites, equals, and immediates. He also introduces loci from the subject and from the predicate into his discussion of categorical arguments. Both moves were controversial. William sometimes refers to the locus as 'the medium', by which he does not necessarily mean the middle term but rather the thing that acts as a link between the extremes of an argument. He also called the locus 'the argument', by which he means that the locus provides the argumentative or inferential force. But whichever name he chooses, William's realist commitment is evident. > The same thing is the argument and the locus according to > Master W. but the locus is the force of the argument [sedes > argumenti], and so this thing is the force of the > argument. (Green-Pedersen 1974: frag 1) In the Introductions, William calls loci "words" [voces], but they are words denoting the things themselves or their relations (habitudines) (IGP I 7). But William is not clear about the importance of this distinction between things and relations. Fragments of his later work report that he was quite careful to distinguish his view from any brand of vocalism. He distinguished the word (vox), the understanding or meaning of the word (intellectus), and the thing itself denoted by the word, and held that the thing itself is the locus. (Green-Pedersen 1974: frag 3,4) In categorical syllogisms, William introduces the locus from subject, predicate, both subject and predicate, and from the whole (ISW II 2.2; IGP I 8ff; see Stump 1989: 117ff). Thus, the argument > > All men are animals > > No animal is a stone > > Therefore, no man is a stone > is warranted by the locus from the predicate. The predicate of the first sentence is the link or medium between the extreme terms in the conclusion. In this case, animal (presumably the universal thing itself) is the locus or seat of the argument. The rule is, if something is predicated of some subject universally, then whatever is removed universally from the predicate is removed universally from the subject (William provides a lengthy set of rules describing the logical relations that hold between subjects and predicates). As described above, William holds that such maximal propositions contain the sense of those propositions that fall under them. The rule contains sentences signifying the relations between man and animal and between animal and stone (IGP I 8.4). The relations of animal to the extremes warrant the inference. William provides a similar account of hypothetical syllogisms. He introduces several loci and rules, but these are not formal rules. What ultimately warrants any hypothetical syllogism is the same sort of derivative signification of an extra-mental thing or relation. Provided Socrates is risible, 'Socrates is a man' is true. The following hypothetical syllogism is warranted by the locus from the consequent: > > (Socrates is a man - Socrates is an animal) - (Socrates is > risible - Socrates is an animal) The rule is: whatever follows from the consequent follows from the antecedent. In this case, the consequent is 'Socrates is a man', and the antecedent is 'Socrates is risible'. This looks like a formal rule: > > (A-B)-((B-C)-(A-C)) > but William does not present it as such. 'Socrates is risible - Socrates is a man' is warranted by the locus from equals, and 'Socrates is man - Socrates is an animal' is warranted by the locus from the part (IGP I 9.2). The conclusion, 'Socrates is risible - Socrates is an animal', is arrived at via topical reasoning, not by formal rules of inference. This is the basis for much of the later criticism of his work. Abelard in particular derives contradictions from his views by treating William's loci as formal rules of inference (see Martin 2004). ### 3.4 Modality Abelard reports that William thought that all modality was de dicto (or, in Abelard's terminology, de sensu) > > > Our teacher taught that modal sentences descend from simple sentences > because modal sentences are about the sense of simple sentences. So > that when we say 'It is possible (or necessary) that Socrates > runs', we are saying that what the sentence 'Socrates > runs' says is possible or necessary. (Abelard D 195.21; see also > Mews 2005: 49; Knuuttila 1993: 87; Kneale and Kneale 1962: > 212). The extent to which this attribution is accurate is unclear. The view is at least superficially inconsistent with William's realism and anti-vocalism in all other areas. Additionally, Abelard was among the first to recognize the importance of the de dicto / de re distinction; the issue may be fairly confused in William's work, as it was in many others. On the other hand, several anonymous texts attribute to William the claim that 'possibly' and other modal terms modify the signification of other words in a sentence. Thus, 'bishop' signifies actual and existing bishops, but 'possible bishop' has a figurative sense signifying all those things whose nature is not repugnant with being a bishop (Iwakuma 1999: 112). This is not necessarily a rejection of de re modality, but again, more of William's work needs to be studied. ## 4. Ethics William's ethics starts with two generally accepted early medieval claims: (1) evil is not any positively existing thing but only the privation of good (Sen 277.1); (2) every act or event that occurs is performed or condoned by God and therefore good (Sen 277.11-15). These lead William, among others, to situate moral good and evil in the human mind. To this end, William names the elements of a fairly complex moral psychology: vice, desire/lust, pleasure, will, intention, and consent. Any or all of these elements might play a role in sinful behavior. Vice itself is not necessarily bad, since the vices are not acquired habits but inborn dispositions. Vices incline us to bad behavior but do not necessarily compel the will: there is no sin unless we consent to the behavior the vice inclines us toward (Sen 278.10). Carnal desire, lust, and pleasure, on the other hand, are always morally bad. William writes extensively about sexual lust and pleasure, especially as they are involved in the transmission of original sin. Before the fall, sex was no more pleasurable than "putting your finger in your mouth" (Sen. 254). In our fallen condition, however, lust and pleasure can never be removed from the act. Because our "ability" to experience this pleasure is the result of original sin, the pleasure itself is always culpable. Consent to a sexual act under the proper circumstances--e.g., between lawful spouses for the purpose of procreation--only lessens the gravity of the sin (Sen 255; 246). In William's moral psychology, will, intention, and consent are the undifferentiated elements of voluntary action. These play the most significant role, but sin is not exclusively in the will, consent, or intention. Our best hope to avoid sin is to will what is right for the right reasons. However, when it comes to discovering what is right, and therefore what ought to be willed, we start out at a disadvantage. The human mind naturally has the power to discern good and evil (Sen 253.10; IGP II 4), but one of the effects of the Fall is that our rational capacities have been clouded and diminished. Before the Fall the senses were subject to reason; after the Fall this is reversed (Sen 253.12-15), for since then the mind has become more closely attached and even enslaved to the body. This is the fomes of sin, a carnal weakness that clouds our reason and makes us more inclined to sin (Sen 246.39). Fear, not reason, is the key for William, since "fear is the beginning of all wisdom" (Sen. 276.1). William describes three kinds of fear (Sen 276.17-34). First is the natural fear of danger and pain; even Jesus was subject to this fear. Second is fear of losing material things or the fear of punishment in hell. This fear shows that one values comfort and material things more that goodness and justice. People motivated by this second fear will act rightly out of fear of losing material goods or fear of the torments of hell, but not out of good or right intention: "he does not merit grace who serves not his love of justice but his love of things or his fear of punishment" (Sen 261.23-25). Third is the fear that arises from respect for God's justice and power. This respectful fear of God, with knowledge of our own weakness and fallibility, "is better called the love of God" (Sen 276.39-43). This humble fear of God should be our guide to what is right. William's influence on Abelard here is obvious. Primarily, William and perhaps others from the school of Anselm of Laon were responsible for the delineating the complex moral psychology that is at the core of Abelard's ethics. The major differences are with regard to reason and responsibility. Because of our fallen state, William has little regard for the ability of human reason to correctly discern what is good. William is also content to prove that evil is no thing by situating its source somehow in the human mind. He is not interested in showing that we are responsible, only that we, and not God, are accountable. Our will, like our mind and body, is defective and we suffer the consequences of its failures. It is Abelard who takes this moral psychology and develops from it a theory of moral responsibility. ## 5. Philosophical Theology William is committed to the belief that the mysteries of faith are beyond the scope of human reason, but this does not prevent him from discussing the issues philosophically and using his skill as a logician to prove his point. The problem of free will and divine foreknowledge is familiar: if God's foreknowledge is infallible, then future events, including the actions of human beings, all happen of necessity; on the other hand, if future events,including the actions of human beings, could occur otherwise, then God's foreknowledge would fallible. William denies both propositions. With regard to the first he offers two arguments. He claims that God foresees the whole range of possible choices and infallibly foreknows which option free creatures will choose (Sen 237; 238). The event itself is not necessary; in fact, God infallibly foresees that the event is not necessary. He then adds an interesting claim about future contingent propositions: they are determinately true or false, but it is the infallibility of divine foreknowledge that makes them so. The future event itself which the proposition is about does not yet exist and is indeterminate (eventus rerum de quibus agitur indeterminate) (Sen 238.36). William seems to hold both that the future is indeterminate and that God's foreknowledge is determinately true or false. The second proposition, 'if future events, including the actions of human beings, could occur otherwise, then God's foreknowledge is fallible', was interpreted in several ways by William's contemporaries. Some held that if the event could occur otherwise God would be fallible. Others held that because the event could occur otherwise God actually is fallible (see Mews 2005: 135). The former would reject the claim that the event could occur otherwise, but the latter would simply reject divine infallibility, arguing rather that God is just very lucky epistemically. Both views are theologically suspect, of course, and William argued that both inferences are invalid. But he does so by introducing an unexplained modal intuition that in context looks question-begging: the inference from 'the event could occur otherwise' to 'God is or could be deceived' is not necessary. It is possible that the former be true and the latter false because "it is never necessary that the impossible follow from the possible, and it is impossible for God to be, or to have been, deceived" (Sen 237.68). William deftly avoids the overreaching that got philosophers like Roscelin and Abelard into such trouble. He outlines a theory of metaphorical language for describing God, and argues that the fundamental relations of the Trinity are beyond the powers of the human mind to comprehend. Predicates such as good, just, etc.,when applied to God have a metaphorical rather than their usual literal meaning. As described above, William argued that all significant words are imposed to name existing things in the world and these words generate sound understandings of those things they were imposed to signify. When we impose the word 'just' or 'good' to name a just or good person, we do so because justice or goodness is in some way a quality of that person or her actions. God, on the other hand, is goodness itself and justice itself. The dignity and power of God cannot be grasped by human reason and so we cannot even conceive of God sufficiently to accurately impose words that name divine attributes: "words imposed for human use are used metaphorically for speaking about God" [ad loquendum de Deo transferuntur] (Sen 236.46). Whenever he discusses the Trinity, William associates the Father with power, the Son with wisdom and the Holy Spirit with love. But this metaphor is not intended to help us toward rational understanding of the Trinity, or to provide a clear delineation of properties of the three ersons in any way that would allow an explanation of their relations and differences. William argues that there is nothing like the Trinity in the created world and so there is nothing the human mind is able to comprehend that could be used as an analogue for explaining the fundamental relation of the Trinity, three persons irreducibly present in one substance: "I do not see by what quality I would be able to explain this, since nothing similar can be found in the nature of any thing" (Sen 236.86). William rejects Augustine's metaphors of the sun and sunshine and the mind and reason. Sunshine is an accident of the air and is no part of the substance of the sun. Likewise, reason is a power of the mind and not its substance (Sen 236.101). William also explains that neither of his theories of universals can be applied to explain the sameness and difference of the divine persons. Material essence realism would imply that the persons in the Godhead are accidentally individuated, and this is unacceptable. The indifference theory would require the non-identity of three separate substances, which is also contrary to the teaching of the faith. What is William's ultimate conclusion? "Because no likeness <to any created thing> could be described, the Trinity must be defended by faith alone" (Sen 236.91).
ockham	## 1. Life Ockham led an unusually eventful life for a philosopher. As with so many medieval figures who were not prominent when they were born, we know next to nothing about the circumstances of Ockham's birth and early years, and have to estimate dates by extrapolating from known dates of events later in his life.[1] Ockham's life may be divided into three main periods. ### 1.1 England (c. 1287-1324) Ockham was born, probably in late 1287 or early 1288, in the village of Ockham (= Oak Hamlet) in Surrey, a little to the southwest of London.[2] He probably learned basic Latin at a village school in Ockham or nearby, but this is not certain.[3] At an early age, somewhere between seven and thirteen, Ockham was "given" to the Franciscan order (the so called "Greyfriars").[4] There was no Franciscan house (called a "convent") in the tiny village of Ockham itself; the nearest one was in London, a day's ride to the northeast. It was there that Ockham was sent. As an educational institution, even for higher education, London Greyfriars was a distinguished place; at the time, it was second only to the full-fledged Universities of Paris and Oxford. At Greyfriars, Ockham probably got most of his "grade school" education, and then went on to what we might think of as "high school" education in basic logic and "science" (natural philosophy), beginning around the age of fourteen. Around 1310, when he was about 23, Ockham began his theological training. It is not certain where this training occurred. It could well have been at the London Convent, or it could have been at Oxford, where there was another Franciscan convent associated with the university. In any event, Ockham was at Oxford studying theology by at least the year 1318-19, and probably the previous year as well, when (in 1317) he began a required two-year cycle of lectures commenting on Peter Lombard's Sentences, the standard theological textbook of the day. Then, probably in 1321, Ockham returned to London Greyfriars, where he remained. Although he had taken the initial steps in the theology program at Oxford (hence his occasional nickname, the Venerabilis Inceptor, "Venerable Beginner"), Ockham did not complete the program there, and never became a fully qualified "master" of theology at Oxford. Nevertheless, London Greyfriars was an intellectually lively place, and Ockham was by no means isolated from the heat of academic controversy. Among his "housemates" were two other important Franciscan thinkers of the day, Walter Chatton and Adam Wodeham, both sharp critics of Ockham's views. It was in this context that Ockham wrote many of his most important philosophical and theological works. In 1323 Ockham was called before the Franciscan province's chapter meeting, held that year in Bristol, to defend his views, which were regarded with suspicion by some of his confreres. About the same time, someone--it is not clear who--went from England to the Papal court at Avignon and charged Ockham with teaching heresy.[5] As a result, a commission of theologians was set up to study the case. Ockham was called to Avignon in May, 1324, to answer the charges. He never went back to England. ### 1.2 Avignon (1324-28) While in Avignon, Ockham stayed at the Franciscan convent there. It has sometimes been suggested that he was effectively under "house arrest," but this seems an exaggeration. On the contrary, he appears to have been free to do more or less as he pleased, although of course he did have to be "on hand" in case the investigating commission wanted to question him about his writings. The investigation must not have demanded much of Ockham's own time, since he was able to work on a number of other projects while he was in Avignon, including finishing his last major theological work, the Quodlibets. It should be pointed out that, although there were some stern pronouncements that came out of the investigation of Ockham, his views were never officially condemned as heretical. In 1327, Michael of Cesena, the Franciscan "Minister General" (the chief administrative officer of the order) likewise came to Avignon, in his case because of an emerging controversy between the Franciscans and the current Pope, John XXII, over the idea of "Apostolic poverty," the view that Jesus and the Apostles owned no property at all of their own but, like the mendicant Franciscans, went around begging and living off the generosity of others. The Franciscans held this view, and maintained that their own practices were a special form of "imitation of Christ." Pope John XXII rejected the doctrine, which is why Michael of Cesena was in Avignon. Things came to a real crisis in 1328, when Michael and the Pope had a serious confrontation over the matter. As a result, Michael asked Ockham to study the question from the point of view of previous papal statements and John's own previous writings on the subject. When he did so, Ockham came to the conclusion, apparently somewhat to his own surprise, that John's view was not only wrong but outright heretical. Furthermore, the heresy was not just an honest mistake; it was stubbornly heretical, a view John maintained even after he had been shown it was wrong. As a result, Ockham argued, Pope John was not just teaching heresy, but was a heretic himself in the strongest possible sense, and had therefore effectively abdicated his papacy. In short, Pope John XXII was no pope at all! Clearly, things had become intolerable for Ockham in Avignon. ### 1.3 Munich (1328/29-47) Under cover of darkness the night of May 26, 1328, Michael of Cesena, Ockham, and a few other sympathetic Franciscans fled Avignon and went into exile. They initially went to Italy, where Louis (Ludwig) of Bavaria, the Holy Roman Emperor, was in Pisa at the time, along with his court and retinue. The Holy Roman Emperor was engaged in a political dispute with the Papacy, and Ockham's group found refuge under his protection. On June 6, 1328, Ockham was officially excommunicated for leaving Avignon without permission.[6] Around 1329, Louis returned to Munich, together with Michael, Ockham and the rest of their fugitive band. Ockham stayed there, or at any rate in areas under Imperial control, until his death. During this time, Ockham wrote exclusively on political matters.[7] He died on the night of April 9/10, 1347, at roughly the age of sixty.[8] ## 2. Writings Ockham's writings are conventionally divided into two groups: the so called "academic" writings and the "political" ones. By and large, the former were written or at least begun while Ockham was still in England, while the latter were written toward the end of Ockham's Avignon period and later, in exile.[9] With the exception of his Dialogue, a huge political work, all are now available in modern critical editions, and many are now translated into English, in whole or in part.[10] The academic writings are in turn divided into two groups: the "theological" works and the "philosophical" ones, although both groups are essential for any study of Ockham's philosophy. Among Ockham's most important writings are: * Academic Writings + Theological Works - Commentary on the Sentences of Peter Lombard (1317-18). Book I survives in an ordinatio or scriptum--a revised and corrected version, approved by the author himself for distribution. Books II-IV survive only as a reportatio--a transcript of the actually delivered lectures, taken down by a "reporter," without benefit of later revisions or corrections by the author. - Seven Quodlibets (based on London disputations held in 1322-24, but revised and edited in Avignon 1324-25). + Philosophical Works - Logical Writings * Expositions of Porphyry's Isagoge and of Aristotle's Categories, On Interpretation, and Sophistic Refutations (1321-24). * Summa of Logic (c. 1323-25). A large, independent and systematic treatment of logic and semantics. * Treatise on Predestination and God's Foreknowledge with Respect to Future Contingents (1321-24). - Writings on Natural Philosophy * Exposition of Aristotle's Physics (1322-24). A detailed, close commentary. Incomplete. * Questions on Aristotle's Books of the Physics (before 1324). Not strictly a commentary, this work nevertheless discusses a long series of questions arising out of Aristotle's Physics. * Political Writings + Eight Questions on the Power of the Pope (1340-41). + The Work of Ninety Days (1332-34). + Letter to the Friars Minor (1334). + Short Discourse (1341-42). + Dialogue (c. 1334-46). Several lesser items are omitted from the above list. ## 3. Logic and Semantics Ockham is rightly regarded as one of the most significant logicians of the Middle Ages. Nevertheless, his originality and influence should not be exaggerated. For all his deserved reputation, his logical views are sometimes derivative[11] and occasionally very idiosyncratic.[12] Logic, for Ockham, is crucial to the advancement of knowledge. In the "Prefatory Letter" to his Summa of Logic, for example, he praises it in striking language: > > For logic is the most useful tool of all the arts. Without it no > science can be fully known. It is not worn out by repeated use, after > the manner of material tools, but rather admits of continual growth > through the diligent exercise of any other science. For just as a > mechanic who lacks a complete knowledge of his tool gains a fuller > [knowledge] by using it, so one who is educated in the firm principles > of logic, while he painstakingly devotes his labor to the other > sciences, acquires at the same time a greater skill at this art. > Ockham's main logical writings consist of a series of commentaries (or "expositions") on Aristotle's and Porphyry's logical works, plus his own Summa of Logic, his major work in the field. His Treatise on Predestination contains an influential theory on the logic of future contingent propositions, and other works as well include occasional discussions of logical topics, notably his Quodlibets. ### 3.1 The Summa of Logic Ockham's Summa of Logic is divided into three parts, with the third part subdivided into four subparts. Part I divides language, in accordance with Aristotle's On Interpretation (1, 16a3-8, as influenced by Boethius's interpretation), into written, spoken and mental language, with the written kind dependent on the spoken, and the spoken on mental language. Mental language, the language of thought, is thus the most primitive and basic level of language. Part I goes on to lay out a fairly detailed theory of terms, including the distinctions between (a) categorematic and syncategorematic terms, (b) abstract and concrete terms, and (c) absolute and connotative terms. Part I then concludes with a discussion of the five "predicables" from Porphyry's Isagoge and of each of Aristotle's categories. While Part I is about terms, Part II is about "propositions," which are made up of terms. Part II gives a systematic and nuanced theory of truth conditions for the four traditional kinds of assertoric categorical propositions on the "Square of Opposition," and then goes on to tensed, modal and more complicated categorical propositions, as well as a variety of "hypothetical" (molecular[13]) propositions. The vehicle for this account of truth conditions is the semantic theory of "supposition," which will be treated below. If Part I is about terms and Part II about propositions made up of terms, Part III is about arguments, which are in turn made up of propositions made up of terms. It is divided into four subparts. Part III.1 treats syllogisms, and includes a comprehensive theory of modal syllogistic.[14] Part III.2 concerns demonstrative syllogisms in particular. Part III.3 is in effect Ockham's theory of consequence, although it also includes discussions of semantic paradoxes like the Liar (the so called insolubilia) and of the still little-understood disputation form known as "obligation." Part III.4 is a discussion of fallacies. Thus, while the Summa of Logic is not in any sense a "commentary" on Aristotle's logical writings, it nevertheless covers all the traditional ground in the traditional order: Porphyry's Isagoge and Aristotle's Categories in Part I, On Interpretation in Part II, Prior Analytics in Part III.1, Posterior Analytics in Part III.2, Topics (and much else) in Part III.3, and finally Sophistic Refutations in Part III.4. ### 3.2 Signification, Connotation, Supposition Part I of the Summa of Logic also introduces a number of semantic notions that play an important role throughout much of Ockham's philosophy. None of these notions is original with Ockham, although he develops them with great sophistication and employs them with skill. The most basic such notion is "signification." For the Middle Ages, a term "signifies" what it makes us think of. This notion of signification was unanimously accepted; although there was great dispute over what terms signified, there was agreement over the criterion.[15] Ockham, unlike many (but no means all) other medieval logicians, held that terms do not in general signify thought, but can signify anything at all (including things not presently existing). The function of language, therefore, is not so much to communicate thoughts from one mind to another, but to convey information about the world.[16] In Summa of Logic I.33, Ockham acknowledges four different kinds of signification. In his first sense, a term signifies whatever things it is truly predicable of by means of a present-tensed, assertoric copula. That is, a term t signifies a thing x if and only if 'This is a t' is true, pointing to x. In the second sense, t signifies x if and only if 'This is (or was, or will be, or can be) a t' is true, pointing to x.[17] These first two senses of signification are together called "primary" signification. In the third sense, terms can also be said to signify certain things they are not truly predicable of, no matter the tense or modality of the copula. For instance, the word 'brave' not only makes us think of brave people (whether presently existing or not); it also makes us think of the bravery in virtue of which we call them "brave." Thus, 'brave' signifies and is truly predicable of brave people, but also signifies bravery, even though it is not truly predicable of bravery. (Bravery is not brave.) This kind of signification is called "secondary" signification. To a first approximation, we can say that a "connotative" term is just a term that has a secondary signification, and that such a connotative term "connotes" exactly what it secondarily signifies; in short, connotation is just secondary signification.[18] The fourth sense, finally, is the broadest one: according to it any linguistic unit, including a whole sentence, can be said to signify whatever things it makes us think of in some way or other. A sentence signifies in this sense whatever it is that its terms primarily or secondarily signify. The theory of supposition was the centerpiece of late medieval semantic theory. Supposition is not the same as signification. First of all, terms signify wherever we encounter them, whereas they have supposition only in the context of a proposition. But the differences go beyond that. Whereas signification is a psychological, cognitive relation, the theory of supposition is, at least in part, a theory of reference. For Ockham, there are three main kinds of supposition[19]: * Personal supposition, in which a term supposits for (refers to) what it signifies (in either of the first two senses of signification described above). For example, in 'Every dog is a mammal', both 'dog' and 'mammal' have personal supposition. * Simple supposition, in which a term supposits for a concept it does not signify. Thus, in 'Dog is a species' or 'Dog is a universal', the subject 'dog' has simple supposition. For Ockham the nominalist, the only real universals are universal concepts in the mind and, derivatively, universal spoken or written terms expressing those concepts. * Material supposition, in which a term supposits for a spoken or written expression it does not signify. Thus, in 'Dog has three letters', the subject 'dog' has material supposition.[20] Personal supposition, which was the main focus, was divided into various subkinds, distinguished in terms of a theory of "descent to singulars" and "ascent from singulars." A quick example will give the flavor: In 'Every dog is a mammal', 'dog' is said to have "confused and distributive" personal supposition insofar as * It is possible to "descend to singulars" as follows: "Every dog is a mammal; therefore, Fido is a mammal, and Rover is a mammal, and Bowser is a mammal ...," and so on for all dogs. * It is not possible to "ascend from any one singular" as follows: "Fido is a mammal; therefore, every dog is a mammal." Although the mechanics of this part of supposition theory are well understood, in Ockham and in other authors, its exact purpose remains an open question. Although at first the theory looks like an account of truth conditions for quantified propositions, it will not work for that purpose. And although the theory was sometimes used as an aid to spotting and analyzing fallacies, this was never done systematically and the theory is in any event ill suited for that purpose.[21] ### 3.3 Mental Language, Connotation and Definitions Ockham was the first philosopher to develop in some detail the notion of "mental language" and to put it to work for him. Aristotle, Boethius and several others had mentioned it before, but Ockham's innovation was to systematically transpose to the fine-grained analysis of human thought both the grammatical categories of his time, such as those of noun, verb, adverb, singular, plural and so on, and -- even more importantly -- the central semantical ideas of signification, connotation and supposition introduced in the previous section.[22] Written words for him are "subordinated" to spoken words, and spoken words in turn are "subordinated" to mental units called "concepts", which can be combined into syntactically structured mental propositions, just as spoken and written words can be combined into audible or visible sentences. Whereas the signification of terms in spoken and written language is purely conventional and can be changed by mutual agreement (hence English speakers say 'dog' whereas in French it is chien), the signification of mental terms is established by nature, according to Ockham, and cannot be changed at will. Concepts, in other words, are natural signs: my concept of dog naturally signifies dogs. How this "natural signification" is to be accounted for in the final analysis for Ockham is not entirely clear, but it seems to be based both on the fact that simple concepts are normally caused within the mind by their objects (my simple concept of dog originated in me as an effect of my perceptual encounter with dogs), and on the fact that concepts are in some way "naturally similar" to their objects.[23] This arrangement provides an account of synonymy and equivocation in spoken and written language. Two simple terms (whether from the same or different spoken or written languages) are synonymous if they are ultimately subordinated to the same concept; a single given term of spoken or written language is equivocal if it is ultimately subordinated to more than one concept. This raises an obvious question: Is there synonymy or equivocation in mental language itself? (If there is, it will obviously have to be accounted for in some other way than for spoken/written language.) A great deal of modern secondary literature has been devoted to this question. Trentman [1970] was the first to argue that no, there is no synonymy or equivocation in mental language. On the contrary, mental language for Ockham is a kind of lean, stripped down, "canonical" language with no frills or inessentials, a little like the "ideal languages" postulated by logical atomists in the first part of the twentieth century. Spade [1980] likewise argued in greater detail, on both theoretical and textual grounds, that there is no synonymy or equivocation in mental language. More recently, Panaccio [1990, 2004], Tweedale [1992] (both on largely textual grounds), and Chalmers [1999] (on mainly theoretical grounds) have argued for a different interpretation, which now tends to be more widely accepted. What comes out at this point is that Ockham's mental language is not to be seen as a logically ideal language and that it does incorporate both some redundancies and some ambiguities. The question is complicated, but it goes to the heart of much of what Ockham is up to. In order to see why, let us return briefly to the theory of connotation.[24] Connotation was described above in terms of primary and secondary signification. But in Summa of Logic I.10, Ockham himself draws the distinction between absolute and connotative terms by means of the theory of definition. For Ockham, there are two kinds of definitions: real definitions and nominal definitions. A real definition is somehow supposed to reveal the essential metaphysical structure of what it defines; nominal definitions do not do that. As Ockham sets it up, all connotative terms have nominal definitions, never real definitions, and absolute terms (although not all of them) have real definitions, never nominal definitions. (Some absolute terms have no definitions at all.[25]) As an example of a real definition, consider: 'Man is a rational animal' or 'Man is a substance composed of a body and an intellective soul'. Each of these traditional definitions is correct, and each in its own way expresses the essential metaphysical structure of a human being. But notice: the two definitions do not signify (make us think of) exactly the same things. The first one makes us think of all rational things (in virtue of the first word of the definiens) plus all animals (whether rational or not, in virtue of the second word of the definiens). The second definition makes us think of, among other things, all substances (in virtue of the word 'substance' in the definiens), whereas the first one does not. It follows therefore that an absolute term can have several distinct real definitions that don't always signify exactly the same things. They will primarily signify--be truly predicable of--exactly the same things, since they will primarily signify just what the term they define primarily signifies. But they can also (secondarily) signify other things as well.[26] Nominal definitions, Ockham says, are different: There is one and only one nominal definition for any given connotative term.[27] While a real definition is expected to provide a structural description of certain things (which can be done in various ways, as we just saw), a nominal definition, by contrast, is supposed to unfold in a precise way the signification of the connotative term it serves to define, and this can only be done, Ockham thinks, by explicitly mentioning, in the right order and with the right connections, which kind of things are primarily signified by this term and which are secondarily signified. The nominal definition of the connotative term "brave", to take a simple example, is "a living being endowed with bravery"; this reveals that "brave" primarily signifies certain living beings (referred to by the first part of the definition) and that it secondarily signifies -- or connotes -- singular qualities of bravery (referred to by the last part of the definition).[28] Any non-equivalent nominal definition is bound to indicate a different signification and would, consequently, be unsuitable if the original one was correct. Now, several commentators, following Trentman and Spade, concluded on this basis that there are no simple connotative terms in Ockham's mental language. They reasoned as follows: a connotative term is synonymous with its nominal definition, but there is no synonymy in mental language according to Ockham; mental language, therefore, cannot contain both a simple connotative term and its complex nominal definition; since it must certainly have the resources for formulating adequate definitions, what must be dispensed with is the defined simple term; and since all connotative terms are supposed to have a nominal definition, it follows that mental language contains only absolute terms (along with syncategorematic ones, of course). It even came to be supposed in this line of interpretation, that the very central point of Ockham's nominalist program was to show that if anything can be truly said about the world, it can be said using only absolute and syncategorematic terms, and that this is precisely what happens in mental language. The consequences were far-reaching. Not only did this interpretation claim to provide an overall understanding of what Ockham was up to, but it also inevitably led to conclude that his whole nominalist program was bound to failure. All relational terms, indeed, are taken to be connotative terms in Ockham's semantics. The program, consequently, was thought to require the semantical reduction of all relational terms to combinations of non-relational ones, which seems hardly possible. Thus, the question whether there are simple connotative terms or not in Ockham's mental language is crucial to our understanding of the success of his overall ontological project. Since spoken and written languages are semantically derivative on mental language, it is vital that we get the semantics of mental language to work out right for Ockham, or else the systematic coherence of much of what he has to say will be in jeopardy. In view of recent scholarship, though, it appears highly doubtful that Ockham's purpose really was to use nominal definitions to eliminate all simple connotative terms from mental language. For one thing, as Spade had remarked himself, Ockham never systematically engages in explicit attempts at such semantical reductions, which would be quite odd if this was the central component of his nominalism. Furthermore it was shown that Ockham did in fact hold that there are simple connotative terms in mental language. He says it explicitly and repeatedly, and in a variety of texts from his earlier to his later philosophical and theological writings.[29] The secondary literature, consequently, has now gradually converged on the view that, for Ockham, there is no synonymy among simple terms in mental language, but that there can be some redundancy between simple terms and complex expressions, or between various complex expressions. If so, nothing prevents a simple connotative concept to coexist in mental language with its nominal definition. Ockham indeed explicitly denies that a complex definition is in general wholly synonymous with the corresponding defined term.[30] His point, presumably, is that the definition usually signifies more things than the defined term. Take "brave" again. Its definition, remember, is "a living being endowed with bravery". Now, the first part of this complex expression makes us think of all living beings, whereas the simple term "brave" has only the brave ones as its primary significates and does not signify in any way the non-brave living beings. This shows in effect that simple connotative terms are not -- at least not always -- shorthand abbreviations for their nominal definitions in Ockham's view. And it must be conjectured that some simple connotative concepts can be directly acquired on the basis of perceptual experiences, just as absolute ones are supposed to be (think of a relational concept like "taller than" or a qualitative one like "white"). Ockham's nominal definitions, then, should not be seen as reductionist devices for eliminating certain terms, but as a privileged means for making conspicuous what the (primary and secondary) significates of the defined terms are. The main point here is that such definitions, when correctly formulated, explicitly reveal the ontological commitments associated with the normal use of the defined terms. The definition of "brave" as "a living being endowed with bravery", for example, shows that the correct use of the term "brave" commits us only to the existence of singular living beings and singular braveries. Ockham's nominalism does not require the elimination of simple connotative concepts after all; its main relevant thesis, on the contrary, is that their use is ontologically harmless since they do not signify (either primarily or secondarily) anything but individual things, as their nominal definitions are supposed to make it clear. ## 4. Metaphysics Ockham was a nominalist, indeed he is the person whose name is perhaps most famously associated with nominalism. But nominalism means many different things: * A denial of metaphysical universals. Ockham was emphatically a nominalist in this sense. * An emphasis on reducing one's ontology to a bare minimum, on paring down the supply of fundamental ontological categories. Ockham was likewise a nominalist in this sense. * A denial of "abstract" entities. Depending on what one means, Ockham was or was not a nominalist in this sense. He believed in "abstractions" such as whiteness and humanity, for instance, although he did not believe they were universals. (On the contrary, there are at least as many distinct whitenesses as there are white things.) He certainly believed in immaterial entities such as God and angels. He did not believe in mathematical ("quantitative") entities of any kind. The first two kinds of nominalism listed above are independent of one another. Historically, there have been philosophers who denied metaphysical universals, but allowed (individual) entities in more ontological categories than Ockham does. Conversely, one might reduce the number of ontological categories, and yet hold that universal entities are needed in the categories that remain. ### 4.1 Ockham's Razor Still, Ockham's "nominalism," in both the first and the second of the above senses, is often viewed as derived from a common source: an underlying concern for ontological parsimony. This is summed up in the famous slogan known as "Ockham's Razor," often expressed as "Don't multiply entities beyond necessity."[31] Although the sentiment is certainly Ockham's, that particular formulation is nowhere to be found in his texts. Moreover, as usually stated, it is a sentiment that virtually all philosophers, medieval or otherwise, would accept; no one wants a needlessly bloated ontology. The question, of course, is which entities are needed and which are not. Ockham's Razor, in the senses in which it can be found in Ockham himself, never allows us to deny putative entities; at best it allows us to refrain from positing them in the absence of known compelling reasons for doing so. In part, this is because human beings can never be sure they know what is and what is not "beyond necessity"; the necessities are not always clear to us. But even if we did know them, Ockham would still not allow that his Razor allows us to deny entities that are unnecessary. For Ockham, the only truly necessary entity is God; everything else, the whole of creation, is radically contingent through and through. In short, Ockham does not accept the Principle of Sufficient Reason. Nevertheless, we do sometimes have sufficient methodological grounds for positively affirming the existence of certain things. Ockham acknowledges three sources for such grounds (three sources of positive knowledge). As he says in Sent. I, dist. 30, q. 1: "For nothing ought to be posited without a reason given, unless it is self-evident (literally, known through itself) or known by experience or proved by the authority of Sacred Scripture." ### 4.2 The Rejection of Universals In the case of universal entities, Ockham's nominalism is not based on his Razor, his principle of parsimony. That is, Ockham does not hold merely that there is no good reason for affirming universals, so that we should refrain from doing so in the absence of further evidence. No, he holds that theories of universals, or at least the theories he considers, are outright incoherent; they either are self-contradictory or at least violate certain other things we know are true in virtue of the three sources just cited. For Ockham, the only universal entities it makes sense to talk about are universal concepts, and derivative on them, universal terms in spoken and written language. Metaphysically, these "universal" concepts are singular entities like all others; they are "universal" only in the sense of being "predicable of many." With respect to the exact ontological status of such conceptual entities, however, Ockham changed his view over the course of his career. To begin with, he adopted what is known as the fictum-theory, a theory according to which universals have no "real" existence at all in the Aristotelian categories, but instead are purely "intentional objects" with a special mode of existence; they have only a kind of "thought"-reality. Eventually, however, Ockham came to think this intentional realm of "fictive" entities was not needed, and by the time of his Summa of Logic and the Quodlibets adopts instead a so called intellectio-theory, according to which a universal concept is just the act of thinking about several objects at once; metaphysically such an "act" is a singular quality of an individual mind, and is "universal" only in the sense of being a mental sign of several things at once and being predicable of them in mental propositions.[32] ### 4.3 Exposition or Parsing Away Entities Thus, Ockham is quite certain there are no metaphysically universal entities. But when it comes to paring down the number of basic ontological categories, he is more cautious, and it is there that he uses his Razor ruthlessly--always to suspend judgment, never to deny. The main vehicle for this "ontological reduction" is the theory of connotation, coupled with the related theory of "exposition." The theory of exposition, which is not fully developed in Ockham, will become increasingly prominent in authors immediately after him. In effect, the theory of connotation is related to the theory of exposition as explicit definition is related to contextual definition. The notion of the "square" of a number can be explicitly defined, for example, as the result of multiplying that number by itself. Contextual definition operates not at the level of terms, but at the level of propositions. Thus, Bertrand Russell famously treated 'The present king of France is bald' as amounting to 'There is an x such that x is a present king of France and x is bald, and for all y if y is a present king of France then y = x'. We are never given any outright definition of the term 'present king of France', but instead are given a technique of paraphrasing away seemingly referential occurrences of that term in such a way that we are not committed to any actually existing present kings of France. So too, Ockham tries to provide us, at the propositional level, with paraphrases of propositions that seem at first to refer to entities he sees no reason to believe in.[33] For example, in Summa of Logic, II.11, among other places, Ockham argues that we can account for the truth of 'Socrates is similar to Plato' without having to appeal to a relational entity called "similarity": > > For example, for the truth of 'Socrates is similar to > Plato', it is required that Socrates have some quality and that > Plato have a quality of the same species. Thus, from the very fact > that Socrates is white and Plato is white, Socrates is similar to > Plato and conversely. Likewise, if both are black, or hot, [then] they > are similar without anything else added. (Emphasis added.) > In this way, Ockham removes all need for entities in seven of the traditional Aristotelian ten categories; all that remain are entities in the categories of substance and quality, and a few entities in the category of relation, which Ockham thinks are required for theological reasons pertaining to the Trinity, the Incarnation and the Eucharist, even though our natural cognitive powers would see no reason for them at all.[34] As is to be expected, the ultimate success of Ockham's program is a matter of considerable dispute.[35] It should be stressed again, however, that this program in no way requires that it should be possible to dispense altogether with terms from any of the ten Aristotelian categories (relational and quantitative terms in particular). Ockham's claim is simply that all our basic scientific terms, whether absolute or connotative, signify nothing but singular substances or qualities (plus some singular relations in certain exceptional theological cases). ## 5. Natural Philosophy Ockham's "physics" or natural philosophy is of a broadly Aristotelian sort, although he interprets Aristotle in his own fashion. Ockham wrote a great deal in this area; indeed his Exposition of Aristotle's Physics is his longest work except for his Commentary on the Sentences.[36] As a nominalist about universals, Ockham had to deal with the Aristotelian claim in the Posterior Analytics that science pertains to certain propositions about what is universal and necessary. He discusses this issue in the Prologue to his Exposition of the Physics,[37] and there agrees with Aristotle. But he interprets Aristotle's dictum as saying that knowledge bears upon certain propositions with general (universal) terms in them; it is only in that sense that science deals with the universal. This of course does not mean that for Ockham our scientific knowledge can never get beyond the level of language to actual things. He distinguishes various senses of 'to know' (scire, from which we get scientia or "science"): * In one sense, to "know" is to know a proposition, or a term in that proposition. It is in this sense that the object of a science is universal, and this is what Aristotle had in mind. * In another sense, we can be said to "know" what the proposition is about, what its terms have supposition for. What we "know" in that sense is always metaphysically individual, since for Ockham there isn't anything else. This is not the sense in which Aristotle was speaking. As described earlier, Ockham holds that we do not need to allow special entities in all ten of Aristotle's categories. In particular, we do not need them in the category of quantity. For Ockham, there is no need for real "mathematical" entities such as numbers, points, lines, and surfaces as distinct from individual substances and qualities. Apparent talk about such things can invariably be parsed away, via the theory of connotation or exposition, in favor of talk about substances and qualities (and, in certain theological contexts, a few relations). This Ockhamist move is illustrative of and influential on an important development in late medieval physics: the application of mathematics to non-mathematical things, culminating in Galileo's famous statement that the "book of nature" is written in the "language of mathematics." Such an application of mathematics violates a traditional Aristotelian prohibition against metabasis eis allo genos, grounded on quite reasonable considerations. The basic idea is that things cannot be legitimately compared in any respect in which they differ in species. Thus it makes little sense to ask whether the soprano's high C is higher or lower than Mount Everest--much less to ask (quantitatively) how much higher or lower it is. But for Aristotle, straight lines and curved lines belong to different species of lines. Hence they cannot be meaningfully compared or measured against one another. The same holds for rectilinear motion and circular motion. Although the basic idea is reasonable enough, Ockham recognized that there are problems. The length of a coiled rope, for example, can straightforwardly be compared to the length of an uncoiled rope, and the one can meaningfully be said to be longer or shorter than, or equal in length to, the other. For that matter, a single rope surely stays the same length, whether it is coiled or extended full-length. Ockham's solution to these problems is to note that, on his ontology, straight lines and curved lines are not really different species of lines--because lines are not extra things in the first place. Talk about lines is simply a "manner of speaking" about substances and qualities. Thus, to compare a "curved" (coiled) rope with a "straight" (uncoiled) one is not really to talk about the lengths of lines in two different species; it is to talk about two ropes. To describe the one as curved (coiled) and the other as straight (uncoiled) is not to appeal to specifically different kinds of entities--curvature and straightness--but merely to describe the ropes in ways that can be expounded according to two different patterns. Since such talk does not have ontological implications that require specifically different kinds of entities, the Aristotelian prohibition of metabasis does not apply. Once one realizes that we can appeal to connotation theory, and more generally the theory of exposition, without invoking new entities, the door is opened to applying mathematical analyses (all of which are exponible, for Ockham) to all kinds of things, and in particular to physical nature. Ockham's contributions were by no means the only factor in the increasing mathematization of science in the fourteenth century. But they were important ones.[38] ## 6. Theory of Knowledge Like most medieval accounts of knowledge, Ockham's is not much concerned with answering skeptical doubts. He takes it for granted that humans not only can but frequently do know things, and focuses his attention instead on the "mechanisms" by which this knowledge comes about. ### 6.1 The Rejection of Species Ockham's theory of knowledge, like his natural philosophy, is broadly Aristotelian in form, although--again, like his natural philosophy--it is "Aristotelian" in its own way. For most Aristotelians of the day, knowledge involved the transmission of a "species"[39] between the object and the mind. At the sensory level, this species may be compared to the more recent notion of a sense "impression." More generally, we can think of it as the structure or configuration of the object, a structure or configuration that can be "encoded" in different ways and found isomorphically in a variety of contexts. One recent author, describing the theory as it occurs in Aquinas, puts it like this:[40] > > Consider, for example, blueprints. In a blueprint of a library, the > configuration of the library itself, that is, the very configuration > that will be in the finished library, is captured on paper but in such > a way that it does not make the paper itself into a library. Rather, > the configuration is imposed on the paper in a different sort of way > from the way it is imposed on the materials of the library. What > Aquinas thinks of as transferring and preserving a configuration we > tend to consider as a way of encoding information. > The configuration of features found in the external object is also found in "encoded" form as a species in the organ that senses the object. (Depending on the sense modality, it may also be found in an intervening medium. For example, with vision and hearing, the species is transmitted through the air to the sense organ.) At the intellectual level, the so called "agent intellect" goes to work on this species and somehow produces the universal concept that is the raw material of intellectual cognition.[41] Ockham rejected this entire theory of species. For him, species are unnecessary to a successful theory of cognition, and he dispenses with them.[42] Moreover, he argues, the species theory is not supported by experience; introspection reveals no such species in our cognitive processes.[43] This rejection of the species theory of cognition, which had been foreshadowed by several previous authors (such as Henry of Ghent in the thirteenth century), was an important development in late medieval epistemology.[44] ### 6.2 Intuitive and Abstractive Cognition One of the more intriguing features of late medieval epistemology in general, and of Ockham's view in particular, is the development of a theory known as "intuitive and abstractive cognition." The theory is found in authors as diverse as Duns Scotus, Peter Auriol, Walter Chatton, and Ockham. But their theories of intuitive and abstractive cognition are so different that it is hard to see any one thing they are all supposed to be theories of. Nevertheless, to a first approximation, intuitive cognition can be thought of as perception, whereas abstractive cognition is closer to imagination or remembering. The fit is not exact, however, since authors who had a theory of intuitive and abstractive cognition usually also allowed the distinction at the intellectual level as well. It is important to note that abstractive cognition, in the sense of this theory, has nothing necessarily to do with "abstraction" in the sense of producing universal concepts from cognitive encounters with individuals. Instead, what abstractive cognition "abstracts" from is the question of the existence or non-existence of the object. By contrast, intuitive cognition is very much tied up with the existence or non-existence of the object. Here is how Ockham distinguishes them:[45] > > For intuitive cognition of a thing is a cognition such that by virtue > of it it can be known whether the thing exists or not, in such a way > that if the thing does exist, the intellect at once judges it to exist > and evidently knows it to exist ... Likewise, intuitive cognition > is such that when some things are known, one of which inheres in the > other or the one is distant in place from the other or is related in > another way to the other, it is at once known by virtue of the > incomplex cognitions of those things whether the thing inheres or does > not inhere, whether it is distant or not distant, and so on for other > contingent truths ... > > > Abstractive cognition, however, is that by virtue of which it cannot > be evidently known of the thing whether it exists or does not exist. > And in this way abstractive cognition, as opposed to intuitive > cognition, "abstracts" from existence and non-existence, > because by it neither can it be evidently known of an existing thing > that it exists, nor of a non-existent one that it does not exist. > > > Ockham's main point here is that an intuitive cognition naturally causes in the mind a number of true contingent judgements about the external thing(s) that caused this intuitive cognition; for example, that this thing exists, or that it is white, and so on. This does not prevent God from deceiving any particular creature if He wants to, even when an intuitive cognition is present, but in such a case, God would have to neutralize the natural causal effect of this intuitive cognition (this is something He can always do, according to Ockham) and directly cause instead a false judgement. Intuitive cognitions, on the other hand, can sometimes induce false beliefs, too, if the circumstances are abnormal (in cases of perceptual illusions in particular), but even then, they would still cause some true contingent judgements. The latter at any rate is their distinctive feature. Abstractive cognitions, by contrast, are not such as to naturally cause true judgements about contingent matters.[46] ## 7. Ethics Ockham's ethics combines a number of themes. For one, it is a will-based ethics in which intentions count for everything and external behavior or actions count for nothing. In themselves, all actions are morally neutral. Again, there is a strong dose of divine command theory in Ockham's ethics. Certain things (i.e., in light of the previous point, certain intentions) becomes morally obligatory, permitted or forbidden simply because God decrees so. Thus, in Exodus, the Israelites' "spoiling the Egyptians" (or rather their intention to do so, which they carried out) was not a matter of theft or plunder, but was morally permissible and indeed obligatory--because God had commanded it. Nevertheless, despite the divine command themes in Ockham's ethics, it is also clear that he wanted morality to be to some extent a matter of reason. There is even a sense in which one can find a kind of natural law theory in Ockham's ethics; one way in which God conveys his divine commands to us is by giving us the natures we have.[47] Unlike Augustine, Ockham accepted the possibility of the "virtuous pagan"; moral virtue for Ockham does not depend on having access to revelation. ### 7.1 The Virtues But while moral virtue is possible even for the pagan, moral virtue is not by itself enough for salvation. Salvation requires not just virtue (the opposite of which is moral vice) but merit (the opposite of which is sin), and merit requires grace, a free gift from God. In short, there is no necessary connection between virtue--moral goodness--and salvation. Ockham repeatedly emphasizes that "God is a debtor to no one"; he does not owe us anything, no matter what we do. For Ockham, acts of will are morally virtuous either extrinsically, i.e. derivatively, through their conformity to some more fundamental act of will, or intrinsically. On pain of infinite regress, therefore, extrinsically virtuous acts of will must ultimately lead back to an intrinsically virtuous act of will. That intrinsically virtuous act of will, for Ockham, is an act of "loving God above all else and for his own sake." In his early work, On the Connection of the Virtues, Ockham distinguishes five grades or stages of moral virtue, which have been the topic of considerable speculation in the secondary literature:[48] 1. The first and lowest stage is found when someone wills to act in accordance with "right reason"--i.e., because it is "the right thing to do." 2. The second stage adds moral "seriousness" to the picture. The agent is willing to act in accordance with right reason even in the face of contrary considerations, even--if necessary--at the cost of death. 3. The third stage adds a certain exclusivity to the motivation; one wills to act in this way only because right reason requires it. It is not enough to will to act in accordance with right reason, even heroically, if one does so on the basis of extraneous, non-moral motives. 4. At the fourth stage of moral virtue, one wills to act in this way "precisely for the love of God." This stage "alone is the true and perfect moral virtue of which the Saints speak." 5. The fifth and final stage can be built immediately on either the third or the fourth stage; thus one can have the fifth without the fourth stage. The fifth stage adds an element of extraordinary moral heroism that goes beyond even the "seriousness" of stage two. The difficulty in understanding this hierarchy comes at the fourth stage, where it is not clear exactly what moral factor is added to the preceding three stages.[49] ### 7.2 Moral Psychology At the beginning of his Nicomachean Ethics, Aristotle remarked that "the good is that at which all things aim." Each thing, therefore, aims at the good, according to the demands of its nature. In the Middle Ages, "Aristotelians" like Thomas Aquinas held that the good for human beings in particular is "happiness," the enjoyment of the direct vision of God in the next life. And, whether they realize it or not, that is what all human beings are ultimately aiming at in their actions. For someone like Aquinas, therefore, the human will is "free" only in a certain restricted sense. We are not free to choose for or against our final end; that is built into us by nature. But we are free to choose various means to that end. All our choices, therefore, are made under the aspect of leading to that final goal. To be sure, sometimes we make the wrong choices, but when that occurs it is because of ignorance, distraction, self-deception, etc. In an important sense, then, someone like Aquinas accepts a version of the so called Socratic Paradox: No one knowingly and deliberately does evil.[50] Ockham's view is quite different. Although he is very suspicious of the notion of final causality (teleology) in general, he thinks it is quite appropriate for intelligent, voluntary agents such as human beings. Thus the frequent charge that Ockham severs ethics from metaphysics by denying teleology seems wrong.[51] Nevertheless, while Ockham grants that human beings have a natural orientation, a tendency toward their own ultimate good, he does not think this restricts their choices. For Ockham, as for Aristotle and Aquinas, I can choose the means to achieve my ultimate good. But in addition, for Ockham unlike Aristotle and Aquinas, I can choose whether to will that ultimate good. The natural orientation and tendency toward that good is built in; I cannot do anything about that. But I can choose whether or not to to act to achieve that good. I might choose, for example, to do nothing at all, and I might choose this knowing full well what I am doing. But more: I can choose to act knowingly directly against my ultimate good, to thwart it.[52] I can choose evil as evil. For Ockham, this is required if I am going to be morally responsible for my actions. If I could not help but will to act to achieve my ultimate good, then it would not be morally praiseworthy of me to do so; moral "sins of omission" would be impossible (although of course I could be mistaken in the means I adopt). By the same token, moral "sins of commission" would be impossible if I could not knowingly act against my ultimate good. But for Ockham these conclusions are not just required by theory; they are confirmed by experience. ## 8. Political Philosophy The divine command themes so prominent in Ockham's ethics are much more muted in his political theory, which on the contrary tends to be far more "natural" and "secular."[53] As sketched above, Ockham's political writings began at Avignon with a discussion of the issue of poverty. But later on the issues were generalized to include church/state relations more broadly. He was one of the first medieval authors to advocate a form of church/state separation, and was important for the early development of the notion of property rights. The Franciscan Order at this time was divided into two parties, which came to be known as the "Conventuals" and the "Spirituals" (or "zealots"). The Spirituals, among whom were Ockham, Michael of Cesena, and the other exiles who joined them in fleeing Avignon, tried to preserve the original ideal of austere poverty practiced and advocated by St. Francis himself (c. 1181-1226). The Conventuals, on the other hand, while recognizing this ideal, were prepared to compromise in order to accommodate the practical needs of a large, organized religious order; they were by far the majority of the order. The issue between the two parties was never one of doctrine; neither side accused the other of heresy. Rather, the question was one of how to shape and run the order--in particular, whether the Franciscans should (or even could) renounce all property rights. ### 8.1 The Ideal of Poverty The ideal of poverty had been (and still is) a common one in religious communities. Typically, the idea is that the individual member of the order owns no property at all. If a member buys a car, for instance, it is not strictly his car, even though he may have exclusive use of it, and it was not bought with his money; he doesn't have any money of his own. Rather it belongs to the order. The original Franciscan ideal went further. Not only did the individual friar have no property of his own, neither did the order. The Franciscans, therefore, were really supposed to be "mendicants," to live by begging. Anything donated to the order, such as a house or a piece of land, strictly speaking remained the property of the original owner (who merely granted the use of it to the Franciscans). (Or, if that would not work--as, for example, in the case of a bequest in a will, after the original owner had died--the ownership would go to the Papacy.) Both the Spirituals and the Conventuals thought this ideal of uncompromising poverty was exhibited by the life of Jesus and the Apostles, who--they said--had given up all property, both individually and collectively. St. Francis regarded this as the clear implication of several Scriptural passages: e.g., Matt. 6:24-34, 8:20, 19:21. In short, the Apostolic (and Franciscan) ideal was, "Live without a safety net." Of course, if everyone lived according to this ideal, so that no one owned any property either individually or collectively, then there would be no property at all. The Franciscan ideal, then, shared by Conventuals and Spirituals alike, entailed the total abolition of all property rights. Not everyone shared this view. Outside the Franciscan order, most theoreticians agreed that Jesus and the Apostles lived without individual property, but thought they did share property collectively. Nevertheless, Pope Nicholas III, in 1279, had officially approved the Franciscan view, not just as a view about how to organize the Franciscan order, but about the interpretation of the Scriptural passages concerning Jesus and the Apostles. His approval did not mean he was endorsing the Franciscan reading as the correct interpretation of Scripture, but only that it was a permissible one, that there was nothing doctrinally suspect about it.[54] Nevertheless, this interpretation was a clear reproach to the Papacy, which at Avignon was wallowing in wealth to a degree it had never seen before. The clear implication of the Franciscan view, therefore, was that the Avignon Popes were conspicuously not living their lives as an "imitation of Christ." Whether for this reason or another, the Avignon Pope John XXII decided to reopen discussion of the question of Apostolic poverty and to come to some resolution of the matter. But, as Mollat [1963] puts it (perhaps not without some taking of sides):[55] > > When discussions began at Avignon, conflicting opinions were freely > put forward. Meanwhile, Michael of Cesena, acting with insolent > audacity, did not await the Holy See's decision: on 30 May 1322 > the chapter-general [of the Franciscan order] at Perugia declared > itself convinced of the absolute poverty of Christ and the Apostles. > It was this act that provoked John XXII to issue his first contribution to the dispute, his bull Ad conditorem in 1322. There he put the whole matter in a legal framework. ### 8.2 The Legal Issues According to Roman law, as formulated in the Code of Justinian, "ownership" and "legitimate use" cannot be permanently separated. For example, it is one thing for me to own a book but to let you use it for a while. Ownership in that case means that I can recall the book, and even if I do not do so, you should return it to me when you are done with it. But it is quite another matter for me to own the book but to grant you permanent use of it, to agree not to recall it as long as you want to keep it, and to agree that you have no obligation to give it back ever. John XXII points out that, from the point of view of Roman law, the latter case makes no sense. There is no practical difference in that case between your having the use of the book and your owning it; for all intents and purposes, it is yours. Notice the criticism here. It is a legal argument against the claim that the Papacy as an institution can own something and yet the Franciscans as an order, collectively, have a permanent right to use it. The complaint is not against the notion that an individual friar might have a right to use something until he dies, at which time use reverts to the order (or as the Franciscans would have it, to the Papacy). This would still allow some distinction between ownership and mere use. Rather the complaint is against the notion that the order would not own anything outright, but would nevertheless have permanent use of it that goes beyond the life or death of any individual friar, so that the ownership somehow remained permanently with the Papacy, even though the Pope could not reclaim it, use it, or do anything at all with it. John XXII argues that this simply abolishes the distinction between use and ownership. ### 8.3 Property Rights Special problems arise if the property involved is such that the use of it involves consuming it--e.g., food. In that case, it appears that there is no real difference between ownership and even temporary use. For things like food, using them amounts for practical purposes to owning them; they cannot be recalled after they are used. In short, for John XXII, it follows that it is impossible fully to live the life of absolute poverty, even for the individual person (much less for a permanent institution like the Franciscan order). The institution of property, and property "rights," therefore began in the Garden of Eden, the first time Adam or Eve ate something. These property rights are not "natural" rights; on the contrary, they are established by a kind of positive law by God, who gave everything in the Garden to Adam and Eve. Ockham disagreed. For him, there was no "property" in the Garden of Eden. Instead, Adam and Eve there had a natural right to use anything at hand. This natural right did not amount to a property right, however, since it could not have been used as the basis of any kind of legal claim. Both John XXII and Ockham seem to agree in requiring that "property" (ownership) be a matter of positive law, not simply of natural law. But John says there was such property in the Garden of Eden, whereas Ockham claims there was not; there was only a natural right, so that Adam and Eve's use of the goods there was legitimate. For Ockham, "property" first emerged only after the Fall when, by a kind of divine permission, people began to set up special positive legal arrangements assigning the legal right to use certain things to certain people (the owners), to the exclusion of anyone else's having a legal right to them. The owners can then give permission to others to use what the owners own, but that permission does not amount to giving them a legal right they could appeal to in a court of law; it can be revoked at any time. For Ockham, this is the way the Franciscans operate. Their benefactors and donors do not give them any legal rights to use the things donated to them--i.e., no right they could appeal to in a court of law. Rather the donation amounts only to a kind of permission that restores the original natural (not legal) right of use in the Garden of Eden.[56]
william-sherwood	## 1. Life and works William of Sherwood was born between 1200 and 1205, probably in Nottinghamshire, and died between 1266 and 1272. His school career is uncertain, but given references to his works and his influence on people such as Peter of Spain, Lambert of Auxerre, Albert the Great, and Thomas Aquinas, it is likely that he was teaching logic at the University of Paris between 1235 and 1250. After that period, he seems to have left logic. He was a master at Oxford in 1252 and treasurer of the cathedral of Lincoln about two years later; still later, he was rector at Aylesbury, in Buckinghamshire, and Attleborough, in Norfolk. He was still alive in 1266 but dead by 1272 (Kretzmann 1968, p. 3). Part of the difficulty with establishing his biography is that he has previously been conflated with various other 13th-century Englishmen named William, such as William of Leicester (d. 1213) and William of Durham (d. 1249) (cf. Grabmann 1937, pp. 11-13; Kretzmann 1966, p. 3; Mullinger 1873, p. 177). He has also sometimes been identified with others bearing the same surname, such as the author of commentary, now lost, on the Sentences of Peter Lombard with the title Shirovodus super sententias, a work that was known to John Leland in the 16th century (Kretzmann 1966, p. 12), or Ricardus de Schirewode, who wrote a treatise on insolubles surviving in Cambridge, St John's MS 100. This confusion, along with the usual problems presented by anonymous material, makes it difficult to identify William's works with certainty. Two works can be confidently ascribed to Sherwood, an Introduction to Logic and a treatise on syncategorematic terms, the Syncategoremata. The Introduction survives in complete form in only one manuscript, Bibliotheque Nationale MS. Lat. 16,617, from the late 13th or 14th century, which begins with the heading Introductiones Magistri Guilli. de Shyreswode in logicam, "Introduction to Logic of Master William of Sherwood". The Syncategoremata immediately follows, and is likewise explicitly attributed to Sherwood. The Syncategoremata also survives in a second manuscript, Bodleian MS Digby 55, where it is again explicitly labeled, Sincategoreumata Magistri Willielmi de Sirewode "[Treatise on] Syncategorematics of Master William of Sherwood", and Chapter Five of the Introduction survives in a second manuscript, MS Worcester Cath. Q. 13, written around 1293/94, and edited by Pinborg & Ebbesen (1984). The Introduction was almost certainly written while William was lecturing on logic at the University of Paris. It was first edited by Grabmann (1937) and translated into English by Kretzmann (1966). A new critical edition was produced by Brands and Kann (1995), accompanied by a translation into German. It has been argued (Mullinger 1873, p. 177) that this book was heavily influenced by a Synopsis treatise written by the 11th-century Byzantine logician Michael Constantine Psellus. The Introduction contains the first appearance of the syllogistic mnemonic verse 'Barbara, Celarent...', which Augustus de Morgan described as "magic words ... more full of meaning than any that ever were made", and Sherwood may even have been the inventor of the verse, though references to individual names appear earlier (Kretzmann 1966, pp. 66-67). Sherwood's treatise on syncategorematic terms has long been held to be if not the earliest such treatise, then the earliest still accessible (Boehner 1952, p. 7). The text was first edited by O'Donnell (1941), and this edition along with the Paris MS was the basis of Kretzmann's 1968 translation into English. O'Donnell's edition has since been superseded by the critical edition of Kann and Kirchhoff (2012), which also includes a translation into German. Kirchhoff (2008) is an extensive discussion and commentary on the Syncategormeta. Three other logical treatises follow the Introductiones and the Syncategoremata in the Paris manuscript, which have as a result been tentatively attributed to Sherwood. The first is a treatise on insolubles, ff. 46-54, edited by Roure (1970), and the second a treatise on obligationes found in Paris, Bibl. Nat. 16617. This text was edited by Green (1963), who tentatively attributed it to Sherwood. The third is the text Petitiones Contrariorum, edited by de Rijk (1976), covering puzzles arising from hidden contradictions in sets of premises. Sherwood may also be the author of the Fallaciae magistri Willelmi "Fallacies, of Master William" preserved in British Museum, King's Library MS 9 E XII, as well as the treatise ascribed to Richard of Sherwood noted above. Three further treatises have previously also been attributed to Sherwood, but the evidence for their attribution is less clear: the commentary on the Sentences of Peter Lombard noted above, a Distictiones theologicae, and a Concionces, or collection of sermons (Kretzmann 1968, p. 4). ## 2. Important doctrines Along with Peter of Spain, Lambert of Auxerre, and Roger Bacon, William of Sherwood was one of the first terminist logicians, working in a context where the newly discovered and translated Aristotelian material of the 12th century had become fully integrated into the logical and philosophical curricula through the works of people such as John le Page (Uckelman & Lagerlund, 2016). The terminist period of logic was marked by the development of logical genres and techniques that go beyond Aristotle and mere commentaries on his works. The two primary branches of terminist logic are the study of the properties of terms and the study of the meaning and function of syncategorematic words. Sherwood's Introduction picks up the first of these two branches in its fifth chapter, and his Syncategoremata deals with the second. The remainder of the Introductiones is devoted to the standard Aristotelian material: Chapter One on statements corresponds to On Interpretation; Chapter Two on predicables to the Categories; Chapter Three on syllogisms to the Prior Analytics; Chapter Four on dialectical reasoning to the Topics; and Chapter Six on sophistical reasoning to the Sophistical Refutations. We will focus our attention in this article on the unusual or distinctive aspects of his logical and semantic theory, as found in Chapter Five of the Introduction and in the Syncategoremata. ### 2.1. Modality In his Introduction, Sherwood gives two definitions for 'mode', depending on whether the term is being used broadly or strictly: "Broadly speaking, a mode is the determination of an act, and in this respect it goes together with every adverb", while strictly speaking we count not all adverbs as modes but only those which determine the inherence of the predicate in the subject (Kretzmann 1966, p. 40). Thus, on the broad conception of 'mode', both "Socrates is necessarily running" and "Socrates is swiftly running" count as modal statements, but on the narrow conception, only the former does. The six modes are 'true', 'false', 'possible', 'contingent', 'impossible', and 'necessary', though "the first two do not distinguish a modal proposition from an assertoric statement" (Kretzmann 1966, pp. 40-41). The other four modes give rise to properly modal propositions, and in statements expressing these propositions, the associated modal adverbs are 'possibly', 'contingently', 'impossibly', and 'necessarily'. Each of these modal adverbs is a syncategorematic term; and in the Syncategoremata, Sherwood devotes an entire chapter to necessario ('necessarily') and contingenter ('contingently'). Here it must be noted that Sherwood's views of modality differ significantly from modern views. Modern modal logic interprets modes in a nominal way, e.g., 'necessary:', 'impossible:', etc. But Sherwood is reluctant to admit a statement such as "That Socrates is running is necessary" as genuinely modal (Kretzmann 1966, p. 43), instead viewing this as an assertoric categorical statement predicating 'necessary' of "that Socrates is running". Similarly, "Socrates is running necessarily" does not count as a modal statement, because here 'necessarily' modifies 'running', not "the inherence of the predicate in the subject". The only genuinely modal construction is that exhibited by "Socrates is necessarily running" (Kretzmann 1966, p. 42). The importance of the adverbial construction, where the modal adverb modifies the inherence of the predicate with the subject, is emphasized in the discussions of modal adverbs in Syncategoremata, which include not only the chapter mentioned above but also analyses of how modal adverbs interact with other syncategorematic terms such as "only", "unless", and "if" (Uckelman 2020, SS5). Since Aristotle it was standard practice to note that 'possible' can be used in two ways. In the first 'possible' is compatible with 'necessary', but in the second, that which is necessary is not (merely) possible. This second way is usually called 'contingent'. Sherwood mentions this distinction, but goes on to say that he will be using it in a broader sense, in which possibility is compatible with necessity (Kretzmann 1966, p. 41). However, he makes a further distinction, which he does maintain, between the two ways that 'impossible' and 'necessary' can be used. These different ways are expressed in temporal notions: > > > [impossible] is used in one way of whatever cannot be true now or in > the future or in the past; and this is 'impossible per > se' ... It is used in the other way of whatever cannot be > true now or in the future although it could have been true in the > past ... and this is 'impossible per accidens'. > Similarly, in case something cannot be false now or in the future or > in the past it is said to be 'necessary per se' ... But it > is 'necessary per accidens' in case something cannot be > false now or in the future although it could have been [false] in the > past (Kretzmann 1966, p. 41). > > > Sherwood says that the reason it is important to separate modal propositions from assertoric ones is that > > > [s]ince our treatment is oriented toward syllogism, we have to > consider them under those differences that make a difference in > syllogism. These are such differences as ... modal, assertoric; and > others of that sort. For one syllogism differs from another as a > result of those differences (Kretzmann 1966, p. 39). > > > Despite this, there is "no treatment of modal syllogisms in any of the works that have been ascribed to Sherwood" (Kretzmann 1966, p. 39, fn. 58). For more on Sherwood's account of modality, see (Uckelman 2008). ### 2.2. Properties of terms The theory of the "properties of terms" forms the basis of medieval semantic theory and provides an account of how terms function within the broader context of sentences, as well as how these different functions can account for the different rules governing inference. Sherwood identifies the four main properties of terms as (1) signification; (2) supposition; (3) copulation; and (4) appellation (Kretzmann 1966, p. 105). In this classification he differs from other accounts which mention (1) signification; (2) supposition and copulation; (3) ampliation and restriction; (4) appellation; and (5) relation. Sherwood defines 'signification' as "a presentation of the form of something to the understanding", a definition harking back to 12th-century semantic theory (Kretzmann 1966, p. 105, fn. 2). Supposition and copulation are symmetrically related to each other; 'supposition' is "an ordering of the understanding of something under something else" while 'copulation' is "an ordering of the understanding of something over something else" (Kretzmann 1966, p. 105). Only substantive nouns, pronouns, and definite descriptions can have supposition, while only adjectives, participles, and verbs can have copulation (Kretzmann 1966, p. 106). Finally, 'appellation' is "the present correct application of a term -- i.e., the property with respect to which what the term signifies can be said of something through the use of the verb 'is'" (Kretzmann 1966, p. 106). That is, the appellata of a term are the presently existing things of which the term can currently be truly predicated. Appellation is between supposition and copulation: Substantive nouns, adjectives, and participles can have appellation, but pronouns and verbs cannot (Kretzmann 1966, p. 106). The central notion of these three is supposition, and the majority of the chapter is devoted to the division of supposition into its various kinds: ![Tree diagram: At the top is Supposition with 2 branches: Material and Formal; Formal has 2 branches: Simple and Personal; Simple has 3 leaves: Manerial, Reduplicative, and Unfixed; Personal has 2 branches: Determinate and Confused; Confused has 2 branches: Merely confused and Distributive confused; Distributive confused has 2 leaves: Mobile and Immobile.](sherwood-supposition.jpg) For a discussion of most of the notions in this division as well as a comparison of Sherwood's views to those of his 13th-century contemporaries and his 14th-century posterity, see the article Medieval Theories: Properties of Terms. We briefly comment on 'manerial' supposition, a type which does not appear outside of Sherwood. Some have considered manerialis an error for materialis, and considered Sherwood to have been talking here about material supposition (Grabmann 1937). However, the manuscript shows n rather than t, and the type of supposition is not the same as the material supposition, which is distinguished from formal supposition, but is itself a type of formal supposition. Sherwood says that manerial supposition is a type of simple supposition in which "a word can be posited for its significatum without any connection with things", and he gives as an example "Man is a species". The explanation is that this is manerial supposition because 'man' "supposits for the specific character in itself" (Kretzmann 1966, p. 111). The notion of maneries meaning 'character, way, mode, manner' can be found in 12th-century discussions of universals; for example, John of Salisbury notes that Joscelin, Bishop of Soissons, says that sometimes the words genus and species "are to be understood as things of a universal kind" and sometimes he "interprets them as modes (maneries) of things" (Hall, Book II, ch. 17, p. 84). John says he does not know where Joscelin got such a distinction, but it occurs in Abelard and in the 12th-century Fallacie Parvipontane (Hall, Book II, ch. 17, p. 84, fnn. a, b; de Rijk 1962, pp. 139, 562). ### 2.3. Semantics of universal affirmatives Statements are composed of two types of parts: principal and secondary. The principal parts of a statement are the nouns and verbs, that is, the subjects and predicates whose properties are the focus of the first branch of terminist logic. The secondary parts are those that remain: "the adjectival name, the adverb, and conjunctions and prepositions, for they are not necessary for the statement's being" (Kretzmann 1968, p. 13). These are words which do not have signification in themselves, but only in conjunction with other significative words, namely the subject and predicate (Kretzmann 1966, p. 24). Because these words do not have signification in themselves, they do not have any of the usual properties of terms -- supposition, copulation, and ampliation. Nevertheless, they still have a significative function. Explaining this function is the focus of Sherwood's treatise on syncategorematic terms. It is in this treatise that we can see evidence for what some have said is a distinctively English approach to syncategorematic terms (Uckelman & Lagerlund 2016). Sherwood and his English contemporary Bacon both consider distributive terms such as 'every' to be syncategorematic, while in continental works such as those by John le Page, Peter of Spain, and Nicholas of Paris, distributive terms are usually considered in summary treatises in the chapter on distribution (a chapter found in no English summary) (Braakhuis 1981, pp. 138-139). In his consideration of 'every', Sherwood argues that propositions involving 'all' require that there be at least three things for which the predicate term truly stands: > > > Rule [II] The sign 'every' or 'all' requires > that there be at least three appellata (Kretzman 1968, p. 23). > > > (Remember that an appellatum is something which both exists and which the subject stands for or refers to.) The justification for this is taken from Aristotle, who Sherwood quotes as saying "Of two men we say that they are two, or both, and not that they are all" (Kretzmann 1968, p. 23). That is, if there were fewer than three, it would be more appropriate to say 'Both' or 'One' rather than 'All'. Sherwood is also noteworthy for being the first logician we know of to treat 'is' as a syncategorematic term (Kretzmann 1968, p. 90). He argues that 'is' is equivocal, in that it can indicate either (1) actual being (esse actuale) or (2) habitual being (esse habituale) (Kretzmann 1968, p. 92, Kretzmann 1966, p. 125; Kretzmann translates habituale as 'relational' in Kretzmann 1966 and as 'conditional' in Kretzmann 1968). As a result, the sentence "Every man is an animal" is ambiguous. When 'is' is taken to indicate actual being, "Every man is an animal" "is false when no man exists" (Kretzmann 1968, p. 93). When 'is' is taken to indicate habitual being, then "insofar as ["Every man is an animal"] is necessary it has the force of this conditional: 'if it is man it is an animal'" (Kretzmann 1966, p. 125). In analyzing universal affirmative categorical sentences in this way, Sherwood can be seen as an 'early adopter' of our modern truth conditions for these sentences. A consequence of this view is a commitment to the existence of possibilia, things that could exist but which do not; such possibilia have a diminished sort of being (esse diminutum) (Kretzmann 1968, p. 93). This was an unusual position in the thirteenth century (Knuuttila 1993), and it was later denounced by William of Ockham (Freddoso & Schuurman 1980, p. 99). ## 3. Influence Bacon's approbation of Sherwood is not the only explicit reference we have to him by his contemporaries. The copy of Chapter Five of the Introduction found in MS Worcester Cath. Q. 13 mentioned earlier is followed by a collection of anonymous notes (also edited in Pinborg & Ebbesen 1984 and there called the Dubitationes). Pinborg and Ebbesen argue that these lecture notes, composed either for a teacher's or a student's use, are unlikely to have been written in response to lectures given in the 1290s, i.e., when the manuscript was copied, but rather date from the 1260s or the 1270s, favoring a date around 1270 (Pinborg and Ebbesen 1984, p. 104). This text thus provides evidence that Sherwood's works continued to be taught and circulated. Beyond these explicit mentions, it is clear that Sherwood's works were widely influential among logicians at the University of Paris in the second half of the 13th century. The most direct line of influence runs from William of Sherwood to his much better-known colleague Peter of Spain. It was for many years believed that Peter was the first person to translate Psellus's text (Mullinger 1873, p. 176), prior to the rediscovery of Sherwood's Introduction, which pre-dates Peter by two decades and which is closer to Psellus's work. Peter's Summary of Logic is clearly indebted to Sherwood. Kretzmann notes that "many of the features of the Summulae that made it far and away the most popular logic textbook in the Middle Ages seem quite plainly to have developed as modifications of more difficult discussions and less felicitous mnemonic devices in the Introductiones" (Kretzmann 1968, p. 6). On the other hand, Sherwood appears to have been influenced by Peter's treatise on syncategorematic terms in writing his own; Kretzmann provides a summary of passages in Sherwood's treatise which "can be read as allusions to positions taken by Peter" (1968, p. 7). Other influences can be found in Albert the Great, who used Sherwood's notion of appellation rather than Peter of Spain's (Kretzmann 1966, p. 5), and Thomas Aquinas, whose analysis of modal propositions follows Sherwood (Kretzmann 1966, p. 5; Uckelman 2008). Though Sherwood's indirect influence on the development of logic and semantics, via his influence on Peter of Spain, should not be underestimated, his direct influence beyond the 13th century was more minimal, in part because the influence of Peter of Spain was so great. In the 14th century, any influence Sherwood's approach to the properties of terms might have still had was almost entirely eclipsed by figures such as William of Ockham, John Buridan, and Albert of Saxony (Bos 2011, p. 773).
williams-bernard	## 1. Biography Bernard Williams was born in Essex in 1929, and educated at Chigwell School and Balliol College, Oxford, where he read Greats, the uniquely Oxonian degree that begins with Homer and Vergil and concludes with Thucydides, Tacitus, and (surprisingly perhaps) the latest in contemporary philosophy. Both Williams' subject of study and his tutors, especially Richard Hare, remained as influences throughout his life: the Greeks' sort of approach to philosophy never ceased to attract him, Hare's sort of approach never ceased to have the opposite effect. (Williams' contemporaries at Balliol, John Lucas for example, still report their mischievous use of "combined tactics" in philosophy tutorials with Hare; or perhaps the relevant preposition is "against".) Early in his career, Williams sat on a number of British government committees and commissions, most famously chairing the Committee on Obscenity and Censorship of 1979, which applied Mill's "harm principle" to the topic, concluding that restrictions were out of place where no harm could reasonably be thought to be done, and that by and large society has other problems which are more worth worrying about. At this time, he also began to publish books. His first book, Morality: an introduction to ethics (1972), already announced many of the themes that were to be central to his work. Already evident, in particular, were his questioning attitude to the whole enterprise of moral theory, his caution about the notion of absolute truth in ethics, and his hostility to utilitarianism and other moral theories that seek to systematise moral life and experience on the basis of such an absolute; as he later put it, "There cannot be any very interesting, tidy or self-contained theory of what morality is... nor... can there be an ethical theory, in the sense of a philosophical structure which, together with some degree of empirical fact, will yield a decision procedure for moral reasoning" (1981: ix-x). His second book, Problems of the Self (= PS; 1973), was a collection of his philosophical papers from 1956 to 1972; his further collections of essays (Moral Luck, 1981, and Making Sense of Humanity, 1998) were as much landmarks in the literature as this first collection. (Posthumously three further collections appeared: In the Beginning was the Deed (ed. Geoffrey Hawthorn), 2005, A Sense of the Past, 2005, and Philosophy as a Humanistic Discipline (2006); at least the second and third of these three collections are already having a considerable impact on philosophy, partly because they include essays that were already well-known and widely discussed in their original places of appearance.) In 1973 Williams also brought out a co-authored volume, Utilitarianism: For and Against, with J.J.C.Smart (= UFA); his contribution to this (the Against bit) being, in the present writer's view, a tour de force of philosophical demolition. Then in 1978 Williams produced Descartes: The Project of Pure Enquiry. This study could be described as his most substantial work outside ethics, but for the fact that the key theme of the book is the impossibility of Descartes' ambition to give a foundation, in the first-personal perspective, to the "absolute conception" of the world, a representation of the world "as it is anyway" that includes, explains, and rationally interrelates all other possible representations of the world (Williams 1968: 65)--a theme that is in an important sense not outside ethics at all. Williams worked in Britain until 1987, when he left for Berkeley in protest at the impact of the Thatcher government's policies on British universities. In 1985, he published the book that offers the most unified and sustained presentation of what Williams had to say about ethics and human life: Ethics and the Limits of Philosophy. On his return to Britain in 1990 (incidentally the year of Mrs. Thatcher's resignation) he succeeded his old tutor Richard Hare as White's Professor of Moral Philosophy at Oxford. While in the Oxford chair he produced Shame and Necessity (1993), a major study of Greek ethics which aims to distinguish what we think about ethics "from what we think that we think" (1993: 91): Williams' thesis is that our deepest convictions are often more like classical Greek ethical thought, and less like the post-Enlightenment "morality system", as Williams came to call it, than most of us have yet realised. (More about the morality system in sections 2 and 3.) In 1999 he published an introductory book on Plato (Routledge). After 1999--when he was knighted--he began to be affected by the cancer which eventually killed him, but was still able to bring out Truth and Truthfulness in 2002. In this Williams argues, against such deniers of the possibility or importance of objective truth as the pragmatist Richard Rorty and the deconstructionist Jacques Derrida, that it is indispensable to any human society to accept both truth and truthfulness as values, and sincerity and accuracy as corresponding virtues. Nor need such beliefs imply anything disreputably "metaphysical", in the Nietzschean sense that they lead us into a covert worship of what Williams takes to be the will o' the wisps of theism or Platonism. On the contrary, Williams argues, Nietzsche is on his side, not the deniers', because Nietzsche himself believes that, while a vindicatory history of the notions of truth and truthfulness certainly has to be a naturalistic one, that is not to say that such a history is impossible. We can write this history if we can supply a "potential explanation", to use Robert Nozick's term (Nozick 1974: 7-9), of how these notions could have arisen. Williams himself attempts to provide such a potential explanation, which if plausible will--given the impossibility of recovering the actual history--provide us with as much insight as we can reasonably hope for into how the notions of truth and truthfulness did in fact arise. Such an understanding of truth and truthfulness, Williams concludes, cannot lead us back into the pre-modern philosophical Edens where truth and truthfulness are taken to have their origin in something entirely transcendent, such as Plato's Forms, or God, or the cognitive powers of the Kantian subject; but it can lead us to the less elevated and more realistic hope that truth, as a human institution, will continue to sustain the virtues of truth "in something like the more courageous, intransigent, and socially effective forms that they have acquired over their history... and that the ways in which future people will come to make sense of things will enable them to see the truth and not be broken by it" (2002: 269). However, it should be noted that Williams, perhaps confusingly, claims to have vindicated the idea that truthfulness is an intrinsic value, while at the same time admitting that his genealogical explanation for the emergence of truthfulness only makes reference to what truthfulness effects or accomplishes. Difficult questions remain about his use of the Platonic category of 'intrinsic value' here (see Rorty 2002, Queloz 2018). Some of Williams' critics have complained that his work is largely "destructive" or "negative". Part of Williams' reply is that his nuanced and particularistic approach to ethics--via the detail of ethical questions--is negative only from the point of view of those operating under a completely undefended assumption. This is the assumption that, if there is to be serious ethical thought, then it must inevitably take the form of moral theory. The impression that any other approach could not be more than "negative" is itself part of the mindset that he is attacking. Williams often also meets the charge of negativity with a counter-offensive, which can be summarised as the retort that there's plenty to be negative about (1995: 217). "Often, some theory has been under criticism, and the more particular material [e.g. Williams' famous examples (UFA: 93-100) of George and Jim: see section 4 below] has come in to remind one of the unreality and, worse, distorting quality of the theory. The material... is itself extremely schematic, but... it at least brings out the basic point that... the theory is frivolous, in not allowing for anyone's experience, including the author's own. Alternatively, the theory does represent experience, but an impoverished experience, which it holds up as the rational norm--that is to say, the theory is stupid." But Williams did publish positive and constructive philosophy, most notably in Truth and Truthfulness itself. Moreover, this conception of Williams as a negative thinker is contestible. Miranda Fricker argues that Williams work represents an affirmation of what she calls "ethical freedom", which follows from the recognition that rationality itself significantly underdetermines how we should live (Fricker 2020). If the Platonic dream is that objective reason can always give us sufficient practical guidance, then Williams negative position is that this ambitious form of rationalism must fail. The correlative positive idea is that each agent has a certain kind of subjective freedom to shape their own conceptions of successful action and of the good life, conceptions which are not subject to censure or approval by some kind of universal rationality. It is important to notice that this only appears to be a negative thesis because we are assuming that a positive contribution to moral philosophy consists in establishing universal limits or boundaries on justified action. But from the perspective of the deciding agent, Williams' thesis is anything but negative; rather, it is positively liberating, freeing agents to shape their own projects in accordance with their own character. Since one of Williams' main objectives is to demonstrate the frivolity and/ or stupidity of too much contemporary moral theory, it is natural to structure our more detailed examination of his contributions to philosophy by beginning with its critical side. The first two of the three themes from Williams that we pick for closer attention are both campaigns of argument against positions: respectively, against the "morality system" (sections 2 and 3), and against utilitarianism (section 4). The aptness of this arrangement comes out in the fact that, as we shall see, most of the constructive positions that Williams adopts can be seen as the "morals" of these essentially destructive stories. Even what we take to be Williams' single most important positive thesis, a view about the nature of motivation and reasons for action, emerges from his critique of other people's views about reasons for action; more about that, his famous "internal reasons" argument, in section 5. ## 2. Williams and Moral Philosophy In the Preface to Williams' first book he notes the charge against contemporary moral philosophy "that it is peculiarly empty and boring". [1]. Moral philosophy, he claims, has found an original way of being boring: and this is "by not discussing moral issues at all." Certainly, this charge is no longer as fair now as it was in 1972. Today there is an entire discipline called "applied" or "practical" ethics, not to mention sub-disciplines called environmental, business, sport, media, healthcare, and medical ethics, to the extent that hardly any moral issues are not discussed by philosophers nowadays. However, while some or even many philosophers today do applied ethics by applying some general, abstract theory, a problem with many of them, as Williams pointed out in an interview in 1983, is that those who proceed in this way often seem to lose any real interest in the perspectives of the human beings who must actually live with a moral problem: > > I do think it is perfectly proper for some philosophers all of the > time and for other philosophers some of the time to be engaged in > technical issues, without having to worry all the time whether their > work is going to revolutionise our view of the employment situation, > or something of that kind. Indeed, without criticising any particular > thinkers or publicists, a problem with "applied ethics" is > that some people have a bit of ready-made philosophical theory, and > they whiz in, a bit like hospital auxiliary personnel who aren't > actually doctors. That kind of applied philosophy isn't even > half-interesting...[2] > He continues: > > ...the temptation is to find a way to apply philosophy to > immediate and practical problems and to do so by arguing about those > problems in a legalistic way. You are tempted to make your moral > philosophy course into a quasi-legal course... All the > philosophical journals are full of issues about women's rights, > abortion, social justice, and so on. But an awful lot of it consists > of what can be called in the purely technical sense a kind of > casuistry, an application of certain moral systems or principles or > theories to discussing what we should think about abortion. > We are now able to see how Williams conceived of the relation between philosophy and lived ethical experience. He firmly believed that philosophy should speak to that experience, on pain of being "empty and boring". But he did not think that we should follow the moral philosophers of his day in preferring the schematic over the detailed, or the general over the particular. Here, Williams joins a long critical tradition that stretches at least back to G.W.F. Hegel, whose own claim that Kant's moral theory was "empty" is importantly related to Williams' own charge against moral theorizing. Moreover, as a general criticism of moral philosophy, this point arguably remains quite correct even today.[3] Bearing this general orientation in mind, we now turn to a discussion of Williams' more determinate charges against various types of moral theory. ## 3. Against the "peculiar institution" The unwillingness to be drawn into discussing particular ethical issues that Williams complains of was a reflection of earlier developments. In particular, it was a reflection of the logical positivists' disdain for "moralising", a disdain which arose naturally from the emotivist conviction of philosophers such as A.J.Ayer that to utter one's first-order moral beliefs was to say nothing capable of truth or falsehood, but simply to express one's attitudes, and hence not a properly philosophical activity at all. More properly philosophical, on emotivist and similar views, was a research-programme that became absolutely dominant during the 1950s and 1960s in Anglophone philosophy, including moral philosophy. This was linguistic analysis in the post-Wittgensteinian style of J.L.Austin, who hoped, starting from an examination of the way we talk (whoever "we" may be: more on that in a minute), to reveal the deep structure of a wide variety of philosophically interesting phenomena: among the most successful applications of Austin's method were his studies of intention, other minds, and responsibility. When Ayer's dislike of preaching and Austin's method of linguistic analysis were combined in moral philosophy, one notable result[4] was Richard Hare's "universal presciptivism", a moral system which claimed to derive the form of all first-order moral utterances simply from linguistic analysis of the two little words "ought" and "good". Hare argued that it followed from the logic of these terms, when used in their full or specially moral sense, that moral utterances were (1) distinct from other utterances in being, not assertions about how the world is, but prescriptions about how we think it ought to be; and (2) distinct from other prescriptions in being universalisable, by which Hare meant that anyone who was willing to make such a prescription about any agent, e.g. himself, should be equally willing to make it about any other similarly-placed agent. In this way Hare's theory preserved the important emotivist thesis that a person's moral commitments are not rationally challengeable for their content, but only for their coherence with that person's other moral commitments--and thus tended to keep philosophical attention away from questions about the content of such commitments.[5] At the same time, his system was also able to accommodate a central part of the Kantian outlook, because it gave a rationale[6] for the twin views that moral commitments are overriding as motivations (so that they will motivate if present), and that they are overriding as rational justifications (so that they rationally must motivate if they are present). Hence cases like akrasia, where a moral commitment appears to be present in an agent but gets overridden by something else along the way to action, must on Hare's view be cases where something has gone wrong: either the agent is irrational, or else she has not really uttered a full-blown moral ought, a properly moral commitment, either because (1) the prescription that she claims to accept is not really one that she accepts at all, or (2) because although she does sincerely accept this prescription, she is not prepared to give it a fully universalised form, and hence does not accept it as a distinctively moral prescription. In assessing a position like Hare's, Williams and other critics often begin with the formidable difficulties involved in the project of deducing anything much about the structure of morality from the logic of moral language: see e.g., Geach, "Good and Evil", Analysis 1956, and Williams 1972: 52-61. These difficulties are especially acute when the moral language we consider is basically just the words "ought" and "good" and their opposites. "If there is to be attention to language, then there should be attention to more of it" (Williams 1985: 127); the closest Williams comes to inheriting the ambitions of linguistic analysis is his defence of the notion of morally "thick concepts" (1985: 140-143)[7]. These--Williams gives coward, lie, brutality and gratitude as examples--are concepts that sustain an ethical load of a culturally-conditioned form, and hence succeed both in being action-guiding (for members of that culture), and in making available (to members of that culture) something that can reasonably be described as ethical knowledge. Given that my society has arrived at the concept of brutality, that is to say has got clear, at least implicitly, about the circumstances under which it is or is not applicable, there can be facts about brutality (hence, ethical facts) and also justified true beliefs[8] about brutality (hence, ethical knowledge). Moreover, this knowledge can be lost, and will be lost, if the concept and its social context is lost. (For a strikingly similar philosophical project to that suggested by this talk of thick concepts, cp. Anscombe 1958a, and Philippa Foot's papers "Moral Beliefs" and "Moral Arguments", both in her Virtues and Vices.) Before we even get to the problem how the structure of morality is supposed to follow from moral language, there is the prior question "Whose moral language?"; and this is a deeper question. We do not suppose that all moral language (not even--to gesture towards an obviously enormous difficulty--all moral language in English) has always and everywhere had exactly the same presuppositions, social context, or cultural significance. So why we should suppose that moral language has always and everywhere had exactly the same meaning, and has always been equally amenable to the analysis of its logical structure offered by Hare? (Or by anyone else: it can hardly be insignificant that when G.E.Moore (Principia Ethica sections 17, 89) anticipated Hare by offering a linguistic analysis of "good", his analysis of this term was on the face of it quite different from Hare's, despite Moore's extreme historical and cultural proximity to Hare.) Basing moral objectivism on the foundations of a linguistic approach leaves it more vulnerable to relativistic worries than other foundations do. For on the linguistic approach, we also face a question of authority, the question why, even if something like the offered analysis of our moral language were correct, that should license us to think that the moral language of our society has any kind of universal jurisdiction over any society's. In its turn, this question is very apt to breed the further question how, if our moral language lacks this universal jurisdiction over other societies, it can make good its claim to jurisdiction even in our society. These latter points about authority are central to Williams' critique of contemporary moral philosophy. Like Anscombe before him, Williams argues that the analysts' tight focus on such words as "ought", "right", and "good" has come, in moral theory, to give those words (when used in their alleged "special moral sense") an air of authority which they could only earn against a moral and religious backdrop--roughly, the Christian world-view--that is nowadays largely missing. What Williams takes to be the correct verdict on modern moral theory is therefore rather like Nietzsche's on George Eliot:[9] the idea that morality can and will go on just as before in the absence of religious belief is simply an illusion that reflects a lack of "historical sense". As Anscombe[10] puts it (1958: 30), "it is not possible to have a [coherent law conception of ethics] unless you believe in God as a law-giver... It is as if the notion 'criminal' were to remain when criminal law and criminal courts had been abolished and forgotten." And as Williams puts it (1985: 38), the "various features of the moral judgement system support each other, and collectively they are modelled on the prerogatives of a Pelagian God." What then are these features? That is a big question, because Williams spent pretty well his whole career describing and criticising them. But he gives his most straightforward, and perhaps the definitive, summary of what the "morality system" comes to in the last chapter of Ethics and the Limits of Philosophy. (The chapter's title provocatively describes morality as "the peculiar institution", this phrase being the American Confederacy's standard euphemism for slavery.[11]) Following this account, we may venture to summarise the "morality system" in nine leading theses.[12] First, the morality system is essentially practical: my moral obligations are always things that I can do, so that "if my deliberation issues in something that I cannot do, then I must deliberate again" (1985: 175). This implies, second, that moral obligations cannot (really) conflict (185: 176). Third, the system includes a pressure towards generalisation which Williams calls "the obligation out-obligation in principle": this is the view that every particular moral obligation needs the logical backing of a general moral obligation, of which it is to be explained as an instance. Fourth, "moral obligation is inescapable" (185: 177): "the fact that a given agent would prefer not to be in [the morality] system will not excuse him", because moral considerations are, in some sense like the senses sharpened up by Kant and by Hare, overriding considerations. In any deliberative contest between a moral obligation and some other consideration, the moral obligation will always win out, according to the morality system. The only thing that can trump an obligation is another obligation (1985: 180); this is a fifth thesis of the morality system, and it creates pressure towards a sixth, that as many as possible of the considerations that we find practically important should be represented as moral obligations, and that considerations that cannot take the form of obligations cannot really be important after all (1985: 179). Seventh, there is a view about the impossibility of "moral luck" that we might call, as Williams calls it, the "purity of morality" (1985: 195-6): "morality makes people think that, without its very special obligation, there is only inclination; without its utter voluntariness, there is only force; without its ultimately pure justice, there is no justice"; whereas "in truth", Williams insists, "almost all worthwhile human life lies between the extremes that morality puts before us" (1985: 194). Eighth, "blame is the characteristic reaction of the morality system" to a failure to meet one of its obligations (1985: 177); and "blame of anyone is directed to the voluntary" (1985: 178). Ninth, and finally, the morality system is impersonal. We shall set this last feature of the system aside until section 4, and focus, for now, on the other eight. For each of the theses, Williams has something (at least one thing) of deep interest to say about why we should reject it. The first and second--about the practicality of morality and the impossibility of real conflict--are his target in his well-known early paper "Ethical Consistency" (PS: 166-186). In real life, Williams argues, there surely are cases where we find ourselves under ethical demands which conflict. These conflicts are not always eliminable in the way that the morality system requires them always to be--by arguments leading to the conclusion that one of the oughts was only prima facie (in Ross's terminology: see Williams 1985: 176-177), or pro tanto (in a more recent terminology: see Kagan 1989), or in some other way eliminable from our moral accounting. But, Williams argues, "it is surely falsifying of moral thought[13] to represent its logic as demanding that in a conflict... one of the conflicting oughts must be totally rejected [on the grounds that] it did not actually apply" (PS: 183-4).[14] For the fact that it did actually apply is registered by all sorts of facts in our moral experience, including the very important phenomenon of ineliminable agent-regret, regret not just that something happened, but that it was me who made it happen (1981: 27-30). Suppose for example[15] that I, an officer of a wrecked ship, take the hard decision to actively prevent further castaways from climbing onto my already dangerously overcrowded lifeboat. Afterwards, I am tormented when I remember how I smashed the spare oar repeatedly over the heads and hands of desperate, drowning people. Yet what I did certainly brought it about that as many people as possible were saved from the shipwreck, so that a utilitarian would say that I brought about the best consequences, and anyone might agree that I found the only practicable way of avoiding a dramatically worse outcome. Moreover, as a Kantian might point out, there was nothing unfair or malicious about what I did in using the minimum force necessary to repel further boarders: my aim, since I could not save every life, was to save those who by no choice of mine just happened to be in the lifeboat already; this was an aim that I properly had, given my role as a ship's officer; and it was absolutely not my intention to kill or (perhaps) even to injure anyone. So what will typical advocates of the morality system have to say to me afterwards about my dreadful sense of regret?[16] If they are--as perhaps they had better not be--totally consistent and totally honest with me, what they will have to say is simply "Don't give it a second thought; you did what morality required, so your deep anguish about it is irrational." And that, surely, cannot be the right thing for anyone to say. My anguish is not irrational but entirely justified. Moreover, it is justified simply as an ex post facto response to what I did: it does not for instance depend for its propriety upon the suggestion--a characteristic one, for many modern moral theorists--that there is prospective value for the future in my being the kind of person who will have such reactions. The third thesis Williams mentions as a part of the morality system is the obligation out-obligation in principle, the view that every particular moral obligation needs the backing of a general moral obligation, of which it is to be explained as an instance. Williams argues that this thesis will typically engage the deliberating agent in commitments that he should not have. For one thing, the principle commits the agent to an implausibly demanding view of morality (1985: 181-182): > > The immediate claim on me, "In this emergency, I am under an > obligation to help", is thought to come from, "One is > under this general obligation: to help in an emergency"... > But once the journey into more general obligations has started, we may > begin to get into trouble--not just philosophical trouble, but > conscience trouble--with finding room for morally indifferent > actions... if we have accepted general and indeterminate > obligations to further various moral objectives... they will be > waiting to provide work for idle hands, and the thought can gain a > footing that... I am under an obligation not to waste time in > doing things that I am under no obligation to do. At this stage, > certainly, only an obligation can beat an obligation [cp. the > fourth thesis], and in order to do what I wanted to > do, I shall need one of those fraudulent items, a duty to myself. > It is only the pressure to systematise that leads us to infer that, if it is X's particular obligation in S to ph, then this must be because there is a general obligation, on any X-like agent, to ph in any S-like situation.[17] Unless some systematic account of morality is true--as Williams of course denies--there is no obvious reason why this inference must hold in any more than trivial sense. But even if it does hold, it is not clear how the general duty explains the particular one; why are general obligations any more explanatory than particular ones? Certainly anyone who is puzzled as to why there is this particular obligation, say to rescue one's wife, is unlikely to find it very illuminating to be pointed towards the general obligation of which it is meant to be an instance. (Williams' closeness to certain particularist strategies should be obvious here: cp. Dancy 2004, and Chappell 2005.) Another inappropriate commitment arising from the obligation out-obligation in principle, famously spelled out at 1981: 18, is the agent's commitment to a "thought too many". If an agent is in a situation where he has to choose which of two people to rescue from some catastrophe, and chooses the one of the two people who is his wife, then "it might have been hoped by some people (for instance, by his wife) that his motivating thought, fully spelled out, would be the thought that it was his wife, not that it was his wife and that in situations of this kind it is permissible to save one's wife." The morality system, Williams is suggesting, makes nonsense of the agent's action in rescuing his wife: its insistence on generality obscures the particular way in which this action is really justified for the agent. Its real justification has nothing to do with the impersonal and impartial standards of morality, and everything to do with the place in the agent's life of the person he chooses to rescue. For Williams, the standard of "what makes life meaningful" is always deeper and more genuinely explanatory than the canon of moral obligation; the point is central, and we shall come back to it below in sections 3 and 4. Williams' opposition to the fourth thesis, about the inescapability of morality, rests on the closely-related contrast he draws between moral considerations, and considerations about "importance": "ethical life is important, but it can see that things other than itself are important" (1985: 184). This notion of importance is grounded, ultimately, in the fact "that each person has a life to lead" (1985: 186). What is important, in this sense, is whatever humans need to make it possible to lead what can reasonably be recognised as meaningful lives; the notion of importance is of ethical use because, and insofar as, it reflects the facts about "what we can understand men as needing and wanting" (1972: 95). The notion that moral obligation is inescapable is undermined by careful attention to this concept of importance, simply because reflection shows that the notion of moral obligation will have to be grounded in the notion of importance if it is to be grounded in anything that is not simply illusory. But if it is grounded in that, then it cannot itself be the only thing that matters. Hence moral obligation cannot be inescapable, which refutes the fourth thesis of the morality system; other considerations can sometimes override or trump an obligation without themselves being obligations, which refutes the fifth; and there can be no point in trying to represent every practically important consideration as a moral obligation, so that it is for instance a distortion for Ross (The Right and The Good, 21 ff.) to talk of "duties of gratitude" (1985: 181); which refutes the sixth. It is worth noting that the fourth thesis,that morality is inescapable for all agents in all situations, has implications beyond the realm of personal ethics. Williams' most enduring contribution to political philosophy is his denial of political moralism, the view that that politics is always and everywhere regulated by morality. The result is his celebrated political realist position, which denies that legitimate politics can just consist in the systematic application of some moral theory or principle. Rather, for Williams, the basic political question is: can the state secure the bare conditions of "order, protection, safety, trust, and the conditions of cooperation"? If so, it has met the Basic Legitimation Demand, which is not a moral demand but rather a kind of precondition for the existence of politics at all. For Williams, all of this means that political normativity stands outside of the morality system (IBD, ch.1). More concretely, this means that certain kinds of considerations distinctive to politics must be allowed to retain self-standing practical significance. We might illustrate this thought by noting that in the realm of ordinary interpersonal ethics, it seems perfectly reasonable to try to reduce or minimize coercive relations, whereas in politics this demand is nonsensical, since politics begins with the question of how a governing body's coercive power ought to be deployed. Political philosophy aside, Williams also denies that personal decision-making must always and everywhere be regulated by moral normativity. Another vivid instance of the escapability of moral obligations is Williams' own example of "Gauguin", a (fictionalised) artist who deliberately rejects a whole host of moral obligations (to his family, for instance) because he finds it more "important", in this sense, to be a painter. As Williams comments (1981: 23), "While we are sometimes guided by the notion that it would be the best of worlds in which morality were universally respected and all men were of a disposition to affirm it, we have, in fact, deep and persistent reasons to be grateful that that is not the world we have"; in other words, moral obligation is escapable because it is not in the deepest human interest that it should be inescapable. ("Because": the fact that this sort of inference is possible in ethics is itself a revealing fact about the nature of ethics.) Williams' Gauguin example, we have suggested, has force against the thesis that morality is inescapable. It also has force against the seventh thesis of the morality system, its insistence on "purity" and its denial of what Williams calls "moral luck". To understand this notion, begin with the familiar legal facts that attempted murder is a different and less grave offence than murder, and that dangerous driving typically does not attract the same legal penalty if no one is actually hurt. Inhabitants of the morality system will characteristically be puzzled by this distinction. How can it be right to assign different levels of blame, and different punishments, to two agents whose mens rea was exactly the same--it was just that one would-be murderer dropped the knife and the other didn't--or to two equally reckless motorists--one of whom just happened to miss the pedestrians while the other just happened to hit them? One traditional answer--much favoured by the utilitarians--is that these sorts of thoughts only go to show that the point of blame and punishment is prospective (deterrence-based), not retrospective (desert-based). There are reasons for thinking that blame and punishment cannot be made sense of in this instrumental fashion (cp. UFA: 124, 1985: 178). "From the inside", both notions seem essentially retrospective, so that if a correct understanding of them said that they were really fictions serving a prospective social function, no one who knew that could continue to use these notions "from the inside": that is, the notions would have proved unstable under reflection for this person, who would thereby have lost some ethical knowledge. If this gambit fails, another answer--favoured by Kantians, but available to utilitarians too--is that the law would need to engage in an impossible degree of mind-reading to pick up all and only those cases of mens rea that deserve punishment irrespective of the outcomes. Even if this is the right thing to say about the law, the answer cannot be transposed to the case of morality: morality contrasts with the law precisely because it is supposed to apply even to the inner workings of the mind. Thus, morality presumably ought to be just as severe on the attempted murderer and the reckless but lucky motorist as it is on their less fortunate doubles. Williams has a different answer to the puzzle why we blame people more when they are successful murderers, or not only reckless but lethal motorists, despite the fact that they have no voluntary control over their success as murderers or their lethality as motorists. His answer is that--despite what the morality system tells us--our practice of blame is not in fact tied exclusively to voluntary control. We blame people not only for what they have voluntarily done, but also for what they have done as a matter of luck: we might also say, of their moral luck. The way we mostly think about these matters often does not distinguish these two elements of control and luck at all clearly--as is also witnessed by the important possibility of blaming people for what they are. These phenomena, Williams argues, help to reveal the basic unclarity of our notion of the voluntary; they also help to show how "what we think" about blame is not always the same as "what we think we think". Parallel points apply with praise. Someone like the Gauguin of Williams' story can be seen as taking a choice of the demands of art over the obligations of family life which will be praiseworthy or blameworthy depending on how it turns out ("The only thing that will justify his choice will be success itself", 1981: 23). Here success or failure is quite beyond Gauguin's voluntary control, and thus, if the morality system were right, would have to be beyond the scope of praise and blame as well. A fault-line in our notions of praise and blame is revealed by the fact that, intuitively, it is not: the case where Gauguin tries and fails to be an artist is one where we condemn him "for making such a mess of his and others' lives", the case where he tries and succeeds is, very likely, one where we say, a little grudgingly perhaps, "Well, all right then -- well done." We have the morality system's narrow or "pure" versions of these notions, in which they apply only to (a narrow or "pure" version of) the voluntary; but we also have a wider version of the notions of praise and blame, in which they also apply to many things that are not voluntary on any account of the voluntary. Williams' thesis about moral luck is that the wider notions are more useful, and truer to experience. (For a sustained defense of Williams on these basic points, see Joseph Raz "Agency and Luck") Nor is it only praise and blame that are in this way less tightly connected to conditions about voluntariness than the morality system makes them seem. Beyond the notion of blame lie other, equally ethically important, notions such as regret or even anguish at one's actions; and these notions need not show any tight connection with voluntariness either. As we saw in my shipwreck example above, the mere fact that it was unreasonable to expect the ship's officer to do much better than he did in his desperate circumstances does not make it reasonable to fob off his anguish with "Don't give it a second thought". Likewise, to use an example of Williams' own (1981: 28), if you were talking to a driver who through no fault of his own had run over a child, there would be something remarkably obtuse--something irrelevant and superficial, even if correct--about telling him that he shouldn't feel bad about it provided it wasn't his fault. As the Greeks knew, such terrible happenings will leave their mark, their miasma, on the agent. "The whole of the Oedipus Tyrannus, that dreadful machine, moves towards the discovery of just one thing, that he did it. Do we understand the terror of that discovery only because we residually share magical beliefs in blood-guilt, or archaic notions of responsibility? Certainly not: we understand it because we know that in the story of one's life there is an authority exercised by what one has done, and not merely by what one has intentionally done" (1993: 69). This sums up Williams' case for thinking that the wider notion of praise and blame is tenable in a way that the narrower notion is not because of its dependence on a questionably "pure" account of the voluntary (1985: 194; cp. MSH Essays 1-3). In this way, he controverts the eighth thesis of the morality system, its insistence on the centrality of blame; which was the last thesis that we listed apart from impersonality, the discussion of which we have postponed till the next section. So much on Williams' critique of the "morality system". How far our discussion has delivered on its promise to show how Williams' positive views emerge from his negative programmes of argument, we leave, for now, to the reader's judgement: we shall say something more to bring the threads together in section 5. Before that, we turn to Williams' critique of utilitarianism, the view that actions, rules, dispositions, motives, social structures, (...etc.: different versions of utilitarianism feature, or stress, some or all of these things) are to be chosen if and only if they maximally promote utility or well-being. ## 4. "The day cannot be too far off...": Williams against utilitarianism > > [T]he important issues that utilitarianism raises should be discussed > in contexts more rewarding than that of utilitarianism itself... > the day cannot be too far off in which we hear no more of it (UFA: > 150).[18] > Williams opposes utilitarianism partly for the straightforward reason that it is an "ism",[19] a systematisation--often a deliberately brisk or indeed "simple-minded" one (UFA: 149)--of our ethical thinking. As we have already seen, he believes that ethical thinking cannot be systematised without intolerable distortions and losses, because to systematise is, inevitably, to streamline our ethical thinking in a reductionist style: "Theory typically uses the assumption that we probably have too many ethical ideas, some of which may well turn out to be mere prejudices. Our major problem now is actually that we have not too many but too few, and we need to cherish as many as we can" (1985: 117). Again, as a normative system, utilitarianism is inevitably a systematisation of our responses, a way of telling us how we should feel or react. As such it faces the same basic and (for Williams) unanswerable question as any other such systematisation, "by what right does it legislate to the moral sentiments?" (1981: x). Of course, Williams also opposes utilitarianism because of the particular kind of systematisation that it is--namely, a manifestation of the morality system. Pretty well everything said in sections 2 and 3 against morality in general can be more tightly focused to yield an objection to utilitarianism in particular, and sometimes this is all we will need to bear in mind to understand some specific objection to utilitarianism that Williams offers. Thus, for instance, utilitarianism in its classic form is bound to face the objections that face any moral system that ultimately is committed to denying the possibility of real moral conflict or dilemma, and the rationality of agent-regret. Given its insistence on generality, it faces the "one thought too many" objection as well, at least in any version that keeps criterion of rightness and decision procedure in communication with each other. Above all, utilitarianism is in trouble, according to Williams, because of the central theoretical place that it gives to the ninth thesis of the morality system--the thesis that we put on one side earlier, about impersonality. Other forms of the morality system are impersonal too, of course, notably Kantianism: "if Kantianism abstracts in moral thought from the identity of persons,[20] utilitarianism strikingly abstracts from their separateness" (1981: 3). Like Kantianism, but on a different theoretical basis, utilitarianism abstracts from the question of who acts well, which for utilitarianism means "who produces good consequences?". It is concerned only that good consequences be produced, but it does not offer a tightly-defined account of what it is for anything to be a consequence. Or rather it does offer an account, but on this account the notion of a consequence is so loosely defined as to be all-inclusive (1971: 93-94): > > Consequentialism is basically indifferent to whether a state of > affairs consists in what I do, or is produced by what I do, where that > notion is itself wide... All that consequentialism is interested > in is the idea of these doings being consequences of what I > do, and that is an idea broad enough to include [many sorts of] > relations. > This explains why consequentialism has the strong doctrine of negative responsibility that leads it to what Williams regards as its fundamental absurdity. Because, for the utilitarian, it can't matter in itself whether (say) a given death is a result of what I do in that I pull the trigger, or a result of what I do in that I refuse to lie to the gunman who is looking for the person who dies, doing and allowing must be morally on a par for the utilitarian, as also must intending and foreseeing. Williams himself is not particularly impressed by those venerable distinctions;[21] but he does think that there is a real and crucial distinction that is closely related to them, and that it is a central objection to utilitarianism that it ignores this distinction. The distinction in question, which utilitarian ignores by being impersonal, is the distinction between my agency and other people's. It is this distinction, and its fundamental moral importance, that lies at the heart of Williams' famous (but often misunderstood) "integrity objection". In a slogan, the integrity objection is this: agency is always some particular person's agency; or to put it another way, there is no such thing as impartial agency, in the sense of impartiality that utilitarianism requires. The objection is that utilitarianism neglects the fact that "practical deliberation [unlike epistemic deliberation] is in every case first-personal, and the first person is not derivative or naturally replaced by [the impersonal] anyone" (1985: 68). Hence we are not "agents of the universal satisfaction system", nor indeed primarily "janitors of any system of values, even our own" (UFA: 118). No agent can be expected to be what a utilitarian agent has to be--someone whose decisions "are a function of all the satisfactions which he can affect from where he is" (UFA: 115); no agent can be required, as all are required by utilitarianism, to abandon his own particular life and projects for the "impartial point of view" or "the point of view of morality", and do all his decision-making, including (if it proves appropriate) a decision to give a lot of weight to his own life and projects, exclusively from there. As Williams famously puts it (UFA: 116-117): > > The point is that [the agent] is identified with his actions as > flowing from projects or attitudes which... he takes seriously at > the deepest level, as what his life is about... It is absurd to > demand of such a man, when the sums come in from the utility network > which the projects of others have in part determined, that he should > just step aside from his own project and decision and acknowledge the > decision which utilitarian calculation requires. It is to alienate him > in a real sense from his actions and the source of his action in his > own convictions. It is to make him into a channel between the input of > everyone's projects, including his own, and an output of > optimific decision; but this is to neglect the extent to which > his projects and his decisions have to be seen as > the actions and decisions which flow from the projects and attitudes > with which he is most closely identified. It is thus, in the most > literal sense, an attack on his integrity. > Here, Williams' commitment to the importance of subjective authenticity is on full display. "The most literal sense" of "integrity" is, according to Chambers' Dictionary (1977 edition), "entireness, wholeness: the unimpaired state of anything"; then "uprightness, honesty, purity". For our purposes the latter three senses in this dictionary entry should be ignored. It is the first three that are relevant to Williams' argument; the word's historical origin in the Latin in-teger, meaning what is not touched, taken away from, or interfered with, is also revealing. An agent's integrity, in Williams' sense, is his ability to originate actions, to further his own initiatives, purposes or concerns, and thus to be something more than a conduit for the furtherance of others' initiatives, purposes or concerns--including, for example and in particular, those which go with the impartial view. Moreover, integrity is an essential component of character, since, for Williams, an agent's character is identical to their set of deep projects and commitments. Williams' point, then, is that unless any particular agents are allowed to initiate actions and to have "ground projects", then either the agents under this prohibition will be subjects for manipulation by other agents who are allowed to have ground projects--the situation of ideological oppression. Or else, if every agent lies under this prohibition and all agents are made to align themselves only with the ground projects of "the impartial point of view", there will not be any agents. To put it another way, all will be ideologically oppressed, but by the ideology itself rather than by another agent or group of agents who impose this ideology. For all agents will then have lost their integrity, in the sense that no single agent will be an unimpaired and individual whole with projects of his own that he might identify himself with; all agents will have to abandon all "ground projects" except the single project that utilitarianism gives them, that of maximising utility by whatever means looks most efficient, and to order all their doings around no other initiatives except those that flow from this single project. What we previously thought of as individual agents will be subsumed as parts of a single super-agent--the utilitarian collective, if you like--which will pursue the ends of impartial morality without any special regard for the persons who compose it, and which is better understood as a single super-agent than as a group of separate agents who cooperate; rather like a swarm of bees or a nest of ants. It is important not to misunderstand this argument. One important misunderstanding can arise fairly naturally from Williams' two famous examples (UFA: 97-99) of "Jim", who is told by utilitarianism to murder one Amazon villager to prevent twenty being murdered, and "George", who is told by utilitarianism to take a job making weapons of mass destruction, since the balance-sheet of utilities shows that if George refuses, George and his family will suffer poverty and someone else--who will do more harm than George--will take the job anyway. It is easy to think that these stories are simply another round in the familiar game of rebutting utilitarianism by counter-examples, and hence that Williams' integrity objection boils down to the straightforward inference (1) utilitarianism tells Jim to do X and George to do Y, (2) but X and Y are wrong (perhaps because they violate integrity?), so (3) utilitarianism is false. But this cannot be Williams' argument, because in fact Williams denies (2). Not only does he not claim that utilitarianism tells both Jim and George to do the wrong things. He even suggests, albeit rather grudgingly, that utilitarianism tells Jim (at least) to do the right thing. (UFA: 117: "...if (as I suppose) the utilitarian is right in this case...") Counter-examples, then, are not the point: "If the stories of George and Jim have a resonance, it is not the sound of a principle being dented by an intuition" (WME 211). The real point, he tells us, is not "just a question of the rightness or obviousness of these answers"; "It is also a question of what sort of considerations come into finding the answer" (UFA: 99). "Over all this, or round it, and certainly at the end of it, there should have been heard 'what do you think?', 'does it seem like that to you?', 'what if anything do you want to do with the notion of integrity?'" (WME 211). Again, despite Williams' interest in the moral category of "the unthinkable" (UFA: 92-93; cp. MSH Essay 4), it is not Williams' claim that either Jim or George, if they are (in the familiar phrase) "men of integrity", are bound to find it literally unthinkable to work in WMD or to shoot a villager, or will regard these actions as the sort of things that come under the ban of some absolute prohibition that holds (in Anscombe's famous phrase) whatever the consequences: "this is a much stronger position than any involved, as I have defined the issues, in the denial of consequentialism... It is perfectly consistent, and it might be thought a mark of sense, to believe, while not being a consequentialist, that there was no type of action which satisfied [the conditions for counting as morally prohibited no matter what]" (UFA: 90).[22] Nor therefore, to pick up a third misunderstanding of the integrity objection, is Williams offering an argument in praise of "the moral virtue of integrity", where "integrity" is--in jejune forms of this misreading--the virtue of doing the right thing not the wrong thing, or--in more sophisticated forms--a kind of honesty about what one's values really are and a firm refusal to compromise those values by hypocrisy or cowardice (usually, with the implication that one has hold of the right values). An agent can be told by utilitarianism to do something terrible in order to avoid something even worse, as Jim and George are. Williams is not opposing this sort of utilitarian conclusion by arguing that the value of "integrity" in the sense of the word that he anyway does not have in mind--the personal quality--is something else that has to be put into the utilitarian balance-sheet, and that when you put it in, the utilitarian verdict comes out differently. Nor is Williams saying, even, that the value of integrity in the sense of the word that he does have in mind--roughly, allowing agents to be agents--is something else that has to be put into the utilitarian balance-sheet, as it is characteristically put in by indirect utilitarians such as Peter Railton and Amartya Sen: "The point here is not, as utilitarians may hasten to say, that if the project or attitude is that central to his life, then to abandon it will be very disagreeable to him and great loss of utility will be involved. I have already argued in section 4 that it is not like that; on the contrary, once he is prepared to look at it like that, the argument in any serious case is over anyway" (UFA: 116). Williams' point is rather that the whole business of compiling balance-sheets of the utilitarian sort is incompatible with the phenomenon of agency as we know it: "the reason why utilitarianism cannot understand integrity is that it cannot coherently describe the relations between a man's projects and his actions" (UFA: 100). As soon as we take up the viewpoint which aims at nothing but the overall maximisation of utility, and which sees agents as no more than nodes in the causal network that is to be manipulated to produce this consequence, we have lost sight of the very idea of agency. And why should it matter if we lose sight of that? To say it again, the point of the integrity objection is not that the world will be a better place if we don't lose sight of the very idea of agency (though Williams thinks this as well[23]). The point is rather that a world-view that has lost sight of the real nature of agency, as the utilitarian world-view has, simply does not make sense: as Williams puts it in the quotation above, it is "absurd". Why is it absurd? Because the view involves deserting one's position in the universe for "what Sidgwick, in a memorably absurd phrase, called 'the point of view of the universe'" (1981: xi).[24] That this is what utilitarianism's impartial view ultimately requires is argued by Williams in his discussion of Sidgwick at MSH 169-170: > > The model is that I, as theorist, can occupy, if only temporarily and > imperfectly, the point of view of the universe, and see everything > from the outside, including myself and whatever moral or other > dispositions, affections or projects, I may have; and from that > outside view, I can assign to them a value. The difficulty is... > that the moral dispositions... cannot simply be regarded, least > of all by their possessor, just as devices for generating actions or > states of affairs. Such dispositions and commitments will > characteristically be what gives one's life some meaning, and > gives one some reason for living it... there is simply no > conceivable exercise that consists in stepping completely outside > myself and from that point of view evaluating in toto the > dispositions, projects, and affections that constitute the substance > of my own life... It cannot be a reasonable aim that I or any > other particular person should take as the ideal view of the > world... a view from no point of view at all. > As Williams also put it, "Philosophers... repeatedly urge one to view the world sub specie aeternitatis; but for most human purposes"--science is the biggest exception, in Williams' view--"that is not a very good species to view it under" (UFA: 118). The utilitarian injunction to see things from the impartial standpoint is, if it means anything, an injunction to adopt the "absolute conception" of the world (1978: 65-67). But even if such a conception were available--and Williams argues repeatedly that it is not available for ethics, even if it is for science (1985 Ch.8)--there is no reason to think that the absolute conception could provide me with the best of all possible viewpoints for ethical thinking. There isn't even reason to think that it can provide me with a better viewpoint than the viewpoint of my own life. That latter viewpoint does after all have the pre-eminent advantage of being mine, and the one that I already occupy anyway (indeed cannot but occupy). "My life, my action, is quite irreducibly mine, and to require that it is at best a derivative conclusion that it should be lived from the perspective that happens to be mine is an extraordinary misunderstanding" (MSH 170). (Notice that Williams is also making the point here that there is no sense in the indirect-utilitarian supposition that my living my life from my own perspective is something that can be given a philosophical vindication from the impartial perspective, and can then reasonably be regarded (by me or anyone else) as justified. Williams sees an incoherence at the very heart of the project of indirect utilitarianism, because he does not believe that the ambition to justify one's life "from the outside" in the utilitarian fashion can be coherently combined with the ambition to live that life "from the inside".[25] The kind of factors that make a life make sense are so different from the kind of factors that utilitarianism is structurally obliged to prize that we have every reason to hope that people will not think in the utilitarian way. In other words, it will be best even from the utilitarian point of view if no one is actually a utilitarian; which means that, at best, "utilitarianism's fate is to usher itself from the scene" (UFA: 134).) While some utilitarians have claimed to be unfazed by this result--it does not imply the falsity of utilitarianism qua theory of right action--the fact that they continue to publish books and articles defending utilitarianism suggests that they do not really wish for the theory to play no direct role in our moral deliberations. On the issue of impartiality, it will no doubt be objected that Williams overstates his case. It seems possible to engage in the kind of impartial thinking that is needed, not just by utilitarianism, but by any plausible morality, without going all the way to Sidgwick's very peculiar notion of "the point of view of the universe". When ordinary people ask, as they always have asked, the question "How would you like it?", or when Robert Burns utters his famous optative "O wad some pow'r the giftie gie us/ To see oorselves as ithers see us",[26] it does not (to put it mildly) make best sense of what they are saying to attribute to them a faithful commitment to the theoretical extravagances of a high-minded Victorian moralist. Can't morality find a commonsense notion of impartiality that doesn't involve the point of view of the universe? Indeed, if Williams' own views about impartiality are plausible, mustn't he himself use some such notion? To this Williams will reply, we think, that a commonsense notion of impartiality is indeed available--to us, though not to moral theory. The place of commonsense impartiality in our ordinary ethical thought is utterly different from the theoretical role of utilitarianism's notion of impartiality. The commonsense notion of impartiality is not, unlike the utilitarian notion, a lowest common theoretical denominator for notions of rightness, by reference to which all other notions of rightness are to be understood. Rather, commonsense impartiality is one ethical resource among others. (Cp. the quotation above from 1985: 117 about avoiding sparseness and reduction in our ethical thinking, and "cherishing as many ethical ideas as we can".) Moreover, and crucially, Williams' acceptance of "methodological intuitionism" (see MSH essay 15) commits him to saying that the relation of the commonsense notion of impartiality to other ethical resources or considerations is essentially indeterminate: "It may be obvious that in general one sort of consideration is more important than another... but it is a matter of judgement whether in a particular case that priority is preserved: other factors alter the balance, or it may be a very weak example of the kind of consideration that generally wins... there is no reason to believe that there is one currency in terms of which all relations of comparative importance can be represented" (MSH 190). The indeterminacy of the relations between commonsense impartiality and other ethical considerations means that commonsense impartiality resists the kind of systematisation that moral theory demands. Hence, there is indeed a notion of impartiality that makes sense, and there is indeed a notion of impartiality that is available to a moral theory such as utilitarianism; but the impartiality that is available to utilitarianism does not make sense, and the impartiality that makes sense is not available to utilitarianism. Williams argues, then, that the utilitarian world-view is absurd because it requires agents to be impartial, not merely in the weak and everyday sense that they take impartiality to be one ethical consideration among an unsystematic collection of other considerations that they (rightly) recognise, but in the much stronger, reductive and systematising, sense that they adopt the absolute impartiality of Sidgwick's "point of view of the universe". We can also say something that sounds quite different, but which in the end is at least a closely related point. We can say that Williams takes the utilitarian world-view to be absurd, because it requires agents to act on external reasons. I turn to that way of putting the point in section 5. ## 5. Internal and external reasons In his famous paper "Internal and external reasons" (1981: 101-113) Williams presents what I'll call "the internal reasons thesis": the claim that all reasons are internal, and that there are no external reasons. The internal reasons thesis is a view about how to read sentences of the form "A has reason to ph". We can read such sentences as implying that "A has some motive which will be served or furthered by his phing" (1981: 101), so that, if there is no such motive, it will not be true that "A has reason to ph". This is the internal interpretation of such sentences. We can also read sentences of the form "A has reason to ph" as not implying this, but as saying that A has reason to ph even if none of his motives will be served or furthered by his phing. This is the external interpretation of such sentences, on which, according to Williams, all such sentences are false. Since he is widely misinterpreted on this point, it is important to see that Williams is only offering a necessary condition for the truth of sentences of the form "A has reason to ph". He is not (officially) committed to the stronger claim, that the presence of some motive which is served or furthered by phing is sufficient for the possession of a reason to ph. Officially, then, Williams is not defining or fully analyzing the concept of a reason; rather, the necessary condition itself represents a threat. (We say "officially" because there are indications that Williams unofficially held a stronger view; see the final paragraphs in this piece for elaboration.) Very roughly, then, the basic idea of Williams' internal reasons thesis is that we cannot have genuine reasons to act that have no connection whatever with anything that we care about. His positive defense of the thesis can be roughly stated as follows: since it must be possible for an agent to act for a reason, reasons must be capable of explaining actions. This argument, it should be noted, brings together a normative and a descriptive concept in a robustly naturalistic manner. The notion of a reason, he argues, is inextricably bound up with the notion of explanation. Absent a motive which can be furthered by some action, it seems impossible for an agent to actually perform the action except under conditions of false information. If an external reason is one that is supposed to obtain in the absence of the relevant motive even under conditions of full information, then external reasons can never explain actions, and hence cannot be reasons at all (1981: 107). This thesis presents a challenge to certain natural and traditional ways of thinking about ethics. When we tell someone that he should not rob bank-vaults or murder bank-clerks, we usually understand ourselves to be telling him that he has reason not to rob bank-vaults or murder bank-clerks. If the internal reasons thesis is true, then the bank-robber can prove that he has no such reason simply by showing that he doesn't care about anything that is achieved by abstaining from bank-robbing. So we seem to reach the disturbing conclusion that morality's rules are like the rules of some sport or parlour-game--they apply only to those who choose to join in by obeying them. One easy way out of this is to distinguish between moral demands and moral reasons. If all reasons to act are internal reasons, then it certainly seems that the bank-robber has no reason not to rob banks. It doesn't follow that the bank-robber is not subject to a moral demand not to rob banks. If (as we naturally assume) there is no opting out of obeying the rules of morality, then everyone will be subject to that moral demand, including the bank-robber. In that case, however, this moral demand will not be grounded on a reason that applies universally--to everyone, and hence even to the bank-robber. At most it will be grounded in the reasons that some of us have, to want there to be no bank-robbing, and in the thought that it would be nice if people like the bank-robber were to give more general recognition to the presence of that sort of reason in others--were, indeed, to add it to their own repertoire of reasons. If we take this way out, then the moral demand not to rob banks will turn out to be grounded not on universally-applicable moral reasons, but on something more like Humean empathy. Williams himself thinks that this is, in general, a much better way to ground moral demands than the appeal to reasons ("Having sympathetic concern for others is a necessary condition of being in the world of morality", 1972: 26; cp. 1981: 122, 1985 Ch.2). In this he stands outside the venerable tradition of rationalism in ethics, which insists that if moral demands cannot be founded on moral reasons, then there is something fundamentally suspect about morality itself. It is this tradition that is threatened by the internal reasons thesis. Of course, we might wonder how significant the threat really is. As we paraphrased it, the internal reasons thesis says that "we cannot have genuine reasons to act that have no connection whatever with anything that we care about". Let us take up this notion of "connections". As Williams stresses, the internal reasons thesis is not the view that, unless I actually have a given motive M, I cannot have an internal reason corresponding to M.[27] The view is rather that I will have no internal reason unless either (a) I actually have a given motivation M in my "subjective motivational set" ("my S": 1981: 102), or (b) I could come to have M by following "a sound deliberative route" (MSH 35) from the beliefs and motivations that I do actually have--that is, a way of reasoning that builds conservatively on what I already believe and care about. So, to cite Williams' own example (1981: 102), the internal reasons thesis is not falsified by the case of someone who is motivated to drink gin and believes that this is gin, hence is motivated to drink this--where "this" is in fact petrol. We are not obliged to say, absurdly, that this person has a genuine internal reason to drink petrol, nor to say, in contradiction of the internal reasons thesis, that this person has a genuine external reason not to drink what is in front of him. Rather we should note the fact that, even though he is not actually motivated not to drink the petrol, he would be motivated not to drink it if he realised that it was petrol. He can get to the motivation not to drink it by a sound deliberative route from where he already is; hence, by (b), he has an internal reason not to drink the petrol. It is this notion of "sound deliberative routes" that prompts the question, how big a threat the internal reasons thesis really is to ethical rationalism. Going back to the bank-robber, we might point out how very unlikely it is to be true that he doesn't care about anything that is achieved by not robbing banks, or lost by robbing them. Doesn't the bank-robber want, like anyone else, to be part of society? Doesn't he want, like anyone else, the love and admiration of others? If he has either of these motivations, or any of a galaxy of other similar ones, then there will very probably be a sound deliberative route from the motivations that the bank-robber actually has, to the conclusion that even he should be motivated not to rob banks; hence, that even he has internal reason not to rob banks. But then, of course, it seems likely that we can extend and generalise this pattern of argument, and thereby show that just about anyone has the reasons that (a sensible) morality says they have. For just about anyone will have internal reason to do all the things that morality says they should do, provided only that they have any of the kind of social and extroverted motivations that we located in the bank-robber, and used to ground his internal reason not to rob banks. Hence, we might conclude, the internal reasons thesis is no threat either to traditional ethical rationalism, nor indeed to traditional morality--not at least once this is shorn by critical reflection of various excrescences that really are unreasonable. This line of thought does echo a pattern of argument that is found in many ethicists, from Plato's Republic to Philippa Foot's "Moral Beliefs". However, it does not ward off the threat to ethical rationalism. The threat still lurks in the "if". We have suggested that the bank-robber will have internal reason not to rob banks, if he shares in certain normal human social motivations. But what if he doesn't share in these? The problem is not merely that, if he doesn't, then we won't know what to say to him. The problem is that the applicability of moral reasons is still conditional on people's actual motivations, and local to those people who have the right motivations. But it seems to be a central thought about moral reasons, as they have traditionally been understood, that they should be unconditionally and universally overriding: that it should not be possible even in principle for any rational agent to stand outside their reach, or to elude them simply by saying "Sorry, but I just don't care about morality". On the present line of thought, this possibility remains open; and so the internal reasons thesis remains a threat to ethical rationalism. One way of responding to this continuing threat is to find an argument for saying that every agent has, at least fundamentally, the same motivations: hence moral reasons, being built upon these motivations, are indeed unconditionally and universally overriding, as the ethical rationalist hoped to show. One way of doing this is the Thomist-Aristotelian way, which grounds the universality of our motivations in our shared nature as human beings, and in certain claims which are taken to be essentially true about humans just as such.[28] Another is the Kantian way, which grounds the universality of our motivations in our shared nature as agents, and in certain claims which are taken to be essentially true about agents just as such. It is interesting to note that this sort of ethical-rationalist response to the internal reasons thesis can seem to undercut Williams' distinction between external and internal reasons. For the Thomist/neo-Aristotelian[29] or the Kantian, the point is not that we can truly say, with the external reasons theorist, that an agent has some reasons that bear no relation at all to the motivations in his present S (subjective motivational set), or even to those motivations he might come, by some sound deliberative route, to derive from his present S. The point is rather that there are some motivations which are derivable from any S* whatever.[30] Williams himself recognises this point in the case of Kant (WME 220, note 3): "Kant thought that a person would recognise the demands of morality if he or she deliberated correctly from his or her existing S, whatever that S* might be, but he thought this because he took those demands to be implicit in a conception of practical reason which he could show to apply to any rational deliberator as such. I think that it best preserves the point of the internalism/ externalism distinction to see this as a limiting case of internalism."[31] So for the Kantian and the neo-Aristotelian or Thomist, there are motivations which appear to ground internal reasons only, since the reasons that they ground are always genuinely related to whatever the agent actually cares about. On the other hand, these motivations also appear to ground reasons which have exactly the key features that the ethical rationalist wanted to find in external reasons. Two in particular: first, these reasons are unconditional, because they depend on features of the human being (Aquinas) or the agent (Kant) which are essential features--it is a necessary truth that these features are present; and second, these reasons are universal, because they depend on ubiquitous features--features which are present in every human or agent. So Williams' response to the neo-Aristotelian or the Kantian view of practical reason had better not be (and indeed is not) simply to invoke his internal reasons thesis. As he realises, he also needs to argue that there can't be reasons of the kinds that the neo-Aristotelian and the Kantian posit: reasons which are genuinely unconditional, but also genuinely related to each and every agent's actual motivations. Whatever else may be wrong with the neo-Aristotelian and Kantian theories of practical reason, it won't be simply that they invoke external reasons; for it is fairly clear that they don't (the contemporary Kantian philosopher who has most effectively pushed this point is Christine Korsgaard, see her Sources of Normativity, 1996). If not even Kant counts as an external reasons theorist, who does? That is a natural question at this point, since it is probably Kant who is usually taken to be the main target of Williams' argument against external reasons. This assumption is perhaps based on the evidence of 1981: 106, where (despite the points we have already noted about Kant's theory which Williams recognised at least by 1995) Williams certainly attributes to Kant the view that there can be "an 'ought' which applies to an agent independently of what the agent happens to want". Even here, however, Williams is actually rather cagey about saying that Kant is an external reasons theorist: he tells us that the question 'What is the status of external reasons claims?' is "not the same question as that of the status of a supposed categorical imperative"; "or rather, it is not undoubtedly the same question", since the relation between oughts and reasons is a difficult issue, and anyway there are certainly external reasons claims which are not moral claims at all, such as Williams' own example of Owen Wingrave's family's pressure on him to follow his father and grandfather into the army (1981: 106). In any case, it is important to see that there do not have to be any examples of philosophers who clear-headedly and definitely espouse an external reasons theory. The point is rather that no one could be a clear-headed and definite external reasons theorist if Williams is right, because, in that case, the notion of external reasons is basically unintelligible (MSH 39: "mysterious", "quite obscure"). Williams' internal reasons thesis is that it is unintelligible to suppose that something could genuinely be a reason for me to act which yet had no relation either to anything I care about, nor to anything that I might, without brainwashing or other violence to my deliberative capacities, come to care about.[32] If this thesis is true, then perhaps we should not expect to find any definite examples of clear-headed external reasons theorists. It will be no surprise if someone who tries to develop a clear-headed external reasons theory turns out not to be definitely an external reasons theorist: thus for example John McDowell's theory in WME Essay 5, even though it is explicitly presented as an example of external reasons theory, is probably not best understood that way. (Very quickly, this is because McDowell wants to develop an external reasons theory as a view about moral perception, "the acquisition of a way of seeing things" (WME 73). But literal perception does not commit us to external reasons. When I literally "just see" something, my visual perception--even my well-habituated and skilful perception--adds something to my stock of internal, not external, reasons. If we take the perceptual analogy seriously in ethics, it is hard to see why we can't say the same about moral perceptions.) Nor, conversely, will it be surprising if someone who tries to develop what is definitely an external reasons theory turns out not to be, so far forth, very clear-headed. Thus Peter Singer's exhortations to us to take up the moral point of view (see e.g. Practical Ethics 10-11[33]) give us perhaps the most definite example available of an external reasons theory in contemporary moral philosophy--but are also one of the least clearly-explained or justified parts of Singer's position. The notion of an external reason is, basically, a confused notion, and Williams' fundamental aim is to expose the confusion.[34] The fact that there can be no clear and intelligible account of external reasons has important consequences, consequences which go to the heart of the morality system discussed in sections 2 and 3, and which also relate back to the critique of utilitarianism that we saw Williams develop in section 4. If there can be no external reasons, then there is no possibility of saying that the same set of moral reasons is equally applicable to all agents. (Not at least unless some universalising system like Kantianism or neo-Aristotelianism can be vindicated without recourse to external reasons; Williams, as we've seen, rejects these systems on other grounds.) Deprived of this possibility, we are thrown immediately into a historicised way of doing ethics--a project with roots in both the Hegelian and Nietzschean traditions. No absolute conception of ethics will be available to us; hence, neither will the kind of impartiality that utilitarianism depends upon. Agents' reasons, and what agents' reasons can become, will always be relativised to their particular contexts and their particular lives; and that fact too will be another manifestation of "moral luck". Furthermore--a consequence that Williams particularly emphasises--without external reasons, or alternatively something like Kantianism or neo-Aristotelianism, there will be no possibility of deploying the notion of blame in the way that the morality system wants to deploy it. "Blame involves treating the person who is blamed like someone who had a reason to do the right thing but did not do it" (MSH 42). But in cases where someone had no internal reason to do (what we take to be) the right thing that they did not do, it was not in fact true that they had any reason to do that thing; for internal reasons are the only reasons. Typical cases of blaming people will, then, often have an unsettling feature closely related to one that we noted at the beginning of this section. They will rest on the fiction that the people blamed had really signed up for the standards whereby they are blamed. And so, once again, there will seem to be something optional about adherence to the standards of morality: morality will seem to be escapable in just the sense that the morality system denies. Williams' denial of the possibility of external reasons--understood in light of his naturalism and his anti-systematic outlook--thus underwrites and supports his views on a whole range of other matters. And though the internal reasons thesis too is, in an important way, a negative thesis, it is arguably the cornerstone of a more robust, positive conception of our practical lives. While he only defended the negative condition in print, there is evidence that Williams actually believed that the right kind of strong desire is a sufficient condition for the possession of a practical reason. In Ethics and the Limits of Philosophy, he briefly opined that "desiring to do something is of course a reason for doing it," (1985:19), and he developed a theory of practical necessity according to which our deep commitments ("ground projects") necessarily constrain and direct our practical rationality (MSH 17). Seen in this light, the internal reasons thesis is the seed out of which most of Williams' ethical ideas grow. At the outset of his writing career, he took for his own "a phrase of D.H. Lawrence's in his splendid commentary on the complacent moral utterances of Benjamin Franklin: 'Find your deepest impulse, and follow that'" (1972: 93). Thirty years later he added, when looking back over his career, "If there's one theme in all my work it's about authenticity and self-expression... It's the idea that some things are in some real sense really you, or express what you and others aren't.... The whole thing has been about spelling out the notion of inner necessity."[35]
williams-dc	## 1. Life Donald Williams was born on 28 May 1899 in Crows Landing, California, at that time a strongly rural district, and died in Fallbrook, also in his beloved California, and also at that time far from cities, on 16 January 1983. His father was Joseph Cary Williams, who seems to have been a jack of all countrymen's trades; his mother Lula Crow, a local farmer's daughter. Donald was the first in his family to pursue an academic education. After studies in English Literature at Occidental College (BA 1923), he went to Harvard for his Masters, this time in Philosophy (AM 1925). He then undertook further graduate study in philosophy, first at the University of California at Berkeley (1925-27), then at Harvard, where he took his PhD in 1928. Also in 1928 he married Katherine Pressly Adams, from Lamar, Colorado, whom he had met at Berkeley, where she was something of a pioneer--a woman graduate student in psychology. In time, there were two sons to the marriage. The couple spent a year in Europe in 1928-29 ("immersing himself in Husserl's Phenomenology to the point of immunization", Firth, Nozick and Quine 1983, p 246). Then Donald began his life's work as a Professor of Philosophy, first spending ten years at the University of California, Los Angeles, and then from 1939 until his retirement in 1967, at Harvard. Throughout his long and distinguished career in philosophy he retained a down-to-earth realism and naturalism in metaphysics, and a conservative outlook on moral and political issues, characteristic of his origins. A stocky, genial, and cheerful man, he found neither the content nor the validation of ethics to be problematic. Although originally a student of literature, whose first publication was a book of poetry, he had not the slightest tincture of literary or academic bohemianism. He was among the very least alienated academics of his generation; which is not surprising, as his career was indeed one version of The American Dream. ## 2. The Nature of Philosophy The traditional ambition of philosophy, in epistemology and metaphysics, is to provide a systematic account of the extent and reliability of our knowledge, and on that basis, to provide a synoptic and well-based account of the main features of Reality. When Williams was in his prime, this ambition was largely repudiated as inappropriate or unattainable, and a much more modest role for philosophy was proposed. In setting forth his own position, in the preface to the collection of his selected essays (1966, p.viii), he lists some of these fashionable philosophies from the mid-twentieth century: > > > ...logical positivism, logical behaviorism, operationalism, > instrumentalism, the miscellaneous American linguisticisms, the > English varieties of Wittgensteinism, the Existentialisms, and Zen > Buddhism... > > > Each of these is, in its own way, a gospel of relaxation. They all propose that, in place of the struggle to uncover how things are, careful descriptions of how things appear will suffice. None is ambitious enough to set about constructing a positive and systematic epistemology and metaphysics. Undeterred by this spirit of the age, Williams continued to insist that philosophical issues, including those of traditional metaphysics, are real questions with genuine answers (Fisher 2017). Conceptual analysis, concentration on phenomenological description, or exploration of the vagaries of language, may have their (subordinate) place, but to elevate them to a central position is an evasion of philosophy's main task. Still worse was the suggestion that philosophical questions are mere surface expressions of a philosopher's underlying psycho-pathology. The claim of Morris Lazerowitz to that effect, suggesting that Bradley's Absolute Idealism was no more than an intellectual's poorly expressed death wish, or that McTaggart's argument against the reality of time a panic fear of change, he met with a stern rebuke (1959, pp 133-56). He set forth a Realist philosophy on traditional empiricist principles: "He thought that practically everything was right out there where it belonged" (Firth, Nozick and Quine 1983, p 246). He maintained that while all knowledge of fact rests on perceptual experience, it is not limited to the perceptually given, but can be extended beyond that by legitimate inference (1934a). In this way his Realism can develop the breadth and depth required to do justice to all the scientific techniques which so far surpass mere perception. Williams's empiricism extended to philosophy itself. He challenged the prevailing orthodoxy that philosophy is a purely a priori discipline. He emphasized the provisional character of much philosophizing, and the striking absence of knock-down arguments in philosophic controversy ("Having Ideas In The Head" 1966, 189-211). ## 3. Metaphysics--Cosmology Following his own prescription for an affirmative and constructive philosophy, Williams worked steadily towards the development of his own distinctive position in metaphysics. He introduced the useful division of the subject into Speculative Cosmology, which deals with the basic elements making up the world we live in, such as matter, mind, and force, together with the relations between them, and Analytic Ontology, which explores the fundamental categories, such as Substance and Property, and how they relate to one another. Speculative Cosmology, in particular, needs to be open to developments in the fundamental sciences, and so needs to be seen as always provisional and a posteriori. Analytic Ontology, the exploration of the categories of being, is a more purely reflective and a priori discipline aiming to elucidate the range of different elements in any universe. To begin with Speculative Cosmology, Williams's position has three main features. It is Naturalistic. The natural world of space, time, and matter, with all its constituents, is a Reality in its own right. Contrary to all Idealist metaphysics, this world, except for the finite minds that are to found within it, is independent of any knowing mind. Moreover, it is the only world. There are no divinities or supernatural powers beyond the realm of Nature ("Naturalism and the Nature of Things" 1966, pp 212-238). Even mathematical realities belong in the natural world: numbers--at least natural ones--as abstractions from clusters, and geometrical objects as abstractions from space. The philosophy of mathematics was an aspect of his position that was never fully worked through. Second, his position is not only Naturalist but also Materialist. This Materialism is not of the rather crude kind that supposes that every reality is composed of a solid, crunchy substance, the stuff that makes up the atoms of Greek speculation. Any spatio-temporal reality, whether an 'insubstantial' property, such as a color, or something as abstract as a relation, such as farther-away-than, or faster-than, so long as it takes its place as a spatio-temporal element at home in the world of physics, is accepted as part of this one great spatio-temporal world. Williams's Materialism is thus one which can accept whatever physical theory posits as the most plausible foundation for the natural sciences, provided that it specifies a world which develops according to natural law, without any teleological (final) causes. The Mental is accommodated in the same way. Although Mind is unquestionably real, mental facts are as spatio-temporally located as any others ("The Existence of Consciousness", "Mind as a Matter of Fact" 1966, pp 23-40, 239-261). The Mental is not an independent realm parallel to and equal to the Material, but rather a tiny, rather insignificant fragment of Being, dependent upon, even if not reducible to, the physical or biological nature of living beings. Williams has a capacious conception of the Material--his position could perhaps be better described as Spatio-Temporal Naturalism. Thirdly, Williams's metaphysics is 4-dimensional. Or, more precisely, given the development of multi-dimensional string theories since his time, it takes Time to be a dimension in the same way that Space has dimensions. The first step is to insist, against Aristotle and his followers, that statements about the future, no less than those concerning the present and the past, are timelessly true or false. They need not await the event they refer to, in order to gain a truth value ("The Sea Fight Tomorrow" 1966, pp 262-288.) This encourages the further view that the facts that underpin truths about the future are (timelessly) Real. All points in time are (timelessly) Real, as are all points on any dimension of our familiar Space. Whatever account is to be given of Change, it does not consist in items gaining or losing Reality. This stance receives powerful support from physical theory. Williams embraced and argued for the conception of Time as a fourth dimension introduced by Minkowski's 'Block Universe' interpretation of Einstein's Theory of Special Relativity. A consequence of this is that the experience of the flow or passage of Time must be some sort of illusion. Williams embraced that consequence, and argued for it in a celebrated paper ('The Myth of Passage', 1951). ## 4. Metaphysics--Ontology Apart from his Materialistic or Spatio-Temporal Naturalism, Williams's major contributions to metaphysics lie in the realm of ontology, and concern the fundamental constituents of Being. His key proposal is that properties are indeed real--in fact Reality consists in nothing but properties--but that these properties are not Universals, as commonly supposed, but particulars with unique spatio-temporal locations. The structure of Reality comprises a single fundamental category, Abstract Particulars, or "tropes". Tropes are particular cases of general characteristics. A general characteristic, or Universal, such as redness or roundness, can occur in any one of indefinitely many instances. Williams' focus was on the particular case of red which occurs as the color, for example, of a particular rose at a specific location in space and time, or the particular case of circularity presented by some particular coin in my hand on a single, particular occasion. These tropes are as particular, and as grounded in place and time, as the more familiar objects, the rose and the coin, to which they belong. These tropes are the building blocks of the world. In his analogy, they provide 'the Alphabet of Being' from which the entities belonging to more complex categories--objects, properties, relations, events--can be constructed, just as words and sentences can be built using the letters of the alphabet. Familiar objects such as shoes and ships and lumps of sealing wax, and their parts as revealed by empirical scientific investigation, such as crystals, molecules and atoms, are concrete particulars or things. In Williams's scheme, each of these consists in a compresent cluster of tropes--the particular thing's particular shape, size, temperature, and consistency, its translucency, or acidity, or positive charge, and so on. All the multitude of different tropes that comprise some single complex particular do so by virtue of their sharing one and the same place, or sequence of places, in Space-Time. That is what 'compresent' means. There is no inner substratum or individuator to hold all the tropes together. The tropes are individuals in their own right, and do not inhere in any thing-like particular. So Williams's view is a No-Substance theory, or, otherwise described, a theory in which each trope is itself a simple Humean substance, capable of independent existence. Universal properties and quantities such as acidity and velocity, which are common to many objects, are not beings in their own right, but resemblance classes of individual tropes. If two objects match in color, both being red, for example, the tropes of color belonging to each are separate tropes, both being members of the class of similar color tropes which constitutes Redness. Relations are treated along the same lines. If London is Larger-than Edinburgh, and Dublin Larger-than Belfast, we have two instances of the Larger-than relation, two relational tropes. And they, along with countless other cases, all belong to the resemblance class whose members are all and only the cases of Larger-than. This account denies that, literally speaking, there is any single entity which is simultaneously fully present in two different cases of the same color, or temperature, or whatever. So it is a No-Universals view, and often described as a version of Nominalism. This is understandable, but it is better to confine the term 'Nominalism' to the denial of the reality of properties at all. So far from denying properties, on Williams's theory the entire world consists in nothing but tropes, which are properties construed as particulars. So his position is better described not as Nominalist but as Particularist. Tropes provide an elegant and economical base for an ontology. Unlike almost all competing ontologies, that of Williams rests on just one basic category, which can be used in the construction not just of things and their properties and relations, but of further categories, such as events and processes. Events are replacements of the tropes that are to be found in a given location. Processes are sequences of such changes. Trope theory is well placed to furnish an attractive analysis of causality, as involving power tropes that govern and drive the transformations to be found in events and processes. It can also be of use in other areas of philosophy, for example in valuation theory, where the existence of many tropes, rather than one single unitary reality, can explain our sometimes divided attitudes toward what, on a Substance ontology, we would regard as the same thing. Something can be good in some respects (tropes), but not in others. To view the manifest world as comprising, for the most part, clusters of compresent tropes makes explicit the complexity of the realities with which we are ordinarily in contact. ## 5. Objections to Williams's Trope Ontology Many philosophers have admitted tropes into their scheme of things: Aristotle, Locke, Spinoza and Leibniz, for example. What is distinctive in Williams is not that tropes are admitted as a category, but as the only fundamental category, a trope-based form of what Schaffer calls "property primitivism." (Schaffer 2003, 125). All else is constructed out of tropes, including concrete particulars and general properties, whereas tropes themselves are not constructed out of anything else, for example, out of a substance, a universal and the relation of exemplification. Not surprisingly, various aspects of Williams' trope primitivism have been subjected to serious philosophical challenges either directly or indirectly. Some philosophers reject all forms of property primitivism, including that of Williams, on the grounds that properties cannot serve as the only independent elements of being. Properties must be had by objects and a property cannot be instantiated in isolation from wholly distinct properties so they lack the requisite independence, the capacity of existing in any combination with "wholly distinct existences." (Some philosophers go so far as to claim that if a property P is a trope, then P is dependent on the specific thing that has it and so could not have existed without that thing existing; see Mulligan, Simons, & Smith 1984, Heil 2003.) There are two ways for this objection to go. (1) Armstrong takes the ostensive fact that properties must be had by objects to establish the dependence of properties on non-property particulars, substrata, and, thus, the falsity of property primitivism (Armstrong 1989, 115). (2) Alternately, one can take the apparent fact that there can be no properties that are not clustered with other properties to show that properties cannot play this role. In response to (1), one can challenge the assumption that if properties must be had by objects, then they are not capable of independent existence. Ross Cameron, for example, suggests that properties might both be capable of independent existence--including existing without substrata--and not be capable of existing without being the property of something. (2006, 104). The dependence of properties on objects is compatible with property primitivism if one adopts a bundle theory of concrete particulars. In that case, properties are always had by objects since they are always found in bundles, even if only a bundle of one, but exist without substrata. As for (2), some philosophers reject the requirement that if properties are the only ultimate constituents of reality, then they must be capable of existing in isolation from all other wholly distinct properties (Simons 1994; Denkel 1997). Properties can be both the only ultimate constituents of reality and inter-dependent. This response requires the rejection of the Humean principle that the basic independent units of being can exist in any combination, including unaccompanied (Schaffer 2003, 126). A very different response to (2) rejects the necessity of trope clustering altogether (Williams 1966, 97; Campbell 1981, 479; Schaffer 2003). > > > Plausible though it be, however, that a color or a shape cannot exist > by itself, I think we have to reject the notion of a standard of > concreteness. ... (Williams 1966, 97) > > > At best, it is a contingent matter that there are no tropes that are not compresent with any other tropes--for example, a mass trope on its own. (Campbell goes so far as to suggest that there is reason to think that there are actual cases of free-floating tropes (1981, 479)). Even if the trope primitivist can get around these quite general objections, there remains the most serious objection to Williams' brand of trope primitivism, an objection that is specific to Williams' conception of tropes, depending on more than the assumption that tropes are properties. The charge is that a Williamsonian trope is not genuinely simple, but complex embracing, at least, an element that furnishes the nature or content of the trope, and an element providing its particularity (Hochberg 1988; Armstrong 2005; Moreland 1985; Ehring 2011). In short, Williamsonian tropes are constructed out of something else, making them incompatible with trope primitivism. This objection is based on two assumptions. First, under Williams' conception, the nature of a trope is a non-reducible, intrinsic matter that is not determined by relations to anything else, including resemblance relations to other tropes or memberships in various natural classes of tropes. And, second, anything that stands in more than one arbitrarily different relation, each of which is grounded intrinsically in that entity, must be complex since that entity will have intrinsic "aspects" that are not identical to each other. The two relevant relations are numerical difference from other tropes and resemblance to other tropes, each of which is grounded intrinsically in the trope relata under the Williams conception. Hence, there are "intrinsic aspects" of each trope that are not identical to each other, a particularity-generating component and a nature-generating component. In response to this "complexity" objection, Campbell claims that the distinction between a trope's nature and its particularity is merely a "formal" distinction, a product of different levels of abstraction, and not a real distinction between different components of a trope. One should no more distinguish a particularity-component from a nature-component of a trope than distinguish components of warmth and orangeness in an orange trope: > > > To recognize the case of orange as warm is not to find a new feature > in it, but to treat it more abstractly, less specifically, than in > recognising it as a case of orange. (Campbell 1990, 56-7; > further in Maurin 2005; Fisher 2020) > > > Alternately, Ehring suggests that we grant this objection, but preserve trope simplicity by switching to a non-Williamsonian conception of tropes, according to which a trope's nature is determined by its memberships in various natural classes of tropes rather than intrinsically, thereby sidestepping one of the assumptions operative in the objection (2011; discussion in Hakkarainen and Keinanen 2017). Coming under criticism as well is Williams' resemblance-based account of general characteristics and property agreement. According to Williams, a fully determinate general color characteristic--say, the shade of red that characterizes this shirt and this chair---is just the set of all tropes that exactly resemble the red trope of this shirt. Different objects of that "same" shade of red each possess a different trope from this set of exactly similar red tropes. But this analysis seems to generate an infinite regress. If trope t1 is related to trope t2 by resemblance trope r1, t2 is related to trope t3 by resemblance trope r2, and t3 is related to t1 by resemblance trope r3, then these resemblance tropes will also resemble each other, giving rise to further resemblance tropes, and so on. To stop this vicious regress, the objection continues, resemblance must be taken to be a universal (Daly 1997, 150). One response to this objection tries to stop the regress before it starts by denying that there are any resemblance trope-relations holding between tropes. In particular, the trope theorist might follow Oliver's advise > > > to avoid saying that when two tropes are exactly similar ..., > there exists a relation-trope of exact similarity ... holding > between the two tropes. (1996, 37) > > > There are no resemblance-tropes corresponding to these resemblance predicates. Another response denies that resemblance relations mark an addition to our ontology and, hence, there can be no regress of resemblance relations. Campbell, for example, argues that the successive resemblance relations in the regress are nothing over and above the non-relational tropes that ultimately ground these relations. Since resemblance is an internal relation, it supervenes on these ground-level non-resemblance tropes, but supervenient "additions" are not real additions to one's ontology (Campbell 1990, 37). There is also a whole host of objections in the literature to Williams' trope bundle theory. According to Williams, concrete particulars are not substrata instantiating various properties. They are wholly constructed out of tropes, forming bundles of tropes, the trope constituents of which are pairwise tied together by a compresence relation. Compresence, in turn, is collocation for Williams, "the unique congress in the same volume." (In order to allow for non-spatial objects, Williams grants the possibility of "locations" in systems analogous to space (1966, 79).) One immediate worry, raised by Campbell, is the possibility that there may be cases of overlapping objects demonstrating that collocation is not sufficient for compresence (1990, footnote 5, 175). In response, the trope bundle theorist can opt for the view that compresence is non-reducible. However, even with this revision there remain significant objections concerning the possibility of accidental properties, the possibility of change, and an apparent vicious regress of compresence relations. Bundle theory has been charged with ruling out the possibility of accidental properties in concrete particulars. Bundles of properties have all of their constituent properties essentially. Objects generally do not. This chair could have been blue instead of red, but the bundle of properties that characterize the chair could not have failed to include that red property. One way around this objection is proposed by O'Leary-Hawthorne and Cover: combine bundle theory (although for them properties are universals) with a specific account of modality, a counterpart semantics for statements about ordinary particulars (1998). What makes it true that a particular object o could have had different properties is that a non-identical counterpart to o, n, in another possible world has different properties than does o. As long as there is a possible world in which there is a object-bundle, n, that is non-identical counterpart to o and differs from o with respect to its properties, then o could have differed in just that way. Simons suggests a very different approach. He proposes to replace unstructured bundles with "nucleus theory." An object consists of an inner core of essential tropes and an outer band of accidental tropes, but no non-property substratum (1994). In like manner, bundle theory has been charged with ruling out the possibility of change in objects. The same bundle complex cannot be composed of one set of property at one time, but a different set at a different time. Concrete objects, on the other hand, can and do change. In response, it has been suggested that this objection loses its force if bundle theory is combined with a four-dimensionalist account of object persistence (Casullo 1988; see also Ehring 2011). For example, if ordinary objects are spacetime worms made up of appropriately related instantaneous temporal parts that are themselves complete bundles of compresent tropes, then change can be read as a matter of having different temporal parts that differ in their constitutive properties. Another response to the change-is-impossible objection is to adopt Simons' "nucleus theory" in place of traditional bundle theory. Nucleus theory seems to allow for change in the outer band of accidental properties (Simons 1994). Trope bundle theory has also been accused of giving rise to Bradley-style vicious regress. According to trope bundle theory, for an object o to exist, its tropes must be mutually compresent. However, it would seem, for trope t1 to be compresent with trope t2, they must be linked by a compresence trope, say, c1. But, the existence of t1, t2, and c1 is insufficient to make it the case that t1 and t2 are compresent since these tropes could each be parts of different, non-overlapping bundles. So for t1 and t2 to be linked by compresence trope c1, c1 must be compresent with t1 by way of a further compresence relation, say c2 (and with t2 by, say, c3) and so on, giving rise to either a vicious, or at least, uneconomical regress (Maurin 2010, 315). In response, one can try to break the link between the relevant predicates and tropes. Oliver suggests that the trope theorist should reject the assumption that there are any relation-tropes corresponding to the predicate "....is compresent with..." even though that predicate has some true applications (1996, 37). This response might be indirectly supported by reference to Lewis's claim that it is an impossible task to give an analysis of all predications since any analysis will bring into play a new predication, itself requiring analysis (Lewis 1983, 353). A second, quite different response is modeled on Armstrong's view that "instantiation" is a "tie," not a relation (since "the thisness and nature are incapable of existing apart"), and, hence, it is not subject to a relation regress. (1978, 109). The idea is that the union of compresent tropes is too intimate to speak of a relation between them since the tropes of the same object could not have existed apart from each other. This response, however, requires more than generic dependencies--for example, that this specific mass trope requires the existence of some solidity trope or other--since generic dependencies would not guarantee that the specific mass and solidity tropes, say, in this particular bundle could not have existed without being compresent. Maurin provides an alternative response that grants the existence of compresence relation-tropes, but denies the regress on the grounds that relation-tropes, including compresence relations, necessitate the existence of that which they relate. There is no need for further compresence relations holding between a compresence-relation and its terms (2002, 164). A fourth response, from Ehring, suggests that this regress can be stopped once it is recognized that compresence is a self-relating relation, a relation that can take itself as a relatum. The supposed infinite regress for the bundle theorist involves an unending series of compresence tropes, c1, c2, ..., and cn, but the series, c1, c2, ..., and cn is taken to be infinite because it is assumed that each "additional" compresence trope is not identical to the immediate preceding compresence trope in the series. However, if compresence is a "self-relating" relation, this assumption may be false (2011). Finally, there is an objection to the very notion of a trope and, hence, to the foundations of trope primitivism (and, perhaps, to any ontology that includes tropes). The idea is that if properties are tropes, then exactly similar properties can be swapped across objects, but there is no such possibility. > > > If the redness of this rose is exactly similar to but numerically > distinct from the redness of that rose, then the redness of this rose > could have been the redness of that rose and vice versa. But this is > not really a possibility and, thus, properties are not tropes. > (Armstrong 1989, 131-132) > > > A similar argument is based on the possibility of tropes swapping positions in space (Campbell 1990, 71). "Property swapping," it is claimed, is an unreal possibility since property swapping would make no difference to the world. (Note that the cross-object version of this objection cannot get off the ground if tropes are not transferable between objects, a view that is found in (Martin 1980), although Martin rejects trope primitivism since he posits substrata in addition to tropes). One response to the no-swapping objection, given by Campbell and Labossiere, rejects the assumption that trope swaps would make no difference to the world. For example, although the effects of "swapped" situations would be exactly similar in nature, those effects would differ in their causes (Campbell 1990, 72; Labossiere 1993, 262). Schaffer, on the other hand, denies that the trope theory is automatically committed to the possibility of trope swapping (2001). If tropes are individuated by times/locations and a counterpart theory of modality is right, then trope swapping is ruled out: > > > The redness which would be here has exactly the same inter- and > intraworld resemblance relations as the redness which actually is > here, and the same distance relations, and hence it is a better > counterpart than the redness which would be there. (Schaffer > 2001, 253). > > > What is clear is that Williams' trope ontology remains at the center of a vibrant and ongoing debate, counting as a serious option among a small field of contenders. His brand of trope primitivism is certainly of more than merely historical interest. Williams has also left a wider, if less plainly manifest legacy as a metaphysician. For a discussion of his influence on David Lewis, and through him on later thinkers, see Fisher 2015. ## 6. The Ground of Induction The Problem of Induction is the problem of vindicating as rational our unavoidable need to generalize beyond our current evidence to comparable cases that we have not yet observed, or that never will be observed. Without inductive inferences of this kind, not only all science but all meaningful conduct of everyday life is paralyzed. Indeed, Williams prefaced his philosophical treatment with a declaration of the evils of inductive skepticism in eroding rational standards in general, even in politics: > > > In the political sphere, the haphazard echoes of inductive skepticism > which reach the liberal's ear deprive him of any rational right > to champion liberalism, and account already as much as anything for > the flabbiness of liberal resistance to dogmatic encroachments from > the left or the right. (Williams 1947, pp 15-20) > > > David Hume, in the eighteenth century, had shown that all such inferences must involve risk: no matter how certain our premises, no inductively reached conclusion can have the same degree of certainty. Hume went further, and held that inductive reasoning provides no rational support whatever for its conclusions. This is Hume's famous inductive scepticism. Williams was almost alone in his time in holding not only that the problem does admit of a solution, but in presenting a novel solution of his own. To do this, he needed to argue against Hume, and Hume's twentieth century successors Bertrand Russell and Karl Popper, who had declared the problem insoluble, and also against contemporaries such as P. F. Strawson and Paul Edwards, who had claimed that there was no real problem at all (Russell 1912, Chapter 6; Popper 1959; Edwards 1949, pp 141-163; Strawson 1952, pp 248-263). Williams tackled the problem head-on. In The Ground of Induction (1947) he makes original use of results already established in probability theory, whose significance for the problem of induction he was the first to appreciate. He treats inductive inference as a special case of the problem of validating sampling techniques. Among any population, that is, any class of similar items, there will be a definite proportion having any possible characteristic. For example, among the population of penguins, 100% will be birds, about 50% will be female, some 10%, perhaps, will be Emperor penguins, some 35%, perhaps, will be more than seven years old, and so on. This is the complexion of the population, with regard to femaleness, or whatever. Now in the pure mathematics of the relations between samples and populations, Jacob Bernoulli had in the 18th century established the remarkable fact that, for populations of any large size whatever (say above 2500), the vast majority of samples of 2500 or more closely match in complexion the population from which they are drawn. In the case of our penguins, for example, if the population contains 35% aged 7 years or more, well over nine tenths of samples of 2500 penguins will contain between 33% and 37% aged 7 years or more. This is a necessary, purely mathematical, fact. In the language of statistics, the vast majority of reasonably sized samples are representative of their population, that is, closely resemble it in complexion. Bernoulli's result enables us to infer from the complexion of a population to the complexion of most reasonably-sized samples taken from it, since the complexion of most samples closely resembles the complexion of the population. Williams's originality was this: he noticed that resemblance is symmetrical. If we can prove, as Bernoulli did, that most samples resemble the population from which they are drawn, then conversely the population's complexion resembles the complexion of most of the samples. This brings us to the problem of induction. What our observations of the natural world provide us with can be regarded as samples from larger populations. For example the penguins we have observed up to this point provide us with a sample of the wider population of all penguins, at all times, whether observed or not. What can we infer about this wider population from the sample we have? That, in all probability, the population's complexion is close to that of the sample. We may of course, have an atypical sample before us. But with samples of more than 2500, well over 90% represent the population fairly closely, so the odds are against it. Thus Williams assimilates the problem of induction to an application of the statistical syllogism (also called direct inference or the proportional syllogism). A standard syllogism concerns complexions of 100%, and has a determinate conclusion: if all S are P, and this present item is an S, then it must be P. A statistical syllogism deals with complexions of less than 100%, and its conclusion is not definite but only probable: for example, if 95% of S are P, this present S is probably P. Some logicians claim that the probability in question is exactly 0.95, but Williams does not need to rely on that additional claim. It is enough that the probability be high. Applying the statistical syllogism to Williams' reversal of the Bernoulli result, we have: 95% of reasonably-sized samples are closely representative of their population, so the sample we have, provided there are no grounds to think otherwise, is probably one of them. On that basis, we are rationally entitled to infer that the population probably closely matches the sample, whose complexion is known to us. The inference is only probable. Induction cannot deliver certainty. In any given case, it is abstractly possible that our sample may be a misleading, unrepresentative one. But to expect an inference from the observed to the unobserved to yield certainty is to expect the impossible. ## 7. Objections to Williams' proposed solution to the problem of induction Williams' treatment of induction created quite a stir when it appeared, but attracted criticisms of varying power from commentators wedded to more defeatist attitudes, and was eclipsed by the Popperian strategy of replacing an epistemology of confirmation with one that focused on refutation. It thus exercised less influence than it deserves. It is a closely reasoned, deductively argued defense of the rationality of inductive inference, well meriting continued attention. Its reliance on a priori reasoning (as opposed to any contingent principles such as the "uniformity of nature" or the action of laws of nature) means that it should hold in all possible worlds. Williams' argument would thus be easily defeated by the exhibition of a possible world in which inductive reasoning did not work (in the sense of mostly yielding false conclusions). However, critics of Williams have not offered such a possible non-inductive world. Chaotic worlds are not non-inductive (as the induction from chaos to more chaos is correct in them), while a world with an anti-inductive demon, who falsifies, say, most of the inductive inferences I make, is not clearly non-inductive either (since although most of my inductions have false conclusions, it does not follow that most inductions in general have false conclusions). Marc Lange (2011) does however propose a counterexample, arising from the "purely formal" nature of Williams' argument. Should it not apply equally to "grue" as to "green"? Objects are grue if they are green up to some future point in time, and blue thereafter. The problem is to show that our sample, to date, of green things is not a sample of things actually grue. (Williams's own views on the grue problem are given in his "How Reality is Reasonable" in (2018)). Stove (1986, 131-144) argued that his more specific version of Williams's argument (see below) was not subject to the objection, and that the failure of induction in the case of 'grue' showed that inductive logic was not purely formal--but then neither was deductive logic. Other criticisms arise from the suggestion that the proportional syllogism in general is not a justified form of inference without some assumption of randomness. Any proportional syllogism (with exact numbers) is of the form * The proportion of Fs that are G is r. * a is F. * So, the probability that a is G is r. (Or, if we let B be the proposition that the proportion of Fs that are G is r, and we let p(h \| e) be the conditional probability of hypothesis h on evidence e, then we can express the above proportional syllogism in the language of probability as follows: p(Ga \| Fa & B) = r.) Do we not need to assume that a is chosen "randomly", in the sense that all Fs have an equal chance of being chosen? Otherwise, how do we know that a is not chosen with some bias, which would make its probability of being G different from r? Defenders of the proportional syllogism (McGrew 2003; Campbell and Franklin 2004) argue that no assumption of randomness is needed. Any information about bias would indeed change the probability, but that is a trivial fact about any argument. An argument infers from given premises to a given conclusion; a different argument, with a different force, moves from some other (additional) premises to that conclusion. Given just that the vast majority of airline flights land safely, I can have rational confidence that my flight will land safely, even though there are any number of other possible premises (such as that I have just seen the wheels fall off) that would change the probability if I added them to the argument. The fact that the probability of the conclusion on other evidence would be different is no reason to change the probability assessment on the given evidence. Similar reasoning applies to any other property that a may have (or, in the case of the Williams argument, that the sample may have), such as having been observed or being in the past. If there is some positive reason to think that property relevant to the conclusion (that a is G), that reason needs to be explained; if not, there is no reason to believe it affects the argument and the original probability given by the proportional syllogism stands. Serious criticisms specific to Williams's argument have been based on claims that the proportional syllogism, though correct in general, has been misapplied by Williams. Any proportional syllogism, * The proportion of Fs that are G is r. * a is F. * So, the probability that a is G is r, is subject to the objection that, in the case at hand, there is actually further information about the Fs that is relevant to the conclusion Ga. For example, in * The proportion of candidates who will be appointed to the board of Albert Smith Corp is 10%. * Albert Smith Jr is a candidate. * So, the probability that Albert Smith Jr will be appointed is 10%, it is arguable that information is hidden in the proper names that is favorably relevant to the younger Smith's chance of success. The question then is whether the same could happen with the proportional syllogism in Williams' argument for induction. Maher (1996) argues that there is such a problem. In this version of Williams' argument: * The proportion of large samples whose complexion approximately matches the population is over 95%. * S is a large sample. * So, the probability that the complexion of S matches the population is over 95%, is there, typically, any further information about the sample S that is relevant to whether it matches the population? The potentially relevant information one has is the proportion in S of the attribute to be predicted (blue, or whatever it might be). Can that be relevant to matching? It certainly can be relevant. For example, if the proportion of blue items in the sample is 100%, it suggests that the population proportion is close to 100%; that is positively relevant to matching since samples of near-homogeneous populations are more likely to match. (For example, if the population proportion is 100%, all samples match the population.) Conversely, fewer samples match the population when the population proportion is near one half. David Stove (1986), in defending Williams' argument, proposed to avoid this problem by taking a more particular case of the argument which would not be subject to the objection. Stove proposed: > > > If F is the class of ravens, G the class of black > things, S a sample of 3020 ravens, r = 0.95, and > 'match' means having proportion within 3% of the > population proportion, then it is evident that 'The proportion > of Fs that are Gs is x' is not > substantially unfavorable to matching (for all x). > > > For even in the worst case, when the proportion of black things is one half, it is still true that the vast majority of samples match the population (Stove 1986, 66-75). Stove's reply emphasizes how strong the mathematical truth about matching of samples is: it is not merely that, for any population size and proportion, one can find a sample size and degree of match such that samples of that size mostly match; it is the much stronger result that one can fix the sample size and degree of match beforehand, without needing knowledge of the population size and proportion. For example, the vast majority of 3020-size samples match to within 3%--irrespective of the population size (provided of course that it is larger than 3020) and the proportion of objects with the characteristic under investigation.. Maher also objects that the attribute itself, such as "blue", might be a priori relevant to its proportion in the sample and hence to matching. For example, if "blue" is one of a large range of possible colors, then a priori it is unlikely that an individual is blue and unlikely that a sample will have many blue items. As with any Bayesian reasoning, a prior probability close to zero (or one) requires a lot of evidence to overcome; in this case, we would conclude that a sample with a high proportion of blue items was most likely a coincidence, and the posterior probability of the sample matching the population would still not be high. Scott Campbell (2001), in reply to Maher, argues that priors do not dominate observations in the way Maher suggests. By analogy, suppose that, while blindfolded, I throw a dart at a dartboard. I am told that 99 of the 100 spots on the board are the same color, and that there are 145 choices of color for the spots. After throwing, I observe just the spot I have hit and find it is blue. Then (in the absence of further information), the chance is very high that almost all the other spots are blue. The prior improbability of blue does not prevent that. In the same way, the fact that the vast majority of samples match the population gives good reason to suppose that the observed sample does too, irrespective of any prior information of the kind Maher advances. Williams' defense of induction thus has resources to supply answers to the criticisms that have been made of it. It remains the most objectivist and ambitious justification of induction.
wilson	## 1. Life and Work Details about the life of John Cook Wilson, or 'Cook Wilson' as he is commonly called, may be found in a memoir published in 1926 by his pupil A. S. L. Farquharson, in his edition of Cook Wilson's posthumous writings, Statement and Inference with other Philosophical Papers (SI, xii-lxiv). He was born in Nottingham on June 6, 1849, the son of a Methodist minister. Educated at Derby Grammar School, he went up to Balliol College in 1868, elected on an exhibition set up by Benjamin Jowett for students from less privileged schools. At Balliol, Cook Wilson read classics with Henry Chandler, from whom he certainly got his penchant for minutiae, and mathematics with Henry J. S. Smith. He also studied philosophy with Jowett and T. H. Green, who steered him towards Kant and idealism (SI, 880), along with other students from his generation, such as Bernard Bosanquet, F. H. Bradley, Richard Lewis Nettleship and William Wallace. He even went to Gottingen in 1873-74 to attend lectures by Hermann Lotze (SI, xxvii), whose portrait he kept in his study. Cook Wilson wrote late in his life that "from the first I would not commit myself even to the most attractive form of idealism, tho' greatly attracted by it" (SI, 815). Farquharson also mentions Friedrich Ueberweg's System of Logic and History of Logical Doctrines (Ueberweg 1871) as an early 'realist' influence (SI, 880). If we are to follow Prichard, however, Cook Wilson abandoned idealism "with extreme hesitation" (Prichard 1919, 309) and "it was only towards the close of his life that he really seemed to find himself" (Prichard 1919, 318). (See also on this point Farquharson's reminiscence in (SI, xix).) Cook Wilson was elected fellow of Oriel College in 1873 and he succeeded Thomas Fowler as Wykeham Professor of Logic, New College in 1889. Bernard Bosanquet, Thomas Case, and John Venn had been among his rivals. He finally moved to New College in 1901, where he remained until his death from pernicious anemia on 11 August 1915. His wife Charlotte, whom he had met in Germany, predeceased him; they had a son (Joseph 1916b, 557). Cook Wilson lived the uneventful life of an Oxford don. Among his awards, he became Fellow of the British Academy in 1907. A Liberal in his convictions (SI, xxix), he did not get involved in politics, nor did he take a prominent part in the affairs of his university. (He is, for example, seldom mentioned in (Engel 1983).) His most cherished extra-curricular activity appears to have been the development of tactics for military bicycle units. Cook Wilson published little during his lifetime. Setting apart publications on military cycling and other incidental writings, the bulk of his publications were in his chosen fields of study, classics and mathematics. His work in Ancient philosophy, which form the bulk of his publications, is discussed in the next section. In mathematics, he published a strange treatise that arose out of his failed attempt at proving the four-colour theorem, On the Traversing of Geometrical Figures (TGF). It had virtually no echo. Farquharson quoted the mathematician E. W. Hobson explaining that Cook Wilson "hardly gave sufficient time and thought to the subject to make himself really conversant with the modern aspects of the underlying problems" (SI, xxxviii). Cook Wilson also published two short papers on probability (IP, PBT) in which he gave new proofs of the discrete Bayes' formula and of Jacob Bernouilli's theorem (this last being known today as the weak law of large numbers). Edgeworth called the former an "elegant proof" (Edgeworth 1911, 378, n.10), but Cook Wilson was seemingly ignorant of an already better proof of the latter via the Bienayme-Chebyshev inequality (Seneta 2012, 448 & 2013, 1104). His mathematical endeavours were otherwise largely wasted on trying to prove the inconsistency of non-Euclidean geometries. Cook Wilson claimed that he had 'apprehended' and thus 'knows' the truth of Euclid's axiom of the parallels--he held it to be "absolutely self-evident" (SI, 561)--calling the idea of a non-Euclidean space a "chimera" (SI, 456) and non-Euclidean geometries "the mere illusion of specialists" (SI, xxxix). He thus tried in vain to find a contradiction to "convince the rank and file of mathematicians" so that "they would at least not suppose the philosophic criticism, by which I intended anyhow to attack, somehow wrong" (SI, xcvi). (See section 5 for further discussion of the point about knowledge.) Cook Wilson published very little in philosophy: his inaugural lecture, On an Evolutionist Theory of the Axioms (ETA, see also SI, 616-634), a critique of Spencer opening with a very short encomium to Green and Lotze (ETA, 3-4), and a short piece in Mind (CLP) on the 'Barber Shop Paradox'. In the former, he argued that Spencer's claim that the criterion of truth p is the impossibility to think its contradictory and that this criterion is the product of evolution is based on a circular reasoning: to prove that the criterion results from evolution, one must apply it. The 'Barber Shop Paradox'--not to be confused with the 'Barber's Paradox'--is attributed to Lewis Carroll (Carroll 1894), but it originated in a private debate about 'hypotheticals' with Cook Wilson, who attempted his own solution in CLP (Moktefi 2007a, chap. v, sect. 2), (Moktefi 2007b), (Moktefi 2008, sect. 5.2), while Russell had already satisfactorily resolved it in a footnote to The Principles of Mathematics (Russell 1903, 18). Cook Wilson was also involved on the same occasion in the genesis of Carroll's better-known 'paradox of inference', in 'What the Tortoise Said to Achilles' (Carroll 1895), which will be discussed in section 8. Cook Wilson's reluctance to publish was partly caused by the fact that he constantly kept revising his views. As mentioned, he apparently reached a more or less stable viewpoint only late in his life. One of his better-known sayings is that > > > ... the (printed) letter killeth, and it is extraordinary how it > will prevent the acutest from exercising their wonted clearness of > vision. (SI, 872) [See also Collingwood (2013, 19-20).] > > > His argument was that authors that have committed to print their views on a given issue would, should they prove to be erroneous, more often than not feel obliged to defend them and to engage in pointless rhetorical exchanges instead of seeing immediately the validity of arguments against them. As a result, Cook Wilson resorted throughout his career to the printing for private circulation of pamphlets, known as Dictata, which he began revising for publication shortly before his death. Only 11 years later did the two volumes of Statement and Inference appear, in 1926, put together by Farquharson from his lecture notes and Dictata, along with some letters. These volumes are subdivided in five parts and 582 sections. Their arrangement, which is not Cook Wilson's, betrays their origin in his lectures in logic and in the theory of knowledge; also interspersed are texts that originate from his study of and lectures on Plato and Aristotle. As the chronological table of the various sections shows (SI, 888-9), the texts thus assembled were written at different dates and, in light of Cook Wilson's frequent change of mind (including his move away from idealism), they express views that are at times almost contradictory. This makes any study of his philosophy particularly difficult and, more often than not, accounts of his views are influenced by those, equally important, of his pupil H. A. Prichard. ## 2. Ancient Philosophy Cook Wilson published over 50 papers in Ancient philosophy, in scholarly journals such as Classical Review, Classical Quarterly, Transactions of the Oxford Philological Society and Philologische Rundschau, and a few book-length studies on Aristotle's Nicomachean Ethics (AS) and on Plato's Timaeus. His main philological claim concerning the structure of the seventh book of Nicomachean Ethics was that it contained traces of three versions probably written by some Peripatetic later than Eudemus. To this he added a year later a discussion (ASV) of interpolations in Categories, Posterior Analytics and Eudemian Ethics. In a postscript to the revised version of AS (1912), he claimed, however, that the variants were probably different drafts written by Aristotle himself. His pamphlet on the Timaeus (IPT) was mainly polemical, painstakingly detailing R. D. Archer-Hind's 'obligations' in his 1888 edition of the Timaeus to earlier authors, J. G. Stallbaum more than any other. Cook Wilson is mainly remembered today for two contributions to Plato studies, foremost for his paper 'On the Platonist Doctrine of the asumbletoi arithmoi' (OPD), which bears on the debate on 'intermediates' in Plato. This issue originates in Aristotle's claim in Met. A 6 987b14-17 that Plato believed that the objects of mathematics occupy an "intermediate position" between sensible things and Ideas/Forms. Since there is no explicit commitment to this claim in Plato, scholars either rejected Aristotle's testimony or looked for passages where Plato might be said to have implicitly endorsed it, such as the Line at the end of Book VI of the Republic. Cook Wilson argued against reading this passage as implicitly endorsing 'intermediates', claiming that objects of thought (dianoia) are Ideas, because Plato stated in Rep. 511d2-3 that they are "intelligible given a principle" (kaitoi noeton onton met' arkes), and, as Cook Wilson saw the matter: "nothing but an idea can be an object of nous" (OPD, 259). He also argued that universals being 'one' in contrast with the 'many' to which they correspond, there could be only one 'Circularity', one 'number Two', etc. This means that, for example, 'the number Two' or the universal 'twoness' being one, it cannot be a plurality composed of units and thus that 'two and two make four' is not an addition of universals such as 'twoness and twoness make fourness' or 'the number Two added to the number Two makes the number Four'. These expressions have no sense according to Cook Wilson, for whom 'two and two make four' merely means that 'any two things added to any other two things make four things' (OPD, 249). This is why numbers qua universals are 'unaddible' or 'uncombinable' (asumbletoi) as Aristotle put it in, e.g., Met. M 8 1083a34, a view that Cook Wilson takes to be "exactly the Platonic doctrine" (OPD, 250). The need to posit 'intermediates' can be seen as arising from the fact that one cannot perform arithmetical operations on these 'Idea numbers', as Cook Wilson called them, while one would need entities that are 'in between' (ta metaxu) sensible things and Idea numbers to account for arithmetical truths as elementary as 'two and two make four'. Cook Wilson nevertheless argued against commitment to 'intermediates' (OPD, SS4), his view being that arithmetical operations are always on particulars and if the 'monadic numbers' of arithmetic are plurality of units, Idea numbers are properties of these numbers. That Plato had reached this view of Idea numbers as asumbletoi arithmoi at the time he wrote the Republic or later on would be yet another issue. Cook Wilson also argued (OPD, SS5) that Idea numbers form a series ordered by a relation of 'before and after' (proteron kai usteron) and, following Aristotle's testimony in Eth. Nic. I 6, 1096a17-19 and Met. B 3, 999a6-14, that there could be no genus of the species forming such ordered series (for a critique of this reading of Aristotle, see Lloyd (1962)). In other words, there can be no Form or universal of the Idea numbers. He would thus claim that in the proposition 'The number Two is a universal', the expression 'a universal' cannot denote in turn a particularization of 'universalness', because the latter is not a true universal as it lacks an "intrinsic character" (SI, 342 n.1 & 351), and this he took to be the point of Plato's doctrine of "unaddible numbers" (SI, 352). Cook Wilson not only believed that these views are the true interpretation of Plato, but also that they are true simpliciter and he criticized Dedekind's definition of continuity in Stetigkeit und die irrationale Zahlen (Dedekind 1872) for "not realising the truth attained so long ago in Greek philosophy that [numbers] are Universals" and not magnitudes (OPD, 250, n.1 & SI, ciii) (Joseph 1948, 59-60). He never explained in any details how this critique is supposed to work against Dedekind, but provided instead lengthy and unconvincing arguments for also rejecting Russell's logicist definition of natural numbers on similar ground (see section 8). Despite the fact that he drew such preposterous consequences from it, Cook Wilson's reading of Plato and Aristotle in OPD remained influential, albeit controversial, throughout the last century, through Ross' edition of Aristotle's Metaphysics (1924, liii-lvii, ad B 3, 999a6-10 & M 6, 1080a15-b4) and especially through its advocacy by Harold Cherniss in Aristotle's Criticism of Plato and the Academy (Cherniss 1944, App. vi) and The Riddle of the Early Academy (Cherniss 1945, 34-37 & 76). Cherniss' student Reginald Allen still claimed in the 1980s that Cook Wilson's is the "true view of Plato's arithmetic" (Allen 1983, 231-233). (For further endorsements of Cook Wilson's reading see (Joseph 1948, 33 & chap. v), (Klein 1968, 62) or (Taran 1981, 13-18) and for criticisms see (Hardie 1936, chap. vi), (Austin 1979, 302) or (Burnyeat 2012, 166-167).) Cook Wilson's other notable contribution is the provision, in 'On the Geometrical Problem in Plato's Meno, 86e sqq.' (GPP), of a key addition to S. H. Butcher's clarification (1888) of the notoriously obscure geometrical illustration of Meno 86e-87b, showing that Plato did not intend to offer an actual solution to the problem of the inscription of an area as a triangle within a circle, but simply to determine, while alluding to the method of analysis, the possibility of its solution. An analogous explanation of that passage was provided later on by T. L. Heath (1921, 298-303) and A. S. L. Farquharson (1923), while Knorr (1986, 71-74) and Menn (2002, 209-214) defended this interpretation without mention of Cook Wilson. (See also the critical discussion of the 'Cook Wilson/Heath/Knorr interpretation' in (Lloyd 1992), as well as (Scott 2006, 133-137) and (Wolfsdorf 2008, 164-169).) Apart from these, Cook Wilson's impact seems to have been limited to minute points of philology, such as the references to his critical comments on Apelt's edition of De Melisso Xenophane Gorgia (APT) in (Kerferd 1955) or to his pamphlet on the Timaeus in A. E. Taylor's and F. M. Cornford's own commentaries of that dialogue, and the critical discussion of his views in Ross' edition of Aristotle's Prior and Posterior Analytics (Ross 1949, 496-497). Given that Cook Wilson's views on knowledge and belief (see section 5) are linked to Plato's own distinction between episteme and doxa, it is worth mentioning that they also had an impact on the study of Plato's dialogues. For example, although A. D. Woozley held a different view of knowledge as "not something generically different from belief, but as the limited case of belief" (Woozley 1949, 193), Cook Wilson's distinction between knowledge and belief is introduced as an interpretative tool in chapter 8 of R. C. Cross and A. D. Woozley, Plato's Republic. A Philosophical Commentary (Cross & Woozley 1964), while the idea that knowledge involves 'reflection' (the 'accretion' described in section 5) and the concept of 'being under the impression that' are involved in Ryle's discussion of knowledge in Plato's Theaetetus (Ryle 1990, 23 & 27-28). (For a similar claim about Theaetetus, see Prichard (1950, 88), and for Ryle on Cook Wilson on Parmenides, see end of section 7.) The impact of Cook Wilson's views was at any rate not limited to Plato studies: they form, for example, the basis for H. A. Prichard's critique of Kant in his Kant'sTheory of Knowledge (Prichard 1909) or of his lectures on the Descartes, Locke, Berkeley and Hume on knowledge (Prichard 1950, chap. 5). (See in particular Prichard (1909, 245), quoted below or (1950, 86, 88 & 96) for statements of the distinction between knowledge and belief.) ## 3. On Method: Ordinary Language If Cook Wilson's reverence towards ordinary language derived from his training as a philologist, it was not limited to occasional references to instances of usage to buttress his arguments. He believed that in philosophy one must above all "uncompromisingly [...] try to find out what a given activity of thought presupposes as implicit or explicit in our consciousness", i.e., to "try to get at the facts of consciousness and not let them be overlaid as is so commonly done with preconceived theories" (SI, 328). He also spoke of the latter as originating in 'reflective thought' and argued it has two major defects. First, it is based on principles that, for all we know, might be false and, concomitantly, it is too abstract, because it is not based on the consideration of particular concrete examples. Indeed, in a passage where he criticized Bradley's regress arguments against the reality of relations (Bradley 1897, chap. III), Cook Wilson begins by pointing out that "throughout this chapter there is not a single illustration, though it is of the last importance that there should be" (SI, 692). (On this critique of Bradley, see also Joseph (1916a, 37).) This is, however, slightly misleading, given that Bradley opens his discussion in his previous chapter, where a first regress is deployed, with the case of a lump of sugar (Bradley 1897, 16).) As H. H. Price put it later, for Cook Wilson and his epigones "to philosophize without instances would be merely a waste of time" (Price 1947, 336). Secondly, Cook Wilson thought that philosophers are most likely to introduce distinctions of their own that do not correspond to the 'facts of consciousness' and thus distort our understanding of them. He therefore strove to uncover these 'facts of consciousness' through an analysis of concrete examples which would be free of philosophical jargon. This is strongly reminiscent of the 'descriptive psychology' of the Brentano School. As a matter of fact, Gilbert Ryle, who described himself as a "fidgetty Cook Wilsonian" in his youth (Ryle 1993, 106) and who was also probably the only Oxonian who knew something about phenomenology in the 1920s, believed Cook Wilson's descriptive analyses to be as good as any from Husserl (Ryle 1971, vol. I, 176 & 203n.). In what may be deemed a variant of the 'linguistic turn', Cook Wilson believed that an examination of the "verbal form of statement" was needed in order "to see what light the form of expression might throw upon problems about the mental state" (SI, 90), thus that ordinary language would be the guide to the 'facts of consciousness', because it embodies philosophically relevant distinctions: > > It is not fair to condemn the ordinary view wholly, nor is it safe: > for, if we do, we may lose sight of something important behind it. > Distinctions in current language can never safely be neglected. (SI, > 46) > > > The authority of language is too often forgotten in philosophy, with > serious results. Distinctions made or applied in ordinary language are > more likely to be right than wrong. Developed, as they have been, in > what may be called the natural course of thinking, under the influence > of experience and in the apprehension of particular truths, whether of > everyday life or of science, they are not due to any preconceived > theory. In this way the grammatical forms themselves have arisen; they > are not the issue of any system, they are not invented by any one. > They have been developed unconsciously in accordance with distinctions > which we come to apprehend in our experience. (SI, 874) > > > Reflective thought tends to be too abstract, while experience which > has developed the popular distinctions recorded in language is always > in contact with the particular facts. (SI, 875) > For this reason, Cook Wilson considered it "repugnant to create a technical term out of all relation to ordinary language" (SI, 713) and SI is replete with appeals to ordinary language. For example, he argued in support of his views on universals that "ordinary language reflects faithfully a true metaphysics of universals" (SI (208), see section 7). But such appeals were not just meant to undermine 'preconceived theories', they were also constructive, e.g., when he distinguished between the activity of thinking and 'what we think', i.e., between 'act' and 'content' (SI, 63-64), arguing that this is "likely to be right" because "it is the natural and universal mode of expression in ordinary untechnical language, ancient and modern" (SI, 67) and "it comes from the very way of speaking which is natural and habitual with those who do not believe in any form of idealism" (SI, 64). As it turns out, Cook Wilson believed that the 'content', i.e., 'what we think' is not about the thing we think about, but the thing itself (so knowledge contains its object, see section 6). These views were to prove particularly influential in the case of J. L. Austin, who began his studies at Oxford four years after the publication of SI. It is a common mistake to think of Wittgenstein as having had some formative influence on Austin, as he was arguably the least influenced by Wittgenstein of the Oxford philosophers (Hacker 1996, 172). (On this point see also Marion (2011), for the contrary claim, Harris & Unnsteinsson (2018).) At any rate, the evidence adduced here and elsewhere since Marion (2000) ought not to be ignored and, the philosophy of G. E. Moore notwithstanding, it is rather Cook Wilson and epigones such as Prichard that are the source of the peculiar brand of 'analytical philosophy' that was to take root in Oxford in the 1930s, known as 'Oxford philosophy' or 'ordinary language philosophy'. One merely needs here to recall the following well-known passage from Austin's 'A Plea for Excuses', which is almost a paraphrase of Cook Wilson: > > Our common stock of words embodies all the distinctions men have found > worth drawing, and the connections they have found worth marking, in > the lifetimes of many generations: these surely are likely to be more > numerous, more sound, since they have stood up to the long test of the > survival of the fittest, and more subtle, at least in all ordinary and > reasonably practical matters, than any that you or I are likely to > think up in our arm-chairs of an afternoon--the most favoured > alternative method. (Austin 1979, 182) > Neither Cook Wilson nor Austin believed, however, that ordinary language was not open to improvements. Cook Wilson was explicit about this when detailing his procedure: > > > Obviously we must start from the facts of the use of a name, and shall > be guided at first certainly by the name: and so far we may appear to > be examining the meaning of a name. Next we have to think about the > individual instances, to see what they have in common, what it is that > has actuated us. [...] At this stage we must take first what > seems to us common in certain cases before us: next test what we have > got by considering other instances of our own application of > the name, other instances in which has been working in us. Now when > thus thinking of other instances, we may see that they do not come > under the formula that we have generalized. [...] There is a > further stage when we have, or think we have, discovered the nature of > the principle which has really actuated us. We may now correct some of > our applications of the name because we see that some instances do not > really possess the quality which corresponds to what we now understand > the principle to be. This explains how it should be possible to > criticize the facts out of which we have been drawing our data. (SI, > 44-45) > > > Austin's 'linguistic phenomenology' (Austin 1979, 182) was devised along similar lines (on this point, see Longworth 2018a), and he also thought that ordinary language could be improved upon: > > > Certainly ordinary language has no claim to be the last word, if there > is such a thing. [...] ordinary language is not the last > word: in principle it can everywhere be supplemented and improved upon > and superseded. Only remember, it is the first word. (Austin > 1979, 185) > > > It is worth noting that Cook Wilson uses in the above passage the example of Socrates' search for definitions as an illustration of his procedure. This strongly suggests that he derived it from consideration of the method of induction (epagoge) in Ancient philosophy: collect first a number of applications of a given term and, focussing on salient features of these cases, formulate as an hypothesis a general claim covering them, then test it for counterexamples against novel applications. (Longworth (2018a) has also drawn some interesting parallels with 'experimental mathematics' (see Baker (2008) and the entry to this Encyclopedia).) ## 4. Apprehension & Judgement Since Cook Wilson's philosophy was largely defined in opposition to British Idealism, it is worth beginning with some points of explicit disagreement. Roughly put, under the idealist view knowledge is constituted by a coherent set of mutually supporting beliefs, none of which are basic, while others would be derivative. Surprisingly, when H. H. Joachim published The Nature of Truth (1906), perhaps the best statement of the coherence theory of truth usually attributed to the British Idealists, Cook Wilson criticized him for relying on a discredited 'correspondence' theory (SI, 809-810). Cook Wilson did not argue directly against the coherence theory, as Russell did (Russell 1910, 131-146), but simply took the opposite foundationalist stance. He reasoned that the chain of justification ought to come to an end and that this end point is some non-derivative knowledge, which he called 'apprehension' (SI, 816). As he put it: "it becomes evident that there must be apprehensions not got by inference or reasoning" (UL, SS 18). As Farquharson noted (SI, 78 n.), Cook Wilson did not define his key notion of 'apprehension'. (This is related with Cook Wilson's claim discussed in section 5 that knowledge is undefinable.) The notion appears to be at the same time close to Aristotle's 'noesis' and to Russell's 'acquaintance'. Cook Wilson obviously took his lead from a tradition beginning with Posterior Analytics B 19 and his comments are reminiscent of Thomas Reid, who argues in his Essays on the Intellectual Powers of Man (Bk. II, chap. v) that perception involves some conception of the object and the conviction of its existence, this conviction being immediate, non-inferential and not open to doubt. Cook Wilson was not exactly faithful to Aristotle and Reid, however, since he argued that apprehensions can be both perceptual and non-perceptual (SI, 79), and that some are obtained by inference while some are not, the latter being the material of inference (SI, 84-85). Furthermore, he argued that perceptual apprehensions should not be confused with sensations, as the mere having of a sensation is not yet to know what the sensation is, an idea that has echoes in Austin (Austin 1979, 91-94 & 122 n.2). For this, one needs an "originative act of consciousness" that goes beyond mere passivity and compares the sensation in order to apprehend its definite character. As Cook Wilson put it: "we are really comparing but we do not recognize that we are" (SI, 46). Cook Wilson thought it misleading to base logic on 'judgement' instead of 'proposition' or 'statement' (SI, 94) and he questioned the traditional analysis of proposition under the form 'S is P', which he saw has having various meanings (Joseph (1916a, 6), see also section 9 for a related point). In his polemics against idealism, Cook Wilson's main target was the traditional theory of judgement that one finds, e.g., in Bradley's Principles of Logic (Bradley 1928), where the topic is simply divided into 'judgement' and 'inference'. There would thus be a common form of thinking called the 'judgement' that 'S is P', which would include non-inferred knowledge, opinion, and belief, but would exclude inferred knowledge. One would be misled, he argued, by the common verbal form 'S is P' that knowledge, belief, and opinion are species of the same genre called 'judgement' (SI, 86-7). He claimed instead to follow 'ordinary usage' in adopting a 'judicial' account of 'judgement': > > A judgement is a decision. To judge is to decide. It implies previous > indecision; a previous thinking process, in which we were doubting. > Those verbal statements, therefore, which result from a state of mind > not preceded by such doubt, statements which are not decisions, are > not judgements, though they may have the same verbal form as > judgements. (SI, 92-3) > He further argued first that inferring is thus one of the forms of judgement: "if we take judging in its most natural sense, that is as decision on evidence after deliberation, then inferring is just one of those form of apprehending to which the words judging and judgement most properly apply" (SI, 86). Some inferences are, however, immediately apprehended, e.g., when one recognizes that it follows from 'if p, then q' and 'p', that 'q'. Furthermore, the presence of a prior indecision or doubt, as opposed to confidence, is deemed an essential ingredient of judgement. It does not, however, fully put an end to doubt: as a judge may well be mistaken, our ordinary judgements "form fallible opinions only" (Joseph 1916a, 160). Now, if indecision and doubt are involved prior to judgement, apprehension or knowledge (perceptual or not) could not be judgement, because, by definition, there is no room for doubt in these cases. When one is of the opinion that p, one has found the evidence being in favour of p, without being conclusive. But Cook Wilson thought statements of opinion as not involving the expression of a decision, so they not judgement either: > > > It is a peculiar thing--the result of estimate--and we call > it by a peculiar name, opinion. For it, taken in its strict and proper > sense, we can use no term that belongs to knowing. For the opinion > that A is B is founded on evidence we know to be insufficient, whereas > it is of the very nature of knowledge not to make its statements at > all on grounds recognized to be insufficient, nor to come to any > decision except that the grounds are insufficient; for it is here that > in the knowing activity we stop. (SI, 99-100) > > > Moreover, there is no 'common mental attitude' involved in 'knowing' and 'opining': > > > One need hardly add that there is no verbal form corresponding to any > such fiction as a mental activity manifested in a common mental > attitude to the object about which we know or about which we have an > opinion. Moreover it is vain to seek such a common quality in belief, > on the ground that the man who knows that A is B and the > man who has that opinion both believe > that A is B. (SI, 100) > > > It is an important characteristic of 'believing', setting it apart from other 'activities of consciousness', that it is accompanied with a feeling of confidence, greater than in opining: > > To a high degree of such confidence, where it naturally exists, is > attached the word belief, and language here, as not infrequently, is > true to distinctions which have value in our consciousness. It is not > opinion, it is not knowledge, it is not properly even judgement. (SI, > 102) > The upshot of these remarks is that 'knowledge', 'belief', and 'opinion' are not, as idealists would have it, species of the same genre, 'judgement' or 'thinking': these are all distinct and sui generis. This leads to the all-important distinction between 'knowledge' and 'belief', discussed in the next section: they do not merely differ in kind, they are not even two species of the same genus (Prichard 1950, 87). But Cook Wilson also held the view that knowing is more foundational, so to speak, as it is presupposed by other 'activities of thinking' such as judging and opining. For example, opinion involves knowledge, but goes beyond it: > > > There will be something else besides judgement to be recognized in the > formation of opinion, that is to say knowledge, as manifested in such > activities as occur in ordinary perception; activities, in other > words, which are not properly speaking decisions. (SI, > 96) > > > ## 5. Knowing & Believing Given that "our experience of knowing [is] the presupposition of any inquiry we can undertake", Cook Wilson reasoned that "we cannot make knowing itself a subject of inquiry in the sense of asking what knowing is" (SI, 39). It follows immediately from this impossibility of inquiring about the nature of knowledge that a 'theory of knowledge' is itself impossible, a consequence he first expressed in a letter to Prichard in 1904: > > We cannot construct knowing--the act of > apprehending--out of any elements. I remember quite early in my > philosophic reflection having an instinctive aversion to the very > expression 'theory of knowledge'. I felt the > words themselves suggested a fallacy. (SL, 803) > > > Knowledge is sui generis and therefore a 'theory' > of it is impossible. Knowledge is simply knowledge, and an attempt to > state it in terms of something else must end in describing something > which is not knowledge. (Prichard 1909, 245) > Thus, knowledge, as obtained in 'apprehension', could not be defined in terms of belief augmented by some other property or properties, as in the definition of knowledge as 'justified true belief'. Cook Wilson is thus to be counted among early 20th-century opponents of this view (Dutant (2015), Le Morvan (2017) & Antognazza (2020)). This view is known since Timothy Williamson's Knowledge and its Limits as 'knowledge first' (see Williamson (2000, v) and Adam Carter, Gordon & Jarvis (2017) for recent developments of this view). At the turn of the last century, it was also held by the neo-Kantian philosopher Leonard Nelson (Nelson 1908 & 1949), to whom Cook Wilson alludes in SI (872), but it was also held unbeknownst to him by members of the Brentano school, such as Adolf Reinach, Max Scheler, and Edmund Husserl. (See Mulligan (2014) for a survey.) It was to become the central plank of 'Oxford Realism'. For the Oxonians and the Brentanians one knows that p only if one 'apprehends' that p. As Kevin Mulligan put it: "one knows that p in the strict sense only if one has perceived that p and such perceiving is not itself any sort of belief or judging" (Mulligan 2014, 382). A version of this view is defended today by Timothy Williamson, according to whom knowledge is a mental state "being in which is necessary and sufficient for knowing p" (Williamson 2000, 21), and which "cannot be analysed into more basic concepts" (Williamson 2000, 33). The claim is, however, about 'knowledge that p' and not anymore about 'apprehending that p'. If knowledge is indeed distinct from belief, then the difference cannot be one of degree in the feeling of confidence or in the amount of evidential support: > > In knowing, we can have nothing to do with the so-called > 'greater strength' of the evidence on which the opinion is > grounded; simply because we know that this 'greater > strength' of evidence of A's being B is compatible with > A's not being B after all. (SI, 100) > > > To know is not to have a belief of a special kind, differing from > beliefs of other kinds; and no improvement in a belief and no increase > in the feeling of conviction which it implies will convert it into > knowledge. (Prichard 1950, 87) > Austin has a nice supporting example in Sense and Sensibilia, using the fact that 'seeing that' is factive: if no pig is in sight, I might accumulate evidence that one lives here: buckets of pig food, pig-like marks on the ground, the smell, etc. But if the pig suddenly appears: > > > ... there is no longer any question of collecting evidence; its > coming into view doesn't provide me with more evidence > that it's a pig, I can now just see that it is, the > question is settled. (Austin 1962, 115) > > > There is a parallel move by John McDowell in 'Criteria, Defeasibility and Knowledge' against the notion of 'criteria', deployed by some commentators of Wittgenstein as a sort of 'highest common factor' between evidence and proof (McDowell 1998, 369-394). Here, a 'highest common factor' would be a state of mind that would count as knowing, depending on one or more added factors, but would count in their absence as something else such as believing (McDowell 1998, 386). McDowell is now seen as having thus put forward a form of 'disjunctivism' about knowledge and belief (more on disjunctivism in section 6). Although McDowell does not mention these authors, Travis (2005) has shown that this move has roots in both Cook Wilson and Austin. The fact that there is no highest common factor to knowledge and belief entails for Cook Wilson a rejection of 'hybrid' and 'externalist' accounts of knowledge. Any 'hybrid' account would factor knowledge into an internal part, possibly a copy of the object known, and a relation of that copy to the object itself (see immediately below and section 6). Since there is no such highest common factor, this view, integral to 'externalist' accounts of knowledge, had to be rejected (Travis 2005, 287). But Cook Wilson was led here to a further thesis. If one is prepared to say 'I believe p', when one is not sure that evidence already known is sufficient to claim that 'I know that p', then it looks as if one should always be in a position to know if one knows or if one merely believes. He argued, however, that 'knowing that one knows' should not mean that, once a particular piece of knowledge has been obtained, one should then decide if it counts as knowledge or not, because this decision would count again as a piece knowledge and "we should get into an unending series of knowings" (SI, 107). This is why he insisted that knowing that one knows "must be contained within the knowing process itself": > > > Belief is not knowledge and the man who knows does not believe at all > when he knows: he knows it. (SI, 101) > > > > The consciousness that the knowing process is a knowing process must > be contained within the knowing process itself. (SI, 107) > > > This claim was given further emphasis by Prichard, but his own formulation differ in an important respect, since he introduces the idea of a "reflection" in virtue of which when one knows that p, one is able to know that one knows that p and, when one believes that p, one is also able to know that one believes that p, so that it would be impossible to mistake knowledge for belief and vice-versa: > > > We must recognize that whenever we know something we either do, or at > least can, by reflecting, directly know that we are knowing it, and > that whenever we believe something, we similarly either do or can > directly know that we are believing it and not knowing it. (Prichard > 1950, 86 & 88) > > > One can see that this is not quite Cook Wilson's position, given his regress argument. Prichard assumes that, whenever one does not know, one knows that one does not know: > > > When knowing, for example, that the noise we are hearing is loud, we > do or can know that we are knowing this and so cannot be mistaken, and > when believing that the noise is due to a car we know or can know that > we are believing and not knowing this. (Prichard 1950, 89) > > > Charles Travis called the claim that in knowing that p one can always distinguish one's condition from all states in which not-p the 'accretion' (Travis (2005, 290) & Kalderon & Travis (2013, 501)) and he argued that it damages the core of Cook Wilson's and Prichard's positions (Travis 2005, 289-294). To use the above example of the pig suddenly coming into view, the claim is that, although one can know by reflection that one knows that one is seeing a pig, one cannot by mere reflection exclude the possibility that one is in fact seeing some cleverly engineered 'ringer'. The upshot of this argument is not immediately clear: if it is the case that one knows p only if p and one knows that one knows that p, how could one be unable to exclude the case that not-p, hence that one does not know p? At all events, Travis sees this as reinstalling the argument from illusion (Travis 2005, 291), that had already been subjected to numerous critiques from an early paper by Prichard to Austin's Sense and Sensibilia: > > > That statements about appearances imply that we at least know enough > of reality to say that real things have certain possible > predicates, e.g., bent or convergent. To deny this is to be wholly > unable to state how things look. [...] It is only because we know > that our distance from an object affects its apparent size that we can > draw a distinction between the size it looks and the size it is. If we > forget this we can draw no distinction at all. (Prichard 1906, > 225-226) > > > > ... it is important to remember that all talk of deception only > makes sense against a background of general non-deception. > (You can't fool all the people all of the time.) It must be > possible to recognize a case of deception by checking the odd > case against more normal ones. (Austin 1962, 11) > > > Guy Longworth (2019) has detailed how much Austin owed to Cook Wilson on knowledge (see Urmson (1988) for a discussion in relation to Prichard): Austin held the 'knowledge first' view and, concomitantly, rejected the possibility of a theory of knowledge (Austin 1962, 124), he viewed knowledge as akin to proof, thus as being different in kind from belief on accumulation of evidence (see Austin (1962, 115) & (1979, 99)). But Austin dropped the 'accretion' in 'Other Minds' as he shifted the analysis of the claim that 'If I know, I can't be wrong' (SI (69) & Prichard (1950, 88)) to that of 'I know that p' as providing a form of warrant, one's authority for saying that p (Austin 1979, 99-100). Krista Lawlor (2013) recently suggested that Austin introduced here the speech act of 'assurance'. (Although it has been claimed that the idea of 'performatives' originates in an exchange of letters on promises with Prichard (Warnock 1963, 347), that dimension remained unexplored in Oxford Realism before Austin, who was consciously moving away here from strict focus on 'statements'.) The 'accretion' raises indeed an issue concerning 'other minds', given that it is one's reflective view that, supposedly, authoritatively determines that one knows: could there be other ways to determine whether someone else knows? (See Longworth (2019).) This is how Austin, who also viewed knowledge as a state of mind, was arguably led to explore the ramifications the challenge 'How do you know?' to the person claiming 'I know'. Although he warned against the 'descriptive fallacy' (Austin 1979, 103), Austin's claim appears to be, rather, that 'I know' has functions "over and above describing the subjective mental state of knowing" (Longworth (2019, 195), see Austin (1979, 78-79)). To come back to Cook Wilson, he grappled here with related problems. It is implied by the above that knowledge requires a sort of warrant very much akin to a proof and, consequently, "we are forced to allow that we are certain of very much less than we should have said otherwise" (Prichard 1950, 97). Mathematical knowledge appears paradigmatic. As Prichard put it: "In mathematics we have, without real possibility of question, an instance of knowledge; we are certain, we know" (Prichard 1919, 302). (Joseph used this view to argue against Mill's empiricist account of mathematics, using 'intuition' where Cook Wilson would use 'apprehension' (Joseph 1916a, 543-553).) Alas, Cook Wilson put forth the axiom of parallels in Euclidean geometry as an example of knowledge, and dismissed non-Euclidean geometries as inconsistent (see section 1). If for any p such as the axiom of parallels, someone fails to 'apprehend' it, then all he could do is, somehow lamely, ask them to try and "remove [...] whatever confusions or prejudices [...] prevent them from apprehending the truth of the disputed proposition" (Furlong 1941, 128). This reply raises a prima facie problem for Cook Wilson, because he could not have known, in the sense of 'knowing p only if p', that 'non-Euclidean geometries are inconsistent', since it has been proved that they are not. He thus unwittingly provided an illustration of the need to account, from his own internalist standpoint, for the sort of error (or 'false judgement') committed when one claims to know that p, while it is the case that not-p. For someone to know or to be in error while thinking that one knows would just be two undistinguishable states, since which of the two happens to be the case would depend on some external factors: > > ... the two states of mind in which the man conducts his > arguments, the correct and the erroneous one, are quite > undistinguishable to the man himself. But if this is so, as the man > does not know in the erroneous state of mind, neither can he know in > the other state. (SI, 107) > Cook Wilson saw this a threat to the very possibility of demonstrative knowledge, since one would never be sure that any demonstration is true (SI, 107-108). Answering the threat, he would need here is thus to make room for errors--when one thinks that one knows but one does not--but without excluding by the same token the possibility that knowing entails being in a position that one knows that one knows. Cook Wilson was thus led to distinguish, in some of his most intriguing descriptive analyses, a further 'form of consciousness', different from both knowledge and belief (or opinion), which he called 'being under the impression that' (SI, 109-113). A typical example being when one sees the back of Smith on the street and, 'being under the impression that' it is a friend, say Jones, one slaps him on the back, only to realize one's mistake when he turns his head. The "essential feature" of this state of mind is, according to Cook Wilson, "the absence of any sense of uncertainty or doubt, the action being one which would not be done if we felt the slightest uncertainty" (SI, 110). Thus, one did not falsely judge that the man on the street was Jones, because there was no judgement at all: one was merely 'under the impression that' it was him. Maybe one had some evidence that it was Jones, but the point is that one acted on it without questioning the evidence: it was not used as evidence, there was no assessment out of which one may be said to prefer this possibility that other possibility, because no other possibility was entertained. In this state of mind, the possibility of error is somehow excluded, given that it does not occur to one that 'This man is Jones' might be false. It is thus not the case that one thought that one knew but really did not, because it is not true that to begin with one thought one knew, since one had not reflected on one's evidence. The absence of doubt or uncertainty opens the door to the possibility of being mistaken while taking oneself as being certain. The notion of 'being under the impression that' played a significant role in the writings of the Oxford Realists, not just those of Prichard, who also spoke of the 'an unquestioning frame of mind', 'thinking without question' or 'taking for granted' (Prichard 1950, 79 & 96-98), but also those of William Kneale, H. H. Price and J. L. Austin--see, e.g., Kneale (1949, 5 & 18), Price (1935), and Austin (1962, 122). Perhaps most strikingly, Prichard was to drop the 'accretion' and argue in a late essay, 'Perception', that perception is not a kind of knowing: we merely see colour extensions, which we systematically mistake for objects or 'take for granted' to be objects (Prichard 1950, 52-68). Still, the notion of 'being under the impression that' had its critics at Oxford, such as H. P. Grice, who was aware of the difficulties raised by the 'accretion': > > > This difficulty led Cook Wilson and his followers to the admission of > a state of "taking for granted", which supposedly is > subjectively indistinguishable from knowledge but unlike knowledge > carried no guarantee of truth. But the modification amounts to > surrender; for what enables us to deny that all of our so-called > knowledge is really only "taking for granted"? (Grice > 1989, 383-384) > > > One may also justifiably feel that Cook Wilson has not fully explained away cases of error such as his own in being certain that 'non-Euclidean geometries are inconsistent', because his conviction was not the result of merely 'being under the impression that'. Nevertheless, the influence of Cook Wilson's conceptions was felt in a variety of ways in the second half of the last century. H. H. Price offered in Belief an important commentary on Cook Wilson's notion of 'being under the impression that' (Price (1969, 204-220), reprising some of the content of Price (1935)). He deemed it an important addition to the traditional 'occurrence' analysis of belief, as opposed to the 'dispositional' analysis. Price usefully contrasted 'being under the impression that' with 'assent' (Price 1969, 211-212), which is usually said to involve preference and confidence: in prefering p one would decide in favour of p having alternatives q and r in mind, but this is precisely not the case when 'being under the impression that', since in this state one does not entertain alternatives. As Cook Wilson points out, "there is a certain passivity and helplessness" involved (SI, 113). There is no confidence either, since in this state of mind doubt is not an option, hence no degree of certainty is involved. Price also explored (1969, 212-216) connections between this unquestioning state of mind and the notion of 'primitive credulity' (Bain 1888, 511), i.e., the idea harking back to Spinoza (Ethics IIp49s) that one naturally believes in the reality of anything that is presented to one's mind, unless some contradicting evidence is also occurring. In contrast to Price, Jonathan Cohen argued that beliefs are dispositions (Cohen (1989, 368) & (1992, 5)). He also rejected the Cook Wilson's claim that belief involves confidence (see section 4), but, without crediting him (except privately), Cohen made use of 'being under the impression that' to define belief as a disposition to 'normally to feel that p' or 'to feel it true that p' (Cohen (1989, 368) & (1992, 7)). So defined, belief would thus differ from 'acceptance', which results from a conscious and voluntary choice and involves, like Price's 'assent', preference and confidence. Cohen further added to these differences that acceptance is also subjectively closed under deductibility, while this is not the case with belief: > > > ... you may well feel it true that p and that > if p then q, without feeling it true > that q. You will just be failing to put two and two together, > as it were. And detective-story writers, for example, show us how > often and easily we can fail to do this with our beliefs. (Cohen 1992, > 31-32) > > > So acceptance of p involves 'premissing', i.e., the decision to use p as a premise or rule of inference in further reasonings, and Cohen thought that he was carving nature at its joint here: "Belief is a disposition to feel, acceptance a policy for reasoning" (Cohen 1992, 5). This is in many ways not faithful to Cook Wilson's original ideas, but one can sense their presence within the idea of being unreflectively disposed to feel that p being conceptually distinct from accepting p. On another note, John McDowell described knowledge in 'Knowledge and the Internal' as a "standing in the space of reasons" and argued against an "interiorization of the space of reasons" that would occur if one were to think of knowledge as achieving flawless standings in the space of reason, "without needing the world to do us any favours" (McDowell 1998 395-396). If appearances were to be misleading, it would be argued that this not be the result of faulty moves within the space of reason but simply an "unkindness of the world". This conception, McDowell sees as opening the door to an 'hybrid' account of knowledge, with flawless standings in the space of reasons as an internal part, which would provide necessary conditions for knowledge, and favours from the world--when thing are as they appear to be--as an extra condition (McDowell 1998, 400). But, McDowell concludes, "the very idea of reason as having a sphere of operation within which it is capable of ensuring, without being beholden to the world that one's postures are all right [...] has the look of a fantasy" (McDowell 1998, 405). This is so precisely because the resources from the space of reasons could not provide factiveness on their own, so knowledge could not be completely constituted by standings within it. To rid oneself of the fantasy, one needs simply to recognize that on occasions when the world is what it appears to be, this favour is "not extra to the person's standing in the space of reasons" (McDowell 1998, 405) and that "we are vulnerable to the world's playing us false; and when the world does not play us false we are indebted to it" (McDowell 1998, 407). Here too, although the terminology is taken from Wilfrid Sellars, there is a recognizable source in Cook Wilson. Finally, another significant development related to the 'accretion' is its connection with the 'knowing that one knows' principle in epistemic logic, first introduced in Jaakko Hintikka's ground-breaking Knowledge and Belief (Hintikka 1962). Hintikka argued for the equivalence between 'I know that p' and 'I know that I know that p' or, more generally, that 'i knows that p' (Kip) implies 'i knows that i knows that p' (KiKip) (Hintikka 1962, chap. 5). In connection with this, he noticed that Prichard's introduction the idea of "reflection" (see Prichard (1950, 88) quoted above) turns the argument into an argument from introspection that does not sustain the more general claim that 'i believes that p' (Bip) implies 'i knows that i believes that p' (KiBip) (Hintikka 1962, 109-110). Still, Hintikka thought that Cook Wilson and Prichard would be right, if their remarks were to be understood as restricted to the case where i is the first-person pronoun 'I' (Hintikka 1962, 110). Although Williamson has picked up the Oxonian banner of 'knowledge first', this is one point where he did not follow Cook Wilson and Prichard, since he argued against the 'knowing that one knows' principle with help of the notion of 'luminosity'. A condition C is said to be 'luminous' if and only if for every case a, if C obtains in a then in a one is in a position to know that C obtains (Williamson 2000, 95). Williamson (2000, chap. 4) provided arguments against 'luminosity', thus that one can know without being able to know that one knows. ## 6. Perception Cook Wilson also argued against idealism that in apprehension it is neither the case that the object exists only within the apprehending consciousness, nor that it is constituted by it: the object is independent of the apprehending consciousness. As he wrote: "the apprehension of an object is only possible through a being of the object other than its being apprehended, and it is this being, no part itself of the apprehending thought, which is what is apprehended" (SI, 74). This independence, he considered to be presupposed by the very idea of knowledge expounded above. But he further rejected the distinction between act, content and object of perception, first by negating the act-object distinction: > > > In our ordinary experience and in the sciences, the thinker or > observer loses himself in a manner in the particular object he is > perceiving or the truth he is proving. That is what he is thinking > about, and not about himself; and, though knowledge and perception > imply both the distinction of the thinker from the object and the > active working of that distinction, we must not confuse this with the > statement that the thinking subject, in actualizing this distinction, > thinks explicitly about himself, and his own activity, as distinct > from the object. (SI, 79) > > > And, secondly, by rejecting the notion of 'content': "For the only thing that can be found as 'content' of the apprehending thought is the nature of the object apprehended" (SI, 75). Austin echoed this last point saying that "our senses are dumb" (Austin 1962, 11). (There is an extensive literature on this claim, see, e.g., (Travis 2004) or (Massin 2011).) The point is not that perception does not aim at an object, but merely to deny that it does so through a 'content' acting as intermediary: > > what I think of the red object is its own redness, not some mental > copy of redness in my mind. I regard it as having real redness and not > as having my copy of redness. [...] If we ask in any instance > what it is we think of a given object of knowledge, we find it always > conceived as the nature or part of the nature of the thing known. (SI, > 64) > What one apprehends must be the real object itself, not "some mental copy" of it, so Cook Wilson is claiming here that, as a state of mind, knowledge contains its object: "what we apprehend [...] is included in the apprehension as a part of the activity or reality of apprehending" (SI, 70). A long letter to G. F. Stout (SI, 764-800) is of particular importance here, where Cook Wilson criticized Stout on 'Primary and Secondary Qualities' (Stout 1904), with this key diagnosis, of the 'objectification of the appearing as appearance': > > > This is sometimes spoken from the side of the object as the > appearance of the object to us. This 'appearance' > then gets distinguished from the object [...] But next the > appearance, though properly the appearing of the > object, gets itself to be looked on as itself an object and the > immediate object of our consciousness, and being already, as we have > seen, distinguished from the object and related to our subjectivity, > becomes, so to say, a merely subjective > 'object'--'appearance' in that sense. and > so, as appearance of the object, it has now to be represented > not as the object but as some phenomenon caused in our consciousness > by the object. Thus for the true appearance (= appearing) to us of the > object is substituted, through the > 'objectification' of the appearing as appearance, the > appearing to us of an appearance, the appearing of a > phenomenon caused in us by the object. (SI, 796) > > > Cook Wilson's rejection of 'hybrid' accounts of knowledge (see the previous section) is linked to his rejection of epistemological 'intermediaries', so that knowledge could not be of some such 'objectified' appearance. He considered all such intermediaries ('images', 'copies', 'representative', tertium quid, etc.) as "not only useless in philosophy but misleading as tending to obscure the solution of a difficult problem" (SI, 772). In this he stood in the tradition of Thomas Reid, his arguments were as a matter of fact first developed against the empiricism of Locke, Berkeley and Hume (for example in UL (SS 10)). In his letter to Stout, Cook Wilson put it thus: > > > You begin an important section of your argument by assuming the idea > of sensations being representative. > > > > They {represent--express--stand for} something other than > themselves. > > > > Now, I venture to think that the idea of such representation > in philosophy, or psychology rather, is very loose and treacherous > and, if used at all, should be preceded by a 'critique' of > such representative character, and an explanation of the > exact sense in which the word representative is used. (SI, 769) > > > Against views of this kind, Cook Wilson developed three arguments in his letter. First, he pointed out that it is impossible to know anything about the relation between the representative and the object, since one can never truly compare the former to the latter. Secondly, he claimed that representationalist theories are always in danger of leading towards idealism, since one must then somehow 'prove' the existence of the object which is, so to speak, 'behind' its representatives--there might be none. Thirdly, he claimed that all such theories beg the question, since the representative has to be apprehended in turn by the mind, and not only this further 'apprehension' remains unexplained, it would require that the mind be equipped with the very apparatus that the representationalist theories were, to begin with, devised to explain: > > We want to explain knowing an object and we explain it solely in terms > of the object known, and that by giving the mind not the object but > some idea of it which is said to be like it--an image (however > the fact may be disguised). The chief fallacy of this is not so much > the impossibility of knowing such image is like the object, or that > there is any object at all, but that it assumes the very thing it is > intended to explain. The image itself has to be apprehended and the > difficulty is only repeated. (SI, 803) > Cook Wilson also inveighed against Stout's notion of 'sensible extension' pointing out inter alia that it makes no sense to claim that these are extended without being in space (SI, 783) and he tried to explain how a given object may appear to have different shapes from different perspectives, without making an appeal to any representative (SI, 790f.). Stout answered these criticisms in print (for a discussion see Nasim 2008, 30-40 & 94-98). He argued that he had not been holding a view akin to Locke's representationalism, claiming that the 'representative function' of his 'presentations' is of a different nature, more like a memory-image would represent what is remembered (Stout 1911, 14f.), but it is at first blush unclear what he meant by this. Against Cook Wilson's first argument, he claimed that in his conception presentations and presented objects form an "inseparable unity" (Stout 1911, 22), this being, once more, unclear. At all events, both Stout and Russell, in his theory of 'sense-data' as 'objects of perception' (Russell 1912), insisted that the physical object and the representative are 'real'. But one might say that this 'objectification of the appearing as appearance', does not annul Cook Wilson's diagnosis of the difficulties inherent in that position. At least Russell was clearer about its implications, requiring a 'logical construction' of physical objects as functions of 'sense-data'. One way to counter Cook Wilson on the absurdity of the locution 'sensible extensions' is to distinguish between 'private' and 'public' space, as Russell was to do in 'The Relation of Sense-Data to Physics' (Russell 1917, 139-171); as it is well-known, this postulation generates its own set of difficulties, e.g., the claim that space must have 6 dimensions (Russell 1917, 154). Russell on 'sense-data' was to become a favourite target for Prichard's acerbic wit (Prichard 1915 & 1928). Cook Wilson also wove his joint critique of the 'objectification of the appearing as appearance' and of representationalism into a broad historical narrative according to which "empiricism ends in the Subjective Idealism it was intended to avoid" (UL, SS 10, see also SI, 60-63); he spoke of an "insidious and scarcely 'conscious' dialectic" that "has done much mischief in modern metaphysics and theories of perception" (SI, 797). Wilfrid Sellars, who attended Prichard's classes as a Rhodes Scholar at Oxford in the mid-1930s, found the idea of such a 'dialectic' appealing: > > I soon came under the influence of H. A. Prichard and, through him, of > Cook Wilson. I found here, or at least seemed to find, a clearly > articulated approach to philosophical issues which undercut the > dialectic, rooted in Descartes, which led to both Hume and 19th > Century Idealism. At the same time, I discovered Thomas Reid and found > him appealing for much the same reasons. (Sellars 1975, 284) > Although Cook Wilson's philosophy may be construed as a continuation of sorts of the Scottish School of Hutcheson and Reid (see also section 10), it is striking that there are hardly any references to these authors in his writings, even more so given that they contain critiques of Locke, Berkeley and Hume from a similar standpoint. (See Alsaleh (2003), focussing on attacks on Hume and their impact on Price (1932) and Austin (1962), and Marion (2009) for an overview of the 19th-century stages of this 'dialectic'). There was, however, no doctrinal unity on perception among Cook Wilson's epigones. Prichard was probably the first to put Cook Wilson's views into print in 'Appearances and Reality' (Prichard 1906) but, as already pointed out, he ended up arguing that in perception we systematically mistake colour expanses for objects. H. H. Price, who was at first close to Cook Wilson (Price 1924), incorporated a sense-data theory while rejecting phenomenalism in Perception (Price 1932). He became for that reason one of Austin's targets in Sense and Sensibilia (Austin 1962), which remains, for all its novelty, faithful to Cook Wilson's orthodoxy on knowledge. At all events, a form of direct realism in the theory of perception is one of the characteristic features of Oxford Realism. It is an ancestor to contemporary variants such as the position argued for by John McDowell in Mind and World (McDowell 1994) and, after a long eclipse, Cook Wilson's views are once more playing a role in current debates about perception. (See for example references to them in Kalderon (2011, 241; 2018, xv, 49, 88 & 184), Siegel (2018, 2), Stoneham (2008, 319-320).) One topic of particular interest in this respect is 'disjunctivism' (see Soteriou (2016) for an overview). This is the view that in perception one is faced either with cases of genuine perception or with cases of illusion or hallucination, as in 'I see a flash of light or I am having the perfect illusion of doing so' (Hinton 1973, 39). Such disjunctions were first analysed in detail by Michael Hinton (see Hinton (1973) and Snowdon (2008)), but there have been suggestions that disjunctivism harks back to Austin or even Cook Wilson and Prichard (Kalderon & Travis (213, 498-499), for Austin's case, see also Longworth (2019)). Adjudicating such claims depends on one's understanding of disjunctivism itself, but a few points can be adduced. Disjunctivism should not be confused with any 'naive' or 'direct realism', but one may appeal to it to defend such views, therefore one should expect Cook Wilson and Oxford Realists to have taken some steps towards it. The sharp distinction between knowledge and belief (section 5), with the rejection of any 'highest common factor' between the two, should count as a first step. (The above-mentioned critiques of the argument from illusion reinforce this point.) A further step towards a form of disjunctivism is also taken when, having distinguished knowledge from belief, Cook Wilson also claimed that belief presupposes knowledge: it is only when assessing what I may know about some thing that I might realize that it is not sufficient to claim that 'I know that p', so that I remain circumspect and merely claim that 'I believe that p' (see sections 4 & 5). ## 7. Universals In metaphysics too, Cook Wilson had first to cope with idealism, thus with the looming threat of Bradley's regress about relations (Bradley 1897, chaps. II-III). Given a relation R between a and b, a regress arises when one asks what relates the relation to its relata, e.g., what relates a to R(and R to b), and one assumes as an answer that a further relation, say, R\, is needed to tie a* and R, a bit like a string that ties two objects together that needs a further string to both of its ends to tie it to the objects. Since a need to explain the tie between a and R\* arises by the same token, one needs to introduce yet another relation R\\*, and so forth. In a brief chapter (SI, 692-695), Cook Wilson argued that the first move here--supposing that R\** is needed to tie a to R--is simply 'unreal', because there could not be a new relation that relates a relation to one of its terms. Thus, the regress would not be generated. One problem with Cook Wilson's objections is that they admittedly aim at the regress concerning 'external relations' (SI, 692), but they are not fully developed against 'internal relations' that Bradley also entertains (Bradley 1897, 17-18). Cook Wilson may not have a full and appropriate answer to Bradley, but he thought he was thus free to entertain the relation of a subject to its attributes. Although this is rarely noted, Cook Wilson is one of the forerunners of what we now call 'trope theories'. What Edmund Husserl called 'moments' and later on G. F. Stout called 'characters' and D. C. Williams 'tropes', Cook Wilson called instead 'particularization of the universal' (SI, 336) or 'particularized qualities' (SI (713), Mulligan et al. (1984, 293 n.13)). Among his contemporaries, Cook Wilson's ideas stand indeed closest to both Stout's and Husserl's. He seems not to have known about the latter, but one noticeable common feature concerns 'dependence' (Mulligan et al. 1984, 294). Cook Wilson suggested towards the end of his life that 'things' taken in themselves should be called 'existences' (SI, 713) and, keeping close to the language of Aristotle (tode ti, 'a this'), he defined an 'existence' as 'a this such and such' (SI, 713). Starting with the subject-attribute distinction, in the case of an existence where the subject is said to be a 'substance', he argued that the "ordinary conception" of its 'attribute-element' is that it is "always a dependent reality" (SI, 157), with these "dependent existences" having in turn further existences dependent on them (SI, 153). Under his conception 'existences' are, one might say, 'bundles' of particularized qualities. But he did not consider 'substance' as a sort of substrate in which universals are particularized, as many defenders of tropes do: the 'thing' is according to him a mere 'unity' of elements, "not something over and above them. which has them, but their unified existence" (SI, 155). He also expressed this by saying that the universal "covers the whole nature of the substance" (SI, 349), as "the particular does not have the universal in it and something also besides the universal to make it particular" (SI, 336, see also Joseph (1916a, 23)). It contains nothing besides the particularization of the universal: "the particular is not something that has the quality, it is the particularized quality. This animal is particularized animality" (SI, 713). Likewise, "the differentia cannot be separated from the genus as something added on to it" (SI, 359), the species are just the forms that the universal, as genus, takes (SI, 335). Stout, citing Cook Wilson with approval, put it thus: "square shape is not squareness plus shape; squareness itself is a special way of being a shape" (Stout 1930, 398). (The view harks back to Aristotle, Met. I 8 1058a21-26, see also on this point Joseph (1916a, 85-86), Wisdom (1934, 29), Prior (1949, 5-6) and Butcharov (1966, 147-153)). Trope theories usually explain the fact that two things 'share', say, a particular shade of yellow with help of a resemblance class. Here, Cook Wilson parts company, claiming that the universal is "something identical in the particulars, which identity cannot be done away by substituting the term similar for same" (SI, 344 & 347). Although the universal is, according to Cook Wilson, nothing outside its particularizations, it is claimed to be a "unity and identity in particulars" or a "real unity in objects" (SI, 344). It is meant to possess an 'intrinsic character' (SI, 342n. & 351), for which Cook Wilson reluctantly introduced a technical term: "the characteristic being of the universal" (SI, 342). But he introduced no equivalent to Stout's fundamentum relationis or 'distributive unity of a class' (Stout 1930, 388), because he did not define the universal in terms of membership of a class. Although he recognized that universals have an extension, i.e., the "total being of the universal" composed of the "whole of the particulars as the particularization of this unity", he did not identify the universal with it (SI, 338). For this reason and in virtue of the realist epistemology which he tied to this conception (immediately below), Cook Wilson is often read as having held a form of 'immanent realism', as opposed to the moderate forms of nominalism more typical of trope theories (Armstrong (1978, 85) or Moreland (2001, 165 n. 16)). One objection against this stance is that there is not much point having both universals and tropes. As David Armstrong put it, one of the two is bound to be redundant: "Either get rid of universals, embrace a trope version of Resemblance Nominalism or else cut out the middlemen, namely, the tropes" (Armstrong 1989, 17 & 132). This reading cannot be the last word, though, since Cook Wilson also appears only to have admitted one sort of entities, the tropes or 'particularizations of the universal'. Thus, there seems to be a tension, at best unresolved, in Cook Wilson's stance, which could explain why it has found few supporters, one exception being J. R. Jones (1949, 1950 & 1951). Cook Wilson's 'particularizations of the universal' are thus "strictly objective" and "not a mere thought of ours" (SI, 335-336); they thus cannot be phenomenal entities. His thinking here is of apiece with his anti-idealist views on 'apprehension'. He had argued against T. H. Green's neo-Kantian stance, that apprehension has no 'synthetic' character: any synthesis apprehended is attributed to the object and not the result of an activity of the 'apprehending mind'. As he put it, "in the judgement of knowledge and the act of knowledge in general we do not combine our apprehensions, but apprehend a combination" (SI, 279), and it is "the nature of the elements themselves" which "determines which unity they have or can have"; the 'apprehending mind' has "no power whatever to make a complex idea out of simple ones" (SI, 524). This view implies that universals and connections between them, as particularized, are in rebus and to be apprehended as such (Price 1947, 336). The view is, therefore, that there is no possible apprehension of the universal except as particularized: > > > Just as the universal cannot be, except as particularized, so we > cannot apprehend it except in the apprehension of a particular. (SI, > 336) > > > Cook Wilson further reasoned that, when one states that 'a is a triangle', one is predicating of a the universal 'triangularity' and that, analogously, in stating that 'triangularity is a universal' one would then put the universal 'triangularity' in the subject position--the 'nominative case to the verb' as he quaintly puts it (SI, 349)--and treat it as if it were a particular, while putting 'universal' in the predicate position. But to talk about the universal in this way would require per impossibile that one apprehends the universal in abstraction from any of its particularizations. Cook Wilson held his conception as being, if not popular among philosophers, at least in accordance with ordinary language and common sense (SI, 344-345). It is thus the particularization of a given 'characteristic being' which we are said to apprehend, but "neither as universal nor as particularized" (SI, 343). There is no suggestion in Cook Wilson's writings of something akin to Husserl's act of 'categorial intuition' or 'ideational abstraction', in virtue of which the universal would be "brought to consciousness and [...] actual givenness" (Husserl 2001, 292). He believed instead that the 'intrinsic character' of any universal is inexplicable, because the relation between particular and universal, although fundamental and thus presupposed by any explanation (SI, 335 & 345), is sui generis, therefore not explainable in terms of something else: > > > I seem to have discovered that the true source of our metaphysical > difficulties lies in the attempt, a mistaken attempt too frequent in > philosophy, to explain the nature of the universal in terms of > something other than itself. In fact the relation of the universal to > the particular is something sui generis, presupposed in any > explanation of anything. The nature of the universal therefore > necessarily and perpetually eludes any attempt to explain itself. The > recognition of this enable one to elucidate the whole puzzle of the > Parmenides of Plato. (SI, 348, see also SI, 361) > > > Cook Wilson thus believed to have found an answer to the notorious set of regress arguments known as the 'third man' in Parm. 132a-143e. (Ryle argued later on that Cook Wilson's answer would not do, developing his own regress argument against the notion of 'being an instance of', that Ryle read Cook Wilson as presupposing, while merely claiming that it is sui generis (Ryle 1971, vol. I, 9-10).) ## 8. Philosophy of Logic Cook Wilson did not contribute to logic. His inclination was at any rate conservative. For example, there was for him no room even to try and make a case for an alternative logic, since he held the Aristotelian principles (the principle of syllogism, the law of excluded middle and the principle of contradiction) to be "those simple laws or forms of thoughts to which thought must conform to be thought at all. Thought therefore cannot throw any doubt on them without committing suicide" (ETA, 17 & SI, 626). Cook Wilson also rejected new developments in symbolic logic from Boole to Russell. He was not as such adverse to the "symbolization of forms of statements": when criticizing Boole's he alluded to "an improved calculus of [his] own" (SI, 638), which he called "fractional method" (SI, 662). But he did not publish it and all we have is the beginnings of an outline (SI, 192-210). His main objection to Boole--a common one at the time--was to the algebraist's use of equations, perceived as an intrusion of mathematics in logic (SI, 635-636). (See the whole chapter (SI, 635-662).) And the little he knew of mathematical logic, Cook Wilson ferociously opposed. He did think of syllogistic as "a science in the same sense as pure mathematics" (SI, 437), but he was opposed to the very idea of logical foundations for mathematics because he believed that logical inferences are exhausted by syllogistic and, "mathematical inference as such is not syllogistic" (SI, xcvi). This is a misunderstanding, common at the time, of the expressive power of quantification theory as developed among others by Gottlob Frege and C. S. Peirce, of which he was clearly insufficiently cognizant, if not plainly ignorant. Cook Wilson used his interpretation of Plato's doctrine of Idea numbers as asumbletoi arithmoi as a basis for an attack on Russell and the logicist definition of numbers, Plato's doctrine being understood (see section 2) as entailing that there cannot be a universal of the members of the ordered series of universals: 1-ness, 2-ness, 3-ness, etc. because they do not share an 'intrinsic character'. Confusing this series with that of the natural numbers, he concluded that the logicist definition in terms of classes of classes, e.g., of the number 5 as the class of all classes equinumerous with a given quintuplet, is "a mere fantastic chimera" (SI, 352). In line with the argument about putting the universal 'triangularity' in the subject position (previous section), Cook Wilson reasoned for the case of natural numbers that there would be an alleged 'universal of numberness' and that this would lead straight to a contradiction: since all particulars of a universal are said to possess its quality, a group of 5 as a particular of '5-ness' would thus possess the 'universal of numberness', thus contradicting his claim that a particular cannot be a universal (SI, 353). By the same token, Cook Wilson thought that this line of reasoning shows that Russell's paradox of the class of all classes that do not contain themselves (Russell 1903, chap. X) is a "mere fallacy of language" (SI, cx). He thus argued at length (SI, SSSS 422-32), including in his correspondence with Bosanquet (SI, SSSS 477-518) that there can no more be a 'class of classes' than 'universalness' could be a 'universal of universals' and that a class can no more be a member of itself than 'universalness' could be a particular of itself: the implied 'universal of universals' or 'universalness', of which universals would be the particulars, would be a particular of itself, which is, Cook Wilson claims, "obviously absurd" (SI, 350). For bad reasons such as these, Cook Wilson was contemptuous of what he called "the puerilities of certain paradoxical authors" (SI, 348). He even wrote to Bosanquet: > > I am afraid I am obliged to think that a man is conceited as well as > silly to think such puerilities are worthy to be put in print: and > it's simply exasperating to think that he finds a publisher > (where was the publisher's reader?), and that in this way such > contemptible stuff can even find its way into examinations. (SI, 739) > The problem with Cook Wilson's arguments is that they are based on an elementary confusion between membership of a class and inclusion of classes (see for example SI, cx & 733-734). Peter Geach called Cook Wilson "an execrably bad logician" (Geach 1978, 123) for committing blunders such as this. (Cook Wilson's claim in a letter to Lewis Carroll that it is not possible to know that 'Some S is P' without knowing which S it is which is P is another such elementary blunder (Carroll 1977, 376).) Fortunately, Cook Wilson made a more interesting contribution to philosophy of logic in his discussion of Lewis Carroll's paradox of inference (Carroll 1895), of which he gave the following formulation: > > > ... let the argument be A1 = B1, B1= C1, therefore A1 = C1. The > rule which has to be put as the major premiss is, things being equal > to the same thing are equal to one another. Under this we subsume > A1 and C1 are things equal to the same thing, > and so draw the conclusion that they are equal to one another. This is > syllogism I. Now syllogism I, which is of the form MP, SM, SP, in turn > exemplifies another rule of inference which is the so-called > dictum de omni et nullo. This must now appear as a major > premiss. The resulting syllogism may be put variously; the following > short form will serve. Every inference which obey the dictum is > correct; the inference of syllogism I obeys the dictum; therefore it > is correct. This is a new syllogism (II) which again has for rule of > inference the same dictum; hence a new syllogism (III) and so on > in saecula saeculorum. (SI, 444) > > > Leaving aside the incorrect identification of the inference rule as the dictum de omni, this is recognizably the infinite regress in Carroll's paradox and, while Carroll did not provide one, Cook Wilson offered the following diagnosis: > > > ... it is clearly a fallacy to represent the rule according to > which the inference is to be drawn from premisses as one of the > premisses themselves. We should anticipate that this must somehow > produce an infinite regress. (SI, 443) > > > These passages cannot be precisely dated and his correspondence with Carroll for the relevant period is lost, so one cannot tell who framed the paradox first (Moktefi 2007, chap. V, sect. 3.1). They were both anticipated, however, by Bernard Bolzano, who already stated the paradox in his Wissenschaftslehre (1837) and provided a similar diagnosis (Bolzano 1972, SS 199). (For a discussion, see Marion (2016).) Interestingly, Cook Wilson's diagnosis is linked to his own views on the apprehension of universals via their particularizations (see the previous section): > > > A direct refutation may, however, be given as follows. In the above > procedure the rule of inference is made a premiss and a particular > inference is represented as deduced from it. But, as we have seen, > that its an inversion of the true order of thought. The validity of > the general rule of inference can only be apprehended in a particular > inference. If we could not see the truth directly in the particular > inference, we should never get the general rule at all. Thus it is > impossible to deduce the particular inference from the general rule. > (SI, 445) > > > In Lewis Carroll's presentation of the paradox, the Tortoise refuses to infer the conclusion when faced with an instance of the rule of Modus Ponens and Achilles suggests that one should then add the rule as a further premise, but the Tortoise still refuses to infer, so that Achilles then suggest to add the whole formula resulting from adding the rule as a premise as yet a further premise to no avail, and so forth. It is often claimed that the Tortoise's repeated refusals indicates that rules of inference are in themselves normatively inert, so that a further ingredient is needed for one to infer, e.g., a ''rational insight'' (Bonjour 1998, 106-107). Cook Wilson's claim that one can only 'apprehend' the validity of the rule in a particular inference, thus 'seeing the truth directly' or possessing a 'direct intuition' (SI, 441), is an analogous move. The idea that a rule of inference cannot be introduced as a premise in an inference in accordance with it, on pains of an infinite regress, was also reprised by Ryle (Ryle 1971, vol. II, 216 & 238). But he used it to argue for his celebrated distinction between 'knowing how' and 'knowing that': ''Knowing a rule of inference is not possessing a bit of extra information. Knowing a rule is knowing how. It is realised in performances which conform to the rule, not in theoretical citations of it.'' (Ryle 1971, vol. II, 217). However, if 'knowing how' is no longer a state of mind (even dispositionally), then the view is no longer Cook Wilson's. Cook Wilson believed that all statements are 'categorical', arguing away 'hypothetical judgements' with the claim that "in the hypothetical attitude", we apprehend "a relation between two problems" (SI, 542-543 & Joseph (1916a, 185)). In other words, conditionals do not express judgements but connections between questions. This view was elaborated by Ryle into his controversial stance on indicative conditionals as 'inference-tickets' in ''If', 'So', and 'Because'' (Ryle 1971, vol. II, 234-249). Ryle compared conditionals of the form 'If p, then q', to "bills for statements that statements could fill" (Ryle 1971, vol. II, 240) and he rejected the form 'If p then q, but p, therefore q', claiming that in some way the p in the major premise cannot for that reason be the same as p asserted by itself. This, of course, runs afoul of Geach's 'Frege Point' (Geach 1972). ## 9. Philosophy of Language Cook Wilson also questioned superficial uniformity of the form 'S is P', calling the subject of attributes 'metaphysical' (SI, 158) in order to distinguish it from the subject of predication. He thus distinguished the ontological distinction between substance and attribute from the logical subject-predicate distinction. Using the traditional definition of the subject as 'what supports the predicate' and the predicate as 'what is said concerning the subject' (quoting Boethius, SI, 114-115), he noted that a 'statement' such as 'That building is the Bodleian' has different analyses, depending on the occasion in which it is used. If in answer to 'What building is that?', with the 'stress accent' on 'the Bodleian' as in 'That building is the Bodleian', the subject is 'that building' and the predicate as 'what is said concerning the subject' is 'that that building is the Bodleian' (SI, 117 & 158). But, if in answer to the question 'Which building is the Bodleian?' with the 'stress accent' now on 'that' as in 'That building is the Bodleian', then the Bodleian is the subject and the predicate is 'that building pointed out was it' (SI, 119). The same goes for 'glass is elastic', where elasticity is predicated of glass and 'glass is elastic', when it is stated in answer to someone looking for substances that are elastic. This shows that the relation of subject to predicate is somehow symmetric, but it is not the case, however, with the subject/attribute distinction because ''The subject cannot be an attribute of one of its own attributes'' (SI, 158). Cook Wilson also noted that ''the stress accent is upon the part of the sentence which conveys the new information'' (SI, 118), and he would thus say that the subject-predicate relation depends on the subjective order in which we apprehend them (SI, 139), while the relation between 'subject' and 'attribute' is objectivein the sense that it is holding between a particular thing and a 'particularized quality', it is a ''relation between realities without reference to our apprehension of them'' (Robinson 1931, 103). In 'How to Talk' (Austin 1979, 134-153), Austin further developed distinctions akin to Cook Wilson's differentiation between the logical subject/predicate and metaphysical subject/attribute distinctions, and this differentiation was shared by P. F. Strawson (Strawson 1959, 144), who also believed in 'particularized qualities' (Strawson (1959, 168) & (1974, 131)). In a bid to avoid Bradley's regress (Strawson 1959, 167), he introduced the idea of 'non-relational ties' between subject and attribute, leaving 'relations' for the link between logical subject and predicate. Some non-relational ties are thus said to hold between particulars and particulars: to the relation between Socrates and the universal 'dying' corresponds an 'attributive tie' between the particulars that are both Socrates and the event of his death. That such 'ties' are less obscure than 'relations' and that this maneuver actually succeeds in stopping the regress are further issues, but it is interesting to note that Strawson chose the name 'attributive tie' in honour of Cook Wilson (Strawson (1959, 168), see SI, 193). Strawson also noted that Cook Wilson's argument for differentiating between the logical subject/predicate and metaphysical subject/attribute distinctions involves an appeal to ''pragmatic considerations'' (Strawson 1957, 476). Cook Wilson's claim that a sentence such as 'glass is elastic' may state something different depending on the occasion in which it is used also had an important continuation in J. L. Austin's more general point that, although the meaning of words plays a role in determining truth-conditions, it is not an exhaustive one: it "does not fix for them a truth-condition" because that depends on how truth is to be decided on the occasion of their use (see Travis (1996, 451) and Kalderon & Travis (2013, 492 & 496)): > > > ... the question of truth and falsehood does not turn only on > what a sentence is, nor yet on what it means, but > on, speaking very broadly, the circumstances in which it is uttered. > (Austin 1962, 111) > > > This line of thought has been pursued further by Charles Travis under the name of 'occasion-sensitivity', i.e., ''the fact that the same state of the world may require different answers on different occasions to the question of whether what was said in a given statement counts as true'' (Travis (1981, 147), see also (1989, 255)), this being a recurring theme, see the papers collected in Part 1 of (Travis 2008)). Thus, Cook Wilson's above examples, 'That building is the Bodleian' or 'glass is elastic', are genealogically related to what are commonly known as 'Travis cases', i.e., sentences used to make a true statement about an item in one occasion and a false one about the same item in another. (For examples of these, see Travis (1989, 18-19), (1997, 89), (2008, 26 & 111-112).) R. G. Collingwood also took Cook Wilson as putting forth a slightly different thesis, namely that the meaning of a statement is determined by the question to which it is an answer (Collingwood 1938, 265 n.). He used this idea as the basis for his 'logic of questions and answers' (Collingwood 2013, chap. 5) and for his theory of presuppositions (Collingwood 1998, chaps. 3-4), this last being further developed by the French linguist Oswald Ducrot (1980, 42f.). Collingwood was reluctant, however, to recognize Cook Wilson as an inspiration, since he thought ill of the idea of 'apprehensions' as a non-derivative basis for knowledge and he believed instead that knowledge comes from asking questions first (Collingwood 2013, 25), and thus that knowledge depends on a 'complex of questions and answers' (Collingwood 2013, 37). ## 10. Moral Philosophy Cook Wilson hardly wrote on topics outside the theory of knowledge and logic, but two remarks ought to be made concerning moral philosophy. First, the last piece included in Statement and Inference is composed of notes for an address to a discussion society in 1897, that was announced as 'The Ontological Proof for God's Existence'. In this text, which opens with a discussion of Hutcheson and Butler, Cook Wilson argued that in the case of "emotions as are proper to the moral consciousness", such as the feeling of gratitude: > > We cannot separate the judgement from the act as something in itself > speculative and in itself without the emotion. We cannot judge here > except emotionally. This is true also of all moral and aesthetic > judgements. Reason in them can only manifest itself emotionally. (SI, > 860) > He argued further, in what amounts to a form of moral realism, that there must be a real experience, i.e., in the case of gratitude, "Goodwill of a person, then, must here be a real experience" (SI, 861), and that the feeling of "reverence with its solemnity and awe" is in itself "not fear, love, admiration, respect, but something quite sui generis" (SI, 861). It is a feeling that, Cook Wilson argued, "seems directed to one spirit and one alone, and only possible for spirit conceived as God" (SI, 864). In other words, the existence of the feeling of reverence presupposes that God exists. Cook Wilson thus sketched within the span of a few pages a theory of emotions, which is echoed today in the moral realism that has been developed, possibly without knowledge of it, in the wake of David Wiggins' 'Truth, Invention and the Meaning of Life' (Wiggins 1976) and a series of influential papers by John McDowell--now collected in McDowell (2001). Cook Wilson's ideas had a limited impact on Oxford theology. Acknowledging his debt, the theologian and philosopher C. C. J. Webb described religious experience as one that "cannot be adequately accounted for except as apprehension of a real object" (Webb 1945, 38), but he nevertheless chose to describe his standpoint as a form of 'Platonic idealism' (Webb 1945, 35). Cook Wilson's realism also formed part of the philosophical background to C. S. Lewis' "new look" in the 1920s, via E. F. Carritt's teaching. In Surprised by Joy,Lewis also described awe as "a commerce with something which [...] proclaims itself sheerly objective" (Lewis 1955, 221), but he quickly moved away from this position (see Lewis (1955, chaps. XIII-XIV) and McGrath (2014, chap. 2)). Secondly, Prichard is also responsible for an extension of Cook Wilson's conception of knowledge to moral philosophy, with his paper 'Does Moral Philosophy Rest on a Mistake?' (Prichard 1912), whose main argument is analogous to Cook Wilson's argument for the impossibility of defining knowledge. In a nutshell, duty is sui generis and not definable in terms of anything else. The parallel is explicit in Prichard (1912, 21 & 35-36). That we ought to do certain things, we are told, arises "in our unreflective consciousness, being an activity of moral thinking occasioned by the various situations in which we find ourselves", and the demand that it is proved that we ought to do these things is "illegitimate" (Prichard 1912, 36). In order to find out our duty, "the only remedy lies in actually getting into a situation which occasions the obligation" and "then letting our moral capacities of thinking do their work" (Prichard 1912, 37). This paper became so influential that Prichard was elected in 1928 to the White's Chair of Moral Philosophy at Corpus Christi, although his primary domain of competence had been the theory of knowledge. His papers in moral philosophy were edited after his death as Prichard (1949, now 2002). Prichard stands at the origin of the school of 'moral' or 'Oxford intuitionism', of which another pupil of Cook Wilson, the Aristotle scholar W. D. Ross (Ross 1930, 1939) remains the foremost representative, along with H. W. B. Joseph, E. F. Carritt, and J. Laird. Some of the views they expressed have recently gained new currency within 'moral particularism', e.g., in the writings of Jonathan Dancy (Dancy 1993, 2004). ## 11. Legacy The historical importance of Cook Wilson's influence ought not to be underestimated. In his obituary, H. W. B. Joseph described him as being "by far the most influential philosophical teacher in Oxford", adding that no one had held a place so important since T. H. Green (Joseph 1916b, 555). This should be compared with the claim that in the 1950s Wittgenstein was "the most powerful and pervasive influence" (Warnock 1958, 62). The 'realist' reaction against British Idealism at the turn of the 20th century was at any rate not confined to the well-known rebellion of G. E. Moore and Bertrand Russell at Cambridge. There were also 'realisms' sprouting in Manchester (with Robert Adamson and Samuel Alexander), and in Oxford too, where Thomas Case had already argued for realism in Physical Realism (Case 1888) (Marion 2002b), although it is clearly Cook Wilson's influence that swayed Oxford away from idealism. Since he published so little, it is therefore mainly through teaching and personal contact that he made a significant impact on Oxford philosophy, not only through the peculiar tutorial style to which generations of 'Greats' students were subjected--as described in Walsh (2000) or Ackrill (1997, 2-5)--but also through meetings that he initiated, which were to become the 'Philosophers' Teas' under Prichard's tutelage, the 'Wee Teas' under Ryle's and 'Saturday Mornings' under Austin's. His legacy can thus be plotted through successive generations of Oxford philosophers. E. F. Carritt, R. G. Collingwood (who reverted to a form of idealism later on), G. Dawes Hicks, H. W. B. Joseph, H. A. Prichard, W. D. Ross and C. C. J. Webb are among his better-known pupils at the turn of the century. After his death, his influence extended through the teaching of Carritt, Joseph and Prichard, and the posthumous volumes of Statement and Inference to the post-World War I generation of the 1920s, including Frank Hardie, W. C. Kneale, J. D. Mabbott, H. H. Price, R. Robinson and G. Ryle, and the early analytic philosophers of the 1930s, J. L. Austin, I. Berlin, J. O. Urmson, and H. L. A. Hart, in particular. For example, Isaiah Berlin's described Hart as "an excellent solid Cook Wilsonian" in a letter to Price (Berlin 2004, 509), and admitted himself to have been at first an Oxford Realist (Jahanbegloo 1992, 153). (See Marion (2000, 490-508) for further details.) Thus, Oxford Realism first dislodged British Idealism from its position of prominence at Oxford and then transformed itself into ordinary language philosophy and, as pointed out in the previous section, moral intuitionism. In the post-World War II years, Cook Wilson's name gradually faded away, however, while 'ordinary language philosophy', which owed a lot to his constant reliance on ordinary language against philosophical jargon, blossomed. It became one of the strands that go under the name of 'analytic philosophy', so Cook Wilson should perhaps be seen as one of its many ancestors. The only Oxford philosopher of note who opposed the 'realists' before World War II was R. G. Collingwood, who died too soon in 1943. He felt increasingly alienated and ended up reduced to invective, describing their theory of knowledge as "based upon the grandest foundation a philosophy can have, namely human stupidity" (Collingwood 1998, 34) and their attitude towards moral philosophy as a "mental kind of decaudation" (Collingwood 2013, 50). Collingwood objected to Cook Wilson's anti-idealist claim that 'knowing makes no difference to the object known', that in order to vindicate it one would need to compare the object as it is being known with the object independently of its being known, which is the same as knowing something unknown, a contradiction (Collingwood 2013, 44). But Collingwood's argument did not rule out the possibility of coming to know an object, while knowing that it was not altered in the process. (For critical appraisals, see Donagan (1985, 285-289) Jacquette (2006) and Beaney (2013).) In another telling complaint, he criticized the Oxford Realists for being interested in assessing the truth or falsity of specific philosophical theses without paying attention to the fact that the meaning of the concepts involved may have evolved through history, and so there is simply no 'eternal problem' (Collingwood 2013, chap. 7). This points to a lack of historical sensitivity, which is indeed another feature of analytic philosophy that arguably originates in Cook Wilson. There was another deleterious side to Cook Wilson's influence in Oxford: his contempt for mathematical logic. It explains why one had to wait until the appointment of Hao Wang in the 1950s for modern formal logic first to be taught at Oxford. In the 1930s, H. H. Price was still teaching deductive logic from H. W. B. Joseph's An Introduction to Logic (Joseph 1916a) and inductive logic from J. S. Mill's System of Logic. This reactionary attitude towards modern logic and later objections to 'ordinary language philosophy' go a long way to explain why Cook Wilson's reputation dropped significantly in the second half of last century. In the 1950s, Wilfrid Sellars was virtually alone in his praise: > > > I can say in all seriousness that twenty years ago I regarded > Wilson's Statement and Inference as the philosophical > book of the century, and Prichard's lectures on perception and > on moral philosophy, which I attended with excitement, as veritable > models of exposition and analysis. I may add that while my > philosophical ideas have undergone considerable changes since 1935, I > still think that some of the best philosophical thinking of the past > hundred years was done by these two men. (Sellars 1957, 458). > > > As the tide of 'ordinary language philosophy' ebbed, Cook Wilson's views on knowledge showed more resilience. In the 1960s, Phillips Griffiths' anthology on Knowledge and Belief included excerpts from Cook Wilson (Phillips Griffiths 1967, 16-27) and John Passmore was able to write that "Cook Wilson's logic may have had few imitators; but his soul goes marching on in Oxford theories of knowledge" (Passmore 1968, 257). As shown in sections 5 and 6, Cook Wilson's views on knowledge and perception are now once more involved in contemporary debates. They had remained influential all along, although his name was often not mentioned. His peculiar combination of the claims that knowledge is a factive state of mind and that it is undefinable, argued for anew by J. L. Austin, has been taken up and further developed by John McDowell (McDowell 1994, 1998), Charles Travis (Travis 1989, 2008), and Timothy Williamson (Williamson 2000, 2007), who is currently Wykeham Professor of Logic, New College. One has, therefore, what Charles Travis once described as "an Oxford tradition despite itself" (Travis (1989, xii), on this last point, see also Williamson (2007, 269-270n)). During the twentieth century, secondary literature on Cook Wilson's philosophy was not considerable, with a few papers of unequal value by Foster (1931), Furlong (1941), Lloyd Beck (1931) and Robinson (1928a, 1928b), along with a few studies on universals in the post-war years (see section 7), and only one valuable commentary, Richard Robinson's The Province of Logic (Robinson 1931). Interest in the study of his philosophy was only revived at the beginning of this century, with Marion (2000) giving a first overview of Oxford Realism. In a short book, Kohne (2010) charts the views of Cook Wilson, Prichard and Austin on knowledge as a mental state. An important contribution, Kalderon & Travis (2013) secured Oxford Realism's place in the history of analytic philosophy, comparing it with other forms of realism in Frege, Russell and Moore, while drawing links with later developments in the writings of J. L. Austin, J. M. Hinton and John McDowell. As a result of this revival of interest, the philosophies of J. L. Austin (Longworth 2018a, 2018b & 2019, and the entry to this Encyclopedia) and of Wilfrid Sellars (Brandhoff 2020) are now being re-interpreted in light of Cook Wilson's legacy.
wilhelm-windelband	## 1. Biographical Sketch Wilhelm Windelband was born in 1848 in Potsdam, Germany. His father, Johann Friedrich Windelband was a state secretary for the Province of Brandenburg. Windelband studied in Jena, Berlin, and Gottingen, attending lectures by Kuno Fischer (1824-1907) in Jena and studying with Hermann Lotze (1817-1881) in Gottingen. Fischer and Lotze would deeply influence Windelband's philosophical thinking, as well as his work as a historian of philosophy. In 1870, Windelband completed his dissertation on Die Lehren vom Zufall [Doctrines of chance] under Lotze's supervision. The following year he served as a soldier in the German-French war. After his military service, he completed his habilitation in Leipzig and took up a position as "Privatdozent" there. His habilitation was published in 1873 under the title Ueber die Gewissheit der Erkenntniss: eine psychologisch-erkenntnisstheoretische Studie [On the certainty of knowledge: a psychological-epistemological study]. In 1874, Windelband married Martha Wichgraf with whom he would have four children. Two years later, Windelband became professor (ordinarius) of "inductive philosophy" in Zurich. He lectured on psychology before taking up a position as professor of philosophy in Freiburg im Breisgau in 1877. In 1882 he accepted an offer from the University of Strasbourg. While his inaugural lectures in Zurich and Freiburg had centered on the relation between psychology and philosophy, his works from the Strasbourg period develop his core themes in the philosophy of values and the philosophy of history. Windelband served as "Rektor" of the University of Strasbourg in 1894/95 and 1897/98. He remained in Strasbourg until 1903, when he accepted a call from the University of Heidelberg. Between 1905 and 1908 he served as representative of the University of Heidelberg in the Baden "Landtag". He was a member of the Berlin Academy of Sciences, and of the Academies of Sciences of Gottingen, Bayern and Heidelberg. He remained in Heidelberg and taught there until his death in 1915. ## 2. From Kant to the Philosophy of Values Windelband's views on normativity are strongly influenced by his teacher Lotze. In his Logic (1874), Lotze distinguishes between psychological laws which determine how thinking proceeds as a matter of fact, and logical laws, which are normative laws and prescribe how thinking ought to proceed (Lotze 1874: SSx; SS332, SS337). This distinction also corresponds to a distinction between act and content. Lotze observes that "ideas" do occur in us as acts or events of the mind. But their content does not consist in such acts, is not reducible to mental activity, and does not exist in the way empirical processes and entities may be said to exist. It is not real, but "valid" (Lotze 1874: SSSS314-318). The distinction between the factual and the normative would become the cornerstone of Windelband's "philosophy of values". In his early writings, however, Windelband does not yet embrace this distinction. In his habilitation thesis Uber die Gewissheit der Erkenntnis (1873), he argues that logic is a normative discipline, but that it needs to be put on a psychological basis. This is because the justification of our knowledge claims is always dependent on and relative to specific epistemic purposes which, in turn, are given psychologically. Although he criticizes the identification of the conditions of knowledge with psychophysical processes, he does think of psychology and logic or epistemology as continuous with one another. His 1875 "Die Erkenntnislehre unter dem volkerpsychologischen Gesichtspunkte" ["The theory of knowledge from the perspective of folk-psychology"] which was published in Moritz Lazarus' and Heyman Steinthal's Zeitschrift fur Volkerpsychologie und Sprachwissenschaft [Journal of folk psychology and linguistics], is more radical. It denies that logical norms are independent of the conscious mind and claims that the origins of logical norms are to be found in the social history of humankind: the principle of contradiction emerges in situations of social conflict together with the distinction between true and false beliefs; and the law of sufficient reason comes into being when conflicts between rival views are no longer settled by brute force. On this picture, logical principles exist only when humans cognize them. There is no boundary between psychological acts and objective logical laws, or between actual historical acceptance and normative validity. This is precisely the type of thinking that Windelband will later reject as "psychologistic", "historicist", and "relativist" (1883: 116-117, 132). It is not clear what led Windelband to this change of view, but a deeper engagement with the Critique of Pure Reason and with the Kant scholarship of his time, in particular with works by Kuno Fischer (1860), Herman Cohen (1871), and Friedrich Paulsen (1875), seem to have factored in. The clear contours of Windelband's anti-psychologistic interpretation of Kant emerge for the first time in his 1877 paper "Uber die verschiedenen Phasen der Kantischen Lehre vom Ding-an-sich" ["On the different stages of the Kantian doctrine of the thing-in-itself"]. In this essay, Windelband argues against the idea, held by prominent figures of the "back-to-Kant" movement like Friedrich Albert Lange (1828-1875) and Hermann von Helmholtz (1821-1894), that knowledge emerges from an interaction between subject and object. According to their view, the object affects the subject's mind, while the subject provides the a priori cognitive structures that organize representations. These a priori structures are innate as they consist in the psychophysical constitution of the human sensory apparatus and mind. Cohen's critique of this (mis)interpretation focuses on the concept of the a priori and the question of objectivity (Cohen 1871). Windelband, in contrast, takes the problem of the thing-in-itself and the concept of truth as his starting points. His critique of the subject-object-interaction model leads him to an immanent conception of truth, according to which truth consists in the normative rules according to which our judgments ought to be formed. The immanent concept of truth thus shifts the focus of philosophical analysis on the universal and necessary "rules" that ground our judgments. The 1877 essay gives a genetic account of the development of Kant's views about the thing-in-itself from the Inaugural Dissertation (1770) to the second edition of the Critique of Pure Reason (1787). This genetic reconstruction is supposed to reveal the underlying philosophical problems and motivations that drove Kant's thinking, as well as the inner tensions that, according to Windelband, permeate critical philosophy. In particular, Windelband identifies a "gulf" (1877: 225) between Kant's critique of knowledge and the metaphysics of morals, as well as a tension between the psychological account of the faculties, and Kant's mature anti-psychologism. He distinguishes between four phases in Kant's thinking about the thing-in-itself, and argues that residues of the latter three stages are present in both editions of the Critique (1781; 1787) and in the Prolegomena (1783). The first phase, Windelband argues, is that of the Inaugural Dissertation (1770). Kant adopts Leibniz's distinction between noumena and phenomena while formulating a genuinely novel thought: he introduces the psychological distinction between receptive sensibility and the spontaneous intellect. This distinction allows him to maintain the claim that, unlike intuitions, concepts relate to things-in-themselves (1877: 240-241). In the second phase, Kant concerns himself more deeply with the question how the concepts of the understanding can relate to objects. He is driven to the insight that > > > [w]e can only have a priori knowledge of that which we > produce by the lawlike forms of our rational activities > [Vernunfthandlungen]. (1877: 246) > > > Here, Windelband follows Fischer's claim that the categories of the understanding are a priori valid for experience because they produce or "make" it (Fischer 1860). Because the understanding cannot "make" the thing-in-itself, the thing-in-itself cannot be known. According to Windelband, Kant then proceeds to inquire why we are even assuming the existence of things-in-themselves if we cannot know them. Here he enters his third, most radical phase. Kant now thinks of the thing-in-itself as a fiction, an illegitimate hypostasis in which > > > the universal form of the synthetic act of the understanding is seen > as something that exists independently of experience. (1877: 254) > > > Windelband argues that it is only by dismissing the existence of the thing-in-itself and by jettisoning the phenomena-noumena distinction that Kant can undertake his anti-psychologistic turn. With the rejection of the thing-in-itself, the characterization of sensibility as "receptive" also needs to be abandoned. And that makes a psychological construal of the faculties nonsensical. Kant is then also able to abandon the concept of truth as correspondence between representations and objects in favor of a strictly immanent conception. The immanent conception defines truth in terms of the universal and necessary rules that the relations between our representations need to accord to (1877: 259-260). And yet, Kant could not rest with this radical view given his commitments in moral philosophy. Practical reason, and in particular the idea that the moral law does not depend on the qualities of humans as sensuous beings, but only on reason, demands a return to the assumption that the thing-in-itself exists (1877: 262). In the fourth phase, Kant thus reintroduces the thing-in-itself, while maintaining his anti-psychologism. This genetic reconstruction allows Windelband to think of the conflict between the different views of the thing-in-itself that had emerged within the neo-Kantian tradition as reflective of inherent tensions in the Critique. He argues that Kant was especially unclear with respect to psychologism: the distinction between judgment (as a cognitive process) and justification (as logical and normative) is inadequately articulated in the first Critique. Hence Kant is at least partly responsible for the fact that in the first wave of neo-Kantianism represented by Lange and Helmholtz "the new concept of aprioricity was soon dragged down to the old idea of psychological priority" (1883: 101). But Windelband's approach to Kant is not merely historical. His sympathies clearly lie with the Kant of the third phase. For Windelband, the insight that natural psychological processes are "utterly irrelevant" for the truth value of our representations (1882b: 24), and the idea that the ultimate problem of philosophy is that of normativity and justification--not a quaestio facti, but a quaestio juris (1882b: 26)--need to be defended from Kant's own unclarities on these matters. Windelband's "philosophy of values" can thus be understood as an attempt to purify, explicate and develop the radical insights of the anti-psychologistic move in Kant's "third phase". The cornerstone of this endeavor is the immanent conception of truth. Windelband repeatedly returns to discussing the correspondence theory of knowledge with its metaphor of a "mirror relation" between mind and object. He criticizes the misconception that our sensual perceptions are the things-in-themselves and that these things could be compared with our representations. Any comparison must occur between representations, since things and representations are "incommensurable" (1881: 130). According to Windelband, Kant's central innovation consists in the insight that the truth of our judgments, and the relation of our representations to an object are not to be found in correspondence at all. Rather, it consists in the "rules" for combining representations (1881: 134). This leads Windelband to define truth as the "normality of thinking" (1881: 138)--with "normality" meaning that thinking proceeds in accordance with rules or norms. Windelband also conceptualizes the object of knowledge in terms of the rules of judgment. The object of knowledge is nothing other than > > > a rule according to which representational elements ought to organize > themselves, in order for them to be recognized as universally valid in > this organization. (1881: 135). > > > Windelband uses the terms "axioms" and "values" interchangeably to refer to the most fundamental rules, and he uses the term "norms" often, but not fully consistently to refer to values or axioms as they relate to psychological experience and the cultural-historical world. Focusing his interpretation on "values" and "norms", Windelband captures the structural similarity of Kant's three Critiques in terms of immanent truth: if truth is nothing other than accordance with a rule, then there is moral and aesthetic truth in just the same way as there is epistemic truth (1881: 140). In a later text, Windelband describes the unified project of the three Critiques in terms of the necessary and universal relation between thought and object. He writes that the postulates of practical reason relate to intelligible objects (ideas) just "as necessarily" as the intuitions and categories relate to the object of experience, and that teleological judgment constructs the purposive whole of nature just "as universally" as the principles of pure understanding apply the categories to experience (1904a: 151). While this formulation glosses over some of the nuances of the distinction between constitutive and regulative uses of reason, Windelband's main intention is not that of giving a detailed and fully accurate reconstruction of Kant's philosophy. In line with his famous dictum that "understanding Kant means to go beyond him" (1915: iv), he instead seeks to revive the critical project in a manner that allows it to answer the needs of his own time. And Windelband thinks that the critical philosophy required at the time of his writing is a "philosophy of values" which reveals the most fundamental values in epistemology, ethics, and aesthetics. ## 3. The Factual, the Normative, and the Method of Philosophy As indicated above, Windelband bases his "philosophy of values" on Lotze's distinction between the factual and the normative. Throughout his career, he seeks to explicate and clarify this distinction and to illuminate its consequences for philosophical method. In "Was ist Philosophie?" ["What is philosophy?"] (1882), Windelband approaches the factual-normative distinction by identifying two basic and irreducible types of cognitive operations: judgments and evaluations. While judgments relate representations in a synthesis, and thus expand our knowledge about an object, evaluations presuppose an object as given. They do not expand our knowledge. Rather, they express a relation between the "evaluating consciousness" and the represented object in a "feeling" of approval or disapproval (1882b: 29-30). Despite characterizing evaluations in terms of the feelings and subjective attitudes of the evaluating consciousness, Windelband argues that some evaluations are "absolutely valid". Even if they are not embraced by everyone as a matter fact, they entail a normative demand: they ought to be accepted universally according to an absolute value (1882b: 37). The basic idea seems to be that the normative force of any particular evaluation that is carried out by an empirical consciousness is derived from its relation to a non-empirical, absolute value. Windelband argues that even if there is disagreement about which evaluations ought to be embraced as universal and necessary, the demand for absolute validity itself can be recognized by everyone: we all believe in the distinction between that which is absolutely valid and that which is not (1882b: 43). Accordingly, there must be a system of absolute values from which the validity of judgments in epistemology, aesthetics, and ethic derives. Critical philosophy, then, is nothing other than the "science of necessary and universal values" (1882b: 26) that explicates this system and thus reveals the grounds of normative appraisal and valid judgment. Note that for Windelband, normativity and validity are closely linked, if not identical. Absolute values endow our judgments with a normative demand. Our judgments are valid if and only if they raise a normative demand, that is if they ought to be accepted universally and necessarily. Having distinguished the factual and the normative by reference to different cognitive operations, Windelband also seeks to identify the points of contact between the two realms. Ultimately, he wants to explain how empirical beings can recognize absolute values. His idea that there is a system of absolute values is thus accompanied by the conception of a "normal consciousness". In normal consciousness the absolute system of values is, at least partially, represented in the form of norms that are known by "empirical consciousness" and that have an effect on it. Starting from the immanent conception of truth as accordance with a rule, Windelband distinguishes from among the infinite possible combinations of representations that might or might not be formed by empirical consciousness a subset of combinations: the subset that accords with universal and necessary rules, and which hence ought to be formed (1881: 135-139: 1882a: 72-73). This subset is what he calls "normal consciousness" (1881: 139). Thought is related to and valid for an object if of the infinite possible combinations of our representations, our thinking forms exactly those judgments that "ought to be thought" (1881: 135). One might say that empirical consciousness contains (parts) of the system of absolute values as its "normal consciousness". To the extent that philosophy seeks not only to reveal the absolute system of values, but also to inquire into how they can be norms for empirical--embodied, psychological and historically situated--human beings, critical philosophy is not only a "science of values", but also a "science of normal consciousness" (1882b: 46). Although Windelband refers to the "philosophy of values" as a science, he emphasizes that philosophy does not rely on the methods of the empirical sciences. Philosophy is a second-order science that reflects on the methods and results of the various empirical disciplines in order to reveal the values "by virtue of which we can evaluate the form and extent of their validity" (1907: 9). Crucially, this reflective endeavor cannot be carried out by means of empirical investigation. Here, the distinction between the factual and the normative assumes a methodological dimension. In "Kritische oder genetische Methode?" ["Critical or genetic method?"] (1883) Windelband lays out in great detail the differences between the "explanatory" and "genetic" method of the empirical sciences, on the one hand, and the "teleological" or "critical" method of philosophy, on the other. And he warns of the devastating consequences that result if the two are conflated. He singles out two disciplines that might be thought to be relevant to philosophical questions about values: individual psychology and cultural history. He does not call into question the legitimacy of these disciplines or of the "genetic method" in general. He even thinks that the genetic method can be applied to values. That is, individual psychology and cultural history can yield valid genetic theories that explain the actual acceptance and development of values in an individual's mental life and in cultural history. But actual acceptance is not the same as normative validity, and validity proper cannot be found by generalizing from the empirical. A firm boundary separates the genetic method and its approach toward actually accepted values from the critical method of philosophy which concerns values as normative. Windelband's argument against the application of the genetic method to philosophical questions has three components. First, he argues that the genetic method cannot solve philosophical questions about normativity and validity, because there is too much variety regarding the values that have been and are actually accepted. The empirical method will not uncover values that are universally embraced by all cultures (1883: 114-115). Second, the genetic method can show and explain why some values have been accepted by this or that individual, or in this or that culture. But insofar as it is an empirical method, it cannot establish that the values in question are universal and necessary. > > > The universally valid can be found neither by inductive comparison of > all individuals and peoples nor by deductive inference from ... > the 'essence' of man. (1883: 115) > > > Windelband points to the absurdity of trying to justify by empirical means that which is the presupposition of any empirical theory: the axioms upon which the validity of any theory is based (1883: 113). Third, Windelband argues that the genetic method leads to relativism. The argument rests on the two claims just outlined, and can be reconstructed as follows. There is variation between individuals and cultures regarding which beliefs are actually prevalent, and the naturally necessary [naturnotwendig] laws of psychology lead to the formation of both true and false beliefs. As an empirical method with no access to the universal and necessary, the genetic method of cultural history and individual psychology has no criterion for distinguishing between valid and invalid beliefs. This means that it has to treat all beliefs as "equally justified [alle gleich berechtigt]" (1882b: 36). > > > For [the genetic explanation], there is thus no absolute measure; it > must treat all beliefs as equally justified because they are all > equally necessary by nature... [R]elativism is the necessary > consequence of the purely empiricist treatment of philosophy's > cardinal question. (1883: 115-116) > > > Note that Windelband does not differentiate between the idea that all beliefs have only relative validity and the claim that they are all equally justified or equally valid. In his view, the genetic method does not merely render belief relative to individuals and cultures; it also forces us to conclude that all beliefs are equally valid. An empirical psychology or cultural history that oversteps its boundaries and tries to address philosophical questions about normativity and validity leads to "historicism", "psychologism", and "relativism" and destroys the basis of normative appraisal altogether. Having rejected the genetic method in philosophy, Windelband explains that philosophical method is purely "formal". The axioms, or values, on which the validity of our judgments is based upon cannot be proven. But it can be shown that the purposes of recognizing truth, beauty, and can only be achieved if absolute values are presupposed. Hence the critical method of philosophy has a teleological structure: > > > [F]or the critical method these axioms, regardless of the extent to > which they are actually accepted, are norms which ought to be valid if > thinking wants to fulfil the purpose of being true, volition the > purpose of being good, and feeling the purpose of capturing beauty, in > a manner that warrants universal validation. (1883: 109) > > > And yet, while Windelband insists on the distinction between the empirical-genetic method of science and the critical-teleological method of philosophy, he takes his theory of "normal consciousness" to imply that empirical facts do play a role in philosophy. In particular, he wants to maintain that empirical facts about individual psychology and culture can provide the starting points for philosophical reflection. He therefore describes the philosophical method as a method of reflection [Selbstbesinnung], in which the empirical mind becomes aware of its own "normal consciousness". Philosophy examines existing claims to validity in light of teleological considerations, and in this way reveals the "processual forms of psychic life that are necessary conditions for the realization of universal appraisal" (1883: 125). Put differently, teleological considerations allow the empirical consciousness to distinguish within itself between empirical and contingent contents, on the one hand, and the "contents and forms" that "have the value of normal consciousness", on the other (1882b: 45-46; 1881: 139). ## 4. The Problem of Freedom The distinction between the normative and the factual and the question how the two realms are related also structure Windelband's reflections on human freedom. Throughout his intellectual career, Windelband returns to this problem, presenting different strategies for reconciling causal determinism and human freedom. His dissertation, completed under Lotze in 1870, deals with the concepts of chance, causal necessity, and freedom. At that time, Windelband still embraces the Kantian concept of transcendental freedom, according to which the noumenal self is the uncaused cause of all intentional action (1870: 16-19). But after the 1877 essay, which had uncovered the anti-metaphysical rejection of the thing-in-itself in Kant's "third phase" as the radical starting point for the "philosophy of values", Kant's metaphysical solution to the problem of freedom ceased to convince him. Windelband's most fully developed effort to arrive at an alternative solution to the problem of freedom can be found in his 1882 essay "Normen und Naturgesetze" ["Norms and natural laws"] and in his 1904 lectures Uber Willenfreiheit [On freedom of the will]. In these texts, Windelband articulates the following core claims. First, the Kantian dualism between phenomena and noumena needs to be overcome, and we need to think of moral responsibility in a way that does not presuppose a noumenal realm. Second, causal explanation and normative evaluation are two irreducible, but ultimately compatible, ways of viewing, or constructing, the world of appearances. Third, the object of moral evaluation is neither a particular moral action, nor a transcendentally free will, but a "personality"--understood as a set of relatively stable motivations and psychological dispositions--that is the natural cause of our actions. Although articulating these same core thoughts, the two texts differ in how they motivate these ideas, and in the consequences that they draw from them. "Normen und Naturgesetze" begins by postulating a conflict between natural law and moral law: if the moral law demands an action that would also result from natural causes alone, it is superfluous. But if it demands an action that does not accord with natural causes, it is useless, because natural necessity cannot be violated (1882a: 59). Windelband holds that causal determinism extends to mental life and that for this reason the conception of freedom as a fundamental capacity that violates "the naturally necessary functions of psychic life" (1882a: 60) is implausible from the get-go. However, he grants that there are two different and irreducible ways of viewing the same objects: on the one hand, there is psychological science which explains what the facts of mental life are. On the other hand, there are ideal norms, which do not explain what the facts are, but express how they ought to be (1882a: 66-67). The solution to the antagonism between natural law and moral law is to be found in the relation between these two points of view. Here, Windelband introduces the claim that although ideal norms differ from causal laws, they are not incommensurable with them and, in fact, act on us causally. His argument builds on his conception of "normal consciousness" as the representation of the system of absolute values within empirical consciousness. Windelband argues that a mind that becomes aware of its own "normal consciousness" is capable of acting on the basis of and in agreement with the norms that it has discovered within itself: > > > [E]ach norm carries with it a sense that the real process of thinking > or willing ought to form itself in accordance with it. With immediate > evidence a form of psychological coercion attaches itself to the > awareness of the norm. (1882a: 85) > > > The norm becomes a determining factor in and for empirical consciousness. It acts as "part of the causal law" and determines psychological life with natural necessity (1882a: 87; see also 1883: 122). The result is what Windelband calls a "deterministic concept of freedom" (1882a: 88), according to which freedom consists in nothing other than the becoming-aware of the norms that command how we ought to act: our becoming-aware of them determines our actions with natural necessity. Freedom is the "determination of empirical consciousness by normal consciousness" (1882a: 88). But Windelband still needs to explain how it is possible that we can be aware of a moral norm and not act on it. Long passages in the 1904 lectures are devoted to developing the thought that what determines our moral decisions and actions, and hence decisions if and when we act in accordance with the moral law is our "personality". Windelband approaches this as a theoretical, not as a normative, question and concludes that we may well call those decisions and actions "free" that are predominantly determined by our constant personality, as opposed to being determined by external circumstances or contingent affects. Freedom is "the unhindered causality of a pre-existing willing" (1904b: 106). However, Windelband concedes that this analysis does not exhaust our concept of freedom since there is not merely a theoretical, but also a normative use of the concept. In this context, Windelband acknowledges the attraction of the Kantian argument that moral responsibility is possible only if we have transcendental free will and with it the capacity for genuine alternatives: we could act differently given the same circumstances. But Windelband rejects the project of grounding human freedom in a noumenal world. He thinks that Kant's distinction between an intelligible noumenal self that is the uncaused cause of our actions, and a deterministic empirical world as constructed by our understanding reproduces the same problems that earlier metaphysical accounts of freedom had encountered. He discusses two problems in particular. First, on the one hand, the personality that a particular individual has developed is part of the empirical world and therefore, on the metaphysical picture, does not feature into the free decisions of the individual. But, on the other hand, the noumenal self is empty. It is an abstract, general self, uniform in all of us; and thus it cannot account for the differences in the moral life of individuals (1904b: 161-163). Second, transcendental freedom is fundamentally incompatible with the "all- encompassing reality and causality of the deity" (1904b: 187). God is the ultimate uncaused cause, and the only way to understand this thought is by assuming a "timeless causality" between God and the intelligible characters (noumenal selves)--a view which ends up undermining the freedom of the latter (1904b: 186-9) Having concluded that Kantian dualism fails to avoid the pitfalls of earlier metaphysical approaches to freedom, Windelband abandons the concept of transcendental freedom altogether. But while in 1882 his alternative had been the "deterministic concept of freedom", in 1904 Windelband proceeds to articulate the view that the concept of free will is a mere placeholder in our normative discourse. We are not free in the metaphysical sense, but we are perfectly entitled to pass moral judgment. And we use the language of freedom to express the fact that when passing moral judgment we disregard questions of causal determination. To spell out this idea, Windelband takes up the 1882 distinction between the "points of view" of explanatory science and ideal norms. He now argues that there are two ways of constructing the world of appearances: we construct the world of appearances according to causal laws, and we construct it according to our normative evaluations. Evaluation > > > reflects within the manifoldness of the given on those moments only > which can be put in relation to the norm... [O]ne could call the > manner in which the objects of experience, the given manifoldness of > the factual, appear uniformly in light of such an evaluation another > form of "appearance".... (1904b: 195-6) > > > Freedom then means not that the will is an uncaused cause. When speaking of freedom, we appeal to the uncaused merely in the sense that we evaluate matters independently of causal deterministic processes (1904b: 197-198). Windelband believes that his view preserves moral responsibility. As described above, his theoretical investigation had yielded the result that that which determines the extent to which we act in agreement with a moral norm is our personality or character. Personality is the constant cause of voluntary action, it determines our actions necessarily according to general psychological laws. From a practical standpoint, personality then is also the ultimate object of moral appraisal. We hold personality responsible, and we are justified in doing so, even if the formation of personality is itself a causal process over which the individual has no control. Ultimately, the upshot of Windelband's discussion is that moral responsibility does not presuppose a noumenal world and transcendental freedom, because it does not presuppose that we could act otherwise. It merely presupposes that another person in the same circumstances could act otherwise. The idea that one could have acted differently refers > > > not to the concrete human being in these concrete circumstances, but > to the generic concept of the human being. (1904b: 212) > > > ## 5. The Natural and the Historical Sciences In Windelband's view, a reconsideration of the Kantian project is not only necessary because of its inherent tensions; broader developments in nineteenth-century culture and science also necessitate an adaptation of the critical method to changed historical circumstances. One important factor is the professionalization of the historical disciplines that had been underway since the early nineteenth century (1907: 12). One of Windelband's central and best known philosophical contributions concerns the question what distinguishes the "historical sciences", that is, those disciplines that study the human-historical and cultural world, from the natural sciences. Windelband's answer to this question is in line with his formal-teleological conception of philosophy: by explicating the autonomous presuppositions of historical method, critical philosophy safeguards the historical sciences against methodological holism, namely the view that there is only one scientific method, and that this is the method of physics and natural science. Windelband shares the goal of securing the autonomy of the historical disciplines with his contemporary Wilhelm Dilthey. In his 1883 Einleitung in die Geisteswissenschaften [Introduction to the human sciences] Dilthey had founded the distinction between the natural sciences and the human sciences or "sciences of spirit" on a distinction between outer and inner experience. While outer experience forms the basis of hypothetical knowledge in natural science, inner experience discloses "from within" how the individual is an intersection of social and cultural relations (Dilthey 1883: 30-32, 60-61, 88-89). Inner experience is at the same time psychological and socio-historical, and a descriptive psychology capable of grasping the integrated nexus of inner experience can provide the "sciences of spirit" with a solid foundation (Dilthey 1894). Windelband profoundly disagrees with Dilthey's strategy for demarcating the historical disciplines. In his Strasbourg rector's address "Geschichte und Naturwissenschaft" ["History and natural science"] from 1894, he takes issue with the suggestion that the facts of the "sciences of spirit" derive from a particular type of experience. He takes Dilthey to endorse an introspective view of psychological method. To this he objects that the facts of the historical disciplines do not derive from inner experience alone, and that inner perception is a dubious method in the first place. He also classifies psychology with the natural sciences, rejecting Dilthey's idea that the human-historical disciplines could be founded on a non-explanatory and non-hypothetical descriptive psychology. Perhaps most fundamentally, Windelband rejects the term "sciences of spirit" on the ground that it suggests that the distinction between different sciences rests on a material distinction between different objects: spirit and nature (Windelband 1894: 141-143). Windelband seeks an alternative method for science-classification that is purely formal. His reflections take scientific justification in its most abstract form as their starting point: justification in science is either inductive or deductive, Windelband argues, and the basic relation on which all knowledge is based is between the general and the particular (1883: 102-103). The distinction between different empirical sciences is then also to be sought at this level. In particular, Windelband argues that science might pursue one of two different "knowledge goals" (1894: 143): it "either seeks the general in the form of natural law or the particular in the historically determined form" (1894: 145). The former approach is that of the "nomothetic sciences" which seek to arrive at universal apodictic judgments, treating the particular and unique as a mere exemplar or special case of the type or of the generic concept. The "idiographic sciences", in contrast, aim to arrive at singular assertoric judgments that represent a unique object in its individual formation (1894: 150). Windelband emphasizes that the distinction between nomothetic and idiographic sciences is a purely formal and teleological one. One and the same object can be approached from both points of view, and which method is appropriate depends entirely on the goal or purpose of the investigation. Moreover, most sciences will involve both general and particular knowledge. The idiographic sciences in particular depend on general and causal knowledge which has been derived from the nomothetic sciences (1894: 156-157). Note, however, that Windelband is not consistently restricting his analysis to the formal level. He tends to use the terms "nomothetic" and "natural" sciences, and the terms "idiographic" and "historical" sciences interchangeably, which at least suggests a correspondence between scientific goals, methods, and objects. Although Windelband does not spell this out in great detail, he also suggests that values are of integral importance to the idiographic method. First, he argues that the selection of relevant historical facts depends on an assessment of what is valuable to us (1894: 153-154). Second, he suggests that the integration of particular facts into larger wholes is only possible if meaningful, value-laden relations can be established such that "the particular feature is a meaningful part of a living intuition of the whole [Gesamtanschauung]" (1894: 154). And third, he claims that we value the particular, unique, and individual in a way in which we do not value the general and recurrent, and that the experience of the individual is indeed at the root of our "feelings of value" (1894: 155-156). It is Windelband's student Heinrich Rickert who takes up these suggestions and develops them into a systematic account of the "individuating" and "value-relating" "concept-formation" of the "historical sciences of culture" (Rickert 1902). Rickert argues that "scientific concept formation", by which science overcomes the "extensive and intensive manifold" of reality, depends on a principle of selection. The principle of selection at work in the natural sciences is that of "generalization". For this reason, the natural sciences cannot account for the unique and unrepeatable character of reality. The historical sciences, in contrast, form their concepts in a manner that allows them to capture individual realities (Rickert 1902: 225, 236, 250-251). They do so on the basis of values: values guide the selection of which particular historical facts belong to and can be integrated into a specific historical "individuality" (examples being "the Renaissance", "the Reformation" or "the German nation state"). According to Rickert, one can clearly distinguish the theoretical value-relation that is the basis of historical science and practical evaluation. The historian relies on values but does not evaluate his material (Rickert 1902: 364-365). Building on Windelband's core ideas about historical method, Rickert arrives at a more refined and systematized account of how historical science forms its concepts, that ultimately leads to an account of what culture as an object of scientific study amounts to. In some of his later writings, Windelband will pick up the more developed thoughts of his student Rickert. For example, he claims that each science creates its objects according to the manner in which it forms its concepts (1907: 18), and that the historical sciences rely on a system of universal values when making selections about what enters into their concepts (1907: 20). ## 6. The History of Philosophical Problems A large part of Windelband's oeuvre consists of writings in the history of philosophy. Windelband primarily covers modern philosophy from the Renaissance to his own time, but he also published on ancient philosophy. As a historian, Windelband is heavily indebted to Fischer. And yet, he goes significantly beyond his teacher, developing a new method and mode of historical presentation. Windelband conceives of the history of philosophy as a "history of problems". Rather than presenting a chronological sequence of great minds, he organizes the presentation of philosophical ideas according to the fundamental problems around which the philosophical debates and arguments of an age were structured. Windelband also takes a reflective attitude towards his own historiographical practice and seeks to clarify the goals and systematic relevance of the history of philosophy. In these reflections, Windelband seeks to integrate two main thoughts: First, the idea that philosophy is a reflection of its time and age and, second, the conviction that the history of philosophy has systematic significance. Windelband finds both thoughts in Hegel's approach to the history of philosophy. Mirroring Hegel's famous dictum that "philosophy is its own time apprehended in thought", he speaks of philosophy the "self-consciousness of developing cultural life" (1909: 4). He also applauds Hegel's "deep insight" that the history of philosophy realizes reason and thus has intrinsic relevance for systematic philosophy (1883: 133). Windelband thinks of history as the "organon of philosophy" (1905: 184), as a guide to the absolute values that the critical method seeks to reveal. But despite the Hegelian gloss of these two claims, Windelband is critical of Hegelianism. This is primarily because he has a different understanding of the idea that philosophy is "its own time apprehended in thought". For Windelband, this means that philosophical thinking is shaped by historically contingent factors. Accordingly, Windelband formulates two criticisms of the Hegelian conception of the history of philosophy. First, he finds fault with the idea that the order in which successive philosophical ideas emerge is necessary and that--in virtue of being necessary--it has systematic significance: > > > [I]n its essentially empirical determination which is accidental with > respect to the "idea" the historical process of > development cannot have this systematic significance. (1883: 133, see > also 1905: 176-177) > > > The history of philosophy is not only shaped by the necessary self-expression of reason, but also by the causal necessity of cultural history. The cultural determinants of philosophical thinking lead to "problem-convolutions" (1891: 11), in which various, conceptually unrelated philosophical questions merge with one another. Windelband also emphasizes that the "individual factor" of the philosopher's character and personality is relevant for how philosophical problems and concepts are articulated in a given historical moment (1891: 12) Second, although Windelband agrees that the historical development of philosophical ideas is partly driven by critique and self-improvement, he does not think of this process in terms of progress. Windelband probes the often unacknowledged presuppositions of our talk about progress. In "Pessimismus und Wissenschaft" (1876) Windelband argues that science cannot decide between historical optimism and historical pessimism (1876: 243). In "Kritische und genetische Methode" (1883) he gives a more detailed analysis of why this is the case. Historical change itself is not progress, he observes. In order to determine whether a given historical development is progressive, we need to be in possession of "a standard, the idea of a purpose, which determines the value of the change" (1883: 119). Windelband is wary of triumphalist narratives that identify the historical development of present-day values with progress. History is determined by contingent cultural factors and could have produced "delusions and follies ... which we only take to be truths now because we are inescapably trapped in them" (1883: 121). Of progress we can only speak legitimately, if we are in possession of an absolute value that allows us to assess the historical development. At minimum, this means that the appeal to a progressive history of philosophy is of no help when it comes to the systematic task of uncovering the system of absolute values: progress cannot aid in revealing absolute values as it presupposes them. But while the fact that philosophy is at least partly determined by contingent cultural factors undercuts necessity and progress, it also opens the door for a reconceptualization of the "essence" of philosophy. In his Einleitung in die Philosophie [Introduction to philosophy] (1914) Windelband reflects on the fact that everyday life, culture, and science do already contain general concepts of the world. These concepts form the initial content of philosophical reflection (1914: 6). But the business of philosophy only takes off when these initial concepts, and the assumptions that are baked into them, become unstable and collapse. An experience of shock and unsettling prompts philosophy to question, rethink, and critically assess the concepts and ideas of everyday life and science. In this process of critical assessment, philosophy strives to purify these concepts and ideas, and to connect them into a coherent, unified system. According to Windelband, in this process of conceptual reorganization, a rational necessity exerts itself. The "vigorous and uncompromising rethinking of the preconditions of our spiritual life" creates certain philosophical problems "with objective necessity" (1914: 8). Windelband provides neither a very detailed account of what "philosophical problems" exactly are, nor of how precisely they spring from the unsettling of everyday concepts. But he makes three claims that, taken together, establish that philosophy, even when reflecting its particular age, is not solely determined by contingent cultural factors. First, philosophical puzzles stem from the "inadequacy and contradictory imbalance" of the contents that philosophy receives from life and science (1891: 10). That is, true philosophical problems emerge whenever the systematizing drive of philosophy is confronted with the deep incoherence of life. Second, philosophical problems are necessary because the conceptual "material" found in life already contains "the objective presuppositions and logical coercions for all rational deliberation about it" (1891: 10). The necessity of philosophical problems is logical necessity. Third, despite historical change > > > [c]ertain differences of world- and life-attitudes reoccur over and > over again, combat each other and destroy each other in mutual > dialectics. (1914: 10) > > > Because philosophical problems emerge "necessarily" and "objectively", they also reoccur throughout history. Philosophical problems are thus timeless and eternal (1891: 9-10; 1914: 11)). Windelband also believes that the history of philosophy, pursued empirically and scientifically, can disentangle from one another the "temporal causes and timeless reasons" (1905: 189) that together give rise to the emergence of philosophical problems. > > > Only through knowledge of the empirical trajectory that is free of > constructions can come to light .... what is the share of > ... the needs of the age on the one hand...but on the other > hand that of the objective necessities of conceptual progress. (1905: > 189) > > > At this point, Windelband reaches a clear verdict on the goal and systematic significance of the history of philosophy: it lies in disentangling that which is contingently actually accepted from that which is "valid in itself" (1905: 199). History is the "organon of critical philosophy" precisely to the extent that it allows us to distinguish the actually accepted norms of cultural life from that which is absolutely valid (1883: 132). ## 7. Critical Philosophy and World-Views Windelband's "philosophy of values" is also an intervention into debates that had preoccupied the neo-Kantian movement and German academic philosophy more broadly since the materialism controversy [Materialismusstreit]: what the relation between science and world-views is, and whether philosophy is capable of providing a "world-view". Given that he takes the core of the critical project to consist in a quaestio juris--a concern with normativity--it seems surprising that throughout the 1880s Windelband insists that critical philosophy does not provide a "world-view": the project of revealing the highest values of human life has nothing to do with "world-views", he argues, because it does not provide a metaphysical account of the world and our place in it (1881: 140-141, 145). Questions about optimism or pessimism cannot be answered by means of "scientific" philosophy, since they depend on the idea that the world as a whole has a purpose. And claims about the ultimate purpose of the world arise only from an unscientific, subjective, and arbitrary projection of particular purposes onto the universe as such (1876: 231). Philosophy is a science, albeit a second-order formal science, and is thus barred from formulating a world-view that would be metaphysical in character. After 1900, however, Windelband's position on the question of world-views changes. He now claims that it is the aim of philosophy to provide a "world-view" with scientific justification (1907: 1). Given that Windelband holds that philosophy in general and the Kantian project in particular need to be adapted to the developing circumstances of time and culture, this change of position is not inconsequential: the cultural landscape of the early 1900s demands a revision of the neo-Kantian project that highlights not only its "negative" and critical aspects, but also develops its positive implications for cultural life (1907, 8). The shift to a more positive attitude towards philosophy as a world-view corresponds to a re-evaluation of Kant's oeuvre, in which Windelband at least partly suspends the anti-metaphysical rigor of the "third phase" of the genesis of the Critique of Pure Reason, and attributes more relevance to the Critique of the Power of Judgment. This view is taken up by Rickert, who later argues in more detail that the third Critique forms the core of critical philosophy, and that Kant's metaphysical project can provide the basis for an encompassing theory of world-views (Rickert 1924: 153-168). In his 1904 "Nach hundert Jahren" ["After one hundred years"] Windelband provides yet another take on the relation between the natural and the normative. Declaring the question how the realm of natural laws is related to the realm of values to be the "highest" philosophical endeavor (1904a: 162), he now sees in the Critique of the Power of Judgment the possibility of solving this problem: here one finds the idea that the purposive system of nature gives rise to the value-determined process of human history. The central concept that allows for connecting nature and history in this way is that of "realization" (1904a: 162-3). The turn to the concept of "realization" also leads to a shift in how Windelband approaches the problem of science classification. In 1894, Windelband had thought of the nomothetic and idiographic sciences as two fundamental yet disjointed and even incommensurable approaches to seeking knowledge of the word: "Law and event persist next to each other as the last, incommensurable factors of our representation of the world" (1894: 160). But in 1904, he claims that the concept of realization provides a common, unified basis for the natural and the historical sciences (1904a: 163). Windelband's language assumes a Hegelian tone. Critical philosophy equipped with the concept of realization is able to grasp and express the "spiritual value content" of reality (1904a: 165). History is capable of revealing the universal value content in the contingent maze of human interests and desires, and in so doing captures "the progressive realization of rational values" (1907: 20-21). The human as a rational being is not determined psychologically or naturally, but humanity is a historical task: "only as a historical being, as the developing species, do we have a share in world-reason" (1910a: 283: see also 1905: 185). Windelband even suggests that history itself has a goal, namely the realization of a common humanity (1908: 257). These remarks remain cursory however, and there is room for debate over how much of Hegel's ontology of history Windelband takes on board, as well as whether his talk about the progressive realization of reason in history is incompatible with the formal-teleological method that he had endorsed in the 1880s and 1890s. His "Kulturphilosophie und transzendentaler Idealismus" ["Philosophy of culture and transcendental idealism"] of 1910 presents us with a puzzling fusion of Kantian and Hegelian language. The goal of critical philosophy is that of revealing the unity of culture, Windelband now claims, and this unity can only be found > > > by grasping the essence of the function ... which is common for > all the particular ...cultural activities: and this can be > nothing else than the self-consciousness of reason which produces its > objects and in them the realm of validity itself. (1910b: 293). > > >
wisdom	## 1. Wisdom as Epistemic Humility Socrates' view of wisdom, as expressed by Plato in The Apology (20e-23c), is sometimes interpreted as an example of a humility theory of wisdom (see, for example, Ryan 1996 and Whitcomb, 2010). In Plato's Apology, Socrates and his friend Chaerephon visit the oracle at Delphi. As the story goes, Chaerephon asks the oracle whether anyone is wiser than Socrates. The oracle's answer is that Socrates is the wisest person. Socrates reports that he is puzzled by this answer since so many other people in the community are well known for their extensive knowledge and wisdom, and yet Socrates claims that he lacks knowledge and wisdom. Socrates does an investigation to get to the bottom of this puzzle. He interrogates a series of politicians, poets, and craftsmen. As one would expect, Socrates' investigation reveals that those who claim to have knowledge either do not really know any of the things they claim to know, or else know far less than they proclaim to know. The most knowledgeable of the bunch, the craftsmen, know about their craft, but they claim to know things far beyond the scope of their expertise. Socrates, so we are told, neither suffers the vice of claiming to know things he does not know, nor the vice of claiming to have wisdom when he does not have wisdom. In this revelation, we have a potential resolution to the wisdom puzzle in The Apology. Although the story may initially appear to deliver a clear theory of wisdom, it is actually quite difficult to capture a textually accurate and plausible theory here. One interpretation is that Socrates is wise because he, unlike the others, believes he is not wise, whereas the poets, politicians, and craftsmen arrogantly and falsely believe they are wise. This theory, which will be labeled Humility Theory 1 (H1), is simply (see, for example, Lehrer & Smith 1996, 3): > > Humility Theory 1 (H1): > > S is wise iff S believes s/he is not > wise. This is a tempting and popular interpretation because Socrates certainly thinks he has shown that the epistemically arrogant poets, politicians, and craftsmen lack wisdom. Moreover, Socrates claims that he is not wise, and yet, if we trust the oracle, Socrates is actually wise. Upon careful inspection, (H1) is not a reasonable interpretation of Socrates' view. Although Socrates does not boast of his own wisdom, he does believe the oracle. If he was convinced that he was not wise, he would have rejected the oracle and gone about his business because he would not find any puzzle to unravel. Clearly, he believes, on some level, that he is wise. The mystery is: what is wisdom if he has it and the others lack it? Socrates nowhere suggests that he has become unwise after believing the oracle. Thus, (H1) is not an acceptable interpretation of Socrates' view. Moreover, (H1) is false. Many people are clear counterexamples to (H1). Many people who believe they are not wise are correct in their self-assessment. Thus, the belief that one is not wise is not a sufficient condition for wisdom. Furthermore, it seems that the belief that one is not wise is not necessary for wisdom. It seems plausible to think that a wise person could be wise enough to realize that she is wise. Too much modesty might get in the way of making good decisions and sharing what one knows. If one thinks Socrates was a wise person, and if one accepts that Socrates did, in fact, accept that he was wise, then Socrates himself is a counterexample to (H1). The belief that one is wise could be a perfectly well justified belief for a wise person. Having the belief that one is wise does not, in itself, eliminate the possibility that the person is wise. Nor does it guarantee the vice of arrogance. We should hope that a wise person would have a healthy dose of epistemic self-confidence, appreciate that she is wise, and share her understanding of reality with the rest of us who could benefit from her wisdom. Thus, the belief that one is not wise is not required for wisdom. (H1) focused on believing one is not wise. Another version of the humility theory is worth considering. When Socrates demonstrates that a person is not wise, he does so by showing that the person lacks some knowledge that he or she claims to possess. Thus, one might think that Socrates' view could be better captured by focusing on the idea that wise people believe they lack knowledge (rather than lacking wisdom). That is, one might consider the following view: > > Humility Theory 2 (H2): > > S is wise iff S believes S does not know > anything. Unfortunately, this interpretation is not any better than (H1). It falls prey to problems similar to those that refuted (H1) both as an interpretation of Socrates, and as an acceptable account of wisdom. Moreover, remember that Socrates admits that the craftsmen do have some knowledge. Socrates might have considered them to be wise if they had restricted their confidence and claims to knowledge to what they actually did know about their craft. Their problem was that they professed to have knowledge beyond their area of expertise. The problem was not that they claimed to have knowledge. Before turning to alternative approaches to wisdom, it is worth mentioning another interpretation of Socrates that fits with the general spirit of epistemic humility. One might think that what Socrates is establishing is that his wisdom is found in his realization that human wisdom is not a particularly valuable kind of wisdom. Only the gods possess the kind of wisdom that is truly valuable. This is clearly one of Socrates' insights, but it does not provide us with an understanding of the nature of wisdom. It tells us only of its comparative value. Merely understanding this evaluative insight would not, for reasons similar to those discussed with (HP1) and (HP2), make one wise. Humility theories of wisdom are not promising, but they do, perhaps, provide us with some important character traits associated with wise people. Wise people, one might argue, possess epistemic self-confidence, yet lack epistemic arrogance. Wise people tend to acknowledge their fallibility, and wise people are reflective, introspective, and tolerant of uncertainty. Any acceptable theory of wisdom ought to be compatible with such traits. However, those traits are not, in and of themselves, definitive of wisdom. ## 2. Wisdom as Epistemic Accuracy Socrates can be interpreted as providing an epistemic accuracy, rather than an epistemic humility, theory of wisdom. The poets, politicians, and craftsmen all believe they have knowledge about topics on which they are considerably ignorant. Socrates, one might argue, believes he has knowledge when, and only when, he really does have knowledge. Perhaps wise people restrict their confidence to propositions for which they have knowledge or, at least, to propositions for which they have excellent justification. Perhaps Socrates is better interpreted as having held an Epistemic Accuracy Theory such as: > > Epistemic Accuracy Theory 1 (EA1): > > S is wise iff for all p, (S believes > S knows > p iff S knows p.) According to (EA1), a wise person is accurate about what she knows and what she does not know. If she really knows p, she believes she knows p. And, if she believes she knows p, then she really does know p. (EA1) is consistent with the idea that Socrates accepts that he is wise and with the idea that Socrates does have some knowledge. (EA1) is a plausible interpretation of the view Socrates endorses, but it is not a plausible answer in the search for an understanding of wisdom. Wise people can make mistakes about what they know. Socrates, Maimonides, King Solomon, Einstein, Goethe, Gandhi, and every other candidate for the honor of wisdom have held false beliefs about what they did and did not know. It is easy to imagine a wise person being justified in believing she possesses knowledge about some claim, and also easy to imagine that she could be shown to be mistaken, perhaps long after her death. If (EA1) is true, then just because a person believes she has knowledge when she does not, she is not wise. That seems wrong. It is hard to imagine that anyone at all is, or ever has been, wise if (EA1) is correct. We could revise the Epistemic Accuracy Theory to get around this problem. We might only require that a wise person's belief is highly justified when she believes she has knowledge. That excuses people with bad epistemic luck. > > Epistemic Accuracy 2 (EA2): > > S is wise iff for all p, (S believes > S knows > p iff S's belief in p is highly > justified.) (EA2) gets around the problem with (EA1). The Socratic Method challenges one to produce reasons for one's view. When Socrates' interlocutor is left dumbfounded, or reduced to absurdity, Socrates rests his case. One might argue that through his questioning, Socrates reveals not that his opponents lack knowledge because their beliefs are false, but he demonstrates that his opponents are not justified in holding the views they profess to know. Since the craftsmen, poets, and politicians questioned by Socrates all fail his interrogation, they were shown, one might argue, to have claimed to have knowledge when their beliefs were not even justified. Many philosophers would hesitate to endorse this interpretation of what is going on in The Apology. They would argue that a failure to defend one's beliefs from Socrates' relentless questioning does not show that a person is not justified in believing a proposition. Many philosophers would argue that having very good evidence, or forming a belief via a reliable process, would be sufficient for justification. Proving, or demonstrating to an interrogator, that one is justified is another matter, and not necessary for simply being justified. Socrates, some might argue, shows only that the craftsmen, poets, and politicians cannot defend themselves from his questions. He does not show, one might argue, that the poets, politicians, and craftsmen have unjustified beliefs. Since we gain very little insight into the details of the conversation in this dialogue, it would be unfair to dismiss this interpretation on these grounds. Perhaps Socrates did show, through his intense questioning, that the craftsmen, poets, and politicians formed and held their beliefs without adequate evidence or formed and held them through unreliable belief forming processes. Socrates only reports that they did not know all that they professed to know. Since we do not get to witness the actual questioning as we do in Plato's other dialogues, we should not reject (EA2) as an interpretation of Socrates' view of wisdom in The Apology. Regardless of whether (EA2) is Socrates' view, there are problems for (EA2) as an account of what it means to be wise. Even if (EA2) is exactly what Socrates meant, some philosophers would argue that one could be justified in believing a proposition, but not realize that she is justified. If that is a possible situation for a wise person to be in, then she might be justified, but fail to believe she has knowledge. Could a wise person be in such a situation, or is it necessary that a wise person would always recognize the epistemic value of what he or she believes?[1] If this situation is impossible, then this criticism could be avoided. There is no need to resolve this issue here because (EA1) and (EA2) fall prey to another, much less philosophically thorny and controversial problem. (EA1) and (EA2) suffer from a similar, and very serious, problem. Imagine a person who has very little knowledge. Suppose further, that the few things she does know are of little or no importance. She could be the sort of person that nobody would ever go to for information or advice. Such a person could be very cautious and believe that she knows only what she actually knows. Although she would have accurate beliefs about what she does and does not know, she would not be wise. This shows that (EA1) is flawed. As for (EA2), imagine that she believes she knows only what she is actually justified in believing. She is still not wise. It should be noted, however, that although accuracy theories do not provide an adequate account of wisdom, they reveal an important insight. Perhaps a necessary condition for being wise is that wise people think they have knowledge only when their beliefs are highly justified. Or, even more simply, perhaps wise people have epistemically justified, or rational, beliefs. ## 3. Wisdom as Knowledge An alternative approach to wisdom focuses on the more positive idea that wise people are very knowledgeable people. There are many views in the historical and contemporary philosophical literature on wisdom that have knowledge, as opposed to humility or accuracy, as at least a necessary condition of wisdom. Aristotle (Nichomachean Ethics VI, ch. 7), Descartes (Principles of Philosophy), Richard Garrett (1996), John Kekes (1983), Keith Lehrer & Nicholas Smith (1996), Robert Nozick (1989), Plato (The Republic), Sharon Ryan (1996, 1999), Valerie Tiberius (2008), Dennis Whitcomb (2010) and Linda Zagzebski (1996) for example, have all defended theories of wisdom that require a wise person to have knowledge of some sort. All of these views very clearly distinguish knowledge from expertise on a particular subject. Moreover, all of these views maintain that wise people know "what is important." The views differ, for the most part, over what it is important for a wise person to know, and on whether there is any behavior, action, or way of living, that is required for wisdom. Aristotle distinguished between two different kinds of wisdom, theoretical wisdom and practical wisdom. Theoretical wisdom is, according to Aristotle, "scientific knowledge, combined with intuitive reason, of the things that are highest by nature" (Nicomachean Ethics, VI, 1141b). For Aristotle, theoretical wisdom involves knowledge of necessary, scientific, first principles and propositions that can be logically deduced from them. Aristotle's idea that scientific knowledge is knowledge of necessary truths and their logical consequences is no longer a widely accepted view. Thus, for the purposes of this discussion, I will consider a theory that reflects the spirit of Aristotle's view on theoretical wisdom, but without the controversy about the necessary or contingent nature of scientific knowledge. Moreover, it will combine scientific knowledge with other kinds of factual knowledge, including knowledge about history, philosophy, music, literature, mathematics, etc. Consider the following, knowledge based, theory of wisdom: > > Wisdom as Extensive Factual Knowledge (WFK): > > S is wise iff S has extensive factual knowledge about > science, history, philosophy, literature, music, etc. According to (WFK), a wise person is a person who knows a lot about the universe and our place in it. She would have extensive knowledge about the standard academic subjects. There are many positive things to say about (WFK). (WFK) nicely distinguishes between narrow expertise and knowledge of the mundane, from the important, broad, and general kind of knowledge possessed by wise people. As Aristotle puts it, "...we think that some people are wise in general, not in some particular field or in any other limited respect..." (Nicomachean Ethics, Book 6, 1141a). The main problem for (WFK) is that some of the most knowledgeable people are not wise. Although they have an abundance of very important factual knowledge, they lack the kind of practical know-how that is a mark of a wise person. Wise people know how to get on in the world in all kinds of situations and with all kinds of people. Extensive factual knowledge is not enough to give us what a wise person knows. As Robert Nozick points out, "Wisdom is not just knowing fundamental truths, if these are unconnected with the guidance of life or with a perspective on its meaning" (1989, 269). There is more to wisdom than intelligence and knowledge of science and philosophy or any other subject matter. Aristotle is well aware of the limitations of what he calls theoretical wisdom. However, rather than making improvements to something like (WFK), Aristotle distinguishes it as one kind of wisdom. Other philosophers would be willing to abandon (WFK), that is, claim that it provides insufficient conditions for wisdom, and add on what is missing. Aristotle has a concept of practical wisdom that makes up for what is missing in theoretical wisdom. In Book VI of the Nicomachean Ethics, he claims, "This is why we say Anaxagoras, Thales, and men like them have philosophic but not practical wisdom, when we see them ignorant of what is to their own advantage, and why we say that they know things that are remarkable, admirable, difficult, and divine, but useless; viz. because it is not human goods they seek" (1141a). Knowledge of contingent facts that are useful to living well is required in Aristotle's practical wisdom. According to Aristotle, "Now it is thought to be the mark of a man of practical wisdom to be able to deliberate well about what is good and expedient for himself, not in some particular respect, e.g. about what sorts of thing conduce to health or to strength, but about what sorts of thing conduce to the good life in general" (Nichomachean Ethics, VI, 1140a-1140b). Thus, for Aristotle, practical wisdom requires knowing, in general, how to live well. Many philosophers agree with Aristotle on this point. However, many would not be satisfied with the conclusion that theoretical wisdom is one kind of wisdom and practical wisdom another. Other philosophers, including Linda Zagzebski (1996), agree that there are these two types of wisdom that ought to be distinguished. Let's proceed, without argument, on the assumption that it is possible to have a theory of one, general, kind of wisdom. Wisdom, in general, many philosophers would argue, requires practical knowledge about living. What Aristotle calls theoretical wisdom, many would contend, is not wisdom at all. Aristotle's theoretical wisdom is merely extensive knowledge or deep understanding. Nicholas Maxwell (1984), in his argument to revolutionize education, argues that we should be teaching for wisdom, which he sharply distinguishes from standard academic knowledge. Similar points are raised by Robert Sternberg (2001) and Andrew Norman (1996). Robert Nozick holds a view very similar to Aristotle's theory of practical wisdom, but Nozick is trying to capture the essence of wisdom, period. He is not trying to define one, alternative, kind of wisdom. Nozick claims, "Wisdom is what you need to understand in order to live well and cope with the central problems and avoid the dangers in the predicaments human beings find themselves in" (1989, 267). And, John Kekes maintains that, "What a wise man knows, therefore, is how to construct a pattern that, given the human situation, is likely to lead to a good life" (1983, 280). More recently, Valerie Tiberius (2008) has developed a practical view that connects wisdom with well being, requiring, among other things, that a wise person live the sort of life that he or she could sincerely endorse upon reflection. Such practical views of wisdom could be expressed, generally, as follows. > > Wisdom as Knowing How To Live Well (KLW): > > S is wise iff S knows how to live well. This view captures Aristotle's basic idea of practical wisdom. It also captures an important aspect of views defended by Nozick, Plato, Garrett, Kekes, Maxwell, Ryan, and Tiberius. Although giving an account of what it means to know how to live well may prove as difficult a topic as providing an account of wisdom, Nozick provides a very illuminating start. > > > Wisdom is not just one type of knowledge, but diverse. > What a wise person needs to know and understand constitutes a varied > list: the most important goals and values of life - the ultimate > goal, if there is one; what means will reach these goals without too > great a cost; what kinds of dangers threaten the achieving of these > goals; how to recognize and avoid or minimize these dangers; what > different types of human beings are like in their actions and motives > (as this presents dangers or opportunities); what is not possible or > feasible to achieve (or avoid); how to tell what is appropriate when; > knowing when certain goals are sufficiently achieved; what limitations > are unavoidable and how to accept them; how to improve oneself and > one's relationships with others or society; knowing what the true and > unapparent value of various things is; when to take a long-term > view; knowing the variety and obduracy of facts, institutions, and > human nature; understanding what one's real motives are; how to cope > and deal with the major tragedies and dilemmas of life, and with the > major good things > too. (1989, 269) With Nozick's explanation of what one must know in order to live well, we have an interesting and quite attractive, albeit somewhat rough, theory of wisdom. As noted above, many philosophers, including Aristotle and Zagzebski would, however, reject (KLW) as the full story on wisdom. Aristotle and Zagzebski would obviously reject (KLW) as the full story because they believe theoretical wisdom is another kind of wisdom, and are unwilling to accept that there is a conception of one, general, kind of wisdom. Kekes claims, "The possession of wisdom shows itself in reliable, sound, reasonable, in a word, good judgment. In good judgment, a person brings his knowledge to bear on his actions. To understand wisdom, we have to understand its connection with knowledge, action, and judgment" (1983, 277). Kekes adds, "Wisdom ought also to show in the man who has it" (1983, 281). Many philosophers, therefore, think that wisdom is not restricted even to knowledge about how to live well. Tiberius thinks the wise person's actions reflect their basic values. These philosophers believe that being wise also includes action. A person could satisfy the conditions of any of the principles we have considered thus far and nevertheless behave in a wildly reckless manner. Wildly reckless people are, even if very knowledgeable about life, not wise. Philosophers who are attracted to the idea that knowing how to live well is a necessary condition for wisdom might want to simply tack on a success condition to (KLW) to get around cases in which a person knows all about living well, yet fails to put this knowledge into practice. Something along the lines of the following theory would capture this idea. > > Wisdom as Knowing How To, and Succeeding at, Living > Well (KLS): > > S is wise iff (i) S knows how to live well, and > (ii) S is successful at living well. > The idea of the success condition is that one puts one's knowledge into practice. Or, rather than using the terminology of success, one might require that a wise person's beliefs and values cohere with one's actions (Tiberius, 2008). The main idea is that one's actions are reflective of one's understanding of what it means to live well. A view along the lines of (KLS) would be embraced by Aristotle and Zagzebski (for practical wisdom), and by Kekes, Nozick, and Tiberius. (KLS) would not be universally embraced, however (see Ryan 1999, for further criticisms). One criticism of (KLS) is that one might think that all the factual knowledge required by (WFK) is missing from this theory. One might argue that (WFK), the view that a wise person has extensive factual knowledge, was rejected only because it did not provide sufficient conditions for wisdom. Many philosophers would claim that (WFK) does provide a necessary condition for wisdom. A wise person, such a critic would argue, needs to know how to live well (as described by Nozick), but she also needs to have some deep and far-reaching theoretical, or factual, knowledge that may have very little impact on her daily life, practical decisions, or well being. In the preface of his Principles of Philosophy, Descartes insisted upon factual knowledge as an important component of wisdom. Descartes wrote, "It is really only God alone who has Perfect Wisdom, that is to say, who has a complete knowledge of the truth of all things; but it may be said that men have more wisdom or less according as they have more or less knowledge of the most important truths" (Principles, 204). Of course, among those important truths, one might claim, are truths about living well, as well as knowledge in the basic academic subject areas. Moreover, one might complain that the insight left standing from Epistemic Accuracy theories is also missing from (KLS). One might think that a wise person not only knows a lot, and succeeds at living well, she also confines her claims to knowledge (or belief that she has knowledge) to those propositions that she is justified in believing. ## 4. Hybrid Theory One way to try to accommodate the various insights from the theories considered thus far is in the form of a hybrid theory. One such idea is: > > S is wise iff 1. S has extensive factual and theoretical knowledge. > 2. S knows how to live well. > 3. S is successful at living well. > 4. S has very few unjustified beliefs. > > > Although this Hybrid Theory has a lot going for it, there are a number of important criticisms to consider. Dennis Whitcomb (2010) objects to all theories of wisdom that include a living well condition, or an appreciation of living well condition. He gives several interesting objections against such views. Whitcomb thinks that a person who is deeply depressed and totally devoid of any ambition for living well could nevertheless be wise. As long as such a person is deeply knowledgeable about academic subjects and knows how to live well, that person would have all they need for wisdom. With respect to a very knowledgeable and deeply depressed person with no ambition but to stay in his room, he claims, "If I ran across such a person, I would take his advice to heart, wish him a return to health, and leave the continuing search for sages to his less grateful advisees. And I would think he was wise despite his depression-induced failure to value or desire the good life. So I think that wisdom does not require valuing or desiring the good life." In response to Whitcomb's penetrating criticism, one could argue that a deeply depressed person who is wise, would still live as well as she can, and would still value living well, even if she falls far short of perfection. Such a person would attempt to get help to deal with her depression. If she really does not care at all, she may be very knowledgeable, but she is not wise. There is something irrational about knowing how to live well and refusing to try to do so. Such irrationality is not compatible with wisdom. A person with this internal conflict may be extremely clever and shrewd, one to listen to on many issues, one to trust on many issues, and may even win a Nobel Prize for her intellectual greatness, but she is not admirable enough, and rationally consistent enough, to be wise. Wisdom is a virtue and a way of living, and it requires more than smart ideas and knowledge. Aristotle held that "it is evident that it is impossible to be practically wise without being good" (Nicomachean Ethics, 1144a, 36-37). Most of the philosophers mentioned thus far would include moral virtue in their understanding of what it means to live well. However, Whitcomb challenges any theory of wisdom that requires moral virtue. Whitcomb contends that a deeply evil person could nevertheless be wise. Again, it is important to contrast being wise from being clever and intelligent. If we think of wisdom as the highest, or among the highest, of human virtues, then it seems incompatible with a deeply evil personality. There is, however, a very serious problem with the Hybrid Theory. Since so much of what was long ago considered knowledge has been abandoned, or has evolved, a theory that requires truth (through a knowledge condition) would exclude almost all people who are now long dead, including Hypatia, Socrates, Confucius, Aristotle, Homer, Lao Tzu, etc. from the list of the wise. Bad epistemic luck, and having lived in the past, should not count against being wise. But, since truth is a necessary condition for knowledge, bad epistemic luck is sufficient to undermine a claim to knowledge. What matters, as far as being wise goes, is not that a wise person has knowledge, but that she has highly justified and rational beliefs about a wide variety of subjects, including how to live well, science, philosophy, mathematics, history, geography, art, literature, psychology, and so on. And the wider the variety of interesting topics, the better. Another way of developing this same point is to imagine a person with highly justified beliefs about a wide variety of subjects, but who is unaware that she is trapped in the Matrix, or some other skeptical scenario. Such a person could be wise even if she is sorely lacking knowledge. A theory of wisdom that focuses on having rational or epistemically justified beliefs, rather than the higher standard of actually having knowledge, would be more promising. Moreover, such a theory would incorporate much of what is attractive about epistemic humility, and epistemic accuracy, theories. ## 5. Wisdom as Rationality The final theory to be considered here is an attempt to capture all that is good, while avoiding all the serious problems of the other theories discussed thus far. Perhaps wisdom is a deep and comprehensive kind of rationality (Ryan, 2012). > > Deep Rationality Theory (DRT): > > S is wise iff > > 1. S has a wide variety of epistemically justified beliefs > on a wide variety of valuable academic subjects. > 2. S has a wide variety of justified beliefs on how to live > rationally (epistemically, morally, and practically). > 3. S is committed to living rationally. > 4. S has very few unjustified beliefs and is sensitive to > her limitations. > > > In condition (1), DRT takes account of what is attractive about some knowledge theories by requiring epistemically justified beliefs about a wide variety of standard academic subjects. Condition (2) takes account of what is attractive about theories that require knowledge about how to live well. For example, having justified beliefs about how to live in a practically rational way would include having a well-reasoned strategy for dealing with the practical aspects of life. Having a rational plan does not require perfect success. It requires having good reasons behind one's actions, responding appropriately to, and learning from, one's mistakes, and having a rational plan for all sorts of situations and problems. Having justified beliefs about how to live in a morally rational way would not involve being a moral saint, but would require that one has good reasons supporting her beliefs about what is morally right and wrong, and about what one morally ought and ought not do in a wide variety of circumstances. Having justified beliefs about living in an emotionally rational way would involve, not dispassion, but having justified beliefs about what is, and what is not, an emotionally rational response to a situation. For example, it is appropriate to feel deeply sad when dealing with the loss of a loved one. But, ordinarily, feeling deeply sad or extremely angry is not an appropriate emotion to spilled milk. A wise person would have rational beliefs about the emotional needs and behaviors of other people. Condition (3) ensures that the wise person live a life that reflects what she or he is justified in believing is a rational way to live. In condition (4), DRT respects epistemic humility. Condition (4) requires that a wise person not believe things without epistemic justification. The Deep Rationality Theory rules out all of the unwise poets, politicians, and craftsmen that were ruled out by Socrates. Wise people do not think they know when they lack sufficient evidence. Moreover, wise people are not epistemically arrogant. The Deep Rationality Theory does not require knowledge or perfection. But it does require rationality, and it accommodates degrees of wisdom. It is a promising theory of wisdom.
wittgenstein	## 1. Biographical Sketch Wittgenstein was born on April 26, 1889 in Vienna, Austria, to a wealthy industrial family, well-situated in intellectual and cultural Viennese circles. In 1908 he began his studies in aeronautical engineering at Manchester University where his interest in the philosophy of pure mathematics led him to Frege. Upon Frege's advice, in 1911 he went to Cambridge to study with Bertrand Russell. Russell wrote, upon meeting Wittgenstein: "An unknown German appeared ... obstinate and perverse, but I think not stupid" (quoted by Monk 1990: 38f). Within one year, Russell was committed: "I shall certainly encourage him. Perhaps he will do great things ... I love him and feel he will solve the problems I am too old to solve" (quoted by Monk 1990: 41). Russell's insight was accurate. Wittgenstein was idiosyncratic in his habits and way of life, yet profoundly acute in his philosophical sensitivity. During his years in Cambridge, from 1911 to 1913, Wittgenstein conducted several conversations on philosophy and the foundations of logic with Russell, with whom he had an emotional and intense relationship, as well as with Moore and Keynes. He retreated to isolation in Norway, for months at a time, in order to ponder these philosophical problems and to work out their solutions. In 1913 he returned to Austria and in 1914, at the start of World War I (1914-1918), joined the Austrian army. He was taken captive in 1918 and spent the remaining months of the war at a prison camp. It was during the war that he wrote the notes and drafts of his first important work, Tractatus Logico-Philosophicus. After the war the book was published in German and translated into English. In the 1920s Wittgenstein, now divorced from philosophy (having, to his mind, solved all philosophical problems in the Tractatus), gave away his part of his family's fortune and pursued several 'professions' (gardener, teacher, architect, etc.) in and around Vienna. It was only in 1929 that he returned to Cambridge to resume his philosophical vocation, after having been exposed to discussions on the philosophy of mathematics and science with members of the Vienna Circle, whose conception of logical empiricism was indebted to his Tractatus account of logic as tautologous, and his philosophy as concerned with logical syntax. During these first years in Cambridge his conception of philosophy and its problems underwent dramatic changes that are recorded in several volumes of conversations, lecture notes, and letters (e.g., Ludwig Wittgenstein and the Vienna Circle, The Blue and Brown Books, Philosophical Grammar, Philosophical Remarks). Sometimes termed the 'middle Wittgenstein,' this period heralds a rejection of dogmatic philosophy, including both traditional works and the Tractatus itself. In the 1930s and 1940s Wittgenstein conducted seminars at Cambridge, developing most of the ideas that he intended to publish in his second book, Philosophical Investigations. These included the turn from formal logic to ordinary language, novel reflections on psychology and mathematics, and a general skepticism concerning philosophy's pretensions. In 1945 he prepared the final manuscript of the Philosophical Investigations, but, at the last minute, withdrew it from publication (and only authorized its posthumous publication). For a few more years he continued his philosophical work, but this is marked by a rich development of, rather than a turn away from, his second phase. He traveled during this period to the United States and Ireland, and returned to Cambridge, where he was diagnosed with cancer. Legend has it that, at his death in 1951, his last words were "Tell them I've had a wonderful life" (Monk: 579). ## 2. The Early Wittgenstein ### 2.1 Tractatus Logico-Philosophicus Tractatus Logico-Philosophicus was first published in German in 1921 and then translated--by C.K. Ogden (and F. P. Ramsey)--and published in English in 1922. It was later re-translated by D. F. Pears and B. F. McGuinness. Coming out of Wittgenstein's Notes on Logic (1913), "Notes Dictated to G. E. Moore" (1914), his Notebooks, written in 1914-16, and further correspondence with Russell, Moore, and Keynes, and showing Schopenhauerian and other cultural influences, it evolved as a continuation of and reaction to Russell and Frege's conceptions of logic and language. Russell supplied an introduction to the book claiming that it "certainly deserves ... to be considered an important event in the philosophical world." It is fascinating to note that Wittgenstein thought little of Russell's introduction, claiming that it was riddled with misunderstandings. Later interpretations have attempted to unearth the surprising tensions between the introduction and the rest of the book (or between Russell's reading of Wittgenstein and Wittgenstein's own self-assessment)--usually harping on Russell's appropriation of Wittgenstein for his own agenda. The Tractatus's structure purports to be representative of its internal essence. It is constructed around seven basic propositions, numbered by the natural numbers 1-7, with all other paragraphs in the text numbered by decimal expansions so that, e.g., paragraph 1.1 is (supposed to be) a further elaboration on proposition 1, 1.22 is an elaboration of 1.2, and so on. The seven basic propositions are: \| \| \| \| \| --- \| --- \| --- \| \| \| Ogden translation \| Pears/McGuinness translation \| \| 1. \| The world is everything that is the case. \| The world is all that is the case. \| \| 2. \| What is the case, the fact, is the existence of atomic facts. \| What is the case--a fact--is the existence of states of affairs. \| \| 3. \| The logical picture of the facts is the thought. \| A logical picture of facts is a thought. \| \| 4. \| The thought is the significant proposition. \| A thought is a proposition with sense. \| \| 5. \| Propositions are truth-functions of elementary propositions. \| A proposition is a truth-function of elementary propositions. \| \| \| (An elementary proposition is a truth function of itself.) \| (An elementary proposition is a truth function of itself.) \| \| 6. \| The general form of truth-function is $[\bar{p}, \bar{\xi}, N(\bar{\xi})]$. \| The general form of a truth-function is $[\bar{p}, \bar{\xi}, N(\bar{\xi})]$. \| \| \| This is the general form of proposition. \| This is the general form of a proposition. \| \| 7. \| Whereof one cannot speak, thereof one must be silent. \| What we cannot speak about we must pass over in silence. \| Clearly, the book addresses the central problems of philosophy which deal with the world, thought and language, and presents a 'solution' (as Wittgenstein terms it) of these problems that is grounded in logic and in the nature of representation. The world is represented by thought, which is a proposition with sense, since they all--world, thought, and proposition--share the same logical form. Hence, the thought and the proposition can be pictures of the facts. Starting with a seeming metaphysics, Wittgenstein sees the world as consisting of facts (1), rather than the traditional, atomistic conception of a world made up of objects. Facts are existent states of affairs (2) and states of affairs, in turn, are combinations of objects. "Objects are simple" (TLP 2.02) but objects can fit together in various determinate ways. They may have various properties and may hold diverse relations to one another. Objects combine with one another according to their logical, internal properties. That is to say, an object's internal properties determine the possibilities of its combination with other objects; this is its logical form. Thus, states of affairs, being comprised of objects in combination, are inherently complex. The states of affairs which do exist could have been otherwise. This means that states of affairs are either actual (existent) or possible. It is the totality of states of affairs--actual and possible--that makes up the whole of reality. The world is precisely those states of affairs which do exist. The move to thought, and thereafter to language, is perpetrated with the use of Wittgenstein's famous idea that thoughts, and propositions, are pictures--"the picture is a model of reality" (TLP 2.12). Pictures are made up of elements that together constitute the picture. Each element represents an object, and the combination of elements in the picture represents the combination of objects in a state of affairs. The logical structure of the picture, whether in thought or in language, is isomorphic with the logical structure of the state of affairs which it pictures. More subtle is Wittgenstein's insight that the possibility of this structure being shared by the picture (the thought, the proposition) and the state of affairs is the pictorial form. "That is how a picture is attached to reality; it reaches right out to it" (TLP 2.1511). This leads to an understanding of what the picture can picture; but also what it cannot--its own pictorial form. While "the logical picture of the facts is the thought" (3), in the move to language Wittgenstein continues to investigate the possibilities of significance for propositions (4). Logical analysis, in the spirit of Frege and Russell, guides the work, with Wittgenstein using logical calculus to carry out the construction of his system. Explaining that "Only the proposition has sense; only in the context of a proposition has a name meaning" (TLP 3.3), he provides the reader with the two conditions for sensical language. First, the structure of the proposition must conform to the constraints of logical form, and second, the elements of the proposition must have reference (Bedeutung). These conditions have far-reaching implications. The analysis must culminate with a name being a primitive symbol for a (simple) object. Moreover, logic itself gives us the structure and limits of what can be said at all. "The general form of a proposition is: This is how things stand" (TLP 4.5) and every proposition is either true or false. This bi-polarity of propositions enables the composition of more complex propositions from atomic ones by using truth-functional operators (5). Wittgenstein supplies, in the Tractatus, a vivid presentation of Frege's logic in the form of what has become known as 'truth-tables.' This provides the means to go back and analyze all propositions into their atomic parts, since "every statement about complexes can be analyzed into a statement about their constituent parts, and into those propositions which completely describe the complexes" (TLP 2.0201). He delves even deeper by then providing the general form of a truth-function (6). This form, $[\bar{p}, \bar{\xi}, N(\bar{\xi})]$, makes use of one formal operation $(N(\bar{\xi}))$and one propositional variable $(\bar{p})$ to represent Wittgenstein's claim that any proposition "is the result of successive applications" of logical operations to elementary propositions. Having developed this analysis of world-thought-language, and relying on the one general form of the proposition, Wittgenstein can now assert that all meaningful propositions are of equal value. Subsequently, he ends the journey with the admonition concerning what can (or cannot) and what should (or should not) be said (7), leaving outside the realm of the sayable propositions of ethics, aesthetics, and metaphysics. ### 2.2 Sense and Nonsense In the Tractatus Wittgenstein's logical construction of a philosophical system has a purpose--to find the limits of world, thought, and language; in other words, to distinguish between sense and nonsense. "The book will ... draw a limit to thinking, or rather--not to thinking, but to the expression of thoughts .... The limit can ... only be drawn in language and what lies on the other side of the limit will be simply nonsense" (TLP Preface). The conditions for a proposition's having sense have been explored and seen to rest on the possibility of representation or picturing. Names must have a bedeutung (reference/meaning), but they can only do so in the context of a proposition which is held together by logical form. It follows that only factual states of affairs which can be pictured can be represented by meaningful propositions. This means that what can be said are only propositions of natural science and leaves out of the realm of sense a daunting number of statements which are made and used in language. There are, first, the propositions of logic itself. These do not represent states of affairs, and the logical constants do not stand for objects. "My fundamental thought is that the logical constants do not represent. That the logic of the facts cannot be represented" (TLP 4.0312). This is not a happenstance thought; it is fundamental precisely because the limits of sense rest on logic. Tautologies and contradictions, the propositions of logic, are the limits of language and thought, and thereby the limits of the world. Obviously, then, they do not picture anything and do not, therefore, have sense. They are, in Wittgenstein's terms, senseless (sinnlos). Propositions which do have sense are bipolar; they range within the truth-conditions drawn by the truth-tables. But the propositions of logic themselves are "not pictures of the reality ... for the one allows every possible state of affairs, the other none" (TLP 4.462). Indeed, tautologies (and contradictions), being senseless, are recognized as true (or false) "in the symbol alone ... and this fact contains in itself the whole philosophy of logic" (TLP 6.113). The characteristic of being senseless applies not only to the propositions of logic but also to mathematics or the pictorial form itself of the pictures that do represent. These are, like tautologies and contradictions, literally sense-less, they have no sense. Beyond, or aside from, senseless propositions Wittgenstein identifies another group of statements which cannot carry sense: the nonsensical (unsinnig) propositions. Nonsense, as opposed to senselessness, is encountered when a proposition is even more radically devoid of meaning, when it transcends the bounds of sense. Under the label of unsinnig can be found various propositions: "Socrates is identical," but also "1 is a number" and "there are objects." While some nonsensical propositions are blatantly so, others seem to be meaningful--and only analysis carried out in accordance with the picture theory can expose their nonsensicality. Since only what is "in" the world can be described, anything that is "outside" is denied meaningfulness, including the notion of limit and the limit points themselves. Traditional metaphysics, and the propositions of ethics and aesthetics, which try to capture the world as a whole, are also excluded, as is the truth in solipsism, the very notion of a subject, for it is also not "in" the world but at its limit. Wittgenstein does not, however, relegate all that is not inside the bounds of sense to oblivion. He makes a distinction between saying and showing which is made to do additional crucial work. "What can be shown cannot be said," that is, what cannot be formulated in sayable (sensical) propositions can only be shown. This applies, for example, to the logical form of the world, the pictorial form, etc., which show themselves in the form of (contingent) propositions, in the symbolism, and in logical propositions. Even the unsayable (metaphysical, ethical, aesthetic) propositions of philosophy belong in this group--which Wittgenstein finally describes as "things that cannot be put into words. They make themselves manifest. They are what is mystical" (TLP 6.522). ### 2.3 The Nature of Philosophy Accordingly, "the word 'philosophy' must mean something which stands above or below, but not beside the natural sciences" (TLP 4.111). Not surprisingly, then, "most of the propositions and questions to be found in philosophical works are not false but nonsensical" (TLP 4.003). Is, then, philosophy doomed to be nonsense (unsinnig), or, at best, senseless (sinnlos) when it does logic, but, in any case, meaningless? What is left for the philosopher to do, if traditional, or even revolutionary, propositions of metaphysics, epistemology, aesthetics, and ethics cannot be formulated in a sensical manner? The reply to these two questions is found in Wittgenstein's characterization of philosophy: philosophy is not a theory, or a doctrine, but rather an activity. It is an activity of clarification (of thoughts), and more so, of critique (of language). Described by Wittgenstein, it should be the philosopher's routine activity: to react or respond to the traditional philosophers' musings by showing them where they go wrong, using the tools provided by logical analysis. In other words, by showing them that (some of) their propositions are nonsense. "All propositions are of equal value" (TLP 6.4)--that could also be the fundamental thought of the book. For it employs a measure of the value of propositions that is done by logic and the notion of limits. It is here, however, with the constraints on the value of propositions, that the tension in the Tractatus is most strongly felt. It becomes clear that the notions used by the Tractatus--the logical-philosophical notions--do not belong to the world and hence cannot be used to express anything meaningful. Since language, thought, and the world, are all isomorphic, any attempt to say in logic (i.e., in language) "this and this there is in the world, that there is not" is doomed to be a failure, since it would mean that logic has got outside the limits of the world, i.e. of itself. That is to say, the Tractatus has gone over its own limits, and stands in danger of being nonsensical. The "solution" to this tension is found in Wittgenstein's final remarks, where he uses the metaphor of the ladder to express the function of the Tractatus. It is to be used in order to climb on it, in order to "see the world rightly"; but thereafter it must be recognized as nonsense and be thrown away. Hence: "whereof one cannot speak, thereof one must be silent" (7). ### 2.4 Interpretative Problems The Tractatus is notorious for its interpretative difficulties. In the decades that have passed since its publication it has gone through several waves of general interpretations. Beyond exegetical and hermeneutical issues that revolve around particular sections (such as the world/reality distinction, the difference between representing and presenting, the Frege/Russell connection to Wittgenstein, or the influence on Wittgenstein by existentialist philosophy) there are a few fundamental, not unrelated, disagreements that inform the map of interpretation. These revolve around the realism of the Tractatus, the notion of nonsense and its role in reading the Tractatus itself, and the reading of the Tractatus as an ethical tract. There are interpretations that see the Tractatus as espousing realism, i.e., as positing the independent existence of objects, states of affairs, and facts. That this realism is achieved via a linguistic turn is recognized by all (or most) interpreters, but this linguistic perspective does no damage to the basic realism that is seen to start off the Tractatus ("The world is all that is the case") and to run throughout the text ("Objects form the substance of the world" (TLP 2.021)). Such realism is also taken to be manifested in the essential bi-polarity of propositions; likewise, a straightforward reading of the picturing relation posits objects there to be represented by signs. As against these readings, more linguistically oriented interpretations give conceptual priority to the symbolism. When "reality is compared with propositions" (TLP 4.05), it is the form of propositions which determines the shape of reality (and not the other way round). In any case, the issue of realism (vs. anti-realism) in the Tractatus must address the question of the limits of language and the more particular question of what there is (or is not) beyond language. Subsequently, interpreters of the Tractatus have moved on to questioning the very presence of metaphysics within the book and the status of the propositions of the book themselves. 'Nonsense' became the hinge of Wittgensteinian interpretative discussion during the last decade of the 20th century. Beyond the bounds of language lies nonsense--propositions which cannot picture anything--and Wittgenstein bans traditional metaphysics to that area. The quandary arises concerning the question of what it is that inhabits that realm of nonsense, since Wittgenstein does seem to be saying that there is something there to be shown (rather than said) and does, indeed, characterize it as the 'mystical.' The traditional readings of the Tractatus accepted, with varying degrees of discomfort, the existence of that which is unsayable, that which cannot be put into words, the nonsensical. More recent readings tend to take nonsense more seriously as exactly that--nonsense. This also entails taking seriously Wittgenstein's words in 6.54--his famous ladder metaphor--and throwing out the Tractatus itself, including the distinction between what can be said and what can only be shown. The Tractatus, on this stance, does not point at ineffable truths (of, e.g., metaphysics, ethics, aesthetics, etc.), but should lead us away from such temptations. An accompanying discussion must then also deal with how this can be recognized, what this can possibly mean, and how it should be used, if at all. This discussion is closely related to what has come to be called the ethical reading of the Tractatus. Such a reading is based, first, on the supposed discrepancy between Wittgenstein's construction of a world-language system, which takes up the bulk of the Tractatus, and several comments that are made about this construction in the Preface to the book, in its closing remarks, and in a letter he sent to his publisher, Ludwig von Ficker, before publication. In these places, all of which can be viewed as external to the content of the Tractatus, Wittgenstein preaches silence as regards anything that is of importance, including the 'internal' parts of the book which contain, in his own words, "the final solution of the problems [of philosophy]." It is the importance given to the ineffable that can be viewed as an ethical position. "My work consists of two parts, the one presented here plus all that I have not written. And it is precisely this second part that is the important point. For the ethical gets its limit drawn from the inside, as it were, by my book; ... I've managed in my book to put everything firmly into place by being silent about it .... For now I would recommend you to read the preface and the conclusion, because they contain the most direct expression of the point" (ProtoTractatus, p.16). Obviously, such seemingly contradictory tensions within and about a text--written by its author--give rise to interpretative conundrums. There is another issue often debated by interpreters of Wittgenstein, which arises out of the questions above. This has to do with the continuity between the thought of the early and later Wittgenstein. Again, the 'standard' interpretations were originally united in perceiving a clear break between the two distinct stages of Wittgenstein's thought, even when ascertaining some developmental continuity between them. And again, the more recent interpretations challenge this standard, emphasizing that the fundamental therapeutic motivation clearly found in the later Wittgenstein should also be attributed to the early. ## 3. The Later Wittgenstein ### 3.1 Transition and Critique of Tractatus The idea that philosophy is not a doctrine, and hence should not be approached dogmatically, is one of the most important insights of the Tractatus. Yet, as early as 1931, Wittgenstein referred to his own early work as 'dogmatic' ("On Dogmatism" in VC, p. 182). Wittgenstein used this term to designate any conception which allows for a gap between question and answer, such that the answer to the question could be found at a later date. The complex edifice of the Tractatus is built on the assumption that the task of logical analysis was to discover the elementary propositions, whose form was not yet known. What marks the transition from early to later Wittgenstein can be summed up as the total rejection of dogmatism, i.e., as the working out of all the consequences of this rejection. The move from the realm of logic to that of the grammar of ordinary language as the center of the philosopher's attention; from an emphasis on definition and analysis to 'family resemblance' and 'language-games'; and from systematic philosophical writing to an aphoristic style--all have to do with this transition towards anti-dogmatism in its extreme. It is in the Philosophical Investigations that the working out of the transitions comes to culmination. Other writings of the same period, though, manifest the same anti-dogmatic stance, as it is applied, e.g., to the philosophy of mathematics or to philosophical psychology. ### 3.2 Philosophical Investigations Philosophical Investigations was published posthumously in 1953. It was edited by G. E. M. Anscombe and Rush Rhees and translated by Anscombe. It comprised two parts. Part I, consisting of 693 numbered paragraphs, was ready for printing in 1946, but rescinded from the publisher by Wittgenstein. Part II was added on by the editors, trustees of his Nachlass. In 2009 a new edited translation, by P. M. S. Hacker and Joachim Schulte, was published; Part II of the earlier translation, now recognized as an essentially separate entity, was here labeled "Philosophy of Psychology - A Fragment" (PPF). In the Preface to PI, Wittgenstein states that his new thoughts would be better understood by contrast with and against the background of his old thoughts, those in the Tractatus; and indeed, most of Part I of PI is essentially questioning. Its new insights can be understood as primarily exposing fallacies in the traditional way of thinking about language, truth, thought, intentionality, and, perhaps mainly, philosophy. In this sense, it is conceived of as a therapeutic work, viewing philosophy itself as therapy. (Part II (PPF), focusing on philosophical psychology, perception etc., was different, pointing to new perspectives (which, undoubtedly, are not disconnected from the earlier critique) in addressing specific philosophical issues. It is, therefore, more easily read alongside Wittgenstein's other writings of the later period.) PI begins with a quote from Augustine's Confessions which "give us a particular picture of the essence of human language," based on the idea that "the words in language name objects," and that "sentences are combinations of such names" (PI 1). This picture of language cannot be relied on as a basis for metaphysical, epistemic or linguistic speculation. Despite its plausibility, this reduction of language to representation cannot do justice to the whole of human language; and even if it is to be considered a picture of only the representative function of human language, it is, as such, a poor picture. Furthermore, this picture of language is at the base of the whole of traditional philosophy, but, for Wittgenstein, it is to be shunned in favor of a new way of looking at both language and philosophy. The Philosophical Investigations proceeds to offer the new way of looking at language, which will yield the view of philosophy as therapy. ### 3.3 Meaning as Use "For a large class of cases of the employment of the word 'meaning'--though not for all--this word can be explained in this way: the meaning of a word is its use in the language" (PI 43). This basic statement is what underlies the change of perspective most typical of the later phase of Wittgenstein's thought: a change from a conception of meaning as representation to a view which looks to use as the crux of the investigation. Traditional theories of meaning in the history of philosophy were intent on pointing to something exterior to the proposition which endows it with sense. This 'something' could generally be located either in an objective space, or inside the mind as mental representation. As early as 1933 (The Blue Book) Wittgenstein took pains to challenge these conceptions, arriving at the insight that "if we had to name anything which is the life of the sign, we should have to say that it was its use" (BB 4). Ascertainment of the use (of a word, of a proposition), however, is not given to any sort of constructive theory building, as in the Tractatus. Rather, when investigating meaning, the philosopher must "look and see" the variety of uses to which the word is put. An analogy with tools sheds light on the nature of words. When we think of tools in a toolbox, we do not fail to see their variety; but the "functions of words are as diverse as the functions of these objects" (PI 11). We are misled by the uniform appearance of our words into theorizing upon meaning: "Especially when we are doing philosophy!" (PI 12) So different is this new perspective that Wittgenstein repeats: "Don't think, but look!" (PI 66); and such looking is done vis a vis particular cases, not generalizations. In giving the meaning of a word, any explanatory generalization should be replaced by a description of use. The traditional idea that a proposition houses a content and has a restricted number of Fregean forces (such as assertion, question, and command), gives way to an emphasis on the diversity of uses. In order to address the countless multiplicity of uses, their un-fixedness, and their being part of an activity, Wittgenstein introduces the key concept of 'language-game.' He never explicitly defines it since, as opposed to the earlier 'picture,' for instance, this new concept is made to do work for a more fluid, more diversified, and more activity-oriented perspective on language. Hence, indeed, the requirement to define harkens back to an old dogma, which misses the playful and active character of language. ### 3.4 Language-games and Family Resemblance Throughout the Philosophical Investigations, Wittgenstein returns, again and again, to the concept of language-games to make clear his lines of thought concerning language. Primitive language-games are scrutinized for the insights they afford on this or that characteristic of language. Thus, the builders' language-game (PI 2), in which a builder and his assistant use exactly four terms (block, pillar, slab, beam), is utilized to illustrate that part of the Augustinian picture of language which might be correct but which is, nevertheless, strictly limited because it ignores the essential role of action in establishing meaning. 'Regular' language-games, such as the astonishing list provided in PI 23 (which includes, e.g., reporting an event, speculating about an event, forming and testing a hypothesis, making up a story, reading it, play-acting, singing catches, guessing riddles, making a joke, translating, asking, thanking, and so on), bring out the openness of our possibilities in using language and in describing it. Language-games are, first, a part of a broader context termed by Wittgenstein a form of life (see below). Secondly, the concept of language-games points at the rule-governed character of language. This does not entail strict and definite systems of rules for each and every language-game, but points to the conventional nature of this sort of human activity. Still, just as we cannot give a final, essential definition of 'game,' so we cannot find "what is common to all these activities and what makes them into language or parts of language" (PI 65). It is here that Wittgenstein's rejection of general explanations, and definitions based on sufficient and necessary conditions, is best pronounced. Instead of these symptoms of the philosopher's "craving for generality," he points to 'family resemblance' as the more suitable analogy for the means of connecting particular uses of the same word. There is no reason to look, as we have done traditionally--and dogmatically--for one, essential core in which the meaning of a word is located and which is, therefore, common to all uses of that word. We should, instead, travel with the word's uses through "a complicated network of similarities overlapping and criss-crossing" (PI 66). Family resemblance also serves to exhibit the lack of boundaries and the distance from exactness that characterize different uses of the same concept. Such boundaries and exactness are the definitive traits of form--be it Platonic form, Aristotelian form, or the general form of a proposition adumbrated in the Tractatus. It is from such forms that applications of concepts can be deduced, but this is precisely what Wittgenstein now eschews in favor of appeal to similarity of a kind with family resemblance. ### 3.5 Rule-following and Private Language One of the issues most associated with the later Wittgenstein is that of rule-following. Rising out of the considerations above, it becomes another central point of discussion in the question of what it is that can apply to all the uses of a word. The same dogmatic stance as before has it that a rule is an abstract entity--transcending all of its particular applications; knowing the rule involves grasping that abstract entity and thereby knowing how to use it. Wittgenstein begins his exposition by introducing an example: "... we get [a] pupil to continue a series (say '+ 2') beyond 1000--and he writes 1000, 1004, 1008, 1012" (PI 185). What do we do, and what does it mean, when the student, upon being corrected, answers "But I did go on in the same way"? Wittgenstein proceeds (mainly in PI 185-243, but also elsewhere) to dismantle the cluster of attendant questions: How do we learn rules? How do we follow them? Wherefrom the standards which decide if a rule is followed correctly? Are they in the mind, along with a mental representation of the rule? Do we appeal to intuition in their application? Are they socially and publicly taught and enforced? In typical Wittgensteinian fashion, the answers are not pursued positively; rather, the very formulation of the questions as legitimate questions with coherent content is put to the test. For indeed, it is both the Platonistic and mentalistic pictures which underlie asking questions of this type, and Wittgenstein is intent on freeing us from these assumptions. Such liberation involves elimination of the need to posit any sort of external or internal authority beyond the actual applications of the rule. These considerations lead to PI 201, often considered the climax of the issue: "This was our paradox: no course of action could be determined by a rule, because every course of action can be made out to accord with the rule. The answer was: if everything can be made out to accord with the rule, then it can also be made out to conflict with it. And so there would be neither accord nor conflict here." Wittgenstein's formulation of the problem, now at the point of being a "paradox," has given rise to a wealth of interpretation and debate since it is clear to all that this is the crux of the general issue of meaning, and of understanding and using a language. One of the influential readings of the problem of following a rule (introduced by Fogelin 1976 and Kripke 1982) has been the interpretation, according to which Wittgenstein is here voicing a skeptical paradox and offering a skeptical solution. That is to say, there are no facts that determine what counts as following a rule, no real grounds for saying that someone is indeed following a rule, and Wittgenstein accepts this skeptical challenge (by suggesting other conditions that might warrant our asserting that someone is following a rule). This reading has been challenged, in turn, by several interpretations (such as Baker and Hacker 1984, McGinn1984, and Cavell 1990), while others have provided additional, fresh perspectives (e.g., Diamond, "Rules: Looking in the Right Place" in Phillips and Winch 1989, and several in Miller and Wright 2002). Directly following the rule-following sections in PI, and therefore easily thought to be the upshot of the discussion, are those sections called by interpreters "the private-language argument." Whether it be a veritable argument or not (and Wittgenstein never labeled it as such), these sections point out that for an utterance to be meaningful it must be possible in principle to subject it to public standards and criteria of correctness. For this reason, a private-language, in which "words ... are to refer to what only the speaker can know--to his immediate private sensations ..." (PI 243), is not a genuine, meaningful, rule-governed language. The signs in language can only function when there is a possibility of judging the correctness of their use, "so the use of [a] word stands in need of a justification which everybody understands" (PI 261). ### 3.6 Grammar and Form of Life Grammar, usually taken to consist of the rules of correct syntactic and semantic usage, becomes, in Wittgenstein's hands, the wider--and more elusive--notion which captures the essence of language as a special rule-governed activity. This notion replaces the stricter and purer logic, which played such an essential role in the Tractatus in providing a scaffolding for language and the world. Indeed, "Essence is expressed in grammar ... Grammar tells what kind of object anything is. (Theology as grammar)" (PI 371, 373). As opposed to grammar-book rules, the "rules" of grammar are not technical instructions from on-high for correct usage and are not idealized as an external system to be conformed to, independently of context. Therefore, they are not appealed to explicitly in any formulation, but are only used in cases of philosophical perplexity to clarify where language misleads us into false illusions. Thus, for example, "I can know what someone else is thinking, not what I am thinking. It is correct to say 'I know what you are thinking,' and wrong to say 'I know what I am thinking.' (A whole cloud of philosophy condensed into a drop of grammar.)" (Philosophical Investigations 1953, p.222). In this example, being sensitive to the grammatical uniqueness of first-person avowals saves us from the blunders of foundational epistemology. Grammar is hence situated within the regular activity with which language-games are interwoven: "... the word 'language-game' is used here to emphasize the fact that the speaking of language is part of an activity, or of a form of life" (PI 23). What enables language to function and therefore must be accepted as "given" are precisely forms of life. In Wittgenstein's terms, "It is not only agreement in definitions but also (odd as it may sound) in judgments that is required" (PI 242), and this is "agreement not in opinions, but rather in form of life" (PI 241). Used by Wittgenstein sparingly--five times in the Investigations--this concept has given rise to interpretative quandaries and subsequent contradictory readings. Forms of life can be understood as constantly changing and contingent, dependent on culture, context, history, etc.; or as a background common to humankind, "shared human behavior" which is "the system of reference by means of which we interpret an unknown language" (PI 206); or as a notion which can be read differently in different cases - sometimes as relativistic, in other cases as expressing a more universalistic approach. ### 3.7 The Nature of Philosophy In his later writings Wittgenstein holds, as he did in the Tractatus, that philosophers do not--or should not--supply a theory, neither do they provide explanations. "Philosophy just puts everything before us, and neither explains nor deduces anything.--Since everything lies open to view there is nothing to explain" (PI 126). The anti-theoretical stance is reminiscent of the early Wittgenstein, but there are manifest differences. Although the Tractatus precludes philosophical theories, it does construct a systematic edifice which results in the general form of the proposition, all the while relying on strict formal logic; the Investigations points out the therapeutic non-dogmatic nature of philosophy, verily instructing philosophers in the ways of therapy. "The work of the philosopher consists in marshalling reminders for a particular purpose" (PI 127). Working with reminders and series of examples, different problems are solved. Unlike the Tractatus which advanced one philosophical method, in the Investigations "there is not a single philosophical method, though there are indeed methods, different therapies, as it were" (PI 133d). This is directly related to Wittgenstein's eschewal of the logical form or of any a-priori generalization that can be discovered or made in philosophy. Trying to advance such general theses is a temptation which lures philosophers; but the real task of philosophy is both to make us aware of the temptation and to show us how to overcome it. Consequently "a philosophical problem has the form: 'I don't know my way about.'" (PI 123), and hence the aim of philosophy is "to show the fly the way out of the fly-bottle" (PI 309). The refusal of theory goes hand in hand with Wittgenstein's objection to have "anything hypothetical in our [philosophical] considerations" (PI 109): "All explanation must disappear, and description alone must take its place." In The Blue Book, Wittgenstein explained the difficulty encountered by philosophers to work on a minor scale and to take seriously "the particular case" as originating from their "craving for generality." This craving is strongly influenced by philosophers' "preoccupation with the method of science," a reductive and unifying tendency which is "the real source of metaphysics" (BB 18). Wittgenstein's determined anti-scientism should not be read as an opposition to science in itself; it is the insistence that philosophy and science are to be kept apart - that (contrary to the prevalent attitude of modern civilization) the scientific framework is not appropriate everywhere, and certainly not for philosophical investigations. The style of the Investigations is strikingly different from that of the Tractatus. Instead of strictly numbered sections which are organized hierarchically in programmatic order, the Investigations fragmentarily voices aphorisms about language-games, family resemblance, forms of life, "sometimes jumping, in a sudden change, from one area to another" (PI Preface). This variation in style is of course essential and is "connected with the very nature of the investigation" (PI Preface). As a matter of fact, Wittgenstein was acutely aware of the contrast between the two stages of his thought, suggesting publication of both texts together in order to make the contrast obvious and clear. Still, it is precisely via the subject of the nature of philosophy that the fundamental continuity between these two stages, rather than the discrepancy between them, is to be found. In both cases philosophy serves, first, as critique of language. It is through analyzing language's illusive power that the philosopher can expose the traps of meaningless philosophical formulations. This means that what was formerly thought of as a philosophical problem may now dissolve "and this simply means that the philosophical problems should completely disappear" (PI 133). Two implications of this diagnosis, easily traced back in the Tractatus, are to be recognized. One is the inherent dialogical character of philosophy, which is a responsive activity: difficulties and torments are encountered which are then to be dissipated by philosophical therapy. In the Tractatus, this took the shape of advice: "The correct method in philosophy would really be the following: to say nothing except what can be said, i.e. propositions of natural science ... and then whenever someone else wanted to say something metaphysical, to demonstrate to him that he had failed to give a meaning to certain signs in his propositions" (TLP 6.53) The second, more far- reaching, "discovery" in the Investigations "is the one that enables me to break off philosophizing when I want to" (PI 133). This has been taken to revert back to the ladder metaphor and the injunction to silence in the Tractatus. ### 3.8 Interpretative Problems Whereas the Tractatus has been interpreted in different ways right from its publication, for about two decades since the publication of Philosophical Investigations in 1953 there seemed to be wide agreement on the proper way to read the book (along with the vast material from 1930 onwards in Wittgenstein's Nachlass, gradually released over the years). Wittgenstein's turn from his early emphasis on the role of logical analysis in his philosophical method to the later preference of particular grammatical descriptions was read, quite unanimously, as marking the two clearly distinct - "early" and "later" - phases in his thought. The later phase, it was agreed, consisted mainly of particular descriptions of our ordinary uses of words - especially those words which tend to create the illusion that they represent something profound, words which seem to point at the "essence of language." "Whereas, in fact, if the words 'language,' 'experience,' 'world' have a use, it must be as humble a one as that of the words 'table,' 'lamp,' 'door'." (PI 97) With the publication of Stanley Cavell's Must We Mean What We Say? (1969) and especially The Claim of Reason: Wittgenstein, Skepticism, Morality, and Tragedy (1979), it turned out that such notions as 'description,' 'ordinary,' and 'usage' are not as innocuous as they had seemed. Cavell's reading of Philosophical Investigations suggested a more radical emphasis on the particularity of contexts and blocked the way to any general description of "grammatical rules." For Cavell, Wittgenstein's idea that meanings are determined only within language games, i.e., only in particular cases, is the key to reading his philosophy. When ignored, we are bound to import an outside requirement into our investigation and thus, favoring a distilled and rigid account of our linguistic exchanges, miss (and misrepresent) their vividness: "As if the meaning were an aura the word brings along with it and retains in every kind of use" (PI 117). In 1980, Oxford philosophers G.P. Baker and P.M.S. Hacker launched the first volume of an analytical commentary on Wittgenstein's Investigations. This gigantic project, spanning over 4 volumes (the 3rd and 4th written by Hacker alone), advocated and developed the orthodox reading of Wittgenstein's later work, according to which a systematic mapping - an overview, or Ubersicht (Cf. PI 122) - of the use of our words is not only possible but also necessary, if we wish to dissolve the philosophical problems arising from the traditional philosophical inattention to ordinary use, due to a prior metaphysical requirement. For Baker and Hacker, a key notion for reading Wittgenstein's later philosophy is that of philosophical 'grammar.' Acute attention to grammatical rules of use offers a way to elucidate meanings and thus to expose philosophical blunders. Gradually, these alternatives in reading Philosophical Investigations came to form two schools of interpretation that are broadly parallel to the ones we encounter regarding Wittgenstein's Tractatus. As noted above (in section 2.4), there is, first, the question of the continuity between Wittgenstein's earlier and later writings. When emphasis is given - as in the orthodox reading - to Wittgenstein's switch from logic to grammar, the break between the philosophical stages is easily marked. When, on the contrary, it is the resistance to generalization (of any kind) that is emphasized, then the similarity in Wittgenstein's therapeutic motivation throughout his life is much more clearly seen. As in the previous stage, the debate here has to do with the question of ethics and its centrality in reading Wittgenstein's oeuvre. According to the traditional approach, amplified in Hacker and Baker's interpretive project, Wittgenstein's main objective is to resolve or dissolve general philosophical puzzles - concerning, e.g., the mind, the nature of color, perception, and rule-following - by clarifying, arranging, and contrasting different grammatical rules relevant to the notion examined. This is an intellectual investigation and, apart from its being critical of traditional philosophy and its reflection in contemporary science, it has nothing particularly relevant to ethics and should not be read as an ethical endeavor. Readers taking their cue from Cavell, on the other hand, believe that the gist of the Wittgensteinian effort in toto is ethical, cultural, or even political. They point at the Investigations' motto ("The trouble about progress is that it always looks much greater than it really is" - Nestroy) and Wittgenstein's mention of "the darkness of this time" in the book's Preface; but more substantially, they argue that Wittgenstein's insistence on the ordinary, the practice of every-day living, is ethical through and through. Our lives are saturated by ethics; we are members of a historically and culturally conditioned society; and our linguistic practices cannot but have an important ethical aspect. Ignoring this aspect, according to these interpreters, entails a rejection of what is human in our linguistic exchanges. To be sure, the interpretations' map is much more complicated than what is briefly sketched here. There are philosophers who object to the characterization each side in the above debate attaches to the other. There are those who claim that while there is a genuine dispute about the right interpretation of the Tractatus, the various interpreters of Philosophical Investigations are much closer to one another than they are ready to acknowledge; and there are independent readings of Wittgenstein which draw their clues from both sides and attempt a different path altogether. ## 4. The Middle Wittgenstein It is noteworthy that the conventional conceptualization of two Wittgensteins, originally thought to be clearly demarcated between the early and later Wittgensteins, has been repeatedly re-worked in the ensuing interpretative project of understanding the philosopher's writings, thoughts, and ideas, with no show of subsiding. One of the natural outcomes of such intense investigation is the awareness that, indeed, understanding Wittgenstein means recognizing other times and other developments, such as before the early Wittgenstein (of the Tractatus), between the early Wittgenstein and the later Wittgenstein (of Philosophical Investigations), and following that later Wittgenstein. The middle Wittgenstein, he of the period between the early Wittgenstein and the later Wittgenstein, was, early on, identified as worthy of exposure and more interpretative work, but such a venture was clearly a function of two mutually impactful grand questions: What is the relationship between the early and later Wittgenstein (as adumbrated in 2.4 and 3.8) and, following upon answers to that question, what is then the more exact time-frame that is worthy of the independent label "middle"? Thus, there is presently much more work being done on the middle Wittgenstein as the questions of the relationship between the early and the later Wittgenstein have taken center-stage and become a constant in Wittgenstein scholarship. How, then, to best demarcate the middle Wittgenstein? The wide date recognition, all along from 1929 (or even earlier) to 1944 (the time when Wittgenstein began the final revision of PI) might seem too obvious, though it can house readings of the continuity-between-early-and-later-Wittgenstein school persuasively. Such is work by, for instance, Joachim Schulte (1992), who notes a middle Wittgenstein already in the TLP leading all the way to later Wittgenstein's key treatments of grammar or mathematics. The opposite extreme, of pinpointing one point in time as expressing a sole significant transformation, seems too summary. This strategy is adopted, for example, by the Hintikkas (1986b) regarding the year 1929 or by Kienzler (1997) concerning a transitional break in 1931. Then there are the various more moderate determinations, which adopt a time-span of a few years, usually sometime between 1929 and 1935, explaining their delineations by philosophical interpretations of the texts and contexts of those times. Such are: O. K. Bouwsma (1961), who points to the period of the Blue and Brown Books (1933-35); P. M. S. Hacker (1986), who follows a two year period - 1929-1930 - as the middle Wittgenstein "bridge"; Alois Pichler (2004), who recognizes a middle Wittgenstein in 1930-1935 based on Wittgensteinian styles; or David Stern (1991), who briefly gives the demarcation of "the late 1920s and early 1930s." If we adopt that last marker of dates, the late 1920s to the early (or mid-)1930s, we can address, in general, the objects of interpretation that have provided the basis for a middle Wittgenstein. Although everything in the Nachlass is now digitally available to the aspiring interpreter, the published versions of typescripts/manuscripts of talks and notes (sometimes dictated to students) have been the evidence upon which middle Wittgenstein interpretation has flourished. These are (in chronological order of their occurrence, not their publication) "Some Remarks on Logical Form" (1929), "A Lecture on Ethics" (1929), Philosophical Remarks (1929-1930), Philosophical Grammar (1932-33), The Big Typescript (1932-1933); the Blue and Brown Books (1933-1935), and "Notes for Lectures on 'Private Experience' and 'Sense Data'" (1935-36). Just as illuminating, even tantalizing, are the publications of notes taken by participants in Wittgenstein's classes in those philosophically eventful years, since it was in those surroundings that Wittgenstein was working out - live - the deliberations and moves into what would become the later Wittgenstein. Supplied by Alice Ambrose, Desmond Lee, G. E. Moore, Rush Rhees, and Friedrich Waismann (on Wittgenstein's conversations with members of the Vienna Circle), these are volumes replete with questions of philosophy, meta-philosophy, method, and style that provide the Wittgensteinian interpreter with voluminous raw data in order to characterize a middle Wittgenstein. The themes that populate the texts above are wide and varied. Some harken back to Tractarian refrains, more appear to introduce later Wittgensteinian terms; a number of subjects make transitory appearances, never to return. Phenomenology, for one, is a constant Wittgensteinian concern, mostly for interpreters of all Wittgensteins who bicker over the very ascription of phenomenology to Wittgenstein's thought, accompanied by their attempt to define Wittgenstein's use of the term. But it is in the middle Wittgenstein that phenomenology makes a literal appearance, specifically in the "Phenomenology is Grammar" chapter of The Big Typescript. The questions then become whether phenomenology can be ascribed to the early Wittgenstein, whether it was really only adopted in the later Wittgenstein, and most relevant here - whether it can be identified as either vacating the scene in the middle Wittgenstein or, contrarily, there becoming terminologically and therefore significantly present. There is also the option that phenomenology appears exclusively in the middle Wittgenstein. These last options may then be seen as (at least partly) definitive of a middle period. Mathematics, and Wittgenstein's original thoughts on it, can undoubtedly be seen as another investigative anchor of middle Wittgenstein. Although it is clear that mathematics was of great interest to him in TLP, his thoughts on it became more enigmatic precisely in the middle period. Several "isms" are attributed to his view(s) on mathematics here - verificationism, formalism, and finitism, along with recognition of a "calculus conception" of mathematics. (Intuitionism, a favorite label for interpreters of Wittgenstein's mathematics, is only identified in the Remarks on the Foundations of Mathematics (1937-1944) and after, i.e., in the later Wittgenstein.) It appears, however, that middle Wittgenstein's dealings with mathematics in this period are, as claimed by Juliet Floyd (2005), constantly changing - and even that, not in any consistent developmental way but in "piecemeal and uneven" fashion. Indeed, along with written evidence of Wittgenstein's influence on and by mathematicians, we can identify the work on mathematics done particularly by the later Wittgenstein and even he of the post-Investigations period, as a substantial overturning of his early and middle conceptions of mathematics. While phenomenology and mathematics stand out as almost quirky (and perhaps suspiciously unclear) subjects making their idiosyncratic appearance in the middle Wittgenstein, the more evident Wittgensteinian constructs of this period are, of course, logic, language, and method. Unsurprisingly, "Some Remarks on Logical Form" in 1929 heralds the transition from early to later Wittgenstein, moving from (ideal, formal) logic to (the rules of use of) grammar. Language is also viewed as the connecting factor between the early and later Wittgenstein, either by discovering still "representational" aspects in descriptions of its use or by deciphering roots of the private language arguments in this Wittgenstein's musings. Other times, analysis of linguistic phenomena as language games rather than tools of logic provides what we have always acknowledged as paradigmatic transitions from early to later Wittgenstein. These all involve methodology - Wittgenstein espousing a method of doing philosophy that is different from that of TLP, but still not the plurality of methods proposed in PI. Furthermore, the method proposed is the turn to grammar as a back turning on dogmatism (and/of metaphysics). David Stern calls the early Wittgenstein's views on logic, language, and method "logical atomism" and the later Wittgenstein's "practical holism," tagging the middle Wittgenstein as "logical holism." Between logical atomism and practical holism seems to lie a drastic divide, but seeing logical holism as bridging the divide gives us the perception - and understanding - of a natural development between them. As spelled out by Mauro Engelmann (2013), this development can then also encompass all the above elements in a comprehensive transition gesturing towards an "anthropological" view. ## 5. After the Investigations It has been submitted that the writings of the period from 1946 until his death (1951) constitute a distinctive phase of Wittgenstein's thought. These writings include, in addition to the second part of the first edition of the Philosophical Investigations (PPF), texts edited and collected in volumes such as Remarks on Colour, Remarks on the Philosophy of Psychology, Zettel, On Certainty, and parts of Remarks on the Foundations of Mathematics. Part II of the Philosophical Investigations was, in fact, an item of contention ab initio. In Anscombe and Rhees's first edition of 1953 (translated by Anscombe), reasons were given for including it, admittedly as a differentiated part, in the book. In Hacker and Schulte's (translated and edited) fourth edition of 2009, its position was contrarily explained. It now became, instead of Part II, an almost independent entity - Philosophy of Psychology - A Fragment (PPF). It is also emphasized that it was a "work in progress." Be that as it may, that installment of a post-Philosophical Investigations text exhibits all the vagaries of Wittgensteinian interpretation - behooving philosophers to attend to questions of placement (which Wittgenstein is here represented, middle, later, or other?), affinity (how do these comments relate to Wittgenstein's later work in general?), and topic (what is Wittgenstein dealing with here, psychology, epistemology, or the method of philosophy?). PPF is the locus classicus of a key Wittgensteinian term - "seeing aspects" (PPF xi), where "two uses of the word 'see'" are elaborated. The second use, where one "sees" a likeness in two objects, is the one that has given rise to the question of aspect perception and the attendant phenomena of aspect-dawning and change of aspect. "I observe a face, and then suddenly notice its likeness to another. I see that it has not changed; and yet I see it differently. I call this experience 'noticing an aspect'" (113). Aspect seeing involves noticing something about an object - an aspect of the object - that one hadn't noticed before and thereby seeing it as something different. Importantly, it also arises as a result of a change of context of our perceptions. This immensely insightful discovery by Wittgenstein, and its successive development, has been the source of a multitude of discussions dealing with questions of objectivity vs. subjectivity, conception vs. perception, and psychology vs. epistemology. It also highlights the move from dogmatic, formalistic universalism to open, humanistic context-laden behavior, aptly reverberating in the to-and-fro of seeing aspects. On Certainty is generally accepted as a work of epistemology, as opposed to PPF, which is usually recognized as dealing with psychology (though even that distinction is called into question by some interpretations). It tackles skeptical doubts and foundational solutions but is, in typical Wittgensteinian fashion, a work of therapy which discounts presuppositions common to both. As always before in the interpretative game, the general view sees Wittgenstein as landing on the non-skeptical side of the epistemological debate, choosing instead to peruse "hinge propositions." These are propositions about which doubt cannot be entertained, but whether this be due to their being epistemically foundational, naturally certain, or logically unavailable to doubt is still a matter for philosophical explanation; as is the question of Wittgenstein's object of critique - Cartesian or radical skepticism - or, in some quarters, the very assumption of his unequivocal anti-skepticism. This is intimately related to another of On Certainty's themes--the primacy of the deed to the word, or, in Wittgenstein's PI terminology, of form of life to logos. Such a clearly delineated philosophical phase has garnered the recognition of a "third Wittgenstein" (initially by Daniele Moyal-Sharrock). Remarks on Colour is a collection of remarks, composed by Wittgenstein during the last year and a half of his life. In these remarks he attempts to clarify the language we use in describing our experience and impression of colours, sameness of colours, saturated colours, light and shade, brightness and opaqueness of surfaces, transparency, etc. These reflections present us with a "geometry of colours," though Wittgenstein makes it clear that due to "the indefiniteness in the concept of colour" (III-78) the "logic of colour concepts" (I-22) should be given piecemeal. When we consider colours "there is merely an inability to bring the concepts into some kind of order" (II-20). Philosophical attention should be given rather to such particularities as the blending in of white (as opposed to blending in yellow), "the coloured intermediary between two colours" (III-49), " the difference between black-red-gold and black-red-yellow" (III-51), etc. The investigation is therefore directed not merely to colour phenomena and the language we use to describe them but also to philosophical methodology, in comparison with scientific inquiry, and to what is taken to be conceivable and what is not. It is also unique in reminding us of a hint, by Wittgenstein, that it was prompted by Johann Wolfgang von Goethe's Zur Farbenlehre (On the Theory of Colour, 1810)! The general tenor of all the writings of this last period can thence be viewed as, on the one hand, a move away from the critical (some would say destructive) positions of the Investigations to a more positive perspective on the same problems that had been facing him since his early writings; on the other hand, this move does not necessarily bespeak a break from the later period but might be more properly viewed as its continuation, in a new light. In other words, the grand question of interpreting Wittgenstein, i.e., the question of continuities or breaks, remains at the forefront of understanding Wittgenstein.
wittgenstein-aesthetics	## 1. The Critique of Traditional Aesthetics Wittgenstein's opening remark is double-barreled: he states that the field of aesthetics is both very big and entirely misunderstood. By "very big", I believe he means both that the aesthetic dimension weaves itself through all of philosophy in the manner suggested above, and that the reach of the aesthetic in human affairs is very much greater than the far more restricted reach of the artistic; the world is densely packed with manifestations of the aesthetic sense or aesthetic interest, while the number of works of art is very much smaller. There is good reason, in his discussions that follow, to believe that he intends that any comprehensive account of the aesthetic would acknowledge the former, and not just - as a good number of philosophical accounts have so restricted themselves - the latter. By "entirely misunderstood", it emerges that he means both (1) that aesthetic questions are of a conceptual type very distinct from empirical questions and the kind of answer, or conceptual satisfaction, we want is very unlike what we might get from an experiment in empirical psychology, and (2) that the philosophically traditional method of essentialistic definition - determining the essence that all members of the class "works of art" exhibit and by virtue of which they are so classified - will conceal from our view more than it reveals. ### 1.1 Properties and Essence It is also vividly apparent from the outset of these lectures that Wittgenstein is urging a heightened vigilance to the myriad ways in which words can, on their grammatical surface, mislead. If, right at the beginning of the inquiry, we see that we use the word "beautiful" as an adjective, we may well very shortly find ourselves asking what is the essence of the property beauty that this particular example exhibits. Imagining a book of philosophy investigating parts of speech (but very many more parts than we find in an ordinary grammar book) where we would give very detailed and nuanced attention to "seeing", "feeling", along with other verbs of personal experience, and equally lengthy studies of "all", "any", "some", numerals, first person pronoun usages, and so forth, he suggests that such a book would, with sufficient attention paid to the contextual intricacies of the grammar of each usage, lay bare the confusion into which language can lead us. Later, in his Philosophical Investigations (Wittgenstein 1958), he will go on to famously develop the analogy between tools and language as a way of breaking the hold of the conceptual picture that words work in one way (by naming things--including the naming of properties, as in the way we too-quickly think of the problem of beauty above), showing the diversity of kind and of use among the various things we find in the tool box (e.g. hammer, glue, chisel, matches). If we redirect our attention, away from the idee fixe of the puzzle concerning the common property named by the word "beauty" or the description "beautiful", and look to the actual use to which our aesthetic-critical vocabulary is put, we will see that it is not some intrinsic meaning carried internally by the linguistic sign (Wittgenstein 1958a) that makes the word in question function as an aesthetic or critical interjection or expression of approval. We will, rather, be able to focus our redirected attention on what actually does make the word in question function aesthetically, i.e. "on the enormously complicated situation in which the aesthetic expression has a place", and he adds that, with such an enlarged vision, we see that "the expression itself has almost a negligible place" (p. 2). He here mentions (seeing aesthetic issues as interwoven with the rest of philosophy) that if he had to identify the main mistake made in philosophical work of his generation, it would be precisely that of, when looking at language, focusing on the form of words, and not the use made of the form of words. He will go on to imply, if not quite to directly assert, that the parallel holds to the work of art: to see it within a larger frame of reference, to see it in comparison to other works of the artist in question and to see it juxtaposed with still other works from its cultural context, is to see what role itplayed in the dialogically unfolding artistic "language-game"[1] of its time and place. In using language, he says next in the lectures, in understanding each other--and in mastering a language initially--we do not start with a small set of words or a single word, but rather from specific occasions and activities. Our aesthetic engagements are occasions and activities of just this kind; thus aesthetics, as a field of conceptual inquiry, should start not from a presumption that the central task is to analyze the determinant properties that are named by aesthetic predicates, but rather with a full-blooded consideration of the activities of aesthetic life. ### 1.2 Predicates and Rules But the adjectival form of many--not all--critical predicates quickly reinforces the "property-with-name" model, and against this Wittgenstein places examples from musical and poetical criticism, where we simply call attention to the rightness of a transition or to the precision or aptness of an image. And it is here that Wittgenstein reminds us that descriptions such as "stately", "pompous", or "melancholy" (where the latter is said of a Schubert piece) are like giving the work a face (Shiner 1978), or we could instead (or in further specification of such descriptions) use gestures.[2] In cases of re-construing a work (e.g. the meter of a poem) so that we understand its rhythm and structure anew, we make gestures, facial expressions, and non-descriptive-predicate based remarks, where aesthetic adjectives play a diminished role or no role at all. And we show our approval of a tailor's work not by describing the suit, but by wearing it. Occasions and activities are fundamental, descriptive language secondary. Wittgenstein here turns to the subject of rules, and rule-following, in aesthetic decision-making. This stems from his reflection on the word "correct" in aesthetic discourse, and he mentions the case of being drilled in harmony and counterpoint. Such rule-learning, he claims, allows what he calls an interpretation of the rules in particular cases, where an increasingly refined judgment comes from an increasingly refined mastery of the rules in question. It is of particular interest that a qualification is entered here, that such rules in contexts of artistic creativity and aesthetic judgment, "may be extremely explicit and taught, or not formulated at all" (Wittgenstein 1966). This itself strongly suggests that, in this way too, actions come first, where these actions may (to invoke Kant's famous distinction) either explicitly follow from the rule, or stand in accordance with them but (for Wittgenstein) in an inexplicit way. The mastery of a practice can be, but need not be, characterized as a cognitive matter of rule-following. Here we thus find one of the points of intersection between Wittgenstein's work in aesthetics and his work in the philosophy of language: the rule-following considerations (Holtzman and Leich 1981 and McDowell 1998) and the debate concerning non-cognitivism (McDowell 2000) link directly to this discussion. Regrettably, Wittgenstein closes this matter prematurely (claiming that this issue should not come in at this point); the linkage between rule-following in language and in aesthetics is still to this day too little investigated. Yet there is a sense in which Wittgenstein extends the discussion, if only implicitly. In investigating the kinds of things meant by aesthetic "appreciation", he does say that "an appreciator is not shown by the interjections he uses" but rather by his choices, selections, his actions on specific occasions. To describe what appreciation consists in, he claims, would prove an impossibility, for "to describe what it consists in we would have to describe the whole environment" (Wittgenstein 1966, 7), and he returns to the case of the tailor--and thus implicitly to rule-following. Such rules, as we discern them in action ranging on a continuum from the cognitively-explicit and linguistically encapsulated to the non-cognitively implicit and only behaviorally manifested, have a life - have an identity as rules--within, and only within, those larger contexts of engagement, those "whole environment[s]". And those environments, those contexts, those language-games, are not reducible to a unitary kind which we then might analyze for essential properties: "correctness", for example, plays a central role in some cases of aesthetic appreciation and understanding, and it is irrelevant in others, e.g. in garment-cutting versus Beethoven symphonies, or in domestic architecture versus the Gothic cathedral. And he explicitly draws, if too briefly, the analogy that emerges here between ethics and aesthetics: he notes the difference between saying of a person that he behaves well versus saying that a person made a great or profound impression. Indeed, "the entire game is different" (Wittgenstein 1966, 8). ### 1.3 Culture and Complexity The central virtue of these lectures is that Wittgenstein never loses a sense of the complexity of our aesthetic engagements, our language attending and in cases manifesting those engagements, and the contextually embedded nature of the aesthetic actions he is working to elucidate. Nor does he lose a sense of the relation--a relation necessary to the meaning of the aesthetic language we use - between the aesthetically-descriptive expressions we employ within particular contexts and "what we call the culture of the period" (Wittgenstein 1966, 8). Of those aesthetic words, he says, "To describe their use or to describe what you mean by a cultured taste, you have to describe a culture" (Wittgenstein 1966, 8). And again making the relation between his work in the philosophy of language and his work in the philosophy of art explicit (if again too briefly), he adds, "What belongs to a language game is a whole culture" (Wittgenstein 1966, 8). There the link to the irreducible character of rule-following is made again as well: "To describe a set of aesthetic rules fully means really to describe the culture of a period" (Wittgenstein 1966, 8, n. 3). If our aesthetic engagements and the interrelated uses of our aesthetic terms are widely divergent and context-sensitive, so are our aesthetic actions and vocabularies context-sensitive in a larger sense as well: comparing a cultured taste of fin-de-siecle Vienna or early-twentieth-century Cambridge with the Middle Ages, he says--implicitly referring to the radically divergent constellations of meaning-associations from one age to the other, "An entirely different game is played in different ages" (Wittgenstein 1966, p. 8). Of a generic claim made about a person that he or she appreciates a given style or genre of art, Wittgenstein makes it clear that he would not yet know--stated in that generic, case-transcending way--not merely whether it was true or not, but more interestingly and more deeply, what it meant to say this. The word "appreciates" is, like the rest of our aesthetic vocabulary, not detachable from the particular context within which it has its life.[3] If, against this diversity or context-sensitivity, we seek to find and then analyze what all such cases of aesthetic engagement have in common, we might well focus on the word "appreciation". But that word will not have meaning with a bounded determinacy fixed prior to its contextualized use, which means that we would find ourselves, because of this philosophical strategy (the entire field "is very big and entirely misunderstood"), in a double bind: first, we would not know the meaning of the term upon which we were focusing; and second, we would, in trying to locate what all cases of aesthetic engagement have in common, leave out of view what he calls in this connection "an immensely complicated family of cases" (Wittgenstein 1966), blinding ourselves to nuance and complexity in the name of a falsifying neatness and overarching generality. "In order to get clear about aesthetic words you have to describe ways of living. We think we have to talk about aesthetic judgments like 'This is beautiful', but we find that if we have to talk about aesthetic judgments we don't find these words at all, but a word used something like a gesture, accompanying a complicated activity" (Wittgenstein 1966, 11). ## 2. The Critique of Scientism Wittgenstein turns to the idea of a science of aesthetics, an idea for which he has precious little sympathy ("almost too ridiculous for words" [Wittgenstein 1966, 11]). But as is often the case in Wittgenstein's philosophical work, it does not follow from this scornful or dismissive attitude that he has no interest in the etiology of the idea, or in excavating the hidden steps or components of thought that have led some to this idea. In the ensuing discussion he unearths a picture of causation that under-girds the very idea of a scientific explanation of aesthetic judgment or preference. And in working underground in this way, he reveals the analogies to cases of genuine scientific explanation, where the "tracing of a mechanism" just is the process of giving a causal account, i.e. where the observed effect is described as the inevitable result of prior links in the causal chain leading to it. If, to take his example, an architect designs a door and we find ourselves in a state of discontentment because the door, within the larger design of the facade (within its stylistic "language-game", we might say), is too low, we are liable to describe this on the model of scientific explanation. Then, we make a substantive of the discontent, see it as the causal result of the lowness of the door, and in identifying the lowness as the cause, think ourselves able to dislodge the inner entity, the discontent, by raising the door. But this mischaracterizes our aesthetic reactions, or what we might call, by analogy to moral psychology, our aesthetic psychology. ### 2.1 Aesthetic Reactions The true aesthetic reaction--itself rarely described in situ in terms of a proximate cause ("In these cases the word 'cause' is hardly ever used at all" [Wittgenstein 1966, 14]) is far more immediate, and far more intertwined with, and related to, what we see in[4] the work of art in question. "It is a reaction analogous to my taking my hand away from a hot plate" (Wittgenstein 1966). He thus says: > > > To say: "I feel discomfort and know the cause", is > entirely misleading because "know the cause" normally > means something quite different. How misleading it is depends on > whether when you said: "I know the cause", you meant it to > be an explanation or not. "I feel discomfort and know the > cause" makes it sound as if there were two things going on in my > soul--discomfort and knowing the cause (Wittgenstein 1966, > 14). > > But there is, as he next says, a "Why?" to such a case of aesthetic discomfort, if not a cause (on the conventional scientific model). But both the question and its multiform answers will take, indeed, very different forms in different cases. Again, if what he suggested before concerning the significance of context for meaning is right, the very meaning of the "Why?"-question will vary case to case. This is not a weaker thesis concerning variation on the level of inflection, where the underlying structure of the "Why?"-question is causal. No, here again that unifying, model-imposing manner of proceeding would leave out a consideration of the nuances that give the "Why?"-question its determinate sense in the first place. But again, Wittgenstein's fundamental concern here is to point out the great conceptual gulf that separates aesthetic perplexities from the methodology of empirical psychology. To run studies of quantified responses to controlled and isolated aesthetic stimuli, where emergent patterns of preference, response, and judgment are recorded within a given population's sample, is to pass by the true character of the aesthetic issue--the actual puzzlement, such as we feel it, will be conceptual, not empirical. And here again we see a direct link to his work in the philosophy of psychology: the penultimate passage of Part II of Philosophical Investigations (1958, sec. xiv), was "The existence of the experimental method makes us think we have the means of solving the problems which trouble us; though problem and method pass one another by" (Wittgenstein 1958, II, iv, 232). He says, near the close of this part of his lectures on aesthetics, "Aesthetic questions have nothing to do with psychological experiments, but are answered in an entirely different way" (Wittgenstein 1966, 17). A stimulus-response model adapted from scientific psychology--what we might now call the naturalizing of aesthetics--falsifies the genuine complexities of aesthetic psychology through a methodologically enforced reduction to one narrow and unitary conception of aesthetic engagement. For Wittgenstein complexity, and not reduction to unitary essence, is the route to conceptual clarification. Reduction to a simplified model, by contrast, yields only the illusion of clarification in the form of conceptual incarceration ("a picture held us captive").[5] ### 2.2 The "Click" of Coherence Aesthetic satisfaction, for Wittgenstein, is an experience that is only possible within a culture and where the reaction that constitutes aesthetic satisfaction or justification is both more immediate, and vastly larger and more expansive, than any simple mechanistic account could accommodate. It is more immediate in that it is not usually possible to specify in advance the exact conditions required to produce the satisfaction, or, as he discusses it, the "click" when everything falls into place. Such exacting pre-specifications for satisfaction are possible in narrowly restricted empirical cases where, for example, we wait for two pointers in a vision examination to come into a position directly opposite each other. And this, Wittgenstein says, is the kind of simile we repeatedly use, but misleadingly, for in truth "really there is nothing that clicks or that fits anything" (Wittgenstein 1966, 19). The satisfaction is more immediate, then, than the causal-mechanistic model would imply. And it is much broader than the causal-mechanistic model implies as well: there is no direct aesthetic analogue to the matched pointers in the case of a larger and deeper form of aesthetic gratification. Wittgenstein does, of course, allow that there are very narrow, isolated circumstances within a work where we do indeed have such empirical pre-specifiable conditions for satisfaction (e.g. where in a piece we see that we wanted to hear a minor ninth, and not a minor seventh, chord). But, contrary to the empirical-causal account, these will not add up, exhaustively or without remainder, to the experience of aesthetic satisfaction. The problem, to which Wittgenstein repeatedly returns in these lectures, is with the kind of answer we want to aesthetic puzzlement as expressed in a question like "Why do these bars give me such a peculiar impression?" (Wittgenstein 1966, 20) In such cases, statistical results regarding percentages of subjects who report this peculiar impression rather than another one in precisely these harmonic, rhythmic, and melodic circumstances are not so much impossible as just beside the point; the kind of question we have here is not met by such methods. "The sort of explanation one is looking for when one is puzzled by an aesthetic impression is not a causal explanation, not one corroborated by experience or by statistics as to how people react.... This is not what one means or what one is driving at by an investigation into aesthetics" (Wittgenstein 1966, 21). It is all too easy to falsify, under the influence of explanatory models misappropriated from science, the many and varied kinds of things that happen when, aesthetically speaking, everything seems to click or fall into place. ### 2.3 The Charm of Reduction In the next passages of Wittgenstein's lectures he turns to a fairly detailed examination of the distinct charm, for some, of psycho-analytic explanation, and he interweaves this with the distinction between scientific-causal explanation of an action versus motive-based (or personally-generated) explanations of that action. It is easy, but mistaken, to read these passages as simply subject-switching anticipations of his lectures on Freud to follow. His fundamental interest here lies with the powerful charm, a kind of conceptual magnetism, of reductive explanations that promise a brief, compact, and propositionally-encapsulated account of what some much larger field of thought or action "really" is. (He cites the example of boiling Redpath, one of his auditors, down to ashes, etc., and then saying "This is all Redpath really is", adding the remark "Saying this might have a certain charm, but would be misleading to say the least" (Wittgenstein 1966, 24). "To say the least" indeed: the example is striking precisely because one can feel the attraction of an encapsulating and simplifying reduction of the bewildering and monumental complexity of a human being, while at the same time feeling, as a human being, that any such reduction to a few physical elements would hardly capture the essence of a person. This, he observes, is an explanation of the "This is only really this" form. Reductive causal explanations function in just the same way in aesthetics, and this links directly to the problems in philosophical methodology that he adumbrated in the Blue and Brown Books (1958a, 17-19), particularly where he discusses what he there calls "craving for generality" and the attendant "contemptuous attitude toward the particular case". If the paradigm of the sciences (which themselves, as he observes in passing, carry an imprimatur of epistemic prestige and the image of incontrovertibility) is Newtonian mechanics, and we then implant that model under our subsequent thinking about psychology, we will almost immediately arrive at an idea of a science of the mind, where that science would progress through the gradual accumulation of psychological laws. (This would constitute, as he memorably puts it, a "mechanics of the soul" [Wittgenstein 1966, 29]). We then dream of a psychological science of aesthetics, where--although "we'd not thereby have solved what we feel to be aesthetic puzzlement" (Wittgenstein 1966, 29)--we may find ourselves able to predict (borrowing the criterion of predictive power from science) what effect a given line of poetry, or a given musical phrase, may have on a certain person whose reaction patterns we have studied. But aesthetic puzzlement, again, is of a different kind, and--here he takes a major step forward, from the critical, to the constructive, phase of his lectures. He writes, "What we really want, to solve aesthetic puzzlements, is certain comparisons--grouping together of certain cases" (Wittgenstein 1966, 29). Such a method, such an approach, would never so much as occur to us were we to remain both dazzled by the misappropriated model of mechanics and contemptuous of the particular case. ## 3. The Comparative Approach Comparison, and the intricate process of grouping together certain cases--where such comparative juxtaposition usually casts certain significant features of the work or works in question in higher relief, where it leads to the emergence of an organizational gestalt,[6] where it shows the evolution of a style, or where it shows what is strikingly original about a work, among many other things--also focuses our attention on the particular case in another way. In leading our scrutiny to critically and interpretatively relevant particularities, it leads us away from the aesthetically blinding presumption that it is the effect, brought about by the "cause" of that particular work, that matters (so that any minuet that gives a certain feeling or awakens certain images would do as well as any other). ### 3.1 Critical Reasoning These foregoing matters together lead to what is to my mind central to Wittgenstein's thoughts on aesthetics. In observing that on one hearing of a minuet we may get a lot out of it and on the next hearing nothing, he is showing how easy it can be to take conceptual missteps in our aesthetic thought from which it can prove difficult to recover. From such a difference, against how we can all too easily model the matter, it does not follow that what we get out of the minuet is independent of the minuet. That would constitute an imposition of a Cartesian dualism between the two ontologies of mind and matter, but in its aesthetic guise. And that would lead in turn to theories of aesthetic content of a mental kind, where the materials of the art form (materials we would then call "external" materials) only serve to carry the real, internal or non-physical content. Criticism would thus be a process of arguing inferentially from outward evidence back to inward content; creation would be a process of finding outward correlates or carriers for that prior inward content; and artistic intention would be articulated as a full mental pre-conception of the contingently finished (or "externalized", as we would then call it) work. It is no accident that Wittgenstein immediately moves to the analogy to language and our all-too-easy misconception of linguistic meaning, where we make "the mistake of thinking that the meaning or thought is just an accompaniment of the word, and the word doesn't matter" (Wittgenstein 1966, 25). This indeed would prove sufficient to motivate contempt for the particular case in aesthetic considerations, by misleadingly modeling a dualistic vision of art upon an equally misleading dualistic model of language (Hagberg 1995). "The sense of a proposition", he says, "is very similar to the business of an appreciation of art" (Wittgenstein 1966, 29). This might also be called a subtractive model, and Wittgenstein captures this perfectly with a question: "A man may sing a song with expression and without expression. Then why not leave out the song--could you have the expression then?" (Wittgenstein 1966, 32 editorial note and footnote) Such a model corresponds, again, to mind-matter Cartesian dualism, which is a conceptual template that also, as we have just seen above, manifests itself in the philosophy of language where we would picture the thought as an inner event and the external word or sign to which it is arbitrarily attached as that inner event's outward corresponding physicalization. It is no surprise that, in this lecture, he turns directly to false and imprisoning pictures of this dualism in linguistic meaning, and then back to the aesthetic case. When we contemplate the expression of a drawn face, it is deeply misleading--or deeply misled, if the dualistic picture of language stands behind this template-conforming thought--to ask for the expression to be given without the face. "The expression"--now linking this discussion back to the previous causal considerations--"is not an effect of the face". The template of cause-and-effect, and of a dualism of material and expressive content (on the subtractive model), and the construal of the material work of art as a means to the production of a separately-identifiable intangible, experiential end, are all out of place here. All of the examples Wittgenstein gives throughout these lectures combat these pictures, each in their own way. But then Wittgenstein's examples also work in concert: they together argue against a form of aesthetic reductionism that would pretend that our reactions to aesthetic objects are isolable, that they can be isolated as variables within a controlled experiment, that they can be hermetically sealed as the experienced effects of isolatable causes. He discusses our aesthetic reactions to subtle differences between differently drawn faces, and our equally subtle reactions to the height or design of a door (he is known to have had the ceiling in an entire room of the house in Vienna he designed for his sister moved only a few inches when the builders failed to realize his plan with sufficient exactitude). The enormous subtlety, and the enormous complexity, of these reactions, are a part of--and as complicated as--our natural history. He gives as an example the error, or the crudeness, of someone responding to a complaint concerning the depiction of a human smile (specifically, that the smile did not seem genuine), with the reply that, after all, the lips are only parted one one-thousandths of an inch too much: such differences, however small in measure, in truth matter enormously. ### 3.2 Seeing Connections Beneath Wittgenstein's examples, as developed throughout these lectures, lies another interest (which presses its way to the surface and becomes explicit on occasion): he is eager to show the significance of making connections in our perception and understanding of art works--connections between the style of a poet and that of a composer (e.g. Keller and Brahms), between one musical theme and another, between one expressive facial depiction and another, between one period of an artist's work and another. Such connections--we might, reviving a term from first-generation Wittgensteinians, refer to the kind of work undertaken to identify and articulate such connections as "connective analysis"--are, for Wittgenstein, at the heart of aesthetic experience and aesthetic contemplation. And they again are of the kind that reductive causal explanation would systematically miss. In attempting to describe someone's feelings, could we do better than to imitate the way the person actually said the phrase we found emotionally revelatory, Wittgenstein pointedly asks? The disorientation we would feel in trying to describe the person's feeling with subtlety and precision without any possibility of imitating his precise expressive utterance--"the way he said it" - shows how very far the dualistic or subtractive conceptual template is from our human experience, our natural history. Connections, of the kind alluded to here--a web of variously-activated relations between the particular aspect of the work to which we are presently attending and other aspects, other parts of the work, or other works, groups of works, or other artists, genres, styles, or other human experiences in all their particularity--may include what we call associations awakened by the work, but connections are not reducible to them only (and certainly not to undisciplined, random, highly subjective, or free associations).[7] The impossibility of the simplifying subtractive template emerges here as well: "Suppose [someone says]: 'Associations are what matter--change it slightly and it no longer has the same associations'. But can you separate the associations from the picture, and have the same thing?" The answer is clearly, again like the case of the singing with and then without expression above, negative: "You can't say: 'That's just as good as the other: it gives me the same associations'" (Wittgenstein 1966, 34). Here again, Wittgenstein shows the great gulf that separates what we actually do with, what we actually say about, works of art, and how we would speak of them if the conceptual pictures and templates with which he has been doing battle were correct. To extend one of Wittgenstein's examples, we would very much doubt the aesthetic discernment, and indeed the sympathetic imagination and the human connectedness, of a person who said of two poems (each of which reminded him of death) that either will do as well as the other to a bereaved friend, that they would do the same thing (where this is uttered in a manner dismissive of nuance, or as though it is being said of two detergents). Poetry, Wittgenstein is showing, does not play that kind of role in our lives, as the nature and character of our critical verbal interactions about it indicate. And he ascends, momentarily, to a remark that characterizes his underlying philosophical methodology (or one dimension of it) in the philosophy of language that is being put to use here within the context of his lectures on aesthetics: "If someone talks bosh, imagine the case in which it is not bosh. The moment you imagine it, you see at once it is not like that in our case" (Wittgenstein 1966, 34). The gulf that separates what we should say if the generalizing templates were accurate from what we in actual particularized cases do say could further call into question not only the applicability or accuracy, but indeed the very intelligibility, of the language used to express those templates, those explanatory pictures. Wittgenstein leaves that more aggressive, and ultimately more clarifying and conceptually liberating, critique for his work on language and mind in Philosophical Investigations (1958) and other writings, but one can see from these lectures alone how such an aggressive critique might be undertaken. ### 3.3 The Attitude Toward the Work of Art Near the end of his lectures Wittgenstein turns to the question of the attitude we take toward the work of art. He employs the case of seeing the very slight change (of the kind mentioned above) in the depiction of a smile within a picture of a monk looking at a vision of the Virgin Mary. Where the slight and subtle change of line yields a transformation of the smile of the monk from a kindly to an ironic one, our attitude in viewing might similarly change from one in which for some we are almost in prayer to one that would for some be blasphemous, where we are almost leering. He then gives voice to his imagined reductive interlocutor, who says, "Well there you are. It is all in the attitude" (Wittgenstein 1966, 35), where we would then focus, to the exclusion of all of the rest of the intricate, layered, and complex human dimensions of our reactions to works, solely on an analysis of the attitude of the spectator and the isolable causal elements in the work that determine it. But that, again, is only to give voice to a reductive impulse, and in the brief ensuing discussion he shows, once again, that in some cases, an attitude of this kind may emerge as particularly salient. But in other cases, not. And he shows, here intertwining a number of his themes from these lectures, that the very idea of "a description of an attitude" is itself no simple thing. Full-blooded human beings, and not stimulus-response-governed automata, have aesthetic experience, and that experience is as complex a part of our natural history as any other. Wittgenstein ends the lectures discussing a simple heading: "the craving for simplicity" (Wittgenstein 1966, 36). To such a mind, he says, if an explanation is complicated, it is disagreeable for that very reason. A certain kind of mind will insist on seeking out the single, unitary essence of the matter, where--much like Russell's atomistic search for the essence of the logic of language beneath what he regarded as its misleadingly and distractingly variegated surface--the reductive impulse would be given free reign. Wittgenstein's early work in the Tractatus (1961) followed in that vein. But in these lectures, given in 1938, we see a mind well into a transition away from those simplifying templates, those conceptual pictures. Here, examples themselves do a good deal of philosophical work, and their significance is that they give, rather than merely illustrate, the philosophical point at hand. He said, earlier in the lectures, that he is trying to teach a new way of thinking about aesthetics (and indeed about philosophy itself). ## 4. Conclusion The subject, as he said in his opening line, is very big and entirely misunderstood. It is very big in its scope--in the reach of the aesthetic dimension throughout human life. But we can now, at the end of his lectures, see that it is a big subject in other senses too: aesthetics is conceptually expansive in its important linkages to the philosophy of language, to the philosophy of mind, to ethics, and to other areas of philosophy, and it resists encapsulation into a single, unifying problem. It is a multi-faceted, multi-aspected human cultural phenomenon where connections, of diverging kinds, are more in play than causal relations. The form of explanation we find truly satisfying will thus strikingly diverge from the form of explanation in science - the models of explanation in Naturwissenschaften are misapplied in Geisteswissenschaften, and the viewing of the latter through the lens of the former will yield reduction, exclusion, and ultimately distortion. The humanities are thus, for Wittgenstein, in this sense autonomous. All of this, along with the impoverishing and blinding superimposition of conceptual models, templates, and pictures onto the extraordinarily rich world of aesthetic engagement, also now, at the end of his lectures, gives content to what Wittgenstein meant at the beginning with the words "entirely misunderstood". For now, at this stage of Wittgenstein's development, where the complexity-accepting stance of the later Philosophical Investigations (1958) and other work is unearthing and uprooting the philosophical presuppositions of the simplification-seeking earlier work, examples themselves have priority as indispensable instruments in the struggle to free ourselves of misconception in the aesthetic realm. And these examples, given due and detailed attention, will exhibit a context-sensitive particularity that makes generalized pronouncements hovering high above the ground of that detail look otiose, inattentive, or, more bluntly, just a plain falsification of experience. What remains is not, then--and this is an idea Wittgenstein's auditors must themselves have struggled with in those rooms in Cambridge, as many still do today--another theory built upon now stronger foundations, but rather a clear view of our multiform aesthetic practices. Wittgenstein, in his mature, later work, did not generate a theory of language, of mind, or of mathematics. He generated, rather, a vast body of work perhaps united only in its therapeutic and intricately labored search for conceptual clarification. One sees the same philosophical aspiration driving his foray into aesthetics.
wittgenstein-atomism	## 1. Names and Objects The "names" spoken of in the Tractatus are not mere signs (i.e., typographically or phonologically identified inscriptions), but rather signs-together-with-their-meanings -- or "symbols." Being symbols, names are identified and individuated only in the context of significant sentences. A name is "semantically simple" in the sense that its meaning does not depend on the meanings of its orthographic parts, even when those parts are, in other contexts, independently meaningful. So, for example, it would not count against the semantic simplicity of the symbol 'Battle' as it figures in the sentence "Battle commenced" that it contains the orthographic part, "Bat," even though this part has a meaning of its own in other sentential contexts. For Wittgenstein, however, something else does count against this symbol's semantic simplicity, namely, that it is analyzable away in favour of talk of the actions of people, etc. This point suggests that in natural language Tractarian names will be rare and hard to find. Even apparently simple singular terms such as 'Obama,' 'London,' etc., will not be counted as "names" by the strict standards of the Tractatus since they will disappear on further analysis. (Hereafter, 'name' will mean "Tractarian name" unless otherwise indicated.) It is a matter of controversy whether the Tractatus reserves the term 'name' for those semantically simple symbols that refer to particulars, or whether the term comprehends semantically simple symbols of all kinds. Since objects are just the referents of names, this issue goes hand in hand with the question whether objects are one and all particulars or whether they include properties and relations. The former view is defended by Irving Copi (Copi 1958) and Elizabeth Anscombe (Anscombe 1959 [1971, 108 ff]), among others. It is supported by Tractatus 2.0231: "[Material properties] are first presented by propositions -- first formed by the configuration of objects." This might seem to suggest that simple properties are not objects but rather arise from the combining or configuring of objects. The Copi-Anscombe interpretation has been taken to receive further support from Tractatus 3.1432: > > We must not say, "The complex sign 'aRb' > says 'a stands in relation R to > b;'" but we must say, "That > 'a' stands in a certain relation to > 'b' says that aRb." > This has suggested to some commentators that relations are not, strictly speaking, nameable, and so not Tractarian objects (see, for example, Ricketts, 1996, Section III). It may, however, be intended instead simply to bring out the point that Tractarian names are not confined to particulars, but include relations between particulars; so this consideration is less compelling. The opposing view, according to which names include predicates and relational expressions, has been defended by Erik Stenius and Merrill and Jaakko Hintikka, among others (Stenius, 1960, 61-69; Hintikka and Hintikka, 1986, 30-34). It is supported by a Notebooks entry from 1915 in which objects are explicitly said to include properties and relations (NB, 61). It is further buttressed by Wittgenstein's explanation to Desmond Lee (in 1930-1) of Tractatus 2.01: "'Objects' also include relations; a proposition is not two things connected by a relation. 'Thing' and 'relation' are on the same level." (LK, 120). The Anscombe-Copi reading treats the forms of elementary propositions as differing radically from anything we may be familiar with from ordinary -- or even Fregean -- grammar. It thus respects Wittgenstein's warning to Waismann in 1929 that "The logical structure of elementary propositions need not have the slightest similarly with the logical structure of [non-elementary] propositions" (WWK, 42). Going beyond this, Wittgenstein seems once to have held that there can be no resemblance between the apparent or surface forms of non-elementary propositions and the forms of elementary propositions. In "Some Remarks on Logical Form" (1929) he says: "One is often tempted to ask from an a priori standpoint: What, after all, can be the only forms of atomic propositions, and to answer, e.g., subject-predicate and the relational propositions with two or more terms further, perhaps, propositions relating predicates and relations with one another, and so on. But this, I believe, is a mere playing with words" (Klagge and Nordman, 1993, 30). A similar thought already occurs in a more compressed form in the Tractatus itself: "There cannot be a hierarchy of the forms of the elementary propositions. Only that which we ourselves construct can we foresee" (5.556). It is possible, then, that the options we began with represent a false dichotomy. Perhaps Wittgenstein simply did not have an antecedent opinion on the question whether Tractarian names will turn out to be names of particulars only, particulars and universals, or whatnot. And perhaps he believed that the final analysis of language would (or might) reveal the names to defy such classifications altogether. This broader range of interpretive possibilities has only recently begun to receive the attention it deserves (See Johnston 2009). ## 2. Linguistic Atomism By "Linguistic atomism" we shall understand the view that the analysis of every proposition terminates in a proposition all of whose genuine components are names. It is a striking fact that the Tractatus contains no explicit argument for linguistic atomism. This fact has led some commentators -- e.g., Peter Simons (1992) -- to suppose that Wittgenstein's position here is motivated less by argument than by brute intuition. And indeed, Wittgenstein does present some conclusions in this vicinity as if they required no argument. At 4.221, for example, he says: "It is obvious that in the analysis of propositions we must come to elementary propositions, which consist of names in immediate combination" (emphasis added). Nonetheless, some basic observations about the Tractatus's conception of analysis will enable us to see why Wittgenstein should have thought it obvious that analysis must terminate in this way. ### 2.1 Wittgenstein's Early Conception of Analysis A remark from the Philosophical Grammar, written in 1936, throws light on how Wittgenstein had earlier conceived of the process of analysis: > > Formerly, I myself spoke of a 'complete analysis,' and I > used to believe that philosophy had to give a definitive dissection of > propositions so as to set out clearly all their connections and remove > all possibilities of misunderstanding. I spoke as if there was a > calculus in which such a dissection would be possible. I vaguely had > in mind something like the definition that Russell had given for the > definite article (PG, 211). > One of the distinctive features of Russell's definition is that it treats the expression "the x such that Fx" as an "incomplete symbol." Such symbols have no meaning in isolation but are given meaning by contextual definitions that treat of the sentential contexts in which they occur (cf. PM, 66). Incomplete symbols do, of course, have meaning because they make a contribution to the meanings of the sentences in which they occur (cf. Principles, Introduction, x). What is special about them is that they make this contribution without expressing a propositional constituent. (For more on the nature of incomplete symbols, see Pickel 2013) Russell's definition is contained in the following clauses (For the sake of expository transparency, his scope-indicating devices are omitted.). > > > [1] G(the x: Fx) = > [?]x([?]y(Fy-y=x) > & Gx) Df. > > > (cf. Russell 1905b; Russell 1990, 173) > > > > [2] (the x: Fx) exists = > [?]x[?]y(Fy-y=x) > Df. > > > (cf. Russell 1990, 174) > > > The fact that existence is dealt with by a separate definition shows that Russell means to treat the predicate 'exists' as itself an incomplete symbol. One can understand why Wittgenstein should have taken there to be an affinity between the theory of descriptions and his own envisioned "calculus," for one can extract from his remarks in the Tractatus and elsewhere two somewhat parallel proposals for eliminating what he calls terms for "complexes": > > [3] F[aRb] iff Fa & Fb & > aRb > > > [4] [aRb] exists iff aRb > Clauses [1] to [4] share the feature that any sentence involving apparent reference to an individual is treated as false rather than as neither true nor false if that individual should be discovered not to exist. Wittgenstein's first contextual definition -- our [3] -- occurs in a Notebooks entry from 1914 (NB, 4), but it is also alluded to in the Tractatus: > > Every statement about complexes can be analysed into a statement about > their constituent parts, and into those propositions which completely > describe the complexes (2.0201). > In [3] the statement "about [the complex's] constituent parts" is "Fa & Fb," while the proposition which "completely describes" the complex is "aRb." If the propositions obtained by applying [3] and [4] are to be further analysed, a two-stage procedure will be necessary: first, the apparent names generated by the analysis -- in the present case 'a' and 'b' -- will need to be replaced[3] with symbols that are overtly terms for complexes, e.g., '[cSd]' and '[eFg];' second, the contextual definitions [3] and [4] will need to be applied again to eliminate these terms. If there is going to be a unique final analysis, each apparent name will have to be uniquely paired with a term for a complex. So the program of analysis at which Wittgenstein gestures, in addition to committing him to something analogous to Russell's theory of descriptions, also commits him to the analogue of Russell's "description theory of ordinary names" (cf. Russell 1905a). The latter is the idea that every apparent name not occurring at the end of analysis is equivalent in meaning to some definite description. Wittgenstein's first definition, like Russell's, strictly speaking, stands in need of a device for indicating scope, for otherwise it would be unclear how to apply the analysis when we choose, say, "~G" as our instance of "F." In such a case the question would arise whether the resulting instance of [3] is [5]: "~G[aRb] = ~Ga & ~Gb & aRb," which corresponds to giving the term for a complex wide scope with respect to the negation operator, or whether it is: [6] "~G[aRb] = ~[Ga & Gb & aRb]," which corresponds to giving the term for a complex narrow scope. One suspects that Wittgenstein's intention would most likely have been to follow Russell's convention of reading the logical operator as having narrow scope unless the alternative is expressly indicated (cf. PM, 172). Definition [3] has obvious flaws. While it may work for such predicates as "x is located in England," it obviously fails for certain others, e.g., "x is greater than three feet long" and "x weighs exactly four pounds." This problem can hardly have escaped Wittgenstein; so it seems likely that he would have regarded his proposals merely as tentative illustrations, open to supplementation and refinement. Although Wittgenstein's second contextual definition -- our [4] -- does not occur in the Tractatus, it is implied by a remark from the Notes on Logic that seems to anticipate 2.0201: > > Every proposition which seems to be about a complex can be analysed > into a proposition about its constituents and ... the proposition > which describes the complex perfectly; i.e., that proposition > which is equivalent to saying the complex exists (NB, > 93; emphasis > added)[4] > Since the proposition that "describes the complex," [aRb], "perfectly" is just the proposition that aRb, Wittgenstein's clarifying addendum amounts to the claim that the proposition "aRb" is equivalent to the proposition "[aRb] exists." And this equivalence is just our [4]. It turns out, then, that existence is defined only in contexts in which it is predicated of complexes. Wittgenstein proposal thus mirrors Russell's in embodying the idea that it makes no sense to speak of the existence of immediately given (that is, named) simples (cf. PM, 174-5). This is why Wittgenstein was later to refer to his "objects" as "that for which there is neither existence nor non-existence" (PR, 72). His view seems to be that when 'a' is a Tractarian name, what we try to say by uttering the nonsense string "a exists" will, strictly speaking, be shown by the fact that the final analysis of some proposition contains 'a' (cf. 5.535). But of course, the Tractatus does not always speak strictly. Indeed, what is generally taken to be the ultimate conclusion of the Tractatus's so-called "Argument for Substance" (2.021-2.0211) itself tries to say something that can only be shown, since it asserts the existence of objects. The sharpness of the tension here is only partly disguised by the oblique manner in which the conclusion is formulated. Instead of arguing for the existence of objects, the Tractatus argues for the thesis that the world "has substance." However, because "objects constitute the substance of the world" (2.021), and because substance is that which exists independently of what is the case (2.024), this is tantamount to saying that objects exist. So it seems that Wittgenstein's argument for substance must be regarded as a part of the ladder we are supposed to throw away (6.54). But having acknowledged this point, we shall set it aside as peripheral to our main concerns. The most obvious similarity between the two sets of definitions is that each seeks to provide for the elimination of what purport to be semantically complex referring expressions. The most obvious difference consists in the fact that Wittgenstein's definitions are designed to eliminate not definite descriptions, but rather terms for complexes, for example the expression "[aRb]," which, judging by remarks in the Notebooks, is to be read: "a in the relation R to b" (NB, 48) (This gloss seems to derive from Russell's manner of speaking of complexes in Principia Mathematica, where examples of terms for complexes include, in addition to "a in the relation R to b," "a having the quality q", and "a and b and c standing in the relation S" (PM, 44).). One might wonder why this difference should exist. That is to say, one might wonder why Wittgenstein does not treat the peculiar locution "a in the relation R to b" as a definite description -- say, "the complex consisting of a and b, combined so that aRb"? This description could then be eliminated by applying the Tractatus's own variant upon the theory of descriptions: > > The F is G - [?]x(Fx > & Gx) & ~[?]x,y(Fx > & Fy) > > > (cf. 5.5321) > Here the distinctness of the variables (the fact that they are distinct) replaces the sign for distinctness "[?]" (cf. 5.53). Since Wittgenstein did not adopt this expedient, it seems likely that he would have regarded the predicate "x is a complex consisting of a and b, combined so that aRb" as meaningless in virtue of -- among other things -- its containing ineliminable occurrences of the pseudo-concepts "complex," "combination," and "constitution." Only the first of these notions figures on his list of pseudo-concepts in the Tractatus (4.1272), but there is no indication that that list is supposed to be exhaustive. There is a further respect in which Wittgenstein's analytical proposals differ from Russell's. Russell's second definition -- our [2] -- has the effect of shifting the burden of indicating ontological commitment from the word 'exists' to the existential quantifier. In Wittgenstein's definition, by contrast, no single item of vocabulary takes over the role of indicating ontological commitment. Instead, that commitment is indicated only after the final application of the definition, by the meaningfulness of the names in the fully analysed proposition -- or, more precisely, by the fact that certain symbols are names (cf. 5.535). The somewhat paradoxical consequence is that one can assert a statement of the form "[aRb] exists" without thereby manifesting any ontological commitment to the complex [aRb] (cf. EPB, 121). What this shows is that the two theories relieve the assertor of ontological commitments of quite different kinds. In Russell's case, the analysis -- our [2] -- removes a commitment to an apparent propositional constituent -- a "denoting concept" [5] --expressed by the phrase 'the F,' but it does not remove the commitment to the F itself. For Wittgenstein, by contrast, the analysis shows that the assertor never was ontologically committed to the complex [aRb] by an utterance of "[aRb] exists." Russell's conception of analysis at the time of the theory of descriptions -- c.a. 1905 -- is relatively clear: Analysis involves pairing up one sentence with another that expresses the very same Russellian proposition but which does so more perspicuously. The analysans counts as more perspicuous than the analysandum because the former is free of some of the latter's merely apparent ontological commitments. By the time of Principia Mathematica, however, this relatively transparent conception of analysis is no longer available. Having purged his ontology of propositions in 1910, Russell can no longer appeal to the idea that analysans and analysandum express one and the same proposition. He now adopts "the multiple relation theory of judgment," according to which the judgment (say) that Othello loves Desdemona, instead of being, as Russell had formerly supposed, a dyadic relation between the judging mind and the proposition Othello loves Desdemona, is now a non-dyadic, or, in Russell's terminology, "multiple," relation whose relata are the judging mind and those items that were formerly regarded as constituents of the proposition Othello loves Desdemona (Russell 1910 [1994, 155]). After 1910 Russell can say that a speaker who sincerely assertively uttered the analysans (in a given context) would be guaranteed to make the same judgment as one who sincerely assertively uttered the analysandum (in the same context), but he can no longer explain this accomplishment by saying that the two sentences express the same proposition. A further departure from the earlier, relatively transparent conception of analysis is occasioned by Russell's resolution of the set-theoretic version of his paradox. In this resolution one gives an analysis of a sentence whose utterance could not be taken to express any judgment. One argues that the sentence "{x: phx} e {x: phx}" is nonsense because the contextual definitions providing for the elimination of class terms yield for this case a sentence that is itself nonsense by the lights of the theory of types (PM, 76). It's (apparent) negation is, accordingly, also nonsense. In Principia, then, there is no very clear model of what is preserved in analysis. The best we can say is that Russell's contextual definitions have the feature that a (sincere, assertive) utterance of the analysans is guaranteed to express the same judgment as the analysandum, if the latter expresses a judgment at all. Some of the unclarity in the conception of analysis introduced by Russell's rejection of propositions is inherited by Wittgenstein, who similarly rejects any ontology of shadowy entities expressed by sentences. In the Tractatus a "proposition" (Satz) is a "propositional sign in its projective relation to the world" (3.12). This makes it seem as though any difference between propositional signs ought to suffice for a difference between propositions, in which case analysans and analysandum could at best be distinct propositions with the same truth conditions. Enough has now been said to make possible a consideration of Wittgenstein's reasons for describing the position I have been calling "linguistic atomism" as "obvious." Since the model for Tractarian analysis is the replacement of apparent names with (apparently) co-referring "terms for complexes," together with eliminative paraphrase of the latter, it follows trivially that the endpoint of analysis, if there is one, will contain neither "terms for complexes" nor expressions that can be replaced by terms for complexes. Wittgenstein, moreover, thinks it obvious that in the case of every proposition this process of analysis does terminate. The reason he supposes analysis cannot go on forever is that he conceives an unanalyzed proposition as deriving its sense from its analysis. As Tractatus 3.261 puts it: "Every defined sign signifies via those signs by which it is defined" (Cf. NB, 46; PT 3.20102). It follows that no proposition can have an infinite analysis, on pain of never acquiring a sense. So the process of analysis must terminate, and when it does so the product will be propositions devoid of incomplete symbols. That much, at least, is plausibly obvious, but unfortunately it does not follow that the final analysis of language will be wholly devoid of complex symbols. The trouble is that for all we have said so far, a fully analysed proposition might yet contain one or more complex symbols that have meaning in their own right. Clearly, Wittgenstein must have been assuming that all genuine referring expressions must be semantically simple: they must lack anything like a Fregean sense. But why should that be so? The seeds of one answer are contained in Tractatus 3.3, the proposition in which Wittgenstein enunciates his own version of Frege's context principle: "Only the proposition has sense; only in the context of a proposition has a name meaning" (3.3). Wittgenstein's juxtaposition of these two claims suggests that the context principle is supposed to be his ground for rejecting senses for sub-sentential expressions. But just how it could provide such a ground is far from clear. Another, more concrete, possibility is that Wittgenstein simply accepted the arguments Russell had given in "On Denoting" for rejecting senses for sub-sentential expressions. ## 3. Metaphysical Atomism By "Metaphysical atomism" we will understand the view that the semantically simple symbols occurring in a proposition's final analysis refer to simples. The Tractatus does not contain a distinct freestanding argument for this thesis, but, as we will see, the needed argument is plausibly extractable from the famous "Argument for Substance" of 2.0211-2: > > 2.0211 If the world had no substance, then whether a proposition had > sense would depend on whether another proposition was true. > > > 2.0212 It would then be impossible to draw up a picture of the world > (true or false). > To see what precisely is being contended for in this argument one needs to appreciate the historical resonances of Wittgenstein's invocation of the notion of "substance." ### 3.1 Objects as the Substance of the World The Tractatus's notion of substance is the modal analogue of Kant's temporal notion. Whereas for Kant, substance is that which "persists" (in the sense of existing at all times) (Critique, A 182), for Wittgenstein it is that which, figuratively speaking, "persists" through a "space" of possible worlds. (Compare the idea of a road that crosses several U.S. States. Such a road might be said, metaphorically speaking, to "persist" from one State to the next: in such a locution, as in Wittgenstein's, a temporal notion is enlisted to do spatial duty, though in Wittgenstein's case the space in question is logical rather than physical space and persistence amounts to reaching through the whole of logical space.) Tractarian substance is the "unchanging" in the metaphorical sense of that which does not undergo existence change in the passage (also metaphorical) from world to world. Less figuratively, Tractarian substance is that which exists with respect to every possible world. For Kant, to assert that there is substance (in the schematized sense of the category) is to say that that there is some stuff such that every existence change (i.e., origination or annihilation) is necessarily an alteration or reconfiguration of that stuff. For Wittgenstein, analogously, to say that there is substance is to say that there are some things such that all "existence changes" in the metaphorical passage from world to world are reconfigurations of them. What undergo "existence changes" are atomic states of affairs (configurations of objects): a state of affairs exists with respect to one world but fails to exist with respect to another. Those things that remain in existence through these existence changes, and which are reconfigured in the process, are Tractarian objects. It follows that the objects that "constitute the substance of the world" (2.021) are necessary existents. The Tractatus , rather wonderfully, compresses this whole metaphorical comparison into a single remark: "The object is the fixed, the existing [das Bestehende]; the configuration is the changing [das Wechselnde]." (2.0271). "Wechsel," it should be noted, is the word that Kant expressly reserves for the notion of existence change as opposed to alteration (Critique, A 187/B 230). (Unfortunately, however, whether Wittgenstein had read the Critique in time for this circumstance to have influenced his own phrasing in the Tractatus is unknown.) Tractarian objects are what any "imagined"--or, more accurately, conceivable--world has in common with the real world (2.022). Accordingly, they constitute the world's "fixed form" (2.022-3). 'Fixed' because, unlike the world's content, objects (existentially speaking) hold fast in the transition from world to world. 'Form' because they constitute what is shared by all the worlds. (On Wittgenstein's conception of possibility, the notion of an "alien" Tractarian object -- one which is merely possible -- is not even intelligible). If the objects make up the world's form, what makes up its content? The answer, I think, is the various obtaining atomic states of affairs. Distinct worlds differ with respect to content because they differ with respect to which possible states of affairs obtain in them. A complication arises becasue possible atomic states of affairs also have both form and content form and content. Their form is the manner of combination of their components, their content those components themselves (that is, their contained objects). It follows that substance -- the totality of objects -- is indeed, as Wittgenstein says, "both form and content" (2.024-5). It is at once both the form of the world and the content of possible states of affairs (These and further details of this interpretation of Wittgenstein's conception of substance as the fixed or unchanging are provided in Proops 2004; see also Zalabardo 2015, Appendix II for more on simples, names, and necessary existents). ### 3.2 The Argument for Substance As we have seen, the immediate goal of the argument for substance is to establish that there are things that exist necessarily. In the context of the assumption that anything complex could fail to exist through decomposition, this conclusion entails that there are simples (2.021). While the argument is presented as a two-stage modus tollens, it is conveniently reconstructed as a reductio ad absurdum (The following interpretation of the argument is a compressed version of that provided in Proops 2004. For two recent alternatives, see Zalabardo 2015, 243-254 and Morris 2008, ch.1, and 2017; and for criticisms of Morris, see Potter 2009): > > Suppose, for reductio, that > > > [1] There is no substance (that is, nothing exists in > every possible world). > > > Then > > > [2] Everything exists contingently. > > > But then > > > [3] Whether a proposition has sense depends on whether > another proposition is true. > > > So > > > [4] We cannot draw up pictures of the world (true or > false). > > > But > > > [5] We can draw up such pictures. > > > > Contradiction > > So > > > [6] There is substance (that is, some things exist in > every possible world). > > > Our [5] is the main suppressed premise. It means, simply, that we can frame senseful propositions. Let us now consider how we might try to defend the inference from [2] to [3] on Wittgensteinian principles. As a preliminary, note that, given Wittgenstein's equation in the Notes on Logic of having sense with having truth-poles (NB, 99), it seems reasonable to suppose that for a sentence to "have sense" with respect to a given world is for it to have a truth value with respect to that world. Let us assume that this is so. Now suppose that everything exists contingently. Then, in particular, the referents of the semantically simple symbols occurring in a fully analysed sentence will exist contingently. But then any such sentence will contain a semantically simple symbol that fails to refer with respect to some possible world (As we will shortly see, this step is in fact controversial.). Suppose, as a background assumption, that there are no contingent simples. (It will be argued below that this assumption plausibly follows from certain Tractarian commitments.) Then, if we assume that a sentence containing a semantically simple term is neither true nor false evaluated with respect to a world in which its purported referent (namely, a complex existing contingently at the actual world) fails to exist -- and, for now, we do -- then, for any such fully analysed sentence, there will be some world such that the sentence depends for its truth valuedness with respect to that world on the truth with respect to that world of some other sentence, viz., the sentence stating that the constituents of the relevant complex are configured in a manner necessary and sufficient for its existence. It follows that if everything exists contingently, then whether a sentence is senseful with respect to a world will depend on whether another sentence is true with respect to that world. The step from [3] to [4] runs as follows. Suppose that whether any sentence "has sense" (i.e., on our reading, has a truth-value) depends (in the way just explained) on whether another is true. Then every sentence will have an "indeterminate sense" in the sense that it will lack a truth value with respect to at least one possible world. But an indeterminate sense is no sense at all, for a proposition by its very nature "reaches through the whole logical space" (3.42) (i.e., it is truth-valued with respect to every possible world).[6] So if every sentence depended for its "sense" (i.e., truth-valuedness) on the truth of another, no sentence would have a determinate sense, and so no sentence would have a sense. In which case we would be unable to frame senseful propositions (i.e., to "draw up pictures of the world true or false"). One apparent difficulty concerns the assumption that to have sense is just to be true or false. How can such a view be attributed to the Wittgenstein of the Tractatus given his view that tautology, which is true, and contradiction, which is false, are without sense (sinnlos) (4.461)? The seeds of an answer may be contained in a remark from Wittgenstein's lectures at Cambridge during the year 1934-1935. Looking back on what he'd written in the Tractatus, he says: > > When I called tautologies senseless I meant to stress a connection > with a quantity of sense, namely 0. ([AM]), 137) > It is possible, then, that Wittgenstein is thinking of a sinnlos proposition as a proposition that "has sense" but has it to a zero degree. According to this conception, a tautology, being true, is, in contrast to a nonsensical string, in the running for possessing a non-zero quantity of sense, but is so constructed that, in the end, it doesn't get to have one. And, importantly, in virtue of being in the running for having a non-zero quantity of sense its possession of a zero quantity amounts to its, broadly speaking, 'having sense'. Such a view, according to which, for some non-count noun N, an N-less entity has N, but has a zero quantity of it, is not without precedent in the tradition. Kant, for example, regards rest (motionlessness) as a species of motion: a zero quantity of it (Bader, Other Internet Resources, 22-23). If, in a similar fashion, Sinnlosigkeit is a species of Sinn, the equation of having Sinn with being true or false will be preserved. To offer a full defence of this understanding of Sinnlosigkeit would take us too far afield, but I mention it to show that the current objection is not decisive. Another apparent difficulty for this reconstruction arises from its appearing to contradict Tractatus 3.24, which clearly suggests that if the complex entity A were not to exist, the proposition "F[A]" would be false, rather than, as the argument requires, without truth value. But the difficulty is only apparent. It merely shows that 3.24 belongs to a theory that assumes that the world does have substance. On this assumption Wittgenstein can say that whenever an apparent name occurs that appears to mention a complex this is only because it is not, after all, a genuine name -- and this is what he does say. But on the assumption that the world has no substance, so that everything is complex, Wittgenstein can no longer say this. For now he must allow that the semantically simple symbols occurring in a proposition's final analysis do refer to complexes. So in the context of the assumption that every proposition has a final analysis, the reductio assumption of the argument for substance entails the falsity of 3.24. But since 3.24 is assumed to be false only in the context of a reductio, it is something that Wittgenstein can consistently endorse (This solution to the apparent difficulty for the present reconstruction is owed, in its essentials, to David Pears (see Pears 1987 [1989, 78]). To complete the argument it only remains to show that Tractarian commitments extrinsic to the argument for substance rule out contingent simples.[7] Suppose a is a contingent simple. Then "a exists" must be a contingent proposition. But it cannot be an elementary proposition because it will be entailed by any elementary proposition containing 'a,' and elementary propositions are logically independent (4.211). So "a exists" must be non-elementary, and so further analyzable. And yet there would seem to be no satisfactory analysis of this proposition on the assumption that 'a' names a contingent simple -- no analysis, that is to say, that is both intrinsically plausible and compatible with Tractarian principles. Wittgenstein cannot analyse "a exists" as the proposition "[?]x(x = a)" for two reasons. First, he would reject this analysis on the grounds that it makes an ineliminable use of the identity sign (5.534). Second, given his analysis of existential quantifications as disjunctions, the proposition "[?]x(x = a)" would be further analysed as the non-contingent proposition "a = a [?] a = b [?] a = c...". Nor can he analyse "a exists" as "~[ ~Fa & ~Ga & ~Ha...]" -- that is, as the negation of the conjunction of the negations of every elementary proposition involving "a." To suppose that it could, is to suppose that the proposition "~Fa & ~Ga & ~Ha..." means "a does not exist," and yet by the lights of the Tractatus this proposition would show a's existence -- or, more correctly, it would show something that one tries to put into words by saying "a exists" (cf. 5.535, Corr, 126)). So, pending an unforeseen satisfactory analysis of "a exists," this proposition will have to be analysed as a complex of propositions not involving a. In other words, 'a' will have to be treated as an incomplete symbol and the fact of a's existence will have to be taken to consist in the fact that objects other than a stand configured thus and so. But that would seem to entail that a is not simple. The argument for substance may be criticized on several grounds. First, the step leading from [2] to [3] relies on the assumption that a name fails to refer with respect to a possible world at which its actual-world referent does not exist. This amounts to the controversial assumption that names do not function as what Nathan Salmon has called "obstinately rigid designators" (Salmon 1981, 34). Secondly, the step leading from [3] to [4] relies on the assumption that a sentence that is neither true nor false with respect to some possible world fails to express a sense. As Wittgenstein was later to realize, the case of intuitively senseful, yet vague sentences plausibly constitutes a counterexample (cf. PI Section 99). Lastly, one may question the assumption that it makes sense to speak of a final analysis, given that the procedure for analysing a sentence of ordinary language has not been made clear (See PI, Sections 60, 63-4, and Section 91). ## 4. The Epistemology of Logical Atomism How could we possibly know that something is a Tractarian object? Wittgenstein has little or nothing to say on this topic in the Tractatus, and yet it is clear from his retrospective remarks that during the composition of the Tractatus he did think it possible in principle to discover the Tractarian objects (See AM, 11 and EPB, 121). So it seems worth asking by what means he thought such a discovery might be made. Sometimes, it can seem as though Wittgenstein just expected to hit upon the simples by reflecting from the armchair on those items that struck him as most plausibly lacking in proper parts. This impression is most strongly suggested in the Notebooks, and in particular in a passage from June 1915 in which Wittgenstein seems to express confidence that certain objects already within his ken either count as Tractarian objects or will turn out to do so. He says: "It seems to me perfectly possible that patches in our visual field are simple objects, in that we do not perceive any single point of a patch separately; the visual appearances of stars even seem certainly to be so" (NB, 64). By "patches in our visual field" in this context Wittgenstein means parts of the visual field with no noticeable parts. In other words, points in visual space (cf. KL, 120). Clearly, then, Wittgenstein at one stage believed he was in a position to specify some Tractarian objects. However, the balance of the evidence suggests that this idea was short-lived. For he was later to say that he and Russell had pushed the question of examples of simples to one side as a matter to be settled on a future occasion (AM, 11). And when Norman Malcolm pressed Wittgenstein to say whether when he wrote the Tractatus he had decided on anything as an example of a "simple object," he had replied -- according to Malcolm's report -- that "at the time his thought had been that he was a logician; and that it was not his business as a logician, to try to decide whether this thing or that was a simple thing or a complex thing, that being a purely empirical matter" (Malcolm 1989, 70). Wittgenstein was not suggesting that the correct way to establish that something is a Tractarian object is to gather evidence that its decomposition is physically impossible. That reading would only have a chance of being correct if Wittgenstein had taken metaphysical possibility to coincide with physical possibility, and that is evidently not so.[8] His meaning seems rather to be just that the objects must be discovered rather than postulated or otherwise specified in advance of investigation (cf. AM, 11). But since Wittgenstein was later to accuse his Tractarian self of having entertained the concept of a distinctive kind of philosophical discovery (see WVC 182, quoted below), we must not jump -- as Malcolm appears to have done, to the conclusion that he conceived of the discovery in question as "empirical" in anything like the contemporary sense of the word. We know that Wittgenstein denied categorically that we could specify the possible forms of elementary propositions and the simples a priori (4.221, 5.553-5.5541, 5.5571). But he did not deny that these forms would be revealed as the result of logical analysis. In fact, he maintained precisely this view. This idea is not explicit in the Tractatus, but it is spelled out in a later self-critical remark from G. E. Moore's notes of Wittgenstein's 1933 lectures at Cambridge: > > I say in [the] Tractatus that you can't say anything > about [the] structure of atomic prop[osition]s: my idea being the > wrong one, that logical analysis would reveal what it would reveal > (entry for 6 February, 1933, Stern et. al., 2016, 252) > Speaking of Tractarian objects in another retrospective remark, this time from a German version of the Brown Book, Wittgenstein says: "What these [fundamental constituents] of reality are it seemed difficult to say. I thought it was the job of further logical analysis to discover them" (EPB 121). These remarks should be taken at face value: it is logical analysis -- the analysis of propositions -- that is supposed to enable us to discover the forms of elementary propositions and the objects. The hope is that when propositions have been put into their final, fully analysed forms by applying the "calculus" spoken of in the Philosophical Grammar we will finally come to know the names and thereby the objects. Presumably, we will know the latter by acquaintance in the act of grasping propositions in their final analysed forms. Admittedly, Wittgenstein's denial that we can know the objects a priori looks strange given the fact that the analytical procedure described in Section 2 above seems to presuppose that we have a priori knowledge both of the correct analyses of ordinary names and of the contextual definitions by means of which terms for complexes are to beeliminated. But some tension in Wittgenstein's position on this point is just what we should expect in view of his later rather jaundiced assessment of his earlier reliance on the idea of philosophical discovery: > > I [used to believe that] the elementary propositions could be > specified at a later date. Only in recent years have I broken away > from that mistake. At the time I wrote in a manuscript of my book, > "The answers to philosophical questions must never be > surprising. In philosophy you cannot discover anything." I > myself, however, had not clearly enough understood this and offended > against it. (WVC, 182, emphasis added) > The remark that Wittgenstein quotes here from "a manuscript of the Tractatus" did not survive into the final version, but its sentiment is clearly echoed in the related remark that there can: "never be surprises in logic" (6.1251). Wittgenstein is clear that in the Tractatus he had unwittingly proceeded as though there could be such a thing as a philosophical surprise or discovery. His idea that the true objects would be discovered through analysis, but are nonetheless not known a priori, is plausibly one instance of this mistake. On the conception of the Tractatus, objects are to be discovered by grasping fully analysed propositions, presumably with the awareness that they are fully analysed. But since that is so, we shall not have fully explained how we are supposed to be able to discover the simples unless we explain how, in practice, we can know we have arrived at the final analysis of a proposition. But on this point, unfortunately, Wittgenstein has little to say. In fact, the only hint he offers is the rather dark one contained in Tractatus 3.24: > > That a propositional element signifies [bezeichnet] a complex > can be seen from an indeterminateness in the propositions in which it > occurs. We know that everything is not yet determined by this > proposition. (The notation for generality contains a prototype). > (3.24) > It is an indeterminateness in propositions -- whatever this might amount to -- that is supposed to alert us to the need for further analysis. In Wittgenstein's view, then, we possess a positive test for analyzability. However, since the notion of "indeterminateness" in question is unclear, the test is of little practical value. The indeterminateness in question is plainly not the one we considered in section 3: what is in question at the present juncture is the indeterminateness of propositions, not of senses. But what does that amount to? According to one line of interpretation, due originally to W. D. Hart (Hart 1971), a proposition is indeterminate when there is more than one way it can be true. Thus if I say "Barack Obama is in the United States," I leave open where in particular he might be. The source of the indeterminacy is the implied generality of this statement, which is tantamount to: "Obama is somewhere in the United States." This line of interpretation has the merit of promising to make sense of the closing parenthetical remark of 3.24. But it cannot be correct. The kind of indeterminacy that Wittgenstein has in mind at 3.24 is supposed to serve as a sign of further analysability. But Hart's notion cannot play this role, since any disjunctive proposition would be indeterminate in his sense, even a fully analysed proposition consisting of a disjunction of elementary propositions. According to a second line of interpretation, a proposition is indeterminate in the relevant sense if the result of embedding it in some context is structurally ambiguous. Consider, for example, the result of embedding "F [A]" in the context "it is not true that," where 'A' is temporarily treated as a semantically simple term designating a complex (Keep in place the assumption that a sentence containing a non-referring semantically simple term is neither true nor false). In this case the question would arise whether the result of this embedding is neither true nor false evaluated with respect to a world in which A does not exist, or simply true. The first option corresponds to giving the apparent name wide scope with respect to the logical operator, the second to giving it narrow scope. Such a scope ambiguity could not exist if 'A' were a genuine Tractarian name, so its presence could reasonably be taken to signal the need for further analysis. So far, so good, but where does the business about the generality notation "containing a prototype" come in? Nothing in the present explanation has yet done justice to this remark. Nor does the present explanation really pinpoint what it is that signals the need for further analysis. That, at bottom, is the fact that we can imagine circumstances in which the supposed referent of 'A' fails to exist. So, again, there is reason to be dissatisfied with this gloss on indeterminacy. It is hard to resist the conclusion that Wittgenstein never supplied an adequate way of recognizing when a proposition is fully analysed, and consequently that he failed to specify a means for recognizing something as a Tractarian object. ## 5. The Dismantling of Logical Atomism Wittgenstein's turn away from logical atomism may be divided into two main phases. During the first phase (1928-9), documented in his 1929 article "Some Remarks on Logical Form" (Klagge and Nordmann, 1993, 29-35), Wittgenstein exhibits a growing dissatisfaction with certain central details of the Tractatus's logical atomism, and notably with the thesis of the independence of elementary propositions. During this phase, however, he is still working within the broad conception of analysis presupposed, if not fully developed, in the Tractatus. The second phase (1931-2) involves a revolutionary break with that very conception. ### 5.1 First phase: The colour-exclusion problem The so-called "colour-exclusion problem" is a difficulty that arises for the Tractatus's view that it is metaphysically possible for each elementary proposition to be true or false regardless of the truth or falsity of the others (4.211). In view of its generality, the problem might more accurately be termed "the problem of the manifest incompatibility of apparently unanalysable statements." The problem may be illustrated as follows: Suppose that a is a point in the visual field. Consider the propositions P: "a is blue at t" and Q: "a is red at t" (supposing "red" and "blue" to refer to determinate shades). It is clear that P and Q cannot both be true; and yet, on the face of it, it seems that this incompatibility (or "exclusion" in Wittgenstein's parlance) is not a logicalimpossibility. In the Tractatus Wittgenstein's response was to treat the problem as merely apparent. He supposed that in such cases further analysis would reveal the incompatibility to be logical in nature: > > For two colours, e.g., to be at one place in the visual field > is impossible, and indeed logically impossible, for it is excluded by > the logical structure of colour. Let us consider how this > contradiction presents itself in physics. Somewhat as follows: That a > particle cannot at the same time have two velocities, that is, that at > the same time it cannot be in two places, that is, that particles in > different places at the same time cannot be identical (6.3751) > As F. P. Ramsey observes in his review of the Tractatus(Ramsey, 1923), the analysis described here actually fails to reveal a logical incompatibility between the two statements in question; for, even granting the correctness of the envisaged reduction of the phenomenology of colour perception to facts about the velocities of particles, the fact that one and the same particle cannot be (wholly) in two places at the same time still looks very much like a synthetic a priori truth. It turns out, however, that Wittgenstein was well aware of this point. He knew that he had not taken the analysis far enough to bring out a logical contradiction, but he was confident that he had taken a step in the right direction. In a Notebooks entry from August 1916 he remarks that: "The fact that a particle cannot be in two places at the same time does look more like a logical impossibility [than the fact that a point cannot be red and green at the same time]. If we ask why, for example, then straight away comes the thought: Well, we should call particles that were in two places [at the same time] different, and this in its turn all seems to follow from the structure of space and particles" (NB, 81; emphasis added). Here Wittgenstein is conjecturing that it will turn out to be a conceptual (hence, for him logical) truth about particles and space (and presumably also time) that particles in two distinct places (at the same time) are distinct. He does not yet possess the requisite analyses to demonstrate this conjecture, but he is optimistic that they will be found. The article "Some Remarks on Logical Form" (1929) marks the end of this optimism. Wittgenstein now arrives at the view that some incompatibilities cannot, after all, be reduced to logical impossibilities. His change of heart appears to have been occasioned by a consideration of incompatibilities involving the attribution of qualities that admit of gradation -- e.g., the pitch of a tone, the brightness of a shade of colour, etc. Consider, for example, the statements: "A has exactly one degree of brightness" and "A has exactly two degrees of brightness." The challenge is to provide analyses of these statements that bring out the logical impossibility of their being true together. What Wittgenstein takes to be the most plausible suggestion -- or at least a sympathetic reconstruction of it -- adapts the standard definitions of the numerically definite quantifiers to the system described in the Tractatus, analysing these claims as respectively: "[?]x(Bx & A has x) & ~[?]x,y(Bx & By & A has x and A has y)" ("Bx" means "x is a degree of brightness") and "[?]x,y(Bx & By & A has x and A has y) & ~[?]x,y,z(Bx & By & Bz & A has x & A has y & A has z)." But the suggestion will not do. The trouble is that this analysis -- absurdly -- makes it seem as though when something has just one degree of brightness there could be a substantive question about which (if any) of the three mentioned in the analysis of the second claim-- x or y or z -- it was--as if a degree of brightness were a kind of corpuscle whose association with a thing made it bright (cf. Klagge and Nordmann, 33). Wittgenstein concludes that the independence of elementary propositions must be abandoned and that terms for real numbers must enter into atomic propositions, so that the impossibility of something's having both exactly one and exactly two degrees of brightness emerges as an irreducibly mathematical impossibility. This, in turn, contradicts the Tractatus's idea that all necessity is logical necessity (6.37). ### 5.2 Second phase: Generality and Analysis Unlike Frege and Russell, the Tractatus does not treat the universal and existential quantifiers as having meaning in isolation. Instead, it treats them as incomplete symbols to be analysed away according to the following schemata: > > [?]x. Phx - Pha & > Phb & Phc... > > > [?]x. Phx - Pha [?] > Phb [?] Phc... > Universal (existential) quantification is treated as equivalent to a possibly infinite conjunction (disjunction) of propositions. Wittgenstein's later dissatisfaction with this view is expressed most clearly in G. E. Moore's notes of Wittgenstein's lectures from Michaelmas term 1932. > > Now there is a temptation to which I yielded in [the] > Tractatus, to say that > > > > > > (x).fx = logical > > product,[9] > > fa . fb . fc... > > > > > > ([?]x).fx = [logical] sum, fa [?] > > fb [?] fc... > > > > > > > > > > This is wrong, but not as absurd as it looks. (entry for 25 November, > 1932, Stern et. al., 2016, > 215)[10]. > > > > Explaining why the Tractatus's analysis of generality is not palpably absurd, Wittgenstein says: > > Suppose we say that: Everybody in this room has a hat = Ursell has a > hat, Richards has a hat etc. This is obviously false, because you have > to add "& a, b, c,... are > the only people in the room." This I knew and said in [the] > Tractatus. But now, suppose we talk of > "individuals" in R[ussell]'s sense, e.g., atoms or > colours; and give them names, then there would be no prop[osition] > analogous to "And a, b, c are the > only people in the room." (ibid.) > Clearly, in the Tractatus Wittgenstein was not making the simple-minded mistake of forgetting that "Every F is G" cannot be analysed as "Ga & Gb & Gc..." even when a, b, c, etc. are in fact the only Fs. (Unfortunately, his claim that he registered this point in the Tractatus is not borne out by the text). His idea was rather that the Tractatus's analysis of generality is offered only for the special case in which a, b, c, etc, are "individuals" in Russell's sense. Wittgenstein had supposed that in this case there is no proposition to express the supplementary clause that is needed in the other cases. Unfortunately, Wittgenstein does not explain why there should be no such proposition, but the answer seems likely to be the following: What we are assumed to be analysing is actually "Everything is G." In this case any allegedly necessary competing clause -- for example, "a, b, c etc., are the only things" (that is, Tractarian objects) -- would just be a nonsense-string produced in the misfired attempt to put into words something that is shown by the fact that when analysis bottoms out it yields as names only such as figure in the conjunction "Ga & Gb & Gc..." (cf. Tractatus 4.1272). What led Wittgenstein to abandon the Tractatus's analysis of generality was his realization that he had failed adequately to think through the infinite case. He had proceeded as though the finite case could be used as a way of thinking about the infinite case, the details of which could be sorted out at a later date. By 1932 he had come to regard this attitude as mistaken. The point is made in a passage from the Cambridge Lectures whose meaning can only be appreciated after some preliminary explanation. The passage in question makes a crucial claim about something Wittgenstein refers to as "The Proposition". By this phrase in this context he means the joint denial of all the propositions that are values of the propositional function "x is in this room". This proposition can be written: > > (x is in this room) [- - - - - T] > > > (Entry for November 25th, 1932, Compare Stern et. al, 217) > Here the symbol '[- - - - - T]' symbolizes the joint-denial operation, and the whole symbol expresses the result of applying this operation to arbitrarily many values of the propositional function "x is in the room". The dashes in the symbol for joint denial represent rows in the truth-table on which one or more of the truth-arguments--that is values of the propositional function--is true. The result of applying the operation of joint denial to those truth-arguments is accordingly false. (In a variant on this notation each of the dashes could be replaced with 'F'). Wittgenstein is interested in the fact that while we write down finitely many dashes we intend the arguments for the joint-denial operation to be arbitrarily many and possibly infinitely many. His criticism of these conceptions runs as follows: > > There is a most important mistake in [the] Tract[atus]...I > pretended that the Proposition was a logical product; but it > isn't, because "..." don't give you a > logical product. It is [the] fallacy of thinking 1 + 1 + 1 ... is > a sum. It is muddling up a sum with the limit of a sum (ibid.) > His point is that the Proposition does not, despite appearances, express a logical product. It rather, he now seems to be saying, expresses something like an indefinitely extensible process. Wittgenstein came to see his earlier hope that it did express a logical product rested on the mistake of confusing "dots of infinitude" with "dots of laziness.". The upshot could scarely be more important: if Wittgenstein is right, the Tractatus's very conception of the general form of the proposition, because it makes essential appeal to the idea of the joint denial of arbitrarily many values of a propositional function, is itself infected with confusion. Wittgenstein, however, does not think that the confusion of kinds of dots was the deepest mistake he made in the Tractatus. Beyond this: "There was a deeper mistake -- confusing logical analysis with chemical analysis. I thought '([?]x)fx' is a definite logical sum, only I can't at the moment tell you which" (November 25, 1932, ibid.; cf. PG, 210). Wittgenstein had supposed that there was a fact of the matter -- unknown, but in principle knowable -- about which logical sum "([?]x).fx" is equivalent to. But because he had failed to specify the analytical procedure in full detail, and because he had not adequately explained what analysis is supposed to preserve, this idea was unwarranted. Indeed, it exemplified an attitude he was later to characterize as amounting to a kind of unacceptable "dogmatism" (WWK, 182).
wittgenstein-mathematics	## 1. Wittgenstein on Mathematics in the Tractatus Wittgenstein's non-referential, formalist conception of mathematical propositions and terms begins in the Tractatus.[1] Indeed, insofar as he sketches a rudimentary Philosophy of Mathematics in the Tractatus, he does so by contrasting mathematics and mathematical equations with genuine (contingent) propositions, sense, thought, propositional signs and their constituent names, and truth-by-correspondence. In the Tractatus, Wittgenstein claims that a genuine proposition, which rests upon conventions, is used by us to assert that a state of affairs (i.e., an elementary or atomic fact; 'Sachverhalt') or fact (i.e., multiple states of affairs; 'Tatsache') obtain(s) in the one and only real world. An elementary proposition is isomorphic to the possible state of affairs it is used to represent: it must contain as many names as there are objects in the possible state of affairs. An elementary proposition is true iff its possible state of affairs (i.e., its 'sense'; 'Sinn') obtains. Wittgenstein clearly states this Correspondence Theory of Truth at (4.25): > > > If an elementary proposition is true, the state of affairs exists; if > an elementary proposition is false, the state of affairs does not > exist. > > > But propositions and their linguistic components are, in and of themselves, dead--a proposition only has sense because we human beings have endowed it with a conventional sense (5.473). Moreover, propositional signs may be used to do any number of things (e.g., insult, catch someone's attention); in order to assert that a state of affairs obtains, a person must 'project' the proposition's sense--its possible state of affairs--by 'thinking' of (e.g., picturing) its sense as one speaks, writes or thinks the proposition (3.11). Wittgenstein connects use, sense, correspondence, and truth by saying that "a proposition is true if we use it to say that things stand in a certain way, and they do" (4.062; italics added). The Tractarian conceptions of genuine (contingent) propositions and the (original and) core concept of truth are used to construct theories of logical and mathematical 'propositions' by contrast. Stated boldly and bluntly, tautologies, contradictions and mathematical propositions (i.e., mathematical equations) are neither true nor false--we say that they are true or false, but in doing so we use the words 'true' and 'false' in very different senses from the sense in which a contingent proposition is true or false. Unlike genuine propositions, tautologies and contradictions "have no 'subject-matter'" (6.124), "lack sense", and "say nothing" about the world (4.461), and, analogously, mathematical equations are "pseudo-propositions" (6.2) which, when 'true' ('correct'; 'richtig' (6.2321)), "merely mark ... [the] equivalence of meaning [of 'two expressions']" (6.2323). Given that "[t]autology and contradiction are the limiting cases--indeed the disintegration--of the combination of signs" (4.466; italics added), where > > > the conditions of agreement with the world--the representational > relations--cancel one another, so that [they] do[] not stand in > any representational relation to reality, > > > tautologies and contradictions do not picture reality or possible states of affairs and possible facts (4.462). Stated differently, tautologies and contradictions do not have sense, which means we cannot use them to make assertions, which means, in turn, that they cannot be either true or false. Analogously, mathematical pseudo-propositions are equations, which indicate or show that two expressions are equivalent in meaning and therefore are intersubstitutable. Indeed, we arrive at mathematical equations by "the method of substitution": > > > starting from a number of equations, we advance to new equations by > substituting different expressions in accordance with the equations. > (6.24) > > > We prove mathematical 'propositions' 'true' ('correct') by 'seeing' that two expressions have the same meaning, which "must be manifest in the two expressions themselves" (6.23), and by substituting one expression for another with the same meaning. Just as "one can recognize that ['logical propositions'] are true from the symbol alone" (6.113), "the possibility of proving" mathematical propositions means that we can perceive their correctness without having to compare "what they express" with facts (6.2321; cf. RFM App. III, SS4). The demarcation between contingent propositions, which can be used to correctly or incorrectly represent parts of the world, and mathematical propositions, which can be decided in a purely formal, syntactical manner, is maintained by Wittgenstein until his death in 1951 (Zettel SS701, 1947; PI II, 2001 edition, pp. 192-193e, 1949). Given linguistic and symbolic conventions, the truth-value of a contingent proposition is entirely a function of how the world is, whereas the "truth-value" of a mathematical proposition is entirely a function of its constituent symbols and the formal system of which it is a part. Thus, a second, closely related way of stating this demarcation is to say that mathematical propositions are decidable by purely formal means (e.g., calculations), while contingent propositions, being about the 'external' world, can only be decided, if at all, by determining whether or not a particular fact obtains (i.e., something external to the proposition and the language in which it resides) (2.223; 4.05). The Tractarian formal theory of mathematics is, specifically, a theory of formal operations. Over the past 20 years, Wittgenstein's theory of operations has received considerable examination (Frascolla 1994, 1997; Marion 1998; Potter 2000; and Floyd 2002), which has interestingly connected it and the Tractarian equational theory of arithmetic with elements of Alonzo Church's $\lambda$-calculus and with R. L. Goodstein's equational calculus (Marion 1998: chapters 1, 2, and 4). Very briefly stated, Wittgenstein presents: 1. ... the sign '[$a, x, O \spq x$]' for the general term of the series of forms $a$, $O \spq a$, $O \spq O \spq a$.... (5.2522) 2. ... the general form of an operation $\Omega\spq(\overline{\eta})$ [as] \[ [\overline{\xi}, N(\overline{\xi})]\spq (\overline{\eta}) (= [\overline{\eta}, \overline{\xi}, N(\overline{\xi})]). (6.01) \] 3. ... the general form of a proposition ("truth-function") [as] $[\overline{p}, \overline{\xi}, N(\overline{\xi})]$. (6) 4. The general form of an integer [natural number] [as] $[0, \xi , \xi + 1]$. (6.03) adding that "[t]he concept of number is... the general form of a number" (6.022). As Frascolla (and Marion after him) have pointed out, "the general form of a proposition is a particular case of the general form of an 'operation'" (Marion 1998: 21), and all three general forms (i.e., of operation, proposition, and natural number) are modeled on the variable presented at (5.2522) (Marion 1998: 22). Defining "[a]n operation [as] the expression of a relation between the structures of its result and of its bases" (5.22), Wittgenstein states that whereas "[a] function cannot be its own argument,... an operation can take one of its own results as its base" (5.251). On Wittgenstein's (5.2522) account of '[$a, x, O \spq x$]', > > > the first term of the bracketed expression is the beginning of the > series of forms, the second is the form of a term $x$ arbitrarily > selected from the series, and the third [$O \spq x$] is the form of > the term that immediately follows $x$ in the series. > > > Given that "[t]he concept of successive applications of an operation is equivalent to the concept 'and so on'" (5.2523), one can see how the natural numbers can be generated by repeated iterations of the general form of a natural number, namely '[$0, \xi , \xi +1$]'. Similarly, truth-functional propositions can be generated, as Russell says in the Introduction to the Tractatus (p. xv), from the general form of a proposition '[$\overline{p}$, $\overline{\xi}$, $N(\overline{\xi})$]' by > > > taking any selection of atomic propositions [where $p$ "stands > for all atomic propositions"; "the bar over the variable > indicates that it is the representative of all its values" > (5.501)], negating them all, then taking any selection of the set of > propositions now obtained, together with any of the originals [where > $x$ "stands for any set of propositions"]--and so > on indefinitely. > > > On Frascolla's (1994: 3ff) account, > > > a numerical identity "$\mathbf{t} = \mathbf{s}$" is an > arithmetical theorem if and only if the corresponding equation > "$\Omega^t \spq x = \Omega^s \spq x$", which is framed > in the language of the general theory of logical operations, can be > proven. > > > By proving > > > the equation "$\Omega^{2 \times 2}\spq x = \Omega^{4}\spq > x$", which translates the arithmetic identity "$2 \times > 2 = 4$" into the operational language (6.241), > > > Wittgenstein thereby outlines "a translation of numerical arithmetic into a sort of general theory of operations" (Frascolla 1998: 135). Despite the fact that Wittgenstein clearly does not attempt to reduce mathematics to logic in either Russell's manner or Frege's manner, or to tautologies, and despite the fact that Wittgenstein criticizes Russell's Logicism (e.g., the Theory of Types, 3.31-3.32; the Axiom of Reducibility, 6.1232, etc.) and Frege's Logicism (6.031, 4.1272, etc.),[2] quite a number of commentators, early and recent, have interpreted Wittgenstein's Tractarian theory of mathematics as a variant of Logicism (Quine 1940 [1981: 55]; Benacerraf & Putnam 1964a: 14; Black 1964: 340; Savitt 1979 [1986: 34]; Frascolla 1994: 37; 1997: 354, 356-57, 361; 1998: 133; Marion 1998: 26 & 29; and Potter 2000: 164 and 182-183). There are at least four reasons proffered for this interpretation. 1. Wittgenstein says that "[m]athematics is a method of logic" (6.234). 2. Wittgenstein says that "[t]he logic of the world, which is shown in tautologies by the propositions of logic, is shown in equations by mathematics" (6.22). 3. According to Wittgenstein, we ascertain the truth of both mathematical and logical propositions by the symbol alone (i.e., by purely formal operations), without making any ('external', non-symbolic) observations of states of affairs or facts in the world. 4. Wittgenstein's iterative (inductive) "interpretation of numerals as exponents of an operation variable" is a "reduction of arithmetic to operation theory", where "operation" is construed as a "logical operation" (italics added) (Frascolla 1994: 37), which shows that "the label 'no-classes logicism' tallies with the Tractatus view of arithmetic" (Frascolla 1998: 133; 1997: 354). Though at least three Logicist interpretations of the Tractatus have appeared within the last 20 years, the following considerations (Rodych 1995; Wrigley 1998) indicate that none of these reasons is particularly cogent. For example, in saying that "[m]athematics is a method of logic" perhaps Wittgenstein is only saying that since the general form of a natural number and the general form of a proposition are both instances of the general form of a (purely formal) operation, just as truth-functional propositions can be constructed using the general form of a proposition, (true) mathematical equations can be constructed using the general form of a natural number. Alternatively, Wittgenstein may mean that mathematical inferences (i.e., not substitutions) are in accord with, or make use of, logical inferences, and insofar as mathematical reasoning is logical reasoning, mathematics is a method of logic. Similarly, in saying that "[t]he logic of the world" is shown by tautologies and true mathematical equations (i.e., #2), Wittgenstein may be saying that since mathematics was invented to help us count and measure, insofar as it enables us to infer contingent proposition(s) from contingent proposition(s) (see 6.211 below), it thereby reflects contingent facts and "[t]he logic of the world". Though logic--which is inherent in natural ('everyday') language (4.002, 4.003, 6.124) and which has evolved to meet our communicative, exploratory, and survival needs--is not invented in the same way, a valid logical inference captures the relationship between possible facts and a sound logical inference captures the relationship between existent facts. As regards #3, Black, Savitt, and Frascolla have argued that, since we ascertain the truth of tautologies and mathematical equations without any appeal to "states of affairs" or "facts", true mathematical equations and tautologies are so analogous that we can "aptly" describe "the philosophy of arithmetic of the Tractatus... as a kind of logicism" (Frascolla 1994: 37). The rejoinder to this is that the similarity that Frascolla, Black and Savitt recognize does not make Wittgenstein's theory a "kind of logicism" in Frege's or Russell's sense, because Wittgenstein does not define numbers "logically" in either Frege's way or Russell's way, and the similarity (or analogy) between tautologies and true mathematical equations is neither an identity nor a relation of reducibility. Finally, critics argue that the problem with #4 is that there is no evidence for the claim that the relevant operation is logical in Wittgenstein's or Russell's or Frege's sense of the term--it seems a purely formal, syntactical operation. "Logical operations are performed with propositions, arithmetical ones with numbers", says Wittgenstein (WVC 218); "[t]he result of a logical operation is a proposition, the result of an arithmetical one is a number". In sum, critics of the Logicist interpretation of the Tractatus argue that ##1-4 do not individually or collectively constitute cogent grounds for a Logicist interpretation of the Tractatus. Another crucial aspect of the Tractarian theory of mathematics is captured in (6.211). > > > Indeed in real life a mathematical proposition is never what we want. > Rather, we make use of mathematical propositions only in > inferences from propositions that do not belong to mathematics to > others that likewise do not belong to mathematics. (In philosophy the > question, 'What do we actually use this word or this proposition > for?' repeatedly leads to valuable insights.) > > > Though mathematics and mathematical activity are purely formal and syntactical, in the Tractatus Wittgenstein tacitly distinguishes between purely formal games with signs, which have no application in contingent propositions, and mathematical propositions, which are used to make inferences from contingent proposition(s) to contingent proposition(s). Wittgenstein does not explicitly say, however, how mathematical equations, which are not genuine propositions, are used in inferences from genuine proposition(s) to genuine proposition(s) (Floyd 2002: 309; Kremer 2002: 293-94). As we shall see in SS3.5, the later Wittgenstein returns to the importance of extra-mathematical application and uses it to distinguish a mere "sign-game" from a genuine, mathematical language-game. This, in brief, is Wittgenstein's Tractarian theory of mathematics. In the Introduction to the Tractatus, Russell wrote that Wittgenstein's "theory of number" "stands in need of greater technical development", primarily because Wittgenstein had not shown how it could deal with transfinite numbers (Russell 1922[1974]: xx). Similarly, in his review of the Tractatus, Frank Ramsey wrote that Wittgenstein's 'account' does not cover all of mathematics partly because Wittgenstein's theory of equations cannot explain inequalities (Ramsey 1923: 475). Though it is doubtful that, in 1923, Wittgenstein would have thought these issues problematic, it certainly is true that the Tractarian theory of mathematics is essentially a sketch, especially in comparison with what Wittgenstein begins to develop six years later. After the completion of the Tractatus in 1918, Wittgenstein did virtually no philosophical work until February 2, 1929, eleven months after attending a lecture by the Dutch mathematician L.E.J. Brouwer. ## 2. The Middle Wittgenstein's Finitistic Constructivism There is little doubt that Wittgenstein was invigorated by L.E.J. Brouwer's March 10, 1928 Vienna lecture "Science, Mathematics, and Language" (Brouwer 1929), which he attended with F. Waismann and H. Feigl, but it is a gross overstatement to say that he returned to Philosophy because of this lecture or that his intermediate interest in the Philosophy of Mathematics issued primarily from Brouwer's influence. In fact, Wittgenstein's return to Philosophy and his intermediate work on mathematics is also due to conversations with Ramsey and members of the Vienna Circle, to Wittgenstein's disagreement with Ramsey over identity, and several other factors. Though Wittgenstein seems not to have read any Hilbert or Brouwer prior to the completion of the Tractatus, by early 1929 Wittgenstein had certainly read work by Brouwer, Weyl, Skolem, Ramsey (and possibly Hilbert) and, apparently, he had had one or more private discussions with Brouwer in 1928 (Finch 1977: 260; Van Dalen 2005: 566-567). Thus, the rudimentary treatment of mathematics in the Tractatus, whose principal influences were Russell and Frege, was succeeded by detailed work on mathematics in the middle period (1929-1933), which was strongly influenced by the 1920s work of Brouwer, Weyl, Hilbert, and Skolem. ### 2.1 Wittgenstein's Intermediate Constructive Formalism To best understand Wittgenstein's intermediate Philosophy of Mathematics, one must fully appreciate his strong variant of formalism, according to which "[w]e make mathematics" (WVC 34, note 1; PR SS159) by inventing purely formal mathematical calculi, with 'stipulated' axioms (PR SS202), syntactical rules of transformation, and decision procedures that enable us to invent "mathematical truth" and "mathematical falsity" by algorithmically deciding so-called mathematical 'propositions' (PR SSSS122, 162). The core idea of Wittgenstein's formalism from 1929 (if not 1918) through 1944 is that mathematics is essentially syntactical, devoid of reference and semantics. The most obvious aspect of this view, which has been noted by numerous commentators who do not refer to Wittgenstein as a 'formalist' (Kielkopf 1970: 360-38; Klenk 1976: 5, 8, 9; Fogelin 1968: 267; Frascolla 1994: 40; Marion 1998: 13-14), is that, contra Platonism, the signs and propositions of a mathematical calculus do not refer to anything. As Wittgenstein says at (WVC 34, note 1), "[n]umbers are not represented by proxies; numbers are there". This means not only that numbers are there in the use, it means that the numerals are the numbers, for "[a]rithmetic doesn't talk about numbers, it works with numbers" (PR SS109). > > > What arithmetic is concerned with is the schema $\|\|\|\|$.--But > does arithmetic talk about the lines I draw with pencil on > paper?--Arithmetic doesn't talk about the lines, it > operates with them. (PG 333) > > > In a similar vein, Wittgenstein says that (WVC 106) "mathematics is always a machine, a calculus" and "[a] calculus is an abacus, a calculator, a calculating machine", which "works by means of strokes, numerals, etc". The "justified side of formalism", according to Wittgenstein (WVC 105), is that mathematical symbols "lack a meaning" (i.e., Bedeutung)--they do not "go proxy for" things which are "their meaning[s]". > > > You could say arithmetic is a kind of geometry; i.e. what in geometry > are constructions on paper, in arithmetic are calculations (on > paper).--You could say it is a more general kind of geometry. > (PR SS109; PR SS111) > > > This is the core of Wittgenstein's life-long formalism. When we prove a theorem or decide a proposition, we operate in a purely formal, syntactical manner. In doing mathematics, we do not discover pre-existing truths that were "already there without one knowing" (PG 481)--we invent mathematics, bit-by-little-bit. "If you want to know what $2 + 2 = 4$ means", says Wittgenstein, "you have to ask how we work it out", because "we consider the process of calculation as the essential thing" (PG 333). Hence, the only meaning (i.e., sense) that a mathematical proposition has is intra-systemic meaning, which is wholly determined by its syntactical relations to other propositions of the calculus. A second important aspect of the intermediate Wittgenstein's strong formalism is his view that extra-mathematical application (and/or reference) is not a necessary condition of a mathematical calculus. Mathematical calculi do not require extra-mathematical applications, Wittgenstein argues, since we "can develop arithmetic completely autonomously and its application takes care of itself since wherever it's applicable we may also apply it" (PR SS109; cf. PG 308, WVC 104). As we shall shortly see, the middle Wittgenstein is also drawn to strong formalism by a new concern with questions of decidability. Undoubtedly influenced by the writings of Brouwer and David Hilbert, Wittgenstein uses strong formalism to forge a new connection between mathematical meaningfulness and algorithmic decidability. > > > An equation is a rule of syntax. Doesn't that explain why we > cannot have questions in mathematics that are in principle > unanswerable? For if the rules of syntax cannot be grasped, > they're of no use at all.... [This] makes intelligible the > attempts of the formalist to see mathematics as a game with signs. > (PR SS121) > > > In Section 2.3, we shall see how Wittgenstein goes beyond both Hilbert and Brouwer by maintaining the Law of the Excluded Middle in a way that restricts mathematical propositions to expressions that are algorithmically decidable. ### 2.2 Wittgenstein's Intermediate Finitism The single most important difference between the Early and Middle Wittgenstein is that, in the middle period, Wittgenstein rejects quantification over an infinite mathematical domain, stating that, contra his Tractarian view, such 'propositions' are not infinite conjunctions and infinite disjunctions simply because there are no such things. Wittgenstein's principal reasons for developing a finitistic Philosophy of Mathematics are as follows. 1. Mathematics as Human Invention: According to the middle Wittgenstein, we invent mathematics, from which it follows that mathematics and so-called mathematical objects do not exist independently of our inventions. Whatever is mathematical is fundamentally a product of human activity. 2. Mathematical Calculi Consist Exclusively of Intensions and Extensions: Given that we have invented only mathematical extensions (e.g., symbols, finite sets, finite sequences, propositions, axioms) and mathematical intensions (e.g., rules of inference and transformation, irrational numbers as rules), these extensions and intensions, and the calculi in which they reside, constitute the entirety of mathematics. (It should be noted that Wittgenstein's usage of 'extension' and 'intension' as regards mathematics differs markedly from standard contemporary usage, wherein the extension of a predicate is the set of entities that satisfy the predicate and the intension of a predicate is the meaning of, or expressed by, the predicate. Put succinctly, Wittgenstein thinks that the extension of this notion of concept-and-extension from the domain of existent (i.e., physical) objects to the so-called domain of "mathematical objects" is based on a faulty analogy and engenders conceptual confusion. See #1 just below.) These two reasons have at least five immediate consequences for Wittgenstein's Philosophy of Mathematics. 1. Rejection of Infinite Mathematical Extensions: Given that a mathematical extension is a symbol ('sign') or a finite concatenation of symbols extended in space, there is a categorical difference between mathematical intensions and (finite) mathematical extensions, from which it follows that "the mathematical infinite" resides only in recursive rules (i.e., intensions). An infinite mathematical extension (i.e., a completed, infinite mathematical extension) is a contradiction-in-terms 2. Rejection of Unbounded Quantification in Mathematics: Given that the mathematical infinite can only be a recursive rule, and given that a mathematical proposition must have sense, it follows that there cannot be an infinite mathematical proposition (i.e., an infinite logical product or an infinite logical sum). 3. Algorithmic Decidability vs. Undecidability: If mathematical extensions of all kinds are necessarily finite, then, in principle, all mathematical propositions are algorithmically decidable, from which it follows that an "undecidable mathematical proposition" is a contradiction-in-terms. Moreover, since mathematics is essentially what we have and what we know, Wittgenstein restricts algorithmic decidability to knowing how to decide a proposition with a known decision procedure. 4. Anti-Foundationalist Account of Real Numbers: Since there are no infinite mathematical extensions, irrational numbers are rules, not extensions. Given that an infinite set is a recursive rule (or an induction) and no such rule can generate all of the things mathematicians call (or want to call) "real numbers", it follows that there is no set of 'all' the real numbers and no such thing as the mathematical continuum. 5. Rejection of Different Infinite Cardinalities: Given the non-existence of infinite mathematical extensions, Wittgenstein rejects the standard interpretation of Cantor's diagonal proof as a proof of infinite sets of greater and lesser cardinalities. Since we invent mathematics in its entirety, we do not discover pre-existing mathematical objects or facts or that mathematical objects have certain properties, for "one cannot discover any connection between parts of mathematics or logic that was already there without one knowing" (PG 481). In examining mathematics as a purely human invention, Wittgenstein tries to determine what exactly we have invented and why exactly, in his opinion, we erroneously think that there are infinite mathematical extensions. If, first, we examine what we have invented, we see that we have invented formal calculi consisting of finite extensions and intensional rules. If, more importantly, we endeavour to determine why we believe that infinite mathematical extensions exist (e.g., why we believe that the actual infinite is intrinsic to mathematics), we find that we conflate mathematical intensions and mathematical extensions, erroneously thinking that there is "a dualism" of "the law and the infinite series obeying it" (PR SS180). For instance, we think that because a real number "endlessly yields the places of a decimal fraction" (PR SS186), it is "a totality" (WVC 81-82, note 1), when, in reality, "[a]n irrational number isn't the extension of an infinite decimal fraction,... it's a law" (PR SS181) which "yields extensions" (PR SS186). A law and a list are fundamentally different; neither can 'give' what the other gives (WVC 102-103). Indeed, "the mistake in the set-theoretical approach consists time and again in treating laws and enumerations (lists) as essentially the same kind of thing" (PG 461). Closely related with this conflation of intensions and extensions is the fact that we mistakenly act as if the word 'infinite' is a "number word", because in ordinary discourse we answer the question "how many?" with both (PG 463; cf. PR SS142). But "'[i]nfinite' is not a quantity", Wittgenstein insists (WVC 228); the word 'infinite' and a number word like 'five' do not have the same syntax. The words 'finite' and 'infinite' do not function as adjectives on the words 'class' or 'set', (WVC 102), for the terms "finite class" and "infinite class" use 'class' in completely different ways (WVC 228). An infinite class is a recursive rule or "an induction", whereas the symbol for a finite class is a list or extension (PG 461). It is because an induction has much in common with the multiplicity of a finite class that we erroneously call it an infinite class (PR SS158). In sum, because a mathematical extension is necessarily a finite sequence of symbols, an infinite mathematical extension is a contradiction-in-terms. This is the foundation of Wittgenstein's finitism. Thus, when we say, e.g., that "there are infinitely many even numbers", we are not saying "there are an infinite number of even numbers" in the same sense as we can say "there are 27 people in this house"; the infinite series of natural numbers is nothing but "the infinite possibility of finite series of numbers"--"[i]t is senseless to speak of the whole infinite number series, as if it, too, were an extension" (PR SS144). The infinite is understood rightly when it is understood, not as a quantity, but as an "infinite possibility" (PR SS138). Given Wittgenstein's rejection of infinite mathematical extensions, he adopts finitistic, constructive views on mathematical quantification, mathematical decidability, the nature of real numbers, and Cantor's diagonal proof of the existence of infinite sets of greater cardinalities. Since a mathematical set is a finite extension, we cannot meaningfully quantify over an infinite mathematical domain, simply because there is no such thing as an infinite mathematical domain (i.e., totality, set), and, derivatively, no such things as infinite conjunctions or disjunctions (G.E. Moore 1955: 2-3; cf. AWL 6; and PG 281). > > > [I]t still looks now as if the quantifiers make no sense for numbers. > I mean: you can't say '$(n) \phi n$', precisely > because 'all natural numbers' isn't a bounded > concept. But then neither should one say a general proposition follows > from a proposition about the nature of number. > > > > But in that case it seems to me that we can't use > generality--all, etc.--in mathematics at all. There's > no such thing as 'all numbers', simply because there are > infinitely many. (PR SS126; PR SS129) > > > 'Extensionalists' who assert that "$\varepsilon(0).\varepsilon(1).\varepsilon(2)$ and so on" is an infinite logical product (PG 452) assume or assert that finite and infinite conjunctions are close cousins--that the fact that we cannot write down or enumerate all of the conjuncts 'contained' in an infinite conjunction is only a "human weakness", for God could surely do so and God could surely survey such a conjunction in a single glance and determine its truth-value. According to Wittgenstein, however, this is not a matter of human limitation. Because we mistakenly think that "an infinite conjunction" is similar to "an enormous conjunction", we erroneously reason that just as we cannot determine the truth-value of an enormous conjunction because we don't have enough time, we similarly cannot, due to human limitations, determine the truth-value of an infinite conjunction (or disjunction). But the difference here is not one of degree but of kind: "in the sense in which it is impossible to check an infinite number of propositions it is also impossible to try to do so" (PG 452). This applies, according to Wittgenstein, to human beings, but more importantly, it applies also to God (i.e., an omniscient being), for even God cannot write down or survey infinitely many propositions because for him too the series is never-ending or limitless and hence the 'task' is not a genuine task because it cannot, in principle, be done (i.e., "infinitely many" is not a number word). As Wittgenstein says at (PR 128; cf. PG 479): "'Can God know all the places of the expansion of $\pi$?' would have been a good question for the schoolmen to ask", for the question is strictly 'senseless'. As we shall shortly see, on Wittgenstein's account, "[a] statement about all numbers is not represented by means of a proposition, but by means of induction" (WVC 82). Similarly, there is no such thing as a mathematical proposition about some number--no such thing as a mathematical proposition that existentially quantifies over an infinite domain (PR SS173). > > > What is the meaning of such a mathematical proposition as > '$(\exists n) 4 + n = 7$'? It might be a > disjunction--$(4 + 0 = 7) \vee{}$ $(4 + 1 = 7) \vee {}$ etc. ad > inf. But what does that mean? I can understand a proposition with > a beginning and an end. But can one also understand a proposition with > no end? (PR SS127) > > > We are particularly seduced by the feeling or belief that an infinite mathematical disjunction makes good sense in the case where we can provide a recursive rule for generating each next member of an infinite sequence. For example, when we say "There exists an odd perfect number" we are asserting that, in the infinite sequence of odd numbers, there is (at least) one odd number that is perfect--we are asserting '$\phi(1) \vee \phi(3) \vee \phi(5) \vee{}$ and so on' and we know what would make it true and what would make it false (PG 451). The mistake here made, according to Wittgenstein (PG 451), is that we are implicitly "comparing the proposition '$(\exists n)$...' with the proposition... 'There are two foreign words on this page'", which doesn't provide the grammar of the former 'proposition', but only indicates an analogy in their respective rules. On Wittgenstein's intermediate finitism, an expression quantifying over an infinite domain is never a meaningful proposition, not even when we have proved, for instance, that a particular number $n$ has a particular property. > > > The important point is that, even in the case where I am given that > $3^2 + 4^2 = 5^2$, I ought not to say '$(\exists x, > y, z, n) (x^n + y^n = z^n)$', since taken extensionally > that's meaningless, and taken intensionally this doesn't > provide a proof of it. No, in this case I ought to express only the > first equation. (PR SS150) > > > Thus, Wittgenstein adopts the radical position that all expressions that quantify over an infinite domain, whether 'conjectures' (e.g., Goldbach's Conjecture, the Twin Prime Conjecture) or "proved general theorems" (e.g., "Euclid's Prime Number Theorem", the Fundamental Theorem of Algebra), are meaningless (i.e., 'senseless'; 'sinnlos') expressions as opposed to "genuine mathematical proposition[s]" (PR SS168). These expressions are not (meaningful) mathematical propositions, according to Wittgenstein, because the Law of the Excluded Middle does not apply, which means that "we aren't dealing with propositions of mathematics" (PR SS151). The crucial question why and in exactly what sense the Law of the Excluded Middle does not apply to such expressions will be answered in the next section. ### 2.3 Wittgenstein's Intermediate Finitism and Algorithmic Decidability The middle Wittgenstein has other grounds for rejecting unrestricted quantification in mathematics, for on his idiosyncratic account, we must distinguish between four categories of concatenations of mathematical symbols. 1. Proved mathematical propositions in a particular mathematical calculus (no need for "mathematical truth"). 2. Refuted mathematical propositions in (or of) a particular mathematical calculus (no need for "mathematical falsity"). 3. Mathematical propositions for which we know we have in hand an applicable and effective decision procedure (i.e., we know how to decide them). 4. Concatenations of symbols that are not part of any mathematical calculus and which, for that reason, are not mathematical propositions (i.e., are non-propositions). In his 2004 (p. 18), Mark van Atten says that > > ... [i]ntuitionistically, there are four ["possibilities > for a proposition with respect to truth"]: > > > 1. $p$ has been experienced as true > 2. $p$ has been experienced as false > 3. Neither 1 nor 2 has occurred yet, but we know a procedure to > decide $p$ (i.e., a procedure that will prove $p$ or prove $\neg > p)$ > 4. Neither 1 nor 2 has occurred yet, and we do not know a procedure > to decide $p$. > > > What is immediately striking about Wittgenstein's ##1-3 and Brouwer's ##1-3 (Brouwer 1955: 114; 1981: 92) is the enormous similarity. And yet, for all of the agreement, the disagreement in #4 is absolutely crucial. As radical as the respective #3s are, Brouwer and Wittgenstein agree that an undecided $\phi$ is a mathematical proposition (for Wittgenstein, of a particular mathematical calculus) if we know of an applicable decision procedure. They also agree that until $\phi$ is decided, it is neither true nor false (though, for Wittgenstein, 'true' means no more than "proved in calculus $\Gamma$"). What they disagree about is the status of an ordinary mathematical conjecture, such as Goldbach's Conjecture. Brouwer admits it as a mathematical proposition, while Wittgenstein rejects it because we do not know how to algorithmically decide it. Like Brouwer (1948 [1983: 90]), Wittgenstein holds that there are no "unknown truth[s]" in mathematics, but unlike Brouwer he denies the existence of "undecidable propositions" on the grounds that such a 'proposition' would have no 'sense', "and the consequence of this is precisely that the propositions of logic lose their validity for it" (PR SS173). In particular, if there are undecidable mathematical propositions (as Brouwer maintains), then at least some mathematical propositions are not propositions of any existent mathematical calculus. For Wittgenstein, however, it is a defining feature of a mathematical proposition that it is either decided or decidable by a known decision procedure in a mathematical calculus. As Wittgenstein says at (PR SS151), > > > where the law of the excluded middle doesn't apply, no other law > of logic applies either, because in that case we aren't dealing > with propositions of mathematics. (Against Weyl and Brouwer). > > > The point here is not that we need truth and falsity in mathematics--we don't--but rather that every mathematical proposition (including ones for which an applicable decision procedure is known) is known to be part of a mathematical calculus. To maintain this position, Wittgenstein distinguishes between (meaningful, genuine) mathematical propositions, which have mathematical sense, and meaningless, senseless ('sinnlos') expressions by stipulating that an expression is a meaningful (genuine) proposition of a mathematical calculus iff we know of a proof, a refutation, or an applicable decision procedure (PR SS151; PG 452; PG 366; AWL 199-200). "Only where there's a method of solution [a 'logical method for finding a solution'] is there a [mathematical] problem", he tells us (PR SSSS149, 152; PG 393). "We may only put a question in mathematics (or make a conjecture)", he adds (PR SS151), "where the answer runs: 'I must work it out'". At (PG 468), Wittgenstein emphasizes the importance of algorithmic decidability clearly and emphatically: > > > In mathematics everything is algorithm and nothing > is meaning [Bedeutung]; even when it > doesn't look like that because we seem to be using > words to talk about mathematical things. Even these > words are used to construct an algorithm. > > > When, therefore, Wittgenstein says (PG 368) that if "[the Law of the Excluded Middle] is supposed not to hold, we have altered the concept of proposition", he means that an expression is only a meaningful mathematical proposition if we know of an applicable decision procedure for deciding it (PG 400). If a genuine mathematical proposition is undecided, the Law of the Excluded Middle holds in the sense that we know that we will prove or refute the proposition by applying an applicable decision procedure (PG 379, 387). For Wittgenstein, there simply is no distinction between syntax and semantics in mathematics: everything is syntax. If we wish to demarcate between "mathematical propositions" versus "mathematical pseudo-propositions", as we do, then the only way to ensure that there is no such thing as a meaningful, but undecidable (e.g., independent), proposition of a given calculus is to stipulate that an expression is only a meaningful proposition in a given calculus (PR SS153) if either it has been decided or we know of an applicable decision procedure. In this manner, Wittgenstein defines both a mathematical calculus and a mathematical proposition in epistemic terms. A calculus is defined in terms of stipulations (PR SS202; PG 369), known rules of operation, and known decision procedures, and an expression is only a mathematical proposition in a given calculus (PR SS155), and only if that calculus contains (PG 379) a known (and applicable) decision procedure, for "you cannot have a logical plan of search for a sense you don't know" (PR SS148). Thus, the middle Wittgenstein rejects undecidable mathematical propositions on two grounds. First, number-theoretic expressions that quantify over an infinite domain are not algorithmically decidable, and hence are not meaningful mathematical propositions. > > > If someone says (as Brouwer does) that for $(x) f\_1 x = f\_2 x$, > there is, as well as yes and no, also the case of undecidability, this > implies that '$(x)$...' is meant extensionally and > we may talk of the case in which all $x$ happen to have a property. > In truth, however, it's impossible to talk of such a case at all > and the '$(x)$...' in arithmetic cannot be taken > extensionally. (PR SS174) > > > "Undecidability", says Wittgenstein (PR SS174) "presupposes... that the bridge cannot be made with symbols", when, in fact, "[a] connection between symbols which exists but cannot be represented by symbolic transformations is a thought that cannot be thought", for "[i]f the connection is there,... it must be possible to see it". Alluding to algorithmic decidability, Wittgenstein stresses (PR SS174) that "[w]e can assert anything which can be checked in practice", because "it's a question of the possibility of checking" (italics added). Wittgenstein's second reason for rejecting an undecidable mathematical proposition is that it is a contradiction-in-terms. There cannot be "undecidable propositions", Wittgenstein argues (PR SS173), because an expression that is not decidable in some actual calculus is simply not a mathematical proposition, since "every proposition in mathematics must belong to a calculus of mathematics" (PG 376). This radical position on decidability results in various radical and counter-intuitive statements about unrestricted mathematical quantification, mathematical induction, and, especially, the sense of a newly proved mathematical proposition. In particular, Wittgenstein asserts that uncontroversial mathematical conjectures, such as Goldbach's Conjecture (hereafter 'GC') and the erstwhile conjecture "Fermat's Last Theorem" (hereafter 'FLT'), have no sense (or, perhaps, no determinate sense) and that the unsystematic proof of such a conjecture gives it a sense that it didn't previously have (PG 374) because > > > it's unintelligible that I should admit, when I've got the > proof, that it's a proof of precisely this proposition, > or of the induction meant by this proposition. (PR > SS155) > > > > > > Thus Fermat's [Last Theorem] makes no sense until I can > search for a solution to the equation in cardinal numbers. > And 'search' must always mean: search systematically. > Meandering about in infinite space on the look-out for a gold ring is > no kind of search. (PR SS150) > > > > I say: the so-called 'Fermat's Last Theorem' > isn't a proposition. (Not even in the sense of a proposition of > arithmetic.) Rather, it corresponds to an induction. (PR > SS189) > > > To see how Fermat's Last Theorem isn't a proposition and how it might correspond to an induction, we need to examine Wittgenstein's account of mathematical induction. ### 2.4 Wittgenstein's Intermediate Account of Mathematical Induction and Algorithmic Decidability Given that one cannot quantify over an infinite mathematical domain, the question arises: What, if anything, does any number-theoretic proof by mathematical induction actually prove? On the standard view, a proof by mathematical induction has the following paradigmatic form. \| \| \| \| --- \| --- \| \| Inductive Base: \| $\phi(1)$ \| \| Inductive Step: \| $\forall n(\phi(n) \rightarrow \phi(n + 1))$ \| \| Conclusion: \| $\forall n\phi(n)$ \| If, however, "$\forall n\phi(n)$" is not a meaningful (genuine) mathematical proposition, what are we to make of this proof? Wittgenstein's initial answer to this question is decidedly enigmatic. "An induction is the expression for arithmetical generality", but "induction isn't itself a proposition" (PR SS129). > > > We are not saying that when $f(1)$ holds and when $f(c + 1)$ > follows from $f(c)$, the proposition $f(x)$ is therefore > true of all cardinal numbers: but: "the proposition $f(x)$ > holds for all cardinal numbers" means "it holds > for $x = 1$, and $f(c + 1)$ follows from $f(c)$". > (PG 406) > > > In a proof by mathematical induction, we do no actually prove the 'proposition' [e.g., $\forall n\phi(n)$] that is customarily construed as the conclusion of the proof (PG 406, 374; PR SS164), rather this pseudo-proposition or 'statement' stands 'proxy' for the "infinite possibility" (i.e., "the induction") that we come to 'see' by means of the proof (WVC 135). "I want to say", Wittgenstein concludes, that "once you've got the induction, it's all over" (PG 407). Thus, on Wittgenstein's account, a particular proof by mathematical induction should be understood in the following way. \| \| \| \| --- \| --- \| \| Inductive Base: \| $\phi(1)$ \| \| Inductive Step: \| $\phi(n) \rightarrow \phi(n + 1)$ \| \| Proxy Statement: \| $\phi(m)$ \| Here the 'conclusion' of an inductive proof [i.e., "what is to be proved" (PR SS164)] uses '$m$' rather than '$n$' to indicate that '$m$' stands for any particular number, while '$n$' stands for any arbitrary number. For Wittgenstein, the proxy statement "$\phi(m)$" is not a mathematical proposition that "assert[s] its generality" (PR SS168), it is an eliminable pseudo-proposition standing proxy for the proved inductive base and inductive step. Though an inductive proof cannot prove "the infinite possibility of application" (PR SS163), it enables us "to perceive" that a direct proof of any particular proposition can be constructed (PR SS165). For example, once we have proved "$\phi(1)$" and "$\phi(n) \rightarrow \phi(n + 1)$", we need not reiterate modus ponens $m - 1$ times to prove the particular proposition "$\phi(m)$" (PR SS164). The direct proof of, say, "$\phi$(714)" (i.e., without 713 iterations of modus ponens) "cannot have a still better proof, say, by my carrying out the derivation as far as this proposition itself" (PR SS165). A second, very important impetus for Wittgenstein's radically constructivist position on mathematical induction is his rejection of an undecidable mathematical proposition. > > > In discussions of the provability of mathematical propositions it is > sometimes said that there are substantial propositions of mathematics > whose truth or falsehood must remain undecided. What the people who > say that don't realize is that such propositions, if we > can use them and want to call them "propositions", are not > at all the same as what are called "propositions" in other > cases; because a proof alters the grammar of a proposition. > (PG 367) > > > In this passage, Wittgenstein is alluding to Brouwer, who, as early as 1907 and 1908, states, first, that "the question of the validity of the principium tertii exclusi is equivalent to the question whether unsolvable mathematical problems exist", second, that "[t]here is not a shred of a proof for the conviction... that there exist no unsolvable mathematical problems", and, third, that there are meaningful propositions/'questions', such as "Do there occur in the decimal expansion of $\pi$ infinitely many pairs of consecutive equal digits?", to which the Law of the Excluded Middle does not apply because "it must be considered as uncertain whether problems like [this] are solvable" (Brouwer, 1908 [1975: 109-110]). "A fortiori it is not certain that any mathematical problem can either be solved or proved to be unsolvable", Brouwer says (1907 [1975: 79]), "though HILBERT, in 'Mathematische Probleme', believes that every mathematician is deeply convinced of it". Wittgenstein takes the same data and, in a way, draws the opposite conclusion. If, as Brouwer says, we are uncertain whether all or some "mathematical problems" are solvable, then we know that we do not have in hand an applicable decision procedure, which means that the alleged mathematical propositions are not decidable, here and now. "What 'mathematical questions' share with genuine questions", Wittgenstein says (PR SS151), "is simply that they can be answered". This means that if we do not know how to decide an expression, then we do not know how to make it either proved (true) or refuted (false), which means that the Law of the Excluded Middle "doesn't apply" and, therefore, that our expression is not a mathematical proposition. Together, Wittgenstein's finitism and his criterion of algorithmic decidability shed considerable light on his highly controversial remarks about putatively meaningful conjectures such as FLT and GC. GC is not a mathematical proposition because we do not know how to decide it, and if someone like G. H. Hardy says that he 'believes' GC is true (PG 381; LFM 123; PI SS578), we must answer that s/he only "has a hunch about the possibilities of extension of the present system" (LFM 139)--that one can only believe such an expression is 'correct' if one knows how to prove it. The only sense in which GC (or FLT) can be proved is that it can "correspond to a proof by induction", which means that the unproved inductive step (e.g., "$G(n) \rightarrow G(n + 1)$") and the expression "$\forall nG(n)$" are not mathematical propositions because we have no algorithmic means of looking for an induction (PG 367). A "general proposition" is senseless prior to an inductive proof "because the question would only have made sense if a general method of decision had been known before the particular proof was discovered" (PG 402). Unproved 'inductions' or inductive steps are not meaningful propositions because the Law of the Excluded Middle does not hold in the sense that we do not know of a decision procedure by means of which we can prove or refute the expression (PG 400; WVC 82). This position, however, seems to rob us of any reason to search for a 'decision' of a meaningless 'expression' such as GC. The intermediate Wittgenstein says only that "[a] mathematician is... guided by... certain analogies with the previous system" and that there is nothing "wrong or illegitimate if anyone concerns himself with Fermat's Last Theorem" (WVC 144). > > > If e.g. I have a method for looking at integers that satisfy the > equation $x^2 + y^2 = z^2$, then the formula $x^{n} + y^n = z^{n}$ > may stimulate me. I may let a formula stimulate me. Thus I shall say, > Here there is a stimulus--but not a question. > Mathematical problems are always such stimuli. (WVC 144, Jan. > 1, 1931) > > > More specifically, a mathematician may let a senseless conjecture such as FLT stimulate her/him if s/he wishes to know whether a calculus can be extended without altering its axioms or rules (LFM 139). > > > What is here going [o]n [in an attempt to decide GC] is an > unsystematic attempt at constructing a calculus. If the attempt is > successful, I shall again have a calculus in front of me, only a > different one from the calculus I have been using so far. > (WVC 174-75; Sept. 21, 1931; italics added) > > > If, e.g., we succeed in proving GC by mathematical induction (i.e., we prove "$G(1)$" and "$G(n) \rightarrow G(n + 1)$"), we will then have a proof of the inductive step, but since the inductive step was not algorithmically decidable beforehand (PR SSSS148, 155, 157; PG 380), in constructing the proof we have constructed a new calculus, a new calculating machine (WVC 106) in which we now know how to use this new "machine-part" (RFM VI, SS13) (i.e., the unsystematically proved inductive step). Before the proof, the inductive step is not a mathematical proposition with sense (in a particular calculus), whereas after the proof the inductive step is a mathematical proposition, with a new, determinate sense, in a newly created calculus. This demarcation of expressions without mathematical sense and proved or refuted propositions, each with a determinate sense in a particular calculus, is a view that Wittgenstein articulates in myriad different ways from 1929 through 1944. Whether or not it is ultimately defensible--and this is an absolutely crucial question for Wittgenstein's Philosophy of Mathematics--this strongly counter-intuitive aspect of Wittgenstein's account of algorithmic decidability, proof, and the sense of a mathematical proposition is a piece with his rejection of predeterminacy in mathematics. Even in the case where we algorithmically decide a mathematical proposition, the connections thereby made do not pre-exist the algorithmic decision, which means that even when we have a "mathematical question" that we decide by decision procedure, the expression only has a determinate sense qua proposition when it is decided. On Wittgenstein's account, both middle and later, "[a] new proof gives the proposition a place in a new system" (RFM VI, SS13), it "locates it in the whole system of calculations", though it "does not mention, certainly does not describe, the whole system of calculation that stands behind the proposition and gives it sense" (RFM VI, SS11). Wittgenstein's unorthodox position here is a type of structuralism that partially results from his rejection of mathematical semantics. We erroneously think, e.g., that GC has a fully determinate sense because, given "the misleading way in which the mode of expression of word-language represents the sense of mathematical propositions" (PG 375), we call to mind false pictures and mistaken, referential conceptions of mathematical propositions whereby GC is about a mathematical reality and so has just a determinate sense as "There exist intelligent beings elsewhere in the universe" (i.e., a proposition that is determinately true or false, whether or not we ever know its truth-value). Wittgenstein breaks with this tradition, in all of its forms, stressing that, in mathematics, unlike the realm of contingent (or empirical) propositions, "if I am to know what a proposition like Fermat's last theorem says", I must know its criterion of truth. Unlike the criterion of truth for an empirical proposition, which can be known before the proposition is decided, we cannot know the criterion of truth for an undecided mathematical proposition, though we are "acquainted with criteria for the truth of similar propositions" (RFM VI, SS13). ### 2.5 Wittgenstein's Intermediate Account of Irrational Numbers The intermediate Wittgenstein spends a great deal of time wrestling with real and irrational numbers. There are two distinct reasons for this. First, the real reason many of us are unwilling to abandon the notion of the actual infinite in mathematics is the prevalent conception of an irrational number as a necessarily infinite extension. "The confusion in the concept of the 'actual infinite' arises" (italics added), says Wittgenstein (PG 471), > > > from the unclear concept of irrational number, that is, from the fact > that logically very different things are called 'irrational > numbers' without any clear limit being given to the concept. > > > Second, and more fundamentally, the intermediate Wittgenstein wrestles with irrationals in such detail because he opposes foundationalism and especially its concept of a "gapless mathematical continuum", its concept of a comprehensive theory of the real numbers (Han 2010), and set theoretical conceptions and 'proofs' as a foundation for arithmetic, real number theory, and mathematics as a whole. Indeed, Wittgenstein's discussion of irrationals is one with his critique of set theory, for, as he says, "[m]athematics is ridden through and through with the pernicious idioms of set theory", such as "the way people speak of a line as composed of points", when, in fact, "[a] line is a law and isn't composed of anything at all" (PR SS173; PR SSSS181, 183, & 191; PG 373, 460, 461, & 473). #### 2.5.1 Wittgenstein's Anti-Foundationalism and Genuine Irrational Numbers Since, on Wittgenstein's terms, mathematics consists exclusively of extensions and intensions (i.e., 'rules' or 'laws'), an irrational is only an extension insofar as it is a sign (i.e., a 'numeral', such as '$\sqrt{2}$' or '$\pi$'). Given that there is no such thing as an infinite mathematical extension, it follows that an irrational number is not a unique infinite expansion, but rather a unique recursive rule or law (PR SS181) that yields rational numbers (PR SS186; PR SS180). > > > The rule for working out places of $\sqrt{2}$ is itself the numeral > for the irrational number; and the reason I here speak of a > 'number' is that I can calculate with these signs (certain > rules for the construction of rational numbers) just as I can with > rational numbers themselves. (PG 484) > > > Due, however, to his anti-foundationalism, Wittgenstein takes the radical position that not all recursive real numbers (i.e., computable numbers) are genuine real numbers--a position that distinguishes his view from even Brouwer's. The problem, as Wittgenstein sees it, is that mathematicians, especially foundationalists (e.g., set theorists), have sought to accommodate physical continuity by a theory that 'describes' the mathematical continuum (PR SS171). When, for example, we think of continuous motion and the (mere) density of the rationals, we reason that if an object moves continuously from A to B, and it travels only the distances marked by "rational points", then it must skip some distances (intervals, or points) not marked by rational numbers. But if an object in continuous motion travels distances that cannot be commensurately measured by rationals alone, there must be 'gaps' between the rationals (PG 460), and so we must fill them, first, with recursive irrationals, and then, because "the set of all recursive irrationals" still leaves gaps, with "lawless irrationals". > > > [T]he enigma of the continuum arises because language misleads us into > applying to it a picture that doesn't fit. Set theory preserves > the inappropriate picture of something discontinuous, but makes > statements about it that contradict the picture, under the impression > that it is breaking with prejudices; whereas what should really have > been done is to point out that the picture just doesn't > fit... (PG 471) > > > We add nothing that is needed to the differential and integral calculi by 'completing' a theory of real numbers with pseudo-irrationals and lawless irrationals, first because there are no gaps on the number line (PR SSSS181, 183, & 191; PG 373, 460, 461, & 473; WVC 35) and, second, because these alleged irrational numbers are not needed for a theory of the 'continuum' simply because there is no mathematical continuum. As the later Wittgenstein says (RFM V, SS32), "[t]he picture of the number line is an absolutely natural one up to a certain point; that is to say so long as it is not used for a general theory of real numbers". We have gone awry by misconstruing the nature of the geometrical line as a continuous collection of points, each with an associated real number, which has taken us well beyond the 'natural' picture of the number line in search of a "general theory of real numbers" (Han 2010). Thus, the principal reason Wittgenstein rejects certain constructive (computable) numbers is that they are unnecessary creations which engender conceptual confusions in mathematics (especially set theory). One of Wittgenstein's main aims in his lengthy discussions of rational numbers and pseudo-irrationals is to show that pseudo-irrationals, which are allegedly needed for the mathematical continuum, are not needed at all. To this end, Wittgenstein demands (a) that a real number must be "compar[able] with any rational number taken at random" (i.e., "it can be established whether it is greater than, less than, or equal to a rational number" (PR SS191)) and (b) that "[a] number must measure in and of itself" and if a 'number' "leaves it to the rationals, we have no need of it" (PR SS191) (Frascolla 1980: 242-243; Shanker 1987: 186-192; Da Silva 1993: 93-94; Marion 1995a: 162, 164; Rodych 1999b, 281-291; Lampert 2009). To demonstrate that some recursive (computable) reals are not genuine real numbers because they fail to satisfy (a) and (b), Wittgenstein defines the putative recursive real number \[ \substack{5 \rightarrow 3 \\ \sqrt{2}} \] as the rule "Construct the decimal expansion for $\sqrt{2}$, replacing every occurrence of a '5' with a '3'" (PR SS182); he similarly defines $\pi '$ as \[ \substack{7 \rightarrow 3\\ \pi} \] (PR SS186) and, in a later work, redefines $\pi '$ as \[ \substack{777 \rightarrow 000 \\ \pi} \] (PG 475). Although a pseudo-irrational such as $\pi '$ (on either definition) is "as unambiguous as ... $\pi$ or $\sqrt{2}$" (PG 476), it is 'homeless' according to Wittgenstein because, instead of using "the idioms of arithmetic" (PR SS186), it is dependent upon the particular 'incidental' notation of a particular system (i.e., in some particular base) (PR SS188; PR SS182; and PG 475). If we speak of various base-notational systems, we might say that $\pi$ belongs to all systems, while $\pi '$ belongs only to one, which shows that $\pi '$ is not a genuine irrational because "there can't be irrational numbers of different types" (PR SS180). Furthermore, pseudo-irrationals do not measure because they are homeless, artificial constructions parasitic upon numbers which have a natural place in a calculus that can be used to measure. We simply do not need these aberrations, because they are not sufficiently comparable to rationals and genuine irrationals. They are not irrational numbers according to Wittgenstein's criteria, which define, Wittgenstein interestingly asserts, "precisely what has been meant or looked for under the name 'irrational number'" (PR SS191). For exactly the same reason, if we define a "lawless irrational" as either (a) a non-rule-governed, non-periodic, infinite expansion in some base, or (b) a "free-choice sequence", Wittgenstein rejects "lawless irrationals" because, insofar as they are not rule-governed, they are not comparable to rationals (or irrationals) and they are not needed. > > > [W]e cannot say that the decimal fractions developed in accordance > with a law still need supplementing by an infinite set of irregular > infinite decimal fractions that would be 'brushed under the > carpet' if we were to restrict ourselves to those > generated by a law, > > > Wittgenstein argues, for "[w]here is there such an infinite decimal that is generated by no law" "[a]nd how would we notice that it was missing?" (PR SS181; cf. PG 473, 483-84). Similarly, a free-choice sequence, like a recipe for "endless bisection" or "endless dicing", is not an infinitely complicated mathematical law (or rule), but rather no law at all, for after each individual throw of a coin, the point remains "infinitely indeterminate" (PR SS186). For closely related reasons, Wittgenstein ridicules the Multiplicative Axiom (Axiom of Choice) both in the middle period (PR SS146) and in the latter period (RFM V, SS25; VII, SS33). #### 2.5.2 Wittgenstein's Real Number Essentialism and the Dangers of Set Theory Superficially, at least, it seems as if Wittgenstein is offering an essentialist argument for the conclusion that real number arithmetic should not be extended in such-and-such a way. Such an essentialist account of real and irrational numbers seems to conflict with the actual freedom mathematicians have to extend and invent, with Wittgenstein's intermediate claim (PG 334) that "[f]or [him] one calculus is as good as another", and with Wittgenstein's acceptance of complex and imaginary numbers. Wittgenstein's foundationalist critic (e.g., set theorist) will undoubtedly say that we have extended the term "irrational number" to lawless and pseudo-irrationals because they are needed for the mathematical continuum and because such "conceivable numbers" are much more like rule-governed irrationals than rationals. Though Wittgenstein stresses differences where others see similarities (LFM 15), in his intermediate attacks on pseudo-irrationals and foundationalism, he is not just emphasizing differences, he is attacking set theory's "pernicious idioms" (PR SS173) and its "crudest imaginable misinterpretation of its own calculus" (PG 469-70) in an attempt to dissolve "misunderstandings without which [set theory] would never have been invented", since it is "of no other use" (LFM 16-17). Complex and imaginary numbers have grown organically within mathematics, and they have proved their mettle in scientific applications, but pseudo-irrationals are inorganic creations invented solely for the sake of mistaken foundationalist aims. Wittgenstein's main point is not that we cannot create further recursive real numbers--indeed, we can create as many as we want--his point is that we can only really speak of different systems (sets) of real numbers (RFM II, SS33) that are enumerable by a rule, and any attempt to speak of "the set of all real numbers" or any piecemeal attempt to add or consider new recursive reals (e.g., diagonal numbers) is a useless and/or futile endeavour based on foundational misconceptions. Indeed, in 1930 manuscript and typescript (hereafter MS and TS, respectively) passages on irrationals and Cantor's diagonal, which were not included in PR or PG, Wittgenstein says: "The concept 'irrational number' is a dangerous pseudo-concept" (MS 108, 176; 1930; TS 210, 29; 1930). As we shall see in the next section, on Wittgenstein's account, if we do not understand irrationals rightly, we cannot but engender the mistakes that constitute set theory. ### 2.6 Wittgenstein's Intermediate Critique of Set Theory Wittgenstein's critique of set theory begins somewhat benignly in the Tractatus, where he denounces Logicism and says (6.031) that "[t]he theory of classes is completely superfluous in mathematics" because, at least in part, "the generality required in mathematics is not accidental generality". In his middle period, Wittgenstein begins a full-out assault on set theory that never abates. Set theory, he says, is "utter nonsense" (PR SSSS145, 174; WVC 102; PG 464, 470), 'wrong' (PR SS174), and 'laughable' (PG 464); its "pernicious idioms" (PR SS173) mislead us and the crudest possible misinterpretation is the very impetus of its invention (Hintikka 1993: 24, 27). Wittgenstein's intermediate critique of transfinite set theory (hereafter "set theory") has two main components: (1) his discussion of the intension-extension distinction, and (2) his criticism of non-denumerability as cardinality. Late in the middle period, Wittgenstein seems to become more aware of the unbearable conflict between his strong formalism (PG 334) and his denigration of set theory as a purely formal, non-mathematical calculus (Rodych 1997: 217-219), which, as we shall see in Section 3.5, leads to the use of an extra-mathematical application criterion to demarcate transfinite set theory (and other purely formal sign-games) from mathematical calculi. #### 2.6.1 Intensions, Extensions, and the Fictitious Symbolism of Set Theory The search for a comprehensive theory of the real numbers and mathematical continuity has led to a "fictitious symbolism" (PR SS174). > > > Set theory attempts to grasp the infinite at a more general level than > the investigation of the laws of the real numbers. It says that you > can't grasp the actual infinite by means of mathematical > symbolism at all and therefore it can only be described and not > represented. ... One might say of this theory that it buys a pig > in a poke. Let the infinite accommodate itself in this box as best it > can. (PG 468; cf. PR SS170) > > > As Wittgenstein puts it at (PG 461), > > > the mistake in the set-theoretical approach consists time and again in > treating laws and enumerations (lists) as essentially the same kind of > thing and arranging them in parallel series so that one fills in gaps > left by the other. > > > This is a mistake because it is 'nonsense' to say "we cannot enumerate all the numbers of a set, but we can give a description", for "[t]he one is not a substitute for the other" (WVC 102; June 19, 1930); "there isn't a dualism [of] the law and the infinite series obeying it" (PR SS180). "Set theory is wrong" and nonsensical (PR SS174), says Wittgenstein, because it presupposes a fictitious symbolism of infinite signs (PG 469) instead of an actual symbolism with finite signs. The grand intimation of set theory, which begins with "Dirichlet's concept of a function" (WVC 102-03), is that we can in principle represent an infinite set by an enumeration, but because of human or physical limitations, we will instead describe it intensionally. But, says Wittgenstein, "[t]here can't be possibility and actuality in mathematics", for mathematics is an actual calculus, which "is concerned only with the signs with which it actually operates" (PG 469). As Wittgenstein puts it at (PR SS159), the fact that "we can't describe mathematics, we can only do it" in and "of itself abolishes every 'set theory'". Perhaps the best example of this phenomenon is Dedekind, who in giving his 'definition' of an "infinite class" as "a class which is similar to a proper subclass of itself" (PG 464), "tried to describe an infinite class" (PG 463). If, however, we try to apply this 'definition' to a particular class in order to ascertain whether it is finite or infinite, the attempt is 'laughable' if we apply it to a finite class, such as "a certain row of trees", and it is 'nonsense' if we apply it to "an infinite class", for we cannot even attempt "to co-ordinate it" (PG 464), because "the relation $m = 2n$ [does not] correlate the class of all numbers with one of its subclasses" (PR SS141), it is an "infinite process" which "correlates any arbitrary number with another". So, although we can use $m = 2n$ on the rule for generating the naturals (i.e., our domain) and thereby construct the pairs (2,1), (4,2), (6,3), (8,4), etc., in doing so we do not correlate two infinite sets or extensions (WVC 103). If we try to apply Dedekind's definition as a criterion for determining whether a given set is infinite by establishing a 1-1 correspondence between two inductive rules for generating "infinite extensions", one of which is an "extensional subset" of the other, we can't possibly learn anything we didn't already know when we applied the 'criterion' to two inductive rules. If Dedekind or anyone else insists on calling an inductive rule an "infinite set", he and we must still mark the categorical difference between such a set and a finite set with a determinate, finite cardinality. Indeed, on Wittgenstein's account, the failure to properly distinguish mathematical extensions and intensions is the root cause of the mistaken interpretation of Cantor's diagonal proof as a proof of the existence of infinite sets of lesser and greater cardinality. #### 2.6.2 Against Non-Denumerability Wittgenstein's criticism of non-denumerability is primarily implicit during the middle period. Only after 1937 does he provide concrete arguments purporting to show, e.g., that Cantor's diagonal cannot prove that some infinite sets have greater 'multiplicity' than others. Nonetheless, the intermediate Wittgenstein clearly rejects the notion that a non-denumerably infinite set is greater in cardinality than a denumerably infinite set. > > > When people say 'The set of all transcendental numbers is > greater than that of algebraic numbers', that's nonsense. > The set is of a different kind. It isn't 'no longer' > denumerable, it's simply not denumerable! (PR > SS174) > > > As with his intermediate views on genuine irrationals and the Multiplicative Axiom, Wittgenstein here looks at the diagonal proof of the non-denumerability of "the set of transcendental numbers" as one that shows only that transcendental numbers cannot be recursively enumerated. It is nonsense, he says, to go from the warranted conclusion that these numbers are not, in principle, enumerable to the conclusion that the set of transcendental numbers is greater in cardinality than the set of algebraic numbers, which is recursively enumerable. What we have here are two very different conceptions of a number-type. In the case of algebraic numbers, we have a decision procedure for determining of any given number whether or not it is algebraic, and we have a method of enumerating the algebraic numbers such that we can see that 'each' algebraic number "will be" enumerated. In the case of transcendental numbers, on the other hand, we have proofs that some numbers are transcendental (i.e., non-algebraic), and we have a proof that we cannot recursively enumerate each and every thing we would call a "transcendental number". At (PG 461), Wittgenstein similarly speaks of set theory's "mathematical pseudo-concepts" leading to a fundamental difficulty, which begins when we unconsciously presuppose that there is sense to the idea of ordering the rationals by size--"that the attempt is thinkable"--and culminates in similarly thinking that it is possible to enumerate the real numbers, which we then discover is impossible. Though the intermediate Wittgenstein certainly seems highly critical of the alleged proof that some infinite sets (e.g., the reals) are greater in cardinality than other infinite sets, and though he discusses the "diagonal procedure" in February 1929 and in June 1930 (MS 106, 266; MS 108, 180), along with a diagonal diagram, these and other early-middle ruminations did not make it into the typescripts for either PR or PG. As we shall see in Section 3.4, the later Wittgenstein analyzes Cantor's diagonal and claims of non-denumerability in some detail. ## 3. The Later Wittgenstein on Mathematics: Some Preliminaries The first and most important thing to note about Wittgenstein's later Philosophy of Mathematics is that RFM, first published in 1956, consists of selections taken from a number of manuscripts (1937-1944), most of one large typescript (1938), and three short typescripts (1938), each of which constitutes an Appendix to (RFM I). For this reason and because some manuscripts containing much material on mathematics (e.g., MS 123) were not used at all for RFM, philosophers have not been able to read Wittgenstein's later remarks on mathematics as they were written in the manscripts used for RFM and they have not had access (until the 2000-2001 release of the Nachlass on CD-ROM) to much of Wittgenstein's later work on mathematics. It must be emphasized, therefore, that this Encyclopedia article is being written during a transitional period. Until philosophers have used the Nachlass to build a comprehensive picture of Wittgenstein's complete and evolving Philosophy of Mathematics, we will not be able to say definitively which views the later Wittgenstein retained, which he changed, and which he dropped. In the interim, this article will outline Wittgenstein's later Philosophy of Mathematics, drawing primarily on RFM, to a much lesser extent LFM (1939 Cambridge lectures), and, where possible, previously unpublished material in Wittgenstein's Nachlass. It should also be noted at the outset that commentators disagree about the continuity of Wittgenstein's middle and later Philosophies of Mathematics. Some argue that the later views are significantly different from the intermediate views (Frascolla 1994; Gerrard 1991: 127, 131-32; Floyd 2005: 105-106), while others argue that, for the most part, Wittgenstein's Philosophy of Mathematics evolves from the middle to the later period without significant changes or renunciations (Wrigley 1993; Marion 1998). The remainder of this article adopts the second interpretation, explicating Wittgenstein's later Philosophy of Mathematics as largely continuous with his intermediate views, except for the important introduction of an extra-mathematical application criterion. ### 3.1 Mathematics as a Human Invention Perhaps the most important constant in Wittgenstein's Philosophy of Mathematics, middle and late, is that he consistently maintains that mathematics is our, human invention, and that, indeed, everything in mathematics is invented. Just as the middle Wittgenstein says that "[w]e make mathematics", the later Wittgenstein says that we 'invent' mathematics (RFM I, SS168; II, SS38; V, SSSS5, 9 and 11; PG 469-70) and that "the mathematician is not a discoverer: he is an inventor" (RFM, Appendix II, SS2; (LFM 22, 82). Nothing exists mathematically unless and until we have invented it. In arguing against mathematical discovery, Wittgenstein is not just rejecting Platonism, he is also rejecting a rather standard philosophical view according to which human beings invent mathematical calculi, but once a calculus has been invented, we thereafter discover finitely many of its infinitely many provable and true theorems. As Wittgenstein himself asks (RFM IV, SS48), "might it not be said that the rules lead this way, even if no one went it?" If "someone produced a proof [of 'Goldbach's theorem']", "[c]ouldn't one say", Wittgenstein asks (LFM 144), "that the possibility of this proof was a fact in the realms of mathematical reality"--that "[i]n order [to] find it, it must in some sense be there"--"[i]t must be a possible structure"? Unlike many or most philosophers of mathematics, Wittgenstein resists the 'Yes' answer that we discover truths about a mathematical calculus that come into existence the moment we invent the calculus (PR SS141; PG 283, 466; LFM 139). Wittgenstein rejects the modal reification of possibility as actuality--that provability and constructibility are (actual) facts--by arguing that it is at the very least wrong-headed to say with the Platonist that because "a straight line can be drawn between any two points,... the line already exists even if no one has drawn it"--to say "[w]hat in the ordinary world we call a possibility is in the geometrical world a reality" (LFM 144; RFM I, SS21). One might as well say, Wittgenstein suggests (PG 374), that "chess only had to be discovered, it was always there!" At MS 122 (3v; Oct. 18, 1939), Wittgenstein once again emphasizes the difference between illusory mathematical discovery and genuine mathematical invention. > > > I want to get away from the formulation: "I now know more about > the calculus", and replace it with "I now have a different > calculus". The sense of this is always to keep before > one's eyes the full scale of the gulf between a mathematical > knowing and non-mathematical > knowing.[3] > > > And as with the middle period, the later Wittgenstein similarly says (MS 121, 27r; May 27, 1938) that "[i]t helps if one says: the proof of the Fermat proposition is not to be discovered, but to be invented". > > > The difference between the 'anthropological' and the > mathematical account is that in the first we are not tempted to speak > of 'mathematical facts', but rather that in this account > the facts are never mathematical ones, never make > mathematical propositions true or false. (MS 117, 263; March > 15, 1940) > > > There are no mathematical facts just as there are no (genuine) mathematical propositions. Repeating his intermediate view, the later Wittgenstein says (MS 121, 71v; 27 Dec., 1938): "Mathematics consists of [calculi \| calculations], not of propositions". This radical constructivist conception of mathematics prompts Wittgenstein to make notorious remarks--remarks that virtually no one else would make--such as the infamous (RFM V, SS9): "However queer it sounds, the further expansion of an irrational number is a further expansion of mathematics". #### 3.1.1 Wittgenstein's Later Anti-Platonism: The Natural History of Numbers and the Vacuity of Platonism As in the middle period, the later Wittgenstein maintains that mathematics is essentially syntactical and non-referential, which, in and of itself, makes Wittgenstein's philosophy of mathematics anti-Platonist insofar as Platonism is the view that mathematical terms and propositions refer to objects and/or facts and that mathematical propositions are true by virtue of agreeing with mathematical facts. The later Wittgenstein, however, wishes to 'warn' us that our thinking is saturated with the idea of "[a]rithmetic as the natural history (mineralogy) of numbers" (RFM IV, SS11). When, for instance, Wittgenstein discusses the claim that fractions cannot be ordered by magnitude, he says that this sounds 'remarkable' in a way that a mundane proposition of the differential calculus does not, for the latter proposition is associated with an application in physics, > > > whereas this proposition ... seems to > [solely] concern... the natural history of > mathematical objects themselves. (RFM II, SS40) > > > Wittgenstein stresses that he is trying to 'warn' us against this 'aspect'--the idea that the foregoing proposition about fractions "introduces us to the mysteries of the mathematical world", which exists somewhere as a completed totality, awaiting our prodding and our discoveries. The fact that we regard mathematical propositions as being about mathematical objects and mathematical investigation "as the exploration of these objects" is "already mathematical alchemy", claims Wittgenstein (RFM V, SS16), since > > > it is not possible to appeal to the meaning > [Bedeutung] of the signs in > mathematics,... because it is only mathematics that gives them > their meaning [Bedeutung]. > > > Platonism is dangerously misleading, according to Wittgenstein, because it suggests a picture of pre-existence, predetermination and discovery that is completely at odds with what we find if we actually examine and describe mathematics and mathematical activity. "I should like to be able to describe", says Wittgenstein (RFM IV, SS13), "how it comes about that mathematics appears to us now as the natural history of the domain of numbers, now again as a collection of rules". Wittgenstein, however, does not endeavour to refute Platonism. His aim, instead, is to clarify what Platonism is and what it says, implicitly and explicitly (including variants of Platonism that claim, e.g., that if a proposition is provable in an axiom system, then there already exists a path [i.e., a proof] from the axioms to that proposition (RFM I, SS21; Marion 1998: 13-14, 226; Steiner 2000: 334). Platonism is either "a mere truism" (LFM 239), Wittgenstein says, or it is a 'picture' consisting of "an infinity of shadowy worlds" (LFM 145), which, as such, lacks 'utility' (cf. PI SS254) because it explains nothing and it misleads at every turn. ### 3.2 Wittgenstein's Later Finitistic Constructivism Though commentators and critics do not agree as to whether the later Wittgenstein is still a finitist and whether, if he is, his finitism is as radical as his intermediate rejection of unbounded mathematical quantification (Maddy 1986: 300-301, 310), the overwhelming evidence indicates that the later Wittgenstein still rejects the actual infinite (RFM V, SS21; Zettel SS274, 1947) and infinite mathematical extensions. The first, and perhaps most definitive, indication that the later Wittgenstein maintains his finitism is his continued and consistent insistence that irrational numbers are rules for constructing finite expansions, not infinite mathematical extensions. "The concepts of infinite decimals in mathematical propositions are not concepts of series", says Wittgenstein (RFM V, SS19), "but of the unlimited technique of expansion of series". We are misled by "[t]he extensional definitions of functions, of real numbers etc". (RFM V, SS35), but once we recognize the Dedekind cut as "an extensional image", we see that we are not "led to $\sqrt{2}$ by way of the concept of a cut" (RFM V, SS34). On the later Wittgenstein's account, there simply is no property, no rule, no systematic means of defining each and every irrational number intensionally, which means there is no criterion "for the irrational numbers being complete" (PR SS181). As in his intermediate position, the later Wittgenstein claims that '$\aleph\_0$' and "infinite series" get their mathematical uses from the use of 'infinity' in ordinary language (RFM II, SS60). Although, in ordinary language, we often use 'infinite' and "infinitely many" as answers to the question "how many?", and though we associate infinity with the enormously large, the principal use we make of 'infinite' and 'infinity' is to speak of the unlimited (RFM V, SS14) and unlimited techniques (RFM II, SS45; PI SS218). This fact is brought out by the fact "that the technique of learning $\aleph\_0$ numerals is different from the technique of learning 100,000 numerals" (LFM 31). When we say, e.g., that "there are an infinite number of even numbers" we mean that we have a mathematical technique or rule for generating even numbers which is limitless, which is markedly different from a limited technique or rule for generating a finite number of numbers, such as 1-100,000,000. "We learn an endless technique", says Wittgenstein (RFM V, SS19), "but what is in question here is not some gigantic extension". An infinite sequence, for example, is not a gigantic extension because it is not an extension, and '$\aleph\_0$' is not a cardinal number, for "how is this picture connected with the calculus", given that "its connexion is not that of the picture \| \| \| \| with 4" (i.e., given that '$\aleph\_0$' is not connected to a (finite) extension)? This shows, says Wittgenstein (RFM II, SS58), that we ought to avoid the word 'infinite' in mathematics wherever it seems to give a meaning to the calculus, rather than acquiring its meaning from the calculus and its use in the calculus. Once we see that the calculus contains nothing infinite, we should not be 'disappointed' (RFM II, SS60), but simply note (RFM II, SS59) that it is not "really necessary... to conjure up the picture of the infinite (of the enormously big)". A second strong indication that the later Wittgenstein maintains his finitism is his continued and consistent treatment of 'propositions' of the type "There are three consecutive 7s in the decimal expansion of $\pi$" (hereafter 'PIC').[4] In the middle period, PIC (and its putative negation, $\neg$PIC, namely, "It is not the case that there are three consecutive 7s in the decimal expansion of $\pi$") is not a meaningful mathematical "statement at all" (WVC 81-82: note 1). On Wittgenstein's intermediate view, PIC--like FLT, GC, and the Fundamental Theorem of Algebra--is not a mathematical proposition because we do not have in hand an applicable decision procedure by which we can decide it in a particular calculus. For this reason, we can only meaningfully state finitistic propositions regarding the expansion of $\pi$, such as "There exist three consecutive 7s in the first 10,000 places of the expansion of $\pi$" (WVC 71; 81-82, note 1). The later Wittgenstein maintains this position in various passages in RFM (Bernays 1959: 11-12). For example, to someone who says that since "the rule of expansion determine[$s$] the series completely", "it must implicitly determine all questions about the structure of the series", Wittgenstein replies: "Here you are thinking of finite series" (RFM V, SS11). If PIC were a mathematical question (or problem)--if it were finitistically restricted--it would be algorithmically decidable, which it is not (RFM V, SS21; LFM 31-32, 111, 170; WVC 102-03). As Wittgenstein says at (RFM V, SS9): "The question... changes its status, when it becomes decidable", "[f]or a connexion is made then, which formerly was not there". And if, moreover, one invokes the Law of the Excluded Middle to establish that PIC is a mathematical proposition--i.e., by saying that one of these "two pictures... must correspond to the fact" (RFM V, SS10)--one simply begs the question (RFM V, SS12), for if we have doubts about the mathematical status of PIC, we will not be swayed by a person who asserts "PIC $\vee \neg$PIC" (RFM VII, SS41; V, SS13). Wittgenstein's finitism, constructivism, and conception of mathematical decidability are interestingly connected at (RFM VII, SS41, par. 2-5). > > > What harm is done e.g. by saying that God knows all > irrational numbers? Or: that they are already there, even though we > only know certain of them? Why are these pictures not harmless? > > > > For one thing, they hide certain problems.-- (MS 124: 139; March > 16, 1944) > > > > Suppose that people go on and on calculating the expansion of $\pi$. > So God, who knows everything, knows whether they will have reached > '777' by the end of the world. But can his > omniscience decide whether they would have reached > it after the end of the world? It cannot. I want to say: Even God can > determine something mathematical only by mathematics. Even for him the > mere rule of expansion cannot decide anything that it does not decide > for us. > > > > We might put it like this: if the rule for the expansion has been > given us, a calculation can tell us that there is a > '2' at the fifth place. Could God have known this, without > the calculation, purely from the rule of expansion? I want to say: No. > (MS 124, pp. 175-176; March 23-24, 1944) > > > What Wittgenstein means here is that God's omniscience might, by calculation, find that '777' occurs at the interval [$n,n+2$], but, on the other hand, God might go on calculating forever without '777' ever turning up. Since $\pi$ is not a completed infinite extension that can be completely surveyed by an omniscient being (i.e., it is not a fact that can be known by an omniscient mind), even God has only the rule, and so God's omniscience is no advantage in this case (LFM 103-04; cf. Weyl 1921 [1998: 97]). Like us, with our modest minds, an omniscient mind (i.e., God) can only calculate the expansion of $\pi$ to some $n$th decimal place--where our $n$ is minute and God's $n$ is (relatively) enormous--and at no $n$th decimal place could any mind rightly conclude that because '777' has not turned up, it, therefore, will never turn up. ### 3.3 The Later Wittgenstein on Decidability and Algorithmic Decidability On one fairly standard interpretation, the later Wittgenstein says that "true in calculus $\Gamma$" is identical to "provable in calculus $\Gamma$" and, therefore, that a mathematical proposition of calculus $\Gamma$ is a concatenation of signs that is either provable (in principle) or refutable (in principle) in calculus $\Gamma$ (Goodstein 1972: 279, 282; Anderson 1958: 487; Klenk 1976: 13; Frascolla 1994: 59). On this interpretation, the later Wittgenstein precludes undecidable mathematical propositions, but he allows that some undecided expressions are propositions of a calculus because they are decidable in principle (i.e., in the absence of a known, applicable decision procedure). There is considerable evidence, however, that the later Wittgenstein maintains his intermediate position that an expression is a meaningful mathematical proposition only within a given calculus and iff we knowingly have in hand an applicable and effective decision procedure by means of which we can decide it. For example, though Wittgenstein vacillates between "provable in PM" and "proved in PM" at (RFM App. III, SS6, SS8), he does so in order to use the former to consider the alleged conclusion of Godel's proof (i.e., that there exist true but unprovable mathematical propositions), which he then rebuts with his own identification of "true in calculus $\Gamma$" with "proved in calculus $\Gamma$" (i.e., not with "provable in calculus $\Gamma$") (Wang 1991: 253; Rodych 1999a: 177). This construal is corroborated by numerous passages in which Wittgenstein rejects the received view that a provable but unproved proposition is true, as he does when he asserts that (RFM III, SS31, 1939) a proof "makes new connexions", "[i]t does not establish that they are there" because "they do not exist until it makes them", and when he says (RFM VII, SS10, 1941) that "[a] new proof gives the proposition a place in a new system". Furthermore, as we have just seen, Wittgenstein rejects PIC as a non-proposition on the grounds that it is not algorithmically decidable, while admitting finitistic versions of PIC because they are algorithmically decidable. Perhaps the most compelling evidence that the later Wittgenstein maintains algorithmic decidability as his criterion for a mathematical proposition lies in the fact that, at (RFM V, SS9, 1942), he says in two distinct ways that a mathematical 'question' can become decidable and that when this happens, a new connexion is 'made' which previously did not exist. Indeed, Wittgenstein cautions us against appearances by saying that "it looks as if a ground for the decision were already there", when, in fact, "it has yet to be invented". These passages strongly militate against the claim that the later Wittgenstein grants that proposition $\phi$ is decidable in calculus $\Gamma$ iff it is provable or refutable in principle. Moreover, if Wittgenstein held this position, he would claim, contra (RFM V, SS9), that a question or proposition does not become decidable since it simply (always) is decidable. If it is provable, and we simply don't yet know this to be the case, there already is a connection between, say, our axioms and rules and the proposition in question. What Wittgenstein says, however, is that the modalities provable and refutable are shadowy forms of reality--that possibility is not actuality in mathematics (PR SSSS141, 144, 172; PG 281, 283, 299, 371, 466, 469; LFM 139). Thus, the later Wittgenstein agrees with the intermediate Wittgenstein that the only sense in which an undecided mathematical proposition (RFM VII, SS40, 1944) can be decidable is in the sense that we know how to decide it by means of an applicable decision procedure. ### 3.4 Wittgenstein's Later Critique of Set Theory: Non-Enumerability vs. Non-Denumerability Largely a product of his anti-foundationalism and his criticism of the extension-intension conflation, Wittgenstein's later critique of set theory is highly consonant with his intermediate critique (PR SSSS109, 168; PG 334, 369, 469; LFM 172, 224, 229; and RFM III, SS43, 46, 85, 90; VII, SS16). Given that mathematics is a "MOTLEY of techniques of proof" (RFM III, SS46), it does not require a foundation (RFM VII, SS16) and it cannot be given a self-evident foundation (PR SS160; WVC 34 & 62; RFM IV, SS3). Since set theory was invented to provide mathematics with a foundation, it is, minimally, unnecessary. Even if set theory is unnecessary, it still might constitute a solid foundation for mathematics. In his core criticism of set theory, however, the later Wittgenstein denies this, saying that the diagonal proof does not prove non-denumerability, for "[i]t means nothing to say: 'Therefore the X numbers are not denumerable'" (RFM II, SS10). When the diagonal is construed as a proof of greater and lesser infinite sets it is a "puffed-up proof", which, as Poincare argued (1913: 61-62), purports to prove or show more than "its means allow it" (RFM II, SS21). > > > If it were said: Consideration of the diagonal procedure shews you that the > concept 'real number' has much less analogy with > the concept 'cardinal number' than we, being misled by > certain analogies, are inclined to believe, that would have a good and honest sense. But just the > opposite happens: one pretends to compare the > 'set' of real numbers in magnitude with that of cardinal > numbers. The difference in kind between the two conceptions is > represented, by a skew form of expression, as difference of extension. > I believe, and hope, that a future generation will laugh at this hocus > pocus. (RFM II, SS22) > > > > The sickness of a time is cured by an alteration in the mode of life > of human beings... (RFM II, SS23) > > > The "hocus pocus" of the diagonal proof rests, as always for Wittgenstein, on a conflation of extension and intension, on the failure to properly distinguish sets as rules for generating extensions and (finite) extensions. By way of this confusion "a difference in kind" (i.e., unlimited rule vs. finite extension) "is represented by a skew form of expression", namely as a difference in the cardinality of two infinite extensions. Not only can the diagonal not prove that one infinite set is greater in cardinality than another infinite set, according to Wittgenstein, nothing could prove this, simply because "infinite sets" are not extensions, and hence not infinite extensions. But instead of interpreting Cantor's diagonal proof honestly, we take the proof to "show there are numbers bigger than the infinite", which "sets the whole mind in a whirl, and gives the pleasant feeling of paradox" (LFM 16-17)--a "giddiness attacks us when we think of certain theorems in set theory"--"when we are performing a piece of logical sleight-of-hand" (PI SS412; SS426; 1945). This giddiness and pleasant feeling of paradox, says Wittgenstein (LFM 16), "may be the chief reason [set theory] was invented". Though Cantor's diagonal is not a proof of non-denumerability, when it is expressed in a constructive manner, as Wittgenstein himself expresses it at (RFM II, SS1), "it gives sense to the mathematical proposition that the number so-and-so is different from all those of the system" (RFM II, SS29). That is, the proof proves non-enumerability: it proves that for any given definite real number concept (e.g., recursive real), one cannot enumerate 'all' such numbers because one can always construct a diagonal number, which falls under the same concept and is not in the enumeration. "One might say", Wittgenstein says, > > > I call number-concept X non-denumerable if it has been stipulated > that, whatever numbers falling under this concept you arrange in a > series, the diagonal number of this series is also to fall under that > concept. (RFM II, SS10; cf. II, SSSS30, 31, > 13) > > > One lesson to be learned from this, according to Wittgenstein (RFM II, SS33), is that "there are diverse systems of irrational points to be found in the number line", each of which can be given by a recursive rule, but "no system of irrational numbers", and "also no super-system, no 'set of irrational numbers' of higher-order infinity". Cantor has shown that we can construct "infinitely many" diverse systems of irrational numbers, but we cannot construct an exhaustive system of all the irrational numbers (RFM II, SS29). As Wittgenstein says at (MS 121, 71r; Dec. 27, 1938), three pages after the passage used for (RFM II, SS57): > > > If you now call the Cantorian procedure one for producing a new real > number, you will now no longer be inclined to speak of a system of > all real numbers. (italics added) > > > From Cantor's proof, however, set theorists erroneously conclude that "the set of irrational numbers" is greater in multiplicity than any enumeration of irrationals (or the set of rationals), when the only conclusion to draw is that there is no such thing as the set of all the irrational numbers. The truly dangerous aspect to 'propositions' such as "The real numbers cannot be arranged in a series" and "The set... is not denumerable" is that they make concept formation [i.e., our invention] "look like a fact of nature" (i.e., something we discover) (RFM II SSSS16, 37). At best, we have a vague idea of the concept of "real number", but only if we restrict this idea to "recursive real number" and only if we recognize that having the concept does not mean having a set of all recursive real numbers. ### 3.5 Extra-Mathematical Application as a Necessary Condition of Mathematical Meaningfulness The principal and most significant change from the middle to later writings on mathematics is Wittgenstein's (re-)introduction of an extra-mathematical application criterion, which is used to distinguish mere "sign-games" from mathematical language-games. "[I]t is essential to mathematics that its signs are also employed in mufti", Wittgenstein states, for > > > [i]t is the use outside mathematics, and so the meaning > [Bedeutung] of the signs, that makes the > sign-game into mathematics. (i.e., a mathematical > "language-game"; RFM V, SS2, 1942; > LFM 140-141, 169-70) > > > As Wittgenstein says at (RFM V, SS41, 1943), > > > [c]oncepts which occur in 'necessary' propositions > must also occur and have a meaning > [Bedeutung] in non-necessary ones. (italics > added) > > > If two proofs prove the same proposition, says Wittgenstein, this means that "both demonstrate it as a suitable instrument for the same purpose", which "is an allusion to something outside mathematics" (RFM VII, SS10, 1941; italics added). As we have seen, this criterion was present in the Tractatus (6.211), but noticeably absent in the middle period. The reason for this absence is probably that the intermediate Wittgenstein wanted to stress that in mathematics everything is syntax and nothing is meaning. Hence, in his criticisms of Hilbert's 'contentual' mathematics (Hilbert 1925) and Brouwer's reliance upon intuition to determine the meaningful content of (especially undecidable) mathematical propositions, Wittgenstein couched his finitistic constructivism in strong formalism, emphasizing that a mathematical calculus does not need an extra-mathematical application (PR SS109; WVC 105). There seem to be two reasons why the later Wittgenstein reintroduces extra-mathematical application as a necessary condition of a mathematical language-game. First, the later Wittgenstein has an even greater interest in the use of natural and formal languages in diverse "forms of life" (PI SS23), which prompts him to emphasize that, in many cases, a mathematical 'proposition' functions as if it were an empirical proposition "hardened into a rule" (RFM VI, SS23) and that mathematics plays diverse applied roles in many forms of human activity (e.g., science, technology, predictions). Second, the extra-mathematical application criterion relieves the tension between Wittgenstein's intermediate critique of set theory and his strong formalism according to which "one calculus is as good as another" (PG 334). By demarcating mathematical language-games from non-mathematical sign-games, Wittgenstein can now claim that, "for the time being", set theory is merely a formal sign-game. > > > These considerations may lead us to say that $2^{\aleph\_0} \gt > \aleph\_0$. > > > > That is to say: we can make the considerations lead us to > that. > > > > Or: we can say this and give this as our reason. > > > > But if we do say it--what are we to do next? In what practice is > this proposition anchored? It is for the time being a piece of > mathematical architecture which hangs in the air, and looks as if it > were, let us say, an architrave, but not supported by anything and > supporting nothing. (RFM II, SS35) > > > It is not that Wittgenstein's later criticisms of set theory change, it is, rather, that once we see that set theory has no extra-mathematical application, we will focus on its calculations, proofs, and prose and "subject the interest of the calculations to a test" (RFM II, SS62). By means of Wittgenstein's "immensely important" 'investigation' (LFM 103), we will find, Wittgenstein expects, that set theory is uninteresting (e.g., that the non-enumerability of "the reals" is uninteresting and useless) and that our entire interest in it lies in the 'charm' of the mistaken prose interpretation of its proofs (LFM 16). More importantly, though there is "a solid core to all [its] glistening concept-formations" (RFM V, SS16), once we see it as "as a mistake of ideas", we will see that propositions such as "$2^{\aleph\_0} \gt \aleph\_0$" are not anchored in an extra-mathematical practice, that "Cantor's paradise" "is not a paradise", and we will then leave "of [our] own accord" (LFM 103). It must be emphasized, however, that the later Wittgenstein still maintains that the operations within a mathematical calculus are purely formal, syntactical operations governed by rules of syntax (i.e., the solid core of formalism). > > > It is of course clear that the mathematician, in so far as he really > is 'playing a game'...[is] acting in > accordance with certain rules. (RFM V, SS1) > > > > To say mathematics is a game is supposed to mean: in proving, we need > never appeal to the meaning [Bedeutung] of the > signs, that is to their extra-mathematical application. (RFM > V, SS4) > > > Where, during the middle period, Wittgenstein speaks of "arithmetic [as] a kind of geometry" at (PR SS109 & SS111), the later Wittgenstein similarly speaks of "the geometry of proofs" (RFM I, App. III, SS14), the "geometrical cogency" of proofs (RFM III, SS43), and a "geometrical application" according to which the "transformation of signs" in accordance with "transformation-rules" (RFM VI, SS2, 1941) shows that "when mathematics is divested of all content, it would remain that certain signs can be constructed from others according to certain rules" (RFM III, SS38). Hence, the question whether a concatenation of signs is a proposition of a given mathematical calculus (i.e., a calculus with an extra-mathematical application) is still an internal, syntactical question, which we can answer with knowledge of the proofs and decision procedures of the calculus. ### 3.6 Wittgenstein on Godel and Undecidable Mathematical Propositions RFM is perhaps most (in)famous for Wittgenstein's (RFM App. III) treatment of "true but unprovable" mathematical propositions. Early reviewers said that "[t]he arguments are wild" (Kreisel 1958: 153), that the passages "on Godel's theorem... are of poor quality or contain definite errors" (Dummett 1959: 324), and that (RFM App. III) "throws no light on Godel's work" (Goodstein 1957: 551). "Wittgenstein seems to want to legislate [[q]uestions about completeness] out of existence", Anderson said, (1958: 486-87) when, in fact, he certainly cannot dispose of Godel's demonstrations "by confusing truth with provability". Additionally, Bernays, Anderson (1958: 486), and Kreisel (1958: 153-54) claimed that Wittgenstein failed to appreciate "Godel's quite explicit premiss of the consistency of the considered formal system" (Bernays 1959: 15), thereby failing to appreciate the conditional nature of Godel's First Incompleteness Theorem. On the reading of these four early expert reviewers, Wittgenstein failed to understand Godel's Theorem because he failed to understand the mechanics of Godel's proof and he erroneously thought he could refute or undermine Godel's proof simply by identifying "true in PM" (i.e., Principia Mathematica) with "proved/provable in PM". Interestingly, we now have two pieces of evidence (Kreisel 1998: 119; Rodych 2003: 282, 307) that Wittgenstein wrote (RFM App. III) in 1937-38 after reading only the informal, 'casual' (MS 126, 126-127; Dec. 13, 1942) introduction of (Godel 1931) and that, therefore, his use of a self-referential proposition as the "true but unprovable proposition" may be based on Godel's introductory, informal statements, namely that "the undecidable proposition [$R(q);q$] states... that [$R(q);q$] is not provable" (1931: 598) and that "[$R(q);q$] says about itself that it is not provable" (1931: 599). Perplexingly, only two of the four famous reviewers even mentioned Wittgenstein's (RFM VII, SSSS19, 21-22, 1941)) explicit remarks on 'Godel's' First Incompleteness Theorem (Bernays 1959: 2; Anderson 1958: 487), which, though flawed, capture the number-theoretic nature of the Godelian proposition and the functioning of Godel-numbering, probably because Wittgenstein had by then read or skimmed the body of Godel's 1931 paper. The first thing to note, therefore, about (RFM App. III) is that Wittgenstein mistakenly thinks--again, perhaps because Wittgenstein had read only Godel's Introduction--(a) that Godel proves that there are true but unprovable propositions of PM (when, in fact, Godel syntactically proves that if PM is $\omega$-consistent, the Godelian proposition is undecidable in PM) and (b) that Godel's proof uses a self-referential proposition to semantically show that there are true but unprovable propositions of PM. For this reason, Wittgenstein has two main aims in (RFM App. III): (1) to refute or undermine, on its own terms, the alleged Godel proof of true but unprovable propositions of PM, and (2) to show that, on his own terms, where "true in calculus $\Gamma$" is identified with "proved in calculus $\Gamma$", the very idea of a true but unprovable proposition of calculus $\Gamma$ is meaningless. Thus, at (RFM App. III, SS8) (hereafter simply 'SS8'), Wittgenstein begins his presentation of what he takes to be Godel's proof by having someone say: > > > I have constructed a proposition (I will use 'P' to > designate it) in Russell's symbolism, and by means of certain > definitions and transformations it can be so interpreted that it says: > 'P is not provable in Russell's system'. > > > That is, Wittgenstein's Godelian constructs a proposition that is semantically self-referential and which specifically says of itself that it is not provable in PM. With this erroneous, self-referential proposition P [used also at (SS10), (SS11), (SS17), (SS18)], Wittgenstein presents a proof-sketch very similar to Godel's own informal semantic proof 'sketch' in the Introduction of his famous paper (1931: 598). > > > Must I not say that this proposition on the one hand is true, and on > the other hand is unprovable? For suppose it were false; then it is > true that it is provable. And that surely cannot be! And if it is > proved, then it is proved that it is not provable. Thus it can only be > true, but unprovable. (SS8) > > > The reasoning here is a double reductio. Assume (a) that P must either be true or false in Russell's system, and (b) that P must either be provable or unprovable in Russell's system. If (a), P must be true, for if we suppose that P is false, since P says of itself that it is unprovable, "it is true that it is provable", and if it is provable, it must be true (which is a contradiction), and hence, given what P means or says, it is true that P is unprovable (which is a contradiction). Second, if (b), P must be unprovable, for if P "is proved, then it is proved that it is not provable", which is a contradiction (i.e., P is provable and not provable in PM). It follows that P "can only be true, but unprovable". To refute or undermine this 'proof', Wittgenstein says that if you have proved $\neg P$, you have proved that P is provable (i.e., since you have proved that it is not the case that P is not provable in Russell's system), and "you will now presumably give up the interpretation that it is unprovable" (i.e., 'P is not provable in Russell's system'), since the contradiction is only proved if we use or retain this self-referential interpretation (SS8). On the other hand, Wittgenstein argues (SS8), '[i]f you assume that the proposition is provable in Russell's system, that means it is true in the Russell sense, and the interpretation "P is not provable" again has to be given up', because, once again, it is only the self-referential interpretation that engenders a contradiction. Thus, Wittgenstein's 'refutation' of "Godel's proof" consists in showing that no contradiction arises if we do not interpret 'P' as 'P is not provable in Russell's system'--indeed, without this interpretation, a proof of P does not yield a proof of $\neg P$ and a proof of $\neg P$ does not yield a proof of P. In other words, the mistake in the proof is the mistaken assumption that a mathematical proposition 'P' "can be so interpreted that it says: 'P is not provable in Russell's system'". As Wittgenstein says at (SS11), "[t]hat is what comes of making up such sentences". This 'refutation' of "Godel's proof" is perfectly consistent with Wittgenstein's syntactical conception of mathematics (i.e., wherein mathematical propositions have no meaning and hence cannot have the 'requisite' self-referential meaning) and with what he says before and after (SS8), where his main aim is to show (2) that, on his own terms, since "true in calculus $\Gamma$" is identical with "proved in calculus $\Gamma$", the very idea of a true but unprovable proposition of calculus $\Gamma$ is a contradiction-in-terms. To show (2), Wittgenstein begins by asking (SS5), what he takes to be, the central question, namely, "Are there true propositions in Russell's system, which cannot be proved in his system?". To address this question, he asks "What is called a true proposition in Russell's system...?", which he succinctly answers (SS6): "'p' is true = p". Wittgenstein then clarifies this answer by reformulating the second question of (SS5) as "Under what circumstances is a proposition asserted in Russell's game [i.e., system]?", which he then answers by saying: "the answer is: at the end of one of his proofs, or as a 'fundamental law' (Pp.)" (SS6). This, in a nutshell, is Wittgenstein's conception of "mathematical truth": a true proposition of PM is an axiom or a proved proposition, which means that "true in PM" is identical with, and therefore can be supplanted by, "proved in PM". Having explicated, to his satisfaction at least, the only real, non-illusory notion of "true in PM", Wittgenstein answers the (SS8) question "Must I not say that this proposition... is true, and... unprovable?" negatively by (re)stating his own (SSSS5-6) conception of "true in PM" as "proved/provable in PM": > > > 'True in Russell's system' means, as was said: > proved in Russell's system; and 'false in Russell's > system' means: the opposite has been proved in Russell's > system. > > > This answer is given in a slightly different way at (SS7) where Wittgenstein asks "may there not be true propositions which are written in this [Russell's] symbolism, but are not provable in Russell's system?", and then answers "'True propositions', hence propositions which are true in another system, i.e. can rightly be asserted in another game". In light of what he says in (SSSS5, 6, and 8), Wittgenstein's (SS7) point is that if a proposition is 'written' in "Russell's symbolism" and it is true, it must be proved/provable in another system, since that is what "mathematical truth" is. Analogously (SS8), "if the proposition is supposed to be false in some other than the Russell sense, then it does not contradict this for it to be proved in Russell's sense", for "[w]hat is called 'losing' in chess may constitute winning in another game". This textual evidence certainly suggests, as Anderson almost said, that Wittgenstein rejects a true but unprovable mathematical proposition as a contradiction-in-terms on the grounds that "true in calculus $\Gamma$" means nothing more (and nothing less) than "proved in calculus $\Gamma$". On this (natural) interpretation of (RFM App. III), the early reviewers' conclusion that Wittgenstein fails to understand the mechanics of Godel's argument seems reasonable. First, Wittgenstein erroneously thinks that Godel's proof is essentially semantical and that it uses and requires a self-referential proposition. Second, Wittgenstein says (SS14) that "[a] contradiction is unusable" for "a prediction" that "that such-and-such construction is impossible" (i.e., that P is unprovable in PM), which, superficially at least, seems to indicate that Wittgenstein fails to appreciate the "consistency assumption" of Godel's proof (Kreisel, Bernays, Anderson). If, in fact, Wittgenstein did not read and/or failed to understand Godel's proof through at least 1941, how would he have responded if and when he understood it as (at least) a proof of the undecidability of P in PM on the assumption of PM's consistency? Given his syntactical conception of mathematics, even with the extra-mathematical application criterion, he would simply say that P, qua expression syntactically independent of PM, is not a proposition of PM, and if it is syntactically independent of all existent mathematical language-games, it is not a mathematical proposition. Moreover, there seem to be no compelling non-semantical reasons--either intra-systemic or extra-mathematical--for Wittgenstein to accommodate P by including it in PM or by adopting a non-syntactical conception of mathematical truth (such as Tarski-truth (Steiner 2000)). Indeed, Wittgenstein questions the intra-systemic and extra-mathematical usability of P in various discussions of Godel in the Nachlass and, at (SS19), he emphatically says that one cannot "make the truth of the assertion ['P' or 'Therefore P'] plausible to me, since you can make no use of it except to do these bits of legerdemain". After the initial, scathing reviews of RFM, very little attention was paid to Wittgenstein's (RFM App. III and RFM VII, SSSS21-22) discussions of Godel's First Incompleteness Theorem (Klenk 1976: 13) until Shanker's sympathetic (1988b). In the last 22 years, however, commentators and critics have offered various interpretations of Wittgenstein's remarks on Godel, some being largely sympathetic (Floyd 1995, 2001) and others offering a more mixed appraisal (Rodych 1999a, 2002, 2003; Steiner 2001; Priest 2004; Berto 2009a). Recently, and perhaps most interestingly, Floyd & Putnam (2000) and Steiner (2001) have evoked new and interesting discussions of Wittgenstein's ruminations on undecidability, mathematical truth, and Godel's First Incompleteness Theorem (Rodych 2003, 2006; Bays 2004; Sayward 2005; and Floyd & Putnam 2006). ## 4. The Impact of Philosophy of Mathematics on Mathematics Though it is doubtful that all commentators will agree (Wrigley 1977: 51; Baker & Hacker 1985: 345; Floyd 1991: 145, 143; 1995: 376; 2005: 80; Maddy 1993: 55; Steiner 1996: 202-204), the following passage seems to capture Wittgenstein's attitude to the Philosophy of Mathematics and, in large part, the way in which he viewed his own work on mathematics. > > > What will distinguish the mathematicians of the future from those of > today will really be a greater sensitivity, and that > will--as it were--prune mathematics; since people will then > be more intent on absolute clarity than on the discovery of new > games. > > > > Philosophical clarity will have the same effect on the growth of > mathematics as sunlight has on the growth of potato shoots. (In a dark > cellar they grow yards long.) > > > > A mathematician is bound to be horrified by my mathematical comments, > since he has always been trained to avoid indulging in thoughts and > doubts of the kind I develop. He has learned to regard them as > something contemptible and... he has acquired a revulsion from > them as infantile. That is to say, I trot out all the problems that a > child learning arithmetic, etc., finds difficult, the problems that > education represses without solving. I say to those repressed doubts: > you are quite correct, go on asking, demand clarification! > (PG 381, 1932) > > > In his middle and later periods, Wittgenstein believes he is providing philosophical clarity on aspects and parts of mathematics, on mathematical conceptions, and on philosophical conceptions of mathematics. Lacking such clarity and not aiming for absolute clarity, mathematicians construct new games, sometimes because of a misconception of the meaning of their mathematical propositions and mathematical terms. Education and especially advanced education in mathematics does not encourage clarity but rather represses it--questions that deserve answers are either not asked or are dismissed. Mathematicians of the future, however, will be more sensitive and this will (repeatedly) prune mathematical extensions and inventions, since mathematicians will come to recognize that new extensions and creations (e.g., propositions of transfinite cardinal arithmetic) are not well-connected with the solid core of mathematics or with real-world applications. Philosophical clarity will, eventually, enable mathematicians and philosophers to "get down to brass tacks" (PG 467).